CHAPTER 1: INTRODUCTION

1.1 INTRODUCTION

During the practicum at SK Bangsar on Semester 5, I teached three classes of Year 4.

Two classes were average pupils and the other one class was remedial pupils.

Therefore, varies approaches should be designed and worked for them. After discussion,

the mathematic teacher agreed that she will proceed teaching fraction however I will

start teaching a new topic which was money.

I had emphasized on methods and strategies that should be used for specific

pupils (remedial & enrichment) when teaching and learning process. When teaching

money, I realized that the pupils understand the concept of money well. They also can

solve questions on money involving basic operations which are addition, subtraction,

multiplication and division. However, the last class which was 4 Progresif not. Only

few pupil able to solve the question correctly. Besides, I also have to guide them step by

step about how to solve the question. Means, without guidance and support, they can

not solve the question individually.

Besides that, although the two classes can solve question about money involving

basic operation, but when I provide them problem solving question involving basic

operation, only half of class can solve the question. Then I was thinking why they can

do the basic operation on money but they can not solve the problem solving. They

showed the process skill correctly and did the calculation using standard written

method. After asked them personally, then I identified that some of them fail to

understand the problem clearly.

1

Besides, many of them also fail to comprehend the question and doing the

transformation. This means, they do not know what the question asked to solve. They

also do not know how to choose the suitable method (operation) to solve the question. I

had to tell them which operation should be used for this question. I observed that they

just take the number and do the calculation themselves. But, when I asked why they do

like that or choose that notation, they had no answer and can not give the explanation

well.

Furthermore, I analyzed that although they master the basic skill of operation,

they still can not solve the problem solving because it not only requires the final answer

but also the comprehension thinking and cognitive skill to relate the information given

with the suitable operation and strategy method. Therefore, I noticed that the pupils still

not master the purpose of problem solving questions. Then I asked myself, “Why the

pupils can not solve the words problem although they master the basic operations

skills?” Also, “What are the factors of these problems/ difficulties?”

From this experience I was so disappointed to blame the teacher because I got

information that the teacher taught them by recognizing the ‘term’ on words problem.

For example the pupils were taught to use subtract notation when seeing the word

‘different’. Therefore, the pupils actually do not know the concept or situation for in

words problem given. Pupils should be taught to understands the situation well because

it was a real life situation and teacher can do a connection or relationship between the

words problem and the real life situation. Teacher can also do a simple role-play to let

the pupils see the situation of the question given.

2

CHAPTER 2: FOCUS OF STUDY

2.1 RESEARCH’S ISSUES

Many pupils able to do basic mathematical operations. However they do not know

how to solve the problem solving involving basic operation because they have not

master the ability and necessary skill required in solving word problem question

although it is real life situation. Problem solving is not only requiring the final answer

of the question but pupils should understand and comprehend the question correctly.

Besides, pupils should be able to transform the information given into an appropriate

strategy and method.

Therefore, teacher especially should take the opportunity to solve this matter.

Teacher has to know why many pupils faced this problem and has to identify pupils’

problems. Teacher should identified who are the pupils and which part of the problem

solving they have difficulties. Teacher also has to detect these pupils at the beginning

so that teacher can take an action and opportunity to solve pupils’ problems. By doing

the diagnostic test at the first level, many suitable actions and programs can be

worked before they go to the higher level that clearly they can not do correct.

Therefore required actions should be started at the beginning of mathematical

development when teacher noticed their true mistakes and problems. Therefore, the

teaching and learning process can be worked properly. Thus, the pupils will always do

errors or mistakes forever if their problem on words problem not solve yet and finally

3

they have not interested in learning mathematics. That is why mathematics subject

matter has varies statistic in school.

Baretta-Lorton (1997) hardly stated that the pupils faced problems because

words problems need the application of higher order thinking. Besides that, pupils

should understand the context problem clearly and master the calculation process to

solve words problem. However, pupils still faced problems because of their errors and

mistakes in arranging the data and in calculation process.

Furthermore, this study is usability because I can learn well from the findings

of this research especially to improve the weaknesses of my Teaching and Learning

process. Usability, as defined by Joseph Dumas and Janice (Ginny) Redish, means

that people who use the product can do so quickly and easily to accomplish their

tasks. Therefore, I can design and manage the activities and methods for the Teaching

and Learning process that are suitable and appropriate for the pupils. Then also,

pupils’ performance can be improved because no repeated error happens next after

teacher had solved their problems or difficulties.

This study also can be controlled properly because the pre-test can be worked

at a time besides the clinical interview can be done at many times. For example, I can

interview one student a day. So, a month I will get 20 samples to evaluate the pupils.

But, I just need 10 samples for this study. This means the study is reasonable to be

conducted and controlled at the school. Interview process will be also done in a good

environment like at the media room or school library.

4

Then, this study will be also very useful for teacher especially because

mathematics teacher can realize at which level actually pupils doing errors in words

problems. After that, teacher can take opportunity on how to solve this matter to

improve pupils’ understanding in mathematics words problem. So teacher will

emphasize on teaching and learning process at the specific errors. Therefore, this

study has a strong relationship between pupils and teachers besides relevance to

school.

5

2.2 LITERATURE REVIEW

Mathematics words problem is an important component in Mathematics Curriculum

KBSR (Kurikulum Bersepadu Sekolah Rendah). Study shows that pupils have difficulty

in solving mathematics word problems although they have mastered the basic skill of

operation procedural [Mokhtar, Aminah & Lim 2001; Third International Mathematics

and Science Study (TIMSS) 1999; Fatimah 1999; Hassan 1998; Mohd. Daud et al.

1997].

Mathematics word problem causes many difficulties and problems especially

when the first learning development of the pupils. [Verschaffel, De Corte and

Vierstraete (1995), Bransford et al. (1996) and Hegarty, Mayer and Monk (1995).

As education has come under criticism from many sectors, educators have

looked for ways to reform teaching, learning, and the curriculum. Many have argued

that the divorce of content from application has adversely affected our educational

system (Hiebert, 1996). Learners often learn facts and rote procedures with few ties to

the context and application of knowledge. 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.

Today there is a strong movement in education to incorporate problem solving

as a key component of the curriculum. The need for learners to become successful

problem solvers has become a dominant theme in many national standards (AAAS,

1993; NCSS, 1997; NCTE, 1996; NCTM, 1989, 1991). For example, the 1989

Curriculum Standards of the National Council of Teachers of Mathematics (NCTM)

6

states: “Problem solving should be the central focus of the mathematics curriculum. As

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

mathematical 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” (National Council of Teachers of Mathematics, 1989).

The classification of errors is defined by Newman (1977). This error-

classification gives an advantage for the teacher because it provides a clear framework

for questioning the pupils and for analyzing any errors. The teacher can discover where

and why the pupils made a mistake. Newman assumed that associated with any give

word problem are a number of hurdles which have to be overcome if a correct solution

is to be obtained, and that failure on any particular hurdle prevents a person from

progressing to the next hurdle and from obtaining the correct solution. Newman defined

a hierarchy of error causes which, she claimed, applies to one-step written mathematical

problems. The hierarchy has five levels, which are Reading, Comprehension,

Transformation, Process skill and Encoding.

Chart 2.2: The hierarachy of Newman’s Errors Analysis

7

According to Newman a person confronted with one-step written problem has to

read the problem, then comprehend what he has read, the carry out the transformation

from the words to the selection of an appropriate mathematical ‘model’, then apply the

necessary process skills, then encode the answer. Therefore, failure at any level of the

hierarchy prevents the person from obtaining the correct answer.

Two other categories of errors are ‘Carelessness’ and ‘Motivation’. These kinds

of errors separate from the hierarchy although they can be associated with any level of

it. A careless error could be a reading error or a comprehension error, and so on.

Besides, C.S.Rice (1920) concluded that the main difficulty which children in

grades 3 to 8 experienced when attempting verbal arithmetic problems was the choice

of the appropriate mathematical operation. L.John (1930) defined four errors categories

(Reasoning, Fundamentals, Reading, and Miscellaneous). Doty (1940) claimed that the

main factors caused the failure of pupils on verbal arithmetic problem were (i) ability in

computation, (ii) ability to gain mathematical implications from language forms, (iii)

understanding of the various mathematical processes, and (iv) effective procedures with

problems (see Hollander, p.329).

Therefore, in a study in the 1950’s C.G. Corle classified data which were

gathered into three major areas: insight, thought process and number relationships, and

computational skills. According to Corle, with the exception of the ability to solve more

problems accurately, individual characteristic of good and poor problem solvers bear

close resemblance (see Hollander, pp. 1-330). However, the present writer would

disagree strongly with his claim.

8

Casey (1978) pointed out, problem solvers often return to lower stages of the

hierarchy when attempting to solve problems. (For example, in the middle of a

complicated calculation someone might decide to reread the question to check whether

all relevant information has been taken into account.) However, even if some of the

steps are revisited during the problem-solving process, the Newman hierarchy provides

a fundamental framework for the sequencing of essential steps.

9

CHAPTER 3: OBJECTIVE OF STUDY

3.1 OBJECTIVES OF THE STUDY

The aim of this study is to analyze children’ errors in solving words problem. The

study also will recognize why many children do mistakes when facing words problem.

The study focus on the four basic operations involving words problem and

what are the factors faced by the pupils when doing the calculation. Based on

Newman (1977, 1983), 4 steps should be followed when solving words problem

which are; (i) Reading, (ii )Comprehension, (iii)Transformation, (iv) Process skill,

and (v) Encoding.

3.2 RESEARCH QUESTIONS

The Newman Error Analysis emphasized to see which level pupils face

problem in hierarchy form when solving words problem because the failure of first,

second third, forth or last step will conduct pupils to get the wrong final answer.

The study will focus:

I. To analyse the errors done by the pupils to solve the questions.

II. The type of errors students commits errors in mathematics words problem?

10

However, the study just focuses on the children in Year 5 Kreatif at Sekolah

Kebangsaan Bandar Baru Seri Petaling 2. Besides that, the writer does not take the

influence of the children’ biographies. This means the influence of the personal

emotional, family background, school environment or the physical impairments that can

influence the children in solving mathematical words problems.

11

CHAPTER 4: TARGET GROUP

The study was conducted with children Year 5 Kreatif at Sekolah Kebangsaan Bandar

Baru Seri Petaling (2).

NO. NAME GENDER RACE

1 HAMMAD ABDUL AZIM MALE MALAY

2 ISAAC NG ZHENG XIAN MALE CHINEESE

3 JOHN A/L ALBALAHEN MALE INDIAN

4 MOHD AL HAFEEZ BIN ZAKARIA MALE MALAY

5 MOHD IRSYAB FAYAZ BIN ARIF MALE MALAY

6 MUHAMAD SYAFIQ AZHAR MALE MALAY

7 MUHAMMAD IMRAN MOHD NAJEEB MALE MALAY

8 MUHAMMAD ISKANDAR B NORAN MALE MALAY

9 MUHAMMAD MUHSIN B SYED NAZIR MALE MALAY

10 MUHAMMAD NAZRUL B AZARSHAH MALE MALAY

11 MUHAMMAD NUR IZZUL IMAN MALE MALAY

12 MUHAMMAD QATADAH NOR HISHAMUDIN MALE MALAY

13 MUHAMMAD QAYYUM AZWA MD LAZIM MALE MALAY

14 MUHAMMAD REDUAN B MOHD ISARUDIN MALE MALAY

15 NEZAN BIN NASIR MALE MALAY

16 SATHISARAN A/L NATHAN MALE INDIAN

17 VEROZDLY MALUDIN MALE BUMIPUTERA

18 AZEERA ANSANGGOR FEMALE MALAY

19 HILAVARASI A/P SAITHAL FEMALE INDIAN

20 JALINAH JULING FEMALE MALAY

21 NUR NEELAM SARI BTE ASMAWI FEMALE MALAY

22 NURSYAQIRAH KASSIM FEMALE MALAY

23 NURUL ARIQAH FEMALE MALAY

24 NURUL HIDAYAH BT IBRAHIM FEMALE MALAY

25 NURUL SYAFIKAH BTE JASNI FEMALE MALAY

26 SHERILYN LOGATHAN FEMALE MALAY

27 SITI SHIRIN SHABIRA FEMALE INDIAN

28 SOFIAH BTE DUL AJIS FEMALE MALAY

29 SUSANNA MOGANASUNDRAM FEMALE INDIAN

Table 4.1: Name list of children in Year 5 Kreatif

12

These were the characteristics of the respondents:

I. Number of students : 29 students

II. Gender /sex : 17 Male; 12 Female

III. Name of class/position : 5 Kreatif

IV. Academic performance : Intermediate – mix abilities

Table 4.2: Analysis of Respondents

Twenty-nine children were involved in the study to analyze their errors in mathematical

words problems. Clearly Table 4.2 showed that Malays children were the major

children in this class and followed by Chinese children. Besides that, the total number

of boys and girls was quiet equal. Furthermore, the respondents involved about five

advance children, five weak children and the rest children are intermediate cognitive

level.

CHAPTER 5: PROCEDURES OF STUDY

MALAY CHINESE TAMIL OTHERS TOTAL

NO OF

BOYS13 1 2 1 17

NO OF

GIRLS9 - 3 - 12

TOTAL 22 1 5 1 29

13

5.1 PROCEDURE ACTION

The writer worked the study by conducting a test for the children. Based on the

children’s performance on previous test, the writer constructed fifteen items for

mathematical words problems. The writer then divided the questions into three

categories which were easy, medium and hard levels of questions. Besides that, the

writer gave opportunity to weak children to ask teacher when solving these questions.

Hence, the writer put these children into a group so that the writer can work the plan

smoothly.

The writer absolutely used the Newman’s Errors Analysis to analyze children

errors in mathematical words problems. The writer marked children’s answer paper by

marking the first error only done by the children. If the children obtain incorrect at

comprehend level, therefore no frequency at transformation, process skill and onward

level. However, the writer will mark correct answer if the children do careless error if

they answer correctly when interview session.

Furthermore, the writer constructed the questions based on the skill in

curriculum specification. The writer applied all skills in topic Money to see different

view from Newman’s error analysis. From the skills stated in Curriculum Specification,

there were six learning outcomes emphasized:

I. Read and write the value of money in ringgit and sen up to RM100 000

II. Add money in ringgit and sen up to RM100 000

III. Subtract money in ringgit and sen within the range of RM100 000

14

IV. Multiply money in ringgit and sen with a whole number, fraction, or decimal

with product within RM100 000

V. Divide money in ringgit and sen with the divisor up to RM100 000

VI. Perform mixed operation of multiplication and division invovling money in

ringgit and sen up to RM100 000

From these learning outcomes, children were taught to understand and use the

vocabulary related to money such as RM, sen, note, value and total. Furthermore,

children were taught to use and apply mathematics concepts when dealing with

money up to RM100 000. This objective was emphasized to relate the real life

situation in mathematical words problems.

Besides written test, the writer worked an interview session for this study. The

interview worked for children those who got incorrect in the test. However, only

fifteen children were chosen as the samples for the interview session. Newman’s

errors analysis was used to conduct this plan. Interview instrument was emphasized to

see children’s understanding in mathematical words problems. The writer can note

clearly what were the children difficulties and initial errors.

Then, time will be consumed for this plan because the study wants to know

what the major error was done by the children when solving mathematical words

problems. For test items, the children were given two hours to solve the questions.

However, the writer gave no specific time for interview session. They also will be

guided if they doing errors or just a carelessness error.

15

After the test and interview session worked, the writer preceed to gather the

data in suitable form. By the way, the writer used also field notes and recorder to

record the information and feeback for this study. Besides that, some of children’s

works also were gathered to show the proof of the findings. Then, the data was

analyzed and interpreted by using table and graph. From these diagrams, so the writer

can show clearly at which level of errors analysis children do errors in mathematical

words problems. The table was chosen to gather the information in clear picture and

simple form. The graph was very suitable because the writer can see which bar (error)

the highest (total number of children) was.

After the writer worked these plans, a methodology of Polya’s Model was used

to improve children’s understanding in mathematical words problems. The writer

emphasized also on difference format of solution. The children were taught to fill

each specific part for each question before doing the calculation process. The children

had to state:

I. Information given: ...............................................................................................

II. Operation: ............................................................................................................

III. Plan: .....................................................................................................................

From this method, there was improvement of children’s performance because they

had to understand the problem to fill each part given.

16

CHAPTER 6: DATA COLLECTION METHOD

Twenty-nine children in Year 5 Kreatif at Sekolah Kebangsaan Bandar Baru Seri

Petaling (2) were given a test of fifteen words problems, and over the next three weeks

fifteen of the children were interviewed. The Newman error analysis was used in an

attempt to discover which skill the children had not obtained correct solutions to the

problems. The testing and interviewing were done by the writer during the internship

for future teacher preparing. For interview section, the writer chose five advance

students, five intermediate students and five weak students.

6.1 THE TEST

There were fifteen words problems items about Money. These items involved the

process of addition, subtraction, multiplication and division. However, the time

available was limited because all testing and interviewing had to be done in school time

as part of normal lessons. Furthermore, the writer was given three main works during

four weeks in this school.

The children were known to be average readers and some of them were slow

readers. Therefore, sometimes the items were presented orally by the writer and

repeated for specific group. The children were given 2 hours to solve all the words

problems.

The test items were selected based on the learning outcomes of Money in

Curriculum Specification to discover which skill that the children still not master yet.

17

Learning Outcome Items

Read and write the value of

money in ringgit and sen

up to RM100 000

-RM, sen, note, value,

total, amount, range,

dividend, combination

1. Ahmad wrote his cheque “Eighty thousand ninety

ringgit and fifty-sen.” What is the value of money in

numeral?

2. Ahmad wrote the second cheque “Twenty thousand

three ringgit and six sen.” What is the value of money in

numeral?

3. Syafiq having loan RM 71 026 from Maybank to buy a

new car, Toyota Vios. What is the value of money in

words?

4. Sidiq having a loan RM 49008.10 to support his study in

India. What is the value of money in words?

Use and apply mathematics

concepts when dealing

with money up to RM100

000.

5. The first, second and third prize of the table tennis

competition cost RM 2500, RM 1 500 and RM 750

respectively. Three categories involve which are single,

double and group. Draw the situation.

6. Ramu brings three notes of RM 50, four notes of RM 10,

one note of RM 5 and three notes of RM 1. Then, his

brother gives two coins of 50 sen and four coins of 10 sen

to buy a sport shoe. Draw the situation.

Subtract money in ringgit

and sen with the range of

RM100 000

7. Yati wants to buy a car which costs RM40 519. If she

has only RM28 260, how much more money does she need

to buy the car?

18

Add money in ringgit and

sen up to RM100 000

8. Mrs. Lee bought a digital camera for RM1 490.50 and a

television for RM975. How much money did she use to

buy both items?

Add money in ringgit and

sen up to RM100 000

9. A ring costs RM1 296. The price of the ring is RM5 752

less than that of a necklace. Find the price of the necklace.

10. Shahrul has savings of RM31 469.79. Adam has

savings of RM13 461.18 more than Shahrul. Calculate the

amount of savings that Adam has.

Multiply money in ringgit

and sen with a whole

number, fraction or

decimal with products

within RM100 000 &

Divide money in ringgit

and sen with the divisor up

to RM100000.

11. Tony saved RM 18.90 a week. How much money did

he save in a year? Write his saving in words.

12. Timah buys a set of sofa which costs RM 6 079.20. If

she pays by monthly installments for 2 years, how much

does he need to pay each month?

Divide money in ringgit

and sen with the divisor up

to RM100 000.

13. Saidah has RM 18 722 in CIMB bank. Her husband

has RM 24 541.90. They want to share their money

equally among their nine children. How much money will

each child receive?

14. Hamid collects RM 12.50 for each friend of his class

for celebrating a birthday party. How many friends does he

have if the party cost RM 375?

19

Add money in ringgit and

sen up to RM100 000 &

Subtract money in ringgit

and sen with the range of

RM100 000.

15. Julia has RM 2 890.48. Maria has RM 2 230 more than

Julia. However Joseph has RM 789.20 less than Maria.

How much money do they have altogether?

Table 6.1: Test Items

20

6.2 DIAGNOSTIC INTERVIEW

The interviews were done over three weeks in school time by the writer. The fifteen

children chosen for the interviews were selected on their achievements during the test

items. Unlike Newman’s study, where only low achievers were tested, it was decided to

analyze the errors of fifteen children and which skill they start doing inappropriate

working based on Newman’s error analysis.

Besides that, the children were not told that they were reworking only words

problems that they answered wrongly in the test. Each chosen children was given two

items only. The types of errors made were recorded as were any comments of the child

and the manipulations of the counters were noted.

The writer used this diagnostic interview form for this section:

Classification Typical Questions Errors (examples)

1. Reading

Please read the question to me.

(If you don’t know a word or number,

leave it out.)

Do not recognize key words or

symbols.

2. Comprehension

(a) (Point to a word or symbol.) What

does this word/symbol mean?

(b) Tell me what the question is asking

you to do.

(What do you mean when you say …?)

Can read the problems well

but cannot comprehend the

meaning of the words,

symbols or question.

21

3. Transformation Tell or show me how you start to find an

answer to this question.

Cannot transform sentences

into mathematical forms.

4. Processing

skills

Show me how you get the answer.

Tell me what you are doing as you

work.

(Let student work on a piece of paper.)

Can choose an appropriate

operation but cannot complete

the operation accurately.

5. Encoding

ability

Write down the answer to the question.

Can perform the correct

operations but writes the

answer incorrectly.

6. Careless Students spot own mistakes Different from the errors

above.

Table 6.2: Interview Form

22

CHAPTER 7: FINDINGS

7.1 TEST FINDINGS

Table 7.1 shows the number of children not master the skills based on the Newman’s

error analysis. There are six types of errors that are decided to discover children

difficulties in solving words problems. It shows that most of the children doing most

errors at comprehend skill.

Items R C T PS E F

1. Ahmad wrote his cheque “Eighty thousand

ninety ringgit and fifty-sen.” What is the value of

money in numeral?

2 10 6 0 2 0

2. Ahmad wrote the second cheque “Twenty

thousand three ringgit and six sen.” What is the

value of money in numeral?

1 8 5 0 4 0

3. Syafiq having loan RM 71 026 from Maybank to

buy a new car, Toyota Vios. What is the value of

money in words?

0 8 3 0 1 0

4. Sidiq having a loan RM 49008.10 to support his

study in India. What is the value of money in words?

1 9 3 0 3 0

5. The first, second and third prize of the table

tennis competition cost RM 2500, RM 1 500 and

3 16 3 0 2 0

23

RM 750 respectively. Three categories involve

which are single, double and group. Draw the

situation.

6. Ramu brings three notes of RM 50, four notes of

RM 10, one note of RM 5 and three notes of RM 1.

Furthermore, his brother gives two coins of 50 sen

and four coins of 10 sen to buy a sport shoe. Draw

the situation.

2 18 4 0 1 0

7. Yati wants to buy a car which costs RM40 519.

If she has only RM28 260, how much more money

does she need to buy the car?

0 10 7 1 2 1

8. Mrs. Lee bought a digital camera for RM1490.50

and a television for RM975. How much money did

she use to buy both items?

3 7 6 2 0 1

9. A ring costs RM1 296. The price of the ring is

RM5 752 less than that of a necklace. Find the

price of the necklace.

1 12 8 1 1 1

10. Shahrul has savings of RM31 469.79. Adam

has savings of RM13 461.18 more than Shahrul.

Calculate the amount of savings that Adam has.

2 14 8 1 0 2

11. Tony saved RM 18.90 a week. How much

money did he save in a year? Write his saving in

words.

1 20 4 0 0 0

24

12. Timah buys a set of sofa which costs

RM6079.20. If she pays by monthly installments

for 2 years, how much does he need to pay each

month?

2 18 5 2 0 0

13. Saidah has RM 18 722 in CIMB bank. Her

husband has RM 24 541.90. They want to share

their money equally among their nine children.

How much money will each child receive?

2 15 9 0 0 0

14. Hamid collects RM 12.50 for each friend of his

class for celebrating a birthday party. How many

friends does he have if the party cost RM 375?

0 12 8 2 3 1

15. Julia has RM 2 890.48. Maria has RM 2 230

more than Julia. However Joseph has RM 789.20

less than Maria. How much money do they have

altogether?

3 14 5 3 1 3

Total 23 191 84 12 20 9

Table 7.1:

Total numbers of children made errors in Test for twenty-nine children in Year 5Kreatif

Notes;

R: Reading, C: Comprehend, T: Transformation, PS: Process Skill, E: Encoding, and F:

Carelessness

25

7.2 INTERVIEW FINDINGS

Errors Analysis

Reading Compre

hend

Transfor

mation

Process

Skill

Encoding Careless

ness

1st

Item

1 4 4 2 1 0

2nd

Item

1 5 3 2 0 0

Total 2 9 7 4 1 0

Table 7.2:

Total numbers of children made errors in interview session for fifteen children in

Year 5 Kreatif

26

CHAPTER 8: RESULTS AND DISCUSSION

8.1 TEST ITEMS

Item 1 until item 4 emphasized on understands and use the vocabulary related to money.

It emphasized on reading and writing the value of money in ringgit and sen up to

RM100 000. These items ask the children to convert the words into mathematical

sentences and vice versa.

Graph 8.1.1: Total numbers of children made errors for item 1 till item 4

From the analysis, the children made most errors at Comprehend skill which is

35 frequencies. This result shows that they still not master the basic skill on money

which is to read and write the value of money. They able to read the words however

disable to write and transform the words into mathematical sentences. That problem

occurs because they not master the skill of comprehension the question.

27

0

5

10

15

20

25

30

35

40

R C T PS E F

Errors Analysis

Furthermore, transformation skill takes the second place of the most error by the

students in mathematics words problem. Besides that, no children made error at Process

Skill and Carelessness errors. The students fail to comprehend the question because

they do not understand the concept of place value. They also got problem when facing a

value that has “0” in the middle.

Item 5 and item 6 emphasized on use and apply mathematics concepts when dealing

with money up to RM100 000. The function of this question is to see the

comprehension of the student reading. Are they understand or not with the wording

problem. However, they are free to answer this section with diagram, table, or picture.

Graph 8.1.2: Total numbers of children made errors for item 5 and item 6

Based on the graph, I can see that the comprehend skill is the most error done by

the students and followed by the transformation skill. The wording question clearly

troubles the children although the words are just repeating. From question 5 and 6, I

28

0

5

10

15

20

25

30

35

40

R C T PS E F

Errors Analysis

realized that the children easily made addition or subtraction operation for these

questions although it not asked. Means they actually can not understand and

comprehend the meaning of these questions. However, the children able to show the

category of prizes but they forgot to expand the information into single, double and

group.

Item 7 emphasized on subtract money in ringgit and sen with the range of RM100 000.

Graph 8.1.3: Total numbers of children made errors for item 7

From the graph, no children made error at reading level. However, they made

most error at comprehend level. At this question the children have problem on

understanding the words “how much money”. Many of them used addition operation

when solving this question. Therefore, they failed to pass the Comprehend level and so

on.

29

0

2

4

6

8

10

12

R C T PS E F

Errors Analysis

Besides that, the children were too careless when working the calculating

process. Furthermore, they arranged the value of money in the wrong place value.

Therefore their final answer is absolutely incorrect.

Item 8 until item 10 emphasized on add money in ringgit and sen up to RM100000.

Reading Comprehend Transformation Process

Skill

Encoding Carelessness

3 7 6 2 0 1

1 12 8 1 1 1

2 14 8 1 0 2

6 33 22 4 1 4

Table 8.1: The frequencies of children made errors for item 8 until item 10

Graph 8.1.4: Total numbers of children made errors for item 8 until item 10

30

Item 8

Item 9

Item 10

Total

0

5

10

15

20

25

30

35

R C T PS E F

Errors Analysis

From the graph, the children made most error at comprehend level and followed

by the transformation level again. The children face problem to comprehend and

understand the asked question hence, they not pass the levels required. The children felt

confuse for the meaning of “less than” and “more than”. They can not transform this

word into mathematical sentence correctly.

Besides that, the children were so careless when doing the mathematical

calculation. In conclusion, the children still not master the skill on adding money in

ringgit and sen up to RM100 000 in mathematics words problem.

Item 11 and item 12 emphasized on use and apply mathematics concepts when dealing

with money up to RM100 000. These items involved 2 skills which are:

I. multiply money in ringgit and sen with a whole number, fraction or decimal

with products within RM100 000 and

II. divide money in ringgit and sen with the divisor up to RM100 000

Graph 8.1.5: Total numbers of children made errors for item 11 and item 12

31

0

5

10

15

20

25

30

35

40

R C T PS E F

Errors Analysis

Clearly there is a big range of children made error at comprehend level

compared to the other levels. The children were too careless to convert a year into a

month and from a month into a week.

However they were still some children understand and transform the word

problem into appropriate operation. Furthermore, item 12 was also too tricky for

children because they were asked to convert years into months before doing division

calculating. They have to get the idea (comprehend skill) to work the plan in solving

this words problem. Many of them also like to do the division without converting 2

years into months.

Item 13 and item 14 emphasized on use and apply mathematics concepts when dealing

with money up to RM100 000. These items involved 2 skills which are:

I. add money in ringgit and sen up to RM100000 and

II. divide money in ringgit and sen with the divisor up to RM100 000

Graph 8.1.6: Total numbers of children made errors for item 13 and item 14

32

0

5

10

15

20

25

30

R C T PS E F

Errors Analysis

The result shows that the children made most error at comprehend level and

followed by the transformation level. However, below 5 frequencies those children

made error at another four levels. These questions were also a little tricky and require

high level thinking because after doing division calculating, the children have to

proceed the calculation progress by subtracting the answer with 1. Furthermore, they

can not understand and comprehend the meaning of “How many friends does he have”

and then transform the words into appropriate operation. These items then applied

mixed operations to test children’s understanding in words problems.

Item 15 emphasized on use and apply mathematics concepts when dealing with money

up to RM100 000. This words problem also emphasized 2 skills which are:

I. add money in ringgit and sen up to RM100 000 and

II. subtract money in ringgit and sen with the range of RM100 000

Graph 8.1.7: Total numbers of children made errors for item 15

33

0

2

4

6

8

10

12

14

R C T PS E F

Errors Analysis

The result shows that the children made most error at comprehend level with 12

frequencies. Many of them fail to comprehend this mathematic words problem because

they can not relate the information given correctly. Some more, the children face

difficulty to understand the meaning of “more than” and “less than.” Furthermore, this

item requires high level of thinking because it involves mixed operations on words

problem. Children have also to think and relate the information given correctly.

8.2 TEST ITEMS CONCLUSION

Errors AnalysisReading Comprehend Transformation Process

SkillEncoding Carelessness

23 191 84 12 20 9

Table 8.2: The frequencies of children made errors in written test

Graph 8.2: Total numbers of children made errors for Test Items

34

0

40

80

120

160

200

R C T PS E F

Errors Analysis

From the graph, I can conclude that the children do errors most at the second

level of the error analysis which is comprehend skill. It proved since each question

given from number 1 till number 15 show that the comprehend skill bar graph is the

highest frequency compared to the other skill.

The second higher that children commit error in mathematics words problem is

transformation skill. However, the children did little error in words problem at reading,

process skill, encoding and carelessness levels.

The most errors done by the children

Chart 8.2: The most errors made by the children in written test

The chart above shows the most common errors done by the children when solving

words problem.

35

Comprehend

Transformation

Reading

Encoding

Process Skill

Carelessness

8.3 INTERVIEW CONCLUSION

Errors AnalysisReading Comprehend Transformation Process

SkillEncoding Carelessness

2 9 7 4 1 0

Table 8.3: The frequencies of children made errors in interview session

Graph 8.3: Total numbers of children made errors for two items in interview session

I. Comprehend Level

The result proved that the children do most errors at reading comprehension

level. The writer noted that the children had not answer correctly when he asked

the meaning of words/ symbols in the sentences. They also can not state what

the question is asking to do. Means the children can read the problems well but

cannot comprehend the meaning of the words, symbols or question.

36

0

12

3

45

6

7

89

10

R C T PS E F

Errors Analysis

II. Transformation Level

Besides that, the children gave no reason why they chose the operation for the

words problems. They can not explain clearly for their choices. Then, they can

not transform the words problems into appropriate mathematical forms.

III. Process Skill

Children committed errors also at process skill level. They can choose an

appropriate operation but cannot complete the operation accurately. This event

always happen especially when they doing multiplication and division

calculation.

IV. Reading Level

The children did little error for reading level. Only 2 frequencies were stated on

the graph. Therefore children recognized key words or symbols in the questions

given.

V. Encoding Level

Children can perform the correct operations but writes the answer incorrectly.

VI. Carelessness

No children did error for this level. Children worked the interview session and

gave good cooperation. Some more, they were given only two questions

therefore they can focus in doing the calculation and avoid the careless mistake.

37

8.4 CONCLUSION

Based on the results of both instruments, clearly shows that the large majority of initial

errors were made at the levels of Comprehension and Transformation. This result was

caused largely by the written words problems item 8 until item 15. These items applied

understand and comprehension skills to be solved. Children have to think and

understand the situation clearly to transform the words into mathematical form.

Besides, the weak children cannot read the mathematical words problems

fluently. They cannot say the words and symbols correctly. Hence, they did not know

what those questions required them to do.

The average children can understand the words problems and can transform the

words into mathematical form. However, they made error at Process Skill level. Many

of them can do addition and subtraction calculation but they do most errors at

multiplication and division calculation. They also spent a long time when doing these

calculations. Means these children still not master the basic skill operation of

multiplication and division.

Furthermore, one aspect of the result that needs to be considered is the low

proportion of errors due to Carelessness or Motivation. The writer agreed with Newman

that she classified errors in the original test given, and if a child obtained the correct

answer during the interview the error in the test was assumed to be due to Carelessness

or Motivation. Some more, children paid more concentration when solving

mathematical words problems for interview session and teacher gave full of support for

them.

38

CHAPTER 9: SUGGESTIONS FOR FURTHER RESEARCH

The study has shown that written test and an interview using the Newman error analysis

can be used by teacher to recognize which level children do errors in words problems.

According to Newman, failure at any level of the hierarchy prevents the person from

obtaining the correct answer. Therefore, The Newman’s error-classification gives an

advantage for the teacher because it provides a clear framework for questioning the

pupils and for analyzing any errors. The teacher can discover where and why the pupils

made a mistake.

Because of the time pressures teachers may say it is no possible to work an

interview session when teaching and learning process. Undoubtedly it is true but there is

no excuse why restricted interviews could not be equally useful. After a test given for

example, those children who do many errors could then be interviewed. By using two

words problems, this would take about five minutes for each child. But then teacher will

get a clear picture of where these children are making errors. Furthermore, the interview

process will give advantages for the children to understand more the concepts in

mathematical words problems.

The interview is also useful for this study because the writer can see that the

children actually understand the concepts in mathematical words problems but they do

careless error in written test. Therefore, the writer can note that these children master

that appropriate skills although they obtained incorrect answer.

However, the test item should be various levels of questions. Means the items

should involve easy, medium and hard levels because there are various types of children

39

in this world. Teacher therefore can analyze and categorize the children according their

level of cognitive; weak, average and advance children. Some more, the test items

should also apply the skills of Bloom Taxonomy which are knowledge, comprehension,

application, analysis, synthesis and evaluation. Therefore, the items constructed are

validity and reliability for the children.

Furthermore, the writer should interview some more samples like Newman.

Therefore the findings of the data will be more accurate and valid. However the writer

just interview only fifteen children and twenty-nine children for written test. Some

more, the writer should take a note on children’s background because it influences the

result on analyzing their errors. It is noted because children from poor not emphasize on

children’s education.

For better findings, the writer can note about the children’s performances. The

writer can choose only advance children or weak children as the samples to analyze

their errors in words problems. Therefore, the writer will know and recognize the most

errors done by the advance children or the most errors done by the weak children.

Clearly, there will be a difference between these findings. For this reason, teacher can

choose suitable methodologies when working the teaching and learning process.

To improve children’s understanding in mathematical words problems, teacher

can teach them using Polya’s Model which emphasizes on four aspects; understand the

problem, devise a plan, carry-out the plan and looking back. Teacher should note these

four steps so that children obtain correct solutions and understand clearly the concepts

in mathematical words problems.

40

CHAPTER 10: REFERENCES

M. A. (Ken) Clements (1980). Analyzing Children’s Errors on Written Mathematical

Task. Educational Studies in Mathematic (Volume 11, pp 1-21) D. Reidel

Publishing Company. Dordrecht, Holland, and Boston, U.S.A.

Ivan Watson (1980). Investigating Errors of Beginning Mathematicians. Educational

Studies in Mathematic (Volume 11, pp 319-329) D. Reidel Publishing

Company. Dordrecht, Holland, and Boston, U.S.A.

Radiah binti Haji Mohidin, Hajah (1999). The Difficulties faced by the Students of

Brunei Darussalam in Transforming Short Mathematical Word Statements into

Algebraic Expressions. Unpublished M Ed report. Universiti Brunei

Darussalam.

Knifong, J. D. and Holtan B. (1976). An Analysis of Children’s Written Solutions to

Word Problems. Journal for Research in Mathematics Education (Volume 7, pp

106-112)

Allan L. White. Active Mathematics in Classrooms: Finding out Why Children Make

Mistakes and Then Doing Something to Help Them. University of Western

Sydney. Retrieved January 21, 2010, from

41

http://www.curriculumsupport.education.nsw.gov.au/primary/mathematics/

assets/pdf/sqone.pdf

M. A. (Ken) Clements & Nerida F. Ellerton (1996). The Newman Procedure for

Analysing Errors on Written Mathematical Tasks. The University of Newcastle:

Faculty of Education. Retrieved January 21, 2010, from

http://users.tpg.com.au/arnold10/PAGES/newman.htm

Jamie Kirkley (2003). Principles for Teaching Problem Solving. Indiana University.

Plato Learning Inc. Retrieved January 21, 2010, from

http://www.plato.com/media/Technical-White%20Papers/2/Principles%20for

%20Teaching%20Problem%20Solving.pdf

Natcha Prakitipong and Satoshi Nakamura. Analysis of Mathematics Performance of

Grade Five Students in Thailand Using Newman Procedure. Graduate School

for International Development and Cooperation Hiroshima University.

Retrieved January 22, 2010, from http://home.hiroshima-u.ac.jp/cice/9-

1prakitipongnakamura.pdf

Susan Reid and Ginnie Denny (2003). A tutor’s guide: Assisting Learners Solve

Written Numeracy Problems. Published by Workbase from Robyn Pigozzo,

University of Southern Queensland. Retrieved January 22, 2010, from

http://www.workbase.org.nz/Resource.aspx?ID=205

42

Mestre, Jose (1989). Hispanic and Anglo Students' Misconceptions in Mathematics.

ERIC Clearinghouse on Rural Education and Small Schools Charleston WV.

Retrieved January 25, 2010, from

http://www.ericdigests.org/pre-9213/hispanic.htm

Drew Polly, Corey Lock and Barbara Bissell. Mathematical Understanding: Analyzing

Student Thought Processes while Completing Mathematical Tasks. Not

published. Retrieved January 25, 2010, from

http://math.unipa.it/~grim/21_project/21_charlotte_PollyLockBisselPaperEdit.p

df

43

APPENDIXES

44

Appendix 1:

NO. NAME GENDER RACE

1 HAMMAD ABDUL AZIM MALE MALAY

2 ISAAC NG ZHENG XIAN MALE CHINEESE

3 JOHN A/L ALBALAHEN MALE INDIAN

4 MOHD AL HAFEEZ BIN ZAKARIA MALE MALAY

5 MOHD IRSYAB FAYAZ BIN ARIF MALE MALAY

6 MUHAMAD SYAFIQ AZHAR MALE MALAY

7 MUHAMMAD IMRAN MOHD NAJEEB MALE MALAY

8 MUHAMMAD ISKANDAR B NORAN MALE MALAY

9 MUHAMMAD MUHSIN B SYED NAZIR MALE MALAY

10 MUHAMMAD NAZRUL B AZARSHAH MALE MALAY

11 MUHAMMAD NUR IZZUL IMAN MALE MALAY

12 MUHAMMAD QATADAH NOR HISHAMUDIN MALE MALAY

13 MUHAMMAD QAYYUM AZWA MD LAZIM MALE MALAY

14 MUHAMMAD REDUAN B MOHD ISARUDIN MALE MALAY

15 NEZAN BIN NASIR MALE MALAY

16 SATHISARAN A/L NATHAN MALE INDIAN

17 VEROZDLY MALUDIN MALE BUMIPUTERA

18 AZEERA ANSANGGOR FEMALE MALAY

19 HILAVARASI A/P SAITHAL FEMALE INDIAN

20 JALINAH JULING FEMALE MALAY

21 NUR NEELAM SARI BTE ASMAWI FEMALE MALAY

22 NURSYAQIRAH KASSIM FEMALE MALAY

23 NURUL ARIQAH FEMALE MALAY

24 NURUL HIDAYAH BT IBRAHIM FEMALE MALAY

25 NURUL SYAFIKAH BTE JASNI FEMALE MALAY

26 SHERILYN LOGATHAN FEMALE MALAY

27 SITI SHIRIN SHABIRA FEMALE INDIAN

28 SOFIAH BTE DUL AJIS FEMALE MALAY

29 SUSANNA MOGANASUNDRAM FEMALE INDIAN

Appendix 2

45

MONEY

Ahmad wrote his cheque “Eighty thousand ninety ringgit and fifty-sen.”What is the value of money in numeral?

Ahmad wrote the second cheque “Twenty thousand three ringgit and six sen.”What is the value of money in numeral?

Syafiq having loan RM 71 026 from Maybank to buy a new car, Toyota Vios. What is the value of money in words?

Sidiq having a loan RM 49008.10 to support his study in India.What is the value of money in words?

The first, second and third prize of the table tennis competition cost RM 2500, RM 1 500 and RM 750 respectively. Three categories involve which are single, double and group.Draw the situation.

Ramu brings three notes of RM 50, four notes of RM 10, one note of RM 5 and

46

Name: …………………………………................................ Year: ……………………………..

three notes of RM 1. Furthermore, his brother gives two coins of 50 sen and four coins of 10 sen to buy a sport shoe.Draw the situation.

Aminah bought three kilogram of sugar, five kilogram of cooking oil, three kilogram of flour and ten kilogram of rice. The prices of these groceries are RM 1.65, RM 2.40, RM 1.75, and RM 3.18 per kilogram respectively. How much money did she pay?

Tony saved RM 18.90 a week. How much money did he save in a year? Write his saving in words.

Timah buys a set of sofa which costs RM 6 079.20. If she pays by monthly installments for 2 years, how much does he need to pay each month?

Saidah has RM 18 722 in CIMB bank. Her husband has RM 24 541.90. They want to share their money equally among their nine children. How much money will each child receive?

47

Hamid collects RM 12.50 for each friend of his class for celebrating a birthday party. How many friends does he have if the party cost RM 375?

Sabri has 3 baskets of mangosteen. A basket contains 50 mangosteen. He then sells the mangosteen for RM 0.50 for each. How much money does he get if only 15 mangosteen are broken?

Julia has RM 2 890.48. Maria has RM 2 230 more than Julia. However Joseph has RM 789.20 less than Maria. How much money do they have altogether?

48

Appendix 3

Items R C T PS E F

1 2 10 6 0 2 0

2 1 8 5 0 4 0

3 0 8 3 0 1 0

4 1 9 3 0 3 0

5 3 16 3 0 2 0

6 2 18 4 0 1 0

7 0 10 7 1 2 1

8 3 7 6 2 0 1

9 1 12 8 1 1 1

10 2 14 8 1 0 2

11 1 20 4 0 0 0

12 2 18 5 2 0 0

13 2 15 9 0 0 0

14 0 12 8 2 3 1

15 3 14 5 3 1 3

Total 23 191 84 12 20 9

49

Appendix 4

Name: ………………………………………………………

Classification Typical Questions Errors

1. Reading

Please read the question to me.

(If you don’t know a word or number,

leave it out.)

2. Comprehension

(a) (Point to a word or symbol.) What

does this word/symbol mean?

(b) Tell me what the question is asking

you to do.

(What do you mean when you say …?)

3. Transformation Tell or show me how you start to find an

answer to this question.

4. Processing

skills

Show me how you get the answer.

Tell me what you are doing as you

work.

(Let student work on a piece of paper.)

5. Encoding

ability

Write down the answer to the question.

6. Careless Students spot own mistakes

50

Appendix 5

Errors Analysis

Reading Comprehend Transformation Process

Skill

Encoding Carelessness

2 9 7 4 1 0

51

Appendix 6

52

53

54

55

56

57