Preparing a pre-requisite test in algebra. Construction and tryout

Contents

CHAPTER 1 Conceptual Framework
1.1 Introduction
1.2 The Place Of Mathematics In School Curriculum
1.2.1 The Place of Mathematics at Secondary Stage
1.3 Objectives Of Teaching Of Mathematics
1.4 Concept Of Algebra
1.4.1 Importance of Algebra
1.4.2 Objective of Teaching Algebra:
1.5 Mathematical Weaknesses
1.6 Rationale Of The Study
1.7 Research Questions
1.8 Statement of the Problem
1.9 Objectives of the Study
1.10 Explanation of the Terms Used
1.11 Delimitation of the Study

CHAPTER 2 REVIEW OF RELATED LITERATURE
2.1 Introduction
2.2 Review Of Related Literature
2.2.1 Studies related to Diagnosis and Remediation in Mathematics
2.2.2 Studies related to Achievement in Mathematics
2.2.3 Studies related to the Achievement in Algebra
2.2.4 Studies related to the Pre-requisite Tests in Mathematics
2.3 Implication Of Review Of Related Literature

CHAPTER 3 RESEARCH METHODOLOGY
3.1 Introduction
3.2 Methodology
3.2.1 Research Methodology
3.2.2 Population of the Study
3.2.3 Sample of the Study
3.3 Tools For Data Collection
3.3.1 Achievement test
3.3.2 Pre-requisite Test
3.4 Data Collection
3.5 Data Analysis
3.5.1 Analysis of Achievement Test
3.5.2 Analysis of Pre-requisite Test

CHAPTER 4 Data Analysis And Interpretation
4.1 Introduction
4.2 Analysis of four achievement tests
4.2.1 Chapter I: Linear Equation in Two Variables
4.2.2 Detailed Analysis of Items for Chapter I: Linear Equation in two variables
4.2.3 Chapter II: H.C.F and L.C.M of Polynomials
4.2.4 Detailed Analysis of Items for Chapter II: H.C.F and L.C.M of Polynomials
4.2.5 Chapter III: Rational Expressions
4.2.6 Detailed Analysis of Items for Chapter III: Rational Expressions
4.2.7 Chapter IV: Quadratic Equation
4.2.8 Detailed Analysis of Items for Chapter IV: Quadratic Equation
4.2.9 Summary of Basic Concepts where Errors Occurred
4.3 Mean and Standard Deviation of Pre-requisite Test
4.4 Analysis of pre-requisite test
4.4.1 Item-wise Analysis of Pre-requisite Test

CHAPTER 5 MAJOR FINDINGS, DISCUSSION AND SUGGESTION
5.1 Introduction
5.2 major findings of the study
5.2.1 Findings of Achievement Test
5.2.2 Findings of Pre-requisite Test
5.3 Discussion
5.4 Suggestion for the teachers
5.5 Suggestion for further study

BIBLIOGRAHY

APPENDIX I

CHAPTER 1 Conceptual Framework

1.1 Introduction

Education is very important for an individual’s success in life. Education provides learning skills among students that prepare them physically, mentally and socially for the world of work in later life. Education is generally seen as the foundation of the society which brings economic wealth, social prosperity and political stability. It is the education only which transforms a person to live a better life and more importantly in a socially well-being. Education builds a remarkable effect on one’s personality. It plays a vital role in the personal growth and the social development among all of us. It imparts us with all the power and necessities in making a noticeable mark in any of the field. The National Policy on Education (NPE, 1986) refers “In our national perception, education is essentially for all…Education has an acculturating role. It refines sensitivities and perceptions that contribute to national cohesion, a scientific temper and independence of mind and spirit- thus furthering the goals of socialism, secularism and democracy enshrined in our Constitution.”

There are three modes of education i.e. formal education, informal education and non-formal education in which formal education encompasses wider range of activities with reference to the teaching learning process. The most important component of formal education is the teaching learning process. This could be done by teaching-learning process. Teaching includes the various classroom activities carried out by the teachers aiming to bring expected changes in behavior of the students, whereas learning involves modification of behavior, acquisition of skills and competences on values. The teaching learning process draws its outline from the curriculum.

Curriculum is a conceptual plan and dynamic entity to achieve the requirements of the people of the country. The school curriculum consists of various subjects like Hindi, English, Social science, Science and Technology, Gujarati, Mathematics, etc. which are taught with their specific objectives. Out of all these subjects, mathematics is considered as the mother of all sciences, since its fundamentals are needed for the understanding of all the concepts of science. Mathematics plays a vital role in day-to-day life. According to NPE (1986), “mathematics should be visualized as the vehicle to train a child to think, reason, analyse and to articulate logically a part from being a specific subject, it should be treated as a concomitant to any subject involving analysis and reasoning.”

Mathematics is crucial not only for success in school, but in being an informed citizen, being productive in one’s chosen career for personal fulfillment. In today’s technology driven society more strain is there on individuals to interpret and use mathematics to make sense of information and complex situations. Mathematics is humanity’s common heritage. It plays a vital role in all areas of modern science, technology, numerous segments of economic, social and cultural life, including industries, technology, numerous segments of economic, social and cultural life, including industries, telecommunications, education, health, transportation, banking, insurance, development and agriculture.

1.2 The Place Of Mathematics In School Curriculum

Mathematics is the science of number and space. It deals with quantitative nature of our life. It helps us in reaching necessary conclusions and interpreting various ideas with useful meaning. It provides opportunity for intellectual exercise of our mind. Report of Education Commission (1964-66) recommended that science and mathematics should be taught on a compulsory basis to all pupils as a part of general education during the first ten years of schooling.

Understanding mathematics is an important part of understanding our world. The subject and its applications in science, commerce and technology are important. Its important role can be ensured if students understand and appreciate the relationships and patterns of both number and space in their thoughts clearly and concisely. It will also help students to develop their capacity of reasoning so that they will think more logically and independently in making rational decisions.

Today’s mathematics curriculum must prepare students for their future roles in society. It must equip them with essential mathematical knowledge and skills; with skills of reasoning, problem solving, and communication; and, most importantly, with the ability and the incentive to continue learning on their own. This curriculum provides a framework for accomplishing their goals.

1.2.1 The Place of Mathematics at Secondary Stage

The secondary Education serves as a bridge between Primary and Higher Secondary education. Mathematics is taught as a compulsory subject up to the secondary stage. According to Arora (1995), “During the 1920s and 1930s, the place of mathematics in the secondary schools was being questioned. Although enrolment at public schools increased in the early part of the twentieth century, the number of students taking mathematics courses was on the decline. Students were displaying a great dissatisfaction with the mathematics curriculum. There were a large number of students failing in secondary mathematics, and the subject was taught with little insight into its everyday utility.”

With the recent introduction of computers in schools, educational computing and the emergence of learning through the understanding needs to be suitably redesigned to bring it in line with modern technological devices. According to NCERT (2000A),”at the secondary stage, the teaching learning of mathematics has to serve two complementary purpose. Firstly, the aim should be to further enhance the capacity of the students to employ mathematics in solving problems that they face into their day-to-day life. Secondly, a systematic study of mathematics as a discipline has to be started here and continued further.”

Joint Commission of the Mathematical Association of America (MAA) and The National Council of Teachers of Mathematics (NCTM) (1940) made recommendations in foundational fields of secondary mathematics for grades 7-12 which are: (1) Number and Computation, (2) Geometric From and Shape Perception, (3) Graphical Representation, (4) Elementary Analysis, (5) Logical (or Straight) Thinking, (6) Relational Thinking, (7) Symbolic Representation and Thinking.

It implies from the above that the study of mathematics at secondary level encompasses wide range concepts which require abstract and logical thinking. To get better understanding of mathematics taught at secondary level, it requires the understanding of objective of teaching of mathematics.

1.3 Objectives Of Teaching Of Mathematics

According to NCERT (2000B), the mathematics has the following objective at the secondary level:

I. To enhance the capacity of the students to employ mathematics in solving problems that they face in their day-to-day life.
II. To start a systematic study of mathematics as a discipline has to be started here and continued further. The curriculum may include the study of relevant mathematical concepts, number system, algebra, geometry and trigonometry, menstruation, graphs, statistics etc.
III. To develop idea of proofs with thrust on deductive reasoning.
IV. To lay emphasis on wider applications of mathematics by way of making data based problems pertaining to actual data on population, agriculture, environment, industry, physical and biological sciences, engineering, defence etc. also the students should attain proficiency in presenting information available in their environment in the form of graphs and charts and be able to do calculations with speed and accuracy.
V. To enable the students to solve problems using algebraic methods and apply the knowledge of simple trigonometry to solve problems of heights and distances etc.
VI. To appraise the students about the history of mathematics with special reference to India and the nature of mathematical thinking.
VII. To encourage the students to enhance their computational skill by the use of Vedic mathematics.

The secondary school mathematics curriculum precedes the development of the learning of mathematics in the primary school. One of the objectives of mathematics education is to start a systematic study of mathematics, as a discipline has to be started here and continued further. Mathematics has various branches like Algebra, Arithmetic, Geometry and trigonometry. Arithmetic is the science of numbers and art of computing. Algebra is defined as science of numbers with the distinction that here numbers are denoted by letters instead of figures. Geometry is the science of lines and graphs. Trigonometry is used for mensuration, astronomy and navigation purposes. All these branches play their different roles in achieving the objectives of mathematics. Moreover, out of all these branches, algebra has also its own objectives. Algebraic competence is commonly considered as an absolute prerequisite for further studies not only in mathematics, but also in other disciplines, primarily in science and engineering. Hence first it is necessary to know its concept and importance.

1.4 Concept Of Algebra

Algebra is a branch of mathematics in which arithmetic operation are generaliseed by using alphabetical symbols to represent unknown numbers. According to Reeve (1960), “Diaphanous of Alxandria wrote the first treatise in the 3rd century AD. The term ‘Algebra’ is derived from the Arabic word al-jabr that literally means “the reunion of broken parts”. It is part of the title of a book written by Mohammed iben-Musa al-Khwarizmi (approximately 830 AD). The book’s title was ‘Hisab al-jabr w’al-muqabalah’. Al-jabr means something like ‘completion’ or ‘restoration’ and refers to the transposition of subtracted terms to the other side of an equation. Muquabalah refers to ‘reduction’ or ‘balancing’ that is cancelling out like terms on opposite sides of the equal sign in equations.” It reveals from the above that algebra deals with equations made up of many terms which has to be subtracted, reduced and cancelled for making it balanced.

1.4.1 Importance of Algebra

Algebra is the language of generalization. If some calculation has to be performed once, algebra is not needed. However, if it is to be done repeatedly, algebra provides a very simple language for describing what is being done. Algebra is the language which describes the patterns. For example, for multiplication of two fractions, multiply both the numerators to get their product and then multiply both the denominators to get their product. For example,

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Now the same rule is written in the language of algebra

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Therefore, we can say that the study of algebra emphasizes the mathematical relationships between variables. It is through the study of these relationships that students develop the ability to solve problems and analyze situations using mathematical models.

According to Sidhu (1995), Algebra is taught because:

I. It is useful in other branches of mathematics. It has especially simplified for the learner, many problems of arithmetic.
II. It gives compact formula or generalizations to be used in all cases. The solutions of problems by equations and by factorization are example of this.
III. It has practical value in many of the trades and industries.
IV. It provides an effective way for expressing complicated relations.
V. It gives a new, good, approach to the study of abstract mathematical relationships through the use of a new language and a new symbolism.
VI. It inculcates the power analysis.
VII. Verification of result is simpler and more satisfactory in algebra than in any other branch of mathematics. It develops confidence among the students.
VIII. It helps in the generalization of scientific truths in to simple and compact formula.
IX. It is a good instrument for mental training.

1.4.2 Objective of Teaching Algebra:

According to NCERT (2000B) the objectives of teaching algebra are:

I. To enable the learner to apply its knowledge to other branches of mathematics.
II. To make them aware of its practical value in trades and industries.
III. To enable the students to express the complicated problems in an effective way.
IV. To inculcate the power of analysis.
V. To develop confidence in the students.
VI. To give mental training of the mind.
VII. To enable the students to use generalization and there by understand its importance.
VIII. To enable the students to develop confidence by verification of results.

Algebra is found to be difficult among all other branches of mathematics. According Kumar (1993), “The reasons for comparative difficulty in Algebra are, it starts from deductive analysis of facts, all the students may not possess the quick manipulative skills which are essential to solve problems and a sound knowledge of algebra is essential to understand further theories.” The learning of algebra at secondary school forms a base for learning the concepts of mathematics at higher secondary stage. For learning any new concept, students require knowledge some basic concept. If the students have poor command over the basic concepts then it will lead to mathematical weaknesses.

1.5 Mathematical Weaknesses

Mathematics has occupied a very important place in our school curriculum. Still we are not able to reach to the desirable level of performance in the mathematical achievement. Students are weak in mathematics. There could be many reasons for their weaknesses.

Level of achievement in mathematics is not as higher as it should be. Differences in students’ abilities, attitudes, interest and different level of understanding, some of them easily understand the concept when explained once, while others grasp after reflecting upon it. Due to this, many students may not understand several concepts of mathematics, which lead to weaknesses in those in those particular concepts of mathematics.

According to Kumar (1993) “method being one of the reason for which students are afraid of studying mathematics.” Rastogi (1983) stated that, “in spite of such an importance is being given to the study, the performance of students in mathematics tells quite different story. A national survey of achievement in mathematics illustrates that our national performance is quite below the desirable one.” Chel (1990) found that underachievement was caused due to lack of understanding of mathematical concepts of earlier stages. Kasat (1991) found that low intelligence, poor numerical abilities, poor comprehension and recall ability, no interest in mathematics and poor study habits were the causes of large failure in mathematics. Sharma (1978) found that the major factors responsible for low achievement in mathematics were imparting of limited knowledge, blind use of rules, heavy syllabus, lack of natural urge, in sufficient drill at primary and absence of methodical approach of classroom teaching. It implies from the above that low intelligence, poor numerical abilities, poor comprehension and recall ability, poor study habits and lack of proper teaching causes mathematical weaknesses. But these weaknesses hinder the grasp of further mathematical weaknesses. But these weaknesses hinder the grasp of further mathematical concepts. Some mechanisum should be found to deal with these weaknesses hindering the learning of new mathematical concepts.

Interest plays an important role in pupil’s success with the basic skills and abilities. Lack of clarity in basic concepts or lack of prerequisite is one of the most important factors causing mathematical weaknesses of students. Weaknesses of students in mathematics can be a major factor, which causes the gap between the expected achievement and actual achievement in mathematics. Thus, weaknesses of students in mathematics at secondary stage will also hinder their progress in learning mathematics at higher stage of their academic life.

1.6 Rationale Of The Study

Mathematics is an important discipline of learning at the secondary stage. It predominately contributes to the development of precision, rational and analytical thinking, reasoning and scientific temper. One of the basic aims of teaching Mathematics at the Secondary stage is to inculcate the skill of quantification of experience gathered by the learner. But still some of the students find mathematics as very difficult subject.

Rastogi (1983) has said that “in spite of such an importance given to the subject the performance of the students in mathematics illustrates that our national performance is quite below the desirable level.” NCF (2005) has stated “it is in the early years, up to class IV that efforts at diagnosing and addressing remedial work in language and mathematics must be directed.” The studies of Ashar (1972), Rastogi (1983) and George (2003) also show that the poor mathematical concepts, lack of mastery over basic mathematical skills, lack of pre-requisites related to basic principles, fundamentals are some of the most prominent aspects which hamper the students’ progress in learning mathematics. According to Patel (2007), “error analysis and remedial measures also help the students to enhance their achievement to some extent.”

In the present Study, the investigators had taken class X students as the sample for the study because in class X, Mathematics is one of the compulsory and important subjects. It is a subject which covers geometry, algebra, statistics, arithmetic, trigonometry and some concepts of computer. Class X scores provide an entry into the various streams like science/ arts/ commerce. Also some students go for vocational courses like diploma after class X. so the examination of class X and its result has a great importance for higher studies. From eleventh standard, the abstract concepts of mathematics are introduced. So the mathematical concepts learnt at class X construct their base for further mathematical learning which becomes helpful to students of class X to achieve better result. The Education commission (1964-66) recommended mathematics as compulsory subject for students at school level (Education and National Development, 1985).

Mathematics is a part of every school curriculum all over the world and perhaps the only subject occupying unrivalled position. Mathematics has achieved central place in school curriculum. In order to lead an intelligent life, one needs minimum essential knowledge of mathematics. Also investigators have chosen algebra as the area of the study because algebra is used to enable the learner to apply its knowledge to other branches of mathematics. According to Gamoran & Hannigan (2000), “students that successfully complete Algebra rather than a general mathematics track are more likely to pursue advanced mathematics and science courses”. When students make the transition from concrete arithmetic to the symbolic language of algebra; they develop abstract reasoning skills necessary to excel in mathematics and science. But the abstract reasoning skills required to solve algebraic problems develop slowly and hence students find the algebra difficult compared to arithmetic and geometry. For developing these skills an advanced training is required. For solving algebraic numerical one should be fluent in the concept of letters and symbols used to represent quantities. Formulas are a part of our lives. One has to drive a car and need to calculate the distance, or need to work out the volume of milk in a container; algebraic formulas are used every day without conscious efforts. Calculating the price of many commodities or profit in business algebra plays a major role. Margins need to be set and calculations need to be made to do strategic planning and analyzing is the way to do it. According to Mishra (2008), “one of the main defect of the existing curriculum is, there is main emphasis on exercise and revision, but it is not highlighted that which part of the content should be revised.” The pre-requisite test can help the students to know their weaknesses in the basic concepts and it also help the teachers to know the level of their students. On the basis of the errors committee by the students in the pre-requisite test teacher can identify the concepts which causes the low achievement of the students in mathematics. This will enable the teachers to provide remedial teaching for weak pre-requisite concepts that can enhance the learning of the new concepts. After revising the pre-requisite, the leaning of new concepts of algebra at class X will definitely provide a better understanding and will also lead to higher achievement in mathematics. Therefore, the present study has been conducted by the investigators.

1.7 Research Questions

i. What is the achievement of the students of standard X in the algebra?
ii. What are the errors committed by the students of standard X in the algebra?
iii. What is the Performance of the students in the pre-requisite test?

1.8 Statement of the Problem

Construction and Try-out of Pre-requisite Test in Algebra of Standard X

1.9 Objectives of the Study

The study was carried out with a view to achieve the following objective:

i. To construct an achievement test for finding the errors committed by the students of standard X in algebra.
ii. To analyse the errors committed by the students in the achievement test.
iii. To construct a pre-requisite test on the basis of the analysis of the errors omitted by the students in the achievement test.
iv. To analyse the errors committed by the students in the students in the pre-requisite test.

1.10 Explanation of the Terms Used

Achievement test: For the present study, achievement test was prepared on four chapter of algebra: Linear equations in two variables, H.C.F and L.C.M. of Polynomials, Quadratic Equations, Rational Expressions on the basis of the learning outcomes for each chapter. The purpose of the achievement test was to identify the errors only.

Pre-requisite test: A Pre-requisite test contained all the items which require to be mastered by the students before a unit/chapter is learnt. In the present study, pre-requisite test was designed based on the errors committed by the students in the achievement test. The pre-requisite test referred to a test containing all the items related to the basic mathematical concepts required to solve the sums of chapters of algebra.

E.g. while solving the example: simplify[illustration not visible in this excerpt] students must have knowledge of

i) Law of indices, i.e.[illustration not visible in this excerpt]

ii) Addition of fractions i.e. when, where and how to take L.C.M.

1.11 Delimitation of the Study

The study was delimited to the Grant-in-aid, Gujarati medium schools of Vadodara city having secondary level following the GSHEB syllabus.

CHAPTER 2 REVIEW OF RELATED LITERATURE

2.1 Introduction

Review of related studies provides an empirical framework to carry out further study. It gives a reflective thinking from the already available studies. It helps to get insight of the problem through studying the past research work which already has been done, the present work which is already going on had provides a direction that which types of work requires to be done in the Future. It provides an orientation to the research regarding types of the research that has been conducted in the field previously. It is necessary that the investigators are aware of the knowledge generated and ongoing process of knowledge generated for a better clarity of the problem and an insight into its methodology issues. For any investigators, review forms the basis for the problem under investigation and helps researcher to arrive at the proper perspective of the study. The purpose of this chapter is to provide an overview of the related study as a foundation for identifying the plan and procedure adopt for present study.

2.2 Review Of Related Literature

Keeping in view the objectives and focus of the present undertaking, total -- studies have been reviewed in this chapter. An attempt has been made to develop a wholistic perspective of the nature and findings of these studies and to draw implications for the present study. In view of the variation in the focus of the studies reviewed, they have been categorized in terms of the following aspects:

- Studies related to Diagnosis and Remediation in Mathematics
- Studies related to Achievement in Mathematics
- Studies related to Achievement in Algebra
- Studies related to Pre-requisite test in Mathematics
- Studies related to Errors in Algebra

2.2.1 Studies related to Diagnosis and Remediation in Mathematics

Ashar (1972) constructed and standardized a diagnostic test in basic algebraic skills for Gujarati medium pupils of secondary schools. The sample consisted of 268 students in all of eight, nine and ten grade from five different schools. The reliability was established using test-reset, parallel form and spilt-half method. Medium reliability coefficient was 0.90. Concurrent validity coefficients against the scores in mathematics in annual examination at eight, ninth and tenth grades were 0.94, 0.91, and 0.98 respectively. Norms in items of standard scores, percentiles were established. Some of the findings were, pupils committed errors due to lack of systematic approach the errors of conceptual type predominated the computational type and Trends of errors continued to a greater extent in the higher grades.

Bhirud (1975) constructed and standardized a diagnostic test in algebra. The main purpose of the study was to construct and standardize a diagnostic test related to some selected units of factorization of grad nine. The try out test was administrated to 370 pupils. Final test consisted of fifty four items. It was administrated to 1,044 pupils. Test reliability coefficients for students ranged from 0.80 to 0.96. The concurrent validity against the marks obtained in the questions on factorization in the annual exam papers was found to be 0.78 Remedial exercises has been developed and outlined. The study revealed that weakness about signs; coefficients and indices were of the basic hindrances to understand and perform algebraic factorization.

Rastogi (1983) has studied on diagnosis of weaknesses in arithmetic s related to the basic arithmetic skills and their remedial measures. The objectives of the study were to establish a relationship between achievement in mathematics and command over basic arithmetic skills, to establish a relationship between command over basic arithmetic skills and attitude towards mathematics and to develop a diagnostic test to determine specific weaknesses of students backward in basic arithmetic skills. The sample consisted of 406 students of grade eight from nine different schools of Arunachal Pradesh. The major findings were: one of the important causes of backwardness of mathematics was the poor command over basic arithmetic skills, when command over basic arithmetic skills improved, attitude towards mathematics became more achievement in mathematics increased. There were no significant sex differences in either attitude towards mathematics. The course of self-help in basic arithmetic skills was equally effective with either sex.

Bhardwaj (1987) has carried out the study on Standardization of a comprehensive Diagnostic Test and Preparation of Remedial material in Mathematics for middle standard students of Haryana. The objectives of the study were to find out the types of errors committed by the pupils in the context of the nature of the teaching units and to construct and try-out remedial material. The test was standardized on a sample of 1146 students (729 boys and 417 girls) belonging to government and aided high schools of Haryana State. On the basis of the Diagnostic test, 377 programmed self-instructional exercises (117 demonstrated, 117 promoted and 143 released) were prepared. The test consisted of three parts that is arithmetic, algebra and geometry, comprising 202, 138 and 158 items respectively. The reliability established through the test-retest method had a coefficient ranging from 0.82 to 0.91 for each of the three areas and the whole test. Intrinsic validity of the test for all the three areas and the whole test ranged from 0.90 to 0.95. The error rate in all the three that is arithmetic, algebra and geometry came out to be 30.4 percent, 50.6 percent and 51.4 percent respectively.

Jain and Burad (1988) have found the following causes as responsible for low results in secondary mathematics in Rajasthan: non-availability of mathematics teaches due to late appointments and frequent teacher transfers; lack of appropriate classrooms; blackboards and other classroom facilities; irregular attendance of students; low standard in the lower classes; non-availability of textbooks; lack of timely correction of homework; overburdened and uninteresting curriculum; lack of child-centered teaching; insufficient period for teaching mathematic; and lack of suitable teaching aids. They have, however, not analyzed how these causes affect mathematics more than other subjects. The sample of the study consisted of rural and urban boys and girls of 100 government and private schools with lower results than those of the private schools of Rajasthan.

Kasat (1991) has made in-depth study of the causes of failures in the S.S.C. examination of Marathi medium high school students in Palghar Tehsil. The sample included 100 boys and girls of twenty five Marathi medium high schools of Palghar Tehsil from October 88 to October 89. Standardize test of numerical ability and self made questionnaire for the teachers were used as tools for knowing the causes of failures in mathematics. The major findings were that most of them had poor intelligence, poor numerical ability, poor comprehension and recall ability, no interest in mathematics, poor study habits, lack of help from parents and teachers and difficulties in certain topics in the course.

George (1993) has constructed and standardized a test to find he mathematical weakness for standard eight students and found that students were poor as far as adding and subtractions of fractions were concerned. Many did not know finding of L.C.M. in case of dissimilar denominators. Bracket multiplication was felt to be very difficult. Students were found to be poor with regard to signs and indices comparing of fractions with negative signs, arranging of decimals in increasing order were the most difficult items. On the basis of thee observations it is doubtful that the learners would gasp the eight standard topic which are taught with the presumption that the learners possess those prerequisites.

George (2003) studied the mathematical backwardness and provided its remediation in Ga. Main objective of the study was to identify backward students on the basis of the scores in the achievement test further construct a diagnostic test in mathematics for standard VII and to conduct in-depth case studies to locate the causes of backwardness and to formulate remedial programmes for the selected case studies. Forty schools of Ponda Taluka were selected using luster sampling technique. The tools used were standardised mathematics achievement test, diagnostic test, intelligence test; interview schedules, Questionnaire and student’s record were used. The major findings were linear equations even fill in the blanks were not responded correctly, remedial programmes showed improvement in terms of attitude and performance and questionnaire brought forth the background of the backward students.

Trivedi (2004) studied the errors committed by the students VIII standard students in Vadodara city. A sample of 120 students was selected randomly by cluster sampling technique. The second periodical test paper of mathematics of elected school was used as a tool. The major findings were that students were not able to apply laws of indices, formulas like (x – y)3, (x + y +z)3, x2 – y2 and procedure of factorization. They were also found weak in addition and subtraction specially while opening the brackets.

Patel (2005) carried out a study on diagnosis and remediation of the learning difficulties in geometry of class VIII students of Vadodara ity. Two schools selected randomly out of the thirty grant-in-aid schools formed the sample. A test based on the learning difficulties was made and administrated to find the errors omitted and then the errors were analysed for providing remedial measures, the major findings were that students scored better in the post test compared to the pre-test and number of correct response of each item also increased.

Shah (2005) studied difficulties in the instructional process of mathematics faced by teachers and students of upper primary class with the objectives such as to study the average performance of students of class five, six and seven in last two years to study the difficulties faced by students in learning mathematics with respect to content and teacher. She found that the students of class five, six and seven have poor commands over the basic fundamentals of mathematics subject. Their fundamentals in mathematics are not up to the mark. Errors omitted by students: concept of place value is not clear to the students. They are not clear when to take L.C.M., why to take and how to take L.C.M. basic concepts in subtraction are not clear. Students get confused in keeping the signs while opening the brackets, when there is negative sign outside the bracket. Students are more concerned with the number in the power and not the sign. Students are not able to apply formulas.

2.2.2 Studies related to Achievement in Mathematics

Lalithama (1975) studied some factors affecting achievement of secondary school pupils in mathematic. The study was conducted on 732 pupils of standard nine selected on a stratified random basis. The tools used were a standardized achievement in mathematics, a study habit inventory and Raven’s standard progressive Matrices. Major findings of the study were: Achievement in mathematics was positively related to intelligence, interest in mathematics, study habits, socio economic stetus and studying lessons daily, repetition in learning and influenced the achievement in mathematics positively.

Chel (1990) has examined the problem of under achievement in the compulsory mathematics in the Madhyamic examination of West Bengal. The investigators found the following causes responsible for underachievement: gaps in knowledge of concepts, difficulties understanding of mathematical language, lack of openness and flexibility in teaching, difficulty in mathematisation of verbal problems and interpretation of mathematical results, the abstract nature of mathematics, fear and anxiety on the part of the students they suggest greater motivation of the students fr learning mathematic, removal of fear of mathematics and clearer presentation of the subject based on the needs of the children.

Ngailiankim (1991) attempted to identify variables associated with achievement in mathematics. The sample consisted of class ten students studying in the central schools located in the states of Nagaland, Meghalaya and Manipur. The tools used were achievement test in mathematics, educational aspiration scale by Sharma and Gupta, occupational scale of Grewal, Differential aptitude test and cattell’s fourteen high school personality Questionnaire. The study revealed that there was a significant association between attitude towards mathematics, educational aspiration, numerical ability and abstract reasoning and achievement in mathematics.

Stacey and Steinle (1999) have conducted a longitudinal study of children’s thinking about decimals. They carried out this study over three thousand students from Grades four to twenty with 5383 tests. The study indicates that less 70% of year ten students (age about fifteen years) understand the numeration well enough to reliably judge the relative size of decimals. On the other hand, more than 30% of grade five students (age about ten years) exhibit strong understanding of this important concept.

Patel (2007) developed a programme for enhancing achievement of the students of the class X mathematics. The Multistage sampling was used the sample for the first stage consisted of thirty two students selected randomly from the 111 Gujarati medium schools in the second stage seventy students were selected from the 719 students who were low achievers. The third stage sample included the seventy parents of the selected students in the second stage. The tools used were information schedule, the pre-test nd post-test and nineteen pre-requisite tests along with that workshop for teachers was also conducted for making the pre-requisite tests. The major findings of the study were: Students could not find values of powers and indices due to insufficient knowledge. Students could not find L.C.M. of algebraic expression due to lack of clarity between H.C.F. and L.C.M. students were unable to solve examples as were lacking in knowledge of multiplication and division in subjects like factors expansion. Students could not simplify simple rational expressions due to inability to express them with some denominators. Students could not simplify due to their inability to cancel factors in same numerator and denominator in addition or subtraction and also in multiplication and division to invert expression in the topic rational expression.

Janet (2008) investigated the relationship among attributes relating to mathematics, parental influences (perceived pressure, perceived support, and family communication), attributes relating to mathematics (perceive usefulness of mathematic motivation towards mathematics, math self-concept) and mathematics achievement. The focus was on an analysis of different relationship outcomes between gifted and non-gifted boys and girls (n=172). The research design of his study was non-experimental and ex post facto. The result of the study found significant difference in math attributes, math attitudes, math attribution and math achievement between elementary and middle school levels.

2.2.3 Studies related to the Achievement in Algebra

Sharma (1978) studied achievement in mathematic of pupils of secondary schools with particular reference to the state of Assam. The sample included 1295 pupils from ten schools. The study was confined to the areas of arithmetic and algebra of school mathematics. A battery of sequential achievement tests were constructed for standard five to ten. Reliability coefficient attained for the achievement test battery ranged from 0.66 to 0.75. Validity coefficient ranged from 0.43 to 0.93. the major factors found responsible for law achievement in mathematic were imparting of limited knowledge, blind use of rules, heavy syllabus, lack of natural urge, in sufficient drill at primary and absence of methodical approach of classroom teaching.

The purpose of the study of Littles & Valerie (2008) was to examine the effectiveness of vertical team teaching instructional initiatives and the impact of the initiative has had on student achievements in one Minnesota Public school district. The district initiated the vertical team teaching instruction at the elementary and secondary levels. The focus of this study, however, was on the secondary level; especially the eighth grade pre-algebra students transitioning in to the ninth grade. Statistical significance found from the impact of exposure to the vertical team instructional strategic on the pre-algebra eighth grade students performance on the Measurement of Academic process assessment.

2.2.4 Studies related to the Pre-requisite Tests in Mathematics

Patel (2007) has designed a series of pre-requisite tests for class X students in mathematics. Basic purpose of these pre-requisite tests was to identify the errors committed by the students which ultimately helped in designing of a programme for enhancing achievement of the class X students. It was found that achievement on pre-requisite test of students was far from satisfactory.

Jayalakshmi (2010) has constructed and tried-out the pre-requisite test in algebra of standard VIII. The main objectives of the study were to analyse the errors committees by the students in achievement test, to study the achievement of the students in the prerequisite test and to analyse the errors committed by the students in the pre-requisite test. He major findings of the study were: ou of the eleven major subunits identified from the achievement test, 98 percent students were weak in the concepts related to L.C.M., 85.06 percent incorrect responses found for the items related to distributive laws, 81.17 percent incorrect responses were found for the items related to polynomials, 74.79 percent incorrect responses were found for the items related to monomials, 61.50 percent incorrect responses were found for the items related to binomials, 79.96 percent incorrect responses were found in the items related to decimals and 57.90 percent incorrect responses were found in items related o fraction.

2.3 Implication Of Review Of Related Literature

Investigators came across few studies related to the present study which can be divided into four categories i.e. studies related to diagnosis and remediation, studies related to achievement in mathematics, studies related to achievement in algebra and studies related to pre-requisite tests in mathematics.

In the first category, the purpose of the studies was preparation and standardization of the test in terms to find out the weaknesses in mathematics and its reasons. The studies were of Ashar (1972), Bhirud (1975), Rastogi (1983), Bhardwaj (1987), Jain & Burad (1988), Kasat (1991), George (1993), George (2003), Triwedi (2004), Patel (2004), Patel (2005) and Shah (2005). Bhardwaj (1987) and George (1993) have constructed and standardized test to find out the mathematical weaknesses and found that students were poor as far as addition and subtraction of fractions were concerned. The studies of Ashar (1972) and Bhirud (1975) were on construction and standardization of a diagnostic test in algebra. Jain & Burad (1988) and Shah (2005) have studied the difficulties in instructional process of mathematics faced by teachers and students of upper-primary class and found that the students have poor command over the basic fundamentals of mathematics subject. The studies of Rastogi (1983), Kasat (1991), George (2003), Triwedi (2004) and Patel (2005) were conducted on errors in mathematics. But only the study of Patel (2004) was conducted on the errors in Algebra. In all the reviewed studies under this category, samples were drawn from secondary schools students except the study of Shah (2005) which has selected sample from upper primary schools students. The studies found that the students are not clear when, why and how to take L.C.M. and non-availability of mathematic teachers due to late appointments and frequent teacher transfer, lack of appropriate classrooms, irregular attendance of students and insufficient periods for teaching mathematics. Ashar (1972) and Bhirud (1975) found that students committed errors due to lack of systematic approach, the errors of conceptual type pre-dominated the computational type and trends of errors continued to a greater extent in the higher grades. The studies of Rastogi (1983), Kasat (1991), George (2003), Triwedi (2004) and Patel (2005) found that remedial programmes showed importance in terms of attitude and performance. They also found that one of the important causes of backwardness of mathematics was the poor command over basic arithmetic skills and through remedial measures command over basic arithmetic skills improved, attitude towards mathematics become more favourable and achievement increased.

In the second category of the presented review, all the studies were survey type in nature except the experimental study of Patel (2007). The purpose of the reviewed studies was to find out the factors or attributes associated with the achievement in mathematics. All the studies were carried out on the secondary school students. Chel (1990), Ngailiankin (1991) and Janet (2008) have found out the factors affecting achievement in mathematics which are attitude towards mathematics, numerical ability, abstract reasoning, gaps in knowledge of concept, difficulties in understanding of mathematical language, lack of openness and flexibility on the part of students, perceived usefulness of mathematics and motivation towards mathematics. Patel (2007) has developed a programme for enhancing achievement in mathematics and found the effectiveness of the programme through improved achievement of the students of standard X.

In the third category of the reviewed studies related to the achievement in Algebra, the study of Sharma (1978) was on standardization of achievement test battery whereas the purpose of the study of Littles and Valerie (2008) was to measure effectiveness of vertical team teaching. Both the studies have carried out secondary students and found that imparting of limited knowledge, blind use of rules, heavy syllabus, lack of natural urge, insufficient drill at primary level and absence of methodical approach of classroom teaching.

Both the studies reviewed under fourth category constructed pre-requisite test in Mathematics. The study of Patel (2007) has designed a series of pre-requisite tests for class X students in mathematics whereas; the study of Jayalakshmi (2010) has constructed a pre-requisite test on the basis of achievement test of standard VIII students. The major finding of the study of Patel (2007) was achievement on pre-requisite test of students was far from satisfactory and most of the students were weak in the concepts related to L.C.M., distributive laws, polynomials and items related to binomials and fractions (Jayalakshmi, 2010).

Hence in the present study, the investigators tried to construct and standardize the pre-requisite test which covers all the units of algebra of standard X.

CHAPTER 3 RESEARCH METHODOLOGY

3.1 Introduction

The theoretical framework as well as review of related literature to the present study has been described in the previous chapter. This chapter focuses on the methodology of the present study. Any research can hardly be completed without the details of a procedure of study to be adopted by the investigator. In fact, this is the soul of research. Unless an investigator has clearly visualized and definitely outlined the sequential steps by which he/she will study a problem in his/her view, he/she can hardly accomplish the task. Plan and procedure includes method of investigation, selection of sample, tools and/or tests to be used etc. it is important part of the total research design. It bears a very close relationship with the purpose of the study and hypothesis or Question of Investigator. Methodology is the part of the Plan and Procedure. It is regarded as the main body of the research. It is desirable to have a proper methodologically designed research Plan. An appropriate methodology and help in getting proper research outcomes.

As per present study is survey in nature; the procedure of scientific research has been followed. In this Chapter, the different aspects of procedure like methodology, research design, Population, Sample and Phase of Study are discussed.

3.2 Methodology

3.2.1 Research Methodology

The present study is quantitative in nature. It has been conducted by employing the survey method.

3.2.2 Population of the Study

All the Gujarati medium non-grant-in-aid secondary schools of Vadodara city constituted as a population for the present study. There were eighty nine Gujarati medium non-grant-in-aid secondary schools in Vadodara city during the year 2011-12. Thus, all the eighty nine Gujarati medium non-grant-in-aid secondary schools constituted population for the present study. All the students of standard X of the eighty nine non-grant-in-aid, Gujarati medium schools of Vadodara city for the academic year 2011-12 following the syllabus prescribed by GSHEB were the population of the study. There were approximately 4895 students were studying in standard X in all the eighty nine Gujarati medium non-grant-in-aid secondary schools of Vadodara city. Thus 4895 students of standard X were formed population of the students.

3.2.3 Sample of the Study

For the first part of the study, out of the eighty nine non-grant-in-aid secondary schools of Vadodara city, two schools were selected randomly. All the students of Standard X of two selected schools were selected as a sample for the achievement test. During the administration of the achievement test, there were ninety students presented so the sample size was ninety for the first part of the study. For the second part of the study, out of remaining eighty seven non-grant-in-aid secondary schools of Vadodara city, twelve schools were selected randomly by lottery method and all the students of standard X of twelve selected schools were the sample for the present study. Thus, cluster sampling technique was used in selecting sample and the size of the sample of the standard X students was 600. The following table 3.1 presents school wise sample size taken under the present investigation.

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