How To Teach Maths?

A recurring question which keeps coming up in discussions is how mathematics should be taught. There is a strong view that given the computing power which is available to us, we should relook the basic maths curriculum. So i went looking, and found this video which i feel looks at the problem in a logical way.

Conrad Wolfram is giving some compelling reasoning for why maths education should change, and gives a description of how it should change, too.

Hand-culcating the mundane way should not be the focus on the curriculum. Rather, the focus should be on building and enforcing the concepts, and leave the calculating to computers. In other words, as he says, students should be taught the procedures which define fundamental concepts, but the implementation of those should be left to computer programs. For example, students should know what a square root it, how it is calculated, but they shouldnt have to calculate, beyond illustrations. And here is the cool part he says … focus on teaching students how to write programs to calculate square roots, rather than having them mechanically execute the procedure for calculating. This will immensely help students clarify their concepts (how can one write a program without understanding the underlying principles to a very large extent?), while at the same time help them become more comfortable with the concept of application of these concepts. In other words, our mathematics curriculum should stress understanding and application (application to real world problems is a very good way of teaching these concepts) rather than stress on the mundane calculations which stress out students as well as parents alike. After all, why should a child lose marks in an exam (thats what happens) if he or she takes the square of 5 to be 10 when all the conceptual aspects of the solution are correct, and the only mistake is a calculation mistake?

Connect this with the post i had written earlier, and a rather innovative picture of mathematics teaching emerges.

Teaching Maths Without Words

Thats a somewhat crazy question. I came across this video, and i would say its quite a nice watch.

This video makes a compelling case for the why and the how of teaching maths in a visual way. After all, maths is not about words or languages, is it? If so, why should mathematics education be so language heavy? It should be about the concepts of mathematics. And i have found that the visual impact of mathematics is quite powerful. I tried this trick with my then 10-year-old … I introduced him to the concepts of integers, and multiplication with negative numbers without bringing in negative numbers to begin with.

We started with directions. Left and right. So, theres 3 to the left and 2 to the right. Or, 4 to the right and 5 to the right. Now, adding the 4 to the right and the 5 to the right is easy, but how does one add 3 to the left and 2 to the right? This is where the concept of the negative sign came into the picture. And once this was done, then it was a simple extension of this to see how multiplication with negative numbers simply changed the direction and nothing else.

What this did is help him build a mental picture of these concepts. In my experience as a trainer, i have found that these mental pictures are far more enduring than theoretical concepts. To take an example, i used to teach the concept of min-max planning, or the sawtooth curve, using the analogy of mom planning to go out and buy rice. Now, a few days after the class, the students, even if they had forgotten the min-max planning theory, or the sawtooth curve, still remembered the rice story, and this helped them to relate to the concept. The rice story here helped build a mental picture which is more enduring than theoretical concepts. So this isnt just about children learning mathematics, but also about adults learning.

Now, knowing him, he would have built a picture of Ben10 doing something which saves the world from the wildest aliens imaginable while multiplying two negative numbers, but hey, thats a picture i can live with!

Oh by the way, where there are integers, can vectors be far behind?


I have been having discussions with some friends about the gamification of education, and one theme which is coming out is the question … What about learning for the sake of knowledge? After all, with gamification, we are changing the process of learning into achieving some sort of recognition, whether it be in the form of peer recognition, or in the form of reaching some levels in the game model. So some folks are asking whether we shall be able to retain the ideal of learning ofr the sake of learning, of getting to know the world around us.

This, i feel, is an important question. But the answers are not too difficult to find. After all, today the way education is looked at is from the prism of exams, and the marks a student gets in these exams. Arent these marks also a form of recognition? If they are, and i feel so, then maybe its a better idea to change the form of recognition which is inherent as a part of the system to a form which is more fun, and which creates real motivation to learn. After all, how many of us studied in school with the aim of learning for the sake of learning?

Another aspect to understand here is that in schools, students need to learn a wide variety of subjects, from maths and science to history, and the arts. Is it really a fact that students would have an interest in all these subjects? After all, learning for learning’s sake happens when you have an inherent interest in a particular subject, which pulls you to study that subject. This means that learning for learning’s sake typically happens in one or two subjects. On the other hand, in school, students are not meant to be learning only one subject, but rather, gain an appreciation of, and learn about, the world around, which encompasses many subjects, and this is why it isnt possible for teachers to grab the attention of students for everything they teach.

So while i feel there is importance in the ideal of learning for the sake of learning, this can happen only when people have matured and have understood what they feel pulled towards. In schools, it is theoretically impossible for this to be achieved, and this is why gamification, or creating the framework for education which makes it more engaging is very important. After all, how will the child know what interests him unless he or she is exposed to all the different subjects, and this is achieved by the education which is given at schools.

What is Gamification?

This is one of the buzzwords today, and probably not very well understood. What is gamification, and why is it important? Does it mean learning by playing games? Or marketing through games? So heres a short video which i thought covered the ideas quite well, giving some nice examples of the different aspects of what makes gamification important.

The video, i feel, is spot on. Gamification helps create more interest, more engagement from the audience through giving them the sense of participation in something which is fun. This participation leads to a sense of achievement and progress towards something definite (whether it is a badge, or completing a level), and fosters a level of somewhat fun competition.

Though i feel one of the aspects which could be added to the list of things which are important building blocks of gamification is recognition. The competition pillar, i feel, instead of pride, could do better with recognition. This, i feel, is more important when you look at gamification from the perspective of education. While the competition aspect is important, it is more important to build a collaborative environment for students to learn in, because in that way, students can focus more on what they are learning, rather than focusing on a competitive aspect of learning, or focusing only on exam marks and rankings in class.

Of course, an important area where the principles of gamification need to be looked at very seriously is education, because by doing so, we could make education, and the process of learning fun, and student-centric.

Heres a video which gives a very nice and simple way of applying gamification to education.

This video gives a simple framework for applying the principles of gamification to the process of learning. As you can see, in theory, its quite simple to do, but it represents such a significant change from the system that exists today, that doing something like this may require quite a bit of doing.

Instead of having children getting marks which mean something to the children only in a relative sense, and hence make children look at how well they are doing compared to other students, gamification can help shift the focus of children to what they are learning, which level they have reached with respect to every topic, and which topics they need to focus on to reach higher levels.

These principles, coupled with the idea of structuring the education delivery process as i had written about earlier, could bring about a sea-change in the level of engagement, and hence the levels of learning, of students.

Apart from this, the video focuses on the process of education. Another aspect, which is probably for another day, is the gamification of content, which, i feel, is an equally important aspect of making education more engaging, fun, and useful.

Teaching … The Other Way Round?

Came across this really nice video on youtube … about Khan Academy Discovery Lab.

Today, the way children are taught in conventional schools is starting with theory. So, they are taught some theory, and then they get to see the application of this theory. And this is where children sometimes switch off. This is because children are natural at interacting with the world around them, and thats how they learn best. By seeing, by doing, and not by theory.

However, this approach could be a different way of teaching, far more effective. Start with the real world phenomena. Let the children do things which will help them to experience a phenomenon, and once they have had their fun, and are comfortable with the phenomenon, then lead them to the theory behind it, which describes why the phenomenon or experiment they did works in the particular way that it did.

After all, thats what science is all about, isnt it? And by science I mean all subjects which relate to facts, and here observation (whether this be an experiment or discerning patterns in natural phenomena) serves as the basis for theory, which is the tool to explain why things work the way they do. It is a decoding of these phenomena, not their definition. This method, being in synch with the fundamentals

The Tigress …

This one is from Jim Corbett‘s My India:

I once saw a tigress stalking a month-old kid. The ground was very open and the kid saw the tigress while she was still some distance away and started bleating, whereupon the tigress gave up her stalk and walked straight up to it. When the tigress had approached to wihin a few yards, the kid went forward to greet her, and on reaching the tigress it stretched ot its neck and put up its head to smell her. For the duration of a few heartbeats the month-old kid and the Queen of the Forest stood nose to nose, and then the queen walked off in the direction from which she had come.

Wonderful passage … just tells the sheer grace, elegance (a oneness maybe?) which comes with real power.

Stories …

As i am writing this, i am watching an animated movie … Alice in Wonderland, with my son. Not qute sure who between the two of us is enjoying it more though. And this got me thinking. When we were children, folks at home were distressed about the prevalence of comics which were distracting us from reading books, just as now we fret over how TV has taken over the lives of children and is distracting them from some good reading.

Well, fact is, earlier, folks had books, and they read them. Then books got distilled into comics, and we, as children, read them (though not only as children … Quite a few of us read them even now), and learnt quite a bit from them. I remember some really happy times reading Amar Chitra Katha, which introduced us children to the rich beauty of mythology and of history. And today, i feel, movies, especially animated ones, are quite an effective introduction to history, mythology, and heritage. After all, animation adds to the rich beauty of the stories, and makes the stories enjoyable to an audience of children. Till then, a very happy unbirthday to you …

Determinism, Computability, Observer Effect

I am reading a fascinating book, The Emperor’s New Mind, by Sir Roger Penrose. In the book, he has very forcefully brought out the idea that the deterministic view of science, developed over the past five centuries, taken to an extreme area by those supporting Artificial Intelligence as a viable alternative to Human Intelligence, is largely misplaced, and that while there are certain aspects which are deterministic in nature, there are many others which are not and even others which are not computable even though deterministic.

Whats the difference? To my understanding, deterministic is the property of a system which tells us that there is some logic, or rule, or algorithm by which given the current styate of the systesm (in sufficient detail, lets add for good measure), the stateof the system at another point, in space, time, or both, can be determines. Computability, on the other hand, is the property which defines the possibility of this determinism being carried out. In some scenarios, this may not be possible to compute because there may be operations which are mathematically not possible, for example, division by zero. Thats a basic one (and the one I understand), though I am told there are others.

The question of determinism and computability apart, because not what I think I am writing about (though with topics like these ones, can never be sure), what I am writing about is my own puny interpretation of the observer effect. For those not familiar with that (either definitionally or conceptually), let me juwst mention the scenario has has described:

In a two-slit experiment, one possibility to describe the interference effect could be the photons of light get split (or maybe the vibrations of the wave function do, I am not quite sure which … or shall we say I am uncertain which?) and each part goes through one of the slits. This can be found out using particle detectors, which would be able to observe either full or non particles. However, if a particle detector is placed at one of the slits, so the observer can see which slit the electron or photon went through, the interference pattern disappears. For the interference to happen, there must be a lack of knowledge about which slit the electron went through. In other words, an observer, by his very presence can make the experiment unfeasible.

Interesting, this. Lets look at this in detail. Who is the observer here. Or rather, what is the observer here? The particle detector. And what is the cause behind the particle detector being there? The cause was human. So, shall we say that the observer in this scenario is human? If we were to say that, then we would also need to agree that what this means is that the presence of the human observer impacts the outcome of the experiment.

Why would that be? Are subatomic particles or photons allergic to human beings? I dont think so. If they were, we would all be invisible because photons would stay away from us. And nobody would ever get an electric shock because electrons would avoid the human body. Do you think I am rambling? In a word, I am. But not from here.

If these particles or waves (take your pick) are not allergic to observer presence then there must be a rule of nature which leads to the phenomenon observed. Is this rule only with respect to the scenario described or can it be generalized? If we were to generalize it, then would it be an over-generalization to say that there are some aspects of nature which are not meant to be observed or measured? We cannot say that these phenomena dont occur because we cant observe them, because we can observe their manifestation. But we have to accept that we are not meant to observe the phenomenon itself. Question then is are there other such phenomena? There are phenomena which have been predicted but not yet observed, but this doesnt mean they would never be observed. There could, I believe, more such scenarios which would arise from developments in scientific thought. And this is where, I believe, we need to understand that not everything is meant to be observed or measures at the material level. That there are things which lend themselves more to thought than to observation, things study of the material universe may never be able to define.

India Questions

As the new year is here, and i am reading, or rather re-reading a book which i find fascinating, some of the questions which i had asked when i first read this book, and have been asking since, without getting complete replies, are coming back. At a time when the nation looks forward to another year of prosperity and growth, these questions are ones which arent even related to these, but rather, are questions which are slightly more basic. I am not the first one to ask these questions, nor would i be the last. But coming back to the book i am reading, this is The Discovery of India. In this book, Panditji has explored the idea of India as a nation, has asked some questions of her, and tried to find answers to some of these questions. I am asking not necessarily the same question, but some of these questions share a context.

India is a large, highly diverse nation. We are a nation poised on the threshold of leading the world into a wonderful future. And yet, there are these questions. As a nation, we are a political and economic entity. We are also a civilization, though, and it is from this aspect that questions are coming. The question is simple … What is it that defines India? India is a land of diversity. The Meitei are very different from the Gujarati, and the Kashmiri are very different from the Malayali. And this dimension of region is only one dimension which defines the variety of India. Then of course there is religion, and there is caste, and a number of socio-economic dimensions which define the great variety of India. And yet, there is something which underlies all of these dimensions, which threads all of this diversity into the garland of India.

What i am asking is simply this … what is this something? Is it a shared culture? Culture is shared to an extent. Is it a shared history? Again, shared to an extent. Is it s shared cause? To an extent. Is it a combination of all of these? What is it that makes each of us stand to attention at the singing of the National Anthem? What is it that makes us watch the Republic Day Parade? What is it that makes us shout, sing, dance, cheer when India is playing and winning at cricket (more so when against Pakistan)? If all of these dimensions were flowers, then what is that thread that brings these flowers together into the garland we love … India?

Expect more such musings as i read this book. These questions havent yet been answered completely.

2 States

I took some time getting roud to reading it, but now that i have read it, i found the book quite nice. At places its hilarious, at places brings out the struggle of the people trying to overcome stereotypes about others who are different from us, culturally, or in any other way. At the same time, the humour in the book comes from these stereotypes.

Rather than stereotypes, we could look at them as caricatures. What i liked particularly the idiosyncracies of individuals which are then projected as traits of the species. What comes to mind is Krish’s wedding where the bride’s uncles are passing around copies of The Hindu (after all, the most popular newspaper in Chennai), and the groom’s uncle passing around strips of Saridon (a hangover from the drinking of the previous evening). And then of course, the drinking and dancing … Punjabbiyaan noo te nachchan da bahanna chaahida e! Punjabis just need an excuse to dance. Whats hilarious is how these caricatures bring out the contrasts of two sets of people very different from each other, but as Mr. Hindu-addict Grumpyswamy says …

Yes, the Tamilian in me is a little disappointed. But the Indian in me is quite happy. And more than anything, the human being in me is happy. After all, we’ve decided to use this opportunity to create more loved ones for ourselves.

And the fitting reply from Rajji Mama …

Ladies and gentlemen of Tamil Nadu, thnak you very much. Now we invite you to some Punjabi-style dancing with the DJ at the backside.

With more than a little bit of alcohol lubricating the dance steps.

I did particularly like Rajji Mama, Sarita Massi, and definitely Mr. Hindu-addict Grumpyswamy. What do you say … Kirron Kher for Krish’s Mom, Neetu Singh for Shipra Massi, Rishi Kapoor for Krish’s Dad, Pavan Malhotra for Rajji Mama, Anant Mahadevan for Ananya’s Dad? Sounds like could make a nice movie? Suggestions for Ananya’s Mom’s role? Go on, write your comments …