What is Problem?

A problem is defined as a situation in which a person wants something and does not know

immediately what series of actions he can perform to get it. In our current context,

mathematical ideas are involved in the actions to resolve the situation. Thus the four elements

that must exist before we are in a problem solving situation are:

1) a situation must exist involving an initial state and a goal state

2) the situation must involve mathematics

3) a person must desire a solution

4) there must be some blockage between the initial and desired states.

Calculating the means of a set of numbers is an exercise or task, not a problem, for 7th

graders. They know immediately how to proceed, having learned the skill in the fifth grade

according to our scope and sequence. Deciding if there is a direct relationship between two

variables and whether the data is reliable enough (using measures of central tendancy like

means) is likely to be a problem for most 7th graders since those skills are not taught before

the 7th grade.

There are three affective considerations to problem solving: 1) You must desire a solution; 2)

You must feel it is within your ability to solve; 3) you must believe that you can begin to work

on the problem. This third consideration comes from having experience in solving problems

and from having an understanding (explicit or intuitive) in the procedures and processes that

are usually involved in solving problems. The purpose of the math curiculum is to give a

person experience in solving a variety of problems (where math is involved) and several

procedures and a general process for solving problems.

A problem comprises a situation and an objective.

What are Mathematical Problems?

Definitions

Bruner (1961) cited the work of Weldon who claimed that one needs to consider

'troubles', 'puzzles', and 'problems' when defining a problem. A 'trouble' is a circumstance or

situation which makes one upset and uncomfortable. A 'puzzle' has a nice tight form, clear

structure, and a neat solution. A problem is a puzzle placed on top of a trouble. Funkhouser

(1990) referred to this definition as “lighthearted”, and Shulman (1985) called it his “favorite

epigram.” I think Bruner’s citation is interesting.

According to Kantowski (1977), "An individual is faced with a problem when he

encounters a question he cannot answer or a situation he is unable to resolve using the

knowledge immediately available to him. He must then think of a way to use the information

at his disposal to arrive at the goal, the solution of the problem" (p. 163). The author

differentiates between a problem and an exercise. In the case of a problem, an algorithm

which will lead to a solution is unavailable. In an exercise one determines the algorithm and

then does the manipulation. Mervis (1978) defines a problem as "a question or condition that

is difficult to deal with and has not been solved" (p. 27).

Lester (1980) says that "A problem is a situation in which an individual or group is called

upon to perform a task for which there is no readily accessible algorithm which determines

completely the method of solution" (Quoted in Lester, 1980, p. 287 from Lester, 1978).

Buchanan (1987) defines mathematical problems as "non-routine problems that required

more than ready-to-hand procedures or algorithms in the solution process" (p. 402). McLeod

(1988) defines problems as "those tasks where the solution or goal is not immediately

attainable and there is no obvious algorithm for the student to use" (p. 135).

According to Blum and Niss (1991), a problem is a situation which has certain open

questions that "challenge somebody intellectually who is not in immediate possession of

direct methods/procedures/algorithms, etc. sufficient to answer the question" (p. 37). Thus a

problem is relative to the individuals involved; that is, what is a problem for one person may

be an exercise for another. For example, the task 2 + 3 may be a problem for a pre-schooler

but not for a middle-schooler.

A common element in the definitions of Kantowski, Lester, Buchanan, McCleod, and

Blum and Niss is that there is no known algorithm to solve a problem. The problem solver

has to design a method of solution.

In Becoming a better problem solver 1 (Ohio Department of Education, 1980), it is

stated that a mathematical problem has four elements:

1. A situation which involves an initial state and a goal state.

2. The situation must involve mathematics.

3. A person must desire a solution.

4. There must be some blockage between the given and desired states (p. 5).

This definition has an affective component (the desire to find a solution) which is absent in the

previous definitions.

Kilpatrick (1985) defines a problem as "a situation in which a goal is to be attained and

a direct route to the goal is blocked" (p. 2). In a similar way, Mayer (1985) claims that a

problem occurs when one is faced with a "given state" and one wants to attain a "goal state."

The preceding three definitions refer to initial and goal states in a problem situation. The

other definitions do not refer explicitly to goals.

Polya (1985), the father of problem solving, identified two categories of problems:

1. Problems to find, the principal parts of which are the unknown, the data, and the

condition.

2. Problems to prove which comprise a hypothesis and a conclusion.

Blum and Niss (1991) also identified two kinds of mathematical problems. There are

applied mathematical problems in which the situation and question belong to the real-world

(outside of mathematics); and there are pure mathematical problems which are embedded

entirely in mathematics. These appear to be similar to Polya’s categories.

Teachers’ Conceptions of a Mathematical Problem

Studies have been done into teachers’ conceptions of a problem. For example,

Thompson (1988) found that 5 of the 16 teachers whom she studied conceived a problem as

"the description of a situation involving stated quantities, followed by a question of some

relationship among the quantities whose answer called for the application of one or more

arithmetic operations" (p. 235). The teachers’ responses implied that a problem has an

answer, usually a number, and there is a unique procedure to obtain that answer. Thompson

(1988) found that teachers had varying conceptions of problems. For example, some

teachers gave 'story' or 'word' problems as examples of problem tasks.

Students’ Conceptions of a Mathematical Problem

There were also studies into students’ conceptions of a problem. For example, Frank

(1988) conducted a study with 27 mathematically talented middle school students to

investigate their beliefs about mathematics and how these beliefs influence their problem-

solving practices. She used a questionnaire, interviews and observations. She found that

students believed that mathematical problems must be solvable quickly in a few steps and

that mathematical problems were routine tasks which could be done by the application of

known algorithms. They perceived non-routine problems as "extra credit" tasks. Students

believed that if a problem could not be solved in less than 5 to 10 minutes, either something

was wrong with them or the problem. The goal of doing mathematics was to obtain "right

answers." Students focused entirely on answers which to them were either completely right

or completely wrong.

Spangler (1992) used open-ended questions to assess students' beliefs about

mathematics and found that students do have beliefs about certain aspects of mathematics.

Some of her findings concurred with those of other researchers, e.g., Frank (1988).

Spangler (1992) found that one of the common beliefs among students was that a

mathematical problem has only one correct answer. Students were not prepared to accept

that a problem could have different answers, all being correct. They indicated that they

preferred one method to multiple methods for solving a problem because they did not have to

remember much. Students admitted that they could obtain the correct answer to a problem

without understanding what they were doing. Students rarely checked to see if their answers

made sense in the context of the given problem. They verified their answers with the teacher

or by checking the text and they are not inclined to look for multiple solutions or to generalize

their results.

Mtetwa and Garofalo (1989) identified the following unhealthy beliefs which students

have about mathematics and mathematical problem solving:

1. In mathematical word problems the relative size of numbers is more important

than the relationships between the quantities which they represent. For example,

numbers which are to be subtracted are usually close in size, and numbers which are

to be divided are not close in size and are evenly divisible. They claimed that

teachers and textbooks help to perpetuate these beliefs.

2. Computation problems must be solved by using a step-by-step algorithm. This

is a consequence of the instructional practices of teachers.

3. Mathematics problems have only one correct answer. The consequence of

such a belief is that students fail to recognize/consider/accept other valid and

reasonable answers. They contend that such a belief could develop from textbook

answers and classroom experiences.

There is some degree of consistency between teachers’ and students’ conceptions of

a problem. For example, they believe that a problem has one correct answer which is usually

obtained by a step-by-step procedure.

Summary

The essence of these definitions is that a problem is a task or experience which is being

encountered by the individual for the very first time and, therefore, there is no known

procedure for handling it. The individual has to design his/her own method of

solution drawing upon the various skills, knowledge, strategies, and so forth, which have been

previously learned. What the individual does in the process of working towards a solution is

referred to as problem solving; so the emphasis is not on the answer but on the processes

involved. From this perspective many routine word problems which appear in textbooks are

mistakenly designated as problems. They are not; they are merely exercises. A problem is

relative to the individual; what may constitute a problem for one person may not be a problem

for another because he/she might have encountered it before. Teachers and students have

similar conceptions of a problem and these conceptions are sometimes inconsistent with the

literature.

What is mathematical problem solving?

Mathematical problem solving is a complex cognitive activity involving a number of processes

and strategies. Problem solving has two stages: problem representation and problem

execution. Successful problem solving is not possible without first representing the problem

appropriately. Appropriate problem representation indicates that the problem solver has

understood the problem and serves to guide the student toward the solution plan. Students

who have difficulty representing math problems will have difficulty solving them.

One of the most powerful problem representation strategies is visualization. Developmentally,

for most children, visualization matures somewhere between the ages of 8 and 11. Therefore,

students in upper elementary school should be able to use visualization effectively to

represent mathematical problems. Students with LD, however, who have been characterized

as having a variety of strategy deficits and differences, usually have difficulties using

visualization as an effective learning strategy for remembering information and representing

problems. Many students do not develop the ability to use visual representation automatically

during math problem solving. These students need explicit instruction in how to use

visualization to represent problems.

Teaching mathematical problem solving is a challenge for many teachers, many of whom rely

almost exclusively on mathematics textbooks to guide instruction. Most mathematics

textbooks simply instruct students to draw a picture or make a diagram using the information

in the problem. Students with LD at the upper elementary level may be incapable of

developing an appropriate representation of the problem for a variety of reasons. First, they

are generally operating at a fairly concrete level. Second, they are poor at visual

representation. As a result, symbolic representation may not be possible without explicit

instruction that incorporates manipulatives and other materials that will help students move

from a concrete to a more symbolic, schematic level. In other words, teachers must provide

systematic, progressive, and scaffolded instruction that considers the students’ cognitive

strengths and weaknesses.

Students who have difficulty solving math word problems usually draw a picture of the

problem without considering the relationships among the problem components and, as a

result, still do not understand the problem and therefore cannot make a plan to solve it. So, it

is not simply a matter of “drawing a picture or making a diagram;” rather, it is the type of

picture or diagram that is important. Effective visual representations, whether with

manipulatives, with paper and pencil, or in one’s imagination, show the relationships among

the problem parts. These are called schematic representations (van Garderen & Montague,

2003). Poor problem solvers tend to make immature representations that are more pictorial

than schematic in nature. The illustration below shows the difference between a pictorial and

a schematic representation of the mathematical problem presented at the beginning of the

brief.

Other cognitive processes and strategies needed for successful mathematical problem

solving include paraphrasing the problem, which is a comprehension strategy, hypothesizing

or setting a goal and making a plan to solve the problem, estimating or predicting the

outcome, computing or doing the arithmetic, and checking to make sure the plan was

appropriate and the answer is correct (Montague, 2003; Montague, Warger, & Morgan, 2000).

Mathematical problem solving also requires self-regulation strategies. Students with LD are

notoriously poor self-regulators. During this developmental period, it is imperative that they be

explicitly taught how to self-instruct (tell themselves what to do), self-question (ask

themselves questions), and self-monitor (check themselves as they solve the problem).

Descriptive Statement

Problem posing and problem solving involve examining situations that arise in mathematics

and other disciplines and in common experiences, describing these situations mathematically,

formulating appropriate mathematical questions, and using a variety of strategies to find

solutions. By developing their problem-solving skills, students will come to realize the

potential usefulness of mathematics in their lives.

Meaning and Importance

Problem solving is a term that often means different things to different people. Sometimes it

even means different things at different times for the same people! It may mean solving

simple word problems that appear in standard textbooks, applying mathematics to real-world

situations, solving nonroutine problems or puzzles, or creating and testing mathematical

conjectures that may lead to the study of new concepts. In every case, however, problem

solving involves an individual confronting a situation which she has no guaranteed way to

resolve. Some tasks are problems for everyone (like finding the volume of a puddle), some

are problems for virtually no one (like counting how many eggs are in a dozen), and some are

problems for some people but not for others (like finding out how many balloons 4 children

have if each has 3 balloons, or finding the area of a circle).

Problem solving involves far more than solving the word problems included in the students'

textbooks; it is an approach to learning and doing mathematics that emphasizes questioning

and figuring things out. The Curriculum and Evaluation Standards of the National Council of

Teachers of Mathematics considers problem solving as the central focus of the mathematics

curriculum.

"As such, it is a primary goal of all mathematics instruction and an integral part of all

mathematics activity. Problem solving is not a distinct topic but a process that should

permeate the entire program and provide the context in which concepts and skills can be

learned." (p. 23)

Thus, problem solving involves all students a large part of the time; it is not an incidental topic

stuck on at the end of the lesson or chapter, nor is it just for those who are interested in or

have already mastered the day's lesson.

Students should have opportunities to pose as well as to solve problems; not all problems

considered should be taken from the text or created by the teacher. However, the situations

explored must be interesting,engaging, and intellectually stimulating. Worthwhile

mathematical tasks are not only interesting to the students, they also develop the students'

mathematical understandings and skills, stimulate them to make connections and develop a

coherent framework for mathematical ideas, promote communication about mathematics,

represent mathematics as an ongoing human activity, draw on their diverse background

experiences and inclinations, and promote the development of all students' dispositions to do

mathematics (Professional Standards of the National Council of Teachers of Mathematics).

As a result of such activities, students come to understand mathematics and use it effectively

in a variety of situations.

Problem solving

Problem solving is a basic skill needed by today’s learners. The problem-solving skill is an

important intellectual activity for human beings; it is also a very important springhead of

humankind’s knowledge. Guided by recent research in problem solving, changing

professional standards, new workplace demands, and recent changes in learning theory,

educators and trainers are revising curricula to include integrated learning environments

which encourage learners to use higher order thinking skills, and in particular, problem solving

skills.

Problem solving has become the means to rejoin content and application in a learning

environment for basic skills as well as their application in various contexts. Problem solving

also includes attitudinal as well as cognitive components. To solve problems, learners have to

want to do so, and they have to believe they can. Motivation and attitudinal aspects such as

effort, confidence, anxiety, persistence and knowledge about self are important to the problem

solving process.

Problem solving involves both analytical and creative skills: analytical in comprehending the

problem and the relationships within the original situation, and in checking the results of

results of each step, and creative in devising the

solution. Imagination plays a large part in both of these skills: problem solving requires the

ability to imagine a chain of intermediate steps and their consequences.

For example to solve the problem of crossing a river by chopping down a tree

and laying it across the river appears to be quite simple. The ability to imagine the individual

steps in a solution and their results can only be gained through

experience, acquisition of subject specific knowledge and understanding, and practice in

using the necessary tools. True creativity in problem solving lies in lateral thinking that is in

the ability to imagine the results of processes in different contexts to those previously

experienced. This requires the ability to abstract, at least sub-consciously, generalizations,

and while such transfer may be possible between different contexts within one academic

discipline it is not as easy to achieve between contexts in different disciplines.

Models of Problem solving

The most explicit analyses of knowledge and procedural skills required to solve the problems

are provided by the following models. The models are based on similar analyses of

performance, but somewhat different characterizations of knowledge at each stage of

problem solving.

2.1 Early Models of Problem Solving

The first problem solving models broke down into

two distinct approaches:

2.1.1 The traditional scientific method

2.1.2 An introspective creative method

Scientists often report using both methods to enable discovery.

2.1.1 The Scientific Process by Dewey (1910)

• Define the problem

• Suggest possible solutions and identify alternative

• Reason about the solutions and implement

• Test and prove

2.1.2 The Creative Process by Wallas (1926)

• Problem formulation and information gathering

• Incubation - allowing the unconscious to work on it

• Illumination - working to gain insight

• Verification - testing for accuracy

2.2 The Engineering Model

Etter (1995) presented a model used by students to

solve engineering problems.

• Define the problem - state it clearly

• Gather information - describe input and output

• Generate and evaluate potential solutions

• Refine and implement solutions

• Verify and test solution method and result

2.3 Classroom Model

Polya (1945 and 1962) was the first to describe a

problem solving model based on classroom experience:

Four stages of Problem Solving are as follows:

• Understand and explore the problem

• Find a strategy

• Use the strategy to solve the problem

• Look back and reflect on the solution.

Although we have listed the Four Stages of Problem

Solving in order, for difficult problems it may not be possible

to simply move through them consecutively to produce an

answer. It is frequently the case that children move backwards

and forwards between and across the steps. In fact the figure1

below is much more like what happens in practice.

Understand

Look back strategy

Solve

Understand and explore the problem

• State the question

• Identify the goal

• Give known, unknowns and conditions

• Introduce drawings or notations

There is no chance of being able to solve a problem unless you can first understand it. This

process requires not only knowing what you have to find but also the key pieces of

information that somehow need to be put together to obtain the answer. Children will often not

be able to absorb all the important information of a problem in one go. It will almost always be

necessary to read a problem several times, both at the start and during working on it. During

the solution process, children may find that they have to look back at the original question

from time to time to make sure that they are on the right track.

Find a strategy

• Outline a potential solution

• Look at similar problems

• Restate the problem differently

• Break it into sub problems

Finding a strategy tends to suggest that it is a fairly simple matter to think of an appropriate

strategy. However, there are certain problems where children may find it necessary to play

around with the information before they are able to think of a strategy that might produce a

solution. This exploratory phase will also help them to understand the problem better and may

make them aware of some piece of information that they had neglected after the first reading.

Use the strategy to solve the problem

• Refine and transform into a solution

• Relate tasks to givens and unknowns

• Check validity of each step

• Define steps in relations to the whole problem

Having explored the problem and decided on a plan of attack, the third problem-solving step,

solve the problem, can be taken. Hopefully now the problem will be solved and an answer

obtained. During this phase it is important for the children to keep a track of what they are

doing. This is useful to show others what they have done and it is also helpful in finding errors

should the right answer not be found.

Look back and reflect on the solution

• Confirm results and arguments

• Assess effectiveness of solution

• Assess accuracy of results

• Assess usefulness of solution for solving other

Problem-Solving Techniques

It is not enough to describe a problem-solving process and to describe how individuals differ

in their approach to or use of it. It is also necessary to identify specific techniques of attending

to individual differences. Fortunately, a variety of problem-solving techniques have been

identified to accommodate individual preferences. It is important that techniques from both

categories be selected and used in the problem-solving process. Duemler and Mayer (1988)

found that when students used exclusively either reflection or inspiration during problem

solving, they tended to be less successful than if they used a moderate amount of both

processes. This section offers some examples of both types of techniques; the next section

will demonstrate how to integrate them into the problem-solving process to accommodate

individual differences.

The following techniques focus more on logic and critical thinking, especially within the

context of applying the scientific approach:

• Analysis

• Backwards planning

• Categorizing/classifying

• Challenging assumptions

• Evaluating/judging

• Inductive/deductive reasoning

• Thinking aloud

• Network analysis

• Plus-Minus-Interesting

• Task analysis

The following problem-solving techniques focusmore on creative, lateral, or divergent

thinking:

• Brainstorming

• Imaging/visualization

• Incubation

• Outcome psychodrama

• Outrageous provocation

• Overload

• Random word technique

• Relaxation

• Synthesizing

• Taking another's perspective

• Values clarification

Integrating Techniques into the Problem-Solving Process

The problem-solving techniques discussed above are most powerful when combined to

activate both the logical/rational and intuitive/creative parts of the brain. The following

narrative will provide an example of how these techniques can be used at specific points in

the problem solving process to address important individual differences. The techniques will

be presented within the context of a group problem-solving situation but are equally

applicable to an

individual situation.

The Input Phase

The goal of the Input phase is to gain a clearer understanding of the problem or situation. The

first step is to identify the problem and state them clearly and concisely. Identifying the

problem means describing as precisely as possible the gap between one's perception of

present circumstances and what one would like to happen. Problem identification is vital to

communicate to one's self and others the focus of the problem-solving/decision-making

process. The second step of the Input phase is to state the criteria that will be used to

evaluate possible alternatives to the problem as well as the effectiveness of selected

solutions. The third step is to gather information or facts relevant to solving the problem or

making a decision. This step is critical for understanding the initial conditions and for further

clarification of the

perceived gap.

The Processing Phase

In the Processing phase the task is to develop, evaluate, and select alternatives and solutions

that can solve the problem. The first step in this phase is to develop alternatives or possible

solutions. This generation should be free, open, and unconcerned about feasibility. Enough

time should be spent on this activity to ensure that non-standard and creative alternatives are

generated. The next step is to evaluate the generated alternatives via the stated criteria. The

third step of the processing phase is to develop a solution that will successfully solve the

problem. For relatively simple problems, one alternative may be obviously superior. However,

in complex situations several alternatives may likely be combined to form a more effective

solution.

The Output Phase

During the Output phase a plan is developed and the solution actually implemented. The plan

must be sufficiently detailed to allow for successful implementation, and methods of

evaluation must be considered and developed. When developing a plan, the major phases of

implementation are first considered, and then steps necessary for each phase are generated.

It is often helpful to construct a timeline and make a diagram of the most important steps in

the implementation using a technique such as network analysis. Backwards planning and task

analysis are also useful techniques at this point. The plan is then implemented as carefully

and as completely as possible, following the steps as they have been developed and making

minor modifications as appropriate.

The Review Phase

The next step, evaluating implementation of the solution, should be an ongoing process.

Some determination as to completeness of implementation needs to be considered prior to

evaluating effectiveness. The second step of this phase is evaluating the effectiveness of the

solution. It is particularly important to evaluate outcomes in light of the problem statement

generated at the beginning of the process. Affective, cognitive, and behavioral outcomes

should be considered, especially if they have been identified as important criteria. The final

step in the process is modifying the solution in ways suggested by the evaluation process.

Evaluation of the solution implementation and outcomes generally presents additional

problems to be considered and addressed. Issues identified in terms of both efficiency and

effectiveness of implementation should be addressed.

Summary and Conclusions

There is a need to develop and use a problem-solving/decision-making process that is both

scientific and considerate of individual differences and viewpoints. While the scientific process

has provided a method used successfully in a wide variety of situations, researchers have

described individual differences that can influence perspectives and goals related to problem

solving. These differences can be used to identify appropriate problem solving techniques

used in each step of the problem-solving process. The process described in this paper allows

individuals to use a standard method in a variety of situations and to adapt it to meet personal

preferences. The same process can be used in group situations to satisfy the unique

perspectives of individual members. Decisions made in this manner are more likely to be

effective since individuals can consciously attend to both personal strengths and weaknesses.

Problem Solving Terminology

Systems Thinking

Problem Solving is very important but problem solvers often misunderstand it. This report

proposes the definition of problems, terminology for Problem Solving and useful Problem

Solving patterns.

We should define what is the problem as the first step of Problem Solving. Yet problem

solvers often forget this first step.

Further, we should recognize common terminology such as Purpose, Situation, Problem,

Cause, Solvable Cause, Issue, and Solution. Even Consultants, who should be professional

problem solvers, are often confused with the terminology of Problem Solving. For example,

some consultants may think of issues as problems, or some of them think of problems as

causes. But issues must be the proposal to solve problems and problems should be negative

expressions while issues should be a positive expression. Some consultants do not mind this

type of minute terminology, but clear terminology is helpful to increase the efficiency of

Problem Solving. Third, there are several useful thinking patterns such as strategic thinking,

emotional thinking, realistic thinking, empirical thinking and so on. The thinking pattern means

how we think. So far, I recognized fourteen thinking patterns. If we choose an appropriate

pattern at each step in Problem Solving, we can improve the efficiency of Problem Solving.

This report will explain the above three points such as the definition of problems, the

terminology of Problem Solving, and useful thinking patterns.

Terminology of Problem Solving

We should know the basic terminology for Problem Solving. This report proposes seven terms

such as Purpose, Situation, Problem, Cause, Solvable Cause, Issue, and Solution.

Purpose

Purpose is what we want to do or what we want to be. Purpose is an easy term to

understand. But problem solvers frequently forget to confirm Purpose, at the first step of

Problem Solving. Without clear purposes, we can not think about problems.

Situation

Situation is just what a circumstance is. Situation is neither good nor bad. We should

recognize situations objectively as much as we can. Usually almost all situations are not

problems. But some problem solvers think of all situations as problems. Before we recognize

a problem, we should capture situations clearly without recognizing them as problems or non-

problems. Without recognizing situations objectively, Problem Solving is likely to be narrow

sighted, because problem solvers recognize problems with their prejudice.

Problem

Problem is some portions of a situation, which cannot realize purposes. Since problem

solvers often neglect the differences of purposes, they cannot capture the true problems. If

the purpose is different, the identical situation may be a problem or may not be a problem.

Cause

Cause is what brings about a problem. Some problem solvers do not distinguish causes from

problems. But since problems are some portions of a situation, problems are more general

than causes are. In other words causes are more specific facts, which bring about problems.

Without distinguishing causes from problems, Problem Solving can not be specific. Finding

specific facts which causes problems is the essential step in Problem Solving.

Solvable Cause

Solvable cause is some portions of causes. When we solve a problem, we should focus on

solvable causes. Finding solvable causes is another essential step in Problem Solving. But

problem solvers frequently do not extract solvable causes among causes. If we try to solve

unsolvable causes, we waste time. Extracting solvable causes is a useful step to make

Problem Solving efficient.

Issue

Issue is the opposite expression of a problem. If a problem is that we do not have money, the

issue is that we get money. Some problem splvers do not know what Issue is. They may think

of "we do not have money" as an issue. At the worst case, they may mix the problems, which

should be negative expressions, and the issues, which should be positive expressions.

Solution

Solution is a specific action to solve a problem, which is equal to a specific action to realize an

issue. Some problem solvers do not break down issues into more specific actions. Issues are

not solutions. Problem solvers must break down issues into specific action.

Thinking patterns for thinking processes

If we can think systematically, we do not have to be frustrated when we think. In contrast, if

we have no systematic method, Problem Solving frustrate us. This reports lists five systematic

thinking processes such as rational thinking, systems thinking, cause & effect thinking,

contingent thinking, and the Toyota �fs five times WHYs method .

Rational thinking

Rational thinking is one of the most common Problem Solving methods. This report will briefly

show this Problem Solving method.

1. Set the ideal situation

2. Identify a current situation

3. Compare the ideal situation and the current situation, and identify the problem

situation

4. Break down the problem to its causes

5. Conceive the solution alternatives to the causes

6. Evaluate and choose the reasonable solution alternatives

7. Implement the solutions

We can use rational thinking as a Problem Solving method for almost all problems.

Systems thinking

Systems thinking is a more scientific Problem Solving approach than the rational thinking

approach. We set the system, which causes problems and analyze them based on systems �f functions. The following arre the system and how the system works.

System

Purpose

Input

Output

Function

Inside cause (Solvable cause)

Outside cause (Unsolvable cause)

Result

In order to realize Purpose, we prepare Input and through Function we can get Output. But

Output does not necessarily realize Purpose. Result of the Function may be different from

Purpose. This difference is created by Outside Cause and Inside Cause. We can not solve

Outside Cause but we can solve Inside Cause. For example, when we want to play golf,

Purpose is to play golf. If we can not play golf, this situation is Output. If we can not play golf

because of a bad weather, the bad weather is Outside Cause, because we can not change

the weather. In contrast, if we cannot play golf because we left golf bags in our home, this

cause is solvable. Then, that we left bags in our home is an Inside Cause.

Systems thinking is a very clear and useful method to solve problems.

Cause & effect thinking

Traditionally, we like to clarify cause and effect relations. We usually think of finding causes

as solving problems. Finding a cause and effect relation is a conventional basic Problem

Solving method.

Contingent thinking

Game Theory is a typical contingent thinking method. If we think about as many situations as

possible, which may happen, and prepare solutions for each situation, this process is a

contingent thinking approach.

Toyota �fs five times WHYs

At Toyota, employees are taught to think WHY consecutively five times. This is an adaptation

of cause and effect thinking. If employees think WHY and find a cause, they try to ask

themselves WHY again. They continue five times. Through these five WHYS, they can break

down causes into a very specific level. This five times WHYs approach is very useful to solve

problems.

Thinking patterns for efficient thinking

In order to think efficiently, there are several useful thinking patterns. This report lists five

patterns for efficient thinking such as hypothesis thinking, conception thinking, structure

thinking, convergence & divergence thinking, and time order thinking.

Hypothesis thinking

If we can collect all information quickly and easily, you can solve problems very efficiently. But

actually, we can not collect every information. If we try to collect all information, we need so

long time. Hypothesis thinking does not require collecting all information. We develop a

hypothesis based on available information. After we developed a hypothesis, we collect

minimum information to prove the hypothesis. If the first hypothesis is right, you do not have

to collect any more information. If the first hypothesis is wrong, we will develop the next

hypothesis based on available information. Hypothesis thinking is a very efficient problem-

solving method, because we do not have to waste time to collect unnecessary information.

Conception thinking

Problem Solving is not necessarily logical or rational. Creativity and flexibility are other

important aspects for Problem Solving. We can not recognize these aspects clearly. This

report shows only what kinds of tips are useful for creative and flexible conception. Following

are portions of tips.

To be visual.

To write down what we think.

Use cards to draw, write and arrange ideas in many ways.

Change positions, forms, and viewpoints, physically and mentally.

We can imagine without words and logic, but in order to communicate to others, we must

explain by words and logic. Therefore after we create ideas, we must explain them literally.

Creative conception must be translated into reasonable explanations. Without explanations,

conception does not make sense.

Structure thinking

If we make a structure like a tree to grasp a complex situation, we can understand very

clearly.

Upper level should be more abstract and lower level should be more concrete. Dividing

abstract situations from concrete situations is helpful to clarify the complex situations. Very

frequently, problem solvers cannot arrange a situation clearly. A clear recognition of a

complex situation increases efficiency of Problem Solving.

Convergence & divergence thinking

When we should be creative we do not have to consider convergence of ideas. In contrast,

when we should summarize ideas we must focus on convergence. If we do convergence and

divergence simultaneously, Problem Solving becomes inefficient.

Time order thinking

Thinking based on a time order is very convenient, when we are confused with Problem

Solving. We can think based on a time order from the past to the future and make a complex

situation clear.

Problem Solving Skill Development

Routine to Non-Routine

Quantitative skills and methods (mathematics) represents a growing body of rules and

patterns that can be carefully, in other words intelligently,  used one at a time and one after

another, alone and in sequence, to arrive at repeatable, reproducible, observable and hence

verifiable results.  

To develop problem solving skills, and avoid re-invention of the wheel, students will be

exposed to problems and situations in which the mathematical skills and concepts they have

met can be applied in routine or predictable manner. The first aim of mathematics instruction

is to give students those skills and concepts - previously found or hard-won by previous

generations - for solving routine problems and puzzles in a straightforward or combinatorial or

opportunistic manner.  For that, logic mastery would be useful for the development of

precision reading and writing skills.  The well-practiced ability to record problem solving steps

and effort in a clear legible format readable by peers, teachers and themselves would make

aid and speed routine problem solving. 

Routine problem solving (challenging as it may be to some students) in which mathematical

skills and concepts are pieces of a jigsaw puzzle - one whose solution is standard - even on

display - represents a first step in developing the critical thinking and problem solving skills of

students.  It provides a standard for all further problem solving. Seeing what kinds of

problems have been met and/or solve before, and how, provides a model for further problem

solving.  Greater knowledge of the kinds of problems met before and how they have been

solved provides a systematic base for further problem solving. 

Mathematics in the first instance, is an art form, a discipline, with simple and then more

complicated rules, patterns and methods to master. For many routine problems or situations

in daily life and in our cultures that students need to learn to address and solve with ways that

lead to repeatable and reproducible results - reliable results.  Once students have sufficient

drill and practice, sufficient exposure, the use of some skills and concepts should become

familiar, automatic, and their use no longer an adventure.  

Problem solving from a state of ignorance is over-rated.  With a combinatorial or creative

mind, standing on prior knowledge of what has worked or not, is better. While creativity (the

combination of previously mastered skills and concepts, and the invention of new ones) is

possible with any level of knowledge, the ability to be creative and in that produce methods to

solve problems in a verifiable manner - a manner that peers can follow or reproduce -

increases with the level of knowledge and level of skill and competence. Students need to

learn when creativity is required and when previous methods give satisfactory results. 

Problem solving situation with incomplete information of what has been done - a partial state

of ignorance - may be provided to show how a greater knowledge of previous solution

reduces problem solving challenges. 

Problem solving in an society where common problems repeat themselves and thus become

routine should be based routine solutions methods,  methods whose efficacy, suitability and

limitations has been checked and understood by the user.  With practice, solving common

problem should become routine. 

Empirical problem solving aims to find or apply methods with repeatable and reproducible,

and reliable results. That may turn open problems into routine problems.  Practice in solving

problems which have become routine may prepare students for open problems. Practice in

solving routine problems and puzzles in a straightforward or combinatorial or opportunistic

manner when solution methods are not given provides a model for tackling non-routine

problems, a model that stands on and then looks beyond previous methods. 

Remark:  Routines and methods in society for "solving" problems may lead to repeatable,

reproducible and harmful results.  The ability to follow instructions carefully and precisely is a

plus for getting results but not a guarantee that the results will be ethnical or that practices will

be sustainable.  So students should not be trained to follow methods or instructions without

reflection on the benefits and limitations of the methods.  Routine solution methods may be

challenged and should be for the everyone's sake.  But those routine methods cannot be

challenge, cannot be considered and examine carefully if their study  is avoided. 

Routine problem solving

From the curricular point of view, routine problem solving involves using at least one of the

four arithmetic operations and/or ratio to solve problems that are practical in nature. Routine

problem solving concerns to a large degree the kind of problem solving that serves a socially

useful function that has immediate and future payoff. Children typically do routine problem

solving as early as age 5 or 6. They combine and separate things such as toys in the course

of their normal activities. Adults are regularly called upon to do simple and complex routine

problem solving.

Example 1:

A sales promotion in a store advertises a jacket regularly priced at RM125.98 but now selling

for 20% off the regular price. The store also waives the tax. You have RM100 in your pocket

(or RM100 left in your charge account). Do you have enough money to buy the jacket?

Answer:

The original price is RM 125.98

20% off the regular price and I have RM 100.00

RM 100−(RM 125.98×80

100 ) = RM 100 – RM 100.80

= -RM 0.80

No enough because I still need RM 0.80 just can buy that jacket.

Example 2

You may need to use more than one operation to

solve some problems. These are called multiple-step

problems.

The blacksmith could forge 6 swords in two days.

How many swords could he forge in 9 days?

First, divide 2 into 6 swords to see how many swords

he could make in one day.

6 ÷ 2 = 3

Then multiply your answer by 9 to see how many swords he could make in 9 days.

3 x 9 = 27

Example 3

At the tavern, Ralph bought a mug of mead for 1 shilling, bread for 2 shillings, meat for 2 shillings, and a souvenir for his children 5 shillings. How much did Ralph spend on food?

First, find the information you need to solve the problem. Some information is extra. The extra information is:

Ralph bought a souvenir toy for his children for 5 shillings.

I'll solve the problem using only the information I need.

1 + 2 + 2 = 5

Ralph spent 5 shillings on food.

Problem Solving Skill Development Routine to Non-Routine

Quantitative skills and methods (mathematics) represents a growing body of rules and

patterns that can be carefully, in other words intelligently,  used one at a time and one after

another, alone and in sequence, to arrive at repeatable, reproducible, observable and hence

verifiable results.  

To develop problem solving skills, and avoid re-invention of the wheel, students will be

exposed to problems and situations in which the mathematical skills and concepts they have

met can be applied in routine or predictable manner. The first aim of mathematics instruction

is to give students those skills and concepts - previously found or hard-won by previous

generations - for solving routine problems and puzzles in a straightforward or combinatorial or

opportunistic manner.  For that, logic mastery would be useful for the development of

precision reading and writing skills.  The well-practiced ability to record problem solving steps

and effort in a clear legible format readable by peers, teachers and themselves would make

aid and speed routine problem solving. 

Routine problem solving (challenging as it may be to some students) in which mathematical

skills and concepts are pieces of a jigsaw puzzle - one whose solution is standard - even on

display - represents a first step in developing the critical thinking and problem solving skills of

students.  It provides a standard for all further problem solving. Seeing what kinds of

problems have been met and/or solve before, and how, provides a model for further problem

solving.  Greater knowledge of the kinds of problems met before and how they have been

solved provides a systematic base for further problem solving. 

Mathematics in the first instance, is an art form, a discipline, with simple and then more

complicated rules, patterns and methods to master. For many routine problems or situations

in daily life and in our cultures that students need to learn to address and solve with ways that

lead to repeatable and reproducible results - reliable results.  Once students have sufficient

drill and practice, sufficient exposure, the use of some skills and concepts should become

familiar, automatic, and their use no longer an adventure.  

Problem solving from a state of ignorance is over-rated.  With a combinatorial or creative

mind, standing on prior knowledge of what has worked or not, is better. While creativity (the

combination of previously mastered skills and concepts, and the invention of new ones) is

possible with any level of knowledge, the ability to be creative and in that produce methods to

solve problems in a verifiable manner - a manner that peers can follow or reproduce -

increases with the level of knowledge and level of skill and competence. Students need to

learn when creativity is required and when previous methods give satisfactory results. 

Problem solving situation with incomplete information of what has been done - a partial state

of ignorance - may be provided to show how a greater knowledge of previous solution

reduces problem solving challenges. 

Problem solving in an society where common problems repeat themselves and thus become

routine should be based routine solutions methods,  methods whose efficacy, suitability and

limitations has been checked and understood by the user.  With practice, solving common

problem should become routine. 

Empirical problem solving aims to find or apply methods with repeatable and reproducible,

and reliable results. That may turn open problems into routine problems.  Practice in solving

problems which have become routine may prepare students for open problems. Practice in

solving routine problems and puzzles in a straightforward or combinatorial or opportunistic

manner when solution methods are not given provides a model for tackling non-routine

problems, a model that stands on and then looks beyond previous methods. 

Remark:  Routines and methods in society for "solving" problems may lead to repeatable,

reproducible and harmful results.  The ability to follow instructions carefully and precisely is a

plus for getting results but not a guarantee that the results will be ethnical or that practices will

be sustainable.  So students should not be trained to follow methods or instructions without

reflection on the benefits and limitations of the methods.  Routine solution methods may be

challenged and should be for the everyone's sake.  But those routine methods cannot be

challenge, cannot be considered and examine carefully if their study  is avoided. 

 

Non-routine problem solving

Non-routine problem solving serves a different purpose than routine problem solving. While routine problem solving concerns solving problems that are useful for daily living (in the present or in the future), non-routine problem solving concerns that only indirectly. Non-routine problem solving is mostly concerned with developing students’ mathematical reasoning power and fostering the understanding that mathematics is a creative endeavour. From the point of view of students, non-routine problem solving can be challenging and interesting. From the point of view of planning classroom instruction, teachers can use non-routine problem solving to introduce ideas (EXPLORATORY stage of teaching); to deepen and extend understandings of algorithms, skills, and concepts (MAINTENANCE stage of teaching); and to motivate and challenge students (EXPLORATORY and MAINTENANCE stages of teaching). There are other uses as well. Having students do non-routine problem solving can encourage the move from specific to general thinking; in other words, encourage the ability to think in more abstract ways. From the point of view of students growing to adulthood, that ability is becoming more important in today’s technological, complex, and demanding world.

Non-routine problem solving can be seen as evoking an ‘I tried this and I tried that, and eureka, I finally figured it out.’ reaction. That involves a search for heuristics (strategies seeking to discover). There is no convenient model or solution path that is readily available to apply to solving a problem. That is in sharp contrast to routine problem solving where there are readily identifiable models (the meanings of the arithmetic operations and the associated templates) to apply to problem situations.

The following is an example of a problem that concerns non-routine problem solving.

Consider what happens when 35 is multiplied by 41. The result is 1435. Notice that all four digits of the two multipliers reappear in the product of 1435 (but they are rearranged). One could call numbers such as 35 and 41 as pairs of stubborn numbers because their digits reappear in the product when the two numbers are multiplied together. Find as many pairs of 2-digit stubborn numbers as you can. There are 6 pairs in all (not including 35 & 41).

Solving problems like the one above normally requires a search for a strategy that seeks to discover a solution (a heuristic). There are many strategies that can be used for solving unfamiliar or unusual problems. The strategies suggested below are teachable to the extent that teachers can encourage and help students to identify, to understand, and to use them. However, non-routine problem solving cannot be approached in an automatized way as can routine problem solving. To say that another way, we cannot find nice, tidy methods of solution for all problems. Inevitably, we will be confronted with a situation that evokes the response; “I haven't got much of a clue how to do this; let me see what I can try.”

The list below does not contain strategies like: ‘read the question carefully’, ‘draw a diagram’, or ‘make a table’. Those kinds of strategies are not the essence of what it takes to be successful at non-routine problem solving. They are only preliminary steps that help in getting organized. The hard part still remains - to actually solve the problem - and that takes more powerful strategies than drawing a diagram, reading the question carefully, or making a table. The following list of strategies is appropriate for Early and Middle Years students in that the strategies involve ways of thinking that are likely to be comfortable for these students.

Look for a pattern Guess and check Make and solve a simpler problem. Work backwards. Act it out/make a model. Break up the problem into smaller ones and try to solve these first.

It is important that students share how they solved problems so that their classmates are exposed to a variety of strategies as well as the idea that there may be more than one way to reach a solution. It is unwise to force students to use one particular strategy for two important reasons. First, often more than one strategy can be applied to solving a problem. Second, the goal is for students to search for and apply useful strategies, not to train students to make use of a particular strategy.

Finally, non-routine problem solving should not be reserved for special students such as those who finish the regular work early. All students should participate in and be encouraged to succeed at non-routine problem solving. All students can benefit from the kinds of thinking that is involved in non-routine problem solving.

Non routine problem

Problem:   The mean of 29 test scores is 77.8. What is the sum of these test scores?

Solution:   To find the mean of n numbers, we divide the sum of the n numbers by n.

 

  If we let n = 29, we can work backwards to find the sum of these test scores.

 

  Multiplying the mean by the 29 we get:   77.8 x 29 = 2,256.2

Answer:   The sum of these test scores is 2,256.2

In the problem above, we found the sum given the mean and the number of items in the data set (n). It is also possible to find the number of items in the data set (n) given the mean and the sum of the data. This is illustrated in Examples 1 and 2 below.

Example 1:   The mean of a set of numbers is 54. The sum of the numbers is 1,350. How many numbers are in the set?

Solution:  

To find n, we need to divide 1,350 by 54.

Answer:   n = 25, so there are 25 numbers in the set.

Example 2:   The mean of a set of numbers is 0.39. The sum of the numbers is 1.56. How many numbers are in the set?

Solution:  

To find n, we need to divide 1.56 by 0.39.

However, we cannot divide by a decimal divisor. We will multiply both the divisor and the dividend by 100 in order to get a whole number divisor.

Answer:   n = 4, so there are 4 numbers in the set.

In the next few examples, we will be asked to find the missing number in the data set given the other numbers in the data set and the mean. The words mean and average will be used interchangeably.

Example 3:   Gini's test scores are 95, 82, 76, and 88. What score must she get on the fifth test in order to achieve an average of 84 on all five tests?

Solution:   We are given four of the five test scores. The sum of these 4 test scores is 341. If we let x represent the fifth test score, then the expression 341 + x can represent the sum of all five test scores.

 

  If we multiply the divisor by the quotient, we get:

  5 · 84 = 341 + x

  420 = 341 + x

  420 - 341 = x

  x = 79

Answer:   Gini needs a score of 79 on her fifth test in order to achieve an average of 84 on all 5 tests.

Example 4:   The Lachance family must drive an average of 250 miles per day to complete their vacation on time. On the first five days, they travel 220 miles, 300 miles, 210 miles, 275 miles and 240 miles. How many miles must they travel on the sixth day in order to finish their vacation on time?

Solution:   The sum of the first 5 days is 1,245 miles. Let x represent the number of miles traveled on the sixth day. We get:

  If we multiply the divisor by the quotient, we get:

  250 · 6 = 1,245 + x

  1,500 = 1,245 + x

  1,500 - 1,245 = x

  x = 255

Answer:   The Lachance family must drive 255 miles on the sixth day in order to finish their vacation on time.

Comparing routine and non-routine problem solving

To make clearer the distinction between routine and non-routine problem solving, consider

the following two problems. Both are suitable for grade 3.

Problem 1

My mom gave me 35 cents. My father gave me 45 cents. My grandmother gave me

85 cents. How many cents do I have now?

Problem 2

Place the numbers 1 to 9, one in each circle

so that the sum of the four numbers along

any of the three sides of the triangle is 20.

There are 9 circles and 9 numbers to place in

the circles. Each circle must have a different

number in it.

Notice that addition is required for both problems. In problem 1, you need to figure out that

you need to add. Understanding addition as modeling a ‘put together’ action helps you

realize that.

In problem 2, you are told to add by the word ‘sum’. Understanding addition as modeling a

‘put together’ action does not help you with solving problem 2. Being good at arithmetic might

help you a bit, but the matter really concerns a search for strategies to apply to the problem.

Guess and check is a useful strategy to begin with.

George Polya1887 - 1985

George Polya was a Hungarian who immigrated to the United States in 1940. His major contribution is for his work in problem solving.

Growing up he was very frustrated with the practice of having to regularly memorize information. He was an excellent problem solver. Early on his uncle tried to convince him to go into the mathematics field but he wanted to study law like his late father had. After a time at law school he became bored with all the legal technicalities he had to memorize. He tired of that and switched to Biology and the again switched to Latin and Literature, finally graduating with a degree. Yet, he tired of that quickly and went back to school and took math and physics. He found he loved math.

His first job was to tutor Gregor the young son of a baron. Gregor struggled due to his lack of problem solving skills. Polya (Reimer, 1995) spent hours and developed a method of problem solving that would work for Gregor as well as others in the same situation. Polya (Long, 1996) maintained that the skill of problem was not an inborn quality but, something that could be taught.

He was invited to teach in Zurich, Switzerland. There he worked with a Dr. Weber. One day he met the doctor�s daughter Stella he began to court her and eventually married her. They spent 67 years together. While in Switzerland he loved to take afternoon walks in the local garden. One day he met a young couple also walking and chose another path. He continued to do this yet he met the same couple six more times as he strolled in the garden. He mentioned to his wife �how could it be possible to meet them so many times when he randomly chose different paths through the garden�.

He later did experiments that he called the random walk problem. Several years later he published a paper proving that if the walk continued long enough that one was sure to return to the starting point.

In 1940 he and his wife moved to the United States because of their concern for Nazism in Germany (Long, 1996). He taught briefly at Brown University and then, for the remainder of his life, at Stanford University. He quickly became well known for his research and teachings on problem solving. He taught many classes to elementary and secondary classroom teachers on how to motivate and teach skills to their students in the area of problem solving.

In 1945 he published the book How to Solve It which quickly became his most prized publication. It sold over one million copies and has been translated into 17 languages. In this text he identifies four basic principles .

Polya�s First Principle: Understand the Problem

This seems so obvious that it is often not even mentioned, yet students are often stymied in their efforts to solve problems simply because they don�t understand it fully, or even in part. Polya taught teachers to ask students questions such as:

Do you understand all the words used in stating the problem? What are you asked to find or show? Can you restate the problem in your own words? Can you think of a picture or a diagram that might help you understand the problem? Is there enough information to enable you to find a solution?

Polya�s Second Principle: Devise a plan

Polya mentions (1957) that it are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy. A partial list of strategies is included:

Guess and check Look for a pattern

Make and orderly list Eliminate possibilities Use symmetry Consider special cases Use direct reasoning Solve an equation

Draw a picture Solve a simpler problem Use a model Work backward Use a formula Be ingenious

Polya�s third Principle: Carry out the plan

This step is usually easier than devising the plan. In general (1957), all you need is care and patience, given that you have the necessary skills. Persistent with the plan that you have chosen. If it continues not to work discard it and choose another. Don�t be misled, this is how mathematics is done, even by professionals. Polya�s Fourth Principle: Look back

Polya mentions (1957) that much can be gained by taking the time to reflect and look back at what you have done, what worked and what didn�t. Doing this will enable you to predict what strategy to use to solve future problems.

George Polya went on to publish a two-volume set, Mathematics and Plausible Reasoning (1954) and Mathematical Discovery (1962). These texts form the basis for the current thinking in mathematics education and are as timely and important today as when they were written. Polya has become known as the father of problem solving.

Multivariate Polya distribution

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The multivariate Pólya distribution, also called the Dirichlet compound multinomial distribution, is a compound probability distribution, where a probability vector p is drawn from a Dirichlet distribution with parameter vector α, and a set of discrete samples x is drawn from the multinomial distribution with probability vector p. The compounding corresponds to a Polya urn scheme. In document classification, for example, the distribution is used to represent probabilities over word counts for different document types.

The probability of a vector of counts x given the parameter vector α is obtained by integrating out the parameters p of the multinomial distribution:

which results in the following explicit formula:

where Γ is the gamma function, and nk is the number of times the outcome in x was k.

The two-dimensional version of the multivariate Pólya distribution is known as the Beta-binomial model.

The multivariate Pólya distribution is used in automated document classification and clustering, genetics, economy, combat modeling, and quantitative marketing.