learners’ errors and misconceptions associated with …
TRANSCRIPT
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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
19
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
20
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.
21
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.
22
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%)
23
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.
24
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.
25
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)
26
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
27
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:
28
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.
29
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:
30
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
31
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
32
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
33
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
34
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%)
35
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
36
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)
37
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
38
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
39
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)
40
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
41
(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
42
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.
43
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.
44
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
45
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:
46
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.
47
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:
48
“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.
49
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:
50
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.
51
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.
52
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;
53
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
54
process. The skill to impart knowledge can help eliminate learners’ errors in this
regard.
55
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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?
………………………………………………………………………………………………
………………………………………………………………………………………………
………………………………………………………………………………………………
………………………………………………………………………………………………
………………………………………………………………………………………………
………………………………………………………………………………………………
………………………………………………………………………………………………
60
………………………………………………………………………………………………
………………………………………………………………………………………………
………………………………………………………………………………………………
………………………………………………………………………………………………
………………………………………………………………………………………………
……………………
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?
………………………………………………………………………………………………………
………………………………………………………………………………………………………
………………………………………………………………………………………………………
………………………………………………………………………………………………………
………………………………………………………………………………………………………
………………………………………………………………………………………………………
………………………………………………………………………………………………………
………………………………………………………………………………………………………
………………………………………………................................................................
61
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 =
62
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=
63
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=
64
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?
65
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?
66
APPENDIX B: LEARNER AND TEACHER RESPONSES
67
68
69
70
71
72
73
74
75
76
77
78
79
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
80
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
81
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”.
82
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”.