BINDURA UNIVERSITY OF SCIENCE EDUCATION
FACULTY OF SCIENCE EDUCATION
DEPARTMENT OF SCIENCE EDUCATION
AN INVESTIGATION INTO ERRORS MADE BY ORDINARY LEVEL
PUPILS WHEN SOLVING TRIGONOMETRY. A CASE STUDY OF A HIGH SCHOOL, IN MUDZI DISTRICT
BY
CHIPURURA ERNEST
B1441521
A PROJECT SUBMITTED TO BINDURA UNIVERSITY OF SCIENCE EDUCATION IN PARTIAL FULFILMENT OF THE REQUIREMENTS OF THE HONOURS DEGREE IN MATHEMATICS
2017
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BINDURA UNIVERSITY OF SCIENCE EDUCATION
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AN INVESTIGATION INTO ERRORS MADE BY ORDINARY LEVEL PUPILS WHEN SOLVING TRIGONOMETRY. A CASE STUDY OF A HIGH SCHOOL, MUDZI DISTRICT
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BINDURA UNIVERSITY OF SCIENCE EDUCATION
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NAME OF AUTHOR CHIPURURA ERNEST
TITLE OF PROJECT AN INVESTIGATION INTO ERRORS MADE BY ORDINARY LEVEL PUPILS WHEN SOLVINGTRIGONOMETRY. A CASE STUDY OF A HIGH SCHOOL, MUDZI DISTRICT.
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ACKNOWLEDGEMENTS
It gives me immense pleasure to acknowledge my sincere gratitude to my academic supervisor Mr Magondora for his constant help encouragement and support throughout the course of the work. This represents one piece of work that I am proud of, thus I will forever be indebted to them because I could not have made it on my own, as it was a lot of hard work.
I am very grateful to my wife Nomatter and family for all the support they gave me during the study
Also my sincere gratitude goes to the administration of Dendera High School for allowing me to go and carry my investigations even during working hours.
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ABSTRACT
Trigonometry is an inseparable part of mathematics in high school. The purpose of the study was to determine the students’ errors in learning trigonometry with a total sample of forty – five students. The study was also undertaken to establish the causes and ways to remediate such errors. The descriptive survey research design was adopted for this study. The population comprised of two hundred and two form four pupils and six mathematics teachers. All the six mathematics teachers were also part of the sample. The type of error is based on Newman Error Hierarchy Model that includes reading type error, comprehension, transformation, process skill, and encoding error The research instruments which were used to collect data were questionnaire, test and interviews. The questionnaire was designed to collect data from teachers and pupils on errors in trigonometry. The test was administered to the forty-five pupils. Nine pupils were interviewed to establish how they solved given questions on the test. The findings of the study revealed that pupils make some reading, comprehension and presentation errors. Pupils also make some computational errors due to carelessness and also make some errors of commission, omission, translation and rounding off. All these errors are due to some inappropriate strategies used and incomplete simplification. The causes of errors identified were lack of comprehension, lack of strategy inability to translate information, imperfect knowledge, overspecialisation, rote learning and inability to apply the laws of logarithms correctly. Possible remedies to the causes were enlightening the pupils on the errors they make, varying teaching and learning methods, strengthening the foundation by building the basics of the concepts, discourage memorisation and encourage neat presentation and checking of solutions frequently. The researcher recommended that the Ministry of Primary and Secondary education should continuously engage on some in-service staff developments. The in-service staff developments help and equip teachers who are already in the field with the requisite skills and power to manage errors as they surface at early stages.
LIST OF TABLES
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Table 2.1 Newmann hierarchy mathematical problems…………………………………..….14
Table 4.3.1Responses from teachers on types of errors made by pupils……………………..31
Table 4.3.2 Common errors done by pupils……….………………………………………..32
Table 4.3.3 Frequency of errors in solving trigonometry…………………………………..35
Table 4.3.4Comparison of the number of answers to question A2 and B4…………………..35
Table 4.6.1 Pupils response on closed questions…………………………………………41 Table 4.7.1 Pupils evaluation of teachers…………………………………………………….42
LIST OF FIGURES
Fig 4.1 Pie chart (Teacher qualifications)……………………………………………………26
Fig 4.2Bar chart (Teacher responses on errors made by pupils when solvingtrigonometry)..
LIST OF APPENDICES
Appendix 1: Questionnaire to teachers…………………………………………………54
Appendix 2: Questionnaires to pupils………………………………………………….57
Appendix 3: Interview questions to pupils…………………………………………….59
Appendix 4: Interview questions to teachers………………………………………….60
Appendix 5: Test guide to pupils………………………………………………………61
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TABLE OF CONTENTS
CONTENTS PAGE APPROVAL FORM…………………………………………………………………………i RELEASE FORM……………………………………………………………………………ii ACKNOWLEDGEMENTS……………………………………………………………… .iii ABSTACT……………………………………………………………...... iv LIST OF TABLES…………………………………………………………………………...v LIST OF FIGURES………………………………………………………………………….vi LIST OF APPENDICES………………………………………………………………...….vii TABLE OF CONTENTS……………………………………………………………..……viii
CHAPTER ONE……………………………………………………………………..……….1 1.1 Introduction ………………………………………………………………………….…....1 1.2 Background of the study……………………………………………………….………….1 1.3 Statement of problem……………………………………………………….……….….…3 1.4 Research questions……………………………………………………………….…..……4 1.5 Research objectives………………………………………………………….……..……...4 1.6 Assumption of the study……………………………………………………….…..………4 1.7 Significance of the study……………………………………………………….…..……...5 1.8 Delimitation of the study…………………………………….……………………..……...5 1.9 Limitations of the study………………………………………….………………..……….6 1.10 Definition of terms…………………………………………….……………….………...6 1.11 Summary……………………………………………………….…………………….…..7
CHAPTER TWO…………………………………………………………………….……….8 REVIEW OF RELATED LITERATURE………………………………………….….……...8 2.0 Introduction……………………………………………………………….…….……….8 2.1.1 The nature of mathematics……………………………………………….……….……..7 2.1.2 Errors ’classification and their causes in solving trigonometry problems…………..10 2.2 How can these errors be minimized ……………………….……………………………17. 2.3 Nature of solving triangles ……………....…………………………………………….. 18 2.4 Summary…………………………………………………………………….….……...20
CHAPTER THREE…………………………………………………………………...……21 RESEARCH METHODOLOGY……………………………………………………….…...21 3.1 Introduction………………………………………………………………….….……….21 3.2 Research design……………………………………………………………………….…21 3.3 Population…………………………………………………………………………….….23 3.4 Sample……………………………………………………………………………….…..24 3.5 Research instruments…………………………………………….………………….…...24 3.5.1 Test………………………………………………………………………………….....24 3.5.2 Interview………………………………………………………………………….……25 3.5.3 Questionnaires……………………………………………………………………..…...26 3.6 Validity……………………………………………………………………………..…..27 3.7 Data collection procedures…………………………………………………….……..…..28 3.7.1 Test and Interviews………………………………………………..……………….…..28 3.8 Data presentation and analysis procedures……….………………………..………….…29
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3.9 Summary……………………………………………………...…………………….……29
CHAPTER FOUR…………………………………………………………………..……....30 DATA PRESENTATION AND ANALYSIS…………………………………...…….….….30 4.1 Introduction………………………………………………………………...……….……30 4.2 Teacher qualifications……………………………………………………………...…….30 4.3 The types of errors pupils make when solving trigonometry.….………………………..32 4.3.1Pupils’ errors from the test……………………………………………………………..32 4.4 Misused data error………………………………………………………...... 38 4.5 Causes of errors in solving trigonometry………………………………………………..38 4.6 Pupils responses on closed questions……………………………………………..….…..41 4.7 Pupils evaluation on their mathematics teacher………………………………..……..….42 4.8 Possible ways of dealing with errors in solving trigonometry…………..…...... 44 4.9 Summary………………………………………………………………………....……..43
CHAPTER FIVE……………………………………………………………………...…….48 SUMMARY, CONCLUSION AND RECOMMENTATIONS……………………...…..…..48 5.1 Introduction……………………………………………………………………...……….48 5.2 Summary……………………………………………………………………...…….……48 5.3 Conclusion…………………………………………………………………...…….……..49 5.4 Recommendations………………………………………………………………….…….50 5.5 Summary………………………………………………………………………………....52
REFERENCES……………………………………………………………………………...53 APPENDIX……………………………………………….………………………...……….55
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1. CHAPTER ONE 1.1 Introduction
The thrust of this chapter is to find background to the study, statement of the problem, research objectives and purpose of the study. Assumptions, delimitations, limitations and definition of terms were also discussed in this chapter. On background of the study, the researcher discussed on what really stimulates the desire to carry out the research. The research questions were stated in this chapter and were answered at the end of the research. Finally on limitations and delimitations, the researcher highlighted some of the cases which might reduce the correct outcome of the research and gave some ways to limit the cases.
1.2 Background of the study
Trigonometry is an inseparable part of mathematics in high school. As a maths teacher for the past ten years I have seen students struggling to solve trigonometry problems. It appears as if trigonometry is an area of mathematics that students believe to be very difficult and abstract compared with many other subjects. In this area very few pupils enjoy and succeed the majority of the students hate and struggle with it. I saw first-hand students making multiple errors when it came to solving trigonometry problems. In addition, it seemed as though many students were trying to memorise procedures to solve trigonometry for tests but not returning the concepts as the year progressed, hence the need to find out what common errors they make in order for one to become an effective teacher when teaching the concepts. Evidence from other teachers over the years has consistently confirmed that teaching trigonometry in secondary schools is difficult, and even though students can often memorise the rules without making effort to understand them and solve problems correctly, their understanding of the fundamental nature of triangles remains in doubt. It was seen that the subject of trigonometry stands on the level of establishment of relationships between angles and triangles .It has also been noticed that the knowledge of trigonometry which is needed to be successful in “A “level studies is insufficient thereby also contribute to the high failure rate at “A” level. Zakaria (2010) alluded that these errors in solving trigonometry problems often occur directly or inadvertently ranging from written to oral and even computation. ZIMSEC examiner’s report (June 2010) indicates that most students do not show understanding of properties of triangles hence the question on the aspect was poorly done by the majority of pupils .Also according to ZIMSEC’s report (November 2010) further exposes that most pupils have common error on uses of triangle properties on answering given questions triangles. Findings has propounded that students’
1 errors are systematic and it follows a certain trend in trigonometry problems. Keeping on making similar errors will not only affects their academic performances but goes as far as affecting their everyday lives since wrong judgements and choices are detrimental to personal and national development options in their lives Thompson, (1993). Mathematics is one of the core subjects to be offered to all students till the tertiary levels of education Salau, (1995). In Zimbabwe, both primary and secondary teacher training colleges have ordinary level mathematics as requirement for enrolment to prospective teachers. Furthermore in Zimbabwe, Mathematics competence is a critical determinant of the Post-Secondary educational and career options available to young people Barrow and Woods, (1987). Due to this there is need to reduce the errors made by the pupils when calculating trigonometry problems in order to improve the pass rates in mathematics.
Failure to use appropriate teaching methods and learning aids on students may lead to rote learning, using the symbols mechanically without much understanding, appreciation and linking the subject to real life and this will be having a bearing on the error made by the students. The misconception of non-utility of mathematics in actual life arises because of the fact that mathematics taught in the classroom is generally divorced from the mathematics of real life (Sudhir and Ratnalikar 2003: 5) and yet mathematics can lead to organized life. A well- organized life involves fixing time, prices, wages, rates, ratios, fares, percentages, targets exchanges, discounts, areas, volumes and many other measurements without which life would be disorderly and chaotic. As a result there is need to reduce errors in trigonometry especially in angles and triangle in order to improve the pass rate.
There is a general awareness all over the world for the need to pay greater attention to the improvement of mathematical teaching and learning skills in order to minimise errors as possible. The reason is that mathematics is a core subject which applies to all subjects and also applies in real life situations but most of the pupils are not performing well. Kasanda (1997) looked at factors affecting pupils’ performance. Its main findings were lack of suitably qualified teachers, poor pupils’ elementary, mathematics background, lack of text books and lack of teaching aids and also unnecessary errors in other topics. Kasanda’s findings also made the researcher investigate the actual errors made by pupils in trigonometry as one of the topics in mathematics. Pupils often set mathematics aside as a cause for concern despite their limited exposure to it and this negative attitude towards it results in doing it without much attention
2 causing them to make vast errors (Holeys, 1982); as a result they tend to make unnecessary errors. It is a subject unlike most others, since it requires a considerable amount or perseverance from the individual in order to succeed and avoid these errors as much as possible. Also, due to the fact that trigonometry is found in the secondary curricula of secondary schools around the world, they are worth studying. Their importance, combined with perceived difficulty and errors, motivated this research project – project designed to find out more about ways in which students understand triangles and the kinds of errors they make and how they can be minimised. This then was the burning issue in the researcher’s mind, to try and pin point some driving forces behind the above and some possible remedies in a bid to reduce errors made by students in their study of trigonometry.
1.3 purpose of the study
The main purpose of the study was to analyse students‟ error in learning of trigonometry which focused on the manipulation of trigonometrically ratios using formula and the right-angled triangle. Specifically, the study sought to:
1. Determine the extent to which individual students commit error in the learning of trigonometry.
2. Find out the possible categorization of these errors in the learning of trigonometry
1.4 Statement of problem
Trigonometry is an important branch of Mathematics which cuts across all spheres. Pupils are introduced to concepts of solving trigonometry at lower grades and the concepts are continuously developed at advanced level. Ideally pupils are expected to master these concepts they have learnt and apply them in real life situations. However, this is not the situation, as pupils continuously make errors when solving trigonometry at ordinary level hence fail to have a base for mathematics courses including calculus, differential equations and complex analysis. According to Alkan, H and Altun, M. (1998) this has a direct bearing to the future of the pupils. Pupils who fail to comprehend basic concepts on solving triangles will have problems in related areas like astronomy and the study of population growth. Such challenges have promoted the researcher to carry out the study with the aim of exploring pupils’ level of understanding triangles, consequently identify the types of errors they make in order to bridge the gap between learning events and real life situations.
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1.5Research questions
The study attempts to answer the following questions
i. To what extent will individual students’ error influence the learning of trigonometry?
ii. What type of errors do students make when solving problems on trigonometry?
iii. What are the causes of the errors in solving trigonometry problems?
iv. What are the ways of avoiding these errors?
1.6 Research objectives
By the end of the study the research must be able to:
i. Identify the influence of an individual student’ error in learning of trigonometry,
ii. identify the type of error made by students when solving trigonometry problems,
iii. outline causes of the error in trigonometry,
iv. establish ways of reducing errors in solving trigonometry problems.
1.6 Assumptions of study
The assumptions of this study are that the selected sample will wholly represent the population and the research will meet all the required results. Also it will be the researcher’s assumption that respondents will fully cooperate with the researcher. The selected sample is assumed to have covered something to do with trigonometry in the previous form. The researcher also assumes that pupils have done simple arithmetic that every pupil is able to do basic operations in mathematics. The participants are literate and are able to answer the questionnaire in an academic and scholarly way.
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1.8 Delimitations of the study
The research was confined to one school in Mudzi district, Kotwa High School. The study starts from December to September 2017. The research was conducted by form fours. The researcher focused on errors made by ordinary level pupils in solving trigonometry focusing mainly on the ratio and triangles
1.9 Limitations of the study
Due to economic hardships and the nature of the course the research was conducted under very limited time and finances to effectively carry out the research. Money for photocopying and printing questionnaires was quite exorbitant. In addition to that some teacher felt that they were being observed so they did not produce the required information. The researcher got hostile responses from some of the respondents. Questionnaires, interviews and tests which were used as instruments for collecting data have some drawbacks. The disadvantage of interviews is that it takes time to interview pupils one by one and in group interviews the ideas will reflect the views of the subjects who are more verbal and thereby few comments for those who are reserved.
1.10 Definitions of terms
Error
Generally, an error means a simple lapse of care or concentration which almost everyone makes at least occasionally. -the American heritage Dictionary (2009) defines an error as an act, assertion, or belief that unintentionally deviates from what is correct, right or true. According to Webster College Dictionary (2010) defines error as a deviation from accuracy or correctness .In this research the word shall be used to refer to calculation mistakes made by students during problem solving any method.
Word problems
Some of the word problems used in this study lead to the process of inquiry, in which students had to develop methods for exploring unfamiliar situations (Ministry of Education, 2007). In these problems, students had to consider real world situations and represent them in mathematical form.
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Pupil
- a person who is studying, usually in a school.
- is a child or young person in school doing his or her studies.
TRIGONOMETRY
-is the branch of mathematics which deals with the measurement of sides and angles of triangles and their relationship with each other.
1.11 Summary
This chapter clarified the background to the study and its significance to the learning and teaching process. The main focus of the study was on research problem. In the same chapter the researcher outlined the limitations of the study and also its delimitations. Finally major terms used in the study have been defined. In the next chapter the researcher is going to dwell on the literature review put forward by other authorities regarding the concept of the topic.
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CHAPTER TWO
REVIEW OF RELATED LITERATURE
2.0 Introduction
The main aim of this chapter is to have an overview of what other researchers have said about the research topic. Literature review involves reading through and recording work or researches that have been written on the research topic or related area the researcher intends to investigate. The related literature is important as it guides the research topic on how some misconceptions and errors were explained in the context of this study and other studies concerning analysis of errors in solving trigonometry problems. The researcher shall start by outlining the nature of mathematics then identify some causes of general errors in solving trigonometry problems done by other researchers. Literature review on the cause of errors will also be looked at followed by the ways to which these errors could be avoided or minimised. Also the researcher will have a brief overview of nature of solving triangles in particular.
2.1 Conceptual framework of errors in solving mathematics.
2.1.1 The nature of Mathematics
Mathematics is an area that students believe to be particularly difficult and abstract compared with the other subjects taught at schools. It is an area that very few students feel very comfortable with and succeed at, and most students have negative attitude towards it and struggle with the subject. Mathematics concepts do not exist in isolation they intertwine so for students to know this subject they need to know the basics of the whole subject. There are diverse views about the nature of mathematics. Some say mathematics is the science of patterns and relationships (Hersh, 1986). Alludes that, Mathematics can be looked as both an applied and theoretical discipline. According to Blanche (1966), mathematics relies on logic and creativity which includes them among some axioms. Axioms are believed to be basic truths which need no justification beyond their own self evidence. Teachers with instructional beliefs belong to the absolute continuum view mathematics content as static fixed or sacrosanct. They
7 believe that sources of legitimate mathematical processes are correct applications of axioms, definitions and theorems (Davis & Hersh (1981). On the other side other view mathematical knowledge as which can be individually or socially constructed by learners through observations, experimentations and abstractions using senses and can therefore be fallible. The nature of mathematics can be viewed in different ways but the only truth above all is that it is a core subject that needs every person to be aware of it. Thompson (1985) notes that a mathematics curriculum is a collection of activities from which students may construct mathematical knowledge and that it is a sequence of activities, situational context from which students construct a particular way of thinking. The dependence of other mathematical disciplines on trigonometry shows that students can construct knowledge from other mathematical disciplines using it. Bhatia (1997) said, the importance of mathematics is a significant not only to any expert technologist but also for the common man. Ernest (2004) says for others, as well as many scientist and engineers, the principal value of mathematics is how it applies to their own work. As a discipline, mathematics is pursued by people for a variety of practical purposes and for its intrinsic interest and scholarly challenges. At this point we would like to indicate that a good knowledge of trigonometry themes will help students to construct knowledge in other disciplines of mathematics. Due to mathematics’ important role in culture, some fundamental understanding for nature of mathematics is requisite for scientific literacy. For this to be achieved, pupils need to recognise mathematics as part of scientific venture, understand the nature of mathematical philosophy and become acquainted with crucial mathematical facts and skills. Mathematics is described as the backbone of the society, mother of all Sciences and the handmaid to all arts and sciences and thus has recognised its importance.
Macnab and Cummine (1986) alleged that the personality of mathematics starts from the discipline itself. It is how beliefs are transmitted to the children. They also viewed mathematics as abstract and mystery accessible to few on a set of rules and details to be remembered. Such external views of mathematics treat it as an unfamiliar subject. In this view, only a few will make some extra effort in solving mathematical problems hence a large number of pupils only attempt for the sake of formality, leading to a rise in errors and misconceptions. Mathematics education is based on problem –solving, application of knowledge and manipulations problems and when the students meet word-problems, their non-systematically and uncompleted knowledge cause faults and conceptual mistakes. Mathematical knowledge is interrelated and error in one branch of mathematics may be carried into other areas of mathematics. A poor mastery of basic concepts may limit a learner to pursue other areas of study. This has triggered
8 the researcher to carry his research in trying to find means of how to minimise or eradicate errors in trigonometry in particular problems in solving triangles
2.1.2 Errors’ classification and their causes in solving mathematical problems.
According to Li (2006) student errors are the symptom of misunderstanding. He went on further to state the most common error among the students are systematic and non-systematic. Li (2006) alludes that among many different types of errors, systematic errors occur to many students over a long time period. The major cause of systematic error is the procedure and conceptual knowledge. When an error occurs due to a mistake, blunder, miscalculation or misjudge it is known as unsystematic errors .Zakaria (2010) points that many studies concerned with mathematics education explain that students have misconceptions and make errors. He went on further and says these errors in solving mathematical problems often occurs directly or inadvertently ranging from writing to oral and even computation. Ryan and Williams (2000) postulate that of late a few researchers have mentioned students’ misconceptions ,error and related to these ,learning complexities about trigonometry and this has made the researcher interested in trying to find error made by students when solving triangle.Fi (2003) stated that much of the literature on trigonometry has focused on trigonometric expressions and metaphors. This has stimulated the researcher to research about the errors involved when solving triangle problems. Brown (2006) studied students ‘understanding of sine and cosine rules. She reached a framework, called trigonometric connection. The study indicates that many students had an incomplete or fragmented understanding of the three major ways to view sine and cosine: as coordinates of a point on the unit circle, as a horizontal and vertical distances that are graphical entailments of those coordinates, and as ratios of sides of a reference triangle. Orhun (2002) studied the difficulties faced by students in using trigonometry for solving problems in trigonometry. Orhun found that the students did not develop the concepts of trigonometry certainly and that they made some mistakes. This has arouse the researchers’ interest to dig deeper trying to find out what are the major causes of the errors in solving trigonometry problems. Is it the traditional methods used by the teachers or what actually is causing the errors in this particular area?
Different Authors classified errors in different ways but however in their classification there is an element of resemblance or oneness. Radatz (1979) classified errors in terms of (1) language difficulties. He pointed out that as mathematics is a language, students need to understand its concepts, symbols and vocabulary. Failing to understand may cause students to make errors.
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(2) difficulties in processing iconic and visual representation of mathematical knowledge. (3) deficiency in requisite skills, facts and concepts for example students may fail to recall related information in solving the problem. (4) Incorrect association or rigidity that is negative transfer caused by decoding and encoding information. Orton has his way of classifying the error committed by students during answering questions on trigonometry.
Orton (1983) classified errors in into three categories as follows
1) Structural error: an error which arises from some failure to appreciate the relationship involved in the problem or to grasp some principal essential to solution
2) Arbitrary error: is that error in which the student behaves arbitrary and failed to take into account the constraint laid down in what was given.
3) Executive error: is that error where students fails to carry out manipulations through the principle involved may have been understood.
Confrey (1990) says that the basis for research for pupils’ errors and misconceptions comprises three major customs. Each custom has its unique epistemological assumptions. According to Confrey (1990), these epistemological assumptions are; “Piagetian studies on the generic epistemology, applications of philosophy of science in the tradition of conceptual change, and research on systematic errors. Research on systematic errors focuses on mathematics and computer programming whereas Piagetian studies and philosophy of science in the custom of conceptual change focuses on conceptions in mathematics and science”. Piagetian work on pupil conceptions looked at the development of pupil understanding of particular mathematical and scientific concepts over time. Piaget’s major theory was that pupil’s awareness is a process not a state. Thus student knowledge needs to be examined in view of its developmental relations. Hence Piaget looked at pupil’s conceptions not misconceptions. Confrey (1990), says that researchers in the custom of systematic errors have acknowledged that students hold mini theories about mathematical and scientific ideas. Posamentier (1998) points out that several studies have many raw theories, misconceptions or preconceptions about mathematics that meddle with their learning. Pupils are much attached to their misconceptions due to the fact that they are actively constructed from their experiences (Posamentier, 1998). According to Radatz (1979) it is for this reason that they stumble to leave them up. He went on to say that many causes of errors and misconceptions in mathematics can be identified by looking at
10 mechanisms used in processing, obtaining, retaining and remembering the mathematical tasks. Radatz identified four categories of these errors namely error due to lack of mastery of prerequisite skills, errors due to incorrect associations or stiffness of thinking leading to rigidity in decoding and encoding new information and the mortification of processing new information, errors due to the application of un related system or strategies and errors due to dealing out with iconic presentations.
Errors are a great concern on which many authors looked into. Barrera et al (2004) looked into errors caused by lack of meaning and postulates that such errors can be classified into three stages which are errors originating from arithmetic, errors originating in use of formulas or procedural errors and structural errors which are due to properties. A theory in the seminal analysis of students’ performance in mathematics was developed by Fischbein and Barash in 1993. The theory is loosely linked to the three components of knowledge namely algorithm, formal and intuitive. Fischbein and Barash further elaborate that algorithmic knowledge is the ability to use theoretically justified procedures, formal knowledge is the ability to use definitions, theorems, axioms and proofs whereas intuitive is self-evident cognition. Thus students experience some errors and misconceptions in these three areas of knowledge. However intuitive knowledge often hinders and manipulates the formal understanding and (or) the use of triangles and angles when solving trigonometry problems. Errors also occur when solving schema is mistakenly applied despite the correct intuitive way of thinking. Fischbein (1994) says that distortions and correct answers are blocked as a result of intuitive interpretation based on limited and a primitive strongly rooted individual experiences.
If the student does not understand the mathematical concepts meaningfully, it is natural they will be committing errors in performing the operation. Sarala (19900 has analysed the conceptual errors of secondary school student in learning selected areas in modern mathematics has found that the number of errors are quite large and these errors are influenced by sex, locality of the school, management of the school , intelligence ,study habits and socio economic status.
The conceptual framework that is used in this study is based on Newman Error Hierarchical Model. The model of error investigation proposed by Newman (1977) has proved to be a reliable model for mathematics teachers. The framework has six types of errors: reading error, comprehension, transformation, process skill, encoding error and carelessness. The Newman Error Hierarchical Model is suitable to be used in identifying students‟ error in mathematics.
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This model has the hierarchy that classifies types of error based on the problem solving level done by students Newman (1977) asserts that the errors pupils make are systematic and can be classified into various categories in which the students interact with the question. These can be best categorised by the sequence of steps in problem solving. According to him, such errors can be categorised as: reading and comprehension, transformations, process skills and encoding. Clements (1980 .p .4.) has the Newman hierarchy mathematical problems table below.
Table 2.1: Newman hierarchy mathematical problems.
Characteristic of the Interaction between the question and the person. Question. Encoding careless Process skills
Transformations
Comprehension motivation Question form Reading
The substance of Newman’s replica is that it provides an all-inclusive stage wise process to analyse mathematical crisis solving tasks. Using this structure, Newman found that 47% of her low achievers made errors earlier to the process phase, of which almost 12% were at the transformation stage. Casey (1978) and Clements (1998) later adopted the Newman’s model for their studies. For different group Clements found that fewer errors were made at the bottom levels; a quarter of the errors occurred at transformation phase. Clements hinted that errors in the early phase of problem solving may lead to selection of erroneous process later.
The distinctive characteristic of the Newman model is that it is appropriate for word problems even though there is no limitation of its use in other areas as well. The left part of Newman’s diagram represents complications of understanding the question which is named as the question form. This points out to the necessity of providing relevant and appropriate questions. Even if the correct questions are provided, various pupils may read between the lines differently from
12 the inferred meaning. The right part represents five stages in problem solving. Mostly, the model reflects when a student produces an inaccurate answer to a subject matter, the blunder resulting in that response may have taken place at one of some stages in the practice of solving the problem. The pupil might have experienced a reading error by misreading the question, or comprehension error could have occurred by misinterpretation of the question. Even if the learner has accurately comprehended the question, he or she may wrongly transform it into math language. On the other hand, despite accurate transformation, a wrong method may have been used to answer the problem thereby encountering a process error. All the steps above may be correct, however the answer may be wrongly encoded translating into encoding error. The pupil may fail to explain or verify the answer and this result in verification error
Besides all the given scenarios there could also be other possibilities such as the possibility of any combination or interaction of above errors. There could also be some psychological factor besides mathematical factors. These other factors include low attention to the task, carelessness, lack of motivation and/or anxiety. Anxiety is a familiar feature in problem solving circumstances. Where there is anxiety, confidence is encoded and there is interference with thinking. Posamentier (1998) deduced two different components of math anxiety which are cognitive or intellectual and emotional or effective. The cognitive component basically involves nerve-racking about failure and its consequences. The effective factor involves fear, being uncomfortable and feeling nervous. This component has a stronger and greater negative connection to student’s mathematics performance.
Posamentier (1998) also noted that pupil’s lack of academic self-concept as another influence to these errors. This involves one’s inability to have self-confidence and lack of self-assurance in one’s capability to achieve, no self-sufficiency and failure to recognise own strengths and weaknesses. Errors due to learner’s affective attitudes are diverse types. At times lack of concentration is caused by self-importance, blockages, or poor memory. All these issues which were not addressed by previous researchers motivate the researcher to carry on with the research focusing on solving trigonometry problems which is a critical area in the field of mathematics.
After observing that Newman’s model was mostly meant for word problems, Watson (1980) used Newman’s model of the sequence of steps in problem solving including other questions which are not problems: reading and comprehension, transformation, process skills, and encoding, to identify students possible errors. He thought that students’ errors may be due to
13 deficiency in one or several of the above steps. In order to verify those hypothesises about students errors, Watson designed both word and computation problems to compare errors made by the two groups of students, with lesser and greater abilities. He found out that most initial errors made by the more able group were at the stage of reading and comprehension. However, the less able group students made many more errors when applying and selecting mathematics processes. The above classification method was simply used to describe students’ errors, but lacked detailed analysis of why students were unable to perform well in some steps. For example, why did students not select correct mathematics processes or operations? What strategies effectively helped students make correct decisions? Why did students have special difficulty in understanding mathematical language?
Being aware of the shortcomings of classification methods, Ashlock (2002) not only categorised students’ errors in computation, geometry and algebra but also tried to attribute errors to overgeneralising. However all the errors looked at are broad hence the need by the researcher to narrow down to errors specifically on solving trigonometry problems. Biber, Tuner and Korkmaz (2013) looked at errors pupils make in the subject of angles under geometry. It is pointed out that pupils make conceptual errors. More work on the study of errors in geometry was covered by (Presscott, Mitchelmore and White 2002; Thirrumurthy 2003; Mayberry, 2003 and Van Hiele-Geldof 1994). All the mentioned researchers concur that errors made by students in solving geometrical questions are procedural and conceptual. The researcher discovered that a gap was left out by previous researchers in the instrumentation errors which occur during solving trigonometry problems. It is in this light of the above that the study set out to analyse students’ error in learning of trigonometry in a school that is located in Mudzi District area in Mashonaland East province
2.2 How can these errors be minimised?
Gunawardena (2011) says that errors in solving problems cannot be easily dislodged since the misconception and errors are deeply seated in students thinking process. This is largely due to the fact that they become attached to the notions they constructed during the learning process. He further highlighted that students might appear to have overcome problem only to resurface at a later stage. Gunawardena suggests that to effectively eliminate the errors and
14 misconceptions, the students must actively participate in the concepts. The statement of Gunawardena concurred with that of Sidhu (1995) who recommends that children should be active participants in the learning. The NCTM (2000) agrees with this notion as it points out that the process does not entirely depends on the teacher. This implies that pupils have to be aware of the errors they are bound to make. In each case through practice and awareness, they tend to attach the correct procedures in their problem solving situations. Copper (1974) argues that active learning can be supported by the use of relevant media which stimulate interest to learn. When pupils are interested they turn to learn much faster and they will grasp the concept and retain it for a longer period because it will be their thing. This will increase concentration and reduces chances of making errors.
Teachers also play an important role in the total elimination of errors and misconceptions made by the pupils. NCTM suggest that teachers should practice classroom learning environments that help the pupils to develop both conceptual and procedural knowledge so that they construct correct conceptions right from the beginning. Gunawarden (2011) has high hopes that a powerful strategy of exposing pupil’s error and misconceptions is to use methods which are pupil centred. Many teachers prefer to teach the class as a whole all the time. However Cohen and Manion (1986) noted that smaller groups have higher level of participation weak pupils will benefit much from being taught in smaller groups. So when the teacher selects teaching methods for mathematics concepts he or she should allow active learning. The teaching methods should provide opportunities for learners to observe, discover, think reflectively and practice. Sidhu (1995) point out that pupils learn by problem solving its self as a method of learning by self –effort. Teacher should allow pupils to explain their thinking process as they give better insights to teacher of pupils to have a look of how the pupils are thinking and it also gives the teacher some allowances and opportunities to rearrange their misunderstanding. This is a powerful tool to eliminate error since the teacher will be awarded an opportunity to listen to pupils expressing their self and refine their mathematical thinking.
According to Woodward (2004) another way of dealing with the error is to uncover them and find the schema where deep seated misconception lies. If the pupil is able to discover his or her error then it will be easy for the pupil to unearth the error. Also it is crucial for the teacher to assess the error of pupils and engage with pupls in a way which rearranges the concept. When pupils identify error for themselves and fellow students help each other to solve them we will have eradicate the error for ever. This concurred with the findings of NCTM (2000) which say employing diagnostic methods of teaching is central to constructivist methods which calibrate
15 the teacher’ instructions based on the student need. So if the student themselves take an active role in solving problems in trigonometry it will minimise or reduce their chance of committing errors
2.3 Nature of solving triangles
Triangles can be classified as a right angled triangle and non- right angled triangle .Channon (2010) et al say that solving a triangle means finding all the three sides and three angles given some features of the triangle. To do these three items must be given which may include, two sides and an angle, two angles and a side or all the three angles. If only the three angles are known then some several similar triangles with same size of angles and different length can be given therefore one of the lengths should be given. Triangle can be solved in different approaches as they take different forms which include right angled triangles, acute triangles and obtuse triangles. Channon et al (2010) clarifies that given any two of the interior angles, the third angle can be found using the tool of interior angles of a triangle sum to 1800.Another tool to solve right angled triangle is the Pythagoras theorem. The use of trigonometry functions can be employed to solve triangles. These trigonometry functions relate with two other sides. When solving non- right angled triangles there are laws which need to be used? According to Channon et al, the methods used to solve non- right angled triangles are the law of sine and the law of cosine
Law of sine
Sin A = Sin B = Sin C
Law of cosine
1. A2= B2 + C2- 2 BC Cos A
2. B2 =A2 + C2- 2AC Cos B
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3. C2= A2 + B2- 2AB Cos C
Summary
This chapter exposed and dug deeper on what other researcher have said on the related topic. The researcher critically analyses and found out the missing links in the existing knowledge of trigonometry. This literature review has shown the general understanding of the nature of mathematics. It has also shown the types of error and ways to deal with errors discussed by previous researchers at a broad spectrum. In the past the research on student’s error have focused much on misconceptions and related to these learning complexities about trigonometry. A few researchers studied more on specific issues in trigonometry such as understanding errors made in the cosine and sine law when solving triangles. The researcher feels that not much was done pertaining to the specific error sources of the trigonometry. Thus the researcher feel more research could be done by narrowing the causes to specific topics which have proved unfamiliar to the students at ordinary level as highlighted by the ZIMSEC’s report on writing examination. This has brought the zeal for the researcher to carry such a research in errors encountered by the student when solving trigonometry in particularly solving triangles and identify exact errors from the solutions of the students. The researcher has tried to close the identified gap and come out with possible solution. The next chapter will deal with the methodology of the study.
CHAPTER THREE
RESEARCH METHODOLOGY
3.1 Introduction
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In this chapter the researcher will explain the main methodological constructs that were employed in various stages of the study and later unite them together to create an overall summary of the methodology. Research methodology involves the procedures that are taken by the researcher to collect data. It also involves the instruments that are used to collect data and their validity and reliability. The researcher will use tests, questionnaires and interviews to carry out the research. In this chapter targeted population and sampling techniques are discussed. The population for the study consists of all ordinary level pupils at Kotwa high school in Mudzi district .The study was conducted using a sample of forty-five pupils randomly drawn from the ordinary pupils at the school.
3.2 Research design
According to Brew (2001), the general purpose of research is to contribute to the body of knowledge that shapes and guides academic and practice discipline. The research is going to clarify all the errors which we committed by the pupils and enhance knowledge that will overcome them. According to Chiromo (2006), research design is a set of plans and procedures that reduce and simultaneously help the researcher to obtain data about isolated variables of interest. The researcher adopted a mixed method research. According to (Johnson& Christensen, 2008: Creswell, 1998: Gay, Mulls & Airasian, 2006) mixed methods researchers believe that they can get richer data and strong evidence from knowledge claims by mixing qualitative and quantitative methods rather than using a single method. Johnson and Christensen, (2008) allude that a sequential design is necessary if development is an objective of the research design. In this research the researcher employed a sequential design which flowed from quantitative to qualitative method. The quantitative part helped the researcher to understand student errors and misconceptions numerically while the qualitative part helped the researcher to deepen his focus to explain more about those errors through student reasoning process. This idea is further reinforced by the belief that social phenomena are extremely complex and in order to understand them better, we need to employ multiple methods. The researcher employed a sequential explanatory design which is characterized by the collection and analysis of quantitative data followed by the collection and analysis of qualitative data (Creswell, 1998) typically; the purpose of a sequential explanatory design is to use qualitative results to assist in explaining and interpreting the findings of a primarily quantitative design. Also the descriptive survey was used since studying a limited number of items with a view of drawing conclusions on a sample (Borg and Gall: 1993). The descriptive survey design was used since the researcher intended to study limited number of cases in order to draw
18 conclusions that cover the generality of the targeted population. Chiromo (2006) says that descriptive survey describes what we see and allows the researcher to investigate the subjects in their natural situations like schools. Descriptive survey was used because it involves collection and documentation of original data. Descriptive survey also presents a chance to combine both quantitative and qualitative data. Quantitative survey allowed data collection from a larger number of pupils than is generally possible with experimental or quasi– experimental research designs. McMillan and Schumacher (1997:32), point out that apart from being the most commonly used method in educational research, the quantitative survey design was used because it is objective in data collection at a point in time, quantifies variables and describes phenomena using numbers to characterise them. In addition, it examines relationships between variables that are errors and cause. Best and Kahn(1993:121),underscore the point when they say that quantitative studies describe what is with special reference to conditions or relationships that exist, and that it involves the formulation and testing of hypotheses. According to Cohen and Manion (1995:83), the quantitative survey design does not involve manipulation of subjects but simply measures subjects as they are in order to generate generalisations and to add to existing knowledge.
These quantitative results can then be used to guide the purposeful sampling of participants for a primarily qualitative study. The findings of the quantitative study determine the type of data to be collected in the qualitative phase (Gay, Mills & Airasian, 2006). Qualitative research design was also used during interviews. Maxwell (2005) identified five particular research purposes for which qualitative studies are especially suited. They are to understand the meaning of the events, situations and actions involved, to understand the particular context within which the participants act, to identify unanticipated phenomena and to generate new grounded theories, to understand the process by which events and actions take place, and to develop causal explanations. Sometimes, more than one of the above purposes would likely be achieved in one study. However, the validity of the information collected depends on the honesty of the respondents. Cohen and Manion (1995:83) have this to say in support of the survey: ...surveys gather data at a point in time with the intention of describing the nature of existing conditions or identifying standards against which existing conditions can be compared, on determining the relationships that exist between specific events, varying in their levels of complexity from those which provide simple frequency counts to those which present
19 relational analysis. The quantitative design generated numbers, which were analysed statistically making comparisons and correlations possible, Denscombe (2000).
The study seeks to investigate errors made by ordinary level pupils when solving trigonometry problems using a variety of methods which vary from quantitative to qualitative.
3.3 Population
Chawawa (1989) defines population as the totality of individuals under study. Best and Khan (1993:12) define population as “a group of individuals that have one or more characteristics in common that are of interest to the researcher.” A research population is generally a large collection of individuals that is the main focus of a scientific query. It is for the benefit of the population that the researches are done. All individual or objects within a certain population usually have a common binding characteristics or trait. It is from the population where a sample is drawn. In this study, the population covered all form four learners and mathematics teachers at Kotwa high school. The population was made up of six teachers and two hundred and forty pupils.
3.4 Sample
Chiromo (2006) views a sample as a smaller group or a subset of a population selected for study. Wegner (2003) also defined it as a subset of a population, part of the population taken into consideration under statistical inquiry. A small proportion of population was selected for analysis. Thus a sample is a representation of the population which reflects its characteristics. The study was conducted using forty five secondary school students and six teachers drawn from Kotwa high school. The mean age of the student was seventeen
3.6 Research instruments
In this study the researcher used test, questionnaires and interviews as research instruments. According to Khan (1993), research instruments are tools used for collecting data. These are tools used to gather data that assists in finding solutions to the problems. Instruments are different tools used by the researcher to gather data. It is important to collect data using a variety of methods in order to get the best understanding of one’s point and needs. These tools made the research successful.
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3.6.1 Tests
The researcher used a set of trigonometric diagnostic test questions as instrument to collect data. The test instrument was designed by the investigator to identify types of errors committed by pupils. The test was administered to the forty-five pupils selected from each group that is fast, average and slow learners. The test enabled the researcher to identify some of the errors which pupils make under examination conditions. Karasar, N (1995), said that researchers should seek expert opinion to determine whether problems contained are suitable for measurement purpose and represent the domain intended to be measured. The researcher used well certified investigator to set the test for it to meet the domains which is intended to be measured. In the test pupils had to show knowledge of some basic properties of solving trigonometric problems. The test was constructed using items in textbooks and past examination papers and also the syllabus as it is the main guide when teaching.
The test reviewed where errors occurred for instance; whether the pupils could read the questions, comprehend the structure of the question translate and solve it. The test is of paramount importance because it enabled the researcher to interact with the pupils directly; hence the researcher will be able to procedurally follow the works of every individual closely. This also enabled the researcher to identify the exact errors committed by pupils and to deduce some misconceptions from their worked solutions. The test also mapped the way on how the interviews were going to be conducted.
3.6.2 Interviews
An interview according to Hussey et al (1997), gives the researcher an opportunity to probe deeply in order to uncover new clues and open new dimensions of a problem under scrutiny. As a general framework for interviews, the researcher adopted the interview format elucidated by Newman (1977), Casey (1978), and Clements (1980). The questions in this format were divided into three main areas: input, process, and output. In the input stage, the components were: reading the problem, interpreting it, and selecting a strategy to solve it. The process stage contained solving the problem using the selected strategy. The reason that individual interviewing was necessary was that from a constructivist point of view, reflective ability is a major source of knowledge on all levels of mathematics. Students should be allowed to articulate their thoughts and to verbalize their actions which will ensure insights into their thinking processes. The researcher used semi-structured interview in order to allow pupils to communicate their opinions and to express their actions which certify the insights in their
21 thinking process. He also decided to take one student from every five students who wrote the test, in order to select students for interview. Charmaz (2006) describes theoretical sampling as starting with the data, constructing tentative ideas about the data, and then examining these ideas through further empirical inquiry. It is during such interviews when contradictions, insufficiencies or misconceptions can be noted. Interviews gave immediate feedback to the researcher since pupils opened up to explain why certain misconceptions occurred and this helped the researcher to provide some answers on causes of errors in solving trigonometric. A number of questions were asked depending on the solutions obtained from the test given for each respondent to get insights into pupils’ constructions, interpretations and thinking process, Newman, (1977). The main objective of the interview sessions was to gain more insights into why certain types of errors were committed by the pupils and to identify their learning gaps should they have any. The interview questions focused around the pupils’ understanding of trigonometric and their ability to identify errors. Two error categories were distinguished from student responses to the test: non-systematic errors and systematic errors. In the non-systematic category, student errors had no apparent patterns to be identified and they were not connected to other concepts. It was hypothesized that these errors were random or they were made due to some other reasons such as forgetfulness, stress or carelessness.
Interviews were used so as to get adequately answered questions which are not well explained on questionnaires. Interviews are more flexible than questionnaires and provide good opportunity for demonstrations. Results are more accurate due to interviewer control and gestures from the respondents. This method is faster since responses will be there and there other than leaving the respondent with your questionnaire which may take them some time to answer, Hussey et al (1997). However interviews have their own drawbacks in that they are time consuming and sometimes respondents may be too busy to attend to the researcher’s questions, Hussey et al (1997).
3.6.3 Questionnaires.
A questionnaire is a set of questions designed to elicit information appropriate for analysis. Borg and Gall (1993), defines questionnaire as a list of questions aimed at the discovery of particular information like opinions. Farrant (1980:60) says “Questionnaires are sets of carefully constructed question designed to provide systematic information in a particular subject.” Questionnaire is a structured investigation, utilizing a common set of questions about a particular research area or just a group of printed questions used to collect information from
22 the respondents. To be effective the questionnaire should be simple, unambiguous or non- technical.
The questionnaire was administered to the teachers identified in the sample. The researcher used both open and closed form or fixed choice questions. Questionnaire allows the researcher to obtain large amounts of quantifiable data on relatively large number of people. Open ended questions were used for examining the learning levels of pupils on the topic trigonometric. Open ended questions were also used to assess the pattern of errors committed by pupils at various levels in problem solving. Respondents were told not to write their names on the questionnaire.
The researcher used a questionnaire as an instrument because it reaches out to a much wider sample within a short space of time. Questionnaire enables the subjects to freely answer questions without any interference from the researcher. In this case the respondents tend to say the truth since the questionnaire does not divulge names of respondents. Contrary to above Tuekman (1976) is of the view that some respondents may refuse to give correct information for reasons known to themselves. An attempt was made to get to the bottom of this problem by making follow ups with the subjects of the study.
3.6.4 Validity
The validity of a test instrument is of paramount important in any study. If a test does not serve its intended function well, then it is not valid. Borg and Gall (1993) define instrument validity as the extent to which the instrument serves its projected function. Borg and Gall (1993) they further give four main types of validity as concurrent validity, content validity, construct validity and predictive validity. Concurrent validity addresses how well test scores match up to already accepted measures of performances. Content validity measures test correspond to already accepted measures of performance. Predictive deals with how well predictions made from the test are confirmed by subsequent evidence. To enhance the validity of research instruments and design namely tests, interviews and questionnaires, the researcher sought assistance of highly qualified teachers to construct the test instruments. Also in order to preserve content validity, the content of the test was prepared by consulting the mathematics curriculum and the scope and the sequence chart on testing. Pilot studies using test,
23 questionnaire and interview guides were conducted using a group of ten pupils and two teachers at Kotwa High School.
3.7 Data collection procedures.
3.7.1 Tests and Interviews
Chiromo (2006) defines data collection procedures as a series of steps taken when collecting data. Thus for data collection purposes the researcher sought permission from the Minister of Primary and Secondary Education down to the school headmaster and the head of mathematics department at Kotwa High School. All the protocols we highly observed. This study involved a total of forty-five pupils coming from three different classes at Kotwa high school. All pupils previously have studied the topic trigonometry in their previous forms. The pupils sat for a 45- minute written test with 20 items. The pupils were required to show their detailed work and were not allowed to use a calculator. The instrument was pilot-tested with a group of ten pupils and based on the pilot study some items were modified and some sub-parts dropped so that the test could be complete in 45 minutes.
A total of nine pupils from the three groups were interviewed for this study. Three pupils were selected from the top third (Upper Group), three from the middle third (Middle Group) and three from the bottom third (Lower Group), after the students had been rank-ordered from one to forty-five based on the test marks. Each of the nine pupils was assigned an alphabet code from A to I. The main objective of the interview sessions was to gain more insights into why certain types of errors were committed by the students and to identify their learning gaps should they have any. The interview questions focused around the pupils’ understanding of trigonometry and their ability to identify errors in samples provided to them. Three teachers were selected at the researcher’s discretion to be interviewed. The researcher explain to the interviewee what will be happening and why. This is because a person will be focused if s/he has an idea of what type of information is wanted.
3.7 Data presentation and analysis procedures
Best and Khan (1993) define data analysis plan as a technique used by researchers to reduce the volume of information and helps in identifying significant patterns. The statistical procedure chosen for any study depends on the research question, the types of groups one is
24 dealing with, the number of variables and the scale of measurement, Hussey et al (1997).The researcher classified categories of the errors made by the pupils. The coding and tallying system was used to quantify the number of pupils with the same error and opinions. Tables and bar graphs were used to represent information obtained from the tests and interviews. Pie charts were also used to represent data from the questionnaires. Salient features from the graphs were also explained out. In analysing and interpreting quantitative data the researcher used qualitative results to aid in further explanations of the findings. Initial quantitative phase of the study were used to characterise individuals along certain traits of interest related to the research questions. The researcher compared the findings from qualitative and quantitative data to ascertain and answer the set research questions.
3.9 Summary
This research used mixed methods as the overall design and case studies as the main method in the qualitative phase. This chapter also focused on identifying the research methodology, design and instruments to be used in the research. Justifications for selecting instruments and sampling techniques were also highlighted. The chapter also outlined validity, validity data collection plan and data analysis plan of the research. The next chapter will present and analyse the data obtained for the study
CHAPTER FOUR
DATA PRESENTATION AND ANALYSIS
4.1 Introduction
In this chapter, the researcher represents the findings on the types of errors, causes of errors and suggested ways to reduce such errors in solving trigonometry. This chapter focused on the
25 responses obtained from the test given questionnaires and the interviews. Tables, graph and pie chart were used as data presentation technique. The findings of this research answered the research questions in chapter one.
4.2 Teachers’ qualifications at the High School.
Teacher qualifications
diploma degree in education masters other
Fig 4.1: Teachers qualifications
From the data obtained, the study showed that most of the Mathematics teachers are diploma holders from teacher’s colleges and also other qualifications. From the sample only one teacher has done Masters and one has done honours in Bachelor of Science Education majoring in mathematics were teaching Mathematics. Kasanda (1997) looked at factors affecting pupils’ performance in mathematics and its main findings were lack of qualified teachers. This contradicts with this study since about 67% of the teachers under the study are qualified but still there are errors in mathematics. Teachers are qualified in the area but however pupils still make errors on trigonometry as a topic under study and many other topics in mathematics in the syllabus.
4.3. The types of errors pupils make when solving trigonometry
In this research several errors were identified, the table below summarises the responses from the teachers pertaining the errors made by the pupils from the questionnaires
Table 4.3.1 Teachers responses on types of errors made by pupils
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Error type Not at all Less extend Greater extend Total Computational 0 3 (50%) 3 (50%) 6 (100%) Reading 1 (17%) 2 (33%) 3 (50%) 6 (100%) Presentation 0 4 (67%) 2 (33%) 6 (100%) Lack of 0 1 (17%) 5 (83%) 6 (100%) knowledge Conceptual 0 5 (83%) 1 (17%) 6 (100%)
From the table above fifty per cent of the teachers were quite positive that pupils encounter some reading errors. Reading errors occur when pupils misread the question or fail to interpret it. Failure to interpret the question is thus classified as the interpretation error. Presentation errors are also part errors identified by the teachers which pupils can make when solving trigonometry. Pupils may correctly read and interpret the question but however due to improper presentation fail to meet the requirements of the question. From the data collected eighty-three per cent of the teachers strongly agree that pupils make errors due to lack of knowledge. Also lack of concept at all contributed to the errors made by the pupils. This showed that the pupils struggle to understand the concepts on trigonometry demonstrating lack of knowledge.
4.3.2 Pupils errors from the test
From the test given to pupils a lot of information was gathered about pupils’ ability and knowledge when working with trigonometry. The findings were revealed in terms of the common errors made by pupils. Another finding was discovered from specific errors analysed from the steps of the solutions. The table below illustrate some common error made by pupils in different selected questions. The table also try to show which type of error occurs when manipulating the trigonometry problems
Table 4.3.2: Common errors done by pupils.
Question number Student’s answers and their percentages Type of error or possible misconception
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A2 (a) i) C2= a2 + b2 (24%) misinterpretation of the diagram ii) C2 =b –a (20%) Omission
iii) C2 =a + b – 2abCos C (28%) Commission A2(b) i) Right angled triangle (20%) Conceptual error.
ii) Cosine rule (43%)
iii) Pythagoras cosine formula (17%) B4 (a) sin2 x +cos2 x =1 process
(a/c)2 + (b/c)2= (a2+ b2)/c2= c2/c2= 1 (if a, b ,and c indicate the sides of a triangle ) (a2+ b2)/c2= c2/c2= 1 (38%)
B4(b) What is the value of X if Sin 300 = X (43%) concept
None of the pupils obtained full marks in the test given. Each of them made at least one error in answering the questions. From table 4.3.2 pupils faced different errors when working with trigonometry. These results obtained from the test given agreed very well with Chua (2003) and Berezovvski (2004) on pupils’ errors while solving trigonometry. They strongly concluded that pupils need a lot of help when solving trigonometry. From the findings pupils really made a lot of errors in dealing with trigonometry. From the table forty-three pupils skipped certain important steps in solving trigonometry. Pupils could just rush to write the solution without showing all the steps. From the findings pupils made errors when they were working with laws of trigonometry, they were not able to apply them correctly. Pupils also worked algebraically
28 wrong in solving trigonometry function, about twenty-nine pupils made this error. The findings concurred with the results of Radatz (1979) who classified errors in different categories, language difficulties, processing skills ,application of irrelevant rules and encoding. In the table above different type of error where identified by the researcher the errors which include error of commission , error of omission , error of processing ,error of translation and comprehension occur most frequently in word problems.
In addition to the errors identified by the researcher some error of commission and omission were also unveiled. Omission is found when a pupil failed to put the power of 2 on their formula. The correct formulae was supposed to be c2 = b2 – a2 they give it as c2 = b- a .This indicated that the student has the knowhow of the Pythagoras theorem and also they have sound background of the subject of the formula hence able to give the correct sign combining ‘b’ and ‘a ’but have an error of omitting the correct formula. Also pupils committed error of commission when they give the formula for c2 as a +b – 2abCos C. The students were supposed to write the formula as a2 +b 2- 2abCos C
The conceptual error occurred on question B4. Students seem not to know the concepts under which the question is asked. Their response could be a result of guessing since they forget or have no idea of the underlying concepts. On this particular question pupils end up dividing by 30 or wrongly use the tables this show that pupils have errors in concepts covered. Also the conceptual error was found in the question that involve the use of the right angled triangle pupils were not able to use correctly the Pythagoras cosine formula the end up making a lot of errors. 80% of the student were found wanting on this type of error. Finding has found out that most students who got the item by use of right – angled triangle method could not replicate same answer using the formula as a result of appropriate placing and manipulation.
The error in the process skill was also noted when pupils were doing their working. This process skill was more pronounced in the use of formula than the right – angled triangle method. This error dominated very well which shows that pupils failed to understand and describe what is required by the question. The majority didn’t do well about 87% made the same error.
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The appendix F reflects other errors which were encountered by the researcher during the study the errors includes trigonometry ratios, cosine rule, sine rule and miscellaneous concepts. The miscellaneous errors are errors which are directly linked to solving categories but cannot be classified under the four conceptual areas indicated before which are Pythagoras theorem, trigonometry ratios, sine rule and cosine rule. From this category the main errors noted were (1) reading errors (2) conceptual errors (3) process errors (4) transformation error (5) comprehension error. The table below show some error committed by pupils when answering question set by the researcher.
Table 4.3.3 Frequency of errors in solving trigonometry problem
Items Reading Comprehension Transformation Process Encoding
1 3 36 28 33 4 2 0 29 30 28 - 3 0 34 29 36 3 4 0 33 22 30 - 5 0 22 38 25 2 6 0 36 28 34 4 Total 0 190 175 186 13
The researcher has unveiled most of the common errors made by students when solving problems in trigonometry. The errors include reading, comprehension, transformation, process skill and encoding as the last error. From the study it has been reviewed that error comprehension had 60% followed by error of process skill which had 59% and the error of transformation had 55% reading and encoding were the least Comprehension errors occur mostly when students do not understand how to approach a given trigonometric problem from the concept. Usually students misunderstand the demand of the question. This may be caused by lack of emphasis by the teacher in teaching the simplification of concepts as they appear. Also it may be caused by rote learning on the part of the learner. According to Gur (2009) alludes that errors committed by students in learning trigonometry may be useful for the teacher
30 in evaluating his teaching so as to be able to correct the students as appropriate. Hence this must ensure teachers to have a variety of teaching methods which must be balanced with arithmetic skills. The teaching methods must be appropriate for the learner it must be learner centred.
Another error the researcher has encountered is that of transformation, this usually occurs during computation process. It takes place due to computation problems especially among achievers. Many students made the transformation error in the study they unable to transform given operations. The findings of this study concurred of those of Norasiah (2002) who noted that most average students face difficulty in performing trigonometric operations. In this study the researcher made a comparison of two questions under study and find out how students performed
Table 4. 3. 4 Comparison of the Numbers of the Answers to Questions A2 and B4
Q Q u e s t i o n B4 u e Wrong Correct Total s t i Wrong 38 4 42 o n Correct 1 2 3
A2 Total 39 6 45 The researcher does a comparison of two questions and come up with a table of results. The table shows that only 2 (two) pupils answered both questions correctly and 38 (thirty-eight) of the pupils answered both questions wrongly. This shows that 84% of the students answer both questions wrongly and only 16% answers it correctly. A critical examination of pupils’ performance showed that the majority of the pupils had difficulties in understanding the concepts of trigonometry. From the working above the pupils has an idea but are failing to apply the concepts correctly. The researcher scrutinise each and every answer sheet of pupils and found out that pupils were making numeral errors. The error found amongst others includes, computational errors, error in manipulating algebraic symbols, error from extracting data from given information, error in executing algorithm, misinterpretation error , the error of
31 commission and other carelessness errors. When pupils we solving question 2 there was misinterpretation error pupils failed to identify the hypotenuse and end up making errors.
4.4 Misused data error
Misused data error was another error on which the researcher has found among pupils. The funding concurred with Movshovitz- Harder et al (1987) which allude that these errors occurred due to discrepancy between the given data in the item and the way pupils treated them. This error involved pupils misusing the data given in the problem and this problem was common in item 3 of the test given. Pupils were ignoring the cosine, sine and some were copying the numbers wrongly. In this item it was expected that pupils would be able to copy the digits correctly but they could write different value of the given figures. These errors were most common in the below average pupils. The pupils who made these errors did not even complete the test. As a result this is due to ignorance or lack of seriousness.
4.5 Causes of errors in solving trigonometry problems
There are several factors that cause errors when solving trigonometry. Several ideas about trigonometry seem not clear to the pupils as other authors say trigonometry is an abstract area. The interview also demonstrates that a correct response is no guarantee of a lack of misconception. The next graph shows teacher responses on errors made by pupils when solving trigonometry
FIG 4.5.1: Teacher responses on main causes of errors.
32
7 6 5 4 3 2 strongly agree 1 Agree 0 Disagree Strongly Disagree
From the information obtained eighty-three per cent of the teachers agreed that lack of comprehension contribute to errors made by pupils when solving trigonometry. Seventeen per cent strongly agree that lack of comprehension is really an area of concern. On lack of strategy sixty- seven agreed that errors are due to lack of strategy whilst thirty-three per cent strongly agreed that really lack of strategy is the cause. Hundred per cent of the teachers agreed that inappropriate strategy and imperfect knowledge are also cause of errors when solving trigonometry. Teachers also agreed that inability to translate given items and inappropriate computations lead to these error.
An interview with the teachers reflected that the following are the causes of errors in solving trigonometry. From the interview the following were the major findings raised by the teachers interviewed
4.5.1 Lack of comprehension: many pupils encounter errors because they fail to understand the question hence response will divert from the required one. If the pupil fails to comprehend there is no way in which the pupil will answer the question correctly because they will answering something different from the one asked. For example as representing the result of calculation of sin 300 and the value of sin300
4.5.2 Inability to translate given items: A number of pupils are impeded to the correct answers because they cannot translate worded expressions to symbolic expression and failure to represent information symbolically. In the study it has been found out that pupils could not
33 distinguish when to use the cosine rule and the sin rule they end up interchanging the two giving wrong answers at end.
4.5.3 Lapse in concentration: Pupils also encounter errors in the process of their workings. A lapse in concentration would lead pupils to make careless errors as they present their solutions. There are those student who do not concentrate because of mischievous they end up doing the wrong process. The result of the study concurred with the findings of Norasiah (2002) in which he posit that the problematic students failed to translate mathematical problems into mathematical form and also having problem in understanding the special terms in mathematics.
4.5.4 Overspecialisation: Pupils often specialise in certain areas negating other areas. When they overspecialise, they will be exposing themselves to limited knowledge and strategies of tackling questions. For example when the teacher advocate for the use of calculator only, shunning totally to work without using a calculator.
4.5.5 Failure to read instructions: A number of pupils fall prey of this error. Some pupils may have the correct working up to the last stage but fail to give the answer correct to the required degree of accuracy.
4.5.6 Lack of knowledge: Raised to researcher’s attention was the fact that teachers pointed out that pupils suffer from ‘number blindness’. The situation become horrible to such pupils at mathematics time, everything will be fine as long as there is no mathematics.
4.5.7 Rote learning: Pupils tend to learn mathematics by memorisation; hence if a different situation arises pupils become stuck. This is because pupils regard solving trigonometry as challenging topic. Their beliefs tend to impede the zeal to put an extra effort.
4.6 Pupils’ responses on closed questions
34
Table 4.6.1 Responses from pupils on closed questions
ITEM NARRATION YES(%) NO(%) i Do you enjoy learning trigonometric? 64 36 ii Do you have enough materials and resources to learn 34 66 trigonometric? iii Do you think teachers contribute to poor performance in 75 25 trigonometry?
The table above reveals that most pupils enjoy learning trigonometry. The table also shows that most pupils do not have enough materials and resources to learn trigonometry. Awang (2009), states that educational aids and resources increase pupils’ achievement, they found that more than 94% of Singapore students own a computer and 78% of them use computer both at home and school as compared with 56% and 25% respectively for the Malaysian students. Singapore students performed better than their counterparts from Malaysia. Basing on Awang’s findings pupils have negative attitude towards mathematics because most of them do not have enough resources. From the sample 75% of the pupils think teachers contribute to poor performance in trigonometry. Studies done by Thompson (1993), had shown that the teachers’ teaching method of teaching and his/her personality greatly accounted for the students’ performance in mathematics and that without interest and personal effort in learning mathematics by the pupils, they can hardly perform well in the subject. The finding of this study agrees with Thompson since 75% is the majority. In response to the teaching media used by teachers, about 90% indicated that their teachers do not use any media when teaching trigonometry. According to Dale (1954:42), human beings learn and retain information as follows:
10% of what we READ,
20% of what we HEAR,
30% of what we SEE,
50% of what we HEAR and SEE,
Higher levels of retention can be achieved through active involvement in learning.
35
If pupils perform well on the subject they get motivated and develop positive attitude towards the subject. Relating to Dale’s results teaching without aids can cause negative attitudes towards learning trigonometry hence bound to make errors. Dale’s views is supported by (Petty2004:357-358) who state the following advantages of visual aids:
They gain attention,
They add variety,
They aid memory,
They show you care.
According to Petty teaching without media means there will be no attention,no memory hence no learning is taking place. If pupils noticed that the teacher does not care obviously they are bound to dislike the topic hence make a lot of errors in the topic..
4.7 Pupils’ evaluation on their mathematics teacher
Table 4.7:1 Pupils’ evaluation of teachers
ITEM NARRATION AGREE(%) DISAGREE(%) i I want to proceed with Mathematics up to 42 58 highest level,but my teacher discourages me. ii Some mathematics teachers are not 78 12 competent in trigonometry. iii Teachers discourage pupils by saying bad 64 36 comments. iv I enjoy how are trigonometry taught at our 52 48 school.
From the above analysis it shows that most pupils do not want to proceed with Mathematics up to highest level because of some reasons which may include poor performance in mathematics. 78% of the pupils state that some Mathematics teachers are incompetent in logarithms. This is supported by (Roueche and Roueche1995)as cited by Adabor(2008) that
36 the major factor which influence student learning is the teacher. If the student has no confident with the teacher then the interest towards the subject will be lost hence lot of errors are made. Generating positive attitudes towards trigonometry among students is an important goal of mathematics education. Research conducted over the last two decades has shown that positive attitudes can impact on students’ inclination for further studies and careers in math-related fields (Haladyna et al., 1983; Maple and Stage, 1991; Trusty, 2002). For example, a recent study using the Third International Math and Science Study (TIMSS) data from Canada, Norway and the United States found attitudes toward math as the strongest predictor of student participation in advanced math courses (Ercikan, et al., 2005).Therefore teacher’s competence contribute to pupils’ attitudes towards Mathematics. Most respondents also indicated that some teachers discourage pupils by saying bad comments. For students to persist in advanced mathematics, teachers need to develop students’ confidence in mathematics by helping pupils to reduce errors as possible, not just their concepts and skills. Reducing pupils errors creates fertile ground in which teachers can plant the seeds of deeper mathematics learning and cultivate independent, advanced math. Therefore teacher’s effort towards pupils may cause reduction in errors towards the subject hence promote learning in any topic, ie trigonometry. 52% of the pupils enjoy how trigonometry is taught and 48% does not enjoy. From the analysis, much must be done on teachers and their teaching methods so as to reduce the pupils’ errors towards trigonometry. Girl(1998) stressed that ,to make mathematics learning a manageable endeavour for every pupil, teachers should attempt to make pupils like the topic hence promote learning.
4.8 Possible ways of dealing with errors in solving trigonometric problems.
The study has reviewed many errors made by the pupils when solving trigonometry. Also it has found out that there are many causes to the errors on trigonometry. In the study it shows blame shifting pupils are blaming the teacher whilst the teachers on the other side are blaming the learners.
4.8.1 Teaching and learning methods: Teachers in the study suggested many ways to deal with errors in trigonometry, they were of the opinion that pupils should be allowed to help one another on the errors they make. The teachers should expose pupils to a variety of teaching methods. The results of this study concurred with the finding of Felder and Henriques (1995) who suggested that students learn more when information is presented in a variety of modes than when a single mode is used. For example trigonometric problems can be presented either
37 by the use of graphs and algebraic methods. Also on the instruction used by the teacher must involve the learners most .Pupils should be equipped with all possible ways of getting the solutions. An example is exposing students to both the use of mathematical tables and calculators to find some trigonometry values. When using tables emphases must be put on the use of correct tables to avoid errors of using the wrong tables. Other researchers have shown that students do not come to the classroom as blank boxes instead; they come with their own ideals and theories constructed from their everyday experiences. So when teaching teachers has to put that into cognisant and connect old knowledge and new knowledge. Presmeg (2006) emphasizes that connecting old with new and allowing ample time and moving into complexity slowly, connecting visual and no- visual registers like numerical, algebraic and graphical signs are important for students, providing memorable summaries in diagram form, which have the potential of becoming prototypical images of trigonometric objects for the students, because these inscriptions are sign vehicles for these objects in trigonometry teaching.
4.8.2 Awareness campaign: The study suggested that for pupils to minimise errors they need to be aware of their errors. If they are aware of their errors it empowers them to develop strategies of venturing into some grey areas. The awareness campaign is an important tool for students to become aware of the possible errors they are bound to make .When students are aware about their errors they turn to pay more attention to those area there by reducing chance of making errors. . Teachers highlighted that pupils should learn the concepts and know how to apply them in problem solving. Rote learning encourages dependence syndrome and when the situation differs, the pupil become stuck hence run short of strategy to solve a problem.
4.8.3 Discourage rote learning: Pupils should be discouraged from learning by memorisation. They should rather be encouraged to learn the concept and processes that are involved in problem solving. Active participation of pupils will make pupils enjoy the learning process and also will make the process and concepts belong to them there by reducing chance to make errors.
4.8.4 Concept development: The results of this study show that concepts need to be developed from low order to higher order concepts. In a trigonometry example, simple material needs to be used in order to show its spiral function. The teacher needs to know the knowledge gap before introducing a new concept to the learners. If the students do not have enough knowledge, or have any misconceptions the teacher need to iron it out before proceeding to the next concept
38 to avoid errors. For example , before teaching trigonometry, the students need to give example of Pythagoras theorem in different triangles. The students need to learn from simple to complex to avoid numeracy errors. Students can make errors due to lagging in certain aspects not correctly covered in the past, since the learning of mathematics is a gradual integrated process where new concepts are built from old concepts. It is difficult for pupils to avoid errors when their background is weak. Teachers interviewed agreed that other classroom practitioners can take this study as a reference so that they can put great effort to help their pupils from doing the same errors over and over again. Therefore, teacher needs to suggest and implement proper techniques in teaching trigonometric so that pupils can acquire deep understanding in trigonometry. Hiebert and Carpenter (1992, cited in Kastberg, 2002, p. 1) stated “one of the most widely accepted ideas within the mathematics education community is the idea that pupils should understand mathematics”. Therefore, this study gives some valuable suggestion towards teaching and learning trigonometry
4.8.5 Teacher’s instructions Giving instruction is another thing that teacher need to do precisely. Interviewed teacher said that teachers need to clearly give their instructions, what they require pupils to do. This was supported by Berezovski (2000) who said that instructional word and terminologies used while teaching trigonometry needs to be clearly stated and explained to pupils in order to reduce misunderstanding while they complete the trigonometry tasks. He said that teacher needs to clearly state the term, trigonometry term, trigonometry expression and trigonometry equation. Furthermore, teacher need to clearly explain the instructional word in trigonometry task. For instances, express…,convert…,find the value of…,evaluate…, solve…, simplify…, and show….It is also vital for teachers to have some test for seeking some information about pupils level of understanding of logarithms. According to Kastberg (2002), a teacher often assumes a pupil understands a concept that has been presented but the pupils are actually facing problem to recall it. Therefore, the teachers agreed that test of understanding trigonometry is needed in order to investigate pupils difficulties in learning trigonometry before finding a way to help pupils reduce their difficulties.
4.8.6 Presentation of work and checking of solution: From the study it was also noted that some of the errors are a result of poor presentation of work by the pupils. A teacher suggested that pupils should be encouraged on neat presentation and checking of solutions. Pupils should be reminded on clearly presenting work and neatness. This tends to be a powerful tool to avoid careless errors. Teachers need to encourage pupils to check their solutions and continuously refer to the question to check whether the demands of the question are met.
39
The possible ways to deal with the errors in trigonometry according to teachers’ perspective did not contradict with remedies initiated by NCTM (1991) and Gunawardena (2011). However Gunawardena (2011) was looking at remedies to algebraic errors. NCTM (1991) and Gunawardena (2011) suggested that pupils must be actively involved in the process of eliminating the errors by letting them know the errors they are bound to make. This is similar to the identified remedy to errors in solving trigonometry identified as, awareness campaign. NCTM (1991) also said that teachers should provide classroom learning experience that help pupils to develop both conceptual and procedural knowledge so as to construct correct conceptions from the beginning. This can be directly linked to remedies identified by teachers interviewed in this study, as strengthening the base of learners and varying teaching and learning methods. The suggestions point out that the teacher is at the centre of error elimination; hence pupils build conceptions through the way they are taught. All these findings from teacher interviews agree that to eliminate errors there should be a shift to learner centred approaches.
4.9 Summary
The thrust of this chapter was to present, analyse and interpret data from questionnaires, test and interviews from the data collected at the school. Demographic data was presented using graphs and descriptive analysis covered qualitative data. The researcher has presented the finding and discusses causes of errors and also ways to minimise those errors. Graphs and tables were used to present data for easy interpretation as information can be seen by just a glance. Direct quotation and phrases were also used to present the response given by pupils and teachers. This data was grouped according to the research questions so as to make it possible to interpret the given data. The next chapter focuses on summary, conclusions and recommendations.
CHAPTER 5
SUMMARY, CONCLUSION AND RECOMMENDATIONS
5.1 Introduction
40
This chapter will summarize the findings of the study. It will discuss the implications of these findings for teaching and learning of trigonometry. Suggestions for further research in this area will also be discussed.
5.2 Summary
The study focused on investigating the types, causes and ways of minimizing the errors made by Ordinary level pupils when solving trigonometry. The research was based on the fact that pupils are not doing well in trigonometry. This impacts heavily and reduces the achievement of those pupils both in school life and daily life. Less and less pupils are pursuing with the subject to Advanced level due to failure in the subject. The researcher assumed that this study would improve the teaching and learning processes hence bridge the gap between application in other subjects and in daily life hence found it worthy to undertake the study.
Studies in mathematics education have shown that pupils encounter some challenges in understanding trigonometry (Prescott, Mitchelmore and White 2003, Van Hiele-Geldof and Yilmaz et al (1984)). The source of errors in trigonometry was identified as the process to which pupils work out their solutions described by Melberry (2003) as rote learning. Radatz (1979) looked at the procedural causes of errors and identified the errors by looking at the mechanisms used in processing, retaining and remembering mathematical tasks. The findings of this study show that no much investigation was done on the subject of solving trigonometry as well as causes of errors in the topic.
In this study the researcher adopted a descriptive survey research design. The descriptive survey design was favored since it allows subjects to be investigated in their natural settings like schools hence original data is collected. A sample of forty-five pupils was selected from Ordinary level pupils at Kotwa High School and all the six mathematics teachers at the school. The researcher collected data using questionnaires, test and interview instruments for triangulation purpose. The questionnaire was administered in person to the teachers. To the pupils the researcher was helped by the head of department to issue the questionnaire. The test was administered to all pupils in the sample with the help of the head of department. After
41 marking the scripts the researcher selected some pupils to clarify on how they came out with their solutions. All the questionnaires were completed and returned on time. All forty-five pupils were able to participate in the test though some pupils did not attempt all questions and this cause some challenges to ascertain whether pupils encounter errors or maybe was lack of time. Major findings from the study show that pupils make some comprehension and computational errors in solving logarithms at Ordinary level. The errors are mainly attributed to imperfect knowledge and rote learning. The recommended ways to deal with errors are the use of pupil centered teaching approaches and awareness campaigns.
5.3 Conclusion
This study explores the common error that pupils make in trigonometry task. It also revealed that most common error committed by students irrespective of the method used are transformation errors and process skills errors. Pupils common errors in learning trigonometry include failing to remember the definition of trigonometry, wrong application of rule of trigonometry, confusedness with their prior knowledge, did not know the appropriate base to use while changing the base of trigonometry, and did not understand the instructional word in trigonometry task. This study also does support some researchers’ claims about pupils’ difficulties in working with trigonometry.
The causes of these errors were due to lack of comprehension, inability to translate given items, lapse in concentration, lack of strategy, overspecialisation, failure to read instructions, lack of knowledge, use of wrong data, interference with previously learnt concepts, incomplete simplification and flawed conceptual or procedural knowledge.
Results of this study have shown that it is not easy to change pupils vigorous misconceptions when errors they make can be ascribed to flawed procedural or conceptual facts. However several ways can be employed to minimise the occurrences of errors in solving trigonometry at Ordinary level. The remedies include, embarking on awareness campaign, varying teaching and learning methods, strengthening the foundation by taking time to explain new concepts and encourage neat presentations of solutions. Pupils are also encouraged to check their solutions and continuously refer to the questions to check whether the demands of the question are met. Pupils should actively participate in the eradication of errors they make. This whole
42 idea will shift classroom practice from the role of the teacher as an evaluator of pupils’ ideas to the role of pupils as self-evaluators of their emerging ideas. Results from this study also show that teacher should relate teaching aids to real life situations.
5.4 Recommendations
In view of the fact that the study was a case study to identify the errors, causes and remediating measures in solving trigonometry, these results cannot be generalised to all other subtopics in solving mathematical concepts, the researcher suggests that related research should be carried out on other conceptual areas in the subject of mathematics so that we can have a broad literature on error analysis in mathematics.
Teacher should make trigonometry lessons exciting by encouraging group works with frequent activity based on demonstration. Learning should be learner centred. All the activities should be done by the pupils so that they internalised for life.
The researcher also encourage teacher to encourage as well pupils to concentrate on one item at a time and proceed stepwise in a logical manner. The pupils need to have the knowhow of the basic concepts before going further. Also they need to concentrate on school business
Educators need to be prepared to spend time on sharing common errors that are made with pupils in trigonometry. When teaching a new topic to pupils, teachers need to show their pupils the common errors they have seen past pupils make. In addition, teachers should research a topic before teaching it. This will allow the teacher to discover common errors on the specific lesson they are about to teach. If pupils are shown the common errors others have made, they are more likely not make the same mistakes. Pupils learn from their mistakes and can learn from past pupils’ mistakes. The next time I teach trigonometry I am going to show my pupils the common errors that were uncovered in this study.
The researcher recommends that educators should be continuously staff developed. The in- service staff developments help and equip teachers with requisite skills and power to manage errors as they surface at early stages. Teachers should be taken to task so that they understand their pupils’ errors. Teachers should engage in dialogue with their pupils to check on thinking patterns.
Errors and misconceptions should be corrected methodologically. An awareness campaign in classrooms should be done. Teachers should try to make pupils aware of their errors and give
43 them chance to correct them through guided discovery methods. If an individual fails to correct his or her error, teachers should use the assistance of other pupils who have done well. Such an analysis with sharp pupils can be educational to the rest of pupils.
Teachers should vary teaching and learning methods as this equips pupils with all possible ways of getting solutions. Teachers are also encouraged not to save time on forming the mathematics concepts. They are recommended to spend adequate time in order to strengthen the foundation of their learners
The learners should be given enough opportunities to do regular problem exercises as this will go a longer way in assisting them and increasing their reasoning skills. If a concept is done over and over by pupils they turn to internalised it became part of the student and it will be difficult for the learner to repeat an error.
In the midst of the increasing revelations and importance of the ICTs, the Ministry of Primary and Secondary Education is recommended to train its teachers the use and appreciation of computer aided designs such that the day today becomes relevant. Pupils should visualise the learning processes and see mathematics as a human activity. Curriculum developers are recommended to speedily incorporate ICTs in the curriculum as they improve the teaching and learning processes.
This study should be done with different mathematically based concepts so as to compare findings for further use.
5.5 Summary
In this chapter, a summary of the findings of the study has been presented. The implications of the study in the present context have been discussed. Finally suggestions for further research in the area have been provided. The major finding of the research was that pupils make errors when solving trigonometry. Therefore in-service training for teachers is mainly recommended for reduction of these errors in trigonometry.
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APPENDIX 1
QUESTIONNAIRE TO TEACHERS
My name is CHIPURURA ERNEST a student of Bindura University of Science Education. I am carrying a study on the errors which ordinary level pupils make when solving trigonometry. May you kindly complete the given questions? This questionnaire is specifically for study purposes and researcher assures that confidentiality will be maintained on the information
48 gathered. Do not write your names on the questionnaires. All your answers are recognised and shall remain anonymous. Thank you.
1 Indicate your teaching experience
Below five years
Above five years
Below ten years
Above ten
2 Highest level of professional qualification:
Diploma in Education
Degree in Education
Masters
Others(specify)……………………………………………………
3 What subject matter do you emphasise most when teaching mathematics?
A. Geometry
B. Algebra
C. Arithmetic
Other (specify)……………………………………………………………
……………………………………………………………
4 Which part of solving trigonometry do u sees pupils encountering more challenges?
A Evaluation of trigonometric functions
B Solving of trigonometry without using calculators.
49
C Solving trigonometry functions.
Other (specify)……………………………………………………….
………………………………………………………..
…………………………………………………………
5 To what extend do you agree with the following statements?
Strongly Agree Disagree Strongly agree Disagree
Most teachers and students regard solving trigonometry as very difficult topic
Solving trigonometry is primarily abstract topic
Solving trigonometry is primarily practical topic
Pupils mainly learn trigonometry by rote
6. Which instrument do you encourage your pupils to use when solving trigonometry
A. calculator
B. Mathematical tables
C. Both Calculators and Tables.
7 Do you have adequate teaching aids when teaching trigonometry?
50
Yes
No
If no what other teaching aids are missing………………………………………………
…………………………………………………………………………………………….
8 What type of errors do pupils mostly make when solving trigonometry?
Error type Not at all Less extend Greater extend
Computational
Reading
Presentation
Lack of knowledge
Conceptual
9. To what extend do you agree or disagree with the following as the main causes of pupils errors in solving trigonometry at Ordinary level.
Strongly Agree Disagree Strongly agree Disagree
Lack of comprehension
Lack of strategy
51
Inability to translate given items
Inappropriate strategy
Inappropriate computations
Imperfect knowledge
10. Any suggestions on how pupils errors in solving trigonometry can be minimised……………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… …………………………………………………………………………………………….
APPENDIX 2
QUESTIONNAIRE TO PUPILS
My name is CHIPURURA ERNEST a student of Bindura University of Science Education. I am carrying a study on errors which ordinary level pupils make when solving trigonometry. May you kindly complete the following questions? The purpose of this questionnaire is to find ideas on how trigonometry can be learned effectively. Write any answer and it shall remain
52 anonymous. Do not write your names. Every answer that you write is correct, no answer is wrong. Thank you.
1) Tick the appropriate box in the front of the questions below.
i) Female Male
ii) Do you enjoy learning trigonometry
Yes No
iii) Do you have enough materials and resources to learn trigonometry?
Yes No
iv) Do you think teachers contribute to poor performance in trigonometry?
Yes No
2)Indicate your choice in the box provided by writing A,B,C,D
i) Which materials do you use when learning mathematics
A. Calculator
B. Mathematical instruments
C. Computer
D. None of the above
E. Mathematical instruments and calculator
ii) Which teaching aids does your teacher normally use when teaching trigonometry?
53
A. Work cards
B. Projector and computers
C. Charts
D. All of the above
E. None of the above
3. Indicate the extend of your agreement for each item by ticking the appropriate category
Agree Disagree i) I want to proceed with mathematics up to higher level but my teacher discourages me. ii) Some mathematics teachers are not competent in trigonometry. iii) Teachers discourage pupils by saying bad comments. iv) I enjoy how trigonometry are taught at our school.
4. What do you think is the major cause of poor performance in trigonometry by most pupils? What can be done to reduce the poor performance?
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APPENDIX 3
INTERVIEW QUESTIONS TO TEACHERS
1. What time of the year do you normally teach trigonometry?
2. Which area on solving trigonometry do your pupils faces most challenges?
3. Are the problems perennial or they are particular to a specific group?
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4. What type of errors do the pupils really make?
5. What approaches do you use in teaching the topic trigonometry?
6. Do you feel you have all the necessary resources to effectively teach the topic?
7. What do you think are the cause of such errors?
8. How can pupils’ errors in trigonometry be addressed?
APPENDIX 4
INTERVIEW GUIDE QUESTIONS TO PUPILS
The interview guide will guide the researcher on asking questions wrongly answered to determine where the error exist and ascertain the cause of the error. Questions with most incorrect responses from the written test guide will form the bases of the interviews. The interview shall assume the following headings on the identified questions.
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1. Read the question
2. Tell me what the question is asking you to do
3. Tell me how you plan to find the answer
4. Show me and tell me what you are supposed to do to get the answer.
5. What is your answer?
APPENDIX 5
TEST GUIDE FOR PUPILS
TIME 45 MINUTES
Instructions
1. Answer all questions
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2. Show all working
3. Do not use a calculator or mathematical tables in this test.
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