Abstract Thinking

What is abstract thinking, and why is it important in mathematics?   Name at least three advantages.

Let us start with an example:  If there are two rooms, room A and room B that are connected by a door, and a person starts in room A, but changes room every time a bell rings, then what room will the person be in if the bell rings three times?  256 times?  Most people get the answer to the first question by moving their finger three times and correctly conclude the person will be in room B.  Sometimes a person does not move a finger but mentally pictures the person going back and forth.  While this mental picturing is an example of abstract thinking, it only represents one level of abstraction that corresponds closely with a physical operation (going back and forth).  Other levels of abstraction exist.  Most people do not attempt to answer the question of “256 times” by going back and forth physically or mentally.  They rely on an additional layer of abstraction.  They realize that an odd number of rings of the bell puts the person in room B, and an even number of rings puts the person in room A.  Abstract thinking is an objective mental process whereby concepts rather than motion or physical objects or activity (or a mental picture thereof) are used to come to a conclusion.  There are many additional layers of abstraction used by mathematicians.

Abstract thinking is generally quicker than more concrete methods.  It also creates a deeper level of understanding.  Hence, it enables one to understand the essence of problem and from the principles involved, thereby, solve other problems similar to the original.  In addition, use of abstract thinking helps to “objectify” a problem and helps eliminate red-lining.