Abstract
This thesis reports on the development of number sense and mental computation proficiencies and their relationship to selected affective variables of Secondary 1 students, following an instructional intervention that lasted 10-weeks. Four government secondary schools and seven qualified and experienced mathematics teachers, and their 210 Secondary 1 mathematics students in Brunei Darussalam, generated the data for study. Two schools were randomly assigned as Treatment schools (with 118 students) and two other schools were the Control schools (with 92 students). A stratified random sample of 45 students drawn from the High (n=15), Middle (n=15) and Low (n=15) groups were interviewed before and after instructional intervention to determine the thinking strategies the students used in answering selected number sense questions. As part of this study, a group of Primary and a group of Lower Secondary mathematics teachers were also tested on their number sense proficiency.
Answers to the following research questions were sought in the study:
Research Question 1:
To what extent were the instructional interventions effective in helping students improve their number sense and mental computation proficiencies?
Related to this major research question are several sub-questions for which answers were first sought. These are as follows:
Research Question 1a: What were the levels of number sense and mental computation proficiencies, attitudes and confidence of the sample of Bruneian Secondary 1 students prior to the instructional intervention?
Research Question 1b: What were the levels of number sense, mental computation proficiencies, attitudes and confidence of the sample of Bruneian Secondary 1 students immediately after instructional intervention ended?
Research Question 1c: What were the levels of number sense and mental computation proficiencies, and confidence of the sample of Bruneian Secondary 1 students at the retention stage of the study (i.e., six weeks after instructional intervention ended)? Were any gains in number sense ability or mental computation proficiencies maintained six weeks after the intervention programme had been completed?
Research Question 1d: Were there gender differences in number sense and mental computation proficiencies at each of the pre-teaching, post-teaching, and retention stages of the study?
Research Question 1e: Did students’ number sense and mental computation proficiencies, attitudes and confidence, change in educationally significant ways over the period of intervention?
Research Question 1f: Were there differences in cognitive processing methods used to answer number sense questions between High, Medium and Low-ability students, at the pre-teaching, and post-teaching stages of the study?
Research Question 2
What are the relationships among number sense, mental computation and written computation proficiencies of Secondary 1 students at the post-teaching stage of the study?
Research Question 3:
What is the level of number sense proficiency of the sample of Upper Primary and Lower Secondary mathematics teachers in Brunei Darussalam?
Research Question 3a: Were there differences in number sense proficiencies between Primary and Secondary mathematics teachers in Brunei Darussalam?
Research Question 3b: Are there gender differences in number sense proficiencies of Primary and Lower Secondary mathematics teachers?
Pre- and post-intervention data from students’ responses to number sense, mental computation and mathematics attitude tests, and data from interviews with 45 interviewees (high-, middle-, and a low-performers), were analysed. Analyses of data show that the pre-instructional number sense proficiency of students at the beginning of secondary schooling was relatively low. The emphasis on developing of standard written algorithms in the primary years of schooling does not appear to have brought with it the practical understanding for dealing with whole numbers, decimals, fractions and percentages. The students’ exposure to these concepts in the primary schools has not been entirely successful. The study further shows that students were very much dependent on memorized procedures.
Post-instructional measures of number sense and mental computation proficiencies show that more students, particularly those in the High and Middle level sub-groups elected to use strategies said to reflect number sense. Students in the Low group made substantial gains in their number sense proficiency but in general continue to be rule dependent in solving number sense items. Post-intervention measures of Treatment students’ number sense proficiencies were higher than that of Control students who followed “normal” school mathematics instruction. Retention-test measures of Treatment students’ mental computation proficiencies were also higher than that of Control students. These measures together with the results of the gains score analysis, effect size values and their gain in confidence measures, suggest that the instructional intervention was effective in raising the number sense and mental computation proficiencies of the Treatment students. This study has shown that lower secondary mathematics teachers can lead students to use numbers in more meaningful and flexible ways and at the same time complete the existing school mathematics syllabus.
Students who were weak in number sense also displayed several errors and misconceptions in their understanding of mathematical concepts related to specific characteristics of number sense. A possible reason for this could be due to the over emphasis on the attainment of skill efficiency rather than conceptual understanding in school mathematics instruction.
A feature unique to this study was the measurement of number sense proficiency of Primary and Secondary mathematics teachers. The study has shown that the sample of Secondary mathematics teachers in the study displayed a higher level of number sense proficiency than their Primary school counterparts. Although no correlation was computed between teacher and student number sense proficiency, the results show that similar areas of weakness were evident among Secondary 1 students and the Primary school mathematics teachers.