AN error analysis oN SOLVING VERBAL PROBLEMS IN SIMULTANEOUS EQUATION
Tilak Bahadur Khatri
Head Teacher of Shree Shiva Vocational Secondary School
Jajarkot
ABSTRACT
The study was to describe “An error analysis on solving verbal problems in simultaneous equation” in Bherimalika municipality of Jajarkot district of Nepal. The researcher used diagnostic test and interview the main instrument for this study. The data was analyzed by using descriptive statistics frequency distributions, percentage and Chi-square test. Researcher classified errors on seven heading: reading, comprehension, supposition, transformation, process skill, operation and encoding and there is no significant difference between the error committed by boys and girls in different levels. The mathematics teacher should be give the more time in develop pre-knowledge, more practice in class, give more homework and check the homework , provide feedback and the teacher should be taught Mathematics by using students centered teaching methods as well as mathematics learning theories for minimizing this errors.
Verbal problem, Error, Comprehension, Supposition, Transformation, Process, Operation, Encoding, Simultaneous equation
INTRODUCTION
The word ”Mathematics” is etymologically derived from an ancient Greek word ”Manthancian” which means ‘to learn’ (Pandit, 2069:1), Where as in Nepali, it is called ‘Ganit’ which means the science of calculation. According to Oxford Advanced Learner’s Dictionary, ”Mathematics is the science of number, quantity, and space. Algebra, arithmetic, trigonometry, geometry are the branches of mathematics.” According to the Dictionary of Mathematics ” Mathematics is a group of related subjects including algebra , geometry, trigonometry and calculus concerned with the study of numbers ,quantity, shapes and spaces and their interrelationship , application , generalization and abstraction . So that Roger Bacon state ”Mathematics is the gate and key to all sciences.”
The world algebra is etymologically derived from an Arabic word al-jabrmuqubulab. Here “al” means ‘’the’’, ‘Jabr” refers to the operation of transferring quantities just form one side of an equation to another while ‘’maqubulab” means the process of while subtracting similar quantities from both sides of an equation. (http://en.wikipedia.org/Algebra)
Algebra is the generalization of arithmetic. It is also through to be the most difficult and abstract of all branches of mathematics. But, it is primarily through for manipulative skills solution of problem through the equation, a power of generalization and use of formulation and idea of functionality. As it has been called generalized arithmetic so it can be related to geometry by saying that algebra is only written geometry and geometry is merely pictured algebra. (Bhandari , 2014:2)
Simultaneous Equation: Simultaneous equation as a set of expression consisting of two or more variables all take the same value at the same time provided all the equations are independent and a solution is possible then, n simultaneous equations containing n variables will have a unique solution. (Salman, 2004:40)
Error Analysis
Error analysis is a technique that teacher uses a detectives for analyzing clues to solve some of the severe learning problem of their students. It allows the teachers to pinpoint the computational mistakes being made by students and interpret the reason for mistakes. Asking question to compute a problem while explaining to the teacher the processes used can provide excellent information about the nature and causes of the errors or difficulties. Furthermore, the teacher can discover and analyze the students responses to detect learning difficulties through the intelligent use of inventory and diagnostic test long with personal interviews and plane the specific remedial measures to correct errors and remove difficulties.(pandit,2069;246)
By observing the students informally, the researcher found that they have some misconceptions in solving verbal problems of algebra at especially in simultaneous equation. Sometimes, they repeatedly made the same error. Also, through discussion with my fellow teachers, the researcher realized that their explanations for these types of behaviors were surprisingly consistent with mine. However, one thing is clear to me these misconceptions are found at basic level of algebra. i.e. Simultaneous equation. Whatever the reasons may be, there should be a way to identify and remedy these problems. The researcher also observed that these students memorized only a few facts and formulas without understanding them conceptually, This is one of my own explanations for the reasons of student errors solving the verbal problems of simultaneous equation and the researcher realize that this problem is common to many students .The researcher always think that there should be a systematic way of studying the errors and to see what students have to admit about their own mistakes. Thinking along this line, the researcher formed statement of the problem.
Statement of the Problem
The Simultaneous equation is a process to find the value of variable. Most of the students have felt that the Simultaneous equation is difficult to understand so this study concern with the identification and comparison of error committed by the students. What type of error generally they commit while they learning? What are the difficulties faced by the students in learning? So this study is concern with the identification and diagnoses the errors committed by grade X students in solving verbal problems of Simultaneous equation. Hence, this study intent to find the answer of the following research questions:
- What types of error is made by the students while solving verbal problem of Simultaneous equation?
- Is there any difference between the girls and boys in terms of errors in problem solving?
- How to minimize the errors in solving the verbal problem of Simultaneous equation?
Objectives of the Study
The main objectives of this study were to find the types of error committed by grade X students in solving verbal problems of Simultaneous equation, to compare errors made by girls and boys in solving verbal problem of Simultaneous equation and to find the way of minimizing the errors of grade X students in solving verbal problem of Simultaneous equation.
Hypothesis
Only one hypothesis would be formulated and tested in this study.
Ho: There is no significant difference between the error made by boys and girls in solving verbal problem of simultaneous equation.
H1: There is significant difference between the error made by boys and girls in solving verbal problem of simultaneous equation.
REVIEW OF RELATED LITERATURE AND CONCEPTUAL FRAMEWORK
Review of Empirical Literature
The researcher has tried to find out the literature relate to identification and analysis of errors committed by the students some of them are below:
Bhatt (2003) carried out the research entitle “An Error analysis in quadratic equation at grade X’’. The main objective of this study was to find the errors of grade X students in understanding, knowledge of solving and application of Quadratic equation. Furthermore it helps to study the error in the topic of quadratic equation with respect to, “Gender” location of the schools and “Types of schools”. While observing researcher found that Students committed more error in knowledge than understanding of quadratic equation, Students generally committed more errors in application of quadratic equation than understanding of quadratic equation and Students committed more errors in formulation rather than solving the quadratic equation etc.
Bhandari (2014) carried out the research entitle “An Error analysis of grad IX students in factorization.” The main objective of this study was to identify error of grad IX students in factorization. The researcher concluded that 39.47 % errors occurred at the comprehension stage, 10.98% errors occurred at the transformation stage, 32.98% errors occurred at the process skill stage, 12.17% errors occurred at the encoding stage, 4.40 % errors occurred at the carelessness stage and reading error was not found in this study.
Salman (2004), Study on the research ”Analysis of errors committed in word problem s involving simultaneous linear equation by Nigerian secondary school students “. In this study he found that six types of errors committed by Nigerian secondary school students. The researcher concluded that 16.20 % errors occurred at the supposition stage, 29.05% errors occurred at the translation stage, 13.11% errors occurred at the operation stage, 14.65% errors occurred at the elimination stage, 15.94 % errors occurred at the substitution stage and 11.05% errors occurred at the Unit stage. Also the commitment of the errors was not influenced by gender of students. Both sexes committed the errors non-significantly different.
Zakarial (2010) studied on “Analysis of student’s error in learning of quadratic equations”. The purpose of this study was to analyze student’s error in learning quadratic equations which focused on subtopics such as Factorization, Completing the square and Quadratic formula. He found that where the study shows that 42 errors occurred at the comprehension stage, 186 errors occurred at the transformation stage, 187 errors occurred at the process skill, 19 errors occurred at the encoding stage and 6 errors occurred at the carelessness stage.
Conceptual Framework
Relating with the Newman’s hierarchy of errors causes for written mathematical tasks, Researcher would be prepared the following conceptual framework for this study
| Causes of Error
· Carelessness
· Motivation |
| Types of Error
I. Reading II. Supposition III. Comprehension IV. Translation V. Process skills VI. Operation VII. Encoding
|
| Learning of Simultaneous
Equations |
METHODS AND PROCEDURE OF THE STUDY
The research design of the study would be quantitative method followed by qualitative method. This study would be based on descriptive in nature. The population of the study would be taken all the students of secondary level of Bherimalika Municipality of Jajarkot district in which all students belong to grade X in academic year 2072 BS. Therefore, the researcher selected only four different public schools of Bherimalika Municipality of Jajarkot district by purposive sampling methods, 20 students from each schools including equal number of boys and girls would be selected by random sampling method, 4 students from each schools including equal number of boys and girls would be selected by purposively who made maximum error in solving verbal problem of simultaneous equation and the sampled schools mathematics teacher would be selected by purposively for interview.
Data Collection Tools and Techniques
To get the reliable data the researcher were develop a diagnostic test and individual interview schedule of each students and teacher on solving verbal problems of simultaneous equation.
Data Collection Procedures
First of all the researcher was take the interview with sampled students of these four schools. Then the researcher was administered test paper of sampled students in these schools. After diagnostic test, the researcher selected 2 girls and 2 boys for interview by Newman procedure. The researcher asked 5 questions in each student which were related to the error made by students in their answer sheet. The researcher had selected respected 4 mathematics class teacher for interview to find the way of minimizing these error.
Data Analysis and Interpretation Procedure
To analysis the qualitative data of the study descriptive methods had used and for quantitative data statistical tools such as chi-square test, percentage and frequency distribution used. These data had presented in table and the result would be analyzed. Then the collected data would be classified to headings: Reading, Comprehension, Supposition, Transformation, Process skill, Operation and Encoding error. After classifying the error, the researcher had found out the frequencies distributions of each errors and find the percentage to illustrate where the highly error occurred.
ANALYSIS AND INTERPRETATION OF THE RESULT
According to Newman, while solving word problem, error might be committed in five steps. These are reading error, comprehension error, transformation error, process skill error and encoding error. Error found in question while implementing test were categories according to Newman’s technique of error analysis. Errors were collected from test and interview too. The error was kept in reading error where the students unable to read the question properly. This error was found by giving them question to read. The error was kept under the comprehension error when they were unable to receive what the question asked. The error was kept supposition error when they unable to use the letters to stand in for the unknown demanded for in the problems. The error was kept under transformation error, when the students were unable to change word problem into mathematics expression. The error was kept under process error when they committed error in processing the answer. The error was kept under operation error when they unable to do operation correctly. At last, the error was kept in encoding error when they committed the error in verbal answer. The frequency distribution and percentage of error for each conceptual area was calculated. In this way, the errors committed by the students are categories in the following table:
Table no. 1 Error Committed and their Frequency Counts and Percentages
| S.N. | Type of error committed | Frequency Counts | Percentage (%) |
| 1 | Reading Error | 8 times | 3.7% |
| 2 | Comprehension Error | 40 times | 18.54% |
| 3 | Supposition Error | 30 times | 13.89% |
| 4 | Translation Error | 49 times | 22.64% |
| 5 | Process Error | 41 times | 18.98% |
| 6 | Operation Error | 20 times | 9.3% |
| 7 | Encoding Error | 28 times | 12.96% |
| Total | 216 times | 100% |
It could be observed in table no.1, the lowest no. of frequency of error was committed in reading error and the highest no. of error was committed on transformation error. This shows that students committed less error while reading question. There were 8 reading errors out of 216 errors. It is about 3.7 %. There were 40 comprehension errors out of 216 errors .It is about 18.54%. There were 30 supposition errors out of 216 errors .It is about 13.89%. There were 49 transformation errors out of 216 errors .It is about 22.64%. There were 41 process skill errors out of 216 errors .It is about 18.98%. There were 20 operation errors out of 216 errors .It is about 9.3% and there were 28 encoding errors out of 216 errors .It is about 12.96%. This indicates that most of the students were unable to translate the verbal problems in mathematical expression appropriately. Mostly transformation error and process error were found trouble some steps in solving verbal problem of simultaneous equation of algebra.
Distribution of Error Committed by Girls and Boys
The second objective of this study was to compare errors made by girls and boys in solving verbal problems of simultaneous equation of mathematics. To find the error committed by boys and girls, the researcher selected 40 boys and 40 girls in administrate the test. Among them 2 boys and 2 girls were selected for interview from each sampled school. The researcher took interview with these students to find the no of error; level of error and the percentage of errors, which are presented in table below:
Table no: 2
Frequency and Percentage of Errors Committed According to Gender
| S.N. | Type of error committed | Girls | Boys | Total |
| 1 | Reading Error | 5 times (62.5%) | 3 times (37.5%) | 8 times |
| 2 | Comprehension Error | 23 times (57.5%) | 17 times (42.5%) | 40 times |
| 3 | Supposition Error | 14 times (46.66%) | 16 times (53.34%) | 30 times |
| 4 | Transformation Error | 24 times (48.97%) | 25 times (51.03%) | 49 times |
| 5 | Process Skill Error | 18 times (43.90%) | 23 times (56.10%) | 41 times |
| 6 | Operation Error | 5 times (25%) | 15 times (75%) | 20 times |
| 7 | Encoding Error | 14 times (50%) | 14 times (50%) | 28 times |
| Total | 103 times (47.69%) | 113times 52.31%) | 216 times |
It could be observed in table no.2, girls students were committed 103 errors out of 216 errors. It was about 47.69%. And boy students were committed 113 errors out of 216 errors. It was 52.31%. It shows that in whole there is no more difference between errors committed by girls and boys in solving verbal problems of Simultaneous equation. By using Chi-square test, Level of significance at 5%, the tabulated value of 2 =12.59 and calculated value is 7.2056. So the calculated value is less then tabulated value. It shows that null hypothesis is accepted. So that there is no significance difference between errors committed by girls and boys.
The Ways of Minimizing Different Errors
- Before starting the chapter, the teacher has to check the pre-knowledge of students and give the fundamental knowledge about the topic by use material with game and quiz.
- Teachers of mathematics should be aware of the language they use in the classroom. They should use simple language and are mathematical concept should be explained with material with necessary illustration.
- The teacher must use new technology and teaching aids for algebra teaching in time to time and must be appear on the mathematics conference and seminar.
- Teacher should use diagnostic test and most identify the error and must use remedial teaching to avoid the errors.
- Classroom management and teaching material should be managed to minimize the error.
- The teacher should try to find out the reason about committing the errors
- The teacher should discuss with other teacher and exports how to minimize the errors and make the mathematical class effectively.
- School Administration should gather students, teachers and guardians for open interaction so that problem could be identified easily.
After analysis and interpretation of data the study of student committed maximum error in translation after that process error: comprehension; supposition error; encoding; operation and reading are respectively. The role of gender is less important for committing the errors in whole, but reading error, comprehension error and supposition errors were committed higher by boys than girls and operation error, translation error and process errors were committed higher by girls than boys and the boys and girls were committed equal error in encoding error. Students were taken as difficult to solve word problem. Students were unable to give meaning of mathematical term properly and unable to choose appropriate operation to solving word problem. Lack of pre- knowledge not to participant actively in class, carelessness, can’t choose appropriate operation, socio- economic status of students, large class size, study habit of students are cause of error occurrence. To minimize the error, teacher should focus on why and how students make mistake, teach individually and discussion with necessary, try to improve classroom management, to participant in mathematical game and quiz, to help the make home environment for study etc.
REFERENCE
Bhandari ,G.C. (2014) Error analysis of grade IX students in factorization . M.Ed. An unpublished Master Degree Thesis, central Department of Mathematics Education T.U.
Bhatta,D.R. (2000) An error analysis in quadratic equation of grade IX M.Ed. An unpublished Master Degree Thesis, central Department of Mathematics Education T.U.
Clements, M.A. (1980), Analyzing children’s errors on Witten mathematical task, Educational studies in Mathematics, 11(1), 21 CDC curriculum of lower secondary school Mathematics, Kathmandu curriculum development http://en.wikipedia.org/Algebra
Khanal,P (2008). Educational Research Methodology, Kathmandu: Sunlight publication (students Books).
Newman, M.A. (1977A), an analysis of sixth grade publishes error on written mathematics Education tasks, In M.A. Clements and J. Foster Eds. Researching Mathematics Education in Australian, 1977 (vol. 2, pp269-287) Melbourne: Swinburne college press.
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Salman,M.F.(2004), Analysis Of Errors Committed In Word Problems Involving Simultaneous Linear Equation By Nigerian Secondary School Students. , Department of Curriculum Studies, and Educational Technology, I.U.
Zakarial, E. (2010), Analysis of Student’s Error in Learning of Quadratic Equation
