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
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.
Understanding the problem. (Recognizing
what is asked for)
Devising a plan. (Responding to
what is asked for)
3. Carrying out
the plan. (Developing the result of the response)
Looking back. (Checking. What
does the result tell me?)
PRAISE FOR "HOW TO SOLVE IT" FROM
(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
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."
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
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:
you understand all the words used in stating the problem?
are you asked to find or show?
you restate the problem in your own words?
you think of a picture or a diagram that might help you understand the
there enough information to enable you to find a solution?
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
Consider special cases
Use direct reasoning
Solve an equation
Look for a pattern
Draw a picture
Solve a simpler problem
Use a model
Use a formula
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.
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
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."
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"
That Make Us Smart (
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
George Polya -
Esgar W. Dijkstra -
Fred Brooks -
C. J. Date
Joe Celko -
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