Understanding of Trigonometric Ratios in the Enhancement of Students’ Performance in Mathematics (Syllabus D)
Sarina Haji Yahya
Maktab Duli PMAMB
See Kin Hai, Hjh Jamilah bt Hj Mohd Yusof
Universiti
Abstract.
This study hopes to investigate the performance of Form 4 students in trigonometric ratios and its impact on Mathematics (Syllabus D). The study involved 194 Form 4 students in three government secondary schools in
It was found that that there was a significant positive correlation between the students’ understanding of trigonometric ratios and their performance in Mathematics (Syllabus D). The understanding of trigonometric ratios may have helped to improve students’ performance in Mathematics (Syllabus D). The type of school setting was found to significantly influence the performance of Form 4 students’ performance in trigonometric ratios.
Background and Rationale
The essential basis for the economic and social well being of a country lies in an understanding by its people of basic mathematical, scientific and technological knowledge. This is acknowledged in Brunei Darussalam by the importance given to mathematics and science in the school curriculum and the fact that nearly all students are entered for GCE O-level in mathematics.
The Ministry of Education has made trigonometry a core component of the secondary mathematics curriculum for all students in the lower and the upper secondary classes. Each year, there is at least one question on trigonometry in the PMB mathematics examination and in the Mathematics (Syllabus D) examination. Trigonometry, literally the measurement of triangles, has a history going back as far as ancient
Secondary mathematics teachers in Brunei Darussalam have often expressed their concern over their students’ poor performance on tests in trigonometry. Most students find the topic of trigonometry as very difficult to understand and the questions are difficult to answer. Trigonometry is undoubted one of the most difficult topics within mathematics and yet, can also to be, very useful (Tapson, 1984). It is of interest to find out if the poor performances in PMB mathematics and GCE O-level Mathematics (Syllabus D) were due to the lack of understanding of certain topics in mathematics such as trigonometry. Considering the importance of trigonometry and the difficulties faced by students in understanding this topic, it was hoped that the investigation which the researcher carried out would reveal the true picture about the level of understanding of Bruneian students on a core topic in trigonometry—trigonometric ratios and its effectiveness in enhancing the students’ performance in Mathematics (Syllabus D).
Three types of school setting were involved in this study: single-sex all girls’ school, single-sex all boys’ school and a co-educational school. The literature with respect to the effects of single-sex and co-educational schools on performance in mathematics is equivocal. It was reported that females from single-sex schools did significantly better than males from single-sex schools (Blithe, Clark, Forbes, Robinson, and Whitwell, 1990). This was also earlier found by Rowe (1988) who claimed strong support for single-sex classrooms especially for girls who performed better in mathematics. However, result from Blithe et al. (1990) study also suggested that there was no particular evidence in their study to support the view that females and males taking mathematics do well in single-sex schools. They found the difference between the performance of female students in co-educational schools and in single-sex schools were not significant. For males, they also found that there was also no significance difference between performance for those from single-sex or co-educational schools. Dunlap (2000) concluded his master study on single-gender mathematics classrooms that there was no statistical significance in girls’ achievement between a single-gender classroom and a co-educational classroom. Marsh and Rowe (1996) also found in their study that the school type had relative little effect upon performance and indeed found limited support for single-sex school for boys. It was the researcher’s intention to find out if there were any significant differences in the performance of students in trigonometric ratios in these three types of school setting.
Methodology
The Research questions
The research was designed to answer the following research questions:
1. Did the understanding of trigonometric ratios help to improve students’ performance in Mathematics (Syllabus D)?
2. Did the performance of the students in trigonometric ratios have been affected by the type of school setting?
Sample
The study involved 194 Form 4 O-level students in three government secondary schools in
The study made use of the intact classes already set by the schools. The two classes of Form 4 from each school were assigned to be the control and experimental groups. The study made use of the intact classes already set by the schools according to certain criteria. These criteria are generally based on the subjects taken by each class. Besides taking the same core subjects, both the control and experimental groups were taking same subjects options.
Research Instruments
Two pencil-and-paper tests—the Trigonometric Ratios Test (TRT) and the Mathematics (Syllabus D) Test (MSDT)—were used to generate pre-test and post-test performance data. The tests were administered to both the control and experimental groups at the end of June and August 2004. The pre-tests on TRT and MSDT were administered just before the mathematics teachers did the treatment activities on trigonometric ratios and before lessons on trigonometry were given. The two post-tests on TRT and MSDT were administered to both groups at the end of the treatment period. Summary comments on each instrument are now provided.
The Trigonometric Ratios Test (TRT). Although this test was made up of ten main questions, there were actually 25 sub questions altogether. The test consisted entirely of pencil-and-paper questions on trigonometric ratios that had appeared in recent PMB and O-level Mathematics examination papers. The internal and external reliability coefficients of TRT were obtained by test-retest method and from Cronbach alpha. They were obtained by calculating Pearson’s product-moment correlation coefficient between scores of the same students on the same achievement tests at two different times. The TRT was first administered to the Form 4 pilot sample in mid February 2004. The same tests were administered again to the same students one month later. The correlation coefficient of the TRT based on the responses of 32 students in the pilot sample was calculated to be 0.80 and its Cronbach alpha reliability was calculated to be 0.95
The Mathematics (Syllabus D) Test (MSDT). This test comprised 15 questions, all of which were selected from recent O-level examination papers. The questions were selected so that they represented Form 4 Mathematics (Syllabus D) topics. The internal and external reliability coefficients of MSDT were also obtained by test-retest method and from Cronbach alpha. The MSDT was also first administered to the Form 5 pilot sample in mid February 2004. The same tests were administered again to the same students one month later. The correlation coefficients of the MSDT based on the responses of 29 students was 0.87 and its Cronbach alpha reliability was calculated to be 0.87.
As part of the process of developing the TRT and MSDT, five Form 4 O-level mathematics teachers at School A, School B and School C were asked whether the questions on the tests were suitable for their students. The five teachers had more than ten years teaching experience. The teachers thought that all the questions on the TRT and MSDT were suitable for assessing the students’ understanding on trigonometric ratios and Mathematics (Syllabus D).
The Intervention (Treatment Activities)
For six weeks, at the beginning of every normal mathematics lesson, the students in the experimental group were given a short revision on trigonometric ratios in order to improve their understanding on trigonometric ratios. These treatment activities were prepared by the researcher and were closely followed in accordance with the expectation laid out in the Syllabus D mathematics syllabus (UCLES, 2003). The treatment activities for the experimental groups were conducted by their own mathematics teachers. None of the mathematics teachers in the control groups taught the experimental groups.
Summary of findings and Answers to the Research Questions
In this section brief answers are given to the two research questions.
Research Question 1: Did the understanding of trigonometric ratios help to improve students’ performance in Mathematics (Syllabus D)?
Data relevant to this question were obtained from the performance of the students in Pre- and post-tests of TRT and MSDT. The Analysis of Covariance (ANCOVA) was employed to analyse the results of the post-test of TRT using TRT pre-test as the covariate. The ANCOVA results were then used to compare the performance of the control and experimental group within each school and between the three schools. A similar procedure of data analysis was used for the scores of MSDT. A Pearson product-moment correlation was then carried out on the scores of the post-tests of TRT and MSDT to see the relationship between the performance of students in TRT and their performance in MSDT.
Results of ANCOVA on the TRT and MSDT
Table 1 shows there was a significant relationship between the pre-TRT mean scores and post-TRT mean scores of both the control and experimental groups in all the three schools as well as when combining the three schools (p < .05).
Table 1
Analysis of Covariance for the Post-TRT Scores Adjusted by Pre-TRT Scores
| School | df | Mean Square | F ratio |
| All Schools Pre-TRT Group | 1 1 | 12733.95 1632.45 | 111.16* 15.14* |
| School A Pre-TRT Group | 1 1 | 4620.75 275.22 | 75.64* 4.51* |
| School B Pre-TRT Group | 1 1 | 4804.75 1661.05 | 37.52* 12.97* |
| School C Pre-TRT Group | 1 1 | 1112.22 398.32 | 13.41* 4.80* |
*p < .05
Table 1 also indicates that when the pre-TRT score was statistically controlled, there was main effect (p < .05) between the control and the experimental groups for all the three schools. Thus, we can say that the treatment given to the experimental group may have influenced the post-TRT mean scores of the students. Entries in Table 1 also show that the experimental groups of School A, School B and School C improved significantly better than their control groups when the pre-TRT scores were statistically controlled.
Table 2 shows there was a significant relationship between the pre-MSDT mean scores and post-MSDT mean scores of both the control and experimental groups in all the three schools as well as when combining the three schools (p < .05).
Table 2
Analysis of Covariance for the Post-MSDT Scores Adjusted by Pre-MSDT Scores
| School | df | Mean Square | F ratio |
| All Schools Pre-MSDT Group | 1 1 | 4949.56 754.64 | 193.87* 29.56* |
| School A Pre-MSDT Group | 1 1 | 1023.59 47.76 | 51.74* 4.81* |
| School B Pre-MSDT Group | 1 1 | 786.49 528.48 | 31.74* 21.33* |
| School C Pre-MSDT Group | 1 1 | 913.89 94.94 | 52.76* 5.48* |
* p < .05
Table 2 also suggests that when the pre-MSDT score was statistically controlled, there was main effect (p < .05) for group for the all the three schools. Thus, we can say that the treatment given to the experimental group may have influenced the post-MSDT mean scores of the students. If we consider school by school, entries in Table 2 show that there was main effect for treatment at School A, School B and School C. The experimental groups of School A, School B and School C improved more than the control groups when the pre-MSDT scores were statistically controlled (p < .05).
Correlations Between the TRT and the MSDT
The Person product-moment correlation was used to analyse the correlation that describes the relationships between students’ performance in post-TRT and their performance post-MSDT. The correlation coefficient obtained from 194 pairs of scores, is a positive correlation significant at the 0.05 level (r = .67, df = 192, p < .05), indicating a significant positive relationship between the performance of students in post-trigonometric ratios and their performance in post-Mathematics (Syllabus D). This result may have suggested that students who did well in trigonometric ratios also did well in Mathematics (Syllabus D). Entries in Table 3 also present the correlations between the post-TRT scores and the post-MSDT scores of School A, School B and School C. Significant positive relationships were found between the post-TRT scores and the post-MSDT scores of each school.
Table 3
Correlation Between the TRT Scores and the MSDT Scores of Each School and Whole Sample
| Schools | Correlation | Significance |
| School A School B School C | .59* .66* .54* | .000 .000 .000 |
| | .67* | .000 |
*Correlation is significant at p <>
Research Question 2: Did the Performance of the Students in Trigonometric Ratios Have Been Affected by the Type of School Setting?
The effect of the type of school setting on the performance of students in TRT was analysed using Chi-square tests of independence. The students were categorized into high performer, medium performer and low performer based on the scores of their post-test scores of TRT. The differences in students’ performance (high, medium and low) were used to determine if their differences may be affected by the different types of school setting.
Preliminary observation in Table 4 indicates that the observed values (Count) and the expected values (Exp. Count) are not similar. The greatest discrepancies are for School A (medium performer), School B (low performer) and School C (high performer). School C which is a co-educational school had 30 high performers of post-TRT compared to School A and School B. School A and school B had 16 and 6 high performers respectively (See Table 4). School A which is a single-sex girls’ school performed averagely and School B which is a single-sex boys’ school had performed lowly in trigonometric ratios.
Table 4
Type of School Setting by Performance in Post-TRT Crosstabulation
| Type of School setting | High Performer | Medium Performer | Low Performer | Total | |
| A | Count Exp. Count Residual | 16 18.2 -2.2 | 40 32.9 7.1 | 12 16.8 -4.8 | 68 68.0 |
| B | Count Exp. Count Residual | 6 16.4 -10.4 | 26 29.6 -3.6 | 29 15.1 13.9 | 61 61.0 |
| C | Count Exp. Count Residual | 30 17.4 12.6 | 28 31.5 -3.5 | 7 16.1 -9.1 | 65 65.0 |
| Total | Count Exp. Count | 52 52.0 | 94 94.0 | 48 48.0 | 194 194.0 |
Even without looking at the chi-square statistics we would anticipate the observed values and expected values differ significantly (that is type of school setting and performances in post-TRT in this sample are dependent of each other). The results in Table 5 support this observation with a high Chi-square value (37.556) and p < .05. Measures of association are high and approach to significance. The type of school setting may have affected the difference in the performance of high, medium and low performers of trigonometric ratios.
Table 5
Chi-Square Analyses
| | Value | df | Significance |
| Pearson Chi-Square Likehood Ratio N of valid cases | 37.556a 36.789 194 | 4 4 | .000 .000 |
a 3 cells (30%) have expected count less than 5. The minimum expected count is 1.95
Conclusions
Performances of the Form 4 students on the two pre- and two post-performance tests were compared, using quantitative procedures. To see whether the understanding of trigonometric ratios can enhance the students’ performance on Mathematics (Syllabus D), treatments on trigonometric ratios were conducted on the experimental groups. Experimental groups were given short lessons on trigonometric ratios for six weeks in order to improve their understanding on trigonometric ratios. The treatment on trigonometric ratios given to the experimental groups may have influenced significantly the post-TRT mean scores of the students. This influenced could be seen from the significant gains made by the experimental groups as compared to the control groups for the pre- and post-tests on trigonometric ratios. The ANCOVA results on TRT also revealed that the treatments given to the experimental groups may have significantly influenced the post-TRT mean scores of the students when the pre-TRT scores were statistically controlled. The gain between the pre- and post-MSDT mean scores and the ANCOVA results on MSDT also revealed that the experimental groups improved significantly more than the control group.
The two post-tests were correlated together. It was found that there was a significant positive correlation between the students’ understanding of trigonometric ratios and their performance in Mathematics (Syllabus D). The understanding of trigonometric ratios may have enhanced the students’ performance on Mathematics (Syllabus D). This finding is similar to what suggested by the British Committee of Inquiry into the Teaching of Mathematics (Cockcroft, 1982) and the Curriculum and Evaluation Standards for School Mathematics (NCTM, 2000). They suggested it was important to help children to learn mathematics with understanding. The students needs more time to understand the knowledge and skills in trigonometric ratios in order to perform well in trigonometric ratios. The students may improve their performance in Mathematics (Syllabus D) if they understand trigonometric ratios.
The performance of the high performing, medium performing and low performing students in trigonometric ratios may have been affected significantly by the type of school setting. The study revealed that co-educational school had many high performers in trigonometric ratios as compared to the single-sex girls’ or single-sex boys’ school. In Brunei Darussalam male and female students in co-educational schools are more dedicated to their studies compared to their peers in single-sex schools. This could be due to the challenge they feel with the opposite sex.
References
Blithe, T.,
Cockcroft, W. H. (1982). Mathematics counts. Report of the Committee of Inquiry into the Teaching of Mathematics in Schools.
Dunlap, C. E. (2000). An examination of gender differences in today’s mathematics classroom: Exploring single-gender mathematics classrooms. M.Ed dissertation,
Ellis, J. (1990). Towards better trigonometry teaching. In K. Clements (Ed.). Whither mathematics? (pp. 452–453).
Marsh, H. W., & Rowe, K. J. (1996). The effects of single-sex and mixed mathematics classes within a co-educational school: A reanalysis and comment. Australian Journal of Education, 40, 147–162.
National Council of Teachers of Mathematics (NCTM) (2000). Principles and standards for school mathematics.
Rowe, K. J. (1988). Single-sex and mixed-sex classes: The effects of class on student achievement, confidence and participation in mathematics. Australian Journal of Education, 32, 180–202.
Tapson, F. (1984). Get out your tables! Mathematics in School, 13(4), 13–17.
Authors
Dr See Kin Hai and Dr Hjh Jamilah Bt Md Yusof both lecture in the Department of Science and Mathematics Education, Universiti
Dy Sarina Hj Yahya, teaches in Maktab Duli and is very experienced in Mathematics Education.
No comments:
Post a Comment