Dear
All,
A
famous quote about Isidor Isaac Rabi (born American physicist and Nobel
laureate recognized in 1944 for his discovery of nuclear magnetic resonance )
“My
mother made me a scientist without ever intending to. Every other Jewish mother
in Brooklyn would ask her child after school: So? Did you learn anything today?
But not my mother. She would say, "Did you ask a good question
today?" That difference--asking good questions--made me become a
scientist.”
We
all are aware that Problem Solving is the focus of learning and teaching of
Mathematics. However the problems have to be exciting, non-routine and
challenging. In order to get thrill and excitement in Mathematics, students and
teachers have to be trained not only in problem solving, but also in Problem Posing in Mathematics. Posing
problems in Mathematics is not as difficult as it may appear, if students can
learn some techniques for posing problems.
As
suggested by late Prof. J. N. Kapur, some techniques for posing problems are as
follows:
(i)
Generalising of known
results
(ii)
Extending known results
(iii)
Adding or removing some
of the conditions of the theorem and seeing whether the same result or a
modified result continues to hold under the modified conditions.
(iv)
Combining different
known results to get a new result.
(v)
Finding whether the
converse of a result is true.
(vi)
Finding new proofs of
known results or solving problems by alternative methods.
Here
are some examples:
Problems posed by
attempts to generalise or extend known results:
(1)
Known
Result: The
roots of quadratic equations.
Problems Posed:
(i) Can we get similar
expressions for the roots of third degree, fourth degree or higher degree
equations?
(ii) Can we say how many
roots will an n^{th} degree equation have?
(iii) Can we find all the
roots numerically or graphically?
(2)
Known
Result: We can in general construct a
triangle if three elements of the triangle, including length of one side, are
known.
Problems Posed:
(i)
Can we construct a
quadrilateral if the lengths of the four sides are given or if any four
elements out of the lengths of four sides, lengths of two diagonals, the
magnitudes of four angles are given?
(i) Do we require more than 4 elements for
constructing a unique quadrilateral?
(ii)
Similarly for a
pentagon, how many elements should be given?
(3) Known Result: The
sum of squares and cubes of the first n
natural numbers.
Problems Posed:
(i)
Can we find the sum of r^{th} powers of the first n natural numbers where r = 4, 5, 6, …
(4) Known Result: Given
two positive numbers a and b, we can find their HCF (h) and LCM (l).
Problems Posed:
(i)
Given h and l, can we find a and b?
(ii)
Given h and a, we can find l and b?
(iii)
Given any two of a, b, h, l, can we find the other two?
(5) Known Result: The
area of the square on the hypotenuse of a right-angled triangle is equal to sum
of the areas on the two sides of the right-angled triangle.
Problems Posed:
(i)
Will the same result
hold if we construct semicircles or similar triangles or equilateral triangles
or regular hexagons on the three sides?
(ii)
For what other shapes
will the same result hold?
Posing
new problems via Inverse Problems:
(1) Known Result: For
a given value of x the value of sin x, cos x, tan x are known.
Problems Posed:
(i)
Given the value of sin x or cos x or tan x, find the
value of x. (This solution is not
unique and this led to the development of theory of inverse trigonometric
functions.
(2) Known
Result: Given a function f(x),
find its derivative.
Problems Posed:
(i)
Given the derivative,
find the function of which it is the derivative. (Obviously there will be
infinitely many answers are possible).
These
are only sample problems provided. One may now begin to think of similar and
other such problems to arouse curiosity in Mathematics. We may not know all the
answers to the problems posed by us or others. But it is certain in search of
answers there will surely be a great deal of learning.
It
is said that “Focus on the journey, not the destination. Joy is found not in
finishing an activity but in doing it”.
Do give your reflections, ideas, suggestions in comment section.