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:

  1. To test and validate the paper-and-pencil Number Sense Test questions (NST4 and NST5) that will be used in the development of valid and reliable paper-and-pencil Number Sense Tests (NST4 and NST5) for the main study. Cronbach alpha reliabilities for the two tests will be calculated from the data generated. Any questions which are considered confusing and misleading will be identified and rectified. The difficulty index of both tests will be computed. The test items with very high and very low percentages of correct and incorrect responses will be modified or replaced with new items for the main study.
  2. To develop reliable paper-and-pencil Number Sense Test questions (NST4 and NST5) through test-retest method. The correlation coefficient will be calculated between scores of the same pupils on the same tests at two different times.
  3. To develop valid and reliable Teacher’s Perception Questionnaires (TPQ) that may suitable for administration to all teachers teaching mathematics in the four schools teaching at all levels (primary 1 to primary 6). The instrument is in the form of Likert scales.

 

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

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

J

F

M

A

M

J

J

A

S

O

N

D

Literature search / review

 

 

 

 

 

 

 

 

 

 

 

 

Develop NST

 

 

 

 

 

 

 

 

 

 

 

 

Develop Questionnaire

 

 

 

 

 

 

 

 

 

 

 

 

Pilot Study for NST

 

 

 

 

 

 

 

 

 

 

 

 

Pilot Study for Questionnaire

 

 

 

 

 

 

 

 

 

 

 

 

Chapter 1, 2 and 3 write up

 

 

 

 

 

 

 

 

 

 

 

 

Defence proposal

 

 

 

 

 

 

 

 

 

 

 

 

NST administration

 

 

 

 

 

 

 

 

 

 

 

 

Interview administration

 

 

 

 

 

 

 

 

 

 

 

 

Questionnaire administration

 

 

 

 

 

 

 

 

 

 

 

 

Data Analysis

 

 

 

 

 

 

 

 

 

 

 

 

Report writing

 

 

 

 

 

 

 

 

 

 

 

 

Submission of Dissertation

 

 

 

 

 

 

 

 

 

 

 

 

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.