Problems solving

Problems solving- Definition

Early 1900s: Associationists explained problem-solving in terms of finding and strengthening stimulus-response

patterns which would deliver solutions (or not): reproductive solutions.

• 1940s: Gestalt psychologists studied productive problem-solving, believed solution involved identifying the appropriate problem structure for a problem.

• Neither approach had much place for cognitive activity.

• Changed by work of Herbert Simon in 1970s.

According to Mayer, problem solving is, "cognitive processing directed at transforming a given situation into a goal situation when no obvious method of solution is available to the problem solver. This definition suggests that there are three major aspects to problem solving:

It is purposeful (i.e., goal directed)

It involves cognitive rather than automatic processes.

A problem only exists when someone lacks the relevant knowledge to produce an immediate solution. Thus, a problem for most people (e.g., a mathematical calcula­tion) may not be so for someone with relevant expertise (e.g., a professional mathematician).

Problem solving refers to the thinking we do in order to answer a complex question or to figure out how to resolve an unfavorable situation.

Strategies for arriving at solutions include: Trial and error, algorithm, heuristic, and Insight.

Trial and error involves trying various possible solutions, and if that fails, trying others.

An algorithm is a step by step strategy for solving a problem, methodically leading to a specific solution.

A heuristic is a short-cut, step-saving thinking strategy or principle which generates a solution quickly (but possibly in error).

Insight refers to a sudden realization, a leap forward in thinking, that leads to a solution.

The most basic definition is “A problem is any given situation that differs from a desired goal”. This definition is very useful for discussing problem solving in terms of evolutionary adaptation, as it allows to understand every aspect of (human or animal) life as a problem. This includes issues like finding food in harsh winters, remembering where you left your provisions, making decisions about which way to go, learning, repeating and varying all kinds of complex movements, and so on.

Problem-solving is a mental process that involves discovering, analyzing and solving problems. The ultimate goal of problem-solving is to overcome obstacles and find a solution that best resolves the issue.

The Problem-Solving Cycle

The problem-solving cycle includes: problem identification, problem definition, strategy formulation, organization of information, allocation of resources, monitoring, and evaluation (shown in Figure 11.2).

Following is a description of each part of the problem-solving cycle.

1. Problem identification: Do we actually have a problem?

2. Problem definition and representation: What exactly is our problem?

3. Strategy formulation: How can we solve the problem? The strategy may involve analysis—breaking down the whole of a complex problem into manageable elements.

Instead, or perhaps in addition, it may involve the complementary process of synthesis—putting together various elements to arrange them into something useful.

Another pair of complementary strategies involves divergent and convergent thinking. In divergent thinking, you try to generate a diverse assortment of possible alternative solutions to a problem. Once you have considered a variety of possibilities, however, you must engage in convergent thinking to narrow down the multiple possibilities to converge on a single best answer.

4. Organization of information: How do the various pieces of information in the problem fit together?

5. Resource allocation: How much time, effort, money, etc., should I put into this problem?

6. Monitoring: Am I on track as I proceed to solve the problem?

7. Evaluation: Did I solve the problem correctly?

Classification of problems

Types of Problems

Problems can be categorized according to whether they have clear paths to a solution. Well-structured problems have clear paths to solutions. These problems also are termed well-defined problems. An example would be, “How do you find the area of a parallelogram?” Ill-structured problems lack clear paths to solutions. These problems are also termed illdefined problems.

· Well defined Vs ill defined

•         Well-structured problems

–        Clear path to the solution

•         Math problems

•         Anagrams

•         Ill-structured problems

–        Dimensions of problem are not specified or easy to infer

•         Finding an apartment

•         Writing a book

Cognitive psychologists often have studied a particular type of well-structured problem: the class of move problems, so termed because such problems require a series of moves to reach a final goal state. Perhaps the most well known of the move problems is one involving two antagonistic parties, whom we call “hobbits” and “orcs”.

People seem to make three main kinds of errors when trying to solve well-structured problems.

These errors are:

(1)   Inadvertently moving backward: They revert to a state that is further from the end goal. (eg. Problems solving disc experiment).

(2)   Making illegal moves: They make an illegal move—that is, a move that is not permitted according to the terms of the problem.

(3)   Not realizing the nature of the next legal move: They become “stuck”—they do not know what to do next, given the current stage of the problem.

One method for studying how to solve well-defined problems is to develop computer simulations. A problem space is the universe of all possible actions that can be applied to solving a problem, given any constraints that apply to the solution of the problem.

Algorithms are sequences of operations (in a problem space) that may be repeated over and over again and that, in theory, guarantee the solution to a problem.

Routine Vs Non routine

Routine problem solving stresses the use of sets of known or prescribed procedures (algorithms) to solve problems. Initially the problems presented to students are simple one-step situations requiring a simple procedure to be performed. Gradually, students are asked to solve more complex problems that involve multiple steps and include irrelevant data. Commencing with the concrete level, students are asked to develop their own story problem situations and demonstrate the solution process with manipulative and/or pictures and later with symbols.

One-step, two-step, or multiple-step routine problems can be easily assessed with paper and pencil tests typically focusing on the algorithm or algorithms being used.

Nonroutine problem solving stresses the use of heuristics and often requires little to no use of algorithms. Heuristics are procedures or strategies that do not guarantee a solution to a problem but provide a more highly probable method for discovering the solution to a problem.

Other problem-solving heuristics such as describing the problem situation, making the problem simpler, finding irrelevant information, working backwards, and classifying information are also emphasized.

There are two types of nonroutine problem solving situations, static and active. Static nonroutine problems have a fixed known goal and fixed known elements which are used to resolve the problem. Solving a jigsaw puzzle is an example of a static nonroutine problem. Given all pieces to a puzzle and a picture of the goal, learners are challenged to arrange the pieces to complete the picture. Various heuristics such as classifying the pieces by color, connecting the pieces which form the border, or connecting the pieces which form a salient feature to the puzzle, such as a flag pole, are typical ways in which people attempt to resolve such problems. Active nonroutine problem solving may have a fixed goal with changing elements; a changing goal or alternative goals with fixed elements; or changing or alternative goals with changing elements. The heuristics used in this form of problem solving are known as strategies.

Adversary Vs non Adversary

The term "adversary problem-solving" is normally used to describe situations in which two or more opponents are trying to achieve some goal. The passive or defensive side tries to prevent the active or attacking side from doing so. This kind of problem-solving usually appears in situations of human conflict and competition. Adversary situations are common in games and sports, but they may also occur in many fields of practical life.

First, some words on terminology. Researchers have examined both adversary and non-adversary problem solving. Chess play is an example of adversarial prob­lem solving, because the game of chess involves an opponent. Code-breaking, de-bugging computer pro­grams, and medical diagnosis are examples of non-adversarial problem domains. Those engaged in adversary problem solving must consider not only their own possible actions, but also those of an opponent.

Non-adversarial means there is a spirit of co-operation, a passive stance, the parties are willing to reach a mutually satisfying resolution to a problem. There is persuasion rather than coercion.

Knowledge rich Vs knowledge lean problems

There is a further important distinction between knowledge-rich and knowledge-lean problems. Knowledge-rich problems can only be solved by individuals possessing a considerable amount of specific knowledge, whereas knowledge-lean problems do not require the possession of such knowledge. In approximate terms, most traditional research on problem solving has involved the use of knowledge-lean problems, whereas research on expertise (e.g., chess grandmasters) has involved knowledge-rich problems.

Strategies of problems solving

In cognitive psychology, the term problem-solving refers to the mental process that people go through to discover, analyze and solve problems. This involves all of the steps in the problem process, including the discovery of the problem, the decision to tackle the issue, understanding the problem, researching the available options and taking actions to achieve your goals.

There are a number of different mental process at work during problem-solving. These include:

·         Perceptually recognizing a problem

·         Representing the problem in memory

·         Considering relevant information that applies to the current problem

·         Identify different aspects of the problem

·         Labeling and describing the problem

Problem-Solving Strategies

·         Algorithms: An algorithm is a step-by-step procedure that will always produce a correct solution. A mathematical formula is a good example of a problem-solving algorithm. While an algorithm guarantees an accurate answer, it is not always the best approach to problem solving. This strategy is not practical for many situations because it can be so time-consuming. For example, if you were trying to figure out all of the possible number combinations to a lock using an algorithm, it would take a very long time.

·         Heuristics: A heuristic is a mental rule-of-thumb strategy that may or may not work in certain situations. Unlike algorithms, heuristics do not always guarantee a correct solution. However, using this problem-solving strategy does allow people to simplify complex problems and reduce the total number of possible solutions to a more manageable set.

·         Trial-and-Error: A trial-and-error approach to problem-solving involves trying a number of different solutions and ruling out those that do not work. This approach can be a good option if you have a very limited number of options available. If there are many different choices, you are better off narrowing down the possible options using another problem-solving technique before attempting trial-and-error.

·         Insight: In some cases, the solution to a problem can appear as a sudden insight. According to researchers, insight can occur because you realize that the problem is actually similar to something that you have dealt with in the past, but in most cases the underlying mental processes that lead to insight happen outside of awareness.

Search strategy

A search strategy is defined by picking the order of node expansion

Strategies are evaluated along the following dimensions:

n  completeness: does it always find a solution if one exists?

n  time complexity: number of nodes generated

n  space complexity: maximum number of nodes in memory

n  optimality: does it always find a least-cost solution?

Time and space complexity are measured in terms of

n  b: maximum branching factor of the search tree

n  d: depth of the least-cost solution

n  m: maximum depth of the state space (may be ∞)

Uninformed search strategies

Uninformed search strategies use only the information available in the problem definition

n  Breadth-first search (Expand shallowest unexpanded node)

n  Uniform-cost search (Expand least-cost unexpanded node)

n  Depth-first search (Expand deepest unexpanded node)

n  Depth-limited search

n  Iterative deepening search

Means-ends analysis

•         Compare your current state with the goal and choose an action to bring you closer to the goal

•         Break a problem down into smaller sub goals

•         May not work if sub goals cannot be identified  

Means-ends analysis is a problem solving strategy that arose from the work on problem solving of Newell and Simon (1972).  In means-ends analysis, one solves a problem by considering the obstacles that stand between the initial problem state and the goal state.  The elimination of these obstacles (and, recursively, the obstacles in the way of eliminating these obstacles) are then defined as (simpler) subgoals to be achieved.  When all of the subgoals have been achieved – when all of the obstacles are out of the way – then the main goal of interest has been achieved.  Because the subgoals have been called up by the need to solve this main goal, means-ends analysis can be viewed as a search strategy in which the long-range goal is always kept in mind to guide problem solving.  It is not as near-sighted as other search techniques, like hill climbing.

Means-ends analysis is a version of divide-and-conquer.  The difference between the two is that divide-and-conquer is purely recursive: the subproblems that are solved are always of the same type.  Means-ends analysis is more flexible, and less obviously recursive, because the subproblems that are defined for it need not all be of the same type.

Analogical transfer

The transfer of knowledge from one situation to another by finding a set of one-to-one correspondences between aspects of one body of information and aspects of another. Analogical transfer is one method we have of coming up with creative solutions to some of life's problems.

Analogical Transfer

n  People try to solve the Target Problem

n  Some are presented with a Source Problem or Source Story that can help them solve the Target

n  Russian Marriage (Source) -> Checkerboard (Target)

Steps of Analogical Problem Solving

Noticing

n  Seeing that there is a possible analogy between problems

n  Most difficult, especially in the real world

Mapping

n  Connecting elements of the source problem to elements of the target problem

Applying

n  Using the analogy to generate the solution

Improving Analogical Transfer

Two types of features (best when similar)

n  Structural Features

n  Surface Features

Analogical Encoding

n  Strategy for training people to be able to notice and apply analogies

n  Compare different source problems first, then solve Target

Working backward

The problem solver starts at the end and tries to work backward from there.

By applying the working backwards strategy, students find the solution to a problem by starting with the answer and using inverse operations to undo the steps stated in the problem. 

Backtracking 

Backtracking is a general algorithmic technique that considers searching every possible combination in order to solve an optimization problem. Backtracking is also known as depth-first search or branch and bound. By inserting more knowledge of the problem, the search tree can be pruned to avoid considering cases that don't look promising. While backtracking is useful for hard problems to which we do not know more efficient solutions, it is a poor solution for the everyday problems that other techniques are much better at solving.

Two main mechanisms in BT

1. Backtracking:

•          To recover from dead-ends

•          To go back

2. Consistency checking:

•          To expand consistent paths

•          To move forward

Backtracking Methodology

1.   View picking a solution as a sequence of choices

2.   For each choice, consider every option recursively

3.   Return the best solution found

Backtracking can be applied only for problems which admit the concept of a "partial candidate solution" and a relatively quick test of whether it can possibly be completed to a valid solution. Backtracking is an important tool for solving constraint satisfaction problems, such as crosswords, verbal arithmetic, Sudoku, and many other puzzles. It is often the most convenient technique for parsing, for the knapsack problem and other combinatorial optimization problems. It is also the basis of the so-called logic programming languages such as Icon, Planner and Prolog. The term "backtrack" was coined by American mathematician D. H. Lehmer in the 1950s.

Schema based models

A schema is an organized structure “consisting of certain elements and relations” specific to a situation (Mayer, 1999, p. 228). Schemata are the appropriate mechanism for the problem solver to “capture both the patterns of relationships as well as their linkages to operations” (Marshall, 1995, p. 67).

Schema-Based Problem Solving Model

 • Schema knowledge/Problem Schema Identification

• Elaboration knowledge/Representation

• Strategic Knowledge/Planning

• Executive Knowledge/Solution

 One schema-based problem-solving model involves applying four procedural steps—identification, representation, planning, and solution—to already known problem types, or schemas. The student first reads the problem and identifies the problem schema. Next, the student represents the problem by diagramming the key infor­mation. Then the student plans how to solve the problem by selecting the appropriate operation and writing out the math equation. Finally, the student solves the problem.

Explicit instruction in problem-solving rules combined with

Multiple trace Model

Multiple trace theory

Multiple Trace Theory (MTT) builds on the distinction between semantic memory and episodic memory and addresses perceived shortcomings of the standard model with respect to the dependency of the hippocampus. Multiple Trace Theory argues that the hippocampus is always involved in the retrieval and storage of episodic memories.[14] It is thought that semantic memories, including basic information encoded during the storage of episodic memories, can be established in structures apart from the hippocampal system such as the neo-cortex in the process of consolidation.Hence, while proper hippocampal functioning is necessary for the retention and retrieval of episodic memories, it is less necessary during the encoding and use of semantic memories. As memories age there are long-term interactions between the hippocampus and neo-cortex and this leads to the establishment of aspects of memory within structures aside from the hippocampus.MTT thus states that both episodic and semantic memories rely on the hippocampus and the latter becomes somewhat independent of the hippocampus during consolidation. An important distinction between MTT and the standard model is that the standard model proposes that all memories become independent of the hippocampus after several years. However, Nadel and Moscovitch have shown that the hippocampus was involved in memory recall for all remote autobiographical memories no matter of their age. An important point they make while interpreting the results is that activation in the hippocampus was equally as strong regardless of the fact that the memories recalled were as old as 45 years prior to the date of the experiment. This is complicated by the fact that the hippocampus is constantly involved in the encoding of new events and activation due to this fact is hard to separate using baseline measures. Because of this, activation of the hippocampus during retrieval of distant memories may simply be a by-product of the subject encoding the study as an event.

Criticisms of multiple trace theory

Haist, Gore, and Mao, sought to examine the temporal nature of consolidation within the hippocampus to test the multiple trace theory against the standard view.They found that the hippocampus does not substantially contribute to the recollection of remote memories after a period of a few years. They claim that advances in the functional magnetic resonance imaging have allowed them to improve their distinction between the hippocampus and the entorhinal cortex which they claim is more enduring in its activation from remote memory retrieval. They also criticize the use of memories during testing which cannot be confirmed as accurate. Finally, they state that the initial interview in the scanner acted as an encoding event as such differences between recent and remote memories would be obscured.

Factors affecting problem solving

Set Effects

People may become biased by experience to prefer certain approaches to a problem, which may block the solution in a particular case — the einstellung effect (mechanization of thought).

Functional Fixedness: This term refers to the tendency to view problems only in their customary manner. Functional fixedness prevents people from fully seeing all of the different options that might be available to find a solution.

Incubation Effects

o  Problems depending upon insight tend to benefit from interruption.

n  Delay may break set effects.

o  Problems depending on a set of steps or procedures do not benefit from interruption.

n  Subjects forget their plan and must review what was previously done.

o  There is no magical “aha” moment where everything falls into place, even though it feels that way.

n  People let go of poor ways of solving the problem during incubation.

o  Subjects do not know when they are close to a solution, so it seems like insight – but they were working all along.

Incorrect representations

Lack of expertise

Improving problems solving

Alert people affected by the problem, if any. This gives them a stake in resolving it.

·         As you work through possible solutions, keep these people informed of your progress. This lets them know what to expect and when to expect it. Be as optimistic as you can, but also as realistic as you can.

Define the problem clearly. Avoid making snap judgments based on a few symptoms but look for root causes whenever possible. Poor performance may not be caused by an individual's lack of skills but by ineffective communication of expectations and insufficient training in how to meet those expectations.

·         Defining the problem clearly may require looking at it from several angles and perspectives, not just one or two. This will avoid identifying a prospective solution as a problem.

Choose a problem-solving strategy. The approach to solving the problem, once it has been defined, can be handled through a number of methods, some of which are listed below:

·         Brainstorming is the generation and recording of ideas as they occur to you, either alone or in a group. You do this for a set period of time, then go through the list of solutions to evaluate their suitability.

·         Appreciative inquiry develops solutions by analyzing what's currently going right and determining whether it can be applied to solve the problem at hand.

·         Design thinking means thinking like a product designer, observing how people interact with a product or service and noting what problems they are having with it.

·         In some cases, a combination of strategies may be the best approach to solving a problem.

Gather information. Coupled with clearly defining a problem is gathering information about it. This may meaning consulting with people closer to certain aspects of the problem to get a proper grasp of its scope, or researching similar situations elsewhere to see what the root causes of those problems were and how they were resolved, if at all.

·         Gathering information is also essential in directing a seemingly undirected problem-solving strategy such as brainstorming. An informed mind can generate better, more appropriate solutions than an uninformed mind.

Analyze the information. The information needs to be analyzed for its relevance to the problem and its importance. The most critical, or key, information should be drawn upon in formulating a solution, while the remaining information may be need to ranked for its importance and relevance.

·         Sometimes, information needs to be organized graphically to be useful, using tools such as flow charts, system diagrams, cause-and-effect diagrams or other such devices.

Develop possible solutions based on the information you've collected and your strategy.

Evaluate the solutions generated. Just as it was necessary to analyze the information found for its relevance to the problem, prospective solutions must be analyzed for their suitability to determine which is best to handle the problem. In some cases, this means constructing prototypes and testing them; in other cases, this may mean using computer simulations or "thought experiments" to analyze the consequences of using that solution.

Implement your solution. Once the best solution is determined, put it into practice. This may be done on a limited scale at first to verify that the solution is indeed the best, or it may be implemented system-wide if the need for it is critical.

Get feedback. While this step should be implemented while testing prospective solutions, it is also helpful to continue getting feedback to verify that the best solution will perform as expected and to find ways to adjust it if it isn't.

Creative problem solving

“Creative problem solving is -  looking at the same thing as everyone else and thinking something different.”

z         The creative person uses information to form new ideas.

z         The real key to creative problem solving is what you do with the knowledge.

z         Creative problem solving requires an attitude that allows you to search for new ideas and use your knowledge and experience.

z         Change perspective and use knowledge to make the ordinary extraordinary and the usual commonplace.

Componential analysis (feature analysis or contrast analysis) is the analysis of words through structured sets of semantic features, which are given as “present”, “absent” or “indifferent with reference to feature”. The method thus departs from the principle of compositionality. Componential analysis is a method typical of structural semantics which analyzes the structure of a word's meaning. Thus, it reveals the culturally important features by which speakers of the language distinguish different words in the domain.

Examples

man = [+ male], [+ mature] or woman = [– male], [+ mature] or boy = [+ male], [– mature] or girl = [– male] [– mature] or child = [+/– male] [– mature]. In other words, the word girl can have three basic factors (or semantic properties): human, young, and female.

To summarize, one word can have basic underlying meanings that are well established depending on the cultural context.

Componential analysis (feature analysis or contrast analysis) is the analysis of words through structured sets of semantic features, which are given as “present”, “absent” or “indifferent with reference to feature”. The method thus departs from the principle of compositionality. Componential analysis is a method typical of structural semantics which analyzes the structure of a word's meaning. Thus, it reveals the culturally important features by which speakers of the language distinguish different words in the domain