Convergent vs Divergent Thinking - K Bhanumoorthy

 

Thinking Mathematically Part - 3

Convergent vs Divergent Thinking

 “It is clear that the chief end of mathematical study must be to make the students think.”

                 ----- John Wesley Young, American mathematician.

 

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                                                                   Convergent Thinking: 

Converge means coming closer together/come to/ towards the same point, with reference to psychological aspects mean to follow only on well-established facts. In biological aspects means evolutionary convergence. In mathematical terms, given a series approaching a definite limit in case a more number of terms are added. There are logarithmic series, exponential series geometric series, other kind of series. All human beings use convergent thinking in some process of our daily life.

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 An example as x tends to infinity implies 1 / x tends to zero (0).

Y= 1 /x, converges to zero as x increases, the limiting value of y is zero since y can be made as small as one thinks of. The line y = 0 (the x axis is known as asymptote of the function.

Let me give another example. Observe the sequence 128,64, 32, 16, 8,4, 2,1, 0.5,0.25. ---

Every number is half (1/2) of the previous number, the limiting value is zero (0), in other words come closer to zero.

Observe the sequence, 3,6,12,24, 48,96, 192, 384, 768, 1536, ----, every number is double the previous number. In this case the limit tending to infinite.

 Convergent thinking helps an individual or in other terms allowing to connect different elements in a question. It is nothing but choosing a correct/ opt method, to solve the problem. It’s a matter of befitting/ correct strategies.

Divergent thinking: 

Divergent means be different from, varying/ dissimilar/ unlike/ unalike. If we think on psychologically, it is variety of thoughts, assumption more in numbers. Mathematically to say increase indefinitely, moving away from normal path. It’s a process /method used to generate creative ideas, finding out many possible solutions Divergent thinking had been coined by Mr. P Guilford in 1956. 

Christopher Danielson had written a number of books where he mentioned   about divergent thinking. Divergent thinking helps in self-entrepreneurs, and also the people who lead the organizations. Creative thinking helps divergent thinking. Open ended questions, brain storming questions generate divergent thinking. It’s a journey of thinking, how one ‘s mind works. In brief I would say thinking outside the box. In mathematics, activities promote divergent thinking. The ideas that flow through one’s mind are in flexible way and displayed. 

Different age people think /respond in different ways. Divergent questions, have no specific answers, also generate divergent thinking and encourage students to think more. Brain storming sessions in a class shall develop divergent thinking skills. Even this can be tried out, a topic may be given to students, may be asked to submit their ideas, children’s creative play also develop divergent thinking. There is a playful approach in this sort of questions. One who thinks what to count is matter of thinking, rather how does one view.

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An example: we know 1+1 =2.  Suppose I write 1+1 = 6, how,  

In what way it is true.  1car + 1 scooter = 6 wheels.

1 goat + 1 man = 6 legs. In the same manner we can think of plenty, yes, a sort of divergent thinking.

Another example: given 123456789. = 100. One can use any fundamental operations of arithmetic (+, -.X, ÷) How then, the answer is given below.

1+2+3+4+5+6+7+8x9 = 100.

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IN 1887, Emmit invented folding bike. There are quite number of things, to use the optimum space in aa house/any place, many, like, folding chairs folding dining tables, folding cots, folding cup-boards, folding cycles, plenty of things so to say. All these things emerged out, on account of divergent thinking. Could I say! .

To conclude human beings are blessed with thinking skills, one among the 6 senses. Let us utilize them with positive thinking  in a proper way, utilize them for the benefit of humanity and make this beautiful land of earth, sustain peace, happiness prevail in the whole universe.

A quote from Lao Tzu.

“Kindness in words creates confidence. Kindness in thinking creates profoundness. Kindness in giving creates love.”