At
the threshold, our psyche changes, getting prepared for entry into a
venerated new realm. It is the first step into the realm. It must not be
high. Mathematics is as old as the human race, it attracted and was
attended by the best brains and served the humblest through the ages. It
keeps growing and expanding in new directions; new concepts giving
birth to new definitions, more theorems and their proofs following.
There are simple (expected) results not necessarily having easy proofs −
like the four colour problem and Fermat’s last theorem. There are
bewildering results, which likes of Ramanujan alone know how it occurred
to them (like approximations to π). In the textbooks on Mathematics,
and also in the classrooms, big space-time is spent on the proofs of the
theorems. Proofs are like a defence of the theorem to the (ignorant)
judges. But someone will prepare the defence (proof) only if he or she
is able to see the theorem (the truth).
The observations which help a person in seeing and arriving at the truth (theorem) convincingly; in the absence of which the drudgery of building of the proof will not be taken up; deserve greater importance in the learning of Mathematics. The observations (called examples or counter-examples in Mathematics) precede the theorems, definitions and even the concept itself. At the threshold of Mathematics, shall we wait a while, enjoying our first observations!
The
real encounter with great things will take time and effort. If the
teaching of Mathematics follows this sequence, beginning with dis-
cussions on examples leading to the concept, it would provide the
student with the joy of discovery at every level of learning and train
him/her better in Mathematical reasoning. The common practice however
has been that the definitions are presented to the learner as if
obtained from the blue and theorems obtained through a magic wand. And
then examples are presented to illustrate the definitions and theorems.
However, the scientific learning should involve exposure to a variety of
examples and situations that take a student to such a height that the
concept and related theorems twinkle to him simultaneously. The granting
of the incubation period is essential before the natural birth of a
concept or definition and their properties (theorems) to a learner. I do
believe that Mathematics Laboratory work must not be only the Physical
verification of the results but include experiments that lead you to the
results.
At the threshold of Mathematics, in the following few activities, an attempt is made to illustrate the proposal made in the preceding paragraphs. Activities suggested are not meant to be read as a part of text by the students. It is a sharing of one teacher to other fellow-teachers; hoping them to develop and share activities for myriad of concepts and theorems they are coming across. In this attempt, few topics have been taken up at each level; early school, secondary school and college.
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