Lessons from George Polya

George Polya, a Hungarian-American mathematician, remains a towering figure in the world of mathematics and education. His seminal work, particularly the 1945 book "How to Solve It," has transcended the boundaries of mathematics, offering a timeless framework for problem-solving applicable to virtually every field of human endeavor. Polya's insights into the nature of discovery, the importance of heuristic thinking, and the art of teaching continue to inspire and guide students, educators, and innovators.

The Art and Science of Problem-Solving

Polya's most famous contribution is his four-step problem-solving process, a deceptively simple yet powerful guide to tackling challenges.

Polya's Four Principles of Problem Solving:

First Principle: Understand the problem. This involves not just reading the problem statement, but also identifying the unknown, the data, and the conditions. It's about asking: "What is the unknown? What are the data? What is the condition?". [1]
Second Principle: Devise a plan. This is the heart of problem-solving, where one finds the connection between the data and the unknown. Polya suggests strategies like looking for a pattern, drawing a picture, solving a simpler problem, or working backward. [1]
Third Principle: Carry out the plan. This stage requires patience and care. Each step of the plan must be checked for correctness. [1]
Fourth Principle: Look back. Once a solution is found, it is crucial to examine it. This includes checking the result and the argument, and considering if the method can be used for other problems. [1]

On the Nature of Discovery and Invention

Polya viewed problem-solving as a creative act, a "grain of discovery" present in the solution of any problem.

"A great discovery solves a great problem, but there is a grain of discovery in the solution of any problem." [2][3]
"The first rule of discovery is to have brains and good luck. The second rule of discovery is to sit tight and wait till you get a bright idea." [2][4]
"Good problems and mushrooms of certain kinds have something in common; they grow in clusters." [3][5]
"Look around when you have got your first mushroom or made your first discovery: they grow in clusters." [2]
"Nothing is more important than to see the sources of invention which are, in my opinion, more interesting than the inventions themselves." [3]
"An idea which can be used once is a trick. If it can be used more than once it becomes a method." [6]

The Mindset of a Problem Solver

Polya emphasized the importance of attitude, persistence, and intellectual courage in the face of a challenge.

"If there is a problem you can't solve, then there is an easier problem you can solve: find it." [2][5]
"It is better to solve one problem five different ways, than to solve five problems one way." [5][6]
"My method to overcome a difficulty is to go round it." [2][6]
"The open secret of real success is to throw your whole personality into your problem." [4][6]
"It is foolish to answer a question that you do not understand." [5]
"You should not put too much trust in any unproved conjecture, even if it has been propounded by a great authority, even if it has been propounded by yourself. You should try to prove it or disprove it." [7]
"Do not believe anything but doubt only what is worth doubting." [8]
"No idea is really bad, unless we are uncritical. What is really bad is to have no idea at all." [3]
"Success in solving the problem depends on choosing the right aspect, on attacking the fortress from its accessible side." [6]

On Mathematics and its Teaching

For Polya, mathematics was not a spectator sport but an active and creative pursuit. He was a passionate advocate for a more engaging and intuitive approach to teaching.

"Mathematics is not a spectator sport!" [3][6]
"To understand mathematics means to be able to do mathematics." [3]
"Mathematics is the cheapest science. Unlike physics or chemistry, it does not require any expensive equipment. All one needs for mathematics is a pencil and paper." [2][6]
"Mathematics is being lazy. Mathematics is letting the principles do the work for you so that you do not have to do the work for yourself." [2][6]
"Mathematics consists of proving the most obvious thing in the least obvious way." [2][6]
"Geometry is the science of correct reasoning on incorrect figures." [2][6]
"A mathematics teacher is a midwife to ideas." [2][6]
"It may be more important in the mathematics class how you teach than what you teach." [4][6]
"One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else." [2][9]
"The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work." [5]
"The traditional mathematics professor of the popular legend is absentminded. He usually appears in public with a lost umbrella in each hand." [2][9]

Heuristics and the Process of Thinking

Polya championed the use of heuristics—rules of thumb or strategies—to guide the problem-solving process.

"Heuristic reasoning is reasoning not regarded as final and strict but as provisional and plausible only, whose purpose is to discover the solution of the present problem." [7]
"We need heuristic reasoning when we construct a strict proof as we need scaffolding when we erect a building." [7]
"The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing." [4]
"If you cannot solve the proposed problem, try to solve first some related problem." [5][6]
"Look at the unknown! And try to think of a familiar problem having the same or a similar unknown." [1]
"Could you restate the problem? Could you restate it still differently? Go back to definitions." [1]
"It is hard to have a good idea if we have little knowledge of the subject, and impossible to have it if we have no knowledge. Good ideas are based on past experience and formerly acquired knowledge." [7]

On Learning and Life

Polya's wisdom extends beyond mathematics, offering insights into the nature of learning and the pursuit of knowledge.

"If you wish to learn swimming you have to go into the water and if you wish to become a problem solver you have to solve problems." [4][6]
"Solving problems is a practical art, like swimming, or skiing, or playing the piano: you can learn it only by imitation and practice." [4][6]
"The first rule of style is to have something to say. The second rule of style is to control yourself when, by chance, you have two things to say; say first one, then the other, not both at the same time." [3][5]
"Beauty in mathematics is seeing the truth without effort." [5][6]
"I am too good for philosophy and not good enough for physics. Mathematics is in between." [2][6]
"A mathematician who can only generalise is like a monkey who can only climb up a tree, and a mathematician who can only specialise is like a monkey who can only climb down a tree." [2][9]
"The elegance of a mathematical theorem is directly proportional to the number of independent ideas one can see in the theorem and inversely proportional to the effort it takes to see them." [2]
"It is sad to work for an end that you do not desire." [5]
"Pedantry and mastery are opposite attitudes toward rules. To apply a rule to the letter, rigidly, unquestioningly, in cases where it fits and in cases where it does not fit, is pedantry." [5][7]
"Even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else. Doing so, they miss an important and instructive phase of the work." [4]
"A good teacher should understand and impress on his students the view that no problem whatever is completely exhausted." [4]
"The principle is so perfectly general that no particular application of it is possible." [4]
"In order to solve this differential equation you look at it until a solution occurs to you." [6]

Sources:

The majority of George Polya's widely cited quotes originate from his foundational book, "How to Solve It." Other significant sources include his multi-volume works, "Mathematics and Plausible Reasoning" and "Mathematical Discovery."

"How to Solve It: A New Aspect of Mathematical Method" (1945): This book is the primary source for Polya's problem-solving framework and many of his most famous aphorisms on the subject. Goodreads
"Mathematics and Plausible Reasoning, Volume I: Induction and Analogy in Mathematics" (1954): This work delves deeper into the role of plausible reasoning in mathematical discovery. ThriftBooks
"Mathematics and Plausible Reasoning, Volume II: Patterns of Plausible Inference" (1954): The second volume continues the exploration of reasoning patterns. ThriftBooks
"Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving" (1962, 1965, combined 1981): This two-volume work provides a wealth of problems and further elaborates on his pedagogical ideas. Wiley
University of St Andrews, MacTutor History of Mathematics archive: A reliable source for a collection of Polya's quotations. mathshistory.st-andrews.ac.uk
Queen's University: Provides a concise summary of Polya's problem-solving techniques. sass.queensu.ca

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