Make Math Education Fun, Says Fields Medal Winner Manjul Bhargava
How the Rubik’s cube inspired Fields Medal winner Manjul Bhargava to write his thesis...
Researchers don’t have the opportunity to pass on knowledge to the next generation in India because the education system separates research and teaching, said acclaimed mathematician Manjul Bhargava.
It is very important to combine research and teaching in Indian universities, said Bhargava, a winner of Fields Medal which is considered equivalent to the Nobel prize. “You can imagine how much manpower in research we are losing, knowing that 90 percent of faculties at universities are not encouraged to do research, but they have the highest level of knowledge with their PhDs.”
Combine research and teaching again so that students can learn from a place where knowledge creation is taking place so they can become great innovators, the Canadian-American mathematician said in a conversation with Infosys Co-Founder NR Narayana Murthy on the sidelines of the Tenth Infosys Science Foundation Awards.
“Mathematics should be taught in a way which makes it recreational, fun, playful, creative and adventurous,” said the professor at Princeton University. “That is what gives best feeling to students when they are doing mathematics.”
Bhargava won the Fields Medal in 2014 for his work on number theory. He also received the Infosys Prize for his contribution to mathematics in 2012.
Watch the full conversation here:
Here’s the edited transcript of the conversation:
There are lot of apocryphal stories about you that you would sit in your mother’s mathematics class in college as a child and you would correct her sometimes or solve all problems ahead of other students. Can you tell us a few stories?
My mother is a professor at Hofstra University in Long Island where I grew up. Sometimes I didn’t like school much so I would skip school and go with her to the university and sit in her classes. In the beginning, I wouldn’t understand anything. But I was so intrigued by what she said. Overtime, I started understanding things and really enjoyed it so much that I started speaking in her class. My mom told me that I corrected small errors she made in the class. Afterwards, she scolded me for it. That happened a couple of times and my mom was not expecting it at all.
Another thing which is unique about you is that you did your PhD under Andrew Wiles, the famous mathematician at Princeton who proved Fermat’s Last Theorem. It is said that within 18-24 months of finishing your PhD, you were offered a full-tenured professorship at Princeton. Tell us about it.
In the U.S., seniority is not a criterion in hiring, and this has a lot of benefits. I am someone who has benefited from that. After receiving my PhD, I proved some things which were considered important in the mathematics community. There was a healthy competition among universities to hire me. I didn’t understand what was going on and I was offered a post-doctoral position, an Assistant Professor position and a Professor position in various universities. Within few months, I started receiving 10-year professorship positions, which is the highest one can go in the hierarchy of promotions. This was just a year after my PhD. I was not doing any negotiation but just watching in awe. Princeton was the first to offer me a full professorship with tenure and it is also the place where my adviser Andrew Wiles was. So, it is nice to go to a place where you really wanted. I felt excited about Princeton and I ended up accepting it.
You are a great role model for people who are interested in a scholastic career and it is a great motivation for them to hear that within a year or so after completing your PhD at the age of 27-28, you became a full-tenure professor. There is a lesson that we can learn from it in India which is that seniority is not the criterion. Contribution of the candidate is important. Your example must be noted by all the decision makers in this country. Tell us something about your work in your PhD. You are supposed to have improved Carl Friedrich Gauss’ work on high composition by using the work of Brahmagupta in 600 AD as well as some of the theory behind Rubik’s cube. Tell us about it.
One of the interesting things that I learnt as a child was when I was skipping schools sometimes to come visit my grandparents in India. I was looking at one of the books in my grandfather’s shelf which was Brahmagupta’s Brāhmasphuṭasiddhānta. In that book, there is a beautiful theorem which says that if you have a number which is sum of two square numbers and when you multiply with another number which is sum of two square numbers, the result will be a sum of other two square numbers. For example, 10 is the sum of two square numbers nine and one. If we multiply 10 with 10, we get 100. And that is 36 for 64 which is also the sum of two square numbers. Brahmagupta has vast generalisation of this result. For example, if you have a number that is sum of square and twice the square and you multiply with other number which is sum of square and twice a square, you will again get a number which is sum of square and twice a square. And the whole sequence of generalisations leading to statements like this and I found this intriguing as you can try with numbers and see it is working. And that is very non-obvious. It is Brahmagupta’s identity in modern mathematics. This was back in year 628 when he discovered it. Over the next 1400 years, people have studied that law, generalised it and maybe the real great leap in understanding this composition law was achieved by Gauss in 1801 and his disquisitiones is where he discovered the most general form of this kind of statement that you have a number which can be expressed in this kind of quadratic way and you have another number which can be expressed in another quadratic way and you can predict that it will be represented in a third quadratic way and that is called Gauss composition. It is a vast generalisation of Brahmagupta’s identity. It intrigued me when I got to graduate school this book of Disquisitiones of Gauss is the foundation of a theory which is the area I work in. The problem was that this generalisation that Gauss composition which he wrote in the book took 20 pages of complicated Algebra to write out. I thought that is something so fundamental to my field, there must be a simple way of trying to understand it. That was a goal in the first year of graduate school and I was stuck on it for many months. This is what happens in mathematics: You never know when inspiration strikes. And my inspiration stuck when I was playing with Rubik’s cube and I thought what would happen when you put numbers on a Rubik’s cube’s corners? When you put numbers on its eight corners it turns out with certain manipulations, you can get a one or two-page derivation of Gauss’ law. It is a simple explanation of Gauss’ composition law. That formed the first chapter of my thesis. I thought I would graduate by just writing that one chapter of simple Gauss law. But then I realised that that perspective led to many other laws of vast generalisation of Gauss composition. In my thesis, I found 20 other laws like it using that perspective for which the inspiration came from Rubik’s cube. That’s how I ended up writing my thesis.
One of the common things we see among children is the fear of mathematics. I don’t know if it is because the way we teach mathematics, or if we make it more complex than what it is. But you have taught courses on mathematics of magic, tricks of cards at Princeton. Do you think that using recreational mathematics as a tool to teach some of the important principles of mathematics, at least at school or college level, is a good way? What would you prescribe?
The reason students fear mathematics is because of the way it is taught. It’s unnecessarily complicated. There is a lot of memorisation. It has a very little play in recreation. By definition, the most enjoyable things are the things that are recreational. Mathematics should be taught in a way which makes it recreational, fun, playful, creative and adventurous. That is what gives best feeling to students when they are doing mathematics. That is why in Princeton, where I teach the course, I start the class with a magic trick that intrigues everyone. In order for them to perform the trick by the end, they have to learn certain amount of mathematics. By seeing mathematics being applied this way, it makes it exciting for students. It can be done throughout the mathematics curriculum with some imagination. I view all the work as recreational. I enjoy what I do with the playing around with stuff and trying to find patterns, making discoveries and it is a big adventure. Somehow students don’t get to see that in their education or when they are learning mathematics. If we could show what mathematics is truly like and the way mathematicians view the subject, if we change the curriculum and make it more play-based, discovery-based, discussion-based, then it will make mathematics fun for students.
Even the theory of probability started when the mayor of one of the cities in Southern France requested Pascal to solve the problem on how they should divide the putty when their dice game was stopped somewhere in between.
Probability is such an important subject in mathematics and science but it has its origins in gambling and playing dice.
Can you give an example of how mathematics can be used in daily life by youngsters, college students or school children?
Most people don’t realise that mathematics, even the area in which I work, that is number theory, used to be an area where mathematicians though it could never be applied in any area of human endeavour. It was done for its beauty, as an art. And now, the number theory is used in everyone’s life every day. If you have a credit card or if you buy anything online, or you send an email or use your cellphone, you are using number theory because the security of all our communications is based on algorithms that use number theory which was developed by humans over 2,000 years ago, not knowing that it will be so important. During the World War II, mathematicians used to do number theory because they were afraid of developing mathematics that could be used for war. So, they wanted to work in an area where there was no application. In the U.S., number theory is a heavily funded branch in mathematics because it is important in making codes, defense communications, cellphone communications and internet transactions. Mathematics is used in everyone’s daily life in a very important way, even when we use cell phones. But we don’t think about it. Most people don’t explicitly do mathematics every day, but they are using mathematics in a fundamental way every day. But I urge people to think about it. When you are using your cellphone, it is nice to know what is happening? How is my message being encoded so that it is secured? Why is my credit card number and my identity secure when I am using it? It is a good thing for everyone to know the mathematics they are using and why they are using it, and not just use it blindly. Cryptography is based on the fact that if you take two 200-digit prime numbers and you multiply them; a computer can do that in a fraction of a second. But, if you ask another computer to do a reverse process, the best-known algorithm will take billions of years to go backwards. That is the basis of a lot of our cryptography and security.
You did something extraordinary towards your contribution to improve the education system of your country. You were part of a committee which will hopefully announce its recommendations in the weeks to come. At the same time, you had doctoral students at Princeton and you are one of the most respected mathematics professors; How would you guide a student? Is there a methodology that you could offer to people in our institutions to enthuse and provide lot of independent thinking opportunities to students so that at the end of their PhD, they are able to guide many more students and add value to society? Do you have any prescriptions for it?
I took a year to work for an education policy in India. I am grateful to my students for letting me be away that much. But is has benefits. Nowadays, we have e-mail, Skype, video conferencing. So, it is not a big deal to not be in the same place. I can be still be in touch with my students even though I was 10,000 miles away from them. As a PhD student, the goal is to learn to become independent and to be able to think of your own problems and to think of their own solutions. By being away, it encouraged them to learn to do that. That would often be the message in my e-mail to them with a slight direction as to where they should go. And then play around it with yourselves and discover things yourself. That process of discovery is what mathematicians are always engaging in and that is what they have to learn. If you teach one course where you show that process of discovery and then let them lose and try to do it themselves, that’s often the best way of teaching students to become independent thinkers and adventurers. It helped me be away as the dependence was not there.
There is a school of thought which has become predominant nowadays that research and teaching should go together. That’s the model that Harvard, MIT, Princeton, Cambridge, Oxford use. On the other hand, in Germany, they went through Max Planck Research Institute and universities. Before World War II, Germany was the place of producing extraordinary researchers. What is your view on the current model in India where there are research labs and universities? Should they not be combined so that the graduate, under-graduate students and research student work together in the same place?
You are right. It was a mistake post-independence in India to separate research and teaching. It is an artificial division in some sense. As a result, we have developed fantastic labs, research institutions but they are isolated islands. The researchers there don’t have the opportunity to pass on that knowledge to the next generation. Conversely, out of all the under-graduate students across the country, 93 percent go to state universities where essentially no research is happening. There are a few exceptions, but research is very limited in state universities. They don’t get to leave the university, knowing the cutting-edge of what’s happening in knowledge creation. It is very important to combine research and teaching to bring research into state universities. You can imagine how much manpower in research we are losing, knowing that 90 percent of faculties at universities are not encouraged to do research, but they have the highest level of knowledge with their PhDs. We must make it a part of our system like Harvard, Oxford and Princeton and like ancient Indian universities did like Nalanda, Takshashila which were well-known to be multi-disciplinary research universities where students came to from around the world. India started that model and left it and other countries are now benefiting with that model. It is time to bring that model back to India. Combine research and teaching again so that students can learn from a place where knowledge creation is taking place. So, they can become great innovators.
I have heard experts discuss and debate about the kind of inventions and proof that people come out with. Like Andrew Wiles who proved the Fermat’s law and, in the process, created a new field of mathematics. Such a long-complicated proof that he created a new field. There are other inventors, discoverers who come out with the simple Euclidean kind of proof. What is your view on these two models?
We need both. For some problems, there are elegant short solutions that take one moment of inspiration. Even though people did years of work on that problem, in the end, it was solved by that one moment of inspiration with an elegant solution. But I don’t think every problem has such a solution. For problems to solve them, you have to work hard at it for years with great focus as Andrew Wiles did. He had many moments of inspiration throughout the period of seven years when he was working on this problem but at the same time it required very hard work, dedication and focus. That model is needed too. After all, for 400 years people tried to find that one moment of inspiration of simple proof, but it didn’t exist. The way to solve it was hard work, focus and dedication with moments of inspiration. We need both, depending on the problem.
How was being guided by Andrew Wiles?
It was wonderful. He is an incredible mathematician but also a kind person. He was always generous with his time. I didn’t work in his area but having his great global perspective of mathematics and that’s what an advisor is really needed for; to provide a global view of mathematics, and that experience is what I learnt from him. That quality of being dedicated and being focused, looking at problems, not being afraid to think big; Just because a problem has not been solved in years, doesn’t meant that you should think it is impossible. You should still work for it. Those are the lessons I learnt from him. Having him there to tell me those things when I was working on problems, which I thought were impossible, was inspirational.