Psychology Of Problem Solving And Reasoning Research Paper

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This research paper discusses two important aspects of human thinking: how people solve problems and how they use systematic thought to reach conclusions from premises. The two processes are intermingled closely in most everyday human thinking.

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Woodworth (1938) wrote:

Two chapters (of 30!) will not be too many for the large topic of thinking.

Woodworth here understates somewhat the importance he assigned to ‘higher mental processes,’ for another chapter examines reading, and nine others, memory, and learning. In the second half of the twentieth century, the small role of high-level cognition in experimental psychology has been enlarged greatly by treating thinking as symbolic information processing, formulating theories of thinking as nonnumerical computer programs, and introducing new observational methods and instruments to capture increased temporal detail in thought processes (Newell and Simon 1972).

Woodworth (1938) divided his final two chapters ‘according to the historical sources of two streams of experimentation … One stream arose in the study of animal behavior and went on to human problem solving; the other started with human thinking of the more verbal sort.’ Two branches of the first stream investigated (a) search as the basic problem solving process (trial-and-error, then selective, search; e.g., Morgan 1894, Thorndike 1898), and (b) insight (e.g., Kohler 1917). Little attention was given to expert problem solving.

The second stream of research, on verbal thinking, included concept attainment (e.g., Hull 1920), logical deduction (‘reasoning’), creativity (mostly anecdotal reports on inventors, scientists, and artists), and verbal problem solving, the latter initiated by Selz (1913, 1922) and further developed by Duncker (1935) and De Groot (1946). All of these lines of work were important harbingers of the new information processing psychology, but the connections among them only became apparent later. This research paper describes today’s knowledge of thought processes within the information processing framework down to, but not including, the neural implementation.

1. Problem Solving

Modern theories of problem solving have gone through three stages. Most early research viewed problem solving as a search process. In the 1960s, attention began to turn to expert problem solving and its dependence on recognition that evoked knowledge from memory. A decade later, it had become evident that solving complex problems usually also required interacting search in several distinct problem spaces.

1.1 Problem Solving As Search

Problem solving is often described as search: moving in a problem space from an initial situation toward a goal situation that satisfies all the conditions for a solution. Problem difficulty was initially thought to vary with the size of the problem space. A space with b choices at each point in the search, and paths s steps in length, would contain bs different paths.

Choosing a chess move w as very difficult, for from a typical current position 1020 or more branching paths might lead to stronger or weaker positions, and ultimately to won, lost, or drawn games. As a human being could search only an insignificant fraction of these paths, skill depended on discovering just those that might lead to favorable outcomes. DeGroot (1946), showed that stronger players did not search much more extensively than weaker players, and that even grandmasters rarely looked at more than 100 positions before making a move.

These findings led to a theory of problem solving postulating: (a) an ‘understanding’ process that generates a problem space from the problem description, (b) a selective (heuristic) search process that uses rules of thumb to follow the most promising paths, and (c) a position evaluation function to assess the relative merits of the positions reached. The process can be repeated, re-evaluating the situation after each step. Several chess programs that deliberately limit themselves to searches of the magnitudes that humans can make demonstrate that chess of credible quality can be played in this way, confirming and extending experimental findings about human players (Simon and Schaeffer 1992).

The same scheme applies to problems generally, not just to chess. Methods for generating paths include: generate-and-test (i.e., trial-and-error), hillclimbing (moving toward higher values of an evaluation function), and means-ends analysis. At each step in means-ends analysis: (a) one or more differences are discovered between the current situation and the goal conditions; (b) using past knowledge and experience, an operator is applied that usually eliminates or reduces differences of the kind discovered; (c) the process is repeated until all differences have been removed and the goal attained.

Suppose the goal is to solve an equation for x: 7x + 10 = 4x + 22. The operators add or subtract the same expression from both sides of the equation or multiply or divide both by the same expression. The differences between the given expression and the solution (x = N) are the presence of a stand-alone number on the left side of the equation, an expression in x on the right, or a coefficient other than 1 for x on the left. In the equation above, means-ends analysis would discover an unwanted number, 10, on the left, then an expression in x, 4x, on the right, and would subtract both from the equation, leaving 3x = 12. Now it would notice x’s coefficient, 3, and would divide, obtaining x = 4.

1.2 Knowledge For Solving Problems

Some problems are hard even though their problem spaces are small. For example, the Chinese ring puzzle, which requires freeing a metal bar from a series of rings, has an unbranching problem space, so that the solution can be reached simply by never reversing direction. Nevertheless, this problem, with seven rings, is seldom solved in less than half an hour. It has been found that people apply ‘reasonable’ heuristics, e.g., never reconnect a ring that has once been removed. But adherence to this plausible means-ends principle guarantees that the problem will not be solved. Gradually, recognition that rings sometimes have to be reconnected uncovers the solution strategy.

Comparing the problem-solving by experts and novices soon demonstrated that domain-specific knowledge (of which this is a simple example) lay at the basis of expert skills, connecting problem solving with memory. A large discrimination net in the expert’s memory permits sensory recognition of common stimulus patterns in the domain. Thus, a chess grandmaster can recognize perhaps 300,000 configurations of pieces that occur repeatedly in games (e.g., ‘fork,’ ‘open file,’ ‘weak bishop,’ etc.). Associated in memory with each such pattern is knowledge of actions that should be explored whenever the pattern is present. Most ‘leaps’ in thought called intuitive, insightful, or creative are produced by this knowledge-based pattern recognition (Richman et al. 1996).

The expert, then, recognizes familiar patterns, evoking knowledge that guides search to strong moves in a limited time, searching only a little more than the novice, but along productive instead of unproductive paths.

1.3 Multiple Problem Spaces

As empirical studies advanced, it became apparent that problem solving search employs many different spaces, including a space of potential problem spaces. Searching for solutions may also be preceded by searching for effective generators of actions. Problems of scientific discovery require search in spaces of potentially useful instruments, experimental designs, and hypotheses. The Soar program developed by Newell, with Laird and Rosenbloom (Newell 1990), searches in multiple spaces of these kinds.

2. Reasoning

Reasoning encompasses thinking that is orderly or ‘logical’: for example, the processes of formal logic and of concept attainment. Hence, most problem solving processes can also be called ‘reasoning.’ In logic, reasoning begins with premises (a situation), and uses rules of inference (actions) to derive conclusions (attain a goal). It may be inductive or deductive.

2.1 Forms Of Reasoning

In inductive logic, the premises usually are empirical assertions and the rules of inference produce generalizations of these using terms like ‘all,’ ‘most,’ ‘probable.’ For example, from observation of many white swans, and none of other colors, one may reason inductively to: ‘All swans are white,’ ‘Probably all swans are white,’ ‘The probability that any swan is white is 99 in 100,’ or ‘Most swans are white.’ At one time or another, all of these sorts of induction have been proposed.

In deductive logic, the initial premises are taken as axioms and the rules of inference are tautologies (e.g., If all A are B and c is an A, then c is a B), so that, in any possible world in which the premises hold, the conclusions must hold.

Reasoning may be used to discover either a concept, an inductive conclusion, or a deductive conclusion; it may also be used to verify such a concept or conclusion, once reached. Discovery is a wholly different process from verification.

Verifying a proof is a deductive process: simply examining each step to assure that it was obtained from previous steps by an application of the rules of inference. It requires no real search. However, discovering a proof is an inductive problem solving process, which involves searching for a path from the premises to the conclusion, using the rules of inference. (The task of the first computer that simulated problem solving, the Logic Theorist, was to find, inductively, proofs for theorems in symbolic logic (Newell and Simon 1956)).

In concept attainment, the task is to find, by examining instances, a rule that distinguishes instances from noninstances of the concept: swans from nonswans. This is an inductive task, where the search proceeds in two spaces: a space of possible instances and noninstances and a space of rules (e.g., feature descriptions that can distinguish instances from noninstances). Search in the space of rules generates possible concepts; search in the space of instances generates objects for testing the rules.

2.2 Everyday Reasoning

Much remains to be learned about how human beings reason in their everyday affairs. One attractive hypothesis, only partially substantiated by the evidence, is that they use the same processes that they use for structured problems. Each situation a person encounters evokes from memory a constantly changing array of knowledge, beliefs, and goals, with varying degrees of mutual consistency and inconsistency. Current goals and the problem spaces they evoke trigger search that leads to actions and thence again to changes in the situations and the materials evoked from memory.

The difficulty of understanding such a system largely derives from the size and complexity of memory, with its idiosyncrasies of what it does or doesn’t recognize at any moment and does or doesn’t know, believe, like, or dislike about the things recognized. The closest approaches to theories of such systems have been simulations (unified theories) of sizable subsets of the cognitive processes (see Richman et al. 1996).

2.3 Reasoning Mechanisms

Sometimes it has been proposed that specific kinds of reasoning employ specialized machinery in the brain, just as sensory and motor processes have their specialized organs. Chomsky (1976), for example, has claimed that there is a special language facility (not merely language ‘regions,’ but biological structures specialized to processing language). Rips (1994) has made claims for an inherited machinery for logical reasoning. Comparable claims have been made for ‘pictorial’ reasoning, using mental imagery.

The alternative hypothesis does not deny the evidences for brain localization and specialized functions but claims that all these faculties rest upon the same basic associative neural processes and organizations.

The latter alternative has been explored by constructing simulations that are capable of performing these functions without specialized mechanisms. Thus, without any specialization for logic, syllogistic reasoning will be carried out by all systems that employ if- then rules (production systems), because whenever all the conditions (‘ifs’) of a rule are satisfied in memory, the actions (‘then’) will be executed—a generalization of the classical stimulus-response mechanism. Production systems, conjoined with associative memories, provide a completely general architecture for symbolic processing, ‘logical’ or not (Eisenstadt and Simon 1997). They can reason about mental diagrams and can combine verbal with visual reasoning.

Concept attainment, as well as various kinds of verbal learning and expert memory performance, can be achieved by a discrimination net which grows, by exposure to stimuli and feedback of results, a network of tests of stimulus features that classify the stimuli. Linked to the discrimination net, and indexed by it, a large associative (semantic) memory holds information about each of its concepts, permitting the kind of problem solving by recognition that was discussed in Sect. 1.2 (Richman et al. 1996).

The evidence available at the turn of the twenty-first century allows no clear choice between specialized mechanisms for special domains of reasoning vs. regional localizations of functions that all employ production systems capable of reasoning and learning (i.e., acquiring new productions) and discrimination nets for recognition, and share an associative semantic memory of knowledge and beliefs, indexed by the net. Whichever turns out to be the actual architecture, there will remain the important task of finding the neural mechanisms that implement it.


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