20140410040636_topic 7 problem solving strategies

8/10/2019 20140410040636_Topic 7 Problem Solving Strategies

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INTRODUCTION

In Topic 1, we have seen two different guidelines in solving a problem (refer sub-

topic 1.4). These guidelines are based on existing concepts. We will explore otherforms of conceptualisation before looking at a number of strategies in problemsolving. These strategies are meant for the most important step in the problem-solving process; and concerns your action plan. This is an interesting topic thatcan be very useful in your personal life as well.

TTooppiicc

77 Problem-

SolvingStrategies

LEARNING OUTCOMES

By the end of this topic, you should be able to:

1. Describe how problem solving can be conceptualised;

2. Describe the problem decomposition or subgoaling approach inproblem solving;

3. Explain the working backwards and hill climbing approach inproblem solving;

4. Describe the means-end analysis and forward chaining strategies inproblem solving ;

5.

Describe the other problem-solving methods such as use of analogy,specialisation and generalisation; and

6. Explain the use of extreme cases method for problem solving.


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TOPIC 7 PROBLEM-SOLVING STRATEGIES 137

CONCEPTUALISATION IN PROBLEMSOLVING

In general, most authors have conceptualised problem solving as a stepwiseprocess. Plya (1957) identified four important steps in the method by whichproblem solving should be done:

(a) Understanding the problem;

(b) Devising a plan;

(c) Carrying out the plan; and

(d) Looking back.

Subsequently, Hayes (1981) extended this conceptualisation by adding a specificreference to representation and by further breaking down looking back intotwo parts where one stressed on assessing the immediate problem-solving effortand the second one on learning something that may be important in the future.Figure 7.1 illustrates these six steps.

Figure 7 1: Stepwise conceptions of problem solving (Hayes, 1981)

7.1


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TOPIC 7 PROBLEM-SOLVING STRATEGIES138

Bransford and Stein (1984) proposed another conceptualisation that is based onthe acronym below:

I : Identify the problem

D : Define and represent the problem

E : Explore possible strategies

A : Act on the strategies

L : Look back and evaluate the effects of your activities

However, many other authors have come up with comparable stepwiseconceptualisations of the problem-solving process which are variations of theoriginal proposal by Plya (1954).

PROBLEM-SOLVING STRATEGIES

Evidence suggests that there are many strategies that are specific to a particularfield. However, there are also some that can be effective in all fields as well. Ingeneral, there appears to be a trade-off between range of applicability and power.The more widely applicable a particular strategy is, the less field dependent itwill be. It is important to know strategies that can be employed to a certainadvantage across domains. In the following sub-topics, we will look into some ofthe experiential methods that have been discovered by several researchers:

(a) Problem decomposition or subgoaling;

(b) Working backwards;

(c)

Hill climbing;

(d) Means-end analysis;

(e) Forward chaining;

(f) Considering analogous problems;

(g) Specialisation and generalisation;

(h) Considering extreme cases; and

(i) Mixing strategies.

7.2

ACTIVITY 7.1

Make an Internet search and briefly explain George Plya's list ofmental operations involved in problem solving called PUPILS.


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7.2.1 Decomposition or Subgoaling

At certain times, a problem may be too large to be solved. Another suitable

alternative would be to break a complex problem into a set of sequences ofsimpler problems first. Then, solve the complex problem by combining thesolutions to the simpler sub-problems (Figure 7.2).

Figure 7 2: Hierarchy and decomposition of a problem into sub-problems

Adapted from: http://ccs.njit.edu/fadi/public/books/dissert.htm

This application is widely used by computer programmers where complexprograms can be considered as a collection of simpler programs. Thus, theprocess of decomposition can be performed at several levels. When a big problemis broken to smaller components, both the identification of solution becomessimpler and furthermore an individual can understand a problem better (Plya,

1957; Hayes, 1981). Nevertheless, a risk that may be faced in this approach is thatthe nature of the problem may be changed perhaps by eliminating a criticalaspect of it, either while breaking the problem down or combining the sub-problem solutions.


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7.2.2 Working Backwards

Some researchers believe that a problem should be characterised in terms of ajourney where one begins at point A (the initial state) and makes an attempt to

arrive at point B (the goal). The main challenge in this approach is to locate apath that can bring the individual from the starting point to the end point.

There are two ways in solving the problem here:

(a) Find your way from point A to B; or

(b) Work out the journey backwards, from point B to A (as illustrated inFigure 7.3).

In addition, one can work backwards not only from the final goal but also from

an intermediate goal state, especially in circumstances where there is a way toidentify these intermediate states.

Figure 7 3: Working backwards strategy in problem solving

Adapted from:

http://www.1000ventures. com/business_guide/

crosscuttings/problem_solving_workback.html

ACTIVITY 7.2

There are several ways to do problem decompositions, the most well-

known probably being recursive decomposition, data decomposition,functional decomposition, exploratory decomposition and speculativedecomposition. Carry out an Internet search and briefly explain thevarious strategies.


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7.2.3 Hill Climbing

Climbing a hill is another approach in problem solving especially in cases where

ones intention is to get to the top of a hill, but there is no apparent marked pathto follow. Hence, a strategy that can be employed is to move in such a way thatan individual is always moving upwards. The moment the individual realisesthat the steps are descending in nature, one turns around and takes the oppositepath. In other words, when applied in problem solving, one is always takingsteps that bring one nearer to the particular goal. The main disadvantage of thehill climbing approach is that one may get stranded on the top of a small hill inthe vicinity or sometimes on the slope of the large hill that the individual hopesto climb. At times, it may be necessary to take a few steps downhill in order toreach the final destination, the peak (Figure 7.4). In the same way, in problem

solving, it is sometimes essential to take steps that appear far from the goal inorder to finally reach the goal. However, in a real life scenario, many individualsfind this difficult to carry out (Anderson, 1990).

Figure 7 4: Hill climbing strategy

SELF-CHECK 7.1

Joe forgot to check how much money he began the day with. During the

day, he spent $8.00 on breakfast, withdrew $40.00 from the ATM, got hisdry cleaning for $12.00, and bought 5 shirts for $22.00 a piece (plus 8%sales tax). At the end of the day, he had $100.00, how much did he startthe day with? Use the working backwards strategy to solve this problem.


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7.2.4 Means-end Analysis

In general, this approach is rather similar to hill climbing, however, it is moreflexible and adaptable to various types of problem solving. Newell & Simon(1972) explained it thoroughly and used it widely in the development ofproblem-solving computer programs. The first thing an individual should dowhen employing this strategy is to identify the goal state followed by a thoroughlisting of the differences between it and the current state. Finally, one shouldidentify a promising procedure in order to reduce the differences between thecurrent state and goal. Two main actions can be taken to reduce the distancebetween the current state and goal:

(a)

By taking a step that will cause the current state to be as similar to the goal;or

(b) By working backward where one will bring the goal closer to the currentstate.

One can be stranded if one sticks too obsessively to the rule of not taking a stepthat decreases the disparity between the current state and the goal. According toAnderson (1993), means-end analysis is a natural component of the thinkingmachinery for both human beings as well as primates. Means-end analysis isvital to solve daily problems such as getting the right train connection. Basically,first of all, you have to figure out where you catch the first train and where youwant to arrive. Then you have to look for possible changes just in case you do notget a direct connection. Finally, you have to figure out what are the most suitabletimes of departure and arrival, on which platforms you leave and arrive andmake it all fit together.

SELF-CHECK 7.2

You are required to use the hill climbing strategy in the river crossing

problem. There are a number of predators and prey on one side of theriver and there is a boat with a limited capacity. The goal state is tohave everyone on the opposite side of the river. Explain how you willemploy this strategy. You must be able to identify the initial state, thegoal and the fact that the boat used has a limited capacity fortransporting individuals. Will the downhill step be necessary insolving this problem?


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7.2.5 Forward Chaining

In this strategy, one starts with the task and works directly towards the goal(Figure 7.5). Forward chaining begins with the available data and uses inferencerules to extract more data until a goal is attained. However, forward chaining can

only be applied by an individual with adequate and thorough understanding of aproblem in order to be able to come up with its correct concrete representationfrom the problem statement (Simon & Simon, 1978).

Figure 7 5

: Forward chaining strategy

Source: http://www.cs.bham.ac.uk/~mmk/Teaching/AI/l2.html

This strategy is usually used by experts and not beginners who are more inclinedto employ strategies such as means-end analysis and working backwards (Larkinet al., 1980a, 1980b). This is due to the fact that experts have the ability tocategorise problems in terms of basic principles and their knowledge of strategiesthat work for specific problem types that makes this possible. However, whenindividuals were provided with more practice with problems of a particular type,they were inclined to change spontaneously from a means-end strategy that theywere using to the forward-chaining strategy.

ACTIVITY 7.3

Carry out a search in the Internet on the ability of infants to employ

simple forms of means-ends analysis in the second half of the first year.Briefly explain this phenomenon.


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OTHER PROBLEM SOLVING STRATEGIES

Apart from the five strategies you have learnt in the previous section, there

several other strategies that are based on some common concepts: anology,specialisation and generalisation. Extremes cases could also be used as a strategyin problem solving. We shall discovers more about these strategies in thefollowing sub-topic.

7.3.1 Analogy in Problem Solving

There are certain times when an individual may be able to obtain a solution foran analogous but simpler problem. The analogous problem may be naturallyeasier due to the fact that the problem solver may have solved a similar problem

in the past. The following example demonstrates this strategy.

Six people are in an elevator. Can you demonstrate that it must be the case thateither at least three of them are mutual acquaintances or at least three arecomplete strangers to one another?

[Adapted from: Poundstone (1990)]

Figure 7 6:Poundstone's graphical analogue

Poundstone uses a graphical analogue (Figure 7.6) to solve the above problemthat may look logically difficult at a first glance. According to his analogy, let the

six people in the elevator be represented by six different dots on a piece of paper.These dots can be located in any fashion, except that no three of them should beon the same line. Allow a solid line between any two dots to symboliseacquaintance between the people represented by those dots, and let a dashed lineindicate that the individuals are strangers. Using this system, a solid trianglerepresents three mutual acquaintances whereas a dashed triangle symbolises atrio of strangers. The main question at this instance is, using either a solid ordashed line between any given pairs of dots, is it possible to connect every dotwith every other dot in such a fashion that no solid triangles and no dashedtriangles appear in the result? Basically, this particular problem is indeed similar

7.3


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to the elevator problem and that the solution to one will automatically reveal thesolution to the other.

Thus, it can be clearly observed that the dot-connecting problem is much easierto solve than the elevator problem, and this is the main principal of this strategywhere the former is a helpful analogue of the latter. The main disadvantage inusing this strategy lies in the fact that identification of an analogue to theproblem one wishes to solve in certain cases may seem to be analogous in thecorrect way(s) but in actual fact it is not.

7.3.2 Specialisation and Generalisation

According to Mason et al. (1985), specialisation can be defined as considering aconcrete example of an abstract problem. In other words, if one is attempting tosolve a problem that has to do with the properties of parallelograms, then onemay find it important to start by considering a particular parallelogram, orseveral particular parallelograms. On the other hand, these researchers statedthat generalisation reasoning involves focusing on certain aspects common tomany examples, and ignoring other aspects. They further elaborated that theprocess of generalising is that of moving from a few instances to makinginformed guesses about a wide class of cases.

SELF-CHECK 7.3

1. State the three important features of a good analogy.

2.

Analogies make it easier to grasp the underlying idea behindanything. The pattern of understood things in our minds is like ajigsaw puzzle. Analogies help in filling these pieces to make thingsclear. Provide a suitable answer(s) to the following analogies:

(a) Just as the Earth revolves around the Sun,________________________________.

(b) A doctor's diagnostic method is similar to________________________________.

(c) Just as sword is the weapon of a warrior,

________________________________.


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7.3.3 Use of Extreme Cases

This approach is an example of specialisation and is explained well by Plya(1954). The traditional view of the method in which extreme cases contribute tothinking is as a check on a problem solution where when the answer to aproblem is stated in the form of a mathematical function, the accuracy of the

function can be verified by putting in extreme values such as zero or infinity, forthe independent variable (Plya, 1954). In most of the situations, one can deducefrom the physical situation what the answer should be in this case and seewhether the function provides the same prediction.

The comparison of the two lever situations shown in Figure 7.7 explains thispoint. Generally, students based on their intuition, very well know that it will bemuch simpler to lift the load in Case A as compared to Case B. Hence, it can beconcluded that as the distance from the load to the fulcrum decreases, the forcerequired to carry the load will also decrease.

SELF-CHECK 7.4

Two different descriptions are provided below. Identify whichdescription explains the concept of generalisation and specialisationrespectively.

(a) This particular concept is an important way to generatepropositional knowledge, by applying general knowledge, suchas the theory of gravity, to specific instances, such as when Irelease this apple, it will fall to the floor.

(b) It is a foundational element of logic and human reasoning and isthe essential basis of all valid deductive inferences. The processof verification is necessary to determine whether this particular

concept holds true for any given situation.


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Figure 7 7: Extreme cases providing data points

Source: Zietsman and Clement, 1997

It is believed that the extreme case can furnish one of the data points requiredto deduce the more general direction of change relation. This function should bemore useful in mastering the basic ideas in science, in which such relations areoften present. In addition, it can be debated that such relations also give anintuitive foundation for the understanding of mathematical relations in science.

Besides the strategies presented in this topic, there are various other strategiesthat have been developed for problem solving. However, the ones mentionedhere are among those that have received the most amount of attention. Variouswriters have emphasised on the importance of a particular strategy but none ofthem have proposed that one specific strategy is sufficient to ensure successfulproblem solving. In certain cases, it would be more beneficial if one couldemploy a combination of strategies in problem solving. The following areexamples of combinations that complement one another:

ACTIVITY 7.4

Dave goes to a museum and meets a lovely lady. Wasting no time, Daveasks the lady for her phone number. He copies the number down. Buton the walk home, the paper flies from Dave's pocket, blowing thephone number, along with his hopes, into the East River. Dave pondershis situation on the train ride home and manages to remember all sevendigits (4, 3, 7, 8, 2, 6, 5) of the woman's phone number. However, hedoes not remember the sequence of the numbers. Dave is determinedto sit in front of the phone all night if necessary to contact this lovelylady. How many telephone numbers must Dave dial?


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(a) Working backward with means-end analysis; and

(b) Problem decomposition or subgoaling with any other problem-solvingstrategy.

The more widely applicable a particular strategy, the less field dependent thestrategy is.

Problem decomposition or subgoaling involves breaking a complex probleminto a set of sequence of simpler problems and then solving the complexproblem by combining the solutions to the simpler sub-problems.

Working backwards involves solving a problem from the goal to the initialstate.

Climbing a hill is an approach in problem solving where an individual isalways moving upwards.

In means-end analysis, an individual identifies the goal state followed by athorough listing of the differences between it and the current state, andfinally, employs a promising procedure in order to reduce the differencesbetween the current state and goal.

SELF-CHECK 7.5

1. State the risks that may be faced in the problem decomposition orsubgoaling approach.

2. Define the working backwards approach in problem solving.

3. What is the main disadvantage of the hill climbing approach?

4. Identify the two main actions that can be taken to reduce the

distance between the current state and goal in the means-endanalysis method.

5. State the main disadvantage of the considering analogous problemapproach.

6. Define the terms specialisation and generalisation in problemsolving.


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In considering analogous problems, the problem may be naturally easier dueto the fact that the problem solver may have solved a similar problem in thepast.

Specialisation can be defined as considering a concrete example of an abstractproblem whereas generalisation reasoning involves focusing on certainaspects common to many examples, and ignoring other aspects.

Considering extreme cases is a heuristic approach that is frequently used toadvantage in both mathematical and non-mathematical problem solving.

In certain cases of problem solving, it is best to employ a combination ofmixing strategies rather than using a single approach in solving the problem.

Analogous problems

Conceptions

Extreme cases

Forward chaining

GeneralisationHill climbing

Means-end analysis

Problem decomposition

Specialisation

Subgoaling

TheoriesWorking backwards

1. Discuss various concepts that can be used in problem solving.

2. Discuss the advantages and disadvantages of the following problem-solvingstrategies:

(a) Decomposition

(b) Working Backwards

(c) Hill Climbing

(d) Means-end Analysis

(e) Forward Chaining

3. Discuss the use of mixed strategies using an example.


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Anderson, J. R. (1990). Cognitive psychology and its implications (3rd ed.).New York: Freeman.

Anderson, J. R. (1993). Problem solving and learning. American Psychologist, 48:35-44.

Bransford, J. D., & Stein, B. S. (1984). The ideal problem solver: A guider forimproving thinking, learning and creativity. New York: Freeman.

Hayes, J. R. (1981). The complete problem solver. Philadelphia: Franklin InstitutePress.

Kerber, M. (2004). Introduction to AI. Retrieved fromhttp://www.cs.bham.ac.uk/~mmk/Teaching/AI/l2.html

Larkin, J. H., McDermott, J., Simon, D. P., & Simon, H. A. (1980a). Expert andnovice performance in solving physics problems. Science, 208: 1335-1342.

Larkin, J. H., McDermott, J., Simon, D. P. & Simon, H. A. (1980b). Modes ofcompetence in solving physics problems. Cognitive Science, 4: 317-345.

Mason, J., Burton, L. & Stacey, K. (1985). Thinking mathematically. Menlo Park,

CA: Addison-Wesley Publishers.

Newell, A., & Simon, H. A. (1972). Human problem solving. Eaglewood Cliffs,NJ: Prentice hall.

Plya, G. (1954). Mathematics and plausible reasoning: Vol. 1. Induction andanalogy in mathematics. Princeton, NJ: Princeton University Press.

Plya, G. (1957). How to solve it: A new aspect of mathematical method. GardenCity, NY: Doubleday.

Poundstone, W. (1990). Labyrinths of reason. New York: Doubleday.

Simon, D. P., & Simon, H. A. (1978). Individual differences in solving problems.In R. S. Siegler (Ed.), Childrens thinking: What develop? Hillsdale, NJ:Erlbaum.

Zietsman, A., & Clement, J. (1997). The role of extreme case reasoning ininstruction for conceptual change. The Journal of the Learning Sciences6(1):61-89. Lawrence Erlbaum Associates, Inc.

20140410040636_topic 7 problem solving strategies

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