What is the connection of APV’s practical math methods to the deeper question of the link syllabus, the inward journey, the question of what the self is?
I think math can do two things, at least. One is to make you more alert. And it
is a good test of honesty, perseverance, and creativity- making links in new areas. For instance, if you are solving a math problem and you are not very sure about a particular step, you feel that this is probably not 100% right, but you go ahead anyway and the answer is wrong, you learn to be more alert and more honest with yourself. Why did I go ahead when it felt wrong? Let me stop and think here. This training, I think, is very valuable for the inner journey as well. Not to cut things short, not to be evasive, not to avoid something that is unclear but to examine the entire problem, to be patient for an answer to present itself, and to only move ahead when the path appears. You see often in math classes the children become very competitive and just race for the right answer without really thinking about it. Here that is not the point. When a child says ‘okay, I have done it,’ we always ask them first how they did it, instead of just checking the answer. Especially children that have come here from other schools, they have had this training in competitiveness. But here it is always ‘how’- ‘how did you do it?’ is the question. And some children are very bright, but they go slowly through the problem, they are never the first to finish- and yet they discover new ways of approaching the problem.
Generally speaking, math is the most dreaded subject in schools- at least in India it is. Surveys were done, and most of them showed that math was the most feared subject among students, and I think the reason is because it is not genuine math that they teach, but very abstract math. So we say here that the more areas of the brain activated, the more connections the brain is constructing, the more creative it is, then the more conducive it is for the inner journey, for meditation. Then, if you look at math the way it is commonly presented, it seems that it is confined to the left brain, to logic. But what we try to do is to make it more holistic and connect it to the emotional and creative parts of the brain. So whenever you introduce a practical method with a 3-dimensional material, doing something in real space and time, that automatically brings in the right brain, because it is a real experience, a full experience. For instance, we have beads representing the place units, and the introduction of beads changes everything. If the tens unit is red, then your eyes become very important, because the colors become essential to the learning process. Then you pick up the bead and count with your hands, and touch is involved. Now, some people argue that this is only good for basic math operations- adding, subtracting, multiplying, etc.- but that as you move on to fractions and decimals and algebra, more advanced topics, then it becomes almost impossible to teach these things practically in a way that is not prohibitively time-consuming and confusing. But if we apply our mind there, we realize that without the practical methods the children at first don’t really understand what you are talking about with the more advanced topics. For instance, some time ago somebody was questioning the use of learning something like (a+b)^2 = a^2 + b^2 +2ab- in what way is it applicable to real life? Many people argue that math is not useful in real life. Now, it depends on what you mean by real life. If you mean that it is not practical in the sense that it is not useful in most of the ways that we earn money, then you are right, but you are missing the whole point. Why are you a human being, different from animals? Why did nature make you this way? Not just to live your practical life- food and clothing and shelter and entertainment and gratification of the senses, and so on. If that is what you mean by life, then it is not a human life but an animal life that you are talking about. And I agree that for that kind of life math is not necessary. But it is not necessary to study anything to live like that. I might be a good cook, I might break stones on the road, and I could still survive. That is not the point of education. The point is to learn about yourself and the world, to evolve. Sa vidya ya vimuktaye- if education is for the purpose of touching the infinite, that quantum physicists and mystics talk about, then we should talk in terms of expansion of the brain. So if you are approaching math merely logically and abstractly, then of course it is not useful in the so-called practical life, nor is it useful in the inner journey. But you can do math in such a way that you come to see many links in practical life, on the one hand, and it is also expanding the brain. For instance, (a+b)^2 can be introduced in a story. And by introducing a concept with a story you are already touching that part of the brain which deals with emotion, imagination, and creativity.
So if I am introducing it for the first time I might tell the story of this king who tells his gardeners to decorate all around his courtyard with flower pots. And the courtyard is covered in beautiful, rare marble tiles, each one square foot. And one of the gardeners slips and his flowerpot falls and cracks one of the tiles. So the king comes rushing out, and he is so in love with these tiles that he demands to know who did it, and when nobody steps forward he announces he will punish all of them if by sundown they have not replaced the square foot of marble. But he wants to make it difficult for them and says that they cannot bring one tile, but that they have to replace the tile with four tiles, two rectangles of equal size and two squares of unequal size that fit this space exactly. And they do not understand what the king is talking about, how to solve this problem, and so they go to the court mathematician secretly. And the mathematician tells them a simple trick. He gives them a piece of paper one square foot in size, and tells them to pick any point on the edge that is not the midpoint and make a square with the corner, then draw a vertical and a horizontal life from the corner of the square, and thus you have divided the sheet into two equal rectangles and two unequal squares. And you have the equation (a+b)^2. And by sundown they bring all these different combinations that satisfy the king’s problem, and the king is shocked, because he never thought there could be so many possible solutions. And he goes and asks the court mathematician, who says ‘King, there could be an almost infinite number of combinations.’ So when you begin with a story, and you demonstrate the math within the context of the story, you are slowly taking the children towards the practical application of this formula. So through these things you can link different areas of the brain and make math very engaging.
It seems that math is practice for truth-seeking in a closed, controlled system. The student learns how to approach a search for the truth with patience and openness and without ego. To get anywhere, you must engage with the problem on its own terms.
Yes, it is a subject where you cannot boast of your knowledge or give your personal opinion, which people are usually so quick to do. ‘I don’t think this way, I don’t agree’- but in math you must think in a certain way, broadly speaking, otherwise you will never get anywhere. You must give up intellectual vanity. So in this way math becomes a microcosm of the inner journey. And then math becomes very exciting, and in our school it is possibly the most cherished subject. Even though we don’t use practical methods for every subject; it seems that once the children have tasted a little bit of practical math they are ready to proceed in that topic without practical material. They seem to have internalized the memory of these practical methods. For instance, if a child has used beads for a number of years, and then they go to a higher class and do more arithmetic, the memory of that material probably comes back and helps them do the mental arithmetic. And then over time they rely on the practical memory less and less, but it is still there in the foundation.
And they will again benefit from practical methods when they begin a new topic. Like algebra- when a child begins algebra they need a lot of practical methods to communicate the topic to them in a strong, tangible way. What does x mean, for instance, in something like x + 5 = 10? This is a mystery for children. So what we do is we cover the beads. We cover 5 beads and pair them with 5 uncovered beads, and then we put 10 beads on the other side of the sheet. And you tell the children that in these two piles there are exactly the same number of beads. Here are 10, and here also there are 10. So how many must be covered? Let’s call that x, the covered one. And very soon they come to realize very tangibly that x means an unknown quantity, but not an inaccessibly unknown quantity. And they progress to understanding these topics internally very quickly, because in beginning with a practical demonstration there are so many more qualities for memory to hold onto- not only abstract qualities but sight and touch and everything that accompanies a physical object.