Overs mutamiri dissertation

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

i
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

ii
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

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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

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.

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.

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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

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

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

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

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

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

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

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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.

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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

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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

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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.

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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

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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.

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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.

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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

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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.

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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
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.

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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

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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.

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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).

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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

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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

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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.

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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

- 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.

- 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.

- 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.

- 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

- 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
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

- 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.

- 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.

- 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%)

- 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

- 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

- 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,

- 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

- 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.

- 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

- 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.

- 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.

- 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.

- 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

- 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

- 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

- 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.

- 38 -
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APPENDIX 1: PROJECT RESEARCH INTRODUCTION LETTER

- 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

- 44 -
APPENDIX 4: PRE-TEST INSTRUMENTS : An extract of a test instrument to
find out learners ability to solve equations

- 45 -
APPENDIX 4.Marking Guide for pre-test

- 46 -
APPENDIX 6: AN EXTRACT OF POST-TEST

- 47 -
APPENDIX 7: AN EXTRACT OF POST-TEST MARKING GUIDE

- 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 )

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