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.

Image Source :- Lateral Thinking (quality-assurance-solutions.com)

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>