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Based on the findings which have been presented in chapter 4 of this study, this chapter provides summary of the findings based on the previous chapter. The aim of this research is to investigate students' level of performance in simplification in algebraic expression chapter. The research will attempt to answer the following questions.
The following is the explanation on the purpose of study made, samples involved in the study and the instruments used. This study is an observation study using the instrument of test questions to identify the extent of Form 4 students' understanding of the procedural and conceptual related to the simplification topic available in the chapter of algebraic expression. 32 students of form four of Sekolah Menengah Teknik Shah Alam were involved in this study. They consisted of 16 male and 24 female students each. All students were of Malay origin. The purpose of the study was also to see the relationship among the procedural understanding and conceptual understanding with their performance in the TAES.
The instruments used in this study were the Test of Algebraic Expression Simplification (TAES) which covered the secondary syllabus on the algebraic expression topic. The test consists of 7 questions. Question 1 and 5 cover the general subtopic of the introduction of unknowns and like, unlike terms, which covers the concept of basic operational mathematics, namely positive and negative. Question 2 covers solution to the fraction concept that involves simplification method. Question 3 and 7 involve a few processes to arrive at the simplest answers such as factorize, expansion and simplify. Meanwhile, question 4 and 6 involve power and fraction for students to solve to simplest form.
5.2 The Findings
The result of the study showed that:
The level of the students' performance in simplification algebraic expression was moderate.
There was no difference between genders of the students with their performance in simplification of algebraic expression.
There was a difference achievement in simplification of algebraic expression between the highest Grade of Mathematic PMR Examination and the lower grade.
There were common errors that were made by the students during answering simplification test.
5.3 Discussion on Findings of Study
Findings of study reported in Chapter 4 were based on research questions which cover the extent of students' achievement in algebraic expression that uses simplification method can improve their grades in the examination. Other than that, researcher also studied common errors made in the tests given. In this chapter, further discussion will be made on the findings of the study obtained.
5.3.1 The Level of Students' Performance in Simplification of Algebraic Expression.
The level of students' performance in simplification of algebraic expression can be measured through the test of algebraic expression simplification. From the Table 4.1, the researcher found that the level of students' performance is moderate with highest mark is 100 while the lowest mark is 39 marks. The average mark was obtained by using the mean score of the students which is 62.63 marks.
This shows that students' performance is still in ordinary. The low level of knowledge and students' performance on the algebra topic cause mistakes to be made in the concept among the students, as reported by Chan (1999). From the TAES carried out by researcher, it can be seen that students still have problems in terms of basic concepts of mathematics such as fraction, negative number and algebraic expression. Besides that, the students have very low understanding of some concepts in algebra such as expansion, factorization and simplification. It leads to the students to make mistakes in algebra test. The students still did not master the basic operational concepts and this causes them to have difficulties to understand what the question wants.
5.3.2 The Difference between Genders of the Students' Performance in TAES.
According to Table 4.3 on the difference between male and female students' in the test of algebraic expression simplification, it is shown that there is no significant difference between male and female performance in the TAES scores. The students' performance of simplifying algebraic expression was not affected by the students' gender. From the result, it can be concluded that both gender obtain the same knowledge, contents and information about algebraic expression. It can be supported by Fierros, (1999), Zhang & Manon, (2000), Johnson, (2000), Leahe & Guo, (2001), Erickan et. al, (2005), showed that no significant difference in achievement between boys and girls as they start getting acquainted with mathematics.
5.3.3 The Relationship of Performance in Test of Algebraic Expression Simplification between the Highest Grade of Mathematic PMR Examination and the Lower Grade.
From this research, as all the students were already done their PMR examination, therefore the researcher wants to measure the students' basic concept of the algebraic expression that had been taken by the students for three years. For form 4 students, they are supposed to be good since they already have good basic concepts in mathematics subject because algebra is the one topic that was being asked in the PMR examination. From the Table 4.4, the researcher can conclude that most of the respondents obtained A in their PMR result. The researcher wants to see the relationship of achievement in test of algebraic expression simplification and the highest grade of mathematics PMR examination and the lower grade.
Twenty three of the respondents obtained A in PMR while 7 students obtained B and 2 students obtained C. In the beginning, the researcher can say that the respondents (students) have good basic concepts in mathematics operational concepts and calculation concepts. After doing the TAES, it can be concluded that not all students who obtained A in PMR examination manage to get better result in TAES because of some reasons.
The first reason is that the students do not have strong basic concepts in mathematics of lower form because usually the question that was asked in PMR examination was so easy and straight forward but the question in TAES requires the students to use all the variety of concepts in algebraic expression such as factorization, simplification method and expansion. In addition, the questions also include some other concepts such as positive and negative integer and fraction. There were misconceptions as well as the students were weak in mastering the basic operations in the negative as proven by research done by Khamsan Omar (1999). Besides that, the students' weaknesses in mathematics can cause them difficulties to master in few chapters in form 4 and 5 syllabuses in modern mathematics or additional mathematics, Wong report, (1987).
Secondly, it is due to the research being carried out when the students were already in form 4, and that most students already forgotten the procedures or method to solve questions related to simplification in algebraic expression. Continuous drill and practice is very much needed to prevent students from forgetting a certain topic in mathematics. Students are supposed to remember the lessons taught previously since the topic of algebraic expression will be asked again in the SPM later on.
Thirdly, during the level of PMR, all students were in different schools thus it reflects the different condition of learning received by the students. Because of that, the factor of the school's environment plays important role in encouraging the mood of study among the students. Researcher is also of the opinion that the interest to study among the students is also important to encourage them to score excellent results in the examination, especially in mathematics. Deep stress is very need in fostering the spirit of wanting to excel in mathematics.
5.3.4 The Common Errors that had been made by the Students in Test of Algebraic Expression Simplification.
To study on how far the students' understanding in the topic of simplification of algebraic expression, research evaluated each items given in the test of algebraic expression simplification. The following is the discussion on the understanding and common errors that had been made by the students in test.
For Question 1 and 4, it was due to the students' carelessness during solving the question that involves positive and negative value. Besides that, the students were also confused since in this question, it involves a mixture of like, unlike terms and whole number. For example, based on the student's answer in Figure 4.1, the students did not check that even though 2x + 3y - 3x + 6y + 7 will give the simplest answer of 5x + 9y + 5, but the student answered 5x + 9y + 7 probably because students did not read the question properly. For example, in Figure 4.4, the student did not manage to solve the question because it involves bracket and this requires student to consider the positive and negative sign when expanding the expression. From the student's working to answer the question, the researcher can deduce that, the students had failed to choose and apply the right technique to simplify the question. Not only that, when the researcher asked the student the method that he used to answer the question, the answer given by the students shows that he actually does not have a good concept of simplification method. Hence, it can be said that, the student was not able to answer the question due his weaknesses in applying basic mathematical operations and in concept of algebraic expression.
For Question 2, it involves fractional and basic operational between two fraction. For this question, most of the students have problem to solve the question because they have poor understanding on fraction concept. Maznah Mahmood (2000) revealed that the most common mistake conducted by student in answering questions that involve fraction is when the students do not simplify the fraction to the proper fraction. Students also made mistake in the addition and subtraction of fraction operation. Besides that, the students were also worried to solve the question because this question needs the students to solve to the simplest form. Usually, the students just answer this question half way and they do not know to simplify the question if the unknown is situated at denominator and nominator. In Figure 4.2 showed that the students still have problem in numerator and denominator which is the students simply minus the first fraction to the second fraction. Then, the students decided to cancel out the unknown x from the numerator and denominator without simplify the whole number. After that, the students canceled out the negative sign from the expression and yield the final answer is totally wrong. From the students' working, the researcher can conclude that the students still have big problems in mathematics algebraic questions, especially when the question involves many concepts such as fractional, negative and positive integer and algebraic expression. So, from the working, it can be seen that the students have poor basic concept in mathematical operational and algebraic expression. The students had already learnt the basic concepts in mathematical operational during primary school and lower secondary school. Usually, the students can apply the basic concepts because of less of doing exercise and drilling. Therefore, it makes them to have difficulties to understand the requirements of the questions and most of the time they cannot answer the question properly. Doing drills on regular basis can help to students enable to make justification on the selection of certain working to solve the problem, as reported by Dossey et al. (2002),
For Question 3, the students were required to factorize the denominator in order to obtain the solution for this question. After factorizing the denominator, the students can cancel of the same expression from the numerator and denominator. The process of doing this procedure is the students must have a good in factorization and they can apply many ways to find the easy solution. Most students expanded all the expressions at the numerator and denominator without thinking of other ways to solve this question. There are three types of students' common errors for this question, namely the students expanding the numerator, the students does not factorize the denominator and lastly, the students just do it half way. When the students expand the numerator, it can cause them to have difficulties to solve the question, because actually the question wants the students to simplify to the simplest form. If the students factorize the denominator, then they will find out the same expression from the denominator and numerator can be cancelled off. When students expanded the expressions, it caused them to commit carelessness in multiplying between the expressions because students need to be careful in expanding the expression. This is because this question involves 3 terms which is a, b and c. Based on students' answer as in Figure 4.3, students did not factorize the expression at denominator but they had expanded the first expression at numerator which is (a + b)2 and this caused these students to confuse in solving this question. The expansion of the numerator makes the students difficult to find the simplest answer.
The last question for the test of algebraic expression simplification is question number 7. 17 of the students could not answer this question because this question involve x-squared, x, whole number, positive and negative operation and fraction. The students still have difficulties in factorizing the questions and some students can factorize but they do know how to simplify the question. There are two types of answer by the students for this question. The first situation can be seen from the Figure 4.5a and 4.5b. The researcher will explain in detail based on these two students' working. For the first one, the student just simply cancels off the x-squared at the beginning of the question. Then, the students leave the answer like in the example given in the Figure 4.5a. It can be concluded that the students have basic concept problem in algebraic chapter that is factorization. There were misconceptions among many form 2 students in the basic operation of algebra, as reported by Saripah Latipah (2000). The researcher felt curious because supposedly, the students already have the strong basic concept in algebra because they learnt the algebra topic for three years in lower secondary school. In this matter, the students should manage to answer this question because it was so easy. In the second part example in Figure 4.5b, the students manage to factorize the expression but he did wrongly for the final answer because he just canceled off all the unknown in that expression. The last answer that he wrote is 5/4. In the beginning, the students factorize the numerator and denominator, and then the students find out have two same expressions that is (x-2). The students cancel off the expression and yield the answer to (x+5)/(x+4). Besides, the expression is the final answer for this question but the students did the mistake with cancelling of the x from the expression and yield the answer 5/4. It can be concluded that the students are still poor in the basic concept of algebraic expression.
Overall, researches is satisfied with the study conducted. This research gives much valuable experience and new knowledge in the researcher's life. It has been a beneficial exposure to the researcher on the educational research in terms of purpose, method, procedure, academic writing and data analysis. Besides that, it increases the self-confidence to face with the public to present the ideas and opinion. Researcher also discovers the improvement in the interpersonal and intrapersonal skills.
However, researcher admits that there are still weaknesses that can be improved in this study. Only basic and important concepts that were touched on. With solid and right basic concepts on the simplification in algebraic expression, students will better understand this topic thus it will help the students to answer the examination later on.
Researcher feels that this study will be better in terms of reliability and believability should the number of respondents for the first test be increased. In this study, the number of students who took part in the test of algebraic expression simplification (TAES) were 32 students. This number can be increased by taking in all the students of mathematics in the school. Should all the school students be involved, the data for the frequency of students who experience on the simplification method will be more precise and accurate. However, since there were few limitations during conducting this research, researcher was only able to get 32 respondents.
Apart from that, researcher needs to conduct interview to get more accurate and meaningful data. During the interview process for the next research on this topic, researcher needs to see that the questions addressed to the respondents during the session can play important roles in getting more precise and accurate results on the problems in simplification. During the interview, researcher also needs to be more curious in addressing the questions to get accurate information from the respondents. The accurate information provided by the respondents can help and support the statement that they are having misconception for the simplification in algebraic expression.
Students' level of performance is very important to measure students' understanding or mastering the basic concepts of algebra for the simplification subtopic. Learning the basic concepts of algebra that involves types of number and spelling in a certain expression through rote memorization is not enough for an individual to master in simplification method but the students must make an effort to learn the topic in a good way such as drilling and practices of many mathematics question.
Researcher hopes that the findings of this research can be benefitted by involving parties. This is because learning algebraic expression is an important objective in education. Without the knowing this chapter can cause the students have problems for others topic that related to algebraic expression especially simplification.
In the final analysis, continuous research on all aspects of education is one way to further strengthen the quality of the national education. Education is meant to fulfill the needs of the society at present and in the future apart from contributing to the nation. Thus, researcher hopes that this research can be benefitted to further improve the quality of the national education.