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A young man who died at the age of 32 in a foreign land he had travelled to, to pursue his craft. A clumsy eccentric who could visualize his complete work in his head before he put it to canvas. A Russian who shuns the limelight and refuses recognition for his work. A traveller who went from country to country on a whim in order to collaborate with others. A man whose scribblings inspired the life work of hundreds. A woman, who escaped the prejudices against her gender to make a name for herself. A recluse who spent close to ten years working on one piece. A revolutionary child prodigy who died in a gun duel before his twenty-first birthday. What do you picture when you read the above? Artists? Musicians? Writers? Surely not mathematicians?
Srinivas Ramanujan (1887-1920) was a self-taught nobody who, in his short life-span, discovered nearly 3900 results, many of which were completely unexpected, and influenced and made entire careers for future mathematicians. In fact there is an entire journal devoted to areas of study inspired by Ramanujan’s work. Even trying to give an overview of his life’s work would require an entire book.
Henri Poincare (1854-1912) was short-sighted and hence had to learn how to visualise all the lectures he sat through. In doing so, he developed the skill to visualise entire proofs before writing them down. Poincare is considered one of the founders of the field of Topology, a field concerned with what remains when objects are transformed. An oft-told joke about Topologists is that they can’t tell their donut from their coffee cup.
A conjecture of Poincare’s, regarding the equivalent of a sphere in 4-dimensional space, was unsolved till this century when Grigori Perelman (1966- ) became the first mathematician to crack a millenium prize problem, with prize money of $1million. Perelman turned it down. He is also the only mathematician to have turned down the Fields Medal, mathematics’ equivalent of the Nobel Prize.
Have you heard of the Kevin Bacon number? Well mathematicians give themselves an Erdos number after Paul Erdos (1913-96) who, like Kevin Bacon, collaborated with everybody important in the field in various parts of the world. If he heard you were doing some interesting research, he would pack his bags and turn up at your doorstep.
Pierre de Fermat (1601-65) was a lawyer and ‘amateur’ mathematician, whose work in Number Theory has provided some of the greatest tools mathematicians have today, and are integral to very modern areas such as cryptography. He made an enigmatic comment in a margin of his copy of Diaphantus’ ‘Arithmetica’ saying:
‘It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvellous proof of this, which this margin is too narrow to contain.’
Whether he actually had a proof is debatable, but this one comment inspired work for the next 300 years.
In these intervening 300 years, one name has to be mentioned – Sophie Germain (1776-1831). Germain remains one of the few women who have broken the glass ceiling and made significant contributions to mathematics. She was responsible for proving Fermat’s scribblings for a large amount of numbers.
I apologise to Andrew Wiles (1953- ) for calling him a recluse, but he did spend close to 10 years on the proof of Fermat’s Last Theorem, during most of which he did not reveal his progress to anybody.
Saving the best for last, Evariste Galois (1811-32), a radical republican in pre-revolutionary France, died in a duel over a woman at the age of 20. Only the night before, he had finished a manuscript with some of the most innovative and impactful results in mathematics. There is speculation that the resulting lack of sleep caused him to lose the duel. Galois developed what became a whole branch of mathematics to itself – Galois Theory, a sub-discipline which connect two other subdisciplines of abstract algebra. It is the only branch of mathematics I can think of which is named after its creator (apart from Mr. Algebra and Ms. Probability).
This might appear to be anecdotal evidence of the creative spirit of mathematics and mathematicians. However, the same can be said about the evidence given for Artistic genius. In fact there is research which shows that the archetype of a mad artistic genius doesn’t stand on firm ground. So, lets move away from exploring creative mathematicians, to the creativity of the discipline.
Mathematics is a highly creative discipline, by any useful sense of the word ‘creative.’ The study of mathematics involves speculation, risk in the sense of the willingness to follow one’s chain of thought to wherever it leads, innovative arguments, exhilaration at achieving a result and many a time beauty in the result. Unlike scientists, mathematicians do not have our universe as a crutch. Elementary mathematics might be able to get inspiration from the universe, but quickly things change. Mathematicians have to invent conjectures from their imagination. Therefore, these conjectures are very tenuous. Most of them will fail to bear any fruit, but if mathematicians are unwilling to take that risk, they will lose any hope of discovery. Once mathematicians are convinced of the certainty of an argument, they have to present a rigorous proof, which nobody can poke any holes in. Once again, they are not as luck as scientists, who are happy with a statistically significant result or at most a result within five standard deviations. As a result of this, once you prove a mathematical theorem, your name will be associated with it for eternity. Aristotle might have been superseded by Newton and Newton by Einstein, but Euclid’s proof of infinite primes will always be true. As Hardy said, “A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.” The beauty of mathematical results and proofs is a fraught terrain, but there are certain results, great masters such as Euler’s identity and Euclid’s proof, which are almost universally accepted as aesthetically pleasing.
So, why are people so afraid of mathematics? Why do they consider it to be boring and staid? Well, the easy answer is that they are taught shopkeeper mathematics. In school, you are taught to follow rules in order to arrive at an answer. In the better schools, you are encouraged to do so using blocks and toys. However, basically the only skills you are getting are those which help you in commercial transactions. At the most, you get the skills to help you in other disciplines like Economics and the Sciences.
There has been a huge push in the recent past for the Arts to be taught in school ‘for art’s sake.’ There would be uproar tomorrow amongst artists and the liberal elite if art class turned into replicating posters (not even creating them). There would even be a furore if the only art students did was to draw the solar system for Science class and the Taj Mahal for Social Studies. What good art classes involve is teachers introducing concepts such as particular shapes and then encouraging students to experiment and create based on those concepts.
What about ‘maths for maths’ sake?’ Students should be encouraged to come up with their own conjectures based on concepts introduced by the teacher. This class would have to be closely guided by a teacher who is conceptually very strong, so that they can give examples in order to get students to come up with conjectures. They would also be required to provide students with counterexamples to any conjecture they have come up with.
I am not suggesting completely doing away with the current model of mathematics education involving repeated practice of questions. Just as replication probably helps in the arts and the arts can serve as great starting points for concepts in other disciplines, repetition is important in mathematics as it helps you intuit concepts and certain mathematical concepts are important for the conceptual understanding of other disciplines and for life. So, there needs to be a blend of mathematics classes (those which teach mathematics) and shopkeeper classes (those which teach mathematical concepts for other disciplines and for life). These would not work as separate entities and might even be taught at the same time. This requires a complete overhaul of the mathematics curriculum with a much lighter load of topics so that teachers can explore concepts in depth with their students. It also requires a larger emphasis on concepts such as symmetry, graph theory and pixel geometry which are easier to inquire into and form conjectures in than topics like calculus.
Now we come to the logistics. How many teachers are there in the country who have a strong enough conceptual understanding required to engage with mathematics in this manner? I would be pleasantly surprised if that were a long list, but I suspect it isn’t. In order to build up this capability, the emphasis at teacher colleges and in teacher professional development has to move from dull and pointless concepts like classroom management and teaching strategies, to developing conceptual understanding, at least in Mathematics. The amount of knowledge required to teach school mathematics is not all that much. All that is required is a strong conceptual base in a few concepts along with an understanding of mathematics as an endeavor, and a disposition for the eccentricities of the discipline.
Even so, this will not be easy to accomplish and will take time. In the meanwhile, wherever possible, professional mathematicians could come in to schools and work with teachers on their lesson plans. In other cases, these mathematicians could partner with educationalists and come up with material, which can more or less be put to use in any class (this is not ideal as lesson plans should be created by the teachers and evolved based on their understanding of their class, but this will have to do in the interim).
Not only will this help in developing a disposition for mathematics and hopefully churn out mathematicians, but it will also help in the understanding of shopkeeper mathematics. Pedagogy and conceptual understanding are not separate entities. In fact a strong conceptual understanding is a prerequisite for effective pedagogy.
Mathematics is unfortunate in its usefulness to other disciplines and the utility it provides for life. In the meanwhile, the real creative essence of the discipline is lost. I don’t blame students for hating mathematics in school. In fact it is completely justified. Mathematics is missing out. Who knows, one of these students would have proved the Riemann Hypothesis in an alternate reality. Artists have been very successful in campaigning for the creativity of their discipline to be an integral part of schools. Mathematicians, on the other hand, really need to pull up their socks and join the fight for the future of mathematics. In the spirit of Galois, Mathematicians of the World Unite! You have nothing to lose but the chains of countless students!
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