Embedded in our brains is an extraordinary ability: the ability to form concepts; the ability to abstract common features and shared qualities from collections of objects or phenomena. It is this ability that lies behind the creation of language, and it is this that enables us to “invent” numbers. To understand what this means, think of a number, say 3. Is 3 a thing? Can it be located somewhere? No, it cannot; but our brains have the ability to see the quality of “threeness” in collections of objects: three fingers, three birds, three kittens, three puppies, three people – the feature they share is the quality of threeness. This ability is intrinsic to the very structure of our brains. Were it not there, we would never be able to learn the concept of number (or any other such concept, because any concept is essentially an abstraction).
Even in something as simple as tally counting – creating a 1-1 correspondence between a set of objects and a set of tally marks – our brains show an innate ability for abstraction: by willfully disregarding the particularities of the various objects and instead considering them as faceless entities. Realization of this insight has pedagogical consequence; for, as has been wisely said, “Concepts are caught, not taught”. It is only by actual contact with collections of objects that concepts form in one's brain. How exactly this happens is still not well understood, but I recall a comment which goes back to Socrates ( the teacher's role is akin to
that of a midwife who assists in delivery).
The invention of algebra represents one more step up the ladder of abstraction. To illustrate what this means, let us examine these number facts: 1+3 = 4, 3+5 = 8, 5+7 = 12, 7+9 = 16, 9+11 = 20. We see a clear pattern: the sum of two consecutive odd numbers is always a multiple of 4. This statement cannot be verified by listing all the possibilities, for there are too many of them – indeed, infinitely many. But we can use algebraic methods! We only have to translate the observation into the algebraic statement
(2n-1) + (2n+1) = 4n;
this instantly proves the statement. Such is the power of algebra and also the power of abstraction and this ability too is intrinsic to our brains.
Another feature intrinsic to the brain is the desire and capacity for play. Most mammals seem to have it, as we see in the play patterns of their young ones – and what a pleasing sight it can be, to watch kittens or puppies or baby monkeys at play! But human beings have a further ability: that of bringing patterns into their play. When our love of play combines with the number concept and with our love of patterns, Mathematics is born. For Mathematics is essentially the science of pattern. It is crucial to understand the element of play in
Mathematics; for one is told, repeatedly, of the utility of Mathematics, how it plays a central role in so many areas of life, and how it is so important to one's career. But the element of play gets passed over in this viewpoint; the subject becomes something one must know, compulsorily, and the stage is set for a long term fearful relationship with the subject.
From the earliest times – in Babylon, Greece, China, India there has been a playful fascination with number patterns and geometrical shapes one can associate with numbers. From this are born number families – prime numbers, triangular numbers, square numbers, and so on. Let us illustrate what the term “pattern” means in this context. We subdivide the counting numbers 1, 2, 3, 4, 5, 6, 7, 8,... into two families, the odd numbers (1, 3, 5, 7, 9, 11, ...), and the even numbers (2, 4, 6, 8, 10, 12, ...). If we keep a running total of the odd numbers here is what we get: 1, 1+3 = 4, 1+3+5 = 9, 1+3+5+7 = 16, 1+3+5+7+9 = 25. Well! We have obtained the list of perfect squares! There is a wonderful way we can show the connection between sums of consecutive odd numbers and the square numbers; it is pleasing to behold and incisive in its power at the same time. All we have to do is to examine the picture below: this property is closely related to one about the
triangular numbers: the sequence 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ... formed by making a running total of the counting numbers: 1, 1+2 = 3, 1+2+3 = 6, 1+2+3+4 = 10, etc. They are so called because we can associate triangular shapes with these numbers.
There are so many topics in which we can bring out the theme of pattern and play in Mathematics:
1.Magic squares (arranging a given set of 9 numbers in a 3 by 3 array, or 16 numbers in a 4 by 4 array, so that the row sums, column sums, diagonal sums are all the same); not only do these bring out nice number relationships, but in the course of the study one learns about symmetry.
2. Cryptarithms (solving arithmetic problems in which digits have been substituted by letters; for example, ON + ON + ON + ON = GO; many simple but pleasing arithmetical insights emerge from the study of such
problems);
3.Digital patterns in the powers of 2 (list the units digits of the successive powers of 2; what do you notice? Now do the same with the powers of 3; what do you notice?) These examples are woven around the theme of number, but the principle extends to geometry in an obvious way. Here we study topics like rangoli and kolam; paper folding; designs made with circles; and so on.
Alongside such activities, teachers could also raise questions relating to the role of Mathematics in society, for discussion with students and fellow teachers; e.g., questions relating to the use of Mathematics for destructive purposes, or more generally, "When is it appropriate to use Mathematics?"; or the question of why society would want to support mathematical activity. After all, most artists find patrons or buyers for their art work, but mathematicians do not sell theorems for a living! Is it that policy makers see Mathematics as a useful tool, and thus enable people in this field to sustain themselves, by teaching or doing useful Mathematics? The notion of usefulness takes us back to the question of appropriateness of usage. Such questions are not generally seen as fitting into a Mathematics class, but there is clearly a place for them in promoting a culture of discussion and inquiry. We need not try to make a complete listing here – it is not possible, because it is too large a list, and ever on the increase. Instead, we wish only to emphasize here that
pattern and play are crucial to the teaching of Mathematics, for pedagogic as well as psychological reasons. A great opportunity is lost when we make Mathematics into a heavy and serious subject reserved for the highly talented, and done under an atmosphere of heavy competition. It denies the experience of Mathematics to so many.
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