Wilfred Gluud Programmering

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George Pólya (1887-1985)

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 became his most prized publication. Second edition 1956 .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.

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?)

Charles Simonyi
(Microsoft Application Software Group): ”Programmers get a couple of books on their firstday here One og them, called How to Solwe It, is by George Polya, the mathematician.  [Simonyi takes the book from a bookcase next to his desk and opens it to a certain page] These two pages are important. The rest of the book just elaborates on these two pages. This is like a checklist for problem solving. We follow these four steps of problem solving: first, understanding the problem, then devising a plan, carrying out the plan, and, finally, looking back."

Esgar W. Dijkstra: "Craftsman or Scientist?" 1975 (EWD 480)
This, of course, raises the question of feasibility of the teaching of thinking. In order to make this question realistic, we shall qualify somewhat: knowing how to teach thinking will not imply that each student is also able to learn it. This need not deter us: in this respect "thinking" would not differ from any other subject that we try to teach. So, let us consider the question after this qualification: can thinking be taught? The blurb on the backside of my 1957 edition of Polya's "How to Solve It". is quite positive: "Deftly, Polya the teacher shows us how to strip away the irrelevancies which clutter our thinking and guides us toward a clear and productive habit of mind."

Steve McConnel writes in ”Code Complete” page 163-164 about ”Design as a Heuristiv(*) Process” and cites the schema from ”How to Solve It”


Growing up George Polya was very frustrated with the practice of having to regularly memorize information. He was an excellent problem solver. Early on his uncle tried to convince him to go into the mathematics field but he wanted to study law like his late father had. After a time at law school he became bored with all the legal technicalities he had to memorize. He tired of that and switched to Biology and the again switched to Latin and Literature, finally graduating with a degree. Yet, he tired of that quickly and went back to school and took math and physics. He found he loved math.

His first job was to tutor the young son of a baron, who struggled due to his lack of problem solving skills. Polya spent hours and developed a method of problem solving that would work for the barns son, as well as others in the same situation. Polya maintained that the skill of problem was not an inborn quality but, something that could be taught.

In 1940 he and his wife Stella moved to the United States because of their concern for Nazism in Germany. He taught briefly at Brown University and then, for the remainder of his life, at Stanford University. He quickly became well known for his research and teachings on problem solving.
He taught many classes to elementary and secondary classroom teachers on how to motivate and teach skills to their students in the area of problem solving.

Polya’s First Principle: Understand the Problem
This seems so obvious that it is often not even mentioned, yet students are often stymied in their efforts to solve problems simply because they don’t understand it fully, or even in part. Polya taught teachers to ask students questions such as:

  • 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 a 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
Polya mentions that it are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy. A partial list of strategies is included:

  • Guess and check
  • Make and orderly list
  • Eliminate possibilities
  • Use symmetry
  • Consider special cases
  • Use direct reasoning
  • Solve an equation
  • Look for a pattern
  • Draw a picture
  • Solve a simpler problem
  • Use a model
  • Work backward
  • Use a formula
  • Be ingenious




third Principle: Carry out the plan

This step is usually easier than devising the plan. In general, all you need is care and patience, given that you have the necessary skills. Persistent with the plan that you have chosen. If it continues not to work discard it and choose another. Don’t be misled, this is how mathematics is done, even by professionals.

Polya’s Fourth Principle: Look back
Polya mentions that much can be gained by taking the time to reflect and look back at what you have done, what worked and what didn’t. Doing this will enable you to predict what strategy to use to solve future problems.

(*) Heuristic - Problem Solving
Type of word; philosophy of science, logic. Iinformal method for solving problems in the absence of an algorithm for formal proof. Heuristics typically have only restricted applicability and limited likelihood of success but, as George Polya showed, contribute significantly to our understanding of mathematical truths.


How to Solve It by George Polya. 253 pages. $ 13.85
This perennial best seller was written by an eminent mathematician, but it is a book for the general reader on how to think straight in any field. In lucid and appealing prose, it shows how the mathematical method of demonstrating a proof or finding an unknown can be of help in attacking any problem that can be "reasoned" out from building a bridge to winning a game of anagrams. Generations of readers have relished G. Polya's deft - indeed, brilliant - instructions on stripping away irrelevancies and going straight to the heart of the problem. - Translated into 5 langues.
"If you want to learn to think, and solve problems, then this book is for you. It is math based, but is applicable in software engineering."
Foreword - Contents -
Any version of this book is OK, is it all reprints.

Mathematics and Plausible Reasoning [Two Volumes in One]
by George Polya
(498 pages 2014) £ 22.50, 2014 Reprint of 1954 American Edition. Full facsimile of the original edition.
This two volume classic comprises two titles: "Patterns of Plausible Inference" and "Induction and Analogy in Mathematics". Guide to the practical art of plausible reasoning, particularly in mathematics, but also in every field of human activity. Using mathematics as the example par excellence, Polya shows how even the most rigorous deductive discipline is heavily dependent on techniques of guessing, inductive reasoning, and reasoning by analogy.
In solving a problem, the answer must be guessed at before a proof can be given, and guesses are usually made from a knowledge of facts, experience, and hunches. The truly creative mathematician must be a good guesser first and a good prover afterward; many important theorems have been guessed but no proved until much later. In the same way, solutions to problems can be guessed, and a god guesser is much more likely to find a correct solution.
This work might have been called "How to Become a Good Guesser"

Simple Heuristics That Make Us Smart (Evolution and Cognition) by Gerd Gigerenzer amd Peter M. Todd (Oxford) £ 24.74 - buy used
This book invites readers to embark on a new journey into a land of rationality that differs from the familiar territory of cognitive science and economics. Traditional views of rationality tend to see decision makers as possessing superhuman powers of reason, limitless knowledge, and all of eternity in which to ponder choices. To understand decisions in the real world, we need a different, more psychologically plausible notion of rationality, and this book provides it. It is about fast and frugal heuristics―simple rules for making decisions when time is pressing and deep thought an unaffordable luxury. These heuristics can enable both living organisms and artificial systems to make smart choices, classifications, and predictions by employing bounded rationality.

IT-gurus: George Polya - Esgar W. Dijkstra - Fred Brooks - C. J. Date
Charles Simonyi - Joe Celko - Pragmatic Programmers - Steve McConnell

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15-06-2016 14:17