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Ramanujam is a Professor at the Institute of Mathematical Sciences, Chennai in India. His research work is principally in theory of computation and applications of mathematical logic in computer science. He has been involved in mathematics education at school level for nearly three decades and edits a monthly science magazine for children in Tamil called Thulir. Ramanujam was involved in the 2005 curriculum reform process of the Government of India and chaired the National Focus Group for the Teaching of Mathematics. Currently he is Vice President of the Mathematics Teachers Association (India).
1. What’s your earliest memory of doing mathematics?
When I was seven or eight years old – I cannot remember precisely – I used to go with my mother to the vegetable market and you would see somebody (quite often a woman) sitting there with all kinds of vegetables around. You’d ask, “How much is that?” and she would say “This is so much for 1 kilo” or “5 for 2 rupees.” Some would be rated in pieces, some in kilos. Some were grains and would have a different value. So yes, all different units. And my mother would buy a ¼ of a kilo of this, a kilo of that, 1.5 kilos of something, 5 pieces of something, and then the lady would go: “OK, this was so much, this was so much, so this is what you should pay.” Initially, I remember looking at it and being fascinated by it, thinking how does this person do it? I was just starting school and I was learning and being taught all these things, mental arithmetic, etc., 200 grams of something – how do you do it all in your head and then come up with the final number? And watching that also made me realise that people do use these things all the time in their head. Then, what I started doing was, at the time, I would try to do it myself. While the buying was going on, I would be putting things in the bag and also trying to calculate, like a race – can I get the answer at the same time? I really enjoyed that. It was much later that I realised that I was doing (mental) mathematics in my head.
I would say this was, as arithmetic goes, “real life arithmetic.” But when it comes to doing mathematics, with that kind of emphasis on doing mathematics, I think it took me a long time – all through school I was learning mathematics, which was something that was given to me, it was not something that was live. I think much later I realised there was more to doing your own mathematics – that you can actually make something of your own, come across something that is true or false and then you try to prove it, try to come up with a construction – that it is possible for you to do. That was a revelation. And then finally when I came to graduate studies, all the people around were proving theorems, while it was difficult enough for me to read books and papers and follow other people's theorems. It took a while to realise you can do it on your own too. But I think that journey all along – that doing mathematics is something that is much more than calculating, learning formulas, following whatever someone tells you, but actually creating something – that took a long time.
2. How has mathematics education changed in the time you have been involved in it?
I think, elementary education has changed enormously. I think it is a lot more tuned towards the psychology of children's learning now. I think research-based practices have influenced elementary education in terms of textbooks, pedagogy, teacher education. I’m not that sure about secondary education – I see it more or less pretty much as the same thing. In India, it is very complicated; mathematics at higher secondary level is the passport to higher education and everything gets lost in that.
In university education, perhaps the biggest change is due to computers – that’s a huge change from when I was learning to now. Computers have influenced it, and look at what the internet has done for university maths education (and indeed all education). Technology has made a significant difference.
To some extent, looking back, I am a little bit surprised that secondary education has not seen the kind of change one has seen in elementary education. This is (of course I suspect) due to the competitive nature of secondary education, because it has the shadow of the Board exams looming over it – this seems to matter even more for mathematics than for other disciplines. Almost everything is geared towards Board exams and then it becomes a whole bunch of tricks that you learn; the kind of joy you could get from explorations is really missed out on.
One significant change I have seen over the years is that geometry, I think, has become less important over time, I think especially spherical geometry. I recall projective geometry: we learnt a lot of it and enjoyed it as well. I don’t see much of that now. Of course, geometric proofs – where you could try proving things of your own – that is decreasing as well, mainly to accommodate many other things that have become important in the curriculum, I suppose. These are the main changes.
In elementary education, I’m not saying everything is very good, I think a lot needs to change. But one significant change is the kind of emphasis given to memorisation. A lot has changed from when I was young. It is a mixed bag, because while a lot of good has happened – the pluses of de-emphasising blind memorisation and more importance given to constructing meaning, for example – there are still many problems, and more change is needed.
3. Tell me about a time in your career when something totally flabbergasted you.
Many, many times – this happens often enough, I would say. It has happened several times that in research, when you’re reading a paper or following someone's work, which is very exciting, you realise, this is something I have been teaching for years, like a technique, I could have put this together in my head – but I didn’t. And it is in fact something very nice, because you learn along the way. And then there are moments of sheer panic when you see some problem and you think this is not going to work. You put in a lot of effort, build something and then you think it is completely wrong, those are terrible moments. But then (perhaps this is the benefit of age), you wait a little, and something happens, and of course, you learn something in the process.
In teaching, there are moments when you are doing something in the class and then a question comes: “Why this? Why not that?” And you are thrown off! You are really thrown off! Ok...why not that? (you ask yourself). That’s true! Sometimes, you can recover from that, or sometimes you come back to it later.
It also happens during interactions with school children. Oh, I have many, many stories of this kind of thing! I remember one episode – I was interacting with a bunch of 13-year-olds and I was talking about maths; I was trying to say that maths is everywhere, look around and with maths you can do all sorts of stuff. And at the end, this one boy comes up to me and says, “Oh, I really liked what you said, and I want to do a maths project.” And then he goes on, “I like spiders very much, can you tell me some maths project I can do with spiders?” This is a classic example of putting your foot in your mouth. I have been saying that maths is everywhere and only a child can ask such a question! And it's very normal for a child to like spiders or something exotic, and for the child to want to do a maths project on them!
You said flabbergasted, right? And here I am, supposed to be a mathematician talking about maths and here is a student asking me this question. I didn’t know what to say and on the spur of the moment I said, “Oh, have you seen a spider’s web? How does it look?” and he said, “Oh yeah! It is a sequence of hexagons with lines radiating off them...” (he didn’t use these exact words). And I said, “Why don’t you just explore that?” He was a little bit worried, because this hexagon may not have equal sides! In school, when you do things like this, they often have equal sides. So I said, “OK, explore, what do you know?” and he said, “I know area, I know perimeter”. So I said, “Yes, you can try them for this project” and then I went off and forgot about it. This was all part of a festival called Maths Expo. Children demonstrate their projects, and it is a lot of fun. When I went to the Expo and I saw this boy, he had one whole notebook full of things. He drew lots of spider webs, used formulas for the things he knew like perimeter and area, and if you look at spiders' webs, they’re always pulled in the corners. He didn’t know the Cartesian coordinate system, and what to do about fixing points, but then he just drew pictures and he had formulas for all sorts of things. Consider a spider in one place and an insect in another part of the web, what would be the shortest route the spider could follow to catch the insect? As I said, nobody had taught him Cartesian coordinates – he didn’t know it! But he came up with his own way of specifying things, although he didn’t know graphs and shortest path algorithms. But he wanted to create formulas for everything. So, I brought him to our institute to give a talk to a bunch of mathematicians because of what he did.
There are many such examples: initially it throws you – suddenly someone asks a problem, and the problem just surprises you! Then the fun starts and something happens. Sometimes you don’t succeed, but that’s ok.
4. Do you practise mathematics differently in company?
Absolutely – I enjoy mathematics in company, ok, we were talking about doing mathematics... I think one of the best kept secrets from me, all along in my school education and even undergraduate education, was that mathematics is a very social activity. Somehow, I always thought of it as a solitary activity, where you sit and think on your own. And it was only in graduate school that I saw how immensely social it is. It is when you start discussing mathematics, talking mathematics, when you argue, debate things – the way your mind works – drawing pictures on the board for people to comment on. Over the years I have learnt that in fact many problems that I think about, that I work on – they tend to get shelved. But when there is somebody I enjoy interacting with, discussing that problem with, they come alive. Many times, when I do work with my PhD students, it is the personality of the student that drives the problem. I’ve tried to stress the importance of collaborative learning and people working together in my entire professional life.
I mean, I have worked alone and written many papers by myself, so I do have some research experience alone. I could do a lot more on my own, but I don’t think the pleasure of working together matches anything I do alone. I do know certain mathematicians who work best alone, so there are people like that but by and large, mathematics seems to be an immensely social act.
In mathematics education one needs to emphasise this, because it seems to miss out on this social aspect far more than other disciplines do. I find that in humanities education and social sciences, working in company – doing things together – is the norm, yet in mathematics not so much. This enjoyment of working together was the best kept secret when I was young and I must say it still seems to be the best kept secret in school – that’s one thing that doesn’t seem to have changed very much over the years.
5. Do you think a brilliant maths teacher is born or made?
Ok, this is a no-brainer, I would say made – undoubtedly.
For me, it is very clear that practice makes the teacher, and practices get refined as you go along. In terms of delving into a discipline and working with your students and children, and the kind of learning that comes through taking children seriously, the kind of engagement, that is important. And we just talked about mathematics being immensely social – I would say the same for teaching as well. There is a lot of understanding of teaching that comes from a collegiate atmosphere, that comes from learning from each other. It may happen individually and in larger communities, but I think it certainly happens.
So, I do not see a brilliant maths teacher born in the sense of walking out and being brilliant. When could this happen – when you get a maths degree? When you get an education degree? I doubt it, because these are only qualifications. There is still this performance aspect – you stand there in front of students and you are on stage – I don’t see that any of the degrees can prepare you for that.
In fact, in the case of the mathematics teacher, there is what Felix Klein (1908) called the “double discontinuity”: as a student you learn certain maths at school, and then you go to university and the school maths has nothing to do with it – you restart from scratch. Then you go back to school and teach, and when you’re back in school it’s school maths. Students have to learn techniques, so you need to re-learn practices and techniques. Practice is the key – being a reflective teacher, using an opportunity to learn by yourself certainly helps.
If you're a researcher, one very good thing is that in any case, you are also learning all the time and therefore you are also able to appreciate the difficulties of learners. With all that, brilliance in teaching mathematics is about commitment to teaching maths, and you could say a whole bunch of things related to knowledge, practice and to being part of a community – that make a good teacher.
6. What’s the most fun a mathematician can have?
I guess, the same as anyone else. Your capacity for enjoyment and fun is something very human and I’d say it is for anyone. If you’re talking about specifically enjoying mathematics and the fun with mathematics, the great things would be being able to play with ideas and explore ideas, and problem solving. When you think that you have solved a complex problem – at different levels of engagement, ages, what you consider “complex” may change, but at any time when you are solving a complex problem – it brings a kind of intense enjoyment. Even when you're writing – there is a time you see something inside your head, you write it out, then it flows, there is a sense of communication. That is enjoyment. Perhaps it is akin to what musicians, artists, performers feel when they feel a sense of communication at an abstract level.
I’m not sure that fun is a good word for this, for what I’m saying – fun in the sense of engagement and pleasure I think is more common, but there is a certain level of enjoyment specific to maths.
7. Do you have a favourite maths joke?
There is this PhD comics series (xkcd) that I used to enjoy. And I remember one from that:
Two people are talking and one of them says:
“I wish I could do maths like I used to when I was young – it really doesn’t come easy later on in life. You know, maths is a game for the young and I need to just back down.”
The other person says:
"Hey! You're 13!”
And the first replies:
“Oh yeah, it’s time I accept that”.
I enjoyed that because there is such a fuss about maths being only for young people.
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