August 29, 2023

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.

 

Courtesy :- Convergent Vs. Divergent Thinking: Know the Real Difference (pinterest.com)

                                                                   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.


Courtesy:- https://www.sketchbubble.com/en/presentation-convergent-thinking.html


 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.

Courtesy:- Divergent Thinking PowerPoint Template - PPT Slides (sketchbubble.com)

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.

                  Courtesy:-  Convergent Thinking PowerPoint Template (sketchbubble.com)

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.”

 

August 20, 2023

Thinking Mathematically Part - 2 By K Bhanumoorthy

   Thinking Mathematically Part - II

 “No problem can be solved until it is reduced to some simple form. The changing of a vague difficulty into a specific, concrete form is a very essential element in thinking.” ----    J.P Morgan

 Concrete Thinking: The word concrete means definite/ real/ genuine/ factual/substantial/ solid/ tangible so on. It is centered on one’s observations, feeling, and experiences felt in the present non-abstract concepts. It is part of leaning process, possible be developed at workplace. It impacts the way people perceive the world. Jean Piaget, a noted Psychologist, said young children do move gradually from concrete to abstract thinking. It is nothing but reflecting one’s experience; they believe in what do they see and hear, trust the words what they hear. They would accept what one says, there is no question of asking questions. It is also called literal thinking, look for physical objects for proper understanding.

Example: Giving directions to others one has to use the language carefully. “Put the book on the table/ red book on the table.” The complex ideas are divided into small, parts /components and then each is concentrated separately, stay more mindful. problems are solved efficiently. In base ten for addition blocks of ten be used to carry out the addition. Using teaching aids make the learning easy, the ideas are made by simple showing objects. 

Example: To show a cube, defining what is an edge, face, vertices, can easily how many edges, faces, vertices are there in a cube. (E 12, F6, V8); 

Another example 4x3 ie 3 times of 4 can be shown by diagrammatic representation, 4things in a group, for 3 groups as 4 + 4+ 4 = 12. Multiplication as repeated addition.

Abstract Thinking: The word means theoretical /having no material existence/ notional / philosophical. There is abstraction in Mathematics, separated from physical and social world. Algebraic concepts are abstract. We say variables (dependent, independent) functions (onto, into one-one), equations, degree of equation, polynomials so on. When we count a number of objects after counting, we represent by a number (a symbol). The number can be representing the number of animals, tables in a class room, the number of days, the number of fruits, so on.  Abstract thinking is an ability to absorb through our senses, high order reasoning skills. It could be humor, wisdom, success, love, happiness, imagination. In brief a way of thinking not connected with physical objects. An example I would like to give. Suppose a project is given, to be completed. Normally one shall look for required steps and go ahead to complete. The question why this project is carried out, the purpose/need, is abstract thinking.

Lateral Thinking: It means indirect /finding other alternatives. It is essence of creativity. It is nothing but moving from known area to new areas. Lateral thinking is linked with Edward Do Bono. In fact, he popularized this kind of thinking. Another popular mathematician from Hungary George Polya. He had written a book “How to solve it”. A problem can be solved in different ways using different methods. Now a days we call it as an open-ended approach. In brief I would say, finding other alternatives to solve or find the solution.

Let me give some examples: 

Example :- A null among say 8 people. Let us say this game is under elimination. Every player loses one game. The question is how matches are to be played to find the winner. The answer is 7 matches are to be played. How arrived the answer?

Let us say the competitors be named as A, B, C, D, E, F, G, H. 

To start with, games between A&B, C& D, E&F, G&H ----------------------- 4 matches.

 Assuming B, C, F, H wins, now the games between B&C, F&H ------------2matches.

Assuming C, H wins, now the game will be between C & H    -------------- 1match

So total 7 matches are to be played. This kind of problem quoted by Mathematician---George Polya.

Another example: If electricity goes quite often one could think of UPS, Generator, another alternative. In case of un-employment problem, one can think of starting own business. It is the question, how one could be able to solve not only problems in mathematics but in real life situation too. Education, in that way helps an individual, value-oriented education is preparation of one’s life.

Another example:  Given 9 match sticks, one can easily 4 equilateral triangles, it will be diagram in 2D.

                                    

Instead given only 6 sticks, can we form 4 triangles of same shape and size. Yes, possible if one who thinks, beyond.


3 Dimension, forming a tetrahedron.    ---- be continued.

This article is written by K Bhanumoorthy sir, who is a retired KV Principal. He is a passionate writer and loves mathematics deeply. You can contact him by sending your mail to - bhanu_haresh@yahoo.co.in

August 1, 2023

Thinking Mathematically - K Bhanumoorthy

 

THINKING

“To think is easy. To act is difficult. To act as one thinks is the most difficult of all.” 

                                               ---- Johann Wolfgang Von Goethe.

 Introduction: 

Let me write about thinking. Think means as per dictionary, it means use your mind to form ideas, solve problems, imagine, cogitate, ruminate, ponder, consider, reflect, so on. There are various ways of thinking. I would like to discuss (restricted to mathematics). Thinking could be creative, analytical, critical, concrete, abstract, divergent, convergent, lateral. The way one approaches problems and solutions relies more on how the brain manages processes information rather than the facts presented.

I. Creative thinking: This way of thinking allows anyone to generate new ideas, a new design is formulated to get the required results. There is involvement, artistic, original, being resourceful. Students shall be able to frame own questions, open ended approach in solving problems. It’s matter of divergent skills. Self-imposed conditions are removed. I mean to say overcome fixation.



                                                                        IMAGE SOURCE

An example: a question of this kind. 

I) Find two numbers where the sum is 9, the difference is 2. Ans: 3.5, 5.5.


                                                               IMAGE SOURCE CREDIT - VISIT

ii) There are 3 boys, 3 girls. They are standing in a line. How many students are there? Everyone knows there are 6 students in total. The question can be modified. In how many ways the students can stand in a line. (ans.4x3x2x1= 24 ways). If they are arranged in a line with condition no students of same gender stand together. How many ways can be arranged?  Then the answer is 2x2x2= 8ways.


iii) The volume of a tank is given as 30 Sq m. what could be different measurements? 



One answer is 3m, 2m,5m.

II. Analytical Thinking: This is detailed study. This is very much required in mathematics. Analytical thinking simplifies mathematics learning, especially in proving theorems. Mathematic is language of pattern. Concept in mathematics requires analytical thinking.



Example: When we increase the perimeter of a rectangle, does the area increase?

1)      Length 6cm, breadth 4cm. Then Area is 6x4 = 24 sq cm. Perimeter is 2(6+4) 20 cm.

2)      l =8cm, b =3cm, A = 24 sq cm, Perimeter is 2(8+3) 22 cm. Many other possibilities are there.

 We can conclude when we increase the perimeter of rectangle, the area does not increase.

III. Critical thinking: This is an important factor, in solving mathematics problems. It is an ability to analyse the given data and then proceed, to find solution. This kind of thinking we do require in one’s personal life and professional life also. This kind of thinking strengthens an individual to identify the sources and to draw the


conclusions. Helps in Mathematics worksheets, puzzles, multiple choice questions, matching questions so on.

Examples: i) An electrical engineer evaluate materials suitable for particular job. Ii) A manager of a company / shop analyses customer feedback for better customer service.

      ii) What do the 4 animals in common? Cat, fox, squirrel, mouse. (Ans: tail.)

      iii) Find the odd man out? Tea, Diesel, coffee, apple juice. (Ans: diesel)

 

                                               ----- be continued

By - K Bhanumoorthy

 











    

 

 

 

July 10, 2023

What Mathematicians/Scientist say about Indian Mathematics

1.

“India is, the cradle of the human race, the birthplace of human speech, the mother of history, the grandmother of legend, and the great grandmother of tradition. our most valuable and most instructive materials in the history of man are treasured up in India only.” – Mark Twain

2.

“Gravitation was known to the Hindus (Indians) before the birth of Newton. The system of blood circulation was discovered by them centuries before Harvey was heard of.”                                         – P. Johnstone

3

“We owe a lot to the Indians, who taught us how to count, without which no worthwhile scientific discovery could have been made.” – Albert Einstein

4.

It is high time that the full story of Indian mathematics from Vedic times through 1600 became generally known. I am not minimizing the genius of the Greeks and their wonderful invention of pure mathematics, but other peoples have been doing math in different ways, and they have often attained the same goals independently. Rigorous mathematics in the Greek style should not be seen as the only way to gain mathematical knowledge. In India, where concrete applications were never far from theory, justifications were more informal and mostly verbal rather than written. One should also recall that the European Enlightenment was an orgy of correct and important but semi rigorous math in which Greek ideals were forgotten. The recent episodes with deep mathematics flowing from quantum field and string theory teach us the same lesson: that the muse of mathematics can be wooed in many different ways and her secrets teased out of her. And so they were in India … David Mumford 

5.

“After the conversations about Indian philosophy, some of the ideas of Quantum Physics that had seemed so crazy suddenly made much more sense.” – W. Heisenberg (German Physicist, 1901-1976)

6.

“Our present knowledge of the nervous system fits in so accurately with the internal description of the human body given in the Vedas (5000 years ago). Then the question arises whether the Vedas are religious books or books on anatomy of the nervous system and medicine.” – B G Rele (The Vedic Gods)

7.

"How great is the science which revealed itself in the Sulba, and how meagre is my intellect! I have aspired to cross the unconquerable ocean in, a mere raft." ---- B Datta 

8.

 The Vedic Hindu, in his great quest of the Para-vidya, Satyasya Sa tyam , made progress in the Apara-Vidya, including the various arts and sciences, to a considerable extent, and with a completeness which is unparalleled in antiquity." -----B Datta 

9.

The Indians have an extremely subtle and penetrating intellect, and when it comes to arithmetic, geometry and other such advanced disciplines, other ideas must make way for theirs. "I will omit all discussion of the science of the Indians, a people not the same as the Syrians; of their subtle discoveries in astronomy, discoveries that are more ingenious than those of the Greeks and the Babylonians; and of their valuable methods of calculation which surpass description."----- (Severus Sebokht in 662 AD) "

10.

"Indeed, if one understands by algebra. the application of arithmetical operations to complex magnitudes of all sorts, whether rational or irrational numbers or spacemagnitudes, then the learned Brahmins of Hindostan are the real inventors of algebra." ----- H Hanke

11.

"The intellectual potentialities of the Indian nation are unlimited and not many years would perhaps be needed before I ndia can take a worthy place in world Mathematics." ---(Andre Weil in 1936)

12.

"India has given to antiquity the earliest scientific physicians, and, according to Sir William Hunter, she has even contributed to modern medical science by the discovery of various chemicals and by teaching you how to reform misshapen ears and noses. Even more it has done in mathematics, for algebra, geometry, astronomy, and the triumph of modern science - mixed mathematics - were all invented in India, just so much as the ten numerals, the very cornerstone of all present civilization, were discovered in India, and, are in reality, sanskrit words." ----Swami Vivekananda

13.

The Hindus solved problems in interest, discount, partnership, alligation, summation of arithmetical and geometric series, and devised rules for determining the numbers of combinations and permutations. It may here be added that chess, the profoundest of all games, had its origin in India."--- F Cajori

14.

" ... it is remarkable to what extent Indian mathematics enters into the science of our time. Both the form and the spirit of the arithmetic and algebra of modern times are essentially Indian. Think of our notation of numbers, brought to perfection by the Hindus, think of the Indian arithmetical operations nearly as perfect as our own, think of their elegant algebraical methods, and then judge whether the Brahmins on the banks of the Ganges are not entitled to some credit."--- F Cajori 

15.

"Incomparably greater progress than in the solution of determinate equations was made by the Hindus in the treatment of indeterminate equations. Indeterminate analysis was a subject to which the Hindu mind showed a happy adaptation." --   F Cajori 

16.

"Unfortunately, some of the most brilliant results in indeterminate analysis, found in the Hindu works, reached Europe too late to exert the influence they would have exerted, had they come two or three centuries earlier." ---F Cajori

17.

"As I look back upon the history of my country, I do not find in the whole world anothercountry which has done quite so much for the improvement of the human mind. Therefore I have no words of condemnation for my nation. I tell them, 'You have done well; only try to do better.' Great things have been done in the past in this land, and there is both time and room for greater things to be done yet ... Our ancestors did great things in the past, but we have to grow into a fuller life and march beyond even their great achievement. " --- Swami Vivekananda (Complete Works Vol III p.195)

18.

"There is no question of the superiority of the Hindus over their rivals in the perfection to which they brought the science. Not only is Aryabhatta superior to Diaphantus (as is shown by his knowledge of the resolution of equations involving several unknown quantities, and in general method of  resolving all indeterminate problems of at least the first degree), but he and his successors press hard upon the discoveries of algebraists who lived almost in our own time!"
                                           - Sir Mountstuart Elphinstone (1779 - 1859)  

19.

 It is very important to note that some 2,500 years ago at the least Pythagoras went from Samos to the Ganges to learn geometry...But he would certainly not have undertaken such a strange journey had the reputation of the Brahmins' science not been long established in Europe..."

    - Francois Marie Arouet Voltaire (1694-1774) France's greatest writer and philosopher.

20.
     " It is more likely that Pythagoras was influenced by India than by Egypt. Almost all the theories, religions, philosophical and mathematical taught by the Pythagoreans, were known in India in the sixth century B.C., and the Pythagoreans, like the Jains and the Buddhists, refrained from the destruction of life and eating meat and regarded certain vegetables such as beans as taboo. "It seems that the so-called Pythagorean theorem of the quadrature of the hypotenuse was already known to the Indians in the older Vedic times, and thus before Pythagoras.”

Prof. H. G. Rawlinson 19th Century, British historian and author
21.

 “We owe a lot to the Indians, who taught us how to count, without which no worthwhile scientific discovery could have been made.” -    Albert Einstein

22.

It is certain that the Vedic Indians knew something of astronomy and that it had a high utilitarian value for them as it did for all peoples of antiquity. The Vedic priests had to make careful calculations of times for their rituals and sacrifices, and also had to determine the time of sowing and harvest. Moreover, astronomical periods played an important role in Vedic thought for they were successive parts of the ever-returning cosmic cycle.”- Fritjof Capra (Physicist)

23.

It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.”

 Pierre-Simon Laplace

24.

The Indian system of counting is probably the most successful intellectual innovation ever devised by human beings. It has been universally adopted. ...It is the nearest thing we have to a universal language. --- John D. Barrow

        25.
        “Hindu cosmology  gives a time-scale for the earth and the universe which is consonant with that of modern scientific cosmology”, as opposed to the limited Biblical-Quranic cosmology, which was protected against more far-sighted alternatives by a vigilant religious orthodoxy. ---- Carl Sagan

        26.

        It is clear how much we owe to this brilliant civilization, and not only in the field of arithmetic; by opening the way to the generalization of the concept of the number, the Indian scholars enabled the rapid development of mathematics and exact sciences. The discoveries of these men doubtless required much time and imagination and, above all, a great ability for abstract thinking. These major discoveries took place within an environment which was at once mystical, philosophical, religious, cosmological, mythological and metaphysical.-       Georges Ifrah.

        27.

        I once heard, and I think it is true, that only one man in the world—some Indian mathematician—understood the mathematics of string theory in eleven-dimensional space, and he dreamed it. ---- Mullis, Kary B  

        28.

        In the Shulba Sutra appended to Baudhayana’s Shrauta Sutra, mathematical instructions are given for the construction of Vedic altars. One of its remarkable contributions is the theorem usually ascribed to Pythagoras, first for the special case of a square (the form in which it was discovered), then for the general case of the rectangle: “The diagonal of the rectangle produces the combined surface which the length and the breadth produce separately.” --- Elst, Koenraad

         29.

        I shall not now speak of the knowledge of the Hindus … of their suitable discoveries in the science of astronomy—discoveries even more ingenious than those of the Greeks and Babylonians, of their rational system of mathematics, or of their method of calculation which no words can praise strongly enough; I mean the system using nine symbols. --- Severus Sebokht

         

        30.

        When I read the Bhagavad Gita and reflect about how God created this universe everything else seems so superfluous. ---Albert Einstein

        31.

        “Modern physics has shown that the rhythm of creation and destruction is not only manifest in the turn of the seasons and in the birth and death of all living creatures, but is also the very essence of inorganic matter,” and that “For the modern physicists, then, Shiva’s dance is the dance of subatomic matter.”

        “Hundreds of years ago, Indian artists created visual images of dancing Shivas in a beautiful series of bronzes. In our time, physicists have used the most advanced technology to portray the patterns of the cosmic dance. The metaphor of the cosmic dance thus unifies ancient mythology, religious art and modern physics.”-       Fritjof Kapra

        32.

        Everything has come down to us from the banks of Ganges ---- Francois Voltaire

        33.

        If there is a country on earth which can justify the claim the honour of having been the cradle of the Human race or at least the scene of primitive civilization, the successive developments of which carried into all parts of the ancient world and even beyond the blessings of knowledge which is the second life of man, that country is assuredly India. -- Friedrich Creuzer - German philologist and archaeologist.

        34.  

        Access to the Vedas is the greatest privilege this century may claim over all previous centuries.” --- Julius Robert Oppenheime
        3535. The Bhagavad Gita… is the most beautiful philosophical song existing in any known tongue. -- --- Julius Robert Oppenheimer - Inventor of Atom Bomb

        36. 
        A millennium before Europeans were willing to divest themselves of the Biblical idea that the world was a few thousand years old, the Mayans were thinking of millions and the Hindus billions ---Carl Sagan                                                                 

          
        37. That Hindu astronomical lore about ancient times cannot be based on later back-calculation, was also argued by Playfair’s contemporary, the French astronomer jean-Sylvain Bailly: “The motions of the stars calculated by the Hindus before some 4500 years vary not even a single minute from the [modem] tables of Cassini and Meyer. The Indian tables give the same annual variation of the moon as that discovered by Tycho Brahe - a variation unknown to the school of Alexandria and also the Arabs. - Jean Sylvain Bally French Astronomer

        38. 
        Most of my ideas & theories are heavily influenced by Vedanta - Erwin Schrödinger









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