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PROBLEM SOLVINGPROBLEM SOLVING

Prepared by Lo Joo SimPrepared by Lo Joo Sim

Institut Bahasa Melayu MalaysiaInstitut Bahasa Melayu Malaysia

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WHY PROBLEM SOLVING?WHY PROBLEM SOLVING?

Problem solving is the mostProblem solving is the mostbasicbasic ofofmathematicalskillsmathematicalskills-- the reason for studyingthe reason for studyingmathematicsmathematics

Problem solving is an integral part of the largerProblem solving is an integral part of the largerarea ofarea ofcriticalthinkingcriticalthinking, which is universally, which is universallyaccepted goal for all educationaccepted goal for all education

Problem solving shows anProblem solving shows an interactioninteraction

betweenmathematicalideasbetweenmathematicalideas In the classroom can lessen the gap betweenIn the classroom can lessen the gap betweenrealworld problemrealworld problem and the classroom worldand the classroom worldand thus set a more positive mood in theand thus set a more positive mood in theclassroom.classroom.

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COMMON CLASSROOMCOMMON CLASSROOM

PRACTICESPRACTICES Focus onFocus on solutionsolution rather than processrather than process

Insufficient practice on solvingInsufficient practice on solvingproblems particularly non routine typeproblems particularly non routine type

Choice ofChoice ofcomputationscomputations -- tendency totendency togroup similar type of problemsgroup similar type of problems

Not giving the right challenge toNot giving the right challenge tostudentsstudents

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WHAT IS AWHAT IS A PROBLEMPROBLEM??

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TYPES OF PROBLEMSTYPES OF PROBLEMS

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ROUTINE PROBLEMROUTINE PROBLEM

Routine problems are those that merely involved anRoutine problems are those that merely involved anapplication of a mathematical procedure in muchapplication of a mathematical procedure in much

the same way as it was learned.the same way as it was learned. Example:Example:

7800 people watched television on Monday7800 people watched television on Monday

9030 people watched on Tuesday9030 people watched on Tuesday8765 people watched on Wednesday8765 people watched on Wednesday

How many people watched in the three days?How many people watched in the three days?

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NONNON--ROUTINE PROBLEMROUTINE PROBLEM

occurs when the choice of mathematicaloccurs when the choice of mathematicalprocedures to solve it is not obvious.procedures to solve it is not obvious.

NonNon--routine problems are those that call for theroutine problems are those that call for theuse of processes far more than those of routineuse of processes far more than those of routineproblemsproblems

Can be solved in many distinct ways requiringCan be solved in many distinct ways requiringsome analysis or different thinking processes.some analysis or different thinking processes.

Example:Example:Begin with the digits 1,2,3,4,5,6,7,8 and 9.Begin with the digits 1,2,3,4,5,6,7,8 and 9.Using each of them at least once, form threeUsing each of them at least once, form threefourfour--digit numbers with the sum of 9636digit numbers with the sum of 9636

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WHAT MAKES A GOODWHAT MAKES A GOOD

PROBLEM?PROBLEM?Agood problem contains some or allAgood problem contains some or allcharacteristics:characteristics:

The solution to the problem involves theThe solution to the problem involves theunderstanding of distinct mathematical conceptsunderstanding of distinct mathematical conceptsor the use of mathematical skills.or the use of mathematical skills.

The solution of the problem leads to aThe solution of the problem leads to ageneralization.generalization.

The problem is openThe problem is open--ended in that it lead toended in that it lead toextensions.extensions. The problem lends itself to a variety of solutions.The problem lends itself to a variety of solutions. The problem should be interesting andThe problem should be interesting and

challenging to the students.challenging to the students.

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WHAT IS PROBLEMWHAT IS PROBLEM

SOLVING?SOLVING?

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WHAT IS PROBLEM SOLVING?WHAT IS PROBLEM SOLVING?

Problem solving is the process of applyingProblem solving is the process of applyingpreviouslyacquiredknowledge,skills,previouslyacquiredknowledge,skills,

andunderstandingandunderstanding to new and unfamiliarto new and unfamiliarsituations.situations.

Problem solving is theProblem solving is the processprocess used to findused to findan answer to a statement or a questionan answer to a statement or a question

(Hamada,R.Y. & Smith).(Hamada,R.Y. & Smith).

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Example:Example:

There are 8 people in a room. Each personThere are 8 people in a room. Each person

shakes hands with each of the other peopleshakes hands with each of the other peopleonce and only once. How many handshakesonce and only once. How many handshakesare there?are there?

A farmer has some horses and some chickens.A farmer has some horses and some chickens.He finds that together they have 70 headsHe finds that together they have 70 headsand 200 legs. How many horses and howand 200 legs. How many horses and howmany chickens does he have?many chickens does he have?

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HOWCAN PROBLEMHOWCAN PROBLEM

SOLVING BE TAUGHTSOLVING BE TAUGHTEFFECTIVELYEFFECTIVELY

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GENERALIZATIONSGENERALIZATIONS

Problem solving strategies can beProblem solving strategies can bespecifically taughtspecifically taught

No one optimal strategyNo one optimal strategy Vital to provide children with a repertoire ofVital to provide children with a repertoire of

problem solving strategiesproblem solving strategies

Non routine problems will encourageNon routine problems will encourage

children to test many alternative approacheschildren to test many alternative approaches Childrens problem solving achievementChildrens problem solving achievement

related to developmental levelrelated to developmental level

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SOME CONSIDERATIONSSOME CONSIDERATIONS

Time involvedTime involved

Planning aidsPlanning aids

Needed resourcesNeeded resources

Role of technologyRole of technology

Managing classManaging class

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PROBLEM SOLVINGPROBLEM SOLVINGPROCESSPROCESS

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George PolyaGeorge Polya

George Polya identified four steps inGeorge Polya identified four steps inthe problem solving process :the problem solving process :

Understanding the problemUnderstanding the problem

Devising a planDevising a plan

Carrying out the planCarrying out the plan Looking backLooking back

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Burton (1984)Burton (1984)

He identified four phases in the problemHe identified four phases in the problem

solving process :solving process : EntryEntry

AttackAttack

ReviewReview

ExtensionExtension

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The Realityand MathematicsThe Realityand Mathematics

Education (RIME) programEducation (RIME) program The Reality and Mathematics EducationThe Reality and Mathematics Education

(RIME) program recommended the(RIME) program recommended the

following steps for mathematical problemfollowing steps for mathematical problemsolving :solving :

Introduce the problemIntroduce the problem

Pose the problemPose the problem

Allow students to carry out initial investigations

Allow students to carry out initial investigations

Encourage students to check their predictionsEncourage students to check their predictions

Assist students to develop a summary and conclusionAssist students to develop a summary and conclusionto what they have been doingto what they have been doing

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Problem Solving ProcessProblem Solving Process((George PolyaGeorge Polya))

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Understandingthe problemUnderstandingthe problem

Can you read the question?Can you read the question?

Can you state the problem in your ownCan you state the problem in your own

words?words? What are you trying to find or do?What are you trying to find or do?

What information do you obtain from theWhat information do you obtain from the

problem?problem? What are the unknowns?What are the unknowns?

What information, if any, is missing or notWhat information, if any, is missing or notneeded?needed?

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Devisinga planDevisinga plan

Find the connection between the dataFind the connection between the dataand the unknown.and the unknown.

Consider auxiliary problem if anConsider auxiliary problem if animmediate connection can be foundimmediate connection can be found

What strategies do you know?What strategies do you know?

Try a strategy that seems as if it willTry a strategy that seems as if it willwork.work.

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Carryingoutthe planCarryingoutthe plan

Use the strategy you selected andUse the strategy you selected andwork the problem.work the problem.

Check each step of the plan as youCheck each step of the plan as youproceedproceed

Ensure that the steps are correctEnsure that the steps are correct

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Looking BackLooking Back

Reread the questionReread the question

Did you answer the question asked?Did you answer the question asked?

Is your answer correct?Is your answer correct?

Does your answer seems reasonable?Does your answer seems reasonable?

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Teaching problemsolvingTeaching problemsolving

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Before :Before :

Read the problem to the class or have aRead the problem to the class or have astudent read the problem. Discuss words orstudent read the problem. Discuss words orphrases students may not understand.phrases students may not understand.

Use a wholeUse a whole--class discussion aboutclass discussion aboutunderstanding the problems. Ask questionsunderstanding the problems. Ask questionsto help students understand the problem.to help students understand the problem.

Ask students which strategies might beAsk students which strategies might be

helpful for finding a solution. Do nothelpful for finding a solution. Do notevaluate students suggestions. You canevaluate students suggestions. You candirect students attention to the list ofdirect students attention to the list ofstrategies on problem solving when askingstrategies on problem solving when askingfor suggestions.for suggestions.

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During :During :

Observe and question students about theirObserve and question students about theirwork.work.

Give hints for solving the problems asGive hints for solving the problems asneeded.needed.

Require students who obtain a solution toRequire students who obtain a solution tocheck their work and answer the problem.check their work and answer the problem.

Give a problem extension to students whoGive a problem extension to students whocomplete the original problem much soonercomplete the original problem much soonerthan others.than others.

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After :After :

Show and discuss students solutionsShow and discuss students solutionsto the original problem. Have studentsto the original problem. Have students

name the strategies used.name the strategies used. Relate the problem to previousRelate the problem to previous

problems and solve an extension ofproblems and solve an extension ofthe original problem.the original problem.

Discuss special features of the originalDiscuss special features of the originalproblem, if any.problem, if any.

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PROBLEM SOLVING ATPROBLEM SOLVING AT

THE PRIMARY LEVELTHE PRIMARY LEVEL Children begin solving problems long beforeChildren begin solving problems long before

they enter the primary grades. Somethey enter the primary grades. Someproblems faced by preproblems faced by pre--primary childrenprimary children

require sophisticated problem solving skills.require sophisticated problem solving skills.By the time children enter the primary level,By the time children enter the primary level,children have developed their own personalchildren have developed their own personalapproaches and strategies for solving theseapproaches and strategies for solving thesecomplex problems. It is the primarycomplex problems. It is the primaryteachers responsibility to assist children inteachers responsibility to assist children inextending, refining and adding to theirextending, refining and adding to theirproblemproblem--solving approaches and strategiessolving approaches and strategies

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PROBLEM SOLVING ATPROBLEM SOLVING AT

THE PRIMARY LEVELTHE PRIMARY LEVEL Problem solving at the primary levelProblem solving at the primary level

should be the focus of almost allshould be the focus of almost all

mathematics instruction the child willmathematics instruction the child willencounter. Computational skills areencounter. Computational skills aretaught so they can be used as tools intaught so they can be used as tools in

the problemthe problem--solving process.solving process.

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PROBLEM SOLVING ATPROBLEM SOLVING AT

THE PRIMARY LEVELTHE PRIMARY LEVEL It is not enough that teachers teachIt is not enough that teachers teach

problem solving by exposing childrenproblem solving by exposing children

to problems and hoping that somehowto problems and hoping that somehowafter working many problems theafter working many problems thechildren would automatically developchildren would automatically develop

into problem solvers. Children usuallyinto problem solvers. Children usuallysolve problems that used addition,solve problems that used addition,division, fractions, etc.division, fractions, etc.

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PROBLEM SOLVING ATPROBLEM SOLVING AT

THE PRIMARY LEVELTHE PRIMARY LEVEL Teachers should utilize the results ofTeachers should utilize the results of

research on how children learn toresearch on how children learn to

solve problems. Students aresolve problems. Students aresystematically involved in examiningsystematically involved in examiningvarious problemvarious problem--solving strategiessolving strategies

which can be used in a wide variety ofwhich can be used in a wide variety ofproblemproblem--solving situations.solving situations.

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THANK YOUTHANK YOU

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MATHEMATICAL PROBLEMMATHEMATICAL PROBLEM

DEFINEDDEFINED

Aproblem is a task for which the personAproblem is a task for which the person

confronting it wants or needs to find aconfronting it wants or needs to find asolution,solution,

has no readily available procedure forhas no readily available procedure forfinding a solution,finding a solution,

and must make an attempts to find aand must make an attempts to find asolution.solution.

Charles & Lester (1982)Charles & Lester (1982)

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3196 6457 75293196 6457 7529

+ 46+ 46 + 456+ 456 +4321+4321

Ali has 345 red seeds in his bag. JohnAli has 345 red seeds in his bag. Johngives him another 245 more. Howgives him another 245 more. Howmany seeds does he have altogether?many seeds does he have altogether?

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FocusonsolutionsFocusonsolutions

Sharifah has RM15. She needs RM40Sharifah has RM15. She needs RM40to buy a new teddy bear. How muchto buy a new teddy bear. How much

more money does she need?more money does she need?

There were 12 marbles missing fromThere were 12 marbles missing fromEzrins box. Then her brother tookEzrins box. Then her brother took

away 2. How many are missing now?away 2. How many are missing now?