Mathematics Experience

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Mathematics Experience-Based Approach

Introduction© Copyright 1985, 1990, 1999; by Pentathlon Institute, Inc. Mary Gilfeather & John del RegatoMEBA

Routine & Nonroutine

Problem SolvingApplying understanding of mathematical ideas and operations to problem-solving situations

is another component of MEBA. Routine problem solving stresses the useof sets of known or prescribed procedures (algorithms) to solve problems. Initially in

MEBA the problems presented to students are simple one-step situations requiring asimple procedure to be performed. Gradually, students are asked to solve more

complex problems that involve multiple steps and include irrelevant data. Commencingwith the concrete level, students are asked to develop their own story problem

situations and demonstrate the solution process with manipulatives and/or picturesand later with symbols. Such problems are later presented to the class for

solution.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.Because of this, routine problem solving receives a great deal of attention by classroom

teachers. With the advent of computers, which can quickly perform the mostcomplex arrangements of algorithms for multi-step routine problems, the amount of

instructional time and the extent to which these problems are tested is being reassessed.Today¶s typical workplace does not require a high level of proficiency in

solving multi-step routine problems without the use of a calculator or computer. However,an increased need for employees with abilities in nonroutine problem solving

has occurred in today¶s workplace.

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

the solution to a problem. Building a model and drawing a picture of the problemare two basic problem-solving heuristics. SIMs incorporates these heuristics as a

natural part of instruction. 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 areused 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, learnersare 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 changingelements. The heuristics used in this form of problem solving are known as strategies.



A summary chart to contrast routine and nonroutine problem solving can befound on page 22.COMMUNICATION

PROBLEM SOLVINGCONCEPTUALUNDERSTANDING

MEBA


Introduction© Copyright 1985, 1990, 1999; by Pentathlon Institute, Inc. Mary Gilfeather & John del RegatoMEBA

To develop the thought processes that are inherent in active nonroutine problem

solving MEBA incorporates the Mathematics Pentathlon ® which features a program of

strategic and interactive games. The name of the program, Mathematics Pentathlon ® ,was coined to liken it to a worldwide series of athletic events, the Decathlon component

of the Olympics. In the world of athletics the Decathlon is appreciated for valuing and rewarding individuals who have developed a diverse range of athletic

abilities. In contrast, the world of mathematics has not, for many decades, valued or rewarded individuals with a diverse range of mathematics abilities. The MathematicsPentathlon ® games promote diversity in mathematical thinking by integrating spatial/

geometric, arithmetic/computational, and logical/scientific reasoning at each division

level. (The games are organized into four division levels, K-1, 2-3, 4-5, and 6-7with five games at each level.) Since each of the five games requires students to broaden their thought processes, it attracts students from a wide range of ability

levels, from those considered ³gifted and talented´ to ³average´ to ³at-risk.´The format of games was chosen for two reasons. First, games that are of a strategic

nature require students to consider multiple options and formulate strategies basedon expected countermoves from the other player. The Mathematics Pentathlon ® further

promotes this type of thought by organizing students into groups of four and teams of two. Teams alternate taking turns and team partners alternate making decisions

about particular plays by discussing aloud the various options and possibilities. Inthis manner, all group members grow in their understanding of multiple options and

strategies. As students play these games over the course of time, they learn to makea plan based on better available options as well as to reaccess and adjust this plan

based on what the other team acted upon to change their prior ideas. Through thisinteractive process of sharing ideas/possibilities students learn to think many steps

ahead, blending both inductive and deductive thinking. Second, games were chosenas a format since they are a powerful motivational tool that attracts students from a

diverse range of ability and interest level to spend more time on task developing basic skills as well as problem-solving skills. While race-type games based on chance

are commonly used in classrooms, they do not typically capture students¶ curiousityfor long periods of time. Students may play such games once or twice, but then lose

interest since they are not seriously challenged. The active nonroutine nature of

the Mathematics Pentathlon ® have seriously challenged students to mature in their ability to think strategically and resolve problems that are continually undergoingchange.

The Mathematics Pentathlon ® program is comprised of five gameboards and manualswith related concrete/pictorial materials at each of the four division levels. To prepare

students to play these active, nonroutine, problem-solving games, prerequisite

activities are suggested and described in two related books, Adventures in Problem

Solving Books I & II. Since active, nonroutine problem solving cannot be assessed



with conventional methods, assessment ideas for Mathematics Pentathlon ® are provided

in Investigation Exercises Books I & II. These publications provide numerous

non-traditional paper-pencil ideas for assessing students¶ understanding of mathematical

relationships/skills that directly relate to the games and Adventures in Problem

Solving activities. In addition, students are challenged to critically examine

various game-playing options and choose better moves.COMMUNICATION

PROBLEM SOLVING

CONCEPTUALUNDERSTANDING

MEBA


Introduction

© Copyright 1985, 1990, 1999; by Pentathlon Institute, Inc. Mary Gilfeather & John del RegatoMEBA

22

PROBLEM

SOLVINGROUTINE

NONROUTINEstresses the use of

sets of known or

prescribed procedures

(algorithms)

to solve problemsSTRENGTH: easily

assessed by paperpencil

tests



WEAKNESS: least

relevant to human

problem solvingSTRENGTH: most relevant to

human problem solving

WEAKNESS: least able to be

assessed by paper-pencil tests

STATIC

fixed, known

goal and known

elements

ACTIVE fixed goal(s) with

changing elements changing or alternative

goal(s) with

fixed elements

changing or alternative

goal(s) with

changing elements

stresses the use of heuristics which do not

guarantee a solution to a problem but provide

a more highly probable method for solvingproblemsCOMMUNICATION

COOPERATION

CONCEPTUALUNDERSTANDING

MEBA MEBA



POLYA MODEL

Menurut Model Polya, penyelesaian masalah boleh dilaksanakan melalui empat peringkat iaitu,memahami dan mentafsir masalah, merancang strategi penyelesaian, melaksanakan strategi penyelesaian dan menyemak semula penyelesaian. Strategi pengajaran dihuraikan mengikut

model Polya adalah seperti berikut :

1. Memahami dan Mentafsir Sesuatu Masalah

Pada peringkat ini, pelajar akan dibimbing untuk mengenal pasti kata-kata kunci dan

menerangkan masalah. Pelajar juga hendaklah mengaitkan dengan masalah lain yang serupadengan melukis gambarajah dan bertanyakan beberapa soalan.

2. Merancang Strategi Penyelesaian

Selepas pelajar memahami soalan tersebut, guru membimbing pelajar untuk merancang strategiyang sesuai dengan permasalahan yang diberikan. Terdapat beberapa jenis strategi penyelesaian

masalah mengikut Polya. Antaranya ialah membuat simulasi, melukis gambarajah, membuatcarta, mengenal pasti pola, cuba jaya, menggunakan analogi dan sebagainya.

3. Melaksanakan Strategi Penyelesaian

Sebaik sahaja strategi penyelesaian masalah dikenal pasti, pelajar akan melaksanakan strategi

tersebut dengan menggunakan kemahiran mengira, kemahiran geometri, kemahiran algebra

ataupun kemahiran menaakul

4. Menyemak Semula Penyelesaian

Akhirnya, pelajar boleh menyemak semula penyelesaian tersebut untuk menentukan sama ada jawapannya munasabah atau tidak. Di samping itu, pelajar boleh mencari cara penyelesaian yang

lain atau membuat andaian serta membuat jangkaan lanjut kepada masalah tersebut.

STRATEGI PENYELESAIAN MASALAH

Berikut adalah beberapa strategi-strategi penyelesaian masalah:

1. Permudahkan masalah

Kadang kala masalah yang diberikan terlalu rumit dan kompleks. Permudahkan masalah tersebut bermakna mewujudkan masalah yang serupa dengan menggunakan angka-angka yang berbeza

dan mudah. Kemudian buatkan perbandingan dan akhirnya kita yang memperolehi jawapan.

2. Melukis Gambarajah



Dengan melukis gambarajah kita dapat melihat pergerakan masalah tersebut secara tersusun.

3. Memodelkan / Menjalankan simulasi / Melakonkan

Memodelkan atau menjalankan simulasi adalah strategi yang paling berkesan untuk melihat pola

perubahan dan keseluruhan masalah dapat dihayati dengan jelas. Dengan menggunakan modelkonkrit, ianya mempermudahkan penyelesaian masalah tersebut.

4. Mengenal pasti pola

Dalam strategi ini pelajar perlu menganalisa pola dan membuat generalisasi berdasarkan

pemerhatian mereka dan mengujinya dengan menggunakan data yang baru. Pola boleh wujuddalam bentuk gambar atau nombor.

5. Menyenaraikan / Menjadualkan secara sistematik

Jadual yang dibina seharusnya teratur dan tersusun agar maklumat dapat dilihat dengan cepat danmudah. Graf juga boleh boleh digunakan untuk menunjukkan perhubungan di antara

pembolehubah-pembolehubah.

6. Cuba Jaya

Strategi ini adalah cara termudah tetapi ianya memerlukan tekaan yang bijak dan penyemakanyang tersusun boleh membawa kepada jawapan atau penyelesaian.

7. Kerja ke belakang

Bekerja ke belakang antara strategi yang sesuai untuk menyelesaikan masalah µsequence¶, pola, persamaan dan lain-lain.

8. Menaakul secara mantik

Pelajar menganalisa semua syarat-syarat dan memecahkan kepada bahagian-bahagian tertentu.Sebahagian daripada masalah itu boleh diselesaikan dengan penyelesaian bahagian-bahagian

kecil yang akan digabungkan semula untuk membentuk penyelesaian masalah tersebut.

9. Menggunakan kaedah algebra

Kaedah algebra akan membentuk beberapa persamaan dan ia dapat membantu menyelesaikanmasalah matematik tersebut.



George Polya

1887 - 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 memorizeinformation. 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 lawschool 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 adegree. 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 betaught.

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 spent67 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 hemet 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 hislife, 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 onhow 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 texthe identifies four basic principles .

Polyas 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 dont understand it fully, or even in part. Polya

taught teachers to ask students questions such as:

y Do you understand all the words used in stating the problem?

y

What are you asked to find or show?y Can you restate the problem in your own words?

y Can you think of a picture or a diagram that might help you understand the problem?y Is there enough information to enable you to find a solution?

Polyas Second Principle: Devise a plan

Polya mentions (1957) that it are many reasonable ways to solve problems. The skill at choosingan appropriate strategy is best learned by solving many problems. You will find choosing a

strategy increasingly easy. A partial list of strategies is included:

y Guess and check y Make and orderly listy Eliminate possibilities

y Use symmetryy Consider special cases

y Use direct reasoningy Solve an equation

y Look for a patterny Draw a picturey Solve a simpler problem

y Use a modely Work backward

y Use a formulay Be ingenious

Polyas 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. Dont be misled, this is how

mathematics is done, even by professionals. Polyas 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 didnt. 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. Polyahas become known as the father of problem solving.



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. Yetproblem 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.

Definition of problem

Describes the process of working through details of a problem to reach a solution.

An individual seeking to solve a problem will have to identify the most important

elements that influence the answer and then work through a series of operations

to determine a logical solution. Problem solving may include mathematical or

systematic operations and can be a gauge of an individual's critical thinking skills.

A problem is decided by purposes. If someone wants money and when he or shehas little money, he or she has a problem. But if someone does not want money,

little money is not a problem.

For example, manufacturing managers are usually evaluated with line-operation

rate, which is shown as a percentage of operated hours to potential total

operation hours. Therefore manufacturing managers sometimes operate lines



without orders from their sales division. This operation may produce more than

demand and make excessive inventories. The excessive inventories may be a

problem for general managers. But for the manufacturing managers, the

excessive inventories may not be a problem.

If a purpose is different between managers, they see the identical situation in

different ways. One may see a problem but the others may not see the problem.

Therefore, in order to identify a problem, problem solvers such as consultants

must clarify the differences of purposes. But oftentimes, problem solvers

frequently forget to clarify the differences of purposes and incur confusion among

their problem solving projects. Therefore problem solvers should start their

problem solving projects from the definition of purposes and problems

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 aboutproblems.

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 recognizingsituations 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.

C ause

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 C ause

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



This report lists fourteen thinking patters. Problem solvers should choose

appropriate patterns, responding to situations. This report categorized these

fourteen patterns into three more general groups such as thinking patterns for

judgements, thinking patterns for thinking processes and thinking patterns for

efficient thinking. The following is the outlines of those thinking patterns.

Thinking patterns for judgements

In order to create a value through thinking we need to judge whether what we

think is right or wrong. This report lists four judging patterns such as strategic

thinking, emotional thinking, realistic thinking, and empirical thinking.

Strategic t hinking

Focus, or bias, is the criterion for strategic thinking. If you judge whether asituation is right or wrong based on whether the situation is focused or not, your

judgement is strategic. A strategy is not necessarily strategic. Historically, many

strategists such as Sonfucis in ancient China, Naplon, M. Porter proposed strategic

thinking when they develop strategies.

Emotional t hinking

In organizations, an emotional aspect is essential. Tactical leaders judge whether

a situation is right or wrong based on the participantsf emotional commitment.They think that if participants can be positive to a situation, the situation is right.

Realistic t hinking

y Start from what we can do

y Fix the essential problem first

These two criteria are very useful. "Starting" is very important, even if we do very

little. We do not have to start from the essential part. Even if we start from an

easier part, starting is a better judgement than a judgement of not-starting in

terms of the first part of realistic thinking. Further, after we start, we should

search key factors to make the Problem Solving more efficient. Usually, 80 % of

the problems are caused by only 20 % of the causes. If we can find the essential

20 % of the causes, we can fix 80 % of problems very efficiently. Then if we try to



find the essential problem, what we are doing is right in terms of the second part

of realistic thinking.

Empirical t hinking

When we use empirical thinking, we judge whether the situation is right or wrong

based on our past experiences. Sometimes, this thinking pattern persists on the

past criteria too much, even if a situation has changed. But when it comes to our

daily lives, situations do not change frequently. Further, if we have the experience

of the identical situation before, we can utilize the experience as a reliable

knowledge data base.

Thinking patterns for t hinking processes

If we can think systematically, we do not have to be frustrated when we think. Incontrast, 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 Toyotafs five

times WHYs method .

Rational t hinking

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 t hinking



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 systemsf functions. The following arre the system and how the system

works.

System

y Purpose

y Input

y Output

y Function

y Inside cause (Solvable cause)

y Outside cause (Unsolvable cause)

y 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.

C ause & effect t hinking

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.

C ontingent t hinking

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.



T oyota 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 thesefive 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 t hinking

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.

Hypot hesis t hinking

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 nexthypothesis 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.

C onception t hinking

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 andflexible conception. Following are portions of tips.

y To be visual.

y To write down what we think.

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

y 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 t hinking

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.

C onvergence & divergence t hinking

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.

T ime order t hinking

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 Skills

One of the most exciting aspects of life is the array of choices that we have on a dailybasis. Some of our decisions are simple, like deciding what to eat for dinner or what

shirt to wear. However, some choices are challenging and take careful thought andconsideration.

When we are confronted with these types of decisions, it can be very difficult to decideon the best option, and we may be plagued by indecision. We may be forced to choosebetween two equally good options, or perhaps, we may have to pick between twochoices that both have drawbacks. We may waver back and forth between differentalternatives and may feel paralyzed to make the decision.

This is a very normal reaction to tough choices in our lives, and we all, at times,experience a sense of being unable to decide on some option. However, researchers

have developed a technique that many people have found useful when they are tryingto make a difficult decision or solve a problem that seems unsolvable. This procedureinvolves a series of steps that you can go through on your own when you are confrontedwith a decision or problem that needs to be solved. This approach may not workperfectly for all difficulties, but it may help with many of the problems you are confrontedwith in your life.

Step 1: Problem Orientation

This step involves recognizing that a problem exists and that solving the difficulty is aworthwhile endeavor. It is important that you approach the decision-making process

with a positive attitude and view the situation as an opportunity or challenge. You shouldtry to approach the situation with confidence and with a willingness to devote some timeand effort to finding an appropriate solution to your problem. Remember, you are acompetent person, and the problem you are facing can likely be solved with a little hardwork.

Step 2: Problem Definition

Before you start to tackle the current problem, it is important to clearly understand thedifficulty and why you are unhappy with the current situation. This may seem obvious,but it is important that you really think about and gather information about the problem,

and make sure that the problem you are trying to solve is the "real" problem. That is,sometimes people find a different problem than the one that is really distressing them,and focus on this one, since it is easier than dealing with the real problem. This stepreally involves your thinking about the difficulty you are having, understanding theproblem, and contemplating why the situation is distressing. Some people think of problems as a discrepancy between what they want and what the current situation islike. It is useful during this stage to think about how the current situation is different fromhow you would like it to be, and what your goals are for the state of affairs. If you are



currently facing many difficult decisions, it may be helpful to prioritize those problemsand deal wStep 3: Generation of Alternative Solutions

During this stage, you should ask yourself, "What have I done in this situation in thepast, and how well has that worked?" If you find that what you have done in the past

has not been as effective as you would like, it would be useful to generate some other solutions that may work better. Even if your behavior in the past has worked like youwanted it to, you should think of other solutions as well, because you may come up withan even better idea. When you start to think of possible solutions, don't limit yourself;think of as many possible options as you can, even if they seem unrealistic. You canalways discard implausible ideas later, and coming up with these may help generateeven better solutions. You may want to write a list of possible options, or ask otherswhat some solutions they might have for your problem.

Step 4: Decision Making

Now you are ready to narrow down some of the options that you have generated in theprevious step. It is important that you examine each of the options, and think about howrealistic each is, how likely you would be to implement that solution, and the potentialdrawbacks of each. For example, if your solution costs a great deal of money or requires many hours of effort each day, this may be too difficult to implement. Youshould also consider the likelihood that each option has in terms of your being able toachieve the goals that you want regarding the solution. As you start to narrow downyour choices, remember, no problem solution is perfect and all will have drawbacks, butyou can always revise the solution if it does not work the way you want it to work.

Step 5: Solution Implementation and Verification

Once you have examined all your options and decided on one that seems to accomplishyour goals and minimizes the costs, it is time to test it out. Make sure that when youimplement this solution, you do so whole-heartedly and give it your best effort. Duringthis stage, you should continue to examine the chosen solution and the degree to whichit is "solving" your problem. If you find that the solution is too hard to implement or it is

just not working, revise it or try something else. Trying to solve these problems is never an easy task, and it may take several solutions before something works. But, don't giveup hope, because with persistence and your best effort, many difficult decisions andproblems can be made better!ith them one at a time.



Problem Solving Strategies

y There are numerous approaches to solving math problems. 'Model Drawing' is the first one thatwe have introduced because we feel that it has

the greatest impact in building children's

confidence in dealing with math problems. Moststudents enjoy visual effects. Seeing abstract

relationships, represented by concrete and

colourful images, helps in understanding,leading to the solution of the problem. Our

section on Model Drawing is by no means

exhaustive but it will open a new doorway for the student who has been struggling with math

problems.

y Besides the Model-Drawing Approach there are

several other strategies, which are necessary for the student to master, to achieve proficiency in

math problem solving. In our next section, we

introduce the important and most useful ones.

These are:

1) Draw a Picture 2) Look for a Pattern 3) Guess and Check

4) Make a Systematic List 5)

Logical Reasoning 6) Work Backwards

y The student may also come across problemswhich may need the use of more than one

strategy before a solution can be found.



Non-Routine Mean Unit

8

Problem: The mean of 29 test scores is 77.8. What is the sumof 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 thedata 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 boththe 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 setgiven 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 scoremust 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 representthe 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 220miles, 300 miles, 210 miles, 275 miles and 240 miles. How many

miles must they travel on the sixth day in order to finish their vacationon 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 tofinish their vacation on time.

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

In this lesson, we learned how to solve non-routine problems involvingthe mean of a set of data. We found the sum given the mean and the

number of items in the data set. We also found the number of items in thedata set given the mean and the sum of the data. Lastly, we found the

missing number in a set of data given the other numbers in the data setand the mean.

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Mathematics Experience