Thursday, September 26, 2013

Problem Posing in Mathematics

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 nth degree equation have?
(iii)           Can we find all the roots numerically or graphically?
for reference click here and here

(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 rth powers of the first n natural numbers where r = 4, 5, 6, …
for reference click here 

(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?
                               for reference click here

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.

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