Factors affecting Form 4 students’ level of knowledge of solving quadratic equations
SEE KIN HAI AND NORIDAH HAJI IBRAHIM
Universiti Brunei Darussalam; Sekolah Menengah Sayyidina Ali
khsee@shbie.ubd.edu.bn
ABSRACT: The purpose of this study is to investigate the factors affecting the knowledge of 187 Form 4 students on the topic of solving quadratic equations at two secondary government schools in Belait district of Negara Brunei Darussalam. Solving quadratic equations is one of the sub-topics taught in O-level Mathematics Syllabus D subject. This is part of a larger study where investigations in this study will also look into other factor such as students’ confidence that might affect or influences the students’ achievement in solving quadratic equations.
A multiple perspective research design was used. Two instruments were used to collect data which included the Solving quadratic equations test (QE TEST) and Solving quadratic equations Confidence questionnaire. Qualitative and quantitative data were collected from six vantage points. In this paper data from only two vantage points; pre- and post-teaching pencil-and-paper test performance scores and post-teaching student confident scores are reported and analysed. From the stepwise multiple regression analysis it was shown that the component of Solving Quadratic Equation using formula was the highest overall contribution to the students’ achievement of solving quadratic equation. This indicates the students successfully using the quadratic formula to solve the quadratic equations appropriately. One-way ANOVA analysis produced the results that there were significant differences in the post-test mean total scores existing between the different confidence levels of students. From the results of the correlational analysis it was reported that the students’ confidence levels correlated significantly with their achievement scores obtained on the performance test.
Introduction
Solving Quadratic equations is one of the sub-topics in Solutions of equations and inequalities and is taught in Mathematics Syllabus D. Four upper secondary, Forms 4 and 5, students were taught according to the O-level mathematics courses of the University of Cambridge Local Examination Syndicate (UCLES), and the algebra components of these courses are carefully defined and examined.
Table 1 gives a summary of the BGCE O-level (October/November) Mathematics Syllabus D results of all the candidates in Brunei for the last five years (Jabatan Peperiksaan, 2006).
Table 1: The BGCE O-Level (October/November) Mathematics D Results from 2001 to 2005
Total no.
Year No. of Distinction Credit Pass of O-Level Fail
Candidates passes
Grade A Grade B Grade C Grades Grades
(%) (%) (%) D-E(%) A-C(%) U(%)
2001 3863 5.15 10.15 15.20 30.42 30.49 39.09
2002 3934 5.80 11.46 14.67 29.36 31.93 38.71
2003 3946 5.58 9.20 13.40 31.60 28.18 40.22
2004 4165 5.71 11.72 13.64 30.71 31.07 38.22
2005 4288 4.71 10.98 15.56 32.0 31.25 36.75
It is of interest to find out if the poor performance in BGCE O-Level Mathematics D was due to the lack of understanding in certain topic in Mathematics D such as solving quadratic equations and the use of quadratic equations in solving other related questions.
Recent studies by Noridah (1999) and Radiah (1998) have generated data suggesting that algebra components of the O-level curricula are especially problematic for many students in Form 4, 5 and 6 in secondary schools in Brunei Darussalam. This finding has significantly shaped the thinking of the researcher. It was also an interest to the researcher in confirming or disconfirming the findings done in the previous studies (see, e.g., Lim, 2000; Noridah, 1999; Radiah, 1998)on students’ performance in the component of algebra-solving quadratic equations. It was expected that the data from the study would provide a check on the amount and quality of students’ knowledge in solving quadratic equations. Part of the investigations in the study would also look into other factors that might affect or influence the students’ achievement in solving quadratic equations.
Background of the Study
This paper reports on an aspect of an investigation on students’ knowledge and achievement of a core component of algebra in solving quadratic equations from two Form 4 O level government secondary schools in Belait District during the 2006 school year. Four intact classes from each school were selected. Both schools were located in the same area and their students were from the same urban background. A qualitative and quantitative research methodology will be used. Both schools, school A and school B, which were involved in the study, are all secondary (lower and upper) government schools.
Literature Review
Learning of Algebra
In some countries, however, not all secondary students study algebra, and of those that do, many study it for one year only. One such country is the United States of America, but there are many U.S. politicians and educators who are unhappy about that (see, e.g., Moses, 2000).
In Brunei Darussalam, there have been many research studies carried out into the teaching and learning of algebra in schools. Clements (1999a) provided evidence from his observations on the teaching of algebra in Brunei Darussalam that students are taught to solve equation mechanically, and many fail to realize that the numerical answers that they will obtain if substituted into the original equations. Lim (2000) who studied on the teaching and learning of algebraic equations and factorization in O level mathematics reported that most students did not really like algebra and they did not enjoy algebra classes. The confidence data revealed that when the students were attempting algebra questions, even questions of a kind that had been dealt with in earlier years, most of them were not sure whether answers they gave were right or wrong. He also concluded from the evidence of pre-test on Algebra that many of the students who do well on the PMB mathematics examination do not seem to have grasped even the most elementary aspects of algebra. Also, through his findings, it shows that the traditional drill and practice method of teaching and learning, and the “examination-oriented” organization of schooling, are not generating the quality of student understanding, and the level of performance in examinations, that are desired. Radiah’s (1998) study of the performance of 327 Form 4 students in 9 Secondary schools in Brunei Darussalam strongly suggested that there is a problem with the teaching and learning of algebra in secondary schools of Brunei Darussalam.
Thomas and Tall (2001) distinguished, among other things, between “algebra as generalised arithmetic” and “manipulation algebra”, and commented that students who completed secondary education were usually able to substitute values correctly for variables in expressions and equations, and were able to interpret variables in symbolic and graphical contexts. However, student thinking in such contexts appeared to be dominated by a need to achieve procedural mastery, and usually there was no guarantee that relational understanding was achieved.
Stacey, Chick and Kendal’s (2004) landmark volume on The Future of the Teaching and Learning of Algebra provided commentary from scholars from numerous countries on many aspects of algebra education, but no attention was given to the cognitive challenges faced by students trying to solve quadratic equations.
Although many researchers (e.g., Warren & Pierce, 2004) have found that the concept of a variable is central to algebra, the lack of an adequate research base with respect to the teaching and learning of quadratic equations has meant that peculiarities associated with variables in quadratic equations, and in particular with the effects of these on student learning, have remained hidden. Vaiyatavutjamai (2004) stated that the best discussion of the teaching and learning quadratic equations is the U.K. report on the teaching of algebra in schools originally written between 1929 and 1933 by a Committee of the Mathematical Association (1962).
Attitudes Towards Mathematics
Studies have shown that promoting positive attitudes towards mathematic become an important objective in teaching mathematics. The term attitudes have included various types such as self-concept, confidence in mathematics, anxiety in mathematics and enjoyment in mathematics (Klum, 1980; Reys, 1980 & 1984; Leder, 1987; Khoo & Veloo, 1990).
Alrwais (2000) examined the relationship among the factors students’ attitude toward learning mathematics, students’ mathematical creativity and students’ school grades and their effect on achievement in mathematics. He found out that that the best predictor was the students’ attitude toward learning mathematics. McLeod (1992) noted that students’ attitudes play a central role in mathematics achievement. Khoo and Veloo (1996) stated that a positive correlation between mathematics achievement and the affective variables, beliefs and attitudes. Findings from their study, students with a high level of mathematics self-concept, enjoyment of mathematics and confidence in learning mathematics but with a low level of mathematics anxiety tend to have a high level of mathematics achievement.
Fadzil (1998) found that there was a positive correlation between achievement and confidence in learning mathematics, followed by enjoyment in learning mathematics, perceptions on the importance of mathematics and interest in mathematics.
Research Questions
It is hoped that the study would answer the following research questions:
- Did the new knowledge acquired by the students after the lessons on solving quadratic equations in Form 4 influence the students’ achievement in solving quadratic equations?
- Is there any relationship between the Form 4 students’ confidence in answering questions on solving quadratic equations and the overall new knowledge acquired by the students after the lessons in solving quadratic equations?
Methodology
Research design
A multiple perspectives research design was employed in this exploratory study. A combination of qualitative and quantitative methods was used in collection of data. The quantitative data were collected using achievement test (QE TEST) and Confidence Questionnaire. The qualitative approach involved interviews with students of the selected schools and retrieval of official documents. The researcher had interviewed six students consisted of two students who were low achievers, two students who were medium achievers and two who were high achievers. All the post-performance data on the achievement tests were needed in order to determine the range of marks so as to categorise the students into high, medium and low levels of achievement based on the whole sample.
The Sample of Students in the Study
The data of this research was collected from two secondary schools in Belait district that would be involved in the study. The students from these schools were presently studying in Form 4 and learning the O-level Mathematics D subject. The total number of students in the sample is 187 students. Four intact classes from each school were selected. The two schools were selected to contribute to the sample of students in the study as both schools were located in the same area and their students were from same urban background.
Instrumentation of the Study
One of the instruments, Solving Quadratic Equations Test (QE TEST) was the achievement test comprising of 18 questions. Another instrument in the form of “Likert Scales”, Quadratic equations Confidence Scale was attached to the achievement test.
Solving Quadratic Equations Test (QE TEST). This test consisted of 18 pencil-and-paper questions covered the different types of method in solving quadratic equations having the last two questions consisting of problem solving questions, as outlined in Table 3. Based on the various types of questions structures, the researcher could get an insight of how the students managed to solve the quadratic equations.
Table 2: Methods of solving quadratic equations in Solving Quadratic Equations TEST (QE TEST)
Methods Questions
Null factor 1 and 2
Simple factorisation and Null factor 3 and 4
Square root method 5 and 6
Factorisation and Null factor 7, 8, 9 and 10
Expansion, Factorisation and Null Factor 11, 12 and 13
Quadratic Formula 14, 15 and 16
Problem Solving 17 and 18
Solving Quadratic Equations Confidence Scale. The scales, in the form of “Likert scales”, were designed to measure the students’ confidence in answering questions on the Solving Quadratic Equations Test (QE TEST). The maximum and minimum possible scores for the Solving Quadratic equations Confidence Scale were 135 and 27 respectively.
Solving Quadratic equations Confidence Scale is similar to the scales used by Lizawati (2004), Zurina (2003), Khoo (2001) and Lim (2000). After a student responded to a question on Solving Quadratic equations Test (QE TEST), he/she is required to indicate the level of confidence he/she has in his/her answered by putting a tick in one of the five columns. The five columns have headings “I’m certain I’m right,” “I’m think I’m right,” “I’ve got a 50-50 chance of being right,” “I think I’m wrong” and “I’m certain I’m wrong.” A response of “I’m certain I’m right,” was allocated a score of 5, “I’m think I’m right,” was allocated a score of 4, “I’ve got a 50-50 chance of being right,” was scored as 3, “I think I’m wrong” a score of 2 and “I’m certain I’m wrong” a score of 1.
Results and Discussions
Research Question 1: Did the new knowledge acquired by the students after the lessons on solving quadratic equations in Form 4 influence the students’ achievement in solving quadratic equations?
Using the stepwise multiple regression analysis would be able to generate the answer to the first research question. In stepwise multiple regressions, the independent variables as predictors (prediction variables) are entered one step at a time to pick out the best predictor that makes a useful contribution to the overall prediction. For this study, the analysis would indicate which knowledge components significantly predict the overall gain of knowledge (achievement of Solving Quadratic Equations) of the Form 4 students.
The inter-correlations between of knowledge of solving quadratic equations variables and the post-test scores variable are given in Table 3. The students’ overall post-test mean total scores on each knowledge component were also given. It can be seen that the components correlate with each other and with the post-test total scores at p < .05 level.
Table 3: Means, Standard Deviations and Inter-correlations Between the Components of Knowledge of Solving Quadratic Equations Variables and the Total Post-test Scores Variable
| Variable | K1 | K2 | K3 | K4 | K5 | K6 | K7 | Post-test Mean | SD |
| 1 | - | - | - | - | - | - | - | 1.57 | 1.562 |
| 2 | .585 | - | - | - | - | - | - | .89 | 1.307 |
| 3 | .275 | .331 | - | - | - | - | - | .65 | .882 |
| 4 | .408 | .449 | .384 | - | - | - | - | 1.70 | 2.384 |
| 5 | .277 | .353 | .432 | .656 | - | - | - | 1.48 | 2.067 |
| 6 | .318 | .262 | -.028 | .358 | .196 | - | - | 2.65 | 3.916 |
| 7 | .175 | .169 | .022 | .207 | .156 | .537 | - | 1.20 | 2.554 |
Note: Correlation is significant at the 0.05 level. K1- Solving quadratic equations using null factor; K2- Solving quadratic equations using simple factorisation and null factor; K3- Solving quadratic equations using square root method; K4- Solving quadratic equations using factorisation and null factor; K5- Solving quadratic equations using expansion, factorisation and null factor; K6- Solving quadratic equations using quadratic formula; K7- Problem solving
Entries in Table 4 list the results summary of the stepwise multiple regression analysis. The students’ scores on solving quadratic equations using formula knowledge component which accounted for 57.9 percent of the variance in the post-test total scores, entered the equation first. This component of knowledge was highly significant, as indicated by the significant F Change – value.
Table 4: Stepwise Multiple Regression Analysis to Predict Post-test Total Scores
| Predictor Variable | Knowledge Entered | Combination Knowledge | R Square | R Square Change | F Change | Sig. F Change |
| 1 | K6 | K6 | .579 | .579 | 254.393 | .000 |
| 2 | K4 | K6 & K4 | .837 | .258 | 292.551 | .000 |
| 3 | K7 | K6, K4 & K7 | .895 | .058 | 100.843 | .000 |
| 4 | K2 | K6, K4, K7 & K2 | .945 | .050 | 163.380 | .000 |
| 5 | K5 | K6, K4, K7, K2 & K5 | .976 | .031 | 232.608 | .000 |
| 6 | K1 | K6, K4, K7, K2, K5 & K1 | .994 | .018 | 538.800 | .000 |
| 7 | K3 | K6, K4, K7, K2, K5, K1 & K3 | 1.000 | .006 | 19092.433 | .000 |
Note: K1- Solving quadratic equations using null factor; K2- Solving quadratic equations using simple factorisation and null factor; K3- Solving quadratic equations using square root method; K4- Solving quadratic equations using factorisation and null factor; K5- Solving quadratic equations using expansion, factorisation and null factor; K6- Solving quadratic equations using quadratic formula; K7- Problem solving. The mean difference is significant at the .05 level.
The second prediction variable entered was the knowledge on solving quadratic equations using factorization and null factor (K4) which contributed 25.8 percent. The knowledge on Problem solving (K7) contributed 5.8 percent, whereas the knowledge on solving quadratic equations using Simple factorization and Null factor (K2) contributed 5.0 percent. The component knowledge on solving quadratic equations using expansion, factorization and null factor (K5) contributed 3.1%, the knowledge on solving quadratic equations using null factor (K1) contributed 1.8% and the knowledge on solving quadratic equations using Square root method contributed 0.6 percent. Although their contributions are low, they are highly significant.
From the results obtained, it shows that the combination of all the components of new knowledge affected the Form 4 students’ achievement of solving quadratic equations. It also included that the component solving quadratic equations using formula (K6) was the highest predictor to the overall new knowledge and achievement of solving quadratic equations of the students in the study. This indicates the students successfully using the quadratic formula to solve the quadratic equations appropriately.
The ability to factorize and solve by equating each of these factors to zero was successfully used by some of the pupils, solving quadratic equations using factorization and null factor (K4) which contributed 25.8 percent of the variance. The other knowledge components were not really significant, each contributing less than 6 percent. Since the marks obtained by the students in this study were so low generally and since the two significant knowledge skills are both procedural, it showed that little understandings is implied. Through the results, students in this study couldn’t do the test very well, and resorted to using the quadratic formula method for solving quadratic equations with which they had some limited success. A few of the pupils could factorize, and could solve the appropriate questions this way. This revealed the students in this study lack of understanding or breadth of knowledge gained in solving quadratic equations.
Research Question 2: Is there any relationship between the Form 4 students’ confidence in answering questions on solving quadratic equations and the overall new knowledge acquired by the students after the lessons in solving quadratic equations?
Confidence levels of the students were measured in the form of post-teaching mean score on Confidence Questionnaire which associated with the Solving quadratic equations test (QE TEST). Data collected from the administration of the Confidence Questionnaire were used to answer the second research question.
Confidence Questionnaires Data and Students’ Acquired Overall New Knowledge
Data generated from the confidence questionnaires. The students were invited to respond to the Confidence Questionnaires during the administration of the Solving Quadratic Equations test (QE TEST). The total scores of each individual student on the questionnaire were obtained. The three percentiles, the lower quartile (25%), median (50%) and the upper quartile of distribution (75%), were determined (using the SPSS package) based on the total scores of the Confidence Questionnaires for the whole sample. The students were then classified into high, medium and low confidence level students.
Based on the percentiles of the whole sample, the high confidence students scored equal to or above 63 marks on the total scores of the Confidence Questionnaire. The range of marks used to classify the medium confidence students was above 42 marks to below 63 marks. The low confidence students achieved scores of 42 marks and less. Entries in Table 10 provide the statistics of the total scores for the Confidence Questionnaire.
Results in Table 5 indicate that about 26 percent of the students in the study had high confidence level in answering the performance test. Approximately 49 percent had medium confidence level and about 26 percent of the sample of students had low confidence. Overall, the medium confidence students (49%) were the majority group in the study followed by the low and high confidence students (26% for both low and high confidence group).
Table 5: Total Mean Score of the Confidence Questionnaires of the Sample
| | N (%) | Confidence Questionnaires Mean Score | Standard Deviation |
| High Confidence Students | 48 (25.7%) | 76.02 | 10.340 |
| | | | |
| Medium Confidence Students | 91 (48.7%) | 52.53 | 5.890 |
| | | | |
| Low Confidence Students | 48 (25.7%) | 33.58 | 5.649 |
| Total | 187 | 53.70 | 16.898 |
Discussion on the Results and Analysis (one-way ANOVA) of the Confidence Questionnaire Data
One-way analysis of variance (ANOVA) was used to compare the high, medium and low confidence students with their performance on the post-test that tested the different components of knowledge and overall knowledge gained. Any significant results of one-way ANOVA were subjected to Schefffé multiple comparison test to determine the significant differences between the three groups. For reference, the statistics of the mean scores on the post-test of each category of confidence levels of students were previously given in Table 5.
Entries in Table 6 provide the results of the Schefffé test that determined between which groups significant differences in their mean post-test total score occurred.
Table 6: Schefffé Test Multiple Comparisons Results of the Students According to their Different
Confidence Levels on the Post-test Total Scores
| Students’ Confidence level (I) | Vs | Students’ Confidence level (J) | Mean (I – J) Difference | Standard Error | Sig. |
| High | | Medium Low | 6.097* 9.104* | 1.628 1.863 | .001 .000 |
| Medium | | Low | 3.007 | 1.628 | .184 |
Note: * The mean difference is significant at the .05 level.
According to these mean difference data (see Table 6), the high confidence students’ post test mean total scores differed significantly from that of the medium confidence students by 6.097 marks and the low confidence students by 9.104 marks. A significant mean difference was also seen between the post-test mean total scores of the medium confidence students and the low confidence students by 3.007 marks. From these data, the high confidence students scored significantly higher in the post test compared to the medium and low confidence students.
All the results above indicate that the high confidence students had higher acquisition level of overall new knowledge on solving quadratic equations compared to the medium and low confidence students.
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