NDTV: Hello and welcome from me and from students
all over India who are studying in the finest of colleges here in Delhi. We are all here
together to do just one thing, to listen and learn from this bewildering genius Manjul
Bhargava. Manjul is one of the greatest minds India has ever created. He has been awarded
a prize, which I consider even greater than the Nobel Prize. Prof Bhargava has recently
won the Fields Medal in Mathematics. Why do I say it’s possibly even greater than the Nobel
Prize? – because this highest global award in Mathematics is given only once in every 4
years. How about a round of applause for this young gentleman. And actually you will notice,
as the evening goes on, that his modesty and simplicity is really truly Indian. Why do
I call him a bewildering genius? Because of what he did to actually win that mathematics award. It’s simple really. He won it for and I quote developing powerful new methods in the geometry of numbers, which he applied to count rings of small rank and to bound the average rank of elliptical curves. Simple as that right? You got it? Simple stuff. So what has he done? He has counted rings
of small rank and bound the average rank of elliptical curves. I don’t think I would
even ask you to explain that in English. Prof Bhargava: Thanks
NDTV: But in English, just basically what does it mean for an average viewer?
Prof Bhargava: Well the subject that I work in is the area of number theory.
NDTV: Number theory. Prof Bhargava: So, number theory basically
is the art of understanding the whole numbers, the numbers we use to count 1234 and 0 and -1 and -2. So these are the whole numbers and number theory is about the science and art
of understanding those numbers, understanding the special sequences of those numbers, like
square numbers, the prime numbers and finally it’s about solving equations whose solutions are whole numbers. NDTV: So it’s like relationships between numbers?
Prof Bhargava: That’s right. NDTV: Basically that’s number theory
Prof Bhargava: Exactly. Simple NDTV: Simple, all of you can do that, right.
So based on that when you, I just want to get an idea of you in Princeton, sitting and
roaming with a whole lot of other mathematicians. What do you talk about? I mean would any one
of us understand anything? Prof Bhargava: Well, I mean a lot of the mathematics that goes on at Princeton is with the students. So students like all of you and so I think
the audience here, since there are mostly mathematics students here, would understand.
NDTV: They would understand what you are talking about because I have listened to some of your
speeches to mathematicians and you know, I did study maths to a certain level, but it’s
just within 10 seconds I was out of it, but I guess a lot of you here would understand. Prof Bhargava: There are many different levels. NDTV: Of course, yeah. Yes, your Mother taught you maths. She was a mathematician herself.
Prof Bhargava: And is. Yeah. NDTV: Yes still is, of course, and you used
to get bored with other subjects and go into a room and chat with her about maths and stuff?
Prof Bhargava: Yes, I always liked mathematics since very, very small, maybe two or three
years old, and I used to love doing little math puzzles and coming up to all maths questions
and then I’d go and bother her and say.. NDTV: Is this the answer?
Prof Bhargava: Yes, is this the answer? Can you tell me how to do it and her answer always
was you should figure it out yourself and then come tell me. And so, I used to just enjoy
just playing around with mathematics and if I ever got stuck, my mother was there, as
a resource, and that was something very useful. NDTV: But obviously she did something in the
way she taught you, the way she inspired you in maths that created your interest in it,
and you’ve often said, that to some extent, while Indians may be good at maths but the teaching is a bit robotic. How should maths be taught? Because a lot of people say,
“Oh God, maths I can’t stand it.” Prof Bhargava: Well, I sympathise with people
who say that because I myself really didn’t like mathematics class.
NDTV: Class, oh really Prof Bhargava: So I used to do anything I
could to avoid going to mathematics class. I would skip out on any excuse I could find… So I used
to do everything I could to avoid going to mathematics class. One thing I did was just
take some several months out of school and come to Jaipur here in India, where I didn’t
have to go to school. In fact one time, my parents and grandparents put me in school
here, but I dropped out after a couple of weeks.
NDTV: Oh I see… Prof Bhargava: But I knew I loved mathematics,
but I didn’t like the way mathematics was taught in school and so I used to find my
own ways to make mathematics fun. I used to play around with mathematics, do puzzles,
ask my mom for nice suggestions that would be fun for me. My grandfather, who was a Sanskrit
scholar, he had lots of mathematics textbooks on his shelf from Ancient India, so I used
to read those. So, I had a very non-standard mathematics education and I really enjoyed
it. I think that contributed a lot NDTV: So enjoyment is the key…
Prof Bhargava: My suggestion for teaching maths in schools is really make mathematics
fun because mathematics actually is fun. A lot of us really don’t realise that when we go to grade school mathematics classes because it is taught in a robotic way. You’re given a problem and
you’re asked to memorise steps to solve it and then you just blindly apply it and try
to be careful so you don’t make an error, but you don’t really know why you’re doing
those steps. It shouldn’t be like that. It should really about being creative; coming
up with those steps on your own and everyone will come up with a different way to do it.
Mathematics is great because there is always one answer, but there are many ways to come
to that answer. And in school we are taught one way to come to that answer, and mathematics is about coming up with your own creative ways to come to that one right answer. There’s
not one path and everybody has their personal path that they can discover and that’s what
makes it fun. That’s the adventurous part of mathematics, the creative part of mathematics
and we miss that in the way mathematics is taught.
NDTV: So are you saying this teaching of mathematics slightly robotic around the world and worse in India, or India is like the rest of the world, it’s generally taught in a robotic way?
Prof Bhargava: It’s pretty bad throughout the world, but in India, here it’s even more
about memorisation than it is in other parts of the world, so maybe it goes to an even further extent.
NDTV: It’s fascinating that you hated maths class, you just did it yourself, bad lesson
for all of you, maybe it’s a bloody good lesson actually…
If any questions now, we can start taking questions and then… you’re from Rajasthan? Student 1: Yeah… First of all, it’s an honour to be speaking to you Professor, my question is that how do you think we can make maths
classes more fun, because I know you have a class for your first year students in which
you teach maths through magic tricks? So can you recommend some methods for the teachers
here in India, you know, how do we make maths fun for people like me who hate it?
Prof Bhargava: It’s a good question, one of the most common questions I get when I’ve been on tours for these past few months talking about mathematics. The most common question is – why is mathematics class always so dry, so boring, so all about memorisation and rote when we hear mathematics is an art and a creative subject, but why do we not see that in school? One
thing that I have been doing at Princeton University for the past three years – I developed a course, for freshmen, for first years, where we teach mathematics through poetry, classical
music, classical Indian music, magic tricks as you said, and games; so those are the
four ingredients of the course through which I get to teach fundamental mathematical concepts.
So most classes begin with a card trick that’s based on a mathematical concept and when you
see this card trick, there’s nothing you can do but want to know how it worked. And for
the card tricks that I do or for the rope tricks or for the various other kinds of magic
tricks that I do, there is a fundamental math concept that you have to learn in order to
know how it works, and everyone is so immediately excited about knowing how did it work. And
so, while they learn how it works and while they learn how to do the magic trick, they
can’t avoid but also learn in the process the fundamental mathematical concept. And
when you learn it that way, there’s no way that you’ll ever forget it because you have
that visual of the trick, the memory of the trick of how it worked and there’s no way
you’ll forget the mathematical concept. NDTV: So if they come and say how does it
work, you say go figure it out yourself? Prof Bhargava: Oh yes, so I teach in 3 hour
seminars. I start with the trick, then I have everyone discuss, okay, what are the things
that went into the trick? What are the things that could’ve been happening and then I give
little hints and everyone has to think about it themselves, and then everyone has to sort
of come up with their own way of understanding how it went. And by the end of the class,
they know how to do the trick and they know the mathematical concept and since they discovered it on their own, there’s no way they’ll ever forget it.
NDTV: Right, exactly and as you say, it’s a visual cum curiosity cum something practical, so you’ll never forget that. Rather than just on a black board. Prof Bhargava: And the same happens there
with games, lots of fundamental games that have basic mathematic concepts embedded in
them. If you know those math concepts, you can always win the game. So I have these people
play games with each other, and eventually some people start winning because they start
seeing what math concept will allow them to win and eventually, everyone figures it out,
and by the end of the class they can go and play their friends who are not in that class
and beat them, and the reason is because they figured out that math concept on their own
during this class. NDTV: That’s why you’re seen at Las Vegas
all the time at the Black Jack table Prof Bhargava: Actually, that’s another common question I get when I’ve been giving lectured tours about this kind of way of teaching. Can this
be used in Las Vegas? And the answer is yes, but I don’t use it that way.
NDTV: So the answer is yes, so that’s even more worrying, the signals you’re giving.
I don’t know what’s going to happen to this serious bunch of students, any other questions?
Student 2: It’s truly an honour to meet you in person. My question to you is that since
you mentioned that as a student you hated maths classes, could you please tell us about
your typical day in a classroom? Did you always prefer the first seat or you know, just like
any other student, most of the students prefer the last bench, so most of the time where
were you found? Prof Bhargava: I preferred, I should probably
not being say this, I preferred not going at all. When I did go, I would hide. I wouldn’t
go right to the back, but you know, I’d hide somewhere. I shouldn’t say that about all
my classes. There were many classes that I liked and I had some fantastic teachers, both
in New York and in Jaipur, when I went to school here. And when I really did like the
class, I’d want to sit in the front because I was so excited about that and when I was less
excited about the class, I’d prefer to hide, so it really depended on the quality of the
teaching and I think that’s true about everyone. When you’re excited you want to go sit up
front and when you’re less excited, but you have to go, then you try to hide. So a couple
of math teachers that I had were actually really good and those were years when I did go to
every single class, and I was excited about it and I’d sit up closer, but most years I
tried to escape. NDTV: Fall asleep kind of thing. I must
say there are shows where, good and bad. When we had a show where we had doctors do live
phone-ins and answering questions, we had one guy phoned in and said, ‘I have a terrible sleep problem. I can’t sleep. I’ve tried every pill’ asking the doctor, ‘what should I do, there’s only one thing that puts me to sleep and that’s when Prannoy Roy comes on the 9:00 news, live’.
And of course, our producers play this again and again. So in the sense, sometimes you fall
asleep in class or nearly or look the other way and think of something else?
Prof Bhargava: I try not to do that, but I often do think about my own things. And another
way to make maths class, I mean when maths class is sometimes boring, you can make it
fun by instead of those doing those rote memorisation that you’ve been asked to
do, try to come up with the solution on your own. Maybe just take hints by just taking
quick peeks at some of the steps, but just really try to fill it in on your own,
and often you’ll find that you’ll come up with different steps that lead but, of course,
math only has one answer. So if you come up with steps that are correct logically, you’ll always end up with the right answer, but since you came up with it yourself, you’ll never forget it,
and that’s one way to make maths class more fun. Do it on your own.
NDTV: Are you all going to do that? You are also a musician, sorry before I ask you that, I wanted to ask you, when did you actually figure out that you are actually quite good at this, that
you are just outstanding? Which point was your turning point?
Prof Bhargava: I think I’ve never even thought in those terms. I knew I liked it and so,
I would do it whenever I had the chance to. I went to a school that didn’t really have
that much math enrichment, it wasn’t one of the top subjects at the public school that
I went to, so I didn’t really think in those terms. I avoided math class as I said
and I would do math on my own and I did well in all the exams. You know I always got a
perfect score on all the exams but I didn’t know what that meant in particular. But when I really got excited about math and realized that’s definitely what I want to do for a living, when I got
to college at Harvard, and that was the first time when I met lots of students who loved
mathematics, before it was just me, so that was a real eye-opening experience for me.
I was at Harvard, doing mathematics with all the other students at Harvard and then I thought,
maybe this is something I should do because I can do it at the same level as the other students here. NDTV: Yes young man. Student 3: Leonardo Da Vinci quoted that we
need to understand science of art and then art of science to excel in life. So is there
something like mathematics of art and art of mathematics? Prof Bhargava: Definitely. Mathematics should be thought of as an art. Sometimes we don’t see that
when we do mathematics in grade school, but just like any art, music, painting, mathematics
is a creative subject. Coming up with the theorems is an art, finding the way to understand why
that theorem is true is an art, everyone comes up with a different answer to the same question
and the reason is that there are different ways of understanding things, different ways of expressing yourselves, just like in any art. Mathematics is very much like that and one reason why
I teach my course through poetry, through magic, these are all arts, through music,
through games, because these are all different kinds of art and mathematics closely connects
to all of them. It should really be taught that way in my opinion. Mathematics is often taught like a science but a lot of people don’t know that its origins, especially in India, is in the
arts, and we shouldn’t forget that artistic aspect. We should be using both sides of the
brain to understand mathematical problems. NDTV: You have said that there is some mathematical basis to music as well, tabla for example, what is the connection? This is very tough for me to understand, maybe
others do, but what is the connection? Prof Bhargava: Tabla. Tabla is all about rhythms,
time cycles, trying to fit various pieces into a rhythm cycle and so, the arrangements
that one practices as a tabla player uses a lot of mathematics. One example that I always
start with in the class that I teach, just to get people to realise that mathematics
is connected with poetry and music, is an example of Hemachandra. That is always one
that I start with because it brings in tabla and poetry and mathematics right away. So,
in Sanskrit poetry, there are two kinds of syllables, there is a notion of laghu syllable
and guru syllable, how many people have heard of laghu and guru before?
NDTV: That’s about 5 percent, 10 percent. Prof Bhargava: So, yes, that’s one thing I
feel should be there in mathematics. The notion of laghu and guru is something that led to
some of the most fundamental breakthroughs in mathematics, believe it or not, in ancient
times in India. So this notion of a laghu syllable and a guru syllable, a short syllable
and a long syllable, so in any kind, every kind of poetry in the world, there is a notion
of stressed syllable and unstressed syllable. When you say poetry, some syllables are stressed and some are unstressed. In Sanskrit, it goes a step further. Stressed syllables are long
and unstressed syllables are short and short syllables take one beat of time to say and
a long syllable takes two beats of time to say. So what’s peculiar about Sanskrit is
that a long syllable takes exactly twice as long to say as a short syllable. So when you
recite Sanskrit poetry, all the syllables will be like this, some will last one beat
of time and some will last two beats of time. The short are one beat and the long are two
beats, and so this is very peculiar about Sanskrit, and as a result of the set-up of
Sanskrit poetry, lots of ancient poets considered lots of mathematical questions that related
to this one beat, two beat set-up of Sanskrit poetry. So one basic question that would come
up, if you’re writing poetry and you have 8 beats left in your stanza and you need to
fill it with long and short syllables, where a long syllable takes two beats and a short
syllable takes one beat, how many ways can you fill in 8 beats with long syllables and
short syllables, where long syllables are two beats and short syllables are one?
NDTV: Don’t give us the answer, does anyone know?
Prof Bhargava: What would be your guess? NDTV: Oh wow, come on.
Audience: Permutations and combinations? Prof Bhargava: Yes, it is about permutations and combinations, but it’s still beyond what you learned in school. What would be your
guess, 8 beats filling it with longs and shorts, long two beats and short one beat? Student: 4 short beats and 2 long beats. NDTV: How many variations can you make? Prof Bhargava: How many ways can you do it?
So you could do long, long, long, long or you could do short, short, short, short, short, short, short, short NDTV: 8 times.
Prof Bhargava: Or you could do short, short, long short, long, long, long, short, short,
long, so I actually clap these rhythms with my class and we try to figure out how many
there are. NDTV: Sorry, say it again.
Student 4: 8th Fibonacci number. Prof Bhargava: Yes, so he knows the answer.
NDTV: So it just went straight above my head. Prof Bhargava: So the answer is 34. So most
people when they first hear this question, they think oh maybe there are 8 ways, 10 ways,
the answer is actually 34, which is more than what most people expect it to be. And in this
ancient Indian work of Hemachandra, which I read to the class because it’s written in
poetry, the answer, so it’s a poetic question, it’s a question about poetry and the answer
is written in poetry. So it’s something that really excites people, it excited me when
I was a child so I like to share it, and it excites students. So Hemachandra’s answer was as follows: write down the numbers 1 and 2, and then every number you write down subsequently
should be the sum of the previous two numbers that you wrote down. So 1, 2, then 1 plus
2 is 3, then 2 plus 3 is 5, then 3 plus 5 is 8, then 5 plus 8 is 13, then 8 plus 13
is 21 and then 13 plus 21 is 34 and so on. The 8th number that you write down will give
you the number of rhythms that have 8 beats, which is 34.
NDTV: Wow, that’s amazing actually. Prof Bhargava: And this goes back to the year
1050, so these numbers may be familiar to many people, right. They are called the Fibonacci
numbers in India. Even though in India they were discovered pre-Fibonacci, by Hemachandra
in this work on poetry, which I find very amazing. So when we are taught Fibonacci numbers
in school, maybe this is true in India as well, I’ve definitely seen it in some Indian
schools, they teach it the way Fibonacci came up with it, which is about the problem about
rabbits. This incestuous problem about rabbits where brothers and sisters are mating, and
it’s a very unnatural act, but the real place where it came up was actually in poetry, and the reason that it came up is a very natural one. You need to know the answer, if you want
to know how many ways you can fill in the rest of the poem if you have a certain number of beats left. So that’s an example of the kind of way mathematics came up in poetry in ancient times and still comes up today. When people compose Marathi poetry or Kannada poetry, the set up of poetry is very similar in those languages as well.