INTRODUCTION

Problem in Mathematics has been interpreted in various way by mathematics

educators.Problem in Mathematics is a task for which the person confronting it.It

needs the person wants or needs to find a solution for a question that has being

ask.In Mathematics problem,there has no readily available procedure for finding a

solution and must make an attempt to find a solution.Problem solving in Mathematics

is an important way to find the solution for the question given.It is the process of

applying previously acquired knowledge, skills, and understanding to new and

unfamiliar situations.In the other word,problem solving is the process used to find an

answer to a statement or a question. 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. 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

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(where math is involved) and several procedures and a general process for

solving problems.

In Mathematics problem solving,there are two types which are routine and non-

routine problem.Routine problem are those that merely involved an arithmetic

operation with the characteristics presents a question to be answered, gives the facts

or number to be use and in can be solved by directly application of previously learned

algorithms and the basic task is to identify the operation appropriate for solving the

problem.It stresses the use of sets of known or prescribed procedures (algorithms) to

solve problems.Routine problem is actually a type of mechanical mathematics

problem.It aimed at training pupils so that they are able to master basic skills

especially in arithmetic skills involving the four operations or direct application of

using mathematics formulae, laws, theorems or equations.

Non- routine problem occurs when an individual is confronted with an unusual

problem situation, and is not aware of a standard procedure for solving it.The

individual has to create a procedure.To make it the person must become familiar with

the problem situation, collect appropriate information, identify an efficient strategy,

and use the strategy to solve the problem. Non-routine stresses the use of

heuristics and often requires little to no use of algorithms. Heuristics are procedures

or strategies that do not guarantee a solution to a problem but provide a more highly

probable method for discovering the solution to a problem.

Polya’s model had been introduced to solve the problem solving.George Polya had

identified four steps in the problem solving process:

Understanding the problem

Devising a plan

Carrying out the plan

Looking back

1. UNDERSTANDING THE PROBLEM

First. You have to understand the problem.

What is the unknown? What are the data? What is the condition?

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Is it possible to satisfy the condition? Is the condition sufficient to determine

the unknown? Or is it insufficient? Or redundant? Or contradictory?

Draw a figure. Introduce suitable notation.

Separate the various parts of the condition. Can you write them down?

2. DEVISING A PLAN

Second. Find the connection between the data and the unknown. You may

be obliged to consider auxiliary problems if an immediate connection

cannot be found. You should obtain eventually a plan of the solution.

Have you seen it before? Or have you seen the same problem in a slightly

different form?

Do you know a related problem? Do you know a theorem that could be

useful?

Look at the unknown! And try to think of a familiar problem having the

same or a similar unknown.

Here is a problem related to yours and solved before. Could you use it?

Could you use its result? Could you use its method? Should you introduce

some auxiliary element in order to make its use possible?

Could you restate the problem? Could you restate it still differently? Go

back to definitions.

If you cannot solve the proposed problem try to solve first some related

problem. Could you imagine a more accessible related problem? A more

general problem? A more special problem? An analogous problem? Could

you solve a part of the problem? Keep only a part of the condition, drop the

other part; how far is the unknown then determined, how can it vary? Could

you derive something useful from the data? Could you think of other data

appropriate to determine the unknown? Could you change the unknown or

data, or both if necessary, so that the new unknown and the new data are

nearer to each other?

Did you use all the data? Did you use the whole condition? Have you taken

into account all essential notions involved in the problem?

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3. CARRYING OUT THE PLAN

Third. Carry out your plan.

Carrying out your plan of the solution, check each step. Can you see

clearly that the step is correct? Can you prove that it is correct?

4. Looking Back

Fourth. Examine the solution obtained.

Can you check the result? Can you check the argument?

Can you derive the solution differently? Can you see it at a glance?

Can you use the result, or the method, for some other problem?

There are many strategies to solve problems in mathematics.These are the

examples of the strategies:

Making table

Working backwards

Drawing diagram

Guess and check

Acting out

Investigating all possibilities

Thus,problem solving can enable all students to build new mathematical knowledge

through problem solving, solve problems that arise in mathematics and in other

contexts, apply and adapt a variety of appropriate strategies to solve problems, and

monitor and reflect on the process of mathematical problem solving.

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Routine and Non-Routine Problem

Routine and non-routine are one type of problems that we learn in this

semester in Basic Mathematics. As we all know, a problem is a task for which the

person confronting it want or need to find a solution and must make an attempt to

find a solution.

From our discussion and previous lesson that we already learn in classroom,

we conclude that routine problem problems are those that merely involved an

arithmetic operation with the characteristics can be solved by direct application of

previously learned algorithms and the basic task is to identify the operation

appropriate for solving problem, gives the facts or numbers to use and presents a

question to be answered.

In other word, routine problem solving involves using at least one of four

arithmetic operations 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. The critical matter

knows what arithmetic to do in the first place. Actually doing the arithmetic is

secondary to the matter.

For non-routine problem, it occurs when an individual is confronted with an

unusual problem situation, and is not aware of a standard procedure for solving it.

The individual has to create a procedure. To do so, we must become familiar with the

problem situation, collect appropriate information, identify an efficient strategy, and

use the strategy to solve the problem.

Non-routine problem are also those that call for the use of processes far more

than those of routine problems with the characteristics use of strategies involving

some non-algorithmic approaches and can be solved in many distinct in many ways

requiring different thinking process.

This problem solving also 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

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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.

It is important that we share how to solve problems so that our friends 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 other people 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 of us 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.

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Differentiating between routine and non-routine problem

Characteristic

Routine Problem Non Routine Problem

Type Of Problems

1. Drill exercise2. Simple translation3. Complex (or multiple-step)

translation

1. Process problem2. Applied problem3. Puzzles problem

Explanation 1. Drill exercise

Drill exercises provide students with practice in using an algorithm and help maintain mastery of basic computational skills.

2. Simple translation

Simple translation problem provide student with experience in translating real world situation into mathematical models.

3. Complex (or multiple-step) translation

Complex translation problem provide student with the same experience as simple translations problem, except that more than one operation may be involved.

1. Process problem

Process problem lend them to exemplify the process inherent in thinking through the solving of a problem.They serve to develop general strategies for understanding, planning and solving problems, as well as evaluating attempts at solutions.

2. Applied problem

Applied problem provide an opportunity for student to use a variety of mathematical skills, processes, concept and facts to solve realistic problems. They make student aware of the value of usefulness of mathematics in everyday problem situations.

3. Puzzle problem

Puzzle problem allow student an opportunity to engage in potentially enriching recreational mathematics. They highlight the importance of flexibility in attaching a problem.

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Characteristic

Routine Problem Non-Routine Problem

Example Of

Questions

1. Drill exercise

280 x 74

2. Simple translation

Minah has 13 marbles and

Fuad has 5 marbles. How

many more marbles does

Minah as than Fuad?

3. Complex (or multi-step)

translation

A chocolate box contain 60

chocolate. A carton holds 36

packs. If a shop owner

ordered 4320 chocolate,

how many cartons that he

need to order?

1. Process problems

A badminton club held a

tournament for its 25 members.

If every member played one

game against each other

members, how many games

were played?

2. Applied problems

How many papers of all kinds

do your college uses in a

month?

3. Puzzle problems

A ring is in a “cup” formed by

four pencils. Try to get the ring

out of the cup by moving only

two pencils.

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EXAMPLE QUESTIONS OF NON-ROUTINE

QUESTION 1 :

Everyday starting from Monday to Friday,Shikin will stop for a while after coming

back from her school at a shop to buy a sweets for her sister.She buys 4 piece of

sweets on Monday.Each day after Monday till Friday,the number of sweet she buys

is twice as a day before.How many sweets she buys on Friday and how is the total

sweets she buys for the 5 days?

PROBLEM SOLVING STRATEGIES :

A.) MAKE A TABLE

1.) UNDERSTAND

We know that on Monday she buys 4 piece of sweets.Then, for the next day the

number of sweets she buys is twice as a day before.Thus, how many of sweets she

buys on Friday and the total of sweets she buys?

2.) DEVISING A PLAN

How can we solve the problem?

We can make table to solve the problem.List the amount of sweets she buys for

everyday.Remember the number of sweets she buys is twice or double as a day

before.

3.) CARRYING OUT THE PLAN

Day Number of sweets

Monday 4

Tuesday 4 + 4 = 8

Wednesday 8 + 8 = 16

Thursday 16 + 16 = 32

Friday 32 + 32 = 64

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Thus, on Friday she buys 64 pieces of sweets for her sister.

Total sweets she buys = 4 + 8 + 16 + 32 + 64

= 124 sweets.

4.) LOOKING BACK

Monday = 4 sweets

Tuesday = 4 + 4 = 8 sweets

Wednesday = 8 + 8 = 16 sweets

Thursday = 16 + 16 = 32 sweets

Friday = 32 + 32 = 64 sweets

Total number of sweets = 124 sweets

B.) FIND A PATTERN

1.) UNDERSTAND

We have to know that from Monday to Friday there are 5 days.On Monday she buys

4 piece of sweets.The next day till the Friday the number of sweets she buys is twice

as a day before.So,how many sweets she buys on Friday and the total of sweets for

the 5 days?

2.) DEVISING A PLAN

How to solve the problem?

We can find a pattern.Even only a day the number of sweets had been stated we still

can find a pattern.The new number is depend on the number before.

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3.) CARRYING OUT A PLAN

X2 x2 x2 x2

Monday Tuesday Wednesday Thursday Friday

4 8 16 32 64 = 124 Sweets

When looking at the pattern above, we can see that the pattern are twice as a

number before or simply the pattern is time 2.

The pattern for the above question is constant that is x2.

4.) LOOKING BACK

Monday = 4 sweets

Tuesday = 4 x 2 = 8 sweets

Wednesday = 8 x 2 = 16 sweets

Thursday = 16 x 2 = 32 sweets

Friday = 32 x 2 = 64 sweets

Total number of sweets is 124.

COMMENT:

For the above questions,we use a table and look for a pattern strategy.When we use

a table to solve the question the had been given it looks more simply and easy to

understand.This is good especially when being a teacher to teach the primary school

and the can understand easily so that they enjoy their study.

For the second strategy, we use look for a pattern.This is the best strategy that we

can use to teach the primary student.They can see the flow what the question need.

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QUESTION 2:

In a class 2 Bestari of Sekolah Kebangsaan Tok Jembal, the students are divided

into 4 different sports house,in conjuction with the sports day that will be held on 27

April 2010. 5 students are Jupiter house,4 students are Mars house,7 students are

Neptune house and the other Pluto house.If the class has 21 students,how many

students are Pluto house and what is the fraction for the Pluto house?

PROBLEM SOLVING STRATEGIES

A.) DRAW A PICTURE

1.) UNDERSTAND

Students are divided to 4 differents house

5 tudents are Jupiter house

4 students are Mars house

7 students are Neptune house

The other are Pluto house

Total students in the class is 21.

2.) DEVISING A PLAN

How can we solve the problem?

We can make a picture to differentiate the 4 house.Differentiate the pictures with

different colours.

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3.) CARRYING OUT THE PLAN

Mars Neptune

Jupiter Pluto

Since Pluto are the green,thus the number of students which are Pluto huse are 5

students.The fraction for the Pluto house is 5/21.

4.) LOOKING BACK

Total number of students are 21.

Mars = 4 students Jupiter = 5 students Neptune = 7 students

Thus, Pluto = 21 – (4 + 5 + 7)

= 21 -16

= 5 students, and the fraction is 5/21.

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B.) MAKE A TABLE

1.) UNDERSTAND

Students are divided to 4 differents house

5 tudents are Jupiter house

4 students are Mars house

7 students are Neptune house

The other are Pluto house

Total students in the class is 21.

2.) DEVISING A PLAN

How can we solve the problem?

We can make a table to find the number of students which are Pluto house.Make a 4

column table and fill up with the information that had been given.

3.) CARRYING OUT THE PLAN

Mars Jupiter Neptune Pluto

Total number of students = 21

Thus, Pluto has = 5 students

Thus,the fraction of Pluto students is 5/21

4.) LOOKING BACK

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Total number of students are 21.

Mars = 4 students Jupiter = 5 students Neptune = 7 students

Thus, Pluto = 21 – (4 + 5 + 7)

= 21 -16

= 5 students, and the fraction is 5/21

COMMENT:

For the question number 2,we use a diagram and a table to solve the question.When we use the diagram especially with the colour we can see clearly what the question want and we easily solve it.Primary student surely easy to understand when we use colour because our brain are focusing more at the things that have colour.

The second strategy we use is make a table just lke strategy for question 1.The table can make us more understand and easily to interpreted the data that had been given in the question.

CONCLUSION

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We can conclude that there are many type of problem solving that can be

used in solving daily problem. In Question 1, we decide to solve our created problem

by using table and look for a pattern method. This method helps us discover

relationships and patterns among data. It encourages us to organize information in a

logical way and to look critically at the data to find patterns and develop a solution.

Drawing a diagram is the most common problem solving strategy.  We use the

strategy of drawing a diagram again and again as we show in Question 2. First we

need to learn how to interpret a problem and draw a useful diagram.  Very often, we

need to draw a diagram just to understand the meaning of the problem.  The diagram

represents the problem in a way we can “see” it, understand it, and think about it

while we look for the next step. 

Problem solving using tables might seem complicated, but it is easily mastered

with some instruction. Not all of the types of question need us to construct a table. It

depends on the question. For question number 2,besides using drawing a picture,we

also make a table.When using a table we can see clearly what the question need.We

can straight found the answer.In addition, we use a table so that we can solve the

problem easily.

Even that, the selection strategy that we had select of course have

advantages and disadvantages.For the above question, we can see that this strategy

make us or the person more understand about what the question need.When we

doing step by step we can easily found the answer.However,the strategy that we

used need more time than usual and it will take time especially during the

examination.This strategy also need more spaces to do.

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In conclusion, we realize that there are many strategies that we can use in

solving a problem.We can used either guess and check, working backward, draw a

diagram, look for a pattern, make a table, simplify the problem,identify the sub-goal,

act out the situation, use concrete object/ model and investigate all possibilities.But,

we have to understand first the question and use the appropriate strategy.. All of

these strategies can be stretched when combined with other strategies such as

looking for patterns or drawing a picture. By combining this strategy with others, we

can analyze the data that is given to find more complex relationships.

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REFLECTION

Name: Mohd Amin Bin Jamal Abdul Nasir

Unit: PPISMP science/BI/BM 1 semester 2 July intake

Matrix no.:122934

Ic no: 901115-02-5119

This is my reflection on the basic mathematics KKP. Along completing this

task, I gain a lot of information on problem solving and polya’s model. This task has

expose me a lot on problem solving which has teach me on the two types of problem

solving which are routine and non routine problems.

The strength in this task is that it helps me to understand more deeply on

problem solving. It also help me to understand the 4 solving step that was introduces

by polya.the four steps are understand the problem, devising a plan, carry out the

plan and looking back. This step is very useful in solving the mathematical problems.

The information for this task is also a lot available on the internet but me and my

friend also get some information from books.

For the weakness of this task is that it is the time given in completing this task

is seems short. I also see that this task can be completed by two person only rather

than doing in groups of three.

To overcome the weaknesses that I have mentioned above the time for

completing this task should be longer. Reducing the number of group members in

each group can ensure that everyone do their work and not just copying from others.

To conclude, this task should be continues for the coming student because it

can helps the student to increase their understanding on the problem solving and the

polya’s model.

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REFLECTION

NAME : AHMAD FARIS BIN JAMALUDIN

I/C NO : 910913-07-5377

MATRIC/NO : 122929

COURSE : PPISMP SN/BI/BM

UNIT : SCIENCE 1

Assalamualaikum w.b.t…Alhamdulillah and very proud to Allah with His blessing I

and my friends can finish up the Mathemathics short assignment for the topic

Problem Solving.Thousand of thanks are wishes to all Mathematic lecturers

especially En.Mazlan our Mathematics lecturer.Thruthly, for the first action when

adapted this assignment we feel it is so tough for me and friends to finish it.But, when

we struggle finding the information about the task it is become easier and more

easy.Honestly, this assignment give a new experience for me and other friends.

As we know, in doing such assignment surely we have several obstacles but

Alhamdulillah with the aid from lecturers and friends our group can through it softly.In

doing such assignment there are several obstacle whether in commerce, time or

other parts.The most problem that we face is to find the information about the routine

and non-routine.Even,there is an internet but not all information are there.But, we are

not give up easily besides using an internet we find the information in the books.

Alhamdulillah we found many questions and information about the task and we are

not straight copying from the books but we changed shortly.

Besides that, the much problem that we face is time.As we know, besides

mathematics,we have other subject assignment like physics,and chemistry.But,it

does not mean that we can’t do that.We were tried our best to make it completely

and make the best.Alhamdulillah we still can encounter the problem.In doing that,we

managed our time kindly by divided the time to other subject too.We were not

focused at all for one subject,we done it slowly and at the same time we still can do

for the other assignment.

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Strength is an essential things which is need in doing any job.Without strength

we can easily give up.During doing this assignment Alhamdulillah we got a firm

strength.Even,there is a little bit problem we can settle it.This is the advantages when

we doing an assignment by group.Actually, doing something activity by grouping

have a lot of advantages.We can helping each other and advised when someone do

something wrong.We also can detect if someone make a mistake about his duty.In

addition,when we are lack of ideas, the other friends can hel us.We can share our

ideas and yhe task will be more interesting.The most important actually in doing

something by grouping is cooperation among the partner.

As a student all the weaknesses that we face during the first task will not be

repeat to the other task.We have to think how to overcome the problem that we face

and repaired during the other task.Like us,first we were lack of information about the

routine and non-routine problem.We had settle by find it to the other sources like a

book.We were not focused only at the internet.Besides we also use the note that had

been given by the lecturer.So, we can mixed the information from the sources that

we got and it will be more interesting.

As a conclusion the task for the topic Problem Solving give more knowledge to

me and my friends,Before, we just learn how to answer the Mathematics question,

we didn’t learn the way or strategies to answer the question.It is important to me as

well for being a teacher for the coming soon.I can application it during become a

teacher and it can make my student become more interesting and easier in answer

the Mathematics question.At last,truthly this is the new knowledge for me and I hope

thai Ican application it along in my life especially when being a teacher.Thank You.

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REFERENCES

Polya.G.(1957)”How to solve it,” 2nd ed,Princeton University Press,

vvhttp://www.math.utah.edu/~alfeld/math/polya.html.Retrieved on 16 April 2010.

The Math Problem Solving Models,

http://www.camas.wednet.edu/chs/staff/rwright/mathpsmodel.html.

Retrieved on 16 April 2010

Schloeglmann.W.(2004),Routines in Non-Routine Problem Solving Processes

www. emis.de /proceedings/PME28/RR/RR126_Schloeglmann.pdf ,

Johannes Kepler University Linz,Ausria. Retrieved on 10 April 2010.

Pantziara.M,Gagatsis.A,Pitta-Pantazri.D. (2004).

The Use of Diagram in Solving non-routine Problems,

www. emis.de /proceedings/PME28/RR/RR126_Schloeglmann.pdf ,

Department of Education, University of Cyprus. Retrieved on 13 April 2010.

Bernard H. Gundlach, Pattern in Mathematics,

Published by Laidlaw Brothers A Division of Doubleday & Company, Inc

River Forest, Illinois.

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