Those of us who love mathematics and teach it, see that it is intrinsically beautiful. Perhaps we would agree with J. Krishnamurti when he says 'mathematics is infinite order' - Why then are so many children in almost all cultures frightened or bored by mathematics? We as teachers seem to convey to the child a sense of fear and helplessness, instead of beauty, with regard to mathematics. Is it possible to teach in such a way that students learn mathematics with a sense of joy and excitement? Is there a way of teaching mathematics whereby both teacher and student learn also about themselves? In this short paper we explore some of these issues. Although our observations have come out of the teaching of mathematics, perhaps they are true for the teaching of other subjects as well.
We would like to first address the question - what is the right atmosphere for learning to take place? We have found that it is imperative for the child to feel completely secure in a classroom in order that learning takes place. For a child to feel completely secure there has to be an absence of fear, pressure and comparison. Although traditionally authority and comparison have been means by which students are motivated to learn a subject, we find to the contrary that authority breeds fear and comparison leads to a feeling of insecurity. When a child is subject to a great deal of pressure, we find that it kills her natural curiosity to learn and replaces it with a feeling of drudgery. We are then faced with the question — how does one bring about a quality of attention in the child which is not the result of fear? Can a teacher bring to the class a quality of attention in himself which communicates to the child, and which neither demands authority nor creates comparison?
We have found that a feeling of insecurity can become acute in some children when they study mathematics. Mathematics traditionally taught (at all levels) seems to give the impression that there is only one way to solve a given problem. There is really no room for discussion, where the student can contribute at a level comparable to the teacher. Since all information flows from the teacher, students immediately set him up as an authority.
But in fact the very nature of mathematics can be used to negate these impressions. Mathematical truths are not authoritative statements. The student can find out for herself what is true. The teacher can encourage the student to question everything that is said, to look for alternate ways of solving problems, so that the teacher and student can engage in free discussion. In addition, he can expose the students to open-ended problems, i.e., problems to which the teacher does not already know the solution. Fortunately there are many sources where one can find suitable material and we have included a short list at the end of this paper. Working on these problems together with the teacher removes, if only for a brief period, the distinction between the teacher and the taught. It also allows students to watch the problem solving process first-hand.
Mathematics lends itself naturally to a process of inquiry and exploration. The exploration demands clarity of thought, both on the part of the teacher and the student, without which one soon finds oneself in murky waters. It is our experience that students often make assumptions which come in the way of investigating a problem. When this is pointed out the student can see concretely for herself how any conclusion comes in the way of learning. A more general assumption that students often make is that there is but one method of solution to the problem, and that they should know this method before they even attempt the problem. The teacher can help dispel this assumption by encouraging the student to 'play' with a problem, to see what it means to stay with a question and let it unfold (see the books by Serge Lang, for an excellent example of how one can dialogue with students about mathematics). More importantly perhaps the teacher can convey, not just in mathematics but in life itself, that the solution to a problem lies in completely understanding the problem itself.
In almost all cultures students are exposed to mathematics from an early age. Children with a flair for mathematics are identified young and are considered 'bright' or very intelligent. As a result, those for whom mathematics does not come easily are left with a feeling of being inadequate, children are constantly compared, reinforcing this sense of inadequacy. But what is intelligence? Is it merely a matter of being talented at mathematics? To the contrary, we believe that the enjoyment of mathematics is neither the privilege of the few, nor the sole determinant of intelligence. In the 20th Century mathematics has become a tremendously important subject. This has been mainly because, with Galileo, scientists have come to believe that mathematics is the language of reality. However, the work of the logician Kurt Godel shows that modern mathematics with its axiomatic approach cannot be both consistent and complete at the same time. Godel's work seems to hint that if reason and logic are viewed as a system of thinking, then reality cannot be completely comprehended by logical thought. This, to us, is strongly reminiscent of Krishnamurti's assertion that truth is beyond thought. In an age where the prevailing world view is determined by science it is perhaps important for a teacher of mathematics to point out not only that mathematics is limited, but also that all knowledge is.
Does the gathering of knowledge take a great deal of time and effort? Many great mathematicians including Poincare have asserted that despite working very hard at a problem, often the solution comes effortlessly in a flash of insight. The insight in one sweep seems to put together all the pieces of the puzzle. This is something we experience even while teaching and learning elementary mathematics. As a teacher of mathematics one can encourage students, wherever possible, to try and solve problems intuitively rather than by a methodical step by step approach. The book by Martin Gardner is devoted to problems of this kind. One also wonders - is learning and understanding of oneself a matter of time and effort or is it a flash of insight in the present?
 Cohen, et aI, Student Research Projects in Calculus, The Mathematical Association if America, 1991.
 M. Gardner, New Mathematical Diversions from Scientific American, Simon and Schuster, 1966.
 M. Gardner, Aha!lnsight, w.H. Freeman, 1978.
 H.R. Jacobs, Mathematics a human endeavor, w.iI. Freeman, 1970.
 S. Lang, Math! Encounters with High School students, Springer- Verlag, 1984.
 S. Lang, The beauty of doing mathematics: three public dialogues, Springer-Verlag, 1985.