"Reflexion is careful and laborious thought, and watchful attention directed to the agreeable effect of one's plan. Invention, on the other hand, is the solving of intricate problems and the discovery of new principles by means of brilliancy and versatility." (Marcus Vitruvius Pollio, "De architectura" ["On Architecture], cca. 15BC)
"This diagrammatic method has, however, serious
inconveniences as a method for solving logical problems. It does not show how
the data are exhibited by cancelling certain constituents, nor does it show how
to combine the remaining constituents so as to obtain the consequences sought.
In short, it serves only to exhibit one single step in the argument, namely the
equation of the problem; it dispenses neither with the previous steps, i.e.,
'throwing of the problem into an equation' and the transformation of the
premises, nor with the subsequent steps, i.e., the combinations that lead to
the various consequences. Hence it is of very little use, inasmuch as the
constituents can be represented by algebraic symbols quite as well as by plane
regions, and are much easier to deal with in this form." (Louis Couturat,
"The Algebra of Logic", 1914)
"A great discovery solves a great problem but there is
a grain of discovery in the solution of any problem. Your problem may be
modest; but if it challenges your curiosity and brings into play your inventive
faculties, and if you solve it by your own means, you may experience the
tension and enjoy the triumph of discovery." (George Polya, "How to solve
it", 1944)
"[The] function of thinking is not just solving an
actual problem but discovering, envisaging, going into deeper questions. Often,
in great discovery the most important thing is that a certain question is
found." (Max Wertheimer, "Productive Thinking", 1945)
“We can scarcely imagine a problem absolutely new, unlike
and unrelated to any formerly solved problem; but if such a problem could
exist, it would be insoluble. In fact, when solving a problem, we should always
profit from previously solved problems, using their result or their method, or
the experience acquired in solving them.” (George Polya, 1945)
"I believe, that the decisive idea which brings the
solution of a problem is rather often connected with a well-turned word or
sentence. The word or the sentence enlightens the situation, gives things, as
you say, a physiognomy. It can precede by little the decisive idea or follow on
it immediately; perhaps, it arises at the same time as the decisive idea.
[…] The right word, the subtly
appropriate word, helps us to recall the mathematical idea, perhaps less
completely and less objectively than a diagram or a mathematical notation, but
in an analogous way. […] It may contribute to fix it in the mind." (George
Pólya [in a letter to Jaque Hadamard, "The Psychology of Invention in the
Mathematical Field", 1949])
"Solving problems is the specific achievement of intelligence." (George Pólya, 1957)
"Systems engineering embraces every scientific and
technical concept known, including economics, management, operations,
maintenance, etc. It is the job of integrating an entire problem or problem to
arrive at one overall answer, and the breaking down of this answer into defined
units which are selected to function compatibly to achieve the specified
objectives." (Instrumentation Technology, 1957)
"Solving problems can be regarded as the most
characteristically human activity." (George Pólya, "Mathematical
Discovery", 1962)
"The final test of a theory is its capacity to solve
the problems which originated it." (George Dantzig, "Linear
Programming and Extensions", 1963)
"It is a commonplace of modern technology that there is
a high measure of certainty that problems have solutions before there is
knowledge of how they are to be solved." (John K Galbraith, "The New
Industrial State", 1967)
"Every problem-solving effort must begin with creating a representation for the problem - a problem space in which the search for the solution can take place. Of course, for most of the problems we encounter in our daily personal or professional lives, we simply retrieve from memory a representation that we have already stored and used on previous occasions. Sometimes, we have to adapt the representation a bit to the new situation, but that is usually a rather simple matter." (Herbert A Simon, "The Sciences of the Artificial", 1968)
"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)
"In general, complexity and precision bear an inverse
relation to one another in the sense that, as the complexity of a problem
increases, the possibility of analysing it in precise terms diminishes. Thus
'fuzzy thinking' may not be deplorable, after all, if it makes possible the
solution of problems which are much too complex for precise analysis."
(Lotfi A Zadeh, "Fuzzy languages and their relation to human
intelligence", 1972)
"If we deal with our problem not knowing, or pretending
not to know the general theory encompassing the concrete case before us, if we
tackle the problem 'with bare hands', we have a better chance to
understand the scientist's attitude in general, and especially the task of the
applied mathematician." (George Pólya, "Mathematical Methods in
Science", 1977)
“Solving problems can be regarded as the most
characteristically human activity.” (George Polya, 1981)
"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 great many problems are easier to solve rigorously if you know in advance what the answer is.” (Ian Stewart, “From Here to Infinity”, 1987)
"There are many things you can do with problems besides solving them. First you must define them, pose them. But then of course you can also refine them, depose them, or expose them or even dissolve them! A given problem may send you looking for analogies, and some of these may lead you astray, suggesting new and different problems, related or not to the original. Ends and means can get reversed. You had a goal, but the means you found didn’t lead to it, so you found a new goal they did lead to. It’s called play. Creative mathematicians play a lot; around any problem really interesting they develop a whole cluster of analogies, of playthings." (David Hawkins, "The Spirit of Play", Los Alamos Science, 1987)
"A scientific problem can be illuminated by the
discovery of a profound analogy, and a mundane problem can be solved in a
similar way." (Philip Johnson-Laird, "The Computer and the
Mind", 1988)
“A mental model is a knowledge structure that incorporates
both declarative knowledge (e.g., device models) and procedural knowledge
(e.g., procedures for determining distributions of voltages within a circuit),
and a control structure that determines how the procedural and declarative
knowledge are used in solving problems (e.g., mentally simulating the behavior
of a circuit).” (Barbara Y White & John R Frederiksen, “Causal Model Progressions
as a Foundation for Intelligent Learning Environments”, Artificial Intelligence
42, 1990)
"An important symptom of an emerging understanding is the capacity to represent a problem in a number of different ways and to approach its solution from varied vantage points; a single, rigid representation is unlikely to suffice." (Howard Gardner, "The Unschooled Mind", 1991)
“[By understanding] I mean simply a sufficient grasp of concepts, principles, or skills so that one can bring them to bear on new problems and situations, deciding in which ways one’s present competencies can suffice and in which ways one may require new skills or knowledge.” (Howard Gardner, “The Unschooled Mind”, 1991)
"We consider the notion of ‘system’ as an organising
concept, before going on to look in detail at various systemic metaphors that
may be used as a basis for structuring thinking about organisations and problem
situations." (Michael C Jackson, "Creative Problem Solving: Total
Systems Intervention", 1991)
“But our ways of learning about the world are strongly
influenced by the social preconceptions and biased modes of thinking that each
scientist must apply to any problem. The stereotype of a fully rational and
objective ‘scientific method’, with individual scientists as logical (and
interchangeable) robots, is self-serving mythology.” (Stephen Jay Gould, “This
View of Life: In the Mind of the Beholder”, “Natural History”, Vol. 103, No. 2,
1994)
“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)
"The purpose of a conceptual model is to provide a
vocabulary of terms and concepts that can be used to describe problems and/or
solutions of design. It is not the purpose of a model to address specific
problems, and even less to propose solutions for them. Drawing an analogy with
linguistics, a conceptual model is analogous to a language, while design
patterns are analogous to rhetorical figures, which are predefined templates of
language usages, suited particularly to specific problems." (Peter P Chen
[Ed.], "Advances in Conceptual Modeling", 1999)
"The three basic mechanisms of averaging, feedback and
division of labor give us a first idea of a how a CMM [Collective Mental Map]
can be developed in the most efficient way, that is, how a given number of
individuals can achieve a maximum of collective problem-solving competence. A
collective mental map is developed basically by superposing a number of
individual mental maps. There must be sufficient diversity among these
individual maps to cover an as large as possible domain, yet sufficient
redundancy so that the overlap between maps is large enough to make the
resulting graph fully connected, and so that each preference in the map is the
superposition of a number of individual preferences that is large enough to
cancel out individual fluctuations. The best way to quickly expand and improve
the map and fill in gaps is to use a positive feedback that encourages
individuals to use high preference paths discovered by others, yet is not so
strong that it discourages the exploration of new paths." (Francis
Heylighen, "Collective Intelligence and its Implementation on the
Web", 1999)
"What it means for a mental model to be a structural
analog is that it embodies a representation of the spatial and temporal
relations among, and the causal structures connecting the events and entities
depicted and whatever other information that is relevant to the problem-solving
talks. […] The essential points are that a mental model can be nonlinguistic in
form and the mental mechanisms are such that they can satisfy the
model-building and simulative constraints necessary for the activity of mental
modeling." (Nancy J Nersessian, "Model-based reasoning in conceptual
change", 1999)
"A model is an imitation of reality and a mathematical model is a particular form of representation. We should never forget this and get so distracted by the model that we forget the real application which is driving the modelling. In the process of model building we are translating our real world problem into an equivalent mathematical problem which we solve and then attempt to interpret. We do this to gain insight into the original real world situation or to use the model for control, optimization or possibly safety studies." (Ian T Cameron & Katalin Hangos, "Process Modelling and Model Analysis", 2001)
"Good numeric representation is a key to effective thinking that is not limited to understanding risks. Natural languages show the traces of various attempts at finding a proper representation of numbers. [...] The key role of representation in thinking is often downplayed because of an ideal of rationality that dictates that whenever two statements are mathematically or logically the same, representing them in different forms should not matter. Evidence that it does matter is regarded as a sign of human irrationality. This view ignores the fact that finding a good representation is an indispensable part of problem solving and that playing with different representations is a tool of creative thinking." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)
"The key role of representation in thinking is often downplayed because of an ideal of rationality that dictates that whenever two statements are mathematically or logically the same, representing them in different forms should not matter. Evidence that it does matter is regarded as a sign of human irrationality. This view ignores the fact that finding a good representation is an indispensable part of problem solving and that playing with different representations is a tool of creative thinking." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)
"Mathematical modeling is as much ‘art’ as ‘science’:
it requires the practitioner to (i) identify a so-called ‘real world’ problem
(whatever the context may be); (ii) formulate it in mathematical terms (the
‘word problem’ so beloved of undergraduates); (iii) solve the problem thus
formulated (if possible; perhaps approximate solutions will suffice, especially
if the complete problem is intractable); and (iv) interpret the solution in the
context of the original problem." (John A Adam, "Mathematics in
Nature", 2003)
"What is a mathematical model? One basic answer is that
it is the formulation in mathematical terms of the assumptions and their
consequences believed to underlie a particular ‘real world’ problem. The aim of
mathematical modeling is the practical application of mathematics to help
unravel the underlying mechanisms involved in, for example, economic, physical,
biological, or other systems and processes." (John A Adam,
"Mathematics in Nature", 2003)
"Alternative models are neither right nor wrong, just
more or less useful in allowing us to operate in the world and discover more
and better options for solving problems." (Andrew Weil," The Natural
Mind: A Revolutionary Approach to the Drug Problem", 2004)
“A conceptual model is a mental image of a system, its
components, its interactions. It lays the foundation for more elaborate models,
such as physical or numerical models. A conceptual model provides a framework
in which to think about the workings of a system or about problem solving in
general. An ensuing operational model can be no better than its underlying
conceptualization.” (Henry N Pollack, “Uncertain Science … Uncertain World”,
2005)
"In specific cases, we think by applying mental rules,
which are similar to rules in computer programs. In most of the cases, however,
we reason by constructing, inspecting, and manipulating mental models. These
models and the processes that manipulate them are the basis of our competence
to reason. In general, it is believed that humans have the competence to
perform such inferences error-free. Errors do occur, however, because reasoning
performance is limited by capacities of the cognitive system, misunderstanding
of the premises, ambiguity of problems, and motivational factors. Moreover,
background knowledge can significantly influence our reasoning performance.
This influence can either be facilitation or an impedance of the reasoning
process." (Carsten Held et al, "Mental Models and the Mind",
2006)
"Every problem has a solution; it may sometimes just
need another perspective." (Rebecca Mallery et al, "NLP for Rookies", 2009)
"Mental acuity of any kind comes from solving problems
yourself, not from being told how to solve them.” (Paul Lockhart, “A
Mathematician's Lament”, 2009)
"Mental models are formed over time through a deep
enculturation process, so it follows that any attempt to align mental models
must focus heavily on collective sense making. Alignment only happens through a
process of socialisation; people working together, solving problems together,
making sense of the world together." (Robina Chatham & Brian Sutton,
"Changing the IT Leader’s Mindset", 2010)
"Mathematical modeling is the application of mathematics
to describe real-world problems and investigating important questions that
arise from it." (Sandip Banerjee, "Mathematical Modeling: Models,
Analysis and Applications", 2014)
"Mental imagery is often useful in problem solving.
Verbal descriptions of problems can become confusing, and a mental image can
clear away excessive detail to bring out important aspects of the problem.
Imagery is most useful with problems that hinge on some spatial relationship.
However, if the problem requires an unusual solution, mental imagery alone can
be misleading, since it is difficult to change one’s understanding of a mental
image. In many cases, it helps to draw a concrete picture since a picture can
be turned around, played with, and reinterpreted, yielding new solutions in a
way that a mental image cannot." (James Schindler,
"Followership", 2014)
“Framing the right problem is equally or even more important
than solving it.” (Pearl Zhu, “Change, Creativity and Problem-Solving”, 2017)
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