1. Problemløsning: hvordan? PROGRAMMERING SOM HÅNDVÆRK V/ Wilfred Gluud I Samarbejde med: Maj 2003Copyright Wilfred Gluud Programmering How to Solve IT

2. Biografi George Pólya 1887-1985 Maj 2003Copyright Wilfred Gluud Programmering George Polya was a Hungarian who immigrated to the United States in 1940. His major contribution is for his work in problem solving. In 1945 he published the book ’How to Solve It’ which quickly became his most prized publication. It sold over one million copies and has been translated into 17 languages. George Polya went on to publish a two-volume set, Mathematics and Plausible Reasoning (1954) and Mathematical Discovery (1962). These texts form the basis for the current thinking in mathematics education and are as timely and important today as when they were written. Polya has become known as the father of problem solving.

3. Prescription for solving problems Maj 2003Copyright Wilfred Gluud Programmering Polya's prescription for solving problems consists of four steps that use the 3 R's of problems solving, Request-Response-Result, and a verification of the result. 1. Understanding the problem. (Recognizing what is asked for) 2. Devising a plan. (Responding to what is asked for) 3. Carrying out the plan. (Developing the result of the response) 4. Looking back. (Checking. What does the result tell me?)

4. 1. How to Solve It Maj 2003Copyright Wilfred Gluud Programmering Summary taken from G. Polya, "How to Solve It", 2nd ed., Princeton University Press, 1957. UNDERSTANDING THE PROBLEM oFirst. You have to understand the problem. oWhat is the unknown? What are the data? What is the condition? oIs it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory? oDraw a figure. Introduce suitable notation. Separate the various parts of the condition. Can you write them down?

5. 2. How to Solve It Maj 2003Copyright Wilfred Gluud Programmering DEVISING A PLAN oSecond. Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution. oHave you seen it before? Or have you seen the same problem in a slightly different form? oDo you know a related problem? Do you know a theorem that could be useful?

6. 2b. How to Solve It Maj 2003Copyright Wilfred Gluud Programmering DEVISING A PLAN oLook at the unknown! And try to think of a familiar problem having the same or a similar unknown. oHere is a problem related to yours and solved before. Could you use it? Could you use its result? Could you use its method? Should you introduce some auxiliary element in order to make its use possible? oCould you restate the problem? Could you restate it still differently? Go back to definitions.

7. 2c. How to Solve It Maj 2003Copyright Wilfred Gluud Programmering DEVISING A PLAN oIf you cannot solve the proposed problem try to solve first some related problem. Could you imagine a more accessible related problem? A more general problem? A more special problem? An analogous problem? Could you solve a part of the problem? Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary? Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown? Could you change the unknown or data, or both if necessary, so that the new unknown and the new data are nearer to each other? oDid you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem?

8. 3. How to Solve It Maj 2003Copyright Wilfred Gluud Programmering 3. CARRYING OUT THE PLAN oThird. Carry out your plan. oCarrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct?

9. 4. How to Solve It Maj 2003Copyright Wilfred Gluud Programmering LOOKING BACK oFourth. Examine the solution obtained. oCan you check the result? Can you check the argument? oCan you derive the solution differently? Can you see it at a glance? oCan you use the result, or the method, for some other problem?