CHAPTER 3
METHODOLOGY
Introduction
This chapter describes the methodology used in the study.
It comprises, pilot study, the implications of the pilot study results for the
research to test the validity and reliability of the paper-and-pencil Number
Sense test that is going to be used in this study. The researcher also includes
an overview of the research design, the population and samples used in this
study and the instruments for data collection, the procedure of the study and
the details of data analysis of the study. Later in the chapter, the researcher
tries to present the planned schedule of the main study.
The Pilot Study
Before the main study is
carried out, a pilot study will be conducted. The sample for pilot study are
pupils of primary four and primary five from one of the government primary
school (School P) in the same district. The paper-and-pencil Number Sense Tests
(NST) adapted and drafted from McIntosh et al., (1997) instruments will be
used. Result of the pilot study will then be analyzed and revised. At this
stage, only a few pupils will be informally interviewed about their responses.
The Implications of the Pilot Study
The findings of the
pilot study will be important for designing the main study. There are reasons
for conducting the pilot study:
Sample
The pilot study will be conducted in April 2006. Primary
four and primary five from school P will be involved. All primary four and
primary five pupils will be chosen for the pilot study. The paper-and-pencil
number sense tests will be different for both levels.
Reliability and Validity of the Instruments
The
paper-and-pencil Number Sense Tests; NST4 and NST5. Both he
paper-and-pencil Number Sense Tests (NST4 and NST5) adapted from McIntosh et
al., (1997) instruments will be used in this pilot study. The writer had prior
discussion with her supervisor regarding the suitability of the tests to be
used in Brunei context and the pupils’ academic level. The revised
paper-and-pencil number sense test instruments which will be used for the pilot
study were trialled with a small number of primary four and five pupils in
school T at the end of March. Both levels of pupils were asked to answer the
tests according to the Test Administration Protocol (Appendix A). Primary five
pupils commented that some of the words on the tests were difficult to
understand. Primary four pupils had no difficulty in understanding the
questions of NST4 since the translations were given in each question. The
Difficulty Index was done and some of the questions were modified and some of
the words were replaced.
The face and content
validity was obtained by distributing the adapted and modified NST4 and NST5 to
some of the Mathematics lecturers, Malay Language Lecturers as well as
experienced Mathematics and Malay teachers. They were asked to review the NST4
and NST5 whether the questions on the tests are suitable for their pupils and
to check the language used. Upon the reviewing, some of the questions were
modified. It is significant to take into account that the questions to be used
in the NST4 and NST5 are suitable for both levels of the pilot schools as well
as for the other four schools which will be involved in the main study.
The internal and external reliability coefficients of
NST4 and NST5 will be obtained from Cronbach alpha and by test-retest method.
The correlation coefficient will be acquired by calculating Pearson’s
product-moment correlation coefficient between scores of the same pupils on the
same paper-and-pencil tests at two different times. NST4 will be administered
to primary four pupils and NST5 will be conducted to primary five pupils. The
Difficulty Index will then be worked out. The administration will be carried out
during normal lessons in April 2006.
The Newman’s Interview
Procedure.
The interview will be based on the Newman’s interview procedure (Saman, 2000).
This instrument will be used to determine the source of difficulties and the
types of errors the pupils encountered in NST. As part of developing the
instrument, some mathematics lecturers from SHBIE will be asked whether the
questions on the instrument are suitable for the selected pupils to provide the
content and face validity, and to check the language used. The questions asked
on the Newman’s interview procedure will be translated to Malay in case there
will be pupils who needed help during the interviews. The author had a discussion with her supervisor
on the translation of the five questions.
The Teacher’s
Perception Questionnaires (TPQ). This instrument will be used to
investigate the teachers’ perceptions on number sense. The instrument will be
both in English and Malay. The TPQ will be shown to some of the mathematics
lecturers in SHBIE to determine the content and face validity to the teachers.
Language lecturers will involved to check the translation used in the
instrument.
The reliability of the TPQ will be obtained by
calculating the Pearson’s product-moment correlation coefficient between scores
of the same teachers on the same TPQ at two different times. The first
administration will be conducted in April 2006 and the second one will be
administered towards the end of April. Based on the responses of teachers from
TPQ administration, the Cronbach alpha reliabilities of TPQ1, TPQ2 and TPQ3
will be calculated.
Test Administration
The author herself administered the paper-and-pencil
Number Sense Test to primary four and five classes at School P on two different
days in early April and towards the end of April 2006. This instrument was
administered during normal lessons of the classes.
The Main Study
The methodology of the main study will be discussed in
this section with emphasis on outlining the main research design, population
and samples, description of instruments, data collection and procedures,
rationale for using the research methodology, intended analyses and planned
schedule of the research.
Design of the Research
In order to explore the
current level of the number sense amongst the primary school pupils in both
levels, a multiple methods will be used to collect data. For the main study
will use a paper-and-pencil test and will be supported by an interview.
Quantitative data will be gathered using the paper-and-pencil Number Sense Tests
(NST4 and NST5) of the selected pupils and a teachers’ perception questionnaire
(TPQ). The qualitative approach will involved interviewing a sample of selected
pupils in both levels in attempting to probe for evidence of how they come up
with the solutions in the NST given.
Population and Samples
Pupils from primary four
and primary five attending four government primary schools in one of the
districts in Brunei Darussalam in 2006 will be chosen for this study. The four
government schools will include two urban schools (School A and B) and two
sub-urban schools (School C and D). The schools’ academic background do not
differ much as the schools that will be selected based upon the schools’
results of Brunei public examination, the Penilaian Sekolah Rendah (PSR)
examination in the year 2005 which were above national target (> 85%). The
samples will not be randomly chosen. All pupils from primary four and primary
five classes of the selected schools will take part in the paper-and-pencil
number sense tests and only a selected few will be interviewed. Only those
pupils who have low performance (less than 30%) and high performance (more than
80%) in the test will be randomly selected for interview.
The number of pupils for
in the main study based on the estimated total of pupils for the whole schools
in one of the district in Brunei Darussalam by the end of January 2006. The NST
will be conducted for the pupils on the following table.
Schools Category |
Schools |
Primary 4 |
Primary 5 |
Total No. of pupils in
each school |
||
No. of Classes |
No. of Pupils |
No. of Classes |
No. of Pupils |
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Urban |
A |
3 |
49 |
3 |
58 |
107 |
B |
2 |
33 |
2 |
43 |
76 |
|
Semi-Urban |
C |
1 |
19 |
2 |
33 |
52 |
D |
2 |
29 |
2 |
36 |
65 |
|
Total |
4 |
8 |
130 |
9 |
170 |
300 |
Table 3. The pupils in the Main Study Sample
School A. This is a co-educational school. There are three
classes of primary four with a total of 49 pupils. There are three classes of
primary five with a total of 58 pupils.
School B. School B is a co-educational school. There are
two classes of primary four with a total of 33 pupils. There are two classes of
primary five with a total of 43 pupils.
School C. School C has only one class of primary four with
19 pupils and two primary five classes with 33 pupils.
School D. This school has two primary four classes with a
total of 29 pupils only. There are two classes of primary five with 36 pupils.
The total number of
pupils that will be in the sample of the main study is 300 pupils. There are
130 pupils in primary four and 170 pupils in primary five. There are 8 classes
of primary four and 9 classes of primary five pupils.
All teachers teaching
mathematics in these schools teaching at all levels (from primary one to
primary six) will also be required to answer the questionnaires. The
questionnaire is aimed to find out their perspectives towards number sense.
Description of Instruments
The instrument consists
of a paper-and-pencil test on number sense (adapted from McIntosh et al.,
1997), an interview to a sample of pupils that will be conducted in this study
and a Teacher’s Perceptions Questionnaire (TPQ) for all mathematics teachers
teaching in all the selected schools. The administration of the instruments
that will be used in this study will be conducted by the researcher with the
help of mathematics teachers in the schools involved.
The Pencil-and-Paper Test
The items for paper-and-pencil Number Sense test (NST) to
be used in this study will be selected from the Number Sense Item Bank (NSIB)
that was published by McIntosh (McIntosh et al., 1997). The instrument contains
various aspects of number sense to be solved consist of: (a) Whole Numbers -
Primary 4 and 5 (b) Fractions - Primary 4 and 5 (c) Decimals – Primary 5 only.
The categorization of the questions on the NST4 and NST5 are shown in Table 1
and Table 2 below. The instrument that is used in the study was developed on
the basis of discussion of the author with the supervisor, the experienced
primary school mathematics teachers and officers from Curriculum Development
Department (CDD) for comments and suggestions. These officers are familiar with
the upper primary mathematics syllabus. As the result of the discussion, the
Number Sense Test (NST) will be carefully modified and revised to suit the
Bruneian context. The test will provide a valuable source of information on the
present level of number sense of the selected pupils in this study.
Topics |
Q. No. |
Total |
Counting &
Computation – Whole Number |
1 - 8 |
8 |
Effect of Operations –
Whole Number |
9 - 14 |
6 |
Equivalent Expressions
– Whole Number |
15 - 22 |
8 |
Multiple
Representations – Whole Number |
23 - 24 |
2 |
Number Concepts –
Whole Number |
25 - 29 |
5 |
Multiple
Representations - Fractions |
30 - 36 |
7 |
Number Concepts -
Fractions |
37 - 40 |
4 |
Total |
40 |
Table 1. NST4
Topics |
Q. No. |
Total |
Counting &
Computation – Whole Number |
1 - 7 |
7 |
Effect of Operations –
Whole Number |
8 - 12 |
5 |
Equivalent Expressions
– Whole Number |
13 - 19 |
7 |
Multiplr
Representations – Whole Number |
20 |
1 |
Number Concepts –
Whole Number |
21 - 23 |
3 |
Counting &
Computation – Fractions |
24 - 26 |
3 |
Multiple
Representations – Fractions |
27 - 32 |
6 |
Number Concepts –
Fractions |
33 - 36 |
4 |
Counting &
Computation – Decimals |
37 |
1 |
Effect of Operations –
Decimals |
38 |
1 |
Equivalent Expressions
– Decimals & Mixed |
39 |
1 |
Multiple
Representations – Decimals & Mixed |
40 |
1 |
Total |
40 |
Table 2. NST5
The Interview
On the completion of
marking the paper-and-pencil Number Sense test (NST), the second stage of the
main study involved interviewing a sample of pupils in primary four and five in
each of the selected school. The interviews will be based on the Newman’s
interview procedure (Saman, 2000).
1. “Read the question to
me. If you don’t know a word tell me” – To identify READING error.
Baca soalan
ini. Jika ada perkataan yang awak tidak faham, beritahu saya.
2. “Tell me, what the
question asked you to do.” – To identify COMPREHENSION error.
Beritahu
saya, apakah yang dikehendaki / diperlukan oleh soalan ini.
3. “Now tell me what method
you used to find the answer. Why?” – To identify TRANSFORMATION error.
Sekarang
beritahu saya, apakah cara penyelesaian yang awak gunakan untuk mencari
jawapannya.
4. “Now, go over each step
of your working and tell me what you are thinking.” – To identify PROCESS SKILL
error.
Sekarang
teliti setiap langkah yang awak gunakan / buat, dan beritahu saya apa yang awak
fikir.
5. “Tell me what is the answer
to the question? Point to your answer.” – To identify ENCODING error.
Beritahu
saya apakah jawapan bagi soalan ini? Tunjuk jawapan awak.
The interview questions
are also translated to Malay for those pupils who needed help during the
interviews. In order to translate the five questions, the researcher discussed
with the lecturers and teachers who are native Malay speakers in the schools
selected.
The Questionnaire
All mathematics teachers
teaching in the selected schools will be given the questionnaire. The
researcher will develop the Teachers’ Perceptions Questionnaire (TPQ). The
questionnaire will be on teachers’ perception about number sense in
mathematics. This is to determine how far mathematics teachers are aware of the
existence of number sense in every mathematics topic taught in primary schools
pupils.
The questionnaire will
be piloted for its reliability before distributing to the teachers in the
respective schools. As a result of this process, there will be some alterations
in the design of the original questionnaire made.
Data Collection and Procedures
The schedule for
conducting the study will depend on the school teaching time. According to the
school calendar, the pupils are having four school terms in which there are
also four term breaks. The researcher will try to conduct the study in the
third school term which will be around the end of June as the pupils have
covered at least half the year’s school syllabus. The two urban schools are
within the reach of the researcher as the schools are close to the town area in
one of the districts in Brunei Darussalam. The two semi-urban are slightly far
away from the town area.
All the samples from the
selected government primary schools will be given the modified and revised NST
to solve. After their answers scripts are checked, a sample of pupils involved
in this study are interviewed. Questionnaires will also be distributed for the
mathematics teachers, teaching the classes to obtain their own attitudes and
perceptions towards number sense in primary school pupils.
The Paper-and-Pencil Test
The researcher will
carry out the paper-and-pencil Number Sense test (NST) for all pupils selected
in six of the government schools in one of the districts in Brunei Darussalam.
Two number sense test instruments (Appendix) will be constructed and utilised
in the study, one for each of two school levels. They will be given a 40-item
paper-and pencil test on number sense. The protocols for the administration of
the NST adapted from McIntosh (McIntosh et al., 1997) are detailed in Appendix
A. The administration of the test will begin by having pupils answering through
two practise items in the same format as those in the test itself. The practise
items will be put in the front page of the test. The test will be read by the
researcher to the pupils item by item once. The pupils will be given about 45
seconds to respond to each item in the test. The next item number will be read
every 45 seconds. Total administration time will be about 30 minutes. Pupils
will be required to write the answer only on the answer sheet provided. There
will be no written workings allowed to arrive at the solutions. The procedure
will be repeated until all questions have been completed.
The administered
paper-and-pencil Number Sense tests will be scored by the researcher by marking
each item either correct or incorrect. The marks will then be recorded in the
mark sheet developed by the researcher. A score of one will be given to every
correct answer and a zero for an incorrect answer. For the purpose of working out pupils’ level of number sense,
means, frequencies, percentages and total scores by area of schools, level of
schooling and gender will be computed and compared to see if there will be any
significant differences between these groups.
The Interview
Based on the
paper-and-pencil Number Sense test (NST) results, the researcher will call a
sample of pupils to be interviewed regarding their responses from all questions
in the NST. The pupils will be categorised and selected from the high achiever
(more than 80%) and low achiever (less than 30%) from the results of the test
administered before. Each pupil will be interviewed individually immediately
after the NST have been mark on the same day following the test. The pupils
will be taken from class to a quiet room or library in the school for
interview. The interview will be audio-taped and transcribed. The interviews
will be based on the Newman’s interview procedure used in Saman’s study (2000)
mentioned earlier. Two interview schedules, based on the Newman interview
protocol, have been prepared for the interviews with pupils (see Appendix and Appendix ). Although the interview schedule for pupils will be prepared,
additional questions might be asked during interviews, if and when this is
deemed to be necessary by the author.
The questions asked on
the Newman’s interview procedure will be translated to Malay for those pupils
who needed help during the interviews. The translation of the five questions
will discuss with the mathematics lecturers in SHBIE and experienced Malay
teachers (who are native Malay speakers). After the pupils have been asked to
read and explain the meaning of the questions, they will be asked to verbalise
the thinking which generated their answers. In this manner their level of
understanding of the NST will be probed.
The interview is aimed
to determine the source of difficulties and the types of errors they encounter.
Within the interview, further questions will be asked to probe for evidence of
how they go over each step to get the answers and to determine whether they
understand the concepts well or vice-versa. This also will help the researcher
to explore their level of number sense in doing the test.
The Questionnaire
The revised TPQ will be given to all mathematics teachers
in the selected schools. The researcher will distribute the questionnaire to
the teachers on the same day when the paper-and-pencil Number Sense test is
conducted. The items in the questionnaire specifically aimed to probe
mathematics teachers’ perceptions of number sense. The responses given by the
teachers on the questionnaire will be worked out using the computer software
SPSS version 14.
Rationale for using the research methodology
There
will be relations between the research questions and reasons for using the
instruments for data collection and analysis.
Research question 1.
What is the level of number sense possessed by primary 4
school pupils in Tutong district?
The
author tries to get the information through the paper-and-pencil Number Sense
Test, NST4. By analyzing pupils’ responses, it is possible to find out the
general level of number sense possessed by primary four.
Research
question 2.
What is the level of
number sense possessed by primary 5 school pupils in Tutong district?
The
author tries to get the information through the paper-and-pencil Number Sense
Test, NST5. By analyzing pupils’ responses, it is possible to find out the
general level of number sense possessed by primary five.
Research
question 3.
Are there any significance
differences in performance between primary four and primary five pupils?
To
answer this question, there is a need to collect data from question 1 and
question 2. By analyzing pupils’ responses from NST4 and NST5, it is possible
to find the significance differences in their performances between the two
levels.
Research
question 4.
Are there any
significance differences in performance between urban schools and semi-urban
schools?
To
answer this question, there is a need to collect data from question 1 and
question 2 from different school settings. By analyzing pupils’ responses from
NST4 and NST5, it is possible to find the significance differences in their
performances between the two settings.
Research
question 5.
What are mathematics
teachers’ perceptions of number sense in the selected schools?
The
author tries to get the information through the Teacher’s Perceptions
Questionnaires (TPQ). By analyzing teachers’ responses, it is possible to find
out the mathematics teachers’ perceptions of number sense in the selected
schools.
Intended Analysis
The analysis will be
conducted for the three instruments used in the study namely: a
paper-and-pencil Number Sense Test (NST4 and NST5), the Newman’s interview
procedure and the Teachers’ perceptions Questionnaires (TPQ). For the purpose
of finding out pupils’ level of number sense, frequencies, percentages and
total scores will be found. Mean scores of all primary four and five pupils,
schools’ area and of schools’ levels will also be computed and compared to see
if there will be significant differences between these groups.
The interview administered to the sample of pupils will
be transcribed. Since the interview will be structured based on Newman’s
interview procedure, the researcher will be able to get information on the
items that are relevant to the study through the analyses of the interviews.
This will be done for the purpose of ensure their way of working out the
responses on the NST conducted earlier.
The questionnaire used to probing mathematics teachers’
perceptions of number sense in learning and teaching mathematics will be
analysed using the computer software program SPSS version 14. The instrument
administered with the intention to discover primary mathematics teachers’
awareness of the existence of number sense in mathematics topics taught in
primary schools.
Planned Schedule of the Study
The planned schedule of
activities for this study is shown in Figure 3.1. The development and piloting
of paper-and-pencil Number Sense test (NST) and the questionnaire will be
conducted as part of researcher’s coursework requirement in the first half of
the year. Hence, the administration of the instruments used in this study will
be carried out on the second part of the year.
Activities |
2006 |
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Develop Questionnaire |
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Pilot Study for Questionnaire |
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Chapter 1, 2 and 3 write up |
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Defence proposal |
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NST administration |
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Interview administration |
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Questionnaire administration |
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Data Analysis |
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Report writing |
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Figure 3.1. Planned schedule of
activities for this study
Summary
The study that
will be conducted by the researcher aims to discover the present general level
of number sense on two schooling levels; primary four and primary five in
government primary schools. The research questions will lead the researcher in
whole process of administration of the instruments. A multiple research methods
will be applied in this study. A qualitative and quantitative approach will be
utilised to determine the effectiveness of the research.
It is hoped that this study will benefit various agents
especially the educators, administrators and mathematics teachers as well as
communities in order to improve the performance of mathematics in Brunei
Darussalam. The researcher is also hope that this study could answers all the
four questions listed before.