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inquiry in order to develop concept mastery.
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concrete models and practical approaches in problem solving, but they seemed to lack a deep
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Overs mutamiri dissertation
- 1. BINDURA UNIVERSITY OF SCIENCE EDUCATION FACULTY OF SCIENCE EDUCATION AN ANALYSIS OF THE DIFFICULTIES ENCOUNTERED BY FORM 3 LEARNERS WHEN SOLVING EQUATIONS WITH FRACTIONS Submitted By OVERS MUTAMIRI B1441467 Supervisor: Mr DZIVA A DISSERTATION SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS OF THE BACHELOR OF SCIENCE EDUCATION HONOURS DEGREE IN MATHEMATICS 2018
- 2. i BINDURA UNIVERSITY OF SCIENCE EDUCATION RELEASE FORM NAME OF AUTHOR : OVERS MUTAMIRI PROJECT TITLE : AN ANALYSIS OF DIFFICULTIES ENCOUNTERED BY FORM 3 LEARNERS WHEN SOLVING EQUATIONS WITH FRACTIONS DEGREE TITLE : BACHELOR OF SCIENCE EDUCATION . HONOURS DEGREE IN MATHEMATICS YEAR TO BE GRANTED: 2018 SIGNED _____________________________ POSTAL ADDRESS: No 10 COGHLAN AVENUE KUMALO BULAWAYO REG No : B1441467 CELL : 0774296701 Permission is hereby granted to the Bindura University of Science Education Library to produce single copies for scholarly or scientific research purposes only. The author does reserve other publication rights and the project or extensive extracts from it may not be printed or otherwise reproduced without the author’s written permission
- 3. ii BINDURA UNIVERSITY OF SCIENCE EDUCATION APPROVAL FORM The undersigned certify that they have supervised, read and recommend to the Bindura University of Science Education for acceptance a research project entitled: An analysis of the difficulties faced by form 3 learners when solving Equations with fractions submitted by Overs Mutamiri ,in partial fulfilment of the requirements for the Bachelor of Science Education Honours Degree in Mathematics programme. …………………………………… ….…/…………/…………/ (Signature of Student) Date …………………………………… ….…/…………/…………/ (Signature of Supervisor) Date …………………………………… ….…/…………/…………/ (Signature of Chairperson) Date
- 4. iii BINDURA UNIVERSITY OF SCIENCE EDUCATION DECLARATION I, OVERS MUTAMIRI, declare the research project herein is my own work and has not been copied or lifted from any source without the acknowledgement of the source. …………………………………… ….…/…………/…………/ Signed Date
- 5. iv DEDICATION This research is dedicated to my wife Thembie, my kids Phoebe and Deuel who have been a pillar of strength whenever l was leaving home for studies. I also dedicate this research to fellow mathematics teachers who are in every corner of the world providing instruction in the subject.
- 6. v ACKNOWLEDGEMENT My sincere gratitude is expressed to my supervisor Mr Dziva for supervising me throughout the study, and for providing all the support, guidance and encouragement and most of all patience, criticism and opinion based on his valuable outstanding experience. Your advice saved me from many a disaster. Thanks Mr Dziva for all this and your overall broad vision in education. I thank the head of Emsizini Secondary School Mr K Moyo who has been continuously providing checks and balances throughout the duration of the program. Finally I wish to thank my wife Thembie, my mother Claris, for their continued support.
- 7. vi TABLE OF CONTENT Content page Cover page Release form i Approval form ii Declaration iii Dedication iv Table of contents vi-viii Appendix ix List of tables x List of figures xi List of acronyms xii Abstract xiii CHAPTER ONE: INTRODUCTION 1.0 Preamble 1 1.1 Background to the study 1 1.2 Statement of the problem 5 1.3 Aim 6 1.4 Objectives of the study 6 1.5 Research questions 6
- 8. vii 1.6 Hypothesis 6 1.7 Significance of the study 7 1.8 Limitations of the study 7 1.8.1 Delimitation of the study 7 1.9 Summary 9 CHAPTER TWO: LITERATURE REVIEW 2.0 Introduction 10 2.1 Difficulties faced by learners 10 2.1.1 Conceptual challenges 10 2.1.2 Procedural knowledge 12 2.1.3 Teacher content knowledge 13 2.2 Strategies for challenges faced by learners 15 2.3 Problem solving and students mental models 16 2.4 Summary 18 CHAPTER THREE: RESEARCH METHODOLOGY 3.0 Introduction 19 3.1Research design 19 3.2 Population 19 3.2.1Sampling and sampling procedures 19 3.3 Research instrument 20 3.4 data collection procedures 20
- 9. viii 3. 5 Data analysis 21 3.6 Summary 22 CHAPTER FOUR: ANALYSIS OF DATA 4.0 Introduction 23 4.1 Presentation of descriptive characteristics of respondents 23 4.2 Learners errors and misconceptions 27 4.3. Analysis of data 30 4.4 Summary 32 CHAPTER 5: SUMMARY, CONCLUSIONS AND RECOMMENDATIONS 5.0 Introduction 35 5.1 Summary 35 5.2 Findings 36 5.3 Conclusions 36 5.4 Recommendations and future research 37 References 40
- 10. ix Appendix page 1: Letter of Introduction 42 2: 1st Section of Scheme cum plan 43 3. 2nd Section of Scheme cum plan 43 4.Pre-test instrument 44 5: Pre-test marking guide 45 6: Post-test instrument 46 7:Post-test marking guide 47 8: mark-list records for girls 48 6: mark-list records for boys 49
- 11. x LIST OF TABLES Table Page 1 Levels of competency 21 2 Learners demographic data 23 3 Participants ability test performance 24 4 Learners levels of competency 25 5 Learners grouping of percentage marks in to classes 26 6 Learners errors and misconceptions 27 7 Summary of the number of learners under each error category for pre-test and post-test 28
- 12. xi LIST OF FIGURES Figure page 1: Bar graph to show mark rankings 24 2: Bar graph to show learners competency level 25 3: histogram to show distribution of percentage marks 26
- 13. xii LIST OF ACRONYMS Acronym page NDP- Numeracy development Project 2 TCK Teacher Content Knowledge 13 NCMT National Council for Mathematics Teachers 13 NRC National Research Council 14 MPC Mathematics Proficiency with content 15 KDU Key Developmental Understanding 15
- 14. xiii ABSTRACT The aim of the research was to carry out an analysis of the difficulties encountered by form 3 learners in solving equations with fractions. A single group pre-post and post-test experimental design was used. The form three class at Emsizini Secondary in Bulawayo Province comprises of only forty four learners, the whole class was selected as the population size and with progression of the research only thirty learners were eligible to participate as they attended school consistently. Pre-test and Post-test were used to collect and analyse data and to answer the research questions. The collected data was tabulated and thereafter analysed. This study found out that learners have problems in solving equations with fractions especially when the algebraic terms are in the denominator and also if the algebraic denominators are linear expressions. Post data interpretation led to the findings that routine teaching of solving equations with fractions yield little results as it does not enhance learner’s problem solving capabilities. Thorough lesson preparation and child centred methods such as discovery learning, problem solving methods aided by awarding of enough time is a necessary pre requisite for learning of equations with fractions. Basing on these findings the research recommended that staff development to equip Maths practitioners with teaching skills which will enable them to employ child centred teaching methods for conceptual understanding of fractions. Learners need to be given enough time to engage in problem solving in pairs, groups and also be exposed to a balanced teaching using conceptual and procedural methods. There is need to improve the learning of equations with fractions and further studies should be carried out in how learners can be equipped with problem solving skills and focus should be shifted from teacher centred methods to child centred methods.
- 15. - 1 - CHAPTER 1: INTRODUCTION 1.0 PREAMBLE This study focuses on the analysis of difficulties faced by learners in solving equations with fractions at Emsizini Secondary school. This chapter will provide a synthesis of the background of the study, brief discussion on solving equations with fractions, statement of the problem, assumptions of the study, significance of the study, research questions, limitations, delimitations and definition of key terms. 1.1 Background to the study Fractions are an important component in the domain of the number framework and are applicable to everyday activities. People need to have a deep understanding of fractions for them to be able to apply them in everyday situations. This implies that it is appropriate for teachers to use the additive reasoning approaches when teaching fractions at lower levels (primary level and junior level) but needs to shift to the multiplicative approaches at higher levels. Pirie and Kieren (1994) point out that fraction learning involves constructing an ever more elaborate, complex, broad and sophisticated fraction world and developing the capacity to function in more complex and sophisticated ways within it. This implies that students’ understanding of the concept are reflected in their ability to solve real life situations. This can be attained if teachers adopt multiplicative approaches to teaching the concept of fractions. However, multiplicative reasoning is difficult for students and often requires formal instruction. Multiplicative reasoning is the entry point to the world of more complex fractions (Sowder et al., 1998). The teaching strategies aimed at developing multiplicative reasoning in the context of this study are known as multiplicative approaches. The problems students
- 16. - 2 - encounter with understanding of fractions is common and recurrent in most nations the world over (Behr et al., 1993). Difficulty with fractions among teachers is well documented in many countries (Jenny and Gill , 2006).Studies on teacher knowledge about fractions have found both procedural and conceptual understanding among teachers , although procedural understanding dominate in this area (Fuller,1997). Among the representations that teachers use to present fractions to students, circular representations which are unable to illustrate conceptually complex operations, are the commonly used. In New Zealand, the Numeracy Development Project findings indicate that multiplicative and proportional thinking remain areas of difficulty with student performance a “little disappointing”. In Algebra domain disappointing student performance in the multiplicative and proportional domains over recent years has been identified as an area requiring further investigation (Young, 2006). There is ample research on the importance of Algebra and the difficulties students face in learning Algebra (leung etal 2014). In New Zealand, Britt and Irwin’s (2005) investigation of the Numeracy Development Project found that those students who acquired flexibility in using a range of general arithmetical strategies also developed the ability to express the structure of those strategies in symbolic forms. Such a foundation is beneficial for students’ learning about solving linear equations. Some of the difficulties that arise when learning to solve equations are associated with the move to solving them algebraically as opposed to arithmetically. Developing productive ways of thinking about algebraic symbols takes time and is one of the big goals of learning algebra (Blanton etal, 2008). One of the approaches that dominate classroom instruction is to repeatedly change the equation by doing the same to both sides until one has an equation that directly gives the answer. Understanding is built on the assumption that the
- 17. - 3 - integrity of the equation is preserved. Equations with fractions like 𝑥 2 + 2 = 5 − 𝑥 3 can be solved by multiplying every term by 6 to reduce it to the form 3x+12=30-2x.The balance model is applied as students should understand the equal sign as a statement of balance and know what operations to both sides of an equation preserve the balance (Pirie and Kieren 1994). The result will be x=3, 6. Researchers have established that many students have considerable difficulty in understanding fractions (Behr et al., 1992) cited in (Chinyoka, Mutambara and Chagwiza 2012). The sources of the students’ problems of understanding Equations with fractions may be categorized into two major related groups. These are the formal teaching strategies employed by teachers and learners attitude towards the concepts .Most of the times, teacher centred methods are used to teach Equations with fractions and as a result students fail to understand the concepts. A case study carried at Redcliff high school in Zimbabwe show that teacher centred methods like exemplification are believed to enhance students ‘problem solving capabilities (Chinyoka etal 2012). Teacher centred instructional strategies usually divorce the classroom concept from the real life situation and make the concept difficult to comprehend. Real life situations entail what students already know and what they encounter in their day to day life.’ teaching strategies that are influenced by the teachers’ understanding about the students’ prior knowledge, content and how students build productively upon understanding of concepts makes the classroom learning of concepts irrelevant and difficult to comprehend. According to Sowder and others (1998) cited in Chinyoka M.etal 2012, the progression of the concept of fraction from low to high level school curriculum requires a transition from additive reasoning to multiplicative reasoning. Post data findings that were drawn from a case study in Zimbabwe suggested that teachers promote procedural understanding of the concept of fraction (Chinyoka etal, 2012).The study recommended that child centred teaching strategies which result in conceptual understanding of fractions should be reinforced. A research was carried out at Emsizini Secondary School in an effort to provide answers to the questions raised and contribute to empirical evidence on how fractions are taught.
- 18. - 4 - Learning involves the active construction of knowledge through personal experience and influenced by prior knowledge as well as student and teacher attitudes and approaches towards learning. Learning does not occur in isolation and is not fixed, but rather it is socially negotiated and expressed through language that focuses on explanation and clarification (Chinyoka etal 2012). Learning is enhanced through collaboration with more knowledgeable others through a scaffolding process where learners progress from assisted to independent performance and assessment is an integral part of the learning process and should be consistent with learning principles (Clarke, 1997; Yackel et al., 1992 as cited in Chinyoka etal). Classroom teachers seeking to merge their beliefs about learning with their pedagogical practices could benefit from the abstraction of shared practices that reflect the principles of social constructivist learning and teaching. The essences of these principles suggest that students need to make sense of the information they are constructing through socially interactive processes so that understanding becomes the goal of learning. Pelfrey R.(2000p3) say that “ a teacher should create problems to provide additional practice to their students on how to answer open ended questions”. The aforementioned ideas discussed regarding students as active participants in knowledge construction provided underpinnings for understanding the way teachers teach fraction concepts. The pedagogical implications of constructivism are that teachers should act as facilitators who provide appropriate activities and support for students to personally construct meanings, rather than receive them ready made from the teacher (Chinyoka etal 2012). Learning should focus on the development of conceptual understanding on problem solving (Professional development Module 2012, university of Nottingham). Those students’ errors which are due to deep-rooted misconceptions should be exposed and discussed in classroom. Students
- 19. - 5 - should investigate fractions through informal explorations such as using materials (Hunker 2010). When students are solving problems instructors should not be satisfied with numerically correct answers. They should require students to demonstrate their conceptual understanding of every aspect of the problems (Finney, 2000). 1.2 Statement of the problem This study analyses difficulties encountered by learners in solving equations with fractions. The concept of solving equations with fractions belongs to the family of Algebra which is one of the most intellectual constituents in mathematics. Algebra is now a required part of most curricula in Zimbabwe from secondary school right to tertiary level. A firm foundation in solving equations is necessary to build stamina and strong mathematical aptitude in preparation for other math areas. The application of Algebra cuts across a variety of fields where even companies use Algebra to figure out their annual budget and annual expenditure. Algebra also has individual applications in the form of calculation of annual taxable income, bank interest, and instalment loans. The formula 𝐼 = PRT 100 is an equation with fractions where each variable like the principal, rate or time can be made the subject of formula. Thus equations with fractions manifest in some key areas which affect the general populace though in a silent mode .Algebraic expressions and equations serve as models for interpreting and making inferences about data. Further, algebraic reasoning and symbolic notations also serve as the basis for the design and use of computer spreadsheet models. Therefore, mathematical reasoning developed through algebra is necessary all through life, affecting decisions we make in many areas such as personal finance.
- 20. - 6 - Although there are many causes of student difficulties in Algebra, particularly solving equations with fractions, an analysis is necessary in order to develop a clear understanding of what factors help students to be successful in this area and how schools and other systems can assist in achieving this goal. Unearthing challenges faced by learners and developing workable solutions to counter the challenges is the goal of this research .The topic of Equations with Fractions at Form 3 has a bearing on performance strength at Form four as failure to grasp the concepts may lead to poor performance in other Math areas. 1.3 Aim of research The aim of the study was to analyse difficulties encountered by Form three learners when solving equations with fractions. 1.4 Objectives 1. To establish the challenges faced by learners when solving equations with fractions. 2. To compare findings on performance of learners when solving equations with fractions before and after an intervention programme. 3. To find the root cause of learners’ errors and misconceptions in solving equations. 4. To prescribe effective teaching methods to help learners overcome conceptual challenges. 1.5 Research questions 1. What challenges do learners encounter when solving equations with fractions? 2. How do Form three learners solve problems related to equations with fractions? 3. What is the root cause of learners’ errors and misconceptions in solving equations? 4. What effective teaching methods can help learners overcome conceptual challenges? 1.6Hypothesis Ho: learners have the same misconceptions when solving equations with fractions before and after an intervention programme.
- 21. - 7 - H1: Learners misconceptions are dispelled after an intervention leading to better performance. 1.7 Significance of the study Pre-test and post-test data was used to establish the challenges faced by learners in solving equations with fractions. Intervention strategies necessary to eliminate misconceptions on solving equations with fractions were also designed. This helped to develop scientifically based policy and technical guidelines on enabling problem solving capabilities in tackling the problems to improve learners regulatory frame works of solving these equations and to enhance problem solving techniques. Results of this study might be beneficial to maths practitioners as they prescribed child centred methods which are imperative for sustainable development of problem solving skills. Education policy makers and drivers are going to benefit as well by being informed on the special skills that Maths classroom practitioners should employ as well as making further findings on how they can equip teachers with lifelong skills that will enable them to employ child centred methods in teaching. 1.8 Limitations of the study Unavailability of some learners in participating in pre-tests and post-tests instrument directly affected the outcome of the results. However those who had not participated in the first round were given a chance to do so when they availed themselves. 1.8.1 Delimitations of the study Due to the large number of potential participants in the study population, the results of the study are not generalizable. 1.8.2 Assumptions An assumption is hereby made that there is no significant change in learners’ performance that is learners still have the same misconceptions and are susceptible to making same errors in carrying out mathematical calculations. The approach is influenced by the teaching method
- 22. - 8 - that is dominant as the classroom activity. Another assumption is that there is going to be a significant improvement in performance when learners problems are identified, intervention strategies employed in solving equations with fractions 1.8.3 Definition of key terms The following definitions are provided to ensure uniformity and understanding of these terms throughout the study. The researcher developed all definitions not accompanied by a citation. Conceptions and misconceptions…. Refers to the student beliefs, their theories, meanings, and explanations . When those conceptions are deemed to be in conflict with the accepted meanings in mathematics, then a misconception has occurred (Osborne & Wittrock, 1983). There are various terms in the literature that have been used in relation to the discussion of student misconceptions. Some of the terms used are misconceptions, preconceptions, alternative conceptions, naïve beliefs, naïve theories, alternative beliefs, flawed conceptions, buggy algorithms and so on (Smith et al., 1993). Each of these terms conveys an epistemological or a psychological position and some of them even carry the same or similar meanings. While recognizing the substantial theoretical diversity of meanings, I define two overarching terms in my study -- errors and misconceptions that contain many of the above theoretical underpinnings. Diagnosis …… means the identification and characterization of errors or misconceptions of students while they are involved in the mathematical problem solving process (Brueckner & Bond, 1955). Equations kufakowadya M.B (2016 p 18) says that “An equation is formed when an expression is equated to a number or to another expression.” To solve an equation hence is to find the value of the unknown letter which represent a certain quantity. Fractions are central to people’s everyday lives hence learners need to have a deep understanding of fractions for them to be able to apply them in their day –today activities.
- 23. - 9 - Equations with Fractions are an extension of linear equations but the only difference is that fractional algebraic terms are involved. e.g. 𝑥+3 = 1 + 2 𝑥 Also 𝑥+2 3 = 5𝑥+8 12 Conceptual knowledge….. refers to knowledge acquired through personal experience and influenced by prior knowledge as well as student and teacher attitudes and approaches towards learning (Chinyoka etal 2012). Learners develop concepts from what they discover through interaction with problem solving techniques. Problems are tasks that need to be tackled to come up with comprehensive answers (Confrey,1994) Difficulties are consistent problem areas that need to be overcome by a solution process (English, 1996) 1.9 Summary Chapter one introduces this study giving the significance for the study. It has covered background against which the study was carried out. The statement of the problem sought to analyse the difficulties faced by learners in solving equations with fractions. The research questions to investigate the challenges faced by learners are also indicated. The limitations, delimitations and definition of terms have been addressed in chapter one. Chapter two will present related literature to the problem.
- 24. - 10 - CHAPTER 2 REVIEW OF RELATED LITERATURE 2.0. Introduction This chapter reviews the literature that informs the study on the analysis of difficulties faced by form three learners in solving equations with fractions. The chapter gives detailed synthesis of literature on difficulties faced by learners, knowledge construction challenges, procedural or conceptual challenges, solving equations, teacher’s content knowledge, and problem solving strategies implemented to cope with challenges faced. 2.1 Difficulties faced by learners The way students understand an idea can have strong implications on how, or whether, they understand other ideas. This observation is important for thinking on what students have learned or actually understand and it has implications on how instructional and curricular designers think about what they intend that students understand. Designers always intend some understanding whether or not they make it available for public inspection. We contend that mathematics education profits from efforts to both publicize and scrutinize those intentions. Such efforts increase the likelihood that meanings we intend students to develop actually have the potential of being consistent with and supporting meanings, understandings, and ideas we hope they develop from them (Thompson and Saldanha, 2000). Difficulties faced by learners in solving equations with fractions mainly relate to conceptual challenges, procedural knowledge acquisition and teacher content knowledge deficiency. More research need to be done on establishing difficulties faced by learners. 2.1.1 Conceptualchallenges Conceptual knowledge refers to knowledge acquired through personal experience and influenced by prior knowledge as well as student and teacher attitudes and approaches
- 25. - 11 - towards learning (Chinyoka etal 2012).Learning does not occur in isolation and is not fixed but rather it is socially negotiated and is expressed through language that focuses on explanation and clarification. Learners should be able to progress from assisted to independent performance and assessment plays an integral part of the learning process and should be consistent with learning principles (Clarke, 1997). Students need to make sense of the information they are constructing through socially interactive processes so that understanding becomes the goal of learning. (Chinyoka etal 2012). Proportional reasoning is important in students' conceptualizing measured quantities. A more comprehensive approach to this subject is presented by Leung etal (2014) who made a survey of Algebra teaching around the world focusing on Confucian Heritage Culture countries (China, Japan, Korea and Singapore) and Western countries (Czech republic, New Zealand, Norway, Sweden and the U.S.A).Many lessons in both countries stressed conceptual understanding. Huang etal (2004) cited in Leung etal (2014) points out that Chinese lesson can be summarized in a sequence which introduces the concept, explains the meaning discriminates the concept with varying exercises and summarizes. The authors argue that a distinctive characteristic of Chinese lessons is teaching with variation. Since algebra is about generalization and transformation, it can certainly be argued that algebra can be learned effectively through teaching with variation (Marton and Booth, 1997) .Variations of various components within a concept, help students to understand the concept. When looking at equations with fractions. A foundational study of linear equations and expressing algebraic fractions as single numbers help learners to connect the concepts when solving equations with fractions.
- 26. - 12 - Vergnaud (1983; 1988) emphasized on proportional reasoning when he placed single and multiple proportions at the foundation of what he called the multiplicative conceptual field. School mathematics work should be characterized by unimaginative teaching. If the emerging multiplicative approaches to teaching fractions create opportunities for students to competently solve fraction problems then the goal of making school mathematics relevant to the students will be attained. Teachers should therefore create classroom opportunities for students to develop multiplicative reasoning skills. Rittle etal (2002) recommended that research in mathematics education focuses on the conceptual knowledge orientation of teaching strategies. The knowledge attained is an outcome of some reasoning approaches which should be used in solving a mathematical problem (Chinyoka etal 2012). 2.1.2 Procedural knowledge Procedural knowledge refers to knowledge that is acquired through a pattern of rules. The “sum” of fractions: 𝑎 𝑏 + 𝑐 𝑑 = 𝑎+𝑏 𝑐+𝑑 is not something the student proposes to the teacher because he believes it to be true, but because he thinks it may be acceptable by the teacher in terms of its form . In the context of a problem involving fractions, it is illusory to imagine that the student reasons, while choosing the appropriate operation to perform, when it is well known that, by contract, his objective is that of receiving a nod of approval and so is perfectly capable of producing a series of proposals often quite contradictory. The apparent absurdity (from the mathematical point of view) of the series of proposals gains a logic (from the point of view of didactics). Students give up taking risks, abdicate the burden of responsibility for their own learning and act only in terms of the contract. With reference to fractions this is rather evident (Pinilla, 2007).
- 27. - 13 - Knowledge of rules or procedural knowledge will not promote conceptual understanding of fractions. Yin (1994) has observed that quite often students have exhibited more procedural understanding than conceptual understanding. Students should acquire a deep understanding of equations with fractions and be able to use them competently in problem solving. Despite the identified similarities of the algebra lessons in the Confucian –Heritage Culture countries, some major differences are evident. It is acknowledged widely that developing conceptual understanding and procedural fluency are important in mathematics learning (Hiebert and Carpenter, 1992).The classrooms in Korea, Singapore, Hong Kong and Macau seem to align most closely to the procedural model , while the Japanese model exemplify the conceptual model. The typical Japanese problem-solving oriented lesson consists of four components namely presenting one problem for the day, problem solving by students, comparing and discussing and summing up by the teacher. Algebra learning requires striking a balance between conceptual understanding and procedural fluency. 2.1.3 Teachercontentknowledge Accumulated research findings in past decades have led to the understanding that teachers’ knowing mathematics for teaching is essential to effective classroom instruction (e.g., Ma, 1999; RAND Mathematics Study Panel, 2003). Corresponding efforts have also been reflected in teacher preparation programs that call for more emphasis on prospective teachers’ learning of mathematics for teaching (CBMS,2001; NCTM, 2000). Such efforts can presumably increase the quality of pre-service teacher preparation and prospective teachers’ confidence and ultimate success in future teaching careers. Yet, much remains to be learned about the extent of knowledge in mathematics and pedagogy that prospective teachers acquire and need to know for developing effective classroom instruction (Li and Smith ,2007). The
- 28. - 14 - learners understanding of mathematics, their ability to use the knowledge to solve problems and their confidence in and disposition of mathematics are all shaped by the teaching they encounter in a school.(NCMT, 2000). There is a widespread agreement that mathematics teachers need to have a deep understanding of mathematics (Ball, 1993). However, teachers’ knowledge of mathematics alone is insufficient to support their attempts to teach mathematics effectively. There are various ways to define PCK in mathematics. While Ball (1990) differentiated two dimensions of teachers’ content knowledge: teachers’ ability to execute an operation (division by a fraction) and their ability to represent that operation accurately for students, Ma (1999) described “profound understanding of fundamental mathematics” in terms of the connectedness, multiple perspectives, fundamental ideas, and longitudinal coherence. Moreover, the National Research Council [NRC] suggested that mathematics teachers need specialized knowledge that “includes an integrated knowledge of mathematics, knowledge of the development of students’ mathematical understanding, and a repertoire of pedagogical practices. Ma’s (1999) revealed Chinese elementary teachers had a profound understanding of fundamental mathematics concepts in subtraction with regrouping, multi-digit multiplication, division by fractions, , in comparison with U.S. counterparts. After that, several studies have focused on mathematics teacher knowledge in China and the U.S. By comparing pedagogical content knowledge of middle school mathematics teachers between the U.S. and China, An et al. (2004) found that the Chinese mathematics teachers emphasized gaining correct conceptual knowledge by relying on a more rigid development of procedures, while the United States teachers emphasized a variety of activities designed to promote creativity and inquiry in order to develop concept mastery. A study investigating Chinese and US middle school teachers’ ways of solving algebraic problems (She, Lan, & Wilhelm, 2011) revealed that U.S. teachers were more likely to use concrete models and practical approaches in problem solving, but they seemed to lack a deep understanding of underlying mathematical theories. The Chinese teachers were inclined to
- 29. - 15 - utilize general roles/strategies and standard procedures for teaching, and they demonstrated an interconnected knowledge network when solving problems. 2.2 Strategies to overcome challengesfacedby learners A strategy is considered as a goal-directed, domain specific procedure employed to facilitate task performance. It is used to facilitate both knowledge acquisition and utilization. Hence, throughout this study, a strategy is viewed as a goal-directed procedure that facilitates both problem solution and acquisition of domain-specific knowledge. A strategy is also seen as potentially conscious and controllable (English, 1996). Grounded in the concept of mathematics proficiency (Kilpatrick etal., 2001), Kilpatrick, Blume and Allen (2006) proposed a framework for Mathematical Proficiency for Teaching. It suggests that mathematical proficiency with content (MPC) and mathematical proficiency in teaching (MPT) should be the main components for teachers to teach for mathematics proficiency. The mathematical proficiency with content (MPC) includes conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, productive disposition, cultural and historical knowledge, knowledge of structure and conventions, and knowledge of connections within and outside the subject. The mathematical proficiency in teaching (MPT) consists of knowing students as learners, assessing one’s teaching, selecting or constructing examples and tasks, understanding and translating across representations, understanding and using classroom discourse, knowing and using the curriculum, and knowing and using instructional tools and materials. This model illustrates Shulman’s (1986) subject matter knowledge and pedagogical knowledge with a focus on mathematics proficiency. Simon (2006) adopted the idea of a Key Developmental Understanding (KDU) in mathematics, namely, understanding a topic from multiple perspectives, building a well- structured knowledge base. KDUs are regarded as powerful springboards for learning and useful goals of mathematics instruction.
- 30. - 16 - 2.3 Problem solving and students’ mental models Students’ construction of knowledge in mathematical problem solving is reflected in their use of strategies as they attempt to master a problem situation. Various stages of the solving process will bring different sets of challenges to them. It is the construction of cognitive structures that are enabling, generative, and proven successful in problem solving (Confrey, 1991) Students begin by identifying their problems, acting on them, and then reflecting on the results of those actions to create operations. This is followed by checks to determine whether those problems were resolved satisfactorily by reflecting on the problems again, thereby making the process cyclic. They have duration and repetition, and they are more easily examinable than isolated actions (Confrey, 1994). “Assimilating an object into a scheme simultaneously satisfies a need and confers on an action, a cognitive structure” (Thompson, 1994, p. 182). By listening to student explanations, teachers can decode student thinking patterns thereby allowing teachers to identify not only the reasons behind their particular actions but also their misconceptions. Hence, analyzing student data can prompt re-examination of one’s mathematical understanding and their mathematical meaning. A more comprehensive approach to this matter is presented profoundly by the United States Department of Education. Compared to elementary mathematics work like arithmetic, solving algebra problems often requires students to think more abstractly. Algebraic reasoning requires students to process multiple pieces of complex information simultaneously, which can limit students’ capacity to develop new knowledge. Such reasoning is sometimes described as imposing high cognitive load or challenging working memory, which can interfere with students’ ability to learn. Solved problems can minimize the burden of abstract reasoning by allowing students to see the problem and many solution
- 31. - 17 - steps at once—without executing each step—helping students learn more efficiently. (institute of educational Sciences,2015) Analysing and discussing solved problems can also help students develop a deeper understanding of the logical processes used to solve algebra problems. Discussion and the use of incomplete or incorrect solved problems can encourage students to think critically. According to Polya (1957), problem solving is a stage-wise procedure. Polya (1957) presented a four-phase heuristic process of problem solving. The stages under this model are: understanding the problem, devising a plan, carrying out the plan, and looking back. Schoenfeld (1983) devised a model for analyzing problem solving that was derived from Polya's model. This model describes mathematical problem solving in five levels: reading, analysis, exploration, planning/implementation, and verification. In applying this framework, Schoenfeld discovered that expert mathematicians returned several times to different heuristic episodes. English (1996) reviewed the steps on children’s development of mathematical models. According to her empirical findings, children first examine the problem for cues or clues that might guide the retrieval from memory of a relevant mental model of a related problem or situation. After retrieving a model, they attempt to map the model onto the problem data. This mapping may involve rejecting, modifying, or extending the retrieved model or perhaps replacing it with another model. If there is a correspondence between the elements of the mental model and the data of the problem, the model is then used to commence the solution process. However, retrieving an appropriate mental model may not be automatic or easy for children. English (1996) further said that, as children progress on the problem, they may recycle through the previous steps in an effort to construct a more powerful model of the problem situation and its solution process.
- 32. - 18 - 2.4 Summary The literature reviewed in this chapter shows that an important aspect of learning pedagogy in solving equations with fractions. It has also shown the importance of multiplicative reasoning in enhancing the full understanding of the concepts. It also covered the challenges faced by learners which include the formal teaching strategies. Strategies for challenges faced by learners have been discussed as well. The next chapter shows how data is going to be collected, presented and analysed.
- 33. - 19 - CHAPTER 3: RESEARCH METHODOLOGY 3.0 Introduction This chapter describes the research methodology which was used in carrying out an analysis of difficulties faced by Form three learners when solving equations with fractions. The chronology of the chapter is research design, population, sample and sampling procedures, research instruments, data collection procedures and data analysis procedures. 3.1 The research design Pre- test and post- test research designs were conducted so as to generate data for quantitative analysis. The data was taken in the final analysis as it formed the primary data for the purpose of analysis. After a lesson was conducted on solving equations with fractions, a pre-test was given to the learners. An intervention programme in form of revision and advice on problem solving preceded a post –test. Each test dossier had low order and higher order questions hence learners understanding of simply concepts on equations were also tested. Participants ‘answer scripts were marked and misconceptions and errors were noted. 3.2 Population The target population of this research was all thirty Form three learners at Emsizini secondary school in Imbizo District, Bulawayo. Fifteen girls and fifteen boys comprised the population under study and all went through the same treatment. The learners were of mixed ability grouping and had various challenges in the topic of solving equations with fractions. 3.2.1 Sample size and Sampling procedures Convenience sampling technique was used as the school comprised of a single group of Form three learners. The sample was selected on the basis of accessibility or convenience. Form three learners were now in the eve of final Ordinary level stage hence they were seen as the best group for the study. Thirty learners were readily available and were given same treatments so they could undergo pre-test and post-test.
- 34. - 20 - 3.3 Research Instruments As a primary instrument of data collection the researchers collected data through pre- test and post- test. The pre-test consisted of seven questions which had weightage ranging from two to four marks. The post-test consisted of five questions which had weightage ranging from two to four marks. The researcher employed triangulation in which several data collection instruments like pre-tests and post tests are considered to be sufficient multiple data sources. 3.3.1 AN EXTRACT OF THE SCHEME CUM PLAN FOR FIRST SECTION The scheme cum plan was drawn which showed the foundational concepts on algebra like collection of like terms, simplifying algebraic terms, identifying coefficients of given terms were covered. Addition of algebraic fractions was also taught in preparation for students’pre- tests. An extract of the scheme cum plan for next section was drawn. All learners went through the same treatment which exposed them to concepts involving operations on algebraic expressions, factorization among others to prepare them for the grand theme of solving equations with fractions. The scheme cum plan outlines the chronological order of concept coverage from factorization right through to solving equation with fractions. 3.4 Data collectionprocedures An ability test in form of a pre-test was administered to ascertain learners’ difficulties challenges and misconceptions in solving equations with fractions. A number of learners had challenges in this area. Before the learners were given the ability test lessons were done on the aspects as they were all subjected to the same treatment. Before arriving to the concepts of equations with fractions, other aspects which were building blocks to the topic of equations with fractions were covered. Participants answer scripts were marked and errors
- 35. - 21 - and misconceptions identified. These were recorded on a misconception and error table according to how the participants answered the questions. Marks were recorded on a mark list record. The data to be analyzed will be collected from the mark lists which contain the information for the pre-test and post-test records Marked Answer scripts were given back to the participants for feedback in self-addressed A4 envelopes. Self-addressed envelopes were distributed to the participants through the class teacher who marks records of attendance for the learners between 5 July 2017 and 14 July 2017 3.5 Data analysis procedures Items in the assessment tool were categorized into five cognitive levels according to skills they assess in learners as shown in Table 1 below. The cognitive levels have been borrowed from Hart (1981). Key skills and knowledge assessed by the questions are briefly outlined under the heading errors and misconceptions in the next chapter. Items categorized as Level 0 to 2 were not so challenging. The items are easy to solve. Level 3 to 4 items are complex, and thus more challenging as learners are expected to recognize their multiplicative nature. The first step of the analysis was to mark learner scripts. Each correct response was awarded 3 to 5 marks and no mark was awarded for an incorrect response. The framework used in the Concepts in Secondary Mathematics and Science study (Hart, 1981) was adopted and adapted for the analysis of learner performance in the test (see Table.2). Table 1 : Levels of competency Level Criterion 0 Learner obtains less than (30%) in Level 1 items 1 Learner obtains (30%) or more in Level 1 items 2 Learner obtains (45%) or more in Level 2 items 3 Learner obtains (50%) or more in Level 3 items 4 Learner obtains 60% or more in Level 4 items There were two Level 1 questions, and these were therefore marked out of a total of 6
- 36. - 22 - marks. If a learner scored less than 2 mark in this category of questions, the learner was regarded as performing at competency Level 0 (refer to Table 2). Similarly, questions at other levels of difficulty were marked out of a total of five marks, and the benchmark for each level was 60%. The overall performance of learners in the test was then summarised. Thereafter the responses to each item were scrutinized for errors and misconceptions. In the analysis of individual questions, I looked for the cause of the error in each incorrect answer and also explored the underlying reason that caused the learner to make the error. The categorisation of errors was informed by the strategies that the learners used to solve the problems .For reporting purposes and data presentation convenience, and also to observe confidentiality, learners were assigned numbers. Instead of referring to the learners by name, learners have been referred to as Learner 1, learner 2, …, up to Learner 30 3.5.1 Student t-test distribution The student t-test distribution was used to test the hypothesis that learners misconception still exist before and after an intervention is made and answer the research question on what challenges do learners face in solving equations with fractions .Results for pre-test and post-test will be tested. 3.6 Summary Experimental research design was chosen for this study since it has wide coverage. Simple random sampling technique and convenience random sampling were used to select the representative of the total population but however since the sample comprised of a single group, the class was taken as the sample size. Pre-tests and post-tests and interviews were selected as data instruments to solicit information from the respondents. The researcher collected the results of the tests and interviews and collected them for data presentation and analysis. Pre-tests and post tests were done so as to extract data.
- 37. - 23 - CHAPTER 4 DATA PRESENTATION AND ANALYSIS 4.0 Introduction The previous chapter gave a detailed outline of how data was collected. The research tool used to collect data was also elaborated on. In this chapter an analysis of the data that was collected is presented. This was be done by first looking at the general performance of learners in the test. Then the performance of learners was examined to identify prevailing errors and misconceptions so as to identify challenges learners encounter when solving equations with fractions .Lastly, for each question, identified learner errors were presented and interpreted as to find out how learners comprehended solving equations with fractions. 4.1 Presentation of Descriptive Characteristics of Respondents Information on descriptive characteristics was put in tables and bar graphs and pie charts will be drawn to represent the data. 4.1.1 Demographic data Demographic information was collected through completion of pre-test participants’ information section. Table 2: Learners demographic data sex Pre-test Participants Post-test participants males 15 15 females 15 15 Equal number of participants in the pre-test and post-test was recorded.50% of the respondents were males and the other 50% were females. Though the information is of little use in the study it exists to show the population.
- 38. - 24 - Table 3 :Ranking of learners ability test results The table shows participants ability test performance both for the pre-test and post- test.The bar graph below shows mark rankings for the two instruments. FIG 1 : BAR GRAPH TO SHOW MARK RANKINGS Results show that there was a significant improvement in percentage ability perfomance.in Test 1 the performance improved as the percentage of those who were below average decreased by 33%. The percentage rise on those who got above average mark is a welcome development as percentage rose by 27%. 18 5 7 8 7 15 0 2 4 6 8 10 12 14 16 18 20 Below Average Average Above Average NumberofStudents Mark Category BAR GRAPH TO SHOW MARK RANKINGS Test 1 Test 2 Mark ranking Pre-test Post-test Below average 18 (60%) 8 (27%) Average 5 (17%) 7 (23%) Above average 7 (23%) 15 (50%)
- 39. - 25 - 4.1.3 Learner performance in the test In the previous chapter an elucidation of how the test items were categorized into cognitive levels was presented. A framework of Learner Levels of competency was also provided. Level 0 refers to learners that obtained less than 30% in questions categorized as Level 0; Level 1 denotes learners that obtained 30%- 40% in Level 1 items; Level 2 indicates that a learner obtained 40%-45% in Level 2 items; Level 3 refers to learners that obtained 45%-55% in Level 3 items; and Level 4 comprise learners that obtained 60% and above in Level 4 items. The number of learners performing at each cognitive level is reflected in Table 4 Table 4: Learner levels of competency Level Number of learners 0 3 1 9 2 1 3 4 4 13 n=30 FIG 2 : BAR GRAPH TO SHOW LEARNERS’ COMPETENCY LEVEL Only about ten percent of the participants in the tests instruments performed at level 0. These are the learners that could not solve elementary problems, in which equations with 3 9 1 4 13 0 2 4 6 8 10 12 14 0 1 2 3 4 NumberofLearners Level of competency BAR GRAPH TO SHOW LEARNERS' COMPETENCY LEVEL
- 40. - 26 - fractions were given .About thirty percent of the participants were in level 2. The learners struggled with problems that required cross multiplication. About 60% of the learners performed at level 3 and above. This also means that at least half of the participants could correctly solve problems in which cross multiplication and simplifying was needed. Almost 40% of the learners could not adequately solve level 4 problems, that is, problems that required them to recognize that other previously covered concepts like application of quadratic formula need to be applied in problem solving. 4.1.4 Learners’ grouping of percentage marks in to classes Table 5 Class(%) 21- 30 31- 40 41- 50 51- 60 61- 70 71- 80 81- 90 frequenc y 6 7 4 8 3 0 2 FIG 3: HISTOGRAM TO SHOW DISTRIBUTION OF PERCENTAGE MARKS The histogram shows that the class 51-60 has the highest frequency followed by the 31-40 class and 21-30 class respectively. The classes 61-70,81-90 and 71-80 constitute the bottom three as the least number of students are recorded in these classes respectively. Participants also showed some errors and misconceptions when answering questions. These were categorised in to seven areas as shown in the table below. 6 7 4 8 3 0 2 0 1 2 3 4 5 6 7 8 9 FREQUENCY CLASS HISTOGRAM TO SHOWDISTRIBUTION OFPERCENTAGEMARKS 21 - 30 31 - 40 41 - 50 51 - 60 61 - 70 71 - 80 81 - 90
- 41. - 27 - 4.2 Table 6: Learners errors and misconceptions ERRORS OR MISCONCEPTIONS PRE-TEST POST-TEST Assigning addition of algebraic terms and whole numbers Misuse of the “change side- change sign” Incorrect cross multiplication Incomplete simplification Incorrect or omission of common denominator Forming incorrect equations Interference from previously learned methods Q3 eg 3b+4=7 (5) Q3 15b+20=51 NOTsame as 15b=51+20 (10) Q2 Wrong cross dot (5x10=3m instead of 3(10)=5m (8) Q6 y=y/y (3) Q7 2x(x-2) is the common denominator (7) Q2 51b+68=15 resulting from wrong cross multiplying (6) Q7 application of quadratic formula to solve the equation (28) Q5 5x+10=15 (6) Q2 X-7=12 NOT the same as x=12-7 (8) Q3 4(1)=(x-3)(x-3) instead of 4(x-3)=1(x-3) (6) Q1 t=20/5 (5) Q5 (x+8)(x+2) is the correct common denominator (5) Q2 4x-28=3 resulting from wrong cross multiplying (4) Q5 application of quadratic formula (26) Errors as systematic persistent patterns of mistakes performed by learners (Brodie & Berger, 2010) as they solve problems. This study revealed several mistakes that suit this definition. The errors varied with contexts and cognitive levels of the questions. Nesher (1987) pointed out that errors often lurk behind misconceptions. In this section I will outline and explain errors and misconceptions identified by the study. The research will also attempt to provide explanations for why learners commit the identified errors. This section will therefore address the second research questions,
- 42. - 28 - Errors resulting from the incorrect use of the strategy were also observed in solutions to problems at a higher cognitive level. The above examples also demonstrate that learners do not interpret the answers they obtain to see whether they make sense in the particular context. One way to judge the reasonableness of their answers would be to make substitution to check if the answer is correct The errors and misconceptions will be named A,B,C…….G in that order for data analysis purposes. Table seven below is a summary of the learners errors and misconceptions They were variations in learners’ errors and misconceptions as shown by the number of learners under each category. Table 7: Summary of the number of learners under each error category for pre- test and post-test Errors and misconceptions A B C D E F G Pre-test 5 10 8 3 7 6 28 Post-test 6 8 6 5 5 4 26 4.2.1 Assigning addition of algebraic terms and whole numbers In pre-test, five participants had a syndrome of making the error of combining algebraic terms and whole numbers whereas in the post-test six learners made the same error. An assumption is made here that there could be an additional participant who previously had not made the error in the pre-test and upon attempting the question in the post-test made the error. Learners need to understand the idea of like terms.eg in 2x²-4x, the result is not -2x as suggested by some learners. 4.2.2. Misuse of the change side-s change sign concept Students who made the above error had difficulties to perceive that when a whole number or algebraic term crosses the equal sign it has to change a sign. There was a 20% decrease of participants who made this error. The pre-test recorded 33% of learners making this error while in the post-test about 13% of participants made the same error. 4.2.3. Incorrect cross multiplication Another possible misconception for some students was their difficulties in understanding the cross multiplication system when reducing fractions to linear equations. In question 2 of pre- test eight participants made the error while in question 3 of post-test, six learners succumbed to the same error when they multiplied numerators on their own and denominators on their
- 43. - 29 - own when cross multiplying representing 26% and 20% of the learners in this category respectively. 4.2.4 Incomplete simplification Question 6 and question 1 of pre-test and post-test respectively had that need of simplifying improper fractions in to whole number solutions. However some learners might have over looked the concept. About 27% of the participants made the error while a considerable half of the same percentage made the error in the post-test. 4.2.5 Incorrect common denominators Question 6 and question 5 of pre-test and post-test respectively showed gross learners ‘errors a result of failure to apply “ Lowest common multiple” concept where a common denominator is needed. Three learners made the error in the pre-test accounting for 10% as compared to five learners who succumbed to the same error accounting for about 17% of the population. 4.2.6 Formation of incorrect equation Cross multiplying in a wrong way will have an effect of generating incorrect equations. About 23% of learners are in that category of learners who generated incorrect equations in the pre-test with seven learners. Post-test results show that 13% of the same population made the same error with four learners making the mistake. 4.2.7 Interference from previously learned methods This specific misconception originated from a number of students who failed to reduce the equation to a quadratic form as this is mostly unexpected in any normal question. Learners did not expect any interference from previously learned methods. An unexpected 93% of participants were held captive by the error in the pre-test while a considerable 87% of the participants succumbed to the error in the post-test.
- 44. - 30 - 4.3 Analysis of Data The previous subsections highlighted how data was collected from the participants. In this section data analysis will be carried out in line with the aforementioned research questions. Research question 1: What challenges do learners encounter when solving equations with fractions? Table 7 was a summary of the challenges faced by learners when solving equations with fractions. A student t-test for repeated measures or paired observations was used as a statistical analysis tool to see if learners made significant improvements with regards to challenges they faced in solving equations with fractions. Hypothesis 1: learners have the same misconceptions before and after an intervention is made. 4.3.1 Statistical analysis of data The table below shows summary of the number of learners under each error category for pre- test and post-test after a treatment was administered. Table 8: Pre-test and post-test error analysis data Errors and misconceptions A B C D E F G Pre-test 5 10 3 4 7 6 28 Post-test 6 8 5 3 5 4 26 We are to test at 5% level of significance whether there was a significant change in the number of learners who improved in dealing with errors and misconceptions. Ho: There was no significant change (learners had the same errors and misconceptions) H1: There was a significant change α=0.05 α/2=0,025 (this is a two-tailed test) Degrees of freedom=7-1=6 tcrit = t 0,025 (6) =2,447 Reject Ho if tstat > tcrit
- 45. - 31 - Table 9: data on computations of the t-distribution Error type Pre-test Post-test d d² A B C D E F G 5 10 3 4 7 6 28 6 8 5 3 5 4 26 1 -2 2 -1 -2 -2 -2 1 4 4 1 4 4 4 -6 22 đ = tstat= tstat = tstat tstat tstat = -1.3527 tstat ˂ tcrit, we reject Ho null hypothesis and conclude that at α=0.05 there was a significant change signalling learners’ ability to deal with errors and misconceptions after an intervention was made. Results of the t test for paired observations indicated a significant difference in the number of participants shown by the decrease in those who succumbed to errors and misconceptions.
- 46. - 32 - 4.4 Summary The chapter main emphasis was data collection, data presentation and analysis. Data was shown in tables, bar charts and histograms were used to represent descriptive measures of data. The t-test was used as a statistical analysis tool for data analysis purposes. The next chapter’s focus and theme is on findings, conclusions and recommendations for future research.
- 47. - 33 - CHAPTER5: FINDINGS, CONCLUSIONS AND IMPLICATIONS 5.0. Introduction The purpose of this study was to analyse the difficulties faced by Form three learners in solving equations with fractions. The research questions were: What challenges do learners encounter when solving equations with fractions? How do Form three learners solve problems related to equations with fractions? The chapter’s emphasis is on the summary of the study, review of findings from the statistical analysis of data and answerers to the research questions stated in the first chapter. Interpretation of findings and conclusions, implications for the study, recommendations for practice and further study and guidelines do further study will be explored as the theme. 5.1 Summary of the study This research study was carried out to analyse the difficulties faced by Form three learners in solving equations with fractions at Emsizini Secondary School in Bulawayo Province. Simple random sampling and convenience sampling were used in coming up with the sample for the study. Thirty participants were targeted to participate in the study but however only thirty took part in coming up with instruments for data collection. Pre-test and post-test were the data collection instruments used in the study and to answer the research questions. In discussing the learners’ challenges, a closer look at their performance in terms of the different cognitive level rather than looking at the overall score obtained is beneficial to get a proper perspective of the study s main theme. Learner performance based on marks obtained may conceal crucial information which is necessary to extract for teaching and learning purposes.
- 48. - 34 - The cognitive levels that were used to classify the assessment items were outlined in Chapter 3. The cognitive levels were extracted from Key skills and knowledge are assesses based on learners cognitive levels (Hart, 1981).Associated with each cognitive level was a corresponding benchmark that was used to place learners at a competency level. 50% of the learners performed at competency level 1 or lower, which means that these learners could not solve problems in which solution of an equation which needed cross multiplication was involved was not given, but easy to find. A considerable percentage of about 87% to 93% of the learners that participated in the study could not solve problems in which the quadratic formula aspect and its application was required. Learner should “be encouraged to sharpen the ability to estimate and judge the reasonableness of solutions, using a variety of strategies (Hart, 1981).About 50% of the participants in the study did not show any of these qualities because they could only solve level 1 questions. These are the two learners who were placed at competency level 0 and the 9 learners performing at competency level 1. 5.2 Findings . Post data analysis, interpretation and discussion led to the findings that learners cognitive ability are at different levels based on the competency level records. However A significant improvement in percentage ability performance occurred which shows that the learners had developed some multiplicative reasoning in dealing with concepts on solving equations with fractions as a result of intervention strategies like group discussion. Only about ten percent of the participants could not solve elementary problems, in which equations with fractions were given .About thirty percent of the participants were learners who struggled with problems that required cross multiplication. At least half of the participants could correctly solve problems in which cross multiplication and simplifying was needed. Almost 40% of the learners could not adequately solve level complex problems, that is, problems that required them to recognize that other previously covered concepts like application of quadratic formula need to be applied in problem solving. 5.3 Conclusions
- 49. - 35 - The illustrative statistical data and analysis information open a window in to real classrooms providing detail and full picture of the teachers and learners’ experiences with solving equations with fractions. Substantiated findings that have been made show that some learners still have a performance lag being fuelled by lack of development of conceptual understanding of concepts. Learners continue to face challenges in solving complex problems where application of other concepts come in as being the gateway to solutions of problems. In Chapter 2 an outline of similar studies carried out in other parts of the world was presented. Are the findings of this study the same or different from those made by the other studies? My answer to the question is that there is no significant difference. A big concern arising from the results of this study is that learner scripts reflect little understanding of equations with fractions in the learners. For Level 1 questions, most observed errors resulted from the use of an incorrect operation or incorrect use of the cross multiplication strategy. The questions in this category required proper multiplication strategy. Learners did not portray the correct conceptualisation of algebra. A large proportion of the learners could barely set recall the quadratic formula. In the introduction, a concern about performance in mathematics was expressed.The myth that mathematics is for the chosen few is just a fallacy, therefore it is just if some learners fail it. The contributors to the status quo in mathematics is the use of ineffective and inappropriate teaching methods. Errors identified by the study point to lack of even instrumental understanding of the cross multiplication algorithm. Instrumental understanding refers to a situation where a learner knows the rule, can correctly use the rule, but does not know why the rule is used and why the rule works (Skemp, 1976). In this case, learners knew the cross multiplication algorithm, but they used it incorrectly. 5.4. Implications, Recommendations and Future research Learners’ scripts suggest that the use of cross multiplication as a strategy to solve problems have not been deeply conceptualized. Teachers need to ensure that learners can correctly identify the common denominator of algebraic fractions and thus correctly solve equations. A key developmental milestone is the ability of a student to begin to
- 50. - 36 - think of equations as an individual entity, not different from the two quantities that made it up” Teaching should consider multiplicative reasoning (Reins, 2009). A key developmental milestone is the ability of a student to begin to think of an equation as a distinct entity, different from the two quantities that made it up. It is clear that the learners in this group have moved too quickly to using the formal cross multiplication strategy without having these prerequisite understandings that Reins (2009) has outlined. Thus many of the learners carried out the cross multiplication rule without first ensuring that they have conceptualized the method. Learners carrying out algebraic manipulations should first of all conceptualize mathematical ideas being communicated to them. The performance of the learners in this study clearly indicates that the majority of learners could not reason algebraically. Teachers should be reminded that from a constructivist Perspective, errors and misconceptions result from knowledge construction by the learner using prior knowledge. This implies that errors and misconceptions are inevitable. Since errors and misconceptions cannot be avoided, they should not be treated as terrible things to be uprooted, as this may confuse the learner and shake the learner’s confidence in his previous knowledge. Making errors should be regarded as part of the process of learning. Teachers should create classrooms where the atmosphere is tolerant of errors and misconceptions, and exploit them as opportunities to enhance learning (Olivier, 1992). Curriculum developers need to streamline and clearly identify the knowledge and skills that learners should acquire on solving equations with fractions. It is not sufficient just to state that form 3 learners should be able to solve problems on equations with fractions. A more specific breakdown of the types of problems they should be able to solve and the types of situations that they should be exposed to, will help teachers better understand what they need to do with their learners. This in turn will help authors of textbooks to produce materials that do not just focus on elementary formulation of equations and solving equations traditionally. Both authors and teacher trainers need to ensure that teachers know the right pedagogy and teaching strategies to employ like the child centred
- 51. - 37 - methods of teaching similarities and differences between a ratio and a fraction. Learners need to know that equations with are an extension of the simple equations that they are accustomed to.. More than anything else, teachers need to emphasise the multiplicative nature of equations with fractions.. I hope that the outlined misconceptions that were identified in this study are not unique to this particular group of learners. It may help teachers of mathematics to assess their learners for these misconceptions, as that can inform the teaching of the topic. As long as it is understood that learners construct their own knowledge using prior knowledge, mathematics teachers have an obligation to ensure learners construct the new knowledge on the correct conceptions of the topic at hand. While it is necessary for learners to develop problem solving skills by first conceptualizing concepts it is equally important that the classroom practitioners be also equipped to tackle problems arising from learners ‘failure to grasp concepts. Staff development workshops and seminars to equip teachers with skills which will enable them to employ child centred teaching strategies that may result in the conceptual understanding of concepts should be in full swing. Teachers should encourage learners to generate their own examples as this will help them to understand concepts better. Increased time on learning concepts will give learners enough time to conceptualize concepts. Assessment should be school based on the process of teaching rather than the outcome of teaching. Continuous assessment should be in place so as to set in to motion the process of making a mathematician.
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- 56. - 42 - APPENDIX 1: PROJECT RESEARCH INTRODUCTION LETTER
- 57. - 43 - Appendix 2: An extract of the first section of the schemecum plan Appendix 3: Extract of the scheme cum plan for the next section
- 58. - 44 - APPENDIX 4: PRE-TEST INSTRUMENTS : An extract of a test instrument to find out learners ability to solve equations
- 59. - 45 - APPENDIX 4.Marking Guide for pre-test
- 60. - 46 - APPENDIX 6: AN EXTRACT OF POST-TEST
- 61. - 47 - APPENDIX 7: AN EXTRACT OF POST-TEST MARKING GUIDE
- 62. - 48 - Participants ability test performance recorded on marklist APPENDIX 8: AN EXTARCT OF LEARNERS ‘MARKLIST RECORDS (GIRLS) APPENDIX 9: AN EXTRACT OF LEARNERS’MARKLIST RECORDS (BOYS )