On Problem Solving XI: Problem Solvers

"The future mathematician should be a clever problem-solver; but to be a clever problem-solver is not enough. In due time, he should solve significant mathematical problems; and first he should find out for which kind of problems his native gift is particularly suited." (George Pólya, "How to solve it", 1945)

"The intelligent problem-solver tries first of all to understand the problem as fully and as clearly as he can. Yet understanding alone is not enough; he must concentrate upon the problem, he must desire earnestly to obtain its solution. If he cannot summon up real desire for solving the problem he would do better to leave it alone. The open secret of real success is to throw your whole personality into your problem." (George Pólya, "How to Solve It", 1945)

"The mathematical experience of the student is incomplete if he never had an opportunity to solve a problem invented by himself." (George Pólya, "How to Solve It", 1945)

"As long as we try and patiently do our best to solve the problem, although we may not get the answer we are looking for, we always get something - even if it is only the valuable experience." (Charles Kettering, "Short Stories of Science and Invention: A Collection of Radio Talks", 1954)

"An expert problem solver must be endowed with two incompatible qualities, a restless imagination and a patient pertinacity." (Howard W Eves, "In Mathematical Circles", 1969)

"Deep in the human nature there is an almost irresistible tendency to concentrate physical and mental energy on attempts at solving problems that seem to be unsolvable." (Ragnar Frisch, "From Utopian Theory to Practical Applications", [Nobel lecture] 1970)

"The definition of a problem and the action taken to solve it largely depend on the view which the individuals or groups that discovered the problem have of the system to which it refers. A problem may thus find itself defined as a badly interpreted output, or as a faulty output of a faulty output device, or as a faulty output due to a malfunction in an otherwise faultless system, or as a correct but undesired output from a faultless and thus undesirable system. All definitions but the last suggest corrective action; only the last definition suggests change, and so presents an unsolvable problem to anyone opposed to change." (Herbert Brün, "Technology and the Composer", 1971)

"As our ability to solve problems expands. the scale of the problems attacked themselves seems to expand at a similar rate. As a result there always exist over the horizon new categories of problems of greater size to tackle." (David M Himmelbau, "Decomposition Methods", 1977) 

"The problem solver needs to stand back and examine problem contexts in the light of different 'Ws' (Weltanschauungen). Perhaps he can then decide which 'W' seems to capture the essence of the particular problem context he is faced with. This whole process needs formalizing if it is to be carried out successfully. The problem solver needs to be aware of different paradigms in the social sciences, and he must be prepared to view the problem context through each of these paradigms." (Michael C Jackson, "Towards a System of Systems Methodologies", 1984)

"A problem exists when there is a discrepancy between an initial state and a goal state, and there is no ready-made solution for the problem solver." (John D Bransford & Barry S Stein, "A Guide for Improving Thinking, Learning, and Creativity" 2nd Ed., 1993)

"Our ability to solve problems is not simply equivalent to a set of general problem-solving skills. One implication of this conclusion is that the same individual may be both good and poor at problem solving, depending on the nature of the problem." (John D Bransford & Barry S Stein, "A Guide for Improving Thinking, Learning, and Creativity" 2nd Ed., 1993)

"The term mental model refers to knowledge structures utilized in the solving of problems. Mental models are causal and thus may be functionally defined in the sense that they allow a problem solver to engage in description, explanation, and prediction. Mental models may also be defined in a structural sense as consisting of objects, states that those objects exist in, and processes that are responsible for those objects’ changing states." (Robert Hafner & Jim Stewart, "Revising Explanatory Models to Accommodate Anomalous Genetic Phenomena: Problem Solving in the ‘Context of Discovery’", Science Education 79 (2), 1995)

"Often a successful problem-solver is one who creates a new context in which to view the problem. This can often be done by directing one's attention away from the distracting details of the difficulty. From a detached perspective, we may examine the situation in a new or different light and, after exploring information and options, choose an appropriate course of action." (John Templeton, "Wisdom From World Religions: Pathways Toward Heaven on Earth", 2002)

"We are constantly using old knowledge in new situations. When a solver can successfully use a solution procedure used in the past to solve a target problem this is known as positive transfer. […] However, it is also the case that a procedure learned in the past can impede one's learning of a new procedure. This is known as negative transfer. In this case what you have learned prevents you from solving a new problem or at least prevents you from seeing an optimal solution." (S Ian Robertson, "Problem Solving", 2001)

"Mathematics is not a matter of 'anything goes', and every mathematician is guided by explicit or unspoken assumptions as to what counts as legitimate – whether we choose to view these assumptions as the product of birth, experience, indoctrination, tradition, or philosophy. At the same time, mathematicians are primarily problem solvers and theory builders, and answer first and foremost to the internal exigencies of their subject." (Jeremy Avigad, "Methodology and Metaphysics in the Development of Dedekind’s Theory of Ideals", 2006)

"A good problem solver must also be a conceptual mathematician, with a good intuitive grasp of structures. But structures remain tools for the problem solver, instead of the main object of study." (David Ruelle, "The Mathematician's Brain", 2007)

"The fledgling problem solver invariably rushes in with solutions before taking time to define the problem being solved. Even experienced solvers, when subjected to social pressure, yield to this demand for haste. When they do, many solutions are found, but not necessarily to the problem at hand." (Donald C Gause & Gerald M Weinberg, "Are Your Lights On?", 2011)

"Under normal conditions the research scientist is not an innovator but a solver of puzzles, and the puzzles upon which he concentrates are just those which he believes can be both stated and solved within the existing scientific tradition." (Thomas S Kuhn, "The Essential Tension: Selected Studies in Scientific Tradition and Change", 2011)

"Calculus succeeds by breaking complicated problems down into simpler parts. That strategy, of course, is not unique to calculus. All good problem-solvers know that hard problems become easier when they’re split into chunks. The truly radical and distinctive move of calculus is that it takes this divide-and-conquer strategy to its utmost extreme - all the way out to infinity." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

"Diverse groups of problem solvers outperformed the groups of the best individuals at solving complex problems. The reason: the diverse groups got stuck less often than the smart individuals, who tended to think similarly." (Scott E Page)

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