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## Problem Solving

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Problem Solving

Initial

State

Most everything we “do” can be considered as a form of problem solving: it is directed toward achieving a goal and involves removing or circumventing obstacles to achieve the goal.

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Goal

State

Problem Solving (con’t)

We typically don’t call behavior which is well-practiced “problem solving” because we already know the solution. Rather, we refer to the initial stages during which we acquire the solution as problem solving.

- Goal directedness - behavior is organized toward some goal.
- Subgoal decomposition - decomposing the original goal into subtasks.
- Operator application - actions that will transform the problem state into another problem state.

Problem solving involves:

The Problem Space and Search

The “problem space” consists of all the different states (i.e., representations of the problem in some degree of solution) a problem can take. The states can by physical or knowledge states.

Each time an operator (i.e., some behavior) is applied, the problem state changes. Therefore, there are many potential combinations or paths that can be taken to reach the goal.

The task is to search through the many potential problem states by choosing the correct set of operators that will reach the goal.

The Problem Space and Search

A “search tree” is a representation of all the possible states that can be reached from previous states. A complete tree will show all possible solutions, including the shortest sequence of operators between the start and goal states.

But how do we “search” the possible states?

- Algorithms -- trying all possibilities; if a solution exits, an algorithm will assure it is found.
- Heuristics -- general rules of thumb or shortcuts in reasoning.

Problem Solving Operators

What determines the operators available to the problem solver and how are particular operators selected from among those available?

- Discovery -- making a discovery on your own (e.g., learn how to use the COPY/PASTE command and, therefore, acquire a new operator for completing paper work).
- Instruction from another -- being told how to use an operator (e.g., use a circuit tester to check for live circuit).
- Observing another -- watching some apply an operator (e.g., tying a sheet bend knot to link two ropes together).

How do we acquire operators?

Analogy

Another means of acquiring operators is through “analogy.” That is, using the solution or understanding of one problem in the context of another problem.

For example:

The solar system as an analogy for the structure of the atom; using the worked-out problems in a math book to solve the problems at the end of the chapter; others?

- Regardless of the analogy, it is necessary to map the elements from the source example to the target.
- It is important, however, that critical aspects of the current problem match the critical aspects of the example used as the analogy.

How We Choose Operators

- When faced with the option of multiple operators in a given problem state, how do we decide which to use?

- Backup avoidance -- reluctance to apply operators that undo the effect of previous operators.
- Difference reduction -- selecting an operator that reduces the greatest difference between the current problem state and the goal state and thereby making the new problem state resemble more the goal state.

In general, we are guided by three principles:

How We Choose Operators (con’t)

- Means-Ends analysis -- similar to the difference reduction method, except the essential focus is on enabling blocked operators (means) by temporarily making the blocked operator a subgoal (ends) and establishing an “operator subgoal” (i.e., a subgoal whose purpose is to eliminate the difference that is blocking the application of an operator).

It appears the prefrontal cortex plays a major role in maintaining the kinds of complex goal structures seen in means-ends analysis in working memory. For example, Vietnam vets with prefrontal cortex damage have difficulty in the Tower of Hanoi problem, in which subgoals are necessary for its solution.

Problem Representation

- In addition to acquiring operators and selecting the appropriate ones for a problem, it is also important to examine how the problem states of a problem are represented since representation can influence operator selection.

Failure to represent a problem correctly may also interfere with one being able to recognize a new problem is the same type of problem as a previous one for which they have a solution… that is, we may not recognize the analogy, even though it is present.

Cognitive Biases

We bring many cognitive biases with us when we approach problems. That is, thinking in rigid ways that sometimes interfere with us finding a solution:

- Functional fixedness -- the inability to represent or use familiar objects in unique ways.
- Set effects (mental set) -- when knowledge or procedures become more available at the expense of others; the inability to break out of a particular line of reasoning.

Cognitive bias can take many forms:

Cognitive Biases (con’t)

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If the available knowledge or procedures are necessary for solution, then problem solving is facilitated. If the available knowledge or procedures are unnecessary, problem solving will be inhibited.

Incubation

In many cases, a set can be overcome by taking some time away from the problem to increase the chance of approaching it from a different perspective at a later time.

It appears that the time away from the problem helps dissipate the tendency to continue using inappropriate procedures or knowledge, not that we are working on it unconsciously as is sometimes believed.

Insight

Have you ever worked on a problem for some time and, in a moment, suddenly realize the solution? Such an experience is often referred to as “insight” or the “aha” experience.

While the subjective feeling is one of sudden realization of the solution, many argue it is the result of one not knowing how close to the solution they actually are.

Negative Information

- The most difficult kinds of problems to solve seem to be those which involve “negative information” -- a non-occurrence.
- “Is there any point to which you would wish to draw my attention?”
- “To the curious incident of the dog in the night-time.”
- “The dog did nothing in the night-time.”
- “That was the curious incident, “ remarked Sherlock Holmes.
- “… I had grasped the significance of the silence of the dog… The Simpson incident had shown me that a dog was kept in the stables, and yet, though someone had been in and had fetched out a horse, he had not barked enough to arouse the two lads in the loft. Obviously the midnight visitor was someone whom the dog knew well.” -- Silver Blaze, Sir Arthur Conan Doyle

We seem to be geared toward using information that exists, not that which does not.

Problem Solving Strategies

While the means-ends analysis is a very powerful strategy when trying to solve problems, there are two others worthy of mention:

- Working backward – begin at the goal state and identify the problem states leading to the initial state.
- Trial-and-Error – when all else fails…

Good Problem Solving Practice

There are a number of steps that can assist in problem solving:

- Preparation
- state problem clearly (i.e., appropriate representation)
- identify relevant information
- identify constraints on solution
- Production
- generate possible solutions through “brainstorming.”
- Evalution
- test each possible solution and identify suitable solution.
- If no solution is found, examine step 1 again and then repeat steps 2 and 3.

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