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i LEARNERS’ ERRORS AND MISCONCEPTIONS ASSOCIATED WITH COMMON FRACTIONS MDAKA BASANI ROSE A Mini Dissertation submitted in fulfilment of the requirements for the degree of MASTERS IN MATHEMATICS EDUCATION in the FACULTY OF EDUCATION at the UNIVERSITY OF JOHANNESBERG Supervisor: Dr Kakoma Luneta October 2011

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LEARNERS’ ERRORS AND MISCONCEPTIONS ASSOCIATED WITH COMMON

FRACTIONS

MDAKA BASANI ROSE

A Mini Dissertation submitted in fulfilment of the requirements for the degree of

MASTERS IN MATHEMATICS EDUCATION

in the

FACULTY OF EDUCATION

at the

UNIVERSITY OF JOHANNESBERG

Supervisor: Dr Kakoma Luneta October 2011

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ABSTRACT

This research aimed to explore errors associated with the concept of fractions displayed

by Grade 5 learners. This aim specifically relates to the addition and subtraction of

common fractions. In order to realize the purpose of the study, the following objective

was set: To identify errors that learners display when adding and subtracting common

fractions. The causes which led to the errors were also established. Possible ways which

can alleviate learners’ misconceptions and errors associated with them were also

discussed. The study was conducted at Dyondzo (Fictitious name) Primary School,

Vhembe District in Limpopo Province. The constructivist theory of learning was used to

help understand how learners construct their meanings of newly acquired knowledge.

It was a qualitative study where most of the data and findings were presented with

think descriptions using descriptive analysis techniques. A group of forty nine learners

was selected purposively within two classes of Grade 5 to write the class work, home

work and test on addition and subtraction of fractions. Learners were interviewed and

so were two teachers. The five teachers also completed a questionnaire of five questions

to supplement the interviews. The study found that learners made a number of errors in

the addition and subtraction of fractions, including conceptual errors, carelessness

errors, procedural errors and application errors. This finding supports findings that

primary school children experience difficulties when learning the concept of fractions.

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DECLARATION

I declare that this research report is my own, unaided work. It is being submitted in

partial fulfilment of the requirements for the degree of Master of Education to the

University of Johannesburg. It has never been submitted for any degree or examination

in any university.

_______________________________

Basani Rose Mdaka

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DEDICATION

To my family: Mzamani, Nhlovo, Eneto and Andziso.

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ACKNOWLEDGMENTS

I would like to acknowledge the support that I received from my supervisor, family,

colleagues and relatives. I specifically want to give thanks to:

Dr K. Luneta for his wonderful support, knowledge, guidance, time and

patience throughout the discussions that went into the writing of this report.

My husband, Mzamani, for his financial support and words of encouragements

at times when I thought of quitting.

My kids, Nhlovo, Eneto, and Andziso for their understanding when I left for

lectures at the University of Johannesburg.

My brother, Thomas Shirinda, and his wife Rose, for giving me a space to sleep

on weekends.

My sister-in-law, Velly Mohale, and her husband Joel , for providing me

accommodation and transport to and from University.

My daughter Vutomi for her moral support.

My colleagues in the teaching fraternity for allowing me to collect data from

them.

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TABLE OF CONTENTS

Page

CHAPTER 1: The background to the study 1

Introduction 1

Literature review 3

Explanation of concepts: Errors and misconceptions 3

Sources of Misconceptions and errors 4

Acquiring mathematical knowledge 5

What can Mathematics teachers do to minimize mistakes 6

The essence of error in acquiring knowledge 6

The meaning of the concept “fraction” and how it is being taught 8

CHAPTER 2: Research Design 10

Introduction 10

Research Problem 10

Research Question: 12

The aim of the study: 12

CHAPTER 3: Methodology 13

Introduction 13

Research instruments 14

Classwork exercises 15

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Homework exercise 16

The test 17

Probing interviews – Learners 18

Teacher Questionnaires 18

Interview questions for teachers 19

Analysis of data 19

CHAPTER 4: Findings 21

Introduction 21

Class work exercises – Results 22

Homework exercises – Results 26

Test Results 34

Teacher questionnaires 39

Teacher interviews – responses 47

CHAPTER 5: Conclusion and Recommendations 52

Conclusions 52

Recommendations 53

REFERENCES 55

APPENDICES 59

APPENDIX A: INSTRUMENTS 59

Teacher Questionnaire 59

Learners Classwork Task 61

Learners Homework Task 62

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Learner Test 63

Interview Questions for Teachers 64

APPENDIX B: LEARNER AND TEACHER RESPONSES 66

APPENDIX C: TEACHER INTERVIEW RESPONSES 79

Teacher Rito 79

Teacher Ntiyiso 80

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LIST OF TABLES

Page

Table 1: Addition and subtraction of common fractions with

the same denominators 22

Table 2: Addition and subtraction of common fractions with

different denominators 26

Table 3: Addition and subtraction of common fractions

with like and unlike denominators 34

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LIST OF ACRONYMS

1. C2005 : Curriculum 2005

2. LCD : Lowest Common Denominator

3. LCM : Lowest Common Multiple

4. HCF : Highest Common Factor

5. HL : Home Language

6. FAL : First Additional Language

7. GET : General Education Band

8. RQ1 : Research Question 1

9. RQ2 : Research Question 2

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CHAPTER 1

The background to the study

Introduction

I am an educator who has taught Mathematics in the Intermediate and Senior phases of

the General Education Band (GET) for more than 15 years in four different schools of

Limpopo Department of Education in Vhembe and Mopani Districts. I have

experienced the transition from the old national curriculum to the outcomes-based

Curriculum 2005 (C2005). However, teachers were not fully involved in the change

process even though they implemented the new curriculum in their daily classroom

teachings. I have over the years encountered a variety of misconceptions and errors that

learners develop while learning Mathematics

Hansen (2006) state that learning of mathematical concepts requires a conducive

learning environment, where learners are free to interact and share each other’s views.

Learners’ opportunity to communicate their views, including the errors they make

when using mathematical concepts are as a result among others of a conducive

environment.

My concern over the years has been that learners acquire misconceptions and make

errors irrespective of the different school contexts I worked in. These concerns inspired

me to explore common misconceptions and errors that learners in Grade 5 show when

learning to add and subtract common fractions. This study, thus, focussed on two

research questions:

1. To identify errors and misconceptions learners have on when learning the

addition and subtraction of common fractions; and

2. To establish causes of errors and the misconceptions that lead to these errors in

adding and subtracting common fractions.

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A constructivist theory of learning (Olivier, 1989; Suffolk, 2008; Watts and Bentley, 1991;

and Hatano, 1996) was applied in order to understand how learners construct their

meaning, including how learners’ misconceptions and errors associated with learning

common fractions are displayed.

Data for responding to the research questions was collected from two sources: learners’

written class work, homework, and tests, as well as from follow-up probing interviews

of learners, and teacher response to a questionnaires and follow-up interview.

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Literature Review

Explanation of concepts: Errors and misconceptions

Hansen (2006:15) explains errors as mistakes made by learners as a result of

carelessness, misinterpretation of symbols and texts, lack of relevant experience or

knowledge related to a Mathematical topic, learning objective or concept, lack of

awareness, or inability to check the answer given. She found that misconceptions lead

to errors. Drew (2005:15) defines misconceptions as the “misapplication of a rule, an

overgeneralization or under-generalization or an alternative conception of the

situation”. Mistakes displayed due to misconceptions learners have about a topic

indicates incorrect interpretation of a Mathematical idea as a result of a student’s

personal experience or incomplete observation. Luneta and Makonye (2010:35) in their

study defined an error as a mistake, slip, blunder, or inaccuracy and deviation from

accuracy.

Hodes and Nolting (1998) proposed four types of errors and explain them as follows:

1. Careless errors: mistakes made which can be caught automatically upon reviewing

ones’ own work.

2. Conceptual errors: mistakes made when the learner does not understand the

properties or principles covered in the textbook and lecture.

3. Application errors: mistakes that learners make when they know the concept but

cannot apply it to a specific situation or question.

4. Procedural errors: these errors occur when learners skip directions or

misunderstand directions, but answer the question or the problem anyway.

In analysing my data, I will use Hodes and Nolting’s definitions of the various errors to

classify my learners’ errors, to see if I find the same or I find different or additional

types of errors.

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Sources of Misconceptions and errors

Luneta and Makonye (2010:36) found the teaching and learning of Mathematics to be so

difficult and ineffective, that they suspect poor performance in mathematics to be

correlated with learner errors and misconceptions.

Battista (2001) states that the way in which learners construct knowledge is dependent

on the cognitive structures learners have previously developed. This means that there

are conceptions and preconceptions that learners of different ages and backgrounds

bring with them to Mathematics classrooms, and if preconceptions are misconceptions,

teachers need knowledge of strategies most likely to be fruitful in reorganizing the

learners’ understanding.

Shulman (1986) argues that the ability to identify learners’ misconceptions is based on

teacher pedagogical skills or teacher competence. In other words, the teacher’s main

focus is not mainly on classroom management, preparing good lessons and presenting

well-structured tasks, but also on the quality of questions about the content of lesson,

and explanations given to learners.

Higgins et.al (2002) found that possible causes of mistakes learners make may be due to

lapses in concentration, hasty reasoning, memory overloaded or failure to notice

important features of a problem. Bell (1993) found the main cause for many students to

be that they appear to understand a concept at the end of a unit, but do not retain it

after a few months. In another way, they lack long term learning. In contrast, students

with long term learning do not forget the acquired knowledge, and are able to apply it

in real life situations. Askew and William (1995) suggest that a diagnostic teaching

strategy can help promote long-term learning and transfer from the immediate topic to

wider situations.

Bell (1993) argues that students see scores and not weaknesses, because they often want

to know if their answer is correct or what score they got on a test, but don’t want to go

beyond scores to look into why they got the score they did. Yet paradoxically, this is

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one of the many ways to improve scores and acquire new knowledge. This is what

Skemp (1976:77) regards as learners being more dependent on instrumental

understanding which is the following of mathematical rules and procedure without

understanding, as compared to relational understanding which is knowing what to do

in Mathematics and the reasons behind that.

Adler and Setati (2001) state that misconceptions arise when a teacher thinks a learner is

familiar with a concept whereas the learner in fact lacks an understanding of certain

aspects of it. For instance; a learner may be using fractions and obtaining the correct

answers but not aware that fractions are numbers.

Acquiring mathematical knowledge

Higgins et.al (2002) argue that mistakes made in acquiring mathematical knowledge,

may indicate alternative ways of reasoning, and that such mistakes should not be

dismissed as “wrong thinking” but be seen as necessary stages of conceptual

development. Piaget (1972) explains development as comprising of four stages: that is,

the sensory-motor, pre-operational, concrete operation, and formal operations stages.

Piaget (1972) argues that it is through these operational stages that we can understand

the development of knowledge. For instance, the formal operation stage shows that

learners can also reason on hypothesis and not only on objects. Vygotsky (1978:9)

indicates that the essential feature of learning is that it creates the zone of proximal

development, that is, learning awakens a variety of internal development processes that

are able to operate only when the child is interacting with people in his/her

environment and in cooperation with peers. Piaget in Siegler (1991) talks about three

developmental processes of how children progress conceptually from one stage to

another namely: assimilation referring the manner in which learners transform

incoming information so that it fits within their way of thinking, accommodation as a

stage where a learner receives new information which is quite different from the

existing knowledge which s/he then tries to re-construct and re-organise ideas, and

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lastly, equilibration referring to the keystone of developmental change between the

learner’s cognitive system and the external world. In other words, equilibration is the

stage in which learners begin to realize the errors and misconceptions they have

developed and further use these mistakes to restructure their existing knowledge.

What can Mathematics teachers do to minimize mistakes?

Riccomini (2005) and Pimm (1987) state that learners need to acquire proficiency in

Mathematics to enable them to understand and apply concepts, because without

proficiency in Mathematics, learners will likely experience difficulty completing other

more advanced branches of Mathematics like Algebra. Pimm (1987) explains

proficiency in Mathematics as gaining control over the mathematical register so as to

talk like, and more subtly to think like a Mathematician. That is, he argues that learning

Mathematics is far more than knowing mathematical symbols and their meanings,

algorithms and their ordered sequence, describing, and the like, but also going beyond

them and being able to transform them to gain further knowledge of the discipline.

The essence of error in acquiring knowledge

Melis (2008:1) and Schoenfeld (1985) argue that in order for learners to shift from

routine and factual knowledge to, “more emphasis on developing competences such as

solving Mathematics related problems, reasoning, and communicating mathematically,

learners should be encouraged to explore, verbalized ideas, build confidence in them, in

learning Mathematics”. This can be done by accommodating their mistakes. If learners’

mistakes are considered to be a source of their learning, such an understanding should

improve learning and students performance because research shows that learners learn

best if they are involved in the construction of their knowledge (Jonassen, 2000; Suffolk,

2008).Therefore, they are free to show their reasoning and understanding knowing that

their errors are the stepping stones to stimulate their meta-cognition (Schoenfeld, 1985).

Luneta (2008) argues that mistakes are legitimate attempts to understand Mathematics.

This can be evident when learners actively attempt to make sense of their experience by

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trying to link everyday knowledge and school knowledge, for example, when learning

fractions and relating to the concept of sharing at home. However, Olivier (1989:11),

states that, ”concepts are not taken directly from experience, but that a person’s ability

to learn from and what he or she learns from experience depends on the quality of ideas

that he or she is able to bring to that experience”. This means that a person experiences,

learning depends on his or her interpretation and other thoughts about such experience.

Research by William and Ryan (2000) and Suffolk (2008) continues to support the idea

that knowledge of common mathematical errors and misconceptions of children can

provide teachers with focus for teaching and learning. Hansen (2006: 16) argues that,

“the most effective teachers ….. cultivate an ethos where pupils do not mind making

mistakes, because errors are seen as part of learning.” Hansen (2006: 15) drawing from

Swan argues that, “trainee teachers need to be taught to recognize common pupil errors

and misconceptions in mathematics, and to understand how these arise, how they can

be prevented and how to remedy them”. This means that although Mathematics

teachers should not teach to hide or avoid learners’ mistakes, at some stage they need to

minimize them, and to be careful how they handle them. This can be done by applying

various teaching and assessment strategies that take cognizance of the value of

misconceptions in teaching Mathematics.

Hansen (2006) suggests that placing children in situations where they feel in control of

identifying mathematical errors and misconceptions leads to greater openness on the

part of learners to explore and discuss their own misconceptions. Jansen (2001) argues

that, “teachers instead of becoming the dominant force in the classroom ….teachers now

become a guide on the side rather than the sage on the stage.” This suggests that a

teachers’ image should slowly but deliberately move from the centre stage into an

invisible position in the classroom in order to facilitate conditions, where learners take

charge of their learning. Thus, with the help of the teacher as a facilitator, learners’

mathematical understanding is likely to be developed because they are in position to

compare their thinking with that of fellow learners in the context of error analysis.

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The meaning of the concept “fraction” and how it is being taught.

Learners’ textbooks such as Classroom Mathematics, Grade 7 explains the concept

“fractions” as part of a whole (Laridon et.al, 2005: 64) and, emphasising the top part of a

fraction as numerator and the bottom part as denominator. Russell (2007) also explains

fractions as part of a whole, and states that the key to teaching fractions to young

children is to keep it concrete like using fraction strips, fraction circles, and other

manipulatives. Kerslake (1991) explains fractions as numbers and argues that it is

ambiguous to explain fractions as part of geometric shapes. Grade 5 learners’ textbooks

mainly use shapes, real world situations and sharing concepts to explain what fractions

are, Classroom Mathematics (Scheiber, Brown, Lombard, Markide, Mbata, and Noort,

2009) and New Understanding Mathematics (Mopape, Mogashoa, and Taylor, 2004).

The manner in which fractions are taught mainly involve only the following skills :

naming fractions, finding a fraction of a whole, discovering equivalent fractions,

conversions to mixed numbers, proper and improper fractions, addition and

subtraction of fractions.

De Turk (2008) argues that without a foundation in fractions, students who come to the

study of rational expressions in Algebra will be severely handicapped. Denise (2007)

states “that elementary students need to know the following about the fraction concept:

how to read the fraction, how to work with fraction families”. Denise (2007) further

states “that learners need to be taught fractions because fractions are important to

learners' test scores, and scores are important to college admissions officers, and that

fractions are necessary foundation underlying Algebra, and Algebra is the gate away to

all higher fractions”.

I think the way in which learners’ books are written contribute to instrumental teaching

and learning. In such a context, learners might end up developing more errors and

misconceptions in relation to the concept of Fractions because they don’t see fractions as

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numbers, as their understanding of fractions is limited to part of a whole. Some

mistakes I have noticed suggest incorrect models of fractions, such as half of a rectangle

with unequal parts. That is, the meaning of half was not correctly modelled to learners,

and thus contribute to misconception.

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CHAPTER 2

Research Design

Introduction

In this chapter I discuss the research problem and research questions that are focus of

the study. I have chosen a qualitative case study [Silverman (2006)] as this allows in-

depth description of classroom teaching of fractions drawing on human and cultural

evidence. In contrast, quantitative designs use highly structured methods like surveys

and structured observations, quantify variation, to establish causal relationships, and

generalise to a population (Silverman 2006). The question format is closed-ended and

the data format is numerical. The design is stable from the beginning to the end.

Participant responses do not influence how and which questions researchers ask next

and that the study is subject to statistical assumptions and conditions overview

document.

Research Problem

This study focused on identifying the nature of misconceptions and the errors that

Grade 5 learners display when adding and subtracting common fractions. the study

aimed to understand the causes of errors and misconceptions learners display. I had

identified this problem during my years of teaching Mathematics in the intermediate

phase, where I’ve found that most learners perform poorly in their class work and tests

related to the concept of “fractions” and that they dislike this section. Poor performance

on fractions and related tasks is acknowledged by my colleagues in the circuit, and

learners say that fractions are hard to grasp. Teaching the concept of fractions seems the

most worrying part of the Mathematics curriculum to educators, hence the focus in this

study on how best can we teach fractions?

Suhrit and Roma (2010:170) also suggest that understanding the causes of

misconceptions and errors can improve learner performance in fractions. Jamilah and

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John(2010) agrees that the teaching of fractions is both important and challenging at the

lower levels of schooling, and that teachers should provide experiences that involve

other mathematical concepts including number, length, weight, and money, as these

experiences should set in meaningful situations to which children relate. Hansen (2006)

argues to say despite these experiences, learners still experience difficulties because

teachers tend to forget about the analysis of the errors displayed by the learners which

they can use as a source of teaching and learning. Yetkiner and Capraro (2009) state

that middle school teachers need to possess a conceptual understanding of fractional

operations to deliver a sense-making curriculum. They further suggest that teachers

need to focus on students’ attention on multiplicative reasoning, as they fractions

equivalency, and that conceptually based instruction of fractions requires teachers to

have a complete understanding of subject matter. I agree with what Yetkiner and

Capraro (2009) are saying because some teachers failed to respond to some of my

questions. Jamilah and John (2010) argue that though the concept of fractions is often

complex in character, fractions provides pupils with important prerequisite conceptual

foundations for the growth and understanding of other number types and algebraic

operations in the later years of their school experience. Pitkethly and Hunting (1996)

also state acknowledges that the topic fraction continues to present problems and

difficulties for children in primary schools, because children tend to make all sort of

errors, not only in computation of fractions but also the basic concept.

Bezuk and Bieck (1993) have concluded that fractions cause more trouble for elementary

and middle school pupils or the GET band than any other area of Mathematics.

Having studied research findings and observed in my teaching, I wanted to investigate

the type of errors displayed by learners when adding and subtracting common

fractions, and probe further to find out the misconceptions that led to the errors they

displayed.

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Research Question:

The research question is stated as follows: What are the common errors and

misconceptions that learners in Grade 5 display when learning addition and subtraction

of common fractions? The study, thus, focuses on investigating errors and then on the

types of misconceptions that are associated with such errors.

The aim of the study:

This study explores Grades 5 learners’ errors and misconceptions associated with the

concept fractions. This specifically relates to the addition and subtraction of common

fractions. In order to realize the aim of the study, the following objective is set:

To identify errors that learners display when adding and subtracting of common

fractions, and with the use of interviews relate the errors to the misconceptions

guided by the learners’ responses.

Errors indicate that something is not right in the answers given by learners. These errors

are mistakes made by learners when doing their class work, home work, tests or

assessment tasks. Some of these errors are the results of carelessness, misinterpretation

of symbols in texts, or lack of relevant knowledge related to fractions. On the other

hand, misconceptions are the overgeneralization or under-generalization of the

situation. Misconception cannot be easily identified on learners’ written tasks, but by

probing further on their supplied responses. Nesher (1987, argues that misconceptions

can also produce correct contributions, and Hatano (1996) and Smith et.al. (in Brodie,

2005:179) state that “learners’ misconceptions when appropriately coordinated with

other ideas, can and do provides points of continuity for the restructuring of current

knowledge into new knowledge”.

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CHAPTER 3

Methodology

Introduction

This chapter discusses the qualitative research approach used in the study, sample

selection, data collection instruments, and data analysis. Validity and the reliability of

the methods are explained too.

This study was carried out in a Dyondzo primary school in the Vhembe District,

Limpopo Department of Education. It is a qualitative study of learners’ errors and

misconceptions. Creswell, J.W (2008:1) defines qualitative research as: “an inquiry

approach useful for exploring and understanding a central phenomenon. The enquirer

asks participants broad, general questions, collects the detailed views of participants in

the form of words or images, and analyses the information for description and themes.

From this data, the researcher interprets the meaning of the information, drawing on

personal reflections and past research. The final structure of the final report is flexible,

and it displays the researcher’s biases and thoughts.”

In my study the final report is based on my ideas and previous research. Qualitative

research was chosen because:

1. It seeks to produces findings that were not determined in advance,

systematically uses a predefined set of procedures to answer the question: the

findings in this study are being researched for the first time in this school and the

procedures to be used are clearly explained beforehand.

2. It seeks to understand a given research problem or topic from a perspective of

the local population it involves: in this study the participants were free to

provide their views in relation to the designed instruments without any

intervention and their opinions are taken as genuine to help the researcher for

analysis.

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3. The instruments used are more flexible: for instance, in this research

probing informal interviews of learners were not planned, but were decided to

further understand the root cause of errors displayed in relation to their work.

4. The question format is open-ended: the research approach allows a researcher

to design questions that will enhance achieving the focus of the research, and are

relevant to the context.

5. The data format is textual: the data can be found in the form of learners’ scripts

and field notes about teachers.

6. Its analytical objective describes variations and explains relationships: in data

analysis, learners various responses are analyzed separately, described and

compared (Silverman, 2006).

A class of forty nine (49) Grade 5 learners was selected based on their class performance

when compared with the other class to provide data for the study. These learners had

already received lessons on fraction skills, like naming the shaded fraction, finding the

equivalents, conversions. They wrote class work, homework and the test which focused

on the addition and subtraction of common fraction (Appendix B, C and D

respectively). The records of all learners who wrote the task were kept to ensure that all

49 learners completed the test, class work and homework. The researcher checked that

tasks were done as most work was completed during her teaching periods.

Research instruments

Learners were interviewed to find out the cause of errors and pin down the probable

misconceptions responsible for them. The interviews were conducted in English as the

language of learning and translated to Xitsonga their Home Language (HL). The main

reason for translating to HL was to avoid the issue of a language barrier in the

interview. For that reason, they wrote the tasks using either their HL or First Additional

Language (FAL) which is English, and responded to interview questions using the two

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languages. A questionnaire [see Appendix A] was developed and given to five teachers

for completion. This was done to gain insight into causes of misconceptions and errors

learners display from the perspective of Mathematics teachers. Two educators were also

interviewed using questions in the questionnaire. A tape recorder was not used to

record interviews because teachers were not willing to be taped, but did permit the

researcher to quickly jot down their responses. The main purpose of the interview was

to gain more insight on understanding teachers’ perspective regarding misconceptions

and errors learners display during their teaching of addition and subtraction of

common fractions.

Teachers who completed the questionnaire and those interviewed were selected from

schools of the same circuit. These schools were randomly selected. The teachers are

permanently employed, degreed and have experience in teaching mathematics and they

are all currently teaching Grade 5 Mathematics classes.

Classwork exercises

In this task five exercises were set (Appendix B). Learners were requested to add two

common fractions with the same denominators using the diagram method. The main

purpose was to check the errors that learners display when working with diagrams and

compare the results with the arithmetic method. I also intentionally gave learners 7/8 +

2/8 to see how they are going to show 1 1/8 as a mixed number in a diagrammatic form,

that is, if they can combine all the shaded shapes into groups of eighths and leftovers.

For the subtraction of fractions with same denominators, I wanted to check the type of

errors they display when applying the diagrammatic method. For instance, for 5/7 - 4/7

was to see if they can draw a shape divided into sevenths and shade it into five parts

and to remove the 4 parts. In this task, I anticipated that most learners will manage to

get correct answers and be in a position to explain how they got them. The study by

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Suhrit and Roma (2010), and my experience have shown that learners learn fractions

best when using geometric shapes and games.

Learners were then requested to write a rule for adding and subtracting fractions with

the same denominators, the main reason was to find the root cause of the errors

displayed or their level of understanding. In other words, the researcher anticipated

that most learners would be good in telling the rules, but surprisingly fail to apply the

very same rule when working with operations, which is an indication that instrumental

understanding took place.

Homework exercise

In these tasks (Appendix C), I gave learners exercises that deal with the addition and

subtraction of common fractions with different denominators using any arithmetic

method of their choice. The main aim for this task was not to check the quantity of

correct answers as compared to class work task, but to check the type of errors when

learners were working with common fractions with different denominators and

compare the degree of difficulties or errors when dealing with the fractions with the

same denominators. My experience has taught me that not all correct answers provided

by learners mean conceptual knowledge; learners might have acquired procedural

knowledge. This led me to probe a further few learners with their correct answers. The

main error I anticipated in this task would be failure to find the LCM or LCD, because

most of our learners in the Circuit lack the foundation for multiplication tables. For

instance, it is common for a Grade 5 learner to say 3x4=7. For this reason it was always a

challenge for them to list the multiples for the given denominators. Using the method of

their choice was not the main problem as they have mastered the procedure very well. I

went to the extent of giving them one exercise in which they were to add three common

fractions with an idea to see the type of errors they displayed, as compared to the

exercise that requested them to add two common fractions. The purpose behind these

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tasks was to check the type of errors and misconceptions that are displayed when

applying the matching, Lowest Common Denominator (LCD), and Lowest Common

Multiple (LCM) methods. I again requested them to write the rule to see if they have

mastered it correctly since we talked about the rule when we do corrections for the class

work activity.

The test

In these tasks (Appendix D), learners were requested to add and subtract fractions with

the same denominators and those with different denominators, and again to write a

rule they used to find the solutions. An exercise with three common fractions was again

given to check if there is an improvement on the number and type of errors displayed

during the class work and home work tasks. In other words, the test served as the final

session of my data collection regarding the written tasks of learners.

By this stage, I have already rectified the errors displayed in class work and homework

exercises, and also have applied all the teaching strategies that I thought will help

eliminate errors displayed in the previous tasks. A revision lesson on how to find the

multiples or multiplication tables also had been conducted. For these reasons I

anticipated little errors in the test. However, my intention was still to check the type of

common or repeated errors and the newly displayed ones.

At this stage too, I aimed at establishing the root causes of errors that led to

misconceptions through probing interviews with selected learners with incorrect

answers, and from the perspective of teachers.

In addition, the test answers will assist the researcher establish whether the fraction

concept is a complex one for primary school children to learn, despite the best efforts of

the teachers, as Jamilah and John (2010) have found in their studies. These researchers

further stated that if learners are to be successful in tackling mathematical problems

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later in their schooling, the one prerequisite is the mastery of the basic concepts in their

primary Mathematics.

Probing interviews - Learners

Probing follow-up interviews were conducted after completing the identification of

errors displayed in learners’ written work. The interviews were informally conducted

since each learner was to respond to his or her own work. The interview questions were

informal because I did not plan them in advance and emanated from discussions with

an individual learner. It was not easy since I had no tape recorder and had to write

what a learner was saying.

The main reason for follow-up questions was to find the root of errors or learners

conceptual misunderstanding. Without probing further it was not easy to find and

categorize errors. However, as Lannin, Barker and Townsend (2007) argue, mistakes are

almost sacred in nature, and one should never try to correct them. On the contrary, they

argue to rationalize them, to understand them thoroughly, in order thereafter to make it

possible to sublimate them. Some responses were difficult to understand even after

interviews. For instance some learners displayed the same errors in writing and after

probing gave different interpretations of their errors, and some could not respond at all.

Teacher Questionnaires

Five mathematics teachers completed the teacher questionnaire (Appendix A). The

main purpose of the questionnaire was to check teachers’ perceptions of common errors

learners displayed in their respective classrooms. Their perceptions refer to errors noted

from teaching addition and subtraction of common fractions. I also requested them to

indicate how best they teach addition and subtraction of common fractions. With this

question, I wanted to check if we, teachers are not contributing to learners

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conceptualization of these errors or not and to improve on our teaching. Although the

meaning of the concept ‘fraction’ is not one of my research questions, I was interested in

knowing the various ways in which learners explain the fraction concept. Their

explanations seemed to show that they understand what a fraction is. The last question

required teachers to write the possible causes of errors displayed with the addition and

subtraction of fractions, and if Mathematics teachers can remedy them. The main

purpose of this data was to find additional causes of errors, and have recommendations

of how to remedy them.

Interview questions for teachers

Two intermediate phase teachers from two other schools were selected for interview

(Appendix E) to provide additional comparative data. They agreed on condition they

were not recorded, and that be completed in private. The interview process was done

on separate days, after school, in teacher’s respective homes to protect their privacy and

avoid travelling cost. Their responses were summarized into a transcript (Appendix G).

Analysis of data

Exercise and test data was tabulated by question, indicated the 49 learners who wrote

the tasks, the percentage of learners who provided correct answers and those with

incorrect answers. Follow-up probing interviews to gain insight of learners’

understanding which led to misunderstanding and errors were categorized as either

conceptual, application, careless or procedural errors. Teachers’ questionnaires and

interviews were also analysed through grouping response that are similar and

compared to get more clarity on their perspectives.

The written work for learners in the form of class work, homework and test (samples

on Appendix F) were analysed in response to the aim to identify misconceptions and

errors learners display when adding and subtracting common fractions focusing on

research question 1 (RQ1). This was followed by probing interviews to seek

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understanding of learners’ perspectives on how such errors and misconceptions were

committed focusing on research question 2 (RQ2). Teachers’ responses to the

questionnaires were analysed in chapter 6 in response to the second objective of the

study of finding the causes of errors and misconceptions learners display when adding

and subtracting common fractions. Teachers’ perspectives are then added to a response

to RQ2. I did attempt to account for the diversity in the data with the developed

categories of errors, that describes all the research findings as Hodes and Nolding

(1998) put it.

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CHAPTER 4

Findings

Introduction

In this chapter I discuss the findings of the study in response to the research questions

asked.

In brief, 14% of learners displayed errors when adding common fractions with the same

denominators, 18% displayed some errors when subtracting fractions with the same

denominators using the diagram method. Only 29% of learners could not write the rule

for adding and subtracting fractions with the same denominators. In short, the learners

displayed fewer errors in this fraction skill.

Data on the home work activity shows a high percentage of errors by learners when

adding and subtracting fractions with different denominators. 76% of learners provided

incorrect answers when using the arithmetic method. When adding three common

fractions with different denominators, 84% of learners failed to provide correct answers.

Data from the test were similar to the data from class work and home work exercises.

The results indicate that some learners still do not perform well adding and subtracting

fractions as they continue to make many mistakes. For example, 53% of learners failed

to add 2/3 +1/6 +4/12.

Five teachers who responded to questionnaires identified errors they observed when

teaching addition and subtraction of common fractions. Two educators provided

additional causes of misconception and errors they identified in their teaching, as

discussed under teachers’ interview responses.

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Findings of the study are presented in the tables below and discussed next in response

to the research questions, that is: to find the common errors and their causes.

Class work exercises – Results

Table 1: Addition and subtraction of common fractions with the same

denominators.

Questions

No. wrote

Correct

responses

Incorrect responses

A. Use Diagram method

1.3/6+2/6=

2.7/8+2/8=

49 42(86%) 7(14%)

B. Write the rule for adding

and subtracting fractions

with the same denominators

49 35(71%) 14(29%)

C. Use Diagram method

1. ¾-1/4=

2.5/7-4/7=

49 38(78%) 9(18%)

2 no answers

D. Write

down a rule you could use

for subtracting fractions with

the same denominators

49 33(67%) 16(33%)

E. Use Arithmetic method

1.3/5+2/5=

2.13/9-4/9=

49

31(63%)

18(37%)

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The data shows that when learners were asked to use the diagrammatic method to add

and subtract common fractions, 86% of learners got the answers correct and 14% did

not. A supporting classwork exercise is presented and discussed below.

This good performance is supported by Suhrit and Roma (2010) when they state that the

teaching of fractions should be done using simple games and geometry, and by Russell

(2007) who argues that the key to teaching fractions to young children is to keep it

concrete, like using fraction strips, manipulatives and fraction circles. However, for the

14% who displayed errors, Hansen (2006) state that Mathematics teachers need to

understand how these errors arise and how to remedy them.

The mistakes learners make are mainly failure to shade the correct number of parts

taken out of the whole, and to write the fraction names. In the example below, the

fraction 7/8 is properly drawn, but is given an incorrect fraction name. The fraction 2/8

is incorrectly drawn, and the diagram is incorrect which led to the incorrect answer.

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Class work exercise 2

In exercise number 2 above, the learner cannot easily tell what the shaded parts

represent and what the denominator actually stand for. This is a conceptual error

because he failed to show that the denominator indicates the number of equal parts cut

from the whole, and the numerator indicates the number of parts taken. The answer

9/9 is also an error which shows that the learner was unable to represent 11/8 in

diagrammatic form.

In exercise number 2, all 49 learners got it wrong, suggesting that they misinterpreted

the rule, namely, to add up the total number of shaded parts and to combine these

shapes into 9 whole groups of eights and leftovers which is 1.

When I probed to check where 1/7 and 2/6 came from, Wiseman said,

“...one na two ti yimele leti nga xediwangiki, kasi seven na six hi leti nga

xediwa”

[“one and two represent the unshaded parts whereas seven and six represent the

shaded parts”].

When probed further about 9/9 answer, he said,

”Hikuva ti parts hikwato ti xediwile…, but I ani swi twisisi kahle laha mina.”

[“Because all parts are shaded …, but I don’t understand it very well when it

comes to this point”]

Suhrit and Roma (2010) state that the process of memorisation of rules of fractions

without comprehension destroys their enthusiasm to appreciate Mathematics.

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Class work exercise 1

In exercise number 1 above, the same error of failing to write a correct fraction as

shaded occurs, hence the answer 4/4 is incorrect , but not shaded at all. The learner

changed the - sign to + signs wherein numerators 1 and 3 were added together, and the

same applied to the denominators. Further probing this learner, indicates this to be a

careless error as the learner rectified the mistake on her own. Subtracting fractions

using a diagram method seemed to be a challenge because of failure to remove 1 of 3

shaded parts to give 2/4 and further simplifying it into 1/2. On probing still further this

mistake is an application error because of failure to show 2 shaded parts and 2

unshaded parts in this answer. Hodes and Nolting (1998) refer to this as a failure of

learners to “symbolise” correctly.

To verify learners’ errors, I further requested them to write a rule they applied when

adding fractions with different denominators. Surprisingly, 100% of learners responded

as follows below:

“If the denominators are not the same we ask ourselves where does it meet, then

we write and add the numerators”(Wiseman, homework)

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These responses indicate me that it is not always the case, when learners give correct

answers that these reflect an understanding of the concept, because 37% made errors

but cited the correct rule. This is what Skemp (1976 page) calls “instrumental

understanding”. When using the arithmetic method, 63% of learners gave the correct

answer. Probing the 37% who gave the wrong answers, it indicated that most errors

were careless errors. As Hodes & Nolting (1998) put it, careless errors are those which

learners can easily rectify as no deep conceptual structures are associated with them.

Homework exercises - Results

Table 2: Addition and subtraction of common fractions with different

denominators.

Questions

No.

wrote

Correct

responses

Incorrect

responses

A. Calculate to the simplest

form using any arithmetic

method

1.2/3+1/6=

2.4/5+2/3=

3.3/4-1/2=

4. 11/12-1/3=

5. ¾+7/10+1/5=

49

30(61%

12(24%)

16(33%)

21((43%)

8(16%)

19(39%)

37(76%)

33(67%)

28(57%)

41(84%)

B. Write down a rule you could

use on the addition of fractions

with different denominators

49 14(29%) 31(63%)

4 (8%)no answers

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C. Write down a rule you could

use for subtracting unlike

fractions.

49 11(22%) 29(59%)

9 (18%)no

answers

As seen in Table 2 above, when learners were asked to use any arithmetic method to

add or subtract common fractions with different denominators, 16 % of learner answers

to Question 5 were correct, but 84% were not. Suhrit and Roma (2010) found that most

concepts of arithmetic operations with fractions are often clouded with complications in

the eyes of children.

The following are examples of errors learners make when adding or subtracting

common fractions with different denominators

Beauty: Khanyisa: Justice:

Solomon:

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This data was discussed together with the probing responses of learners as they pave

way to understand misconceptions and errors displayed in their written exercises

Khanyisa answered, 32. Probing to find how she reached the answer, she responded:

“Ani ri ni tekile reciprocal ya kona se ni cinca yi va multiplication

,then…then…….uhmm ni eda ni kuma 32”. (Khanyisa, Probing response)

[“Its just that I took its reciprocal then I make multiplication sign,

then…,then…..uhmm I find 32.”]

This data suggests that this might symbolize the procedural error, when learners apply

the matching method to calculate the lowest common denominator. It may also suggest

that instead of the reciprocal of ¾, she wrote 4/5, and she later recognised the error on

her own. In addition, from the previous step one cannot easily identify how she finally

got 32 as answer, but can notice that she added all numerators and the higher

denominator (5+16+4+7=32).

Justice responded to the question, Please explain how you calculated your answer to

13+76?,

“A na ha swi tsundzuki ku ri ni swi endle njhani mina” (Justice, Probing

response)

[“I cannot remember how I did it”].

I regarded Justice’s answer as one of the most complicated to work out as there was no

single step that link to adding three common fractions with different denominators,

¾+7/10+1/5. Justice’s response suggests that the error displayed was conceptual because

he does not show an understanding of the concept of adding common fractions with

different denominators, using any method, which might either be matching or finding

the Lowest Common Denominator.

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Beauty‘s explanation was correct even if her procedure was incorrect. She quickly

explained the procedure she used and did so with confidence. She then insisted that she

was right. Beauty‘s argument of seeing herself as being correct is supported by what

Bell (1993) found, that students see scores not weaknesses, because they always want to

see what score they got in test.

She said:

“for step 1 I have to reduce the number of fractions so that I left with 2 fractions

because is easy for me, I then added as you can see, then I got 28/8,then I divide

and I got 36”. (Beauty, Probing response)

Beauty’s response indicates that it is difficult for most learners (84%) to work out the

answer properly by finding the LCD or LCM. Although wrong, Beauty could have gone

further to find the LCM of 3 and 5 instead of adding them.

Solomon’s work suggests he was trying to use the LCD method, but omitted the third

fraction. Consequently, he worked with large numbers which was confusing even to

him as he got stuck when explaining where 29 came from? His was the procedural

error.

Other samples of errors are discussed below:

Nyeleti: Sylvia:

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Nyeleti and Sylvia also gave their answers as 11/9 below, individually. When asked to

explain how they arrived at the answer 11/9 to find common misconceptions, Nyeleti

said,

“Mina ni lo eda ti numerators na ti denominators because a ni swi twisisi ku ri hi

ti kuma njhani ti LCM.” (Nyeleti, Probing response)

[“I just added the numerators and the denominators because I did not

understand how to get the LCM.”]

Her respond shows a conceptual error, as she did not understand the properties of how

to add three common fractions with different denominators, and decided to do simple

addition. Sylvia explained that she was right because the numerators add to 11 and the

denominators add to 19. Hers is a procedural error, as she didn’t follow the procedure

but answered the problem in anyway.

84% of learners answered this question incorrectly. They managed to write the rule for

adding fractions with different denominators, but could not apply it to a specific

situation. And, 16% found it difficult to add three common fractions with different

denominators. The error indicates that learners encounter problems when requested to

add or subtract three common fractions with different denominators to make it the

same.

In answering Homework Task, Question C, Pretty gave the correct rule of first finding

the LCD before subtracting the numerators, but could not subtract the fractions

correctly, as seen below:

Pretty: Homework exercise

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When asked to explain how she converted the common fractions to natural numbers,

she responded as follows:

“Hikuva swa tika ku khakhuleta ti fractions, se ni lo ncinca ni ta kota ku

multiplaya hi ku olova.” (Pretty, Probing response)

[“Because it is hard to work with fractions, then I change it so that I can multiply

easily.”]

The misconception here is over-generalization, referring to valid knowledge acquired

earlier but applying it wrongly (Drew 2005). Drew points out that there is no

Mathematical rule that allows one to convert a fraction by multiplying its numerator

with its denominator to make it a natural number.

This is what researchers like Hartung (1958), Bezuk and Bieck, (1993), Orton, (1992) and

Pitkethly & Hunting (1996) concluded, namely that the fraction concept is complex to

children, even though they acknowledge its importance.

Question A, number 3 shows another error in learning fractions.

Phumudzo: Homework exercise

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Phumudzo’s answer indicates that he is failing to find the common denominator of 4

and 2. The error shown here is to subtract the numerators and the denominators in

order to get the common denominator. Even if 2/2 is incorrect, the final answer is

wrongly simplified. This data suggests that some learners could not easily simplify

fractions by dividing or using the highest common factor.

The error of incomplete answers can also be seen in Brighton’s work below:

Brighton: Homework exercise

Briton’s answer is in contrast with most learners preferring the matching method when

adding or subtracting fractions and committing procedural error, as they experience

difficulty of finding the HCF of the denominators. The answer shows a conceptual error

as he failed to state his understanding of how to find the highest common factor of 15

and 18. In cases where learners use the lowest common multiple and the lowest

common denominator, they are likely to get correct answers because most answers do

not require simplification.

Another error was found in Question A, number 4, below.

Justice: Homework exercise

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The learner treats fractions as if they were whole numbers. This data indicate the

learners did not apply the rule of finding LCD before subtracting numerators, but

simply subtracted numerators [11-1] and subtracted denominators [12-3]. This was

confirmed when probed as his explanation said that he subtracted top numbers

together and bottom numbers together, and that 9 goes into 10 once, hence the answer

is 1. The final answer is also wrongly simplified as 10/9 should be 11/9.

Another error was found in the same Question A number 4 as below:

Collen: Homework exercise

The error above reflects that the learners failed to convert 1/3 to be 4/12 even though he

managed to correctly identify the LCD. When probed, Collen corrected himself saying

he should have multiplied 1/3 by 4/4 when converting to the denominator of 12. This

shows that the error displayed is of careless nature.

Another error found in the same Question A number 4 shows a learner who seems not

to understand fractions, as seen below.

Beauty: Homework exercise

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This learner could not account for 11/12 – 1/3 = -1; 88 or say any word in relation to

fractions. The learner indicated the she subtracted 11 from 12 to get -1, and 11-3 to get

the first 8, and 12-3-1 to get the second 8. She further went on tell that “+1 – 88 = -87”.

The error displayed can be classified as conceptual as the learner do not understand the

properties or principles covered in the learning process.

Data analysis in response to RQ1 indicates that learners display errors that are

conceptual, careless, procedural and applicational in nature.

Test Results

Table 3: Addition and subtraction of common fractions with like and unlike

denominators.

Question

No. wrote

No. of Correct

Responses

No. of

Incorrect

responses

A. Use either the matching method or

the LCM ,LCD method to find the

answers to these questions

1.11/8-3/8=

2.4/10+2/10=

49

1. 44(90%)

2. 41(83%)

3. 32(65%)

1. 5(10%)

2. 8(17%)

3. 17(35%)

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3. 5/2-3/7=

4. 2/3+1/6+4/12

4. 23(47%)

4. 26(53%)

B. Write down a rule you could use to

add or subtract fractions with the

same denominators.

49 43(88%) 6(12%)

C. Write down a rule you could use

to add or subtract fractions with

different denominators

49 35(71%) 14(29%)

The table above indicate results of learners’ response to questions in the test. Errors

displayed through the task are as discussed below:

Mercy: Test Phumudzo: Test Discharge : Test

Solomon: Test

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90% of learners answered QA.1 correctly, Phumudzo nearly got it right too but his

answer was incomplete. When asked if his final answer is in a simplified form or not,

he replied: “it is in the simplified form” (Phumudzo; Probing response), and symbolises

a lack of conceptual knowledge relating to the simplification of fractions.

Solomon’s answer shows a learner who gets stuck when adding and subtracting

fractions, but he also managed to find the Highest Common Factor (HCF) which is 2.

This error can be referred to as procedural error, because he skipped some steps to

divide both the numerator and the denominator by 2. He then writes the final answer as

2/2. When asked to explain his answer 2/2, he said: “Hikuva hi yona HCF” [“..because it is

the HCF.”],(Solomon, Probing response).

Mercy’s answer suggests an intention to apply the matching method, wherein both the

numerator and denominator were used incorrectly. Another error displayed is the use

of wrong operational signs as addition and multiplication signs which were not needed

was used. This indicates a procedural error as she was unable to explain operational

signs changed. When asked to explain how she obtained 0, 13 as an answer; she said:

“A no divider ku ri 5 yi ya ka ngani ka 11, then ni kuma 0,13”.

[ “I was dividing to find out how many times does 5 goes into 11, then I found

0,13.] (Mercy, Probing response)

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Discharge’s answer shows him correctly applying the matching method even if the

denominators are the same. There was no need to match as the denominators are the

same and this makes the operation difficult as he then had to work with bigger

numbers. The error he displayed was to change the operation sign from negative to

positive which resulted in him getting 112/64 instead of 64/64 which when simplified is 1.

Further discussion of learners’ response to the questions is as below:

Cecilia: Test Prescila: Test

Cecilia’s answer was close to being correct but failed to recognise that 8 divide by 8 is 1

without leaving a remainder. The error is that of indicating a remainder of 0/8, as this is

zero and when a zero is written after a number it changes the decimal place and value

for that number. This is an indication that teachers need to emphasise properties of

multiplication, division, addition and subtraction before they start teaching fractions.

Prescila’s answer shows the use of the matching method, like Discharge, but displayed

an error by subtracting the numerators together and denominators together. She also

displayed the misconception of dividing by zero which is undefined in Mathematics.

In responding to Question A number 4, 47% of learners were able to add correctly when

given three fractions as [2/3+1/6+4/12 ] in the test. This suggests that most learners

encounter difficulties with bigger operations than with adding only two fractions, as

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they were doing the same task for the third time. For these learners, the same errors

which had been rectified before surfaced again in the test. This is seen in the learners

answer in the test, below.

Solomon: Test

Some learners’ answers were difficult to categorize by the type of error displayed, like

the ones below. When asked to explain how you arrived at your answer, these learners

could not explain their answer. These mistakes imply that learner’s do not understands

principles guiding addition and subtraction of fractions, and referred to as conceptual

errors by Hodes and Nolting (1998).

The responses below indicate errors which are difficult to classify. Learners who

displayed these errors were probed but could not clearly explain how they arrived at

their answers. Through probing, learners indicated to have conceptual

misunderstanding.

Netsai: Test

Collen: Test

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Learner’s response to Question D in a test of what a fraction is indicated that they have

an idea of what the concept fraction is. This question was not part of addressing the

research questions of the study, but to make certain of learners’ understanding of the

concept fraction. Russell, (2007) explained fractions as part of a whole, and stated that

the teaching of fractions to young children is to keep it concrete like using fraction

strips, fraction circles and other manipulatives. Scheiber, et. al, (2009) and Mopape et.al

, (2004) found that the manner in which fractions are taught mainly involves the

following skills objectives: naming fractions, finding a fraction of a whole, discovering

equivalent fractions, conversions to mixed numbers, proper and improper fractions and

addition and subtraction of fractions. Thus some learners explain the concept of fraction

in relation to fraction skills. I think teaching the concept fractions or understanding

fractions as limited to part of a whole, and incorrect models of fractions [such as half of

a rectangle with unequal parts], lead to misconceptions about fractions when learning

them. Moreover, in instances where the teaching of fractions involves the relations

between two quantities, an explanation of what is ½? The next question could be ½ of

what?, because of ½ of an apple will differ with ½ of R52,00. This means that teachers

need to go beyond and see fractions as numbers. From the results of this study, only 2

learners explain the concept of fraction as a number.

Teacher questionnaires

Five teachers also completed a 5-item questionnaire based on the teaching of common

fractions, and two more educators were interviewed using the same questions. The

questionnaire aimed at finding answers to both the errors displayed by learners (RQ1)

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and causes of such errors displayed (RQ2) from the perspective of Grade 5 Mathematics

teachers. My findings are discussed by each of the 5 questions in the questionnaire.

1. What are the different ways in which learners explain the concept fraction?

Teachers Ntila and Makungu gave their perception of learners’ explanation of the

concept of fraction, namely that a fraction is part of a whole. This is similar to the

explanations given by Russell (2007) and Laridon (2005). Russell further states that the

key to teaching fraction to young children is to keep it concrete, like using fraction

strips, fraction circles, and other manipulatives. Learners’ response to what a fraction is

indicated that 18 out of 49 learners [38%] also explained the concept of fraction as part

of a whole.

Teacher Khanimamba gave three different explanations of a fraction,

has top and bottom number. This kind of an explanation is preferred by Kerslake

(1991). He states that the fraction concept should not be limited to part of

geometric shapes but also be explained as numbers.

has to do with division and multiplication. This explanation emphasises what is

mainly involved when teaching fractions, that is, the skill objectives like, adding

and subtracting fractions, converting fractions, equivalents and the like.

has to do with sharing, because the concept fraction forms a starting point when

people want to share something. Scheiber et.al. (2009) and Mopape et.al. (2004)

use the sharing concept to explain the concept of fractions.

Teacher Mahlahle explained the fraction concept as, being part of a whole and being

divided into equal parts [for example dividing a rectangle]. This which seemed to be a

similar definition of teachers’ perceptions of fractions, with one difference namely, the

example of a rectangle is given. These perceptions are supported by Suhrit and Roma

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(2010) when they state that the teaching of fractions should be done using simple games

and geometry shapes.

Teacher Yinhla explained the concept, fractions being,

pieces of a whole,

sharing objects into equal parts, and

the only way for all people to share equally

These explanations emphasise, the sharing of things in real life. The last adds the word

“equally”. In my experience the concept ‘half’ has been incorrectly understood when

saying of a square divided unequally, that it has been divided in half.

2. Why do we teach fractions?

Teachers Khanimamba, Yinhla and Makungu indicated that fractions are taught

because learners will be able to further apply the gained knowledge in their day to day

life, such as when budgeting, measuring baking ingredients, sharing objects/things,

and the like. De Turk (2008) argues that without the foundation in fractions, students

who come to study of rational expressions will be severely handicapped. Teachers

Mahlahle and Ntila wrote that fractions are taught to equip learners with the meaning

of fraction, various fraction families and the skills levels involved. Denise (2007) stated

that elementary students need to know how to read the fraction, how to work with

fraction families and that the fraction concept is important to learners’ test scores.

3. How best can we teach the addition and subtraction of common fractions?

Teacher Khanimamba responded that we should start from a practical sharing of bread

and say that a slice is piece or part of a whole and a loaf is a whole. By so doing learners

will have an idea that fractions are parts of a whole. Teacher Mahlahle responded by

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saying she will ask learners: if I have an apple and want to share it equally with my

friend, into how many parts will I cut it? Then s/he will draw a picture, a circle is fine,

divide it and ask: what do we call each piece? [a half]. S/he’ll then ask: what do we call

the top and the bottom halves? and test learners further by asking questions like: what

does the denominator tell us?

For Teacher Yinhla, the best way to teach the addition and subtraction of common

fractions is to use concrete objects of different shapes, colour, or texture. He will write a

number on the chalkboard and divide the object with them, form small groups of 4-6,

and let them have the object on the table to share equally amongst themselves.

Hansen (2006) suggest that placing children in situations where they feel in control of

identifying mathematical errors and misconceptions lead to greater openness on the

part of learners to explore and discuss their own misconceptions. Hansen (2006) further

argues that the most effective teaching strategy is to cultivate an ethos where pupils do

not mind making mistakes, because errors are seen as part of learning.

The three teachers have used their methods in different contexts in similar ways. Their

ideas are also supported by Russell (2007) when he explains that the fraction concept

should be taught using concrete objects. Although Suhrit and Roma (2010) acknowledge

that some very elementary concepts of fractions could be easily explained by simple

geometry, they further state that most concepts of arithmetic operations with fractions

are often clouded with complications in the eyes of children. I suggest this could be one

of the contributing factors why learners displayed a lot of errors in this study. My

experience and teachers’ responses also confirm this finding.

Furthermore, Teacher Makungu stated that learners must know the multiplication

tables and multiples before introducing LCD and LCM. I agree with this and regard

this idea as the second stage after having introduced fractions using concrete objects.

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Teacher Makungu’s method is supported by Kerslake when he argues that the fraction

concept should not be explained as limited to part of geometric shapes or concrete

objects, but is also to be explained as numbers. With Kerslake’s finding in mind,

emphasising multiplication tables before giving much time to addition and subtraction

of common fraction can be of utmost importance for teaching fractions in this school.

4. Have you observed common errors and misconceptions that your learners have

developed when teaching addition and subtraction of common fractions? If so,

please give some examples.

Teachers in the sample say they are grappling with errors learners typically make when

learning fractions. Although learners in this study are in Grade 5, the same problems

are identified by teachers of Mathematics in the Foundation Phase, and include the

following,

changing operation signs inappropriately.

omitting a division line

throwing away the denominators, and ending up multiplying the numerators

with the denominators. Example for this study is by teacher Yinhla:

5/8-3/11

=5(8)-3(11)

=40-33

=7

Adding or subtracting the denominators,

Teacher Mahlahle:

Example: 1.1/3+1/3 =2/6

2. 7/8- 1/2= 6/6

Matching both denominators even if the LCM is one of the denominators, which

make the operation too long and complicated.

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Failure to reach simplified answers.

Example: 8/10 - 2/10 = 6/10 = 2/2 [HCF become the final answer]

Incomplete answers, for example: 7/3 +5/3 = 12/3

Failure to get the LCD when adding 3 common fractions,

for example: 2/3 +1/2+4/12 [LCD becomes 24 instead of 12 which make the

operation to long and complicated]

Difficult to interpret, for example: ¾+1/2 =13

Misunderstanding the meaning of brackets when applying the LCM method,

that is, instead of multiplying they add.

Most of the common errors indicated by teachers above were found in my study. This

confirms Hansen’s (2006) finding that errors are mistakes made by learners as a result

of carelessness, misinterpretation of symbols and texts, lack of relevant experience or

knowledge related to that particular mathematical topic, and that misconceptions as the

misapplication of over-generalisation and under-generalisation of a rule, can lead to

errors. Luneta (2008) argues that mistakes are also legitimate attempts to understand

Mathematics, and thus that Mathematics teachers need to regard these errors as a

powerful source of learning.

5. What do you think are the possible causes of errors and misconceptions, and

how can we remedy them?

Teachers Ntila and Khanimamba responded that,

learners who unable to master the multiplication table, which is the foundation

for mathematics learning, make errors when learning fractions. And, educators

not giving enough effort. Hansen (2006) suggests that placing learners in

situations where they feel in control of identifying mathematical errors and

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misconceptions, leads to greater openness on the part of learners to explore and

discuss their own misconceptions.

it is a misconception that Mathematics is a difficult subject. Riccomini (2005) and

Pimm (1987) argue that learners need to acquire proficiency in Mathematics to

enable them to understand and apply mathematical concepts, because without

proficiency in Mathematics, learners will likely experience difficulty completing

other more advanced branches of Mathematics. In addition, Yetkiner and

Capraro (2009) state that middle school teachers need to possess a conceptual

understanding of fractional operations to deliver a sense-making curriculum.

lack of concentration leads learners to commit careless mistakes. Higgins et.al.

(2002) found that possible causes of the mistakes learners make may be are due

to lapses in concentration, hasty reasoning, memory overloaded or failure to

notice important features of a problem.

Teachers should,

teach and practice multiplication tables 10 minutes before the actual lesson.

give motivation from former Mathematics students.

give more exercises to learners.

allow learners to voice their thoughts. This idea is supported by Melis (2008) and

Schoenfeld (1985) when they argued that in order for learners to shift from

routine and factual knowledge to more emphasis being given to developing

competences, such as solving mathematics related problems, reasoning,

communicate mathematically, encouragement to explore, verbalise their ideas,

build confidence in themselves and that they can learn Mathematics. This can

best be done by accommodating learners’ mistakes.

In responding to Question 5 on teachers’ questionnaires regarding the causes of

misconceptions and how they can be rectified, teachers responded as discussed below:

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Teacher Mahlahle:

Misconception that everything must be added or subtracted. This is what Skemp

(1976) regard as learners being more dependent on instrumental learning, which

includes following mathematical rules and procedures without understanding.

Teachers need to emphasize the importance of finding the common denominator.

Teacher Yinhla:

Failure to check what learners already know from the previous grades.

Before teaching fractions learners must know Mental Mathematics.

Teachers Yinhla further indicated that teachers need:

not to allow learners to use calculators

to teach key concepts before starting a new lesson.

Teacher Makungu responded by indicating that teachers need to start teaching from

simple to complex by using number lines.

Teachers’ responses above serve as evidence that the fraction concept is complicated in

its nature and difficult to learn for primary school children. This is supported by what

Lannin et.al. (2007) have found. Mistakes are of a sacred nature. Teachers should never

try to correct them but try to understand them thoroughly, to sublimit them.

Based on this study, I therefore suggest the following to help remedy errors when

teaching Grade 5 learners fractions.

1. Teachers should use the learners’ errors make as the source of learning.

2. Teachers should revise the four basic operations first, before dealing with the

fractional operations.

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3. Teachers should encourage learners to communicate their mathematical thinking

coherently and clearly.

4. Teachers to have a deep grasp of the content knowledge of Mathematics.

5. Teachers to be problem solvers themselves [workshops in my experience have

failed to solve challenges faced by teachers in the teaching of the concept

fraction].

6. Teachers develop strategies which will be fruitful in reorganising learners’

understanding, because teacher pedagogical skills or competence can help

identify learners’ errors.

Teacher interviews - responses

Probing interviews were conducted with teachers from other schools so as to make

certain that data for responding to RQ2 of the study is authentic, that is getting multiple

views from teachers.

Both teachers in other schools acknowledged that primary school children encounter

difficulties when learning addition and subtraction of common fractions. Teacher B

[fictitious name] said that learners perform poorly particularly when learning to add

and subtract common fractions. This respond confirms Kerslake (1991), Suhrit and

Roma (2010) finding that the fraction concept seems difficult for to primary school

children to learn. This also seems in-line with teachers’ questionnaires data that

common mistakes made by learners indicates the difficulty they have in this section of

the Mathematics curriculum.

Learner difficulties are shown again by Teacher A who indicated that most learners

experience problems when adding or subtracting fractions using the fraction board, and

seen in this example:

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“Learners show a lot of mistakes, for instance when they are adding or

subtracting fractions using the fraction board and are requested to complete

answers like:7/8-3/8=----.Their answers will mainly be 4/0.To them 7/8 does not

mean 1/8 x7 strips, that is why they got a denominator of zero which undefined

in mathematics. For instance, 6/10+2/10 =8/20 which is the same mistake.” (Teacher

Rito: Interview)

This indicates that Mathematics teachers need to go beyond telling learners the concepts

but to attach some meaning to our explanations. In other words the meaning of 7 in the

fraction 7/8 should be clearly explained to them to an extent that learners can even show

7/8 on the fraction board. Such an understanding can help reduce mistakes. Errors

related to the use of fraction board were first mentioned in interviews, but none of the

teachers who completed the questionnaires talked about it. It seemed that fraction

boards were used less by teachers to teach the fractions.

Teacher Rito also said that learners made confusing computations:

“Some learners just produce confusion in their calculation, for example, 5/9-3/9

=5/9, some learners do fail to subtract or add correctly the numerators, for

instance, 3/4+5/4=7/4”. (Teacher Rito: Interview)

The two errors given above seem to be alike in the sense that they both show incorrect

subtraction or addition of numerators. However, I understood him to mean that the

first answer shows confusion because of the number 5 appearing as if the learner just

copied it without any computation. Probing learners further can help better understand

what the actual error was.

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Teacher Ntiyiso talked about the error of falling into the a trap of finding the product of

the two denominators. She put it in this way:

“Of cause yes, when given an operation with unlike denominators, learners

normally fall into a trap of finding the product of the two denominators, this

does not always work when we have the LCD. For instance, where the

denominators are 2 and 6, learners will write 12 as the common denominator

instead of 6 as an LCD.” (Teacher Ntiyiso: Interview)

I did not regard what she said as an error, because using the LCD or the product of the

two denominators always leads to the same correct answer. The only challenge with the

product method is that learners work with bigger numbers which mainly lead to

mistakes. The response by teacher B suggests that teachers need to have adequate

content knowledge and always be prepared to apply alternative teaching strategies so

that they don’t teach misconceptions.

Teacher Ntiyiso give below is also common to my class. In her words,

“Some go to an extent of adding the numerator and denominators in the following

manner: say we have: 2/4 +3/6

=6/10+9/10

=15/10,

In other words, they add the numerators and denominators for each fraction and

make them the new numerators. They again add the denominators to make them

the same” (Teacher Ntiyiso: Interview)

I think this kind of an error is mostly seen in learners who are struggling to apply the

LCM method. For instance, the answer cited below was supposed to use LCM method:

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2/4+3/6

= 3(2) + 2(3)/12

= 6+6/12

=12/12

=1 (Teacher Ntiyiso: Interview)

Another error from teacher Ntiyiso is as follows.

“When learners apply the grouping or matching method, they normally

don’t multiply but add, for instance: 2/3+3/4

= (2/3+4/4) + (3/4+3/3 )

=6/7 +6/7

=12/7

=1 5/7 “

This type of an error is common to what other teachers have said in the questionnaires.

Learners mainly change the multiplication and subtraction sign to addition. This might

imply that to most learners, addition is much simpler than the other operation signs.

One other example of an error was explained by teacher Ntiyiso,

“Lastly, the use of cancellation to them means to remove when finding the HCF.

For instance, 18/24 when each cancelled the answer is likely to be given as 6 not

3/4”. (Teacher Ntiyiso: Interview)

The above example learners tend to write HCF as an answer which was not called for,

instead of using it to simplify their final answer.

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Possible causes suggested by the two teachers interviewed include:

poor knowledge of multiplication tables by learners

poor knowledge of the meaning of the fraction concept

poor teaching methods

instrumental understanding of fractions

the language medium, English as barrier to learning.

Whilst the teachers perception of causes of error in learning fractions might be

persuasive, I would argue that the most important cause supports Luneta’s (2008)

finding, namely that errors are also legitimate attempts to understand Mathematics,

and thus that teachers need to use them as a source of learning. Riccomini (2005)

supports this view, arguing that error analysis, means that teachers need to use

students’ mathematical errors to improve instruction and correct misconceptions.

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CHAPTER 5

Conclusion and Recommendations

The purpose of this study was to explore errors that Grade 5 learners at Dyondzo

Primary School in Limpopo Province show when learning how to add and subtract

common fractions. Two research questions are asked,

1. What are the common errors and misconceptions that learners in Grade 5 display

when learning addition and subtraction of common fractions?

2. What are the causes of errors and misconceptions?

Data comprised written classwork, written homework, and a test, learners thereafter

being probed for causes of error. And, teachers completed a questionnaire, and were

interviewed to confirm errors and find causes.

Conclusions

In responding to the first research question, the study found that 76% of learners

encounter difficulties when adding or subtracting two common fractions with different

denominators, and 84% of learners displayed errors when requested to add three

common fractions with different denominators.

Generally speaking, 4 types of error were found: conceptual, procedural, careless and

applicational.

More specifically, errors included:

adding or subtracting the denominators together, in case they were unable

to find the LCM;

incorrect use of basic operation signs, and changing the given operation

sign without following a rule of operation;

matching both denominators even if the LCM is one of the denominators,

which make the operation too long and complicated;

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failure to reach simplified answers, which mainly involve the use of HCF;

and,

failure to get the LCD when adding 3 common fractions.

In response to the second research question, possible causes include:

poor knowledge of multiplication tables by learners; that is, learners lack

basic Mathematics skills to be able to add/subtract common fractions;

poor knowledge of the meaning of the fraction concept;

poor instructional approaches used to teach the concept of fractions leading

to, teachers unintentionally teaching misconceptions, and to learners gaining

an instrumental rather than relational understanding of this concept; and,

the language of instruction as well as the presentation of the content books

and workbooks was found to be a barrier to teaching fractions and hence

contributed to learners’ mistakes as they, failed to read and interpret the

question on their own.

Recommendations

Taking into consideration the findings, several recommendations need to be considered

to reduce misconceptions and the errors associated with them. These include:

learners being taught the concept of fraction from simple to complex operations.

That is, teaching should start with simple geometric shapes to explain the

concept, and to do simple addition and subtraction of common fractions with the

same denominators; and move to common fractions with different denominators.

learners being taught thoroughly basic mathematical operations, so they can

add/subtract correctly. This should include them being taught: mathematical

tables, multiples, factors, the meanings of the four basic operations, brackets, and

related mathematical operations and,

teachers need to have the relevant content knowledge and relevant pedagogical

expertise for teaching content to learners, as it plays a vital role in the learning

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process. The skill to impart knowledge can help eliminate learners’ errors in this

regard.

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APPENDICES

APPENDIX A: INSTRUMENTS

Teacher Questionnaire

1. What are the different ways in which learners explain the concept of fraction?

………………………………………………………………………………………………

………………………………………………………………………………………………

………………………………………………………………………………………………

………………………………………………………………………………………………

………………………………………………………………………………………………

………………………………………………………………………………………………

2. Why do we teach fractions?

………………………………………………………………………………………………

………………………………………………………………………………………………

………………………………………………………………………………………………

………………………………………………………………………………………………

………………………………………………………………………………………………

………………………………………………………………………………………………

………………………………………………………………………………………………

…………………………………………………………

3. How best can we teach the addition and subtraction of common fractions?

………………………………………………………………………………………………

………………………………………………………………………………………………

………………………………………………………………………………………………

………………………………………………………………………………………………

………………………………………………………………………………………………

………………………………………………………………………………………………

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………………………………………………………………………………………………

………………………………………………………………………………………………

………………………………………………………………………………………………

………………………………………………………………………………………………

………………………………………………………………………………………………

……………………

4. Have you observed the common errors and misconceptions that your learners

have developed when dealing with the addition and subtraction of common

fractions? Please give some examples.

………………………………………………………………………………………………

………………………………………………………………………………………………

………………………………………………………………………………………………

………………………………………………………………………………………………

………………………………………………………………………………………………

………………………………………………………………………………………………

…………

5. What do you think are the possible causes of these errors and misconceptions,

and how can Mathematics educators remedy them?

………………………………………………………………………………………………………

………………………………………………………………………………………………………

………………………………………………………………………………………………………

………………………………………………………………………………………………………

………………………………………………………………………………………………………

………………………………………………………………………………………………………

………………………………………………………………………………………………………

………………………………………………………………………………………………………

………………………………………………................................................................

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Learners Classwork Task

NAME OF LEARNER:…………………………………………..

DATE :…………………………………………………………...

GRADE :…………………………………………………………….

SCHOOL:…………………………………………………………….

TASK 1: CLASSWORK

Skill: Addition and subtraction of fractions with the same denominators

A. Draw your own diagrams and use them to find the answers to the

following sums:

1. 3/6 +2/6 =

2. 7/8 +2/8 =

B. Write down a rule for adding fractions with the same denominators

………………………………………………………………………………………

………………………………………………………………………………………

………………………………………………………………………………………

………

C. Use diagrams to show the subtractions below and write the answer as

fraction:

1. ¾ - ¼ =

2. 5/7-4/7 =

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D. Write down a rule you could use for subtracting fractions.

………………………………………………………………………………………

……………………………………………

E. Use the arithmetic method to work out the following:

1. 3/5 +2/5 =

2. 13/9-4/9 =

Learners Homework Task

NAME OF LEARNER:……………………………………………

DATE :…………………………………………………………...

GRADE :…………………………………………………………….

SCHOOL:…………………………………………………………….

TASK 2: HOMEWORK

Skill: Addition and subtraction of fractions with different denominators

A. Calculate to the simplest form using any arithmetic method.

1. 2/3 + 1/6 =

2. 4/5 + 2/3=

3. 3/4 - ½ =

4.11/12-1/3=

5.3/4+7/10+1/5=

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B. Write a rule you could use on the addition of fractions with different

denominators.

………………………………………………………………………………………………

………………………………………………………………………………………………

………………………………………………………………………………………………

………………………………………………………………………………………………

………………………………………………………………………………………………

Write down a rule you could use for subtracting unlike fractions.

………………………………………………………………………………………………

………………………………………………………………………………………………

………………………………………………………………………………………………

………………………………………………………………………………………………

………………………………………………………………………………………………

Learner Test

NAME OF LEARNER:………………………………………….

DATE :………………………………………………………….....

GRADE :…………………………………………………………….

SCHOOL:…………………………………………………………….

TASK 3:

A. Use either the matching method or the Lowest Common Multiple method to find the

answers to these questions:

1. 11/8-3/8=

2. 4/10+2/10=

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3. 5/2-3/7=

4. 2/3+1/6+4/12=

B. Write down a rule you could use to add or subtract fractions with the same/common

denominator.

………………………………………………………………………………………………………

………………………………………………………………………………………………………

………………………………………………………………………………………………………

…………

C. Write down a rule you could use to add or subtract fractions with different

denominators.

………………………………………………………………………………………………………

…………………………………………………………What is a fraction?

………………………………………………………………………………………………………

………………………………………………………………………………………………………

………………………………………………………………………………………………………

…………..

Interview Questions for Teachers

1. Do you think Primary School children encounter some problems when

adding or subtracting common fractions?

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2. Have you observed some common mistakes or errors, that Grade 5 learners

display when working with the addition and subtraction of common

fractions? Would you mind to give some examples?

3. What do you think are possible causes of these errors?

4. In your opinion, what do you think are the possible ways to help eliminate

these errors?

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APPENDIX B: LEARNER AND TEACHER RESPONSES

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APPENDIX C: TEACHER INTERVIEW RESPONSES

Teacher Rito

1. Do you think Primary School children encounter some problems when

adding or subtracting common fractions?

“Yes...yes...yes, primary school children do have some serious problems when it

comes to fractions”.

2. Have you observed some common mistakes or errors, which Grade 5

learners display when working with the addition and subtraction of

common fractions? Would you mind to give me some examples?

“Learners show a lot of mistakes, for instance when they are adding or subtracting

fractions using the fraction board and are requested to complete answers like:7/8-3/8=----

.Their answers will mainly be 4/0.To them 7/8 does not mean 1/8 x7 strips, that is why they

got a denominator of zero which undefined in mathematics. For 6/10+2/10 =8/20 which is

the same mistake.

Some learners just produce confusion in their calculation, for example, 5/9-3/9 =5/9, some

learners do fail to subtract or add correctly the numerators, for instance, 3/4+5/4=7/4

Other mistakes include the finding of LCM in cases the denominators are different ,

learners mainly add or subtract the denominators depending on the given operation sign.

My learners are good at using fractions circles to add or subtract fractions, and when

they have common denominators.”

3. What do you think are possible causes of these errors?

“There are many causes, to give a few, poor knowledge of multiplication tables by

learners, poor knowledge of the meaning of the concept fraction, here I mean the

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foundation skills related to it, and poor teaching methods all contribute high failure of

learners in fractional activities”.

4. In your opinion, what do you think are the possible ways to help eliminate

these errors?

“Training learners to learn multiplication tables not buy rote but with understanding of

what is actually involved in multiplication operations.

Teachers need to improve their teaching strategies, by attending workshops, furthering

their studies, working as a team, inviting curriculum advisors for assistance.

The use learner-centred approach always work best, errors can clearly be seen.

Teachers to know the content very well, if possible they become specialist in mathematics.

Teachers must always conduct fruitful corrections, ensure that no stone is left unturned.

Prepare their lessons thoroughly.

Always try to teach Mathematics using concrete things”.

Teacher Ntiyiso

1. Do you think Primary School children encounter some problems when

adding or subtracting common fractions?

“Yes, they do experience some challenges. They perform poorly in this section”.

2. Have you observed some common mistakes or errors, that Grade 5 learners

display when working with the addition and subtraction of common

fractions? Would you mind to give me some examples?

“.Of cause yes, when given an operation with unlike denominators, learners normally fall

into a trap of finding the product of the two denominators, this does not always work

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when we have the LCD .For instance, where the denominators are 2 and 6,learners will

write 12 as the common denominator instead of 6 as an LCD.

In addition, instead of finding the multiples of a given denominators, learners just add or

subtract.

Some go to an extent of adding the numerator and denominators in the following

manner: say we have 2/4 +3/6 =6/10+9/10=15/10, in other words they add the numerators and

denominators for each fraction and make them the new numerators. They again add the

denominators to make them the same.

When learners apply the grouping or matching method, they normally don’t multiply

but add, for instance:2/3+3/4

= (2/3+4/4) + (3/4+3/3 )

=6/7 +6/7

=12/7

=1 5/7

Lastly , the use of cancellation to them means to remove when finding the HCF. For

instance, 18/24 when each cancelled the answer is likely to be given as 6 not 3/4”.

3. What do you think are possible causes of these errors?

“Learners don’t know their Multiplication tables very well. They also forget the procedure

on how to calculate fractions. Learners also have the issue of language problem, they can’t

read and understand instruction in English on their own”.

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4. In your opinion, what do you think are the possible ways to help eliminate

these errors?

“They also need to taught very well the simplification of fractions. Teachers must do a lot of

revisions. Teachers must start from the simple to complex when teaching fractions, and re

teach multiplication tables. Encourage the use of any language during the learning process

so that they can freely voice out their opinions”.