George Polya, Hungary-born educationalist and the Stanford Professor, is exalted as the Father of Problem Solving, on account of his path-breaking book, How to solve it?, which was sold over a million copies and got translated into 17 different languages.
An unassuming student as he was, George shifted his discipline of study very often as he disliked rote learning. He started with Law in pursuit of the family tradition of lawyers, but soon got suffocated by the formidable task of memorizing legal technicalities. He then shifted to Biology only to give it up sooner and eventually landed in Latin and Literature before his graduation. But once again he decided to change the subject in view of his newly found love for Physics and Mathematics. He repeated his school to get basics of the new discipline and he finally got graduated in Mathematics. George’s book on problem solving facilitates teachers and students to solve mathematical problems effectively.
Story of ‘How to solve it?’
The story of How to solve it? starts with George’s assignment to teach mathematics to Gregor, the young son of a baron. Gregor did not have any taste for mathematics and he struggled due to his lack of skills of problem solving in mathematics. George Polya developed a course tailor-made to the requirement of Gregor to develop his problem-solving skills. This stepwise programme which proved itself as an excellent tool in teaching mathematical problem solving skills, was published later and became a major masterpiece of George. Polya ultimately proved his view that problem solving is a skill that could be systematically nurtured, if not gifted to the student by nature.
The book “How to solve it?” suggests four steps towards a typical problem solving session: 1) Understanding the Problem; 2) Devising a plan to solve; 3) Carrying out the plan; and 4) Looking back.
Polya’s First Principle: Understand the problem
It is said: “A wise man begins in the end, a fool ends in the beginning.” It is better to start from the end rather than ending in the beginning! It calls for a thorough understanding of the whole problem, before venturing to solve it. Full understanding of the problem at hand is a prerequisite to start solving it.
Often, people jump into conclusion, without even considering the problem in its totality. Polya wants to make sure that students start their effort of problem solving, from a subtle understanding of it. He promoted teachers to ask the following questions to the students before they attempt to solve it:
· Do you understand all the words used in stating the problem?
· What are you asked to find or show?
· Can you restate the problem in your own words?
· Can you think of a picture or diagram that might help you understand the problem?
· Is there enough information to enable you to find a solution?
Polya’s Second Principle: Devise a plan
According to Polya, there are as many ways to solve a problem as the number of problems are. Problems are unique in one sense. However, there are some general methods which could be adapted to fit in to solve the given problem. One has to devise an appropriate strategy to solve a specific problem based on a large number of strategies already available or those derived from them. The skill of devising the strategy could be mastered only by actually implementing them to solve a large number of problems based on them. As experience goes up, it will be made easy to strategize problem solving.
Some general strategies to solve a given problem is listed below:
· Guess and check or trial and error method
· Make an orderly list
· Eliminate possibilities
· Use symmetry
· Use direct reasoning
· Solve an equation
· Look for a pattern
· Draw a picture
· Solve a simpler problem
· Use a model
· Work backwards
Polya’s Third Principle: Carry out the plan
Once you have understood the problem and devised a strategy to solve it, what remains is to actually solve it based on that strategy. It calls forth many unique skills to actually implement a strategy to solve a given problem. It may be important to persevere and to stick to the plans you would have already made to arrive at a solution. Often, your inability to persist may kill your chances to solve an important problem.
Polya’s Fourth Principle: Look back
Finally, it is important to reflect over the entire procedure of problem solving, once you have already solved it. Look back at what you have already done. What all strategies were planned? What actually worked? What did not work? Why did they not work? This exercise will make it easy for a student to solve the problems in future.
A detailed examination to the problem and solution is solicited here. You may start looking at the result and then going to the arguments that delivered the obtained solution. You may also attempt to solve the problem in alternative ways. You can also think that the same strategy is suitable for a different problem too. You may ask the following questions to implement a looking back:
· Did you answer the core question?
· Is your result reasonable?
· Double check to make sure that all of the conditions related to the problem are satisfied.
· Double check any computations involved in finding your solution.
You may further ask more questions to make it relevant to a future problem:
· Can you derive the result differently? Can you see it at a glance?
· Can you use the result, or the method, for some other problem?
Checklist method of problem solving
Even though Polya designed the method for solving problems in mathematics, which calls for logical thinking, it finds application also in taking on problems seeking creative solutions, which requires rather lateral thinking. The four principles of Polya constitute four steps in the checklist of the procedure of creative problem solving. Hence, the creativity method based on Polya Principles is called Method of Check-listing.
In the Check-listing method, all the four steps of Polya’s problem solving method are kept intact. The only difference is that lateral or creative thinking is promoted in each step. In the first step, the problem is perceived in its totality in a creative manner; in the second step, creative strategies are chalked out to tackle the problem; in the third step, creative ways of implementation of that strategy is employed; and, in the fourth step, creative feedback is promoted.
Thus, the method of Check-listing could be considered as a systematic creative problem-solving technique.