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- The Problem of Learning
- Problemistics Courseware
- Corso su Problemistica
- Resources Management
- Manuale/Intellettuale
- Campagna/Città
Problemistics - Problémistique - Problemistica
The Art & Craft of Problem Dealing
Problem Solving
Definition (Herbert A. Simon)
Definition (Russell L. Ackoff and Fred E. Emery)
Characteristics (Max Wertheimer)
Situations (Jean-Michel Hoc)
Stages (Alex F. Osborn)
Phases and tasks (George Pólya)
Methods (Richard E. Mayer)
Methods (Jean-Michel Hoc)
Methods (Herbert A. Simon)
Approaches (George Pólya)
Methods and steps (George Pólya)
Analogy (Albert Einstein and Leopold Infeld)
Concept clarification (George Pólya)
Representation (Max Wertheimer)
Representation (Karl Dunker)
Representation (Herbert A. Simon)
Representation for understanding (Jean-Michel Hoc)
Information and knowledge (Herbert A. Simon)
Knowledge for problem solving (A. J. Repo)
Skill (George Pólya)
Memory (James G. March and Herbert A. Simon)
[1988] Herbert A. Simon, The Sciences of the Artificial
“Problem solving is often described as a search through a vast maze of possibilities, a maze that describes the environment. Successful problem solving involves searching the maze selectively and reducing it to manageable proportions.” (Chapter 3, p. 66)
[1972] Russell L. Ackoff and Fred E. Emery, On Purposeful Systems, Tavistock Publications, London.
“Solving a Problem: selecting one of a set of available courses of action that, as a result of an inquiry, the individual believes in as the most likely to produce a state of satisfaction in him, and that does produce a state of satisfaction.”
“Solving a problem involves answering two questions:
(1) What alternatives are available?
(2) Which one is best or good enough?
Any alternative that replaces dissatisfaction with satisfaction is a satisficing solution to the problem. An available solution that produces as much or more satisfaction than can any other solution not only satisfices but optimizes.” (Chapter II, p. 109)
Characteristics
[1968, First Edition 1945] Max Wertheimer, Productive Thinking
“On the basis of my experience, I would say that the essential features in genuine problem solving are:
- not to be bound, blinded by habit;
- not merely to repeat slavishly what one has been taught;
- not to proceed in a mechanized state of mind;
- in a piecemeal attitude;
- with piecemeal attention;
- by piecemeal operations;
but to look at the situation freely, open-mindedly, viewing the whole,
- trying to discover, to realize how the problem and the situation are related,
- trying to penetrate, to realize and to trace out the inner relation between form and task; in the finest cases getting at the roots of the situation, illuminating and making transparent essential structural features of regular series, in spite of the difficulties.” (Chapter 4, pp. 120-121)
[1988] Jean-Michel Hoc, Cognitive Psychology of Planning
“Problem solving situations, i.e. those situations in which an individual forms a representation of the task without having an adequate procedure immediately available to reach the goal.” (Introduction, p. 4)
[1963, First Edition 1953] Alex F. Osborn, Applied Imagination
The creative problem-solving process ideally comprises these stages:
(1) Fact - Finding
Problem-definition: Picking out and pointing up the problem.
Preparation: Gathering and analysing the pertinent data.
(2) Idea - Finding
Idea-production: Thinking up tentative ideas as possible leads.
Idea-development: Selecting from resultant ideas, adding others, and
reprocessing by means of modification, combination, et cetera.
(3) Solution - Finding
Evaluation: verifying the tentative solutions, by tests and otherwise.
Adoption: deciding on and implementing the final solution.
(from Chapter IX, p. 111)
[1963, First Edition 1953] Alex F. Osborn, Applied Imagination
“The several steps in problem-solving might follow an alternating sequence something like this:
1. Think up all phases of the problem
2. Select the sub-problem to be attacked
3. Think up what data might help
4. Select the most likely sources of data
5. Dream up all possible ideas as keys to the problem
6. Select the ideas most likely to lead to solution
7. Think up all possible ways to test
8. Select the soundest ways to test
9. Imagine all possible contingencies
10. Decide on the final answer.”
(Chapter XIV, pp. 207-208)
[1990, First published 1945] George Pólya, How to Solve It
"1) Understanding the problem. You have to understand the problem.
- What is the unknown?
- What are the data?
- What is the condition?
- It is possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory?
- Draw a figure. Introduce suitable notation.
- Separate the various parts of the condition. Can you write them down?
2) Devising a plan. Find the connection between the data and the unknown.
You may be obliged to consider auxiliary problems if an immediate connection cannot be found.
- Have you seen it before? Or have you seen the same problem in a slightly different form?
- Do you know a related problem? Do you know a theorem that could be useful?
- Look at the unknown! And try to think of a familiar problem having the same or a similar unknown.
- Could you restate the problem? Could you restate it still differently? Go back to definitions.
- Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem?
3) Carrying out the plan. Carrying out your plan of the solution, check each step.
- Can you see clearly that the step is correct? Can you prove that it is correct?
4) Looking back. Examine the solution obtained
- Can you check the result? Can you check the argument?
- Can you derive the result differently? Can you see it at a glance?
- Can you use the result, or the method, for some other problem?"
(from Preface to the Second Edition, p. XXXV)
[1992, Second Edition] Richard E. Mayer, Thinking, Problem Solving, Cognition,
- By Analogy. “An analogy is a problem that contains a similar structure but not necessarily the same story line as another problem; that is, an analogy has
structural similarity but not surface similarity with a target problem.” (p. 419)
- By Models. “A model of a system includes the essential parts of the system as well as the cause-and-effect relations between a change in the status of one part and the change in the status of another part.” (p. 431)
- By Example. “A worked-out example is a problem along with its step-by-step solution from the same domain (i.e. using the same variables and structure) as the target problem but with different specific values for the variables; this contrasts with an analogy, which involves a structurally similar problem from a different domain.” (p. 439)
[1988] Jean-Michel Hoc, Cognitive Psychology of Planning
Transfer. "An individual can transfer knowledge of a procedure to a new situation, but this procedure must be associated with a device identical to the one in the new problem." (p. 70)
Analogy. "Knowledge from another domain can be used to tackle a new problem.
In this case the term analogy is used rather than transfer" (p. 71)
[1988, Second edition] Herbert A. Simon, The Sciences of the Artificial, M.I.T. Press, Cambridge Massachusetts
“... human problem solving, from the most blundering to the most insightful, involves nothing more than varying mixtures of trial and error and selectivity. The selectivity derives from various rules of thumb, or heuristics, that suggest which path should be tried first and which leads are promising.” (p. 207)
“There are two basic kinds of selectivity. One we have already noted: various paths are tried out, the consequences of following them are noted, and this information is used to guide further search.”
“The second source of selectivity in problem solving is previous experience. We see this particularly clearly when the problem to be solved is similar to one that has been solved before. Then, by simply trying again the paths that led to the earlier solution, or their analogues, trial-and-error search is greatly reduced or altogether eliminated.” (pp. 207-208)
[1990, First published 1945] George Pólya, How to Solve It
“Trying to solve a problem, we consider different aspects of it in turn, we roll it over and over incessantly in our mind; variation of the problem is essential to our work. We may vary the problem by decomposing and recombining its elements, or by going back to the definition of certain of its terms, or we may use the great resources of generalization, specialization, and analogy. Variation of the problem may lead us to auxiliary elements, or to the discovery of a more accessible auxiliary problem.”
(Part III, p. 131)
[1990, First published 1945] George Pólya, How to Solve It
“To solve a problem is, essentially, to find the connection between the data and the unknown.” (p. 182)
Possible steps in problem solving:
- To solve an auxiliary problem as a stepping stone to solve a more difficult general problem (p. 196)
- To vary the problem in order to create new possibilities of solving it. (p. 210)
- Working backwards from the desired final situation to the present situation, from the unknown to the data. (p. 227)
- To solve the problem going round it (e.g. reaching the goal going round an obstacle) (p. 232)
[1938] Albert Einstein and Leopold Infeld, The Evolution of Physics
“It has often happened in physics that an essential advance was achieved by carrying out a consistent analogy between apparently unrelated phenomena.”
“The association of solved problems with those unsolved may throw new light on our difficulties by suggesting new ideas. It is easy to find a superficial analogy which really expresses nothing. But to discover some essential common features, hidden beneath a surface of external differences, to form, on this basis, a new successful theory, is important creative work.” (Chapter IV, pp. 286-287)
[1990, First published 1945] George Pólya, How to Solve It
“In solving a practical problem, we are often obliged to start from rather hazy ideas; then, the clarification of the concepts may become an important part of the problem.” (Part III, p. 151)
[1968, First Edition 1945] Max Wertheimer, Productive Thinking
“Two things that are related are decisive here: the structural meaning of the parts and the character of their inner relatedness with reaching the goal, actually, transparently.” (p. 66)
“This [understanding] involves sensible grouping, realization of functional hierarchy, of direction, of the structurally sensible place, role and function of each statement, its meaning in the whole.” (p. 75)
“... we should not forget that often, in productive situations, the sensible view of the whole plays a decisive role: the realization of symmetry and balance in the whole, and of deviations from it, in sensible grouping.” (p. 129)
[1968] Karl Dunker, in P. C. Wason and P. N. Johnson-Laird eds., Thinking and Reasoning (readings)
“It is therefore meaningful to say that what it is really done in any solution of problems consists in formulating the problem more productively.” (p. 34) (“On Problem Solving” first translated in English in “Psychological Monography” vol. 58, 1945 - German Edition 1935)
[1988, Second edition] Herbert A. Simon, The Sciences of the Artificial, M.I.T. Press, Cambridge Massachusetts
“... solving a problem simply means representing it so as to make the solution transparent.”
“... a deeper understanding of how representations are created and how they contribute to the solution of problems will become an essential component in the future theory of design.” (Chapter 5, p. 153)
[1988] Jean-Michel Hoc, Cognitive Psychology of Planning
“Defining a problem as a task representation pinpoints the fact that one of the core issues of problem solving is the building of this representation.”
“Broadly speaking, understanding is a goal-directed activity, which consists of constructing a coherent representation of a set of elements. The goal corresponds to what individuals aim at accomplishing with the representation they have constructed.” (p. 169)
“In other words, all understanding activities require the evaluation of a recovered or constructed representation from the dual standpoint of its coherence and its ability to satisfy a goal.” (p. 170)
“In problem solving, understanding strategies are related to tasks and thus can be linked to sets of extremely heterogeneous elements (states, properties, relations or procedures). Furthermore, understanding strategies do not only operate at the start of problem solving, but are at work during the entire process.” (p. 170)
[1988, Second edition] Herbert A. Simon, The Sciences of the Artificial, M.I.T. Press, Cambridge Massachusetts
“Efforts to solve a problem must be preceded by efforts to understand it.” (p. 111)
“Along one dimension we can distinguish between acquiring information (stored data structures) and acquiring skills (stored procedures).” (pp. 118-119)
[1986] A. J. Repo, The Dual Approach to the Value of Information
"One can find five kinds of knowledge and information which are needed and processed during a problem solving or handling task:
1. Task knowledge - knowledge that determines the problem and tasks in the domain.
2. Domain knowledge - earlier produced and organized data and information about the facts and relations which contribute to the problem.
3. Problem knowledge about the conditions of the problem at hand.
4. Problem solving knowledge - methodological know-how needed.
5. Outcome knowledge - new knowledge produced in the task: part of this is often presented as output information in the form of reports, manuals, etc." (p. 377)
(in "Information Processing & Management", vol. 22, n. 5)
[1990] George Pólya, How to Solve It, first published 1945
“Solving problems is a practical skill like, let us say, swimming. We acquire any practical skill by imitation and practice.” (Part I, p. 4)
[1966, First Edition 1958] James G. March and Herbert A. Simon, Organizations
“In virtually all human problem-solving, memory plays an enormous role. In the memory are stored both repertories of possible solutions to classes of problems that have been encountered in the past and repertories of components of problem solutions.” (Chapter 7, p. 177)