analysing pupils’ errors in operations of integers...
TRANSCRIPT
ANALYSING PUPILS’ ERRORS IN OPERATIONS OF
INTEGERS AMONG FORM 1 PUPILS
BY
ZULMARYAN BINTI EMBONG
A dissertation submitted in fulfilment of the requirement for
the degree of Doctor of Philosophy in Education
Kulliyyah of Education
International Islamic University Malaysia
MARCH 2020
ii
ABSTRACT
Previous studies have shown that pupils have difficulties and errors in dealing with
many areas of mathematics including in various topics of the number system such as
integers. Pupils’ difficulty with the concept of integers causes them to struggle in
solving mathematical problems, especially those involving the four basic operations.
This study aims to diagnose pupils’ errors in the operations of integers, subsequent to
validating the Errors Identification Integers Test (EIIT) which can identify the types of
errors that pupils possess in dealing with the operations of integers. The EIIT which
consists of multiple-choice questions involving different combinations of positive and
negative numbers was adapted to suit the Malaysian context. The population of this
study is all Form One pupils from selected public schools in Peninsular Malaysia. A
total of eight schools were involved in the data collection as samples. Cluster sampling
was employed in order to ensure that the selected schools represented the population.
The Rasch Model was used to improve and validate the instrument used in this study.
In addition, teachers’ and pupils’ interviews were conducted to find and confirm the
errors of the operations of integers. Then, a teaching intervention that was designed to
remedy and improve pupils’ understanding of the operations of integers was
implemented. Sixty pupils who were involved at this stage of the quasi-experimental
study were selected using purposive sampling. The pre and post tests were given to the
pupils to determine the effectiveness of the algebraic tiles method in improving their
performance in the operations of integers. Carelessness, rule mix-up, inability to
assimilate concepts and surface understanding/poor knowledge were identified as the
types of errors in this study. Meanwhile, this study found that parenthesis
misapprehension, poor mathematical language, calculator hooking, superficial
understanding and external limitation were the possible causes that led to the errors.
The results also showed that the intervention using algebraic tiles as a strategy for
teaching operations of integers was successful. The ANCOVA used to calculate the
difference between the post-test as compared to the pre-test further returned a
statistically significant result.
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خلاصة البحث ABSTRACT IN ARABIC
أوضحت الدراسات السابقة أن الطلبة يواجهون صعوبات و مفاهيم خاطئة في التعامل مع الصحيحة. هذه الصعوبات عدة مجالات من الرياضيات بما فيها أنظمة الأرقام مثل الأعداد
مليات الأساسية تؤدي إلى صراع في حل المشكلات الرياضية، وخاصة تلك التي تحتوي على العبعد الأربع. وهذه الدراسة تهدف إلى تشخيص أخطاء الطلبة في عمليات الأعداد الصحيحة
فاهيم الخاطئة التحقق من ثبات اختبار الأعداد الأربعة الصحيحة والتي يمكنها تحديد أنواع المسئلة الخيارات التي يمارسها الطلبة في التعامل مع عمليات الأعداد الصحيحة، ويشمل الاختبار أ الماليزي. وتكون المتعددة التي تتضمن تركيب من الأعداد الإيجابية والسلبية ليتناسب مع السياق
جزيرة ماليزيا. وتمثلت مجتمع الدراسة من طلبة الصف الأول في مدارس حكومية مختارة في شبهن أن المدارس المختارة العينة في ثمان مدارس لجمع البيانات. وتم أخذ عينات عنقودية للتأكد م
داة الدراسة. إضافة تمثل مجتمع الدراسة. وتماستخدام نموذج راش للتحسين و التحقق من ثبات أة لعمليات الأعداد إلى ذلك، تمت مقابلات مع مدرسين وطلبة لإيجاد المفاهيم الخاطئ
الطلبة الصحيحة. بعد ذلك، تم تنفيذ مداخلة للمدرسين صممت خصيصا لعلاج وتحسين فهم العينة لعمليات الأعداد الصحيحة. وتم اختيار ستين طالبا لهذه الدراسة الشبه تجريبية عن طريق
طريقة الأشكال القصدية. كما تم إجراء اختبارات قبلية وبعدية للطلبة من أجل تحديد فعالية ة المعرفة، وعدم الجبرية في تحسين أدائهم في عمليات الأعداد الصحيحة. وتم تحديد اللامبالاة وقل هذه الدراسة. قدرة استيعاب الأفكار والمفاهيم السطحية ضمن أنواع مفاهيم الطلاب الخاطئة فييات الضعيفة، في الوقت نفسه، وجدت هذه الدراسة أن سوء فهم الأقواس، ولغة الرياض
هيم خاطئة. وتثبيتات الآلة الحاسبة، والمفاهيم السطحية والحدود الخارجية كلها تؤدي إلى مفاكما أظهرت النتائج أن إدخال الأشكال الجبرية كخطة لتدريس عمليات الأعداد الصحيحة كانت ناجحة. كما أعطى تحليل التغاير المستعمل في حساب الفرق بين الاختبار البعدي
. والاختبار القبلي نتائج دالة إحصائيا
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APPROVAL PAGE
The dissertation of Zulmaryan binti Embong has been approved by the following:
_____________________________
Madihah Khalid
Supervisor
_____________________________
Joharry Othman
Co-Supervisor
_____________________________
Nik Suryani binti Nik Abd Rahman
Internal Examiner
_____________________________
Zaleha binti Ismail
External Examiner
_____________________________
Sharifa Norul Akmar binti Syed Zamri
External Examiner
_____________________________
Saim Kayadibi
Chairman
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DECLARATION
I hereby declare that this dissertation is the result of my own investigations, except
where otherwise stated. I also declare that it has not been previously or concurrently
submitted as a whole for any other degrees at IIUM or other institutions.
Zulmaryan binti Embong
Signature ........................................................... Date .........................................
vi
COPYRIGHT
INTERNATIONAL ISLAMIC UNIVERSITY MALAYSIA
DECLARATION OF COPYRIGHT AND AFFIRMATION OF
FAIR USE OF UNPUBLISHED RESEARCH
ANALYSING PUPILS’ ERRORS IN OPERATIONS OF
INTEGERS AMONG FORM 1 PUPILS
I declare that the copyright holders of this dissertation are jointly owned by the
student and IIUM.
Copyright © 2020 Zulmaryan binti Embong and International Islamic University Malaysia. All rights
reserved.
No part of this unpublished research may be reproduced, stored in a retrieval system,
or transmitted, in any form or by any means, electronic, mechanical, photocopying,
recording or otherwise without prior written permission of the copyright holder except
as provided below
1. Any material contained in or derived from this unpublished research may
only be used by others in their writing with due acknowledgement.
2. IIUM or its library will have the right to make and transmit copies (print
or electronic) for institutional and academic purposes.
3. The IIUM library will have the right to make, store in a retrieved system
and supply copies of this unpublished research if requested by other
universities and research libraries.
By signing this form, I acknowledged that I have read and understand the IIUM
Intellectual Property Right and Commercialization policy.
Affirmed by Zulmaryan binti Embong
……..…………………….. ………………………..
Signature Date
DEDICATION
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This dissertation is dedicated to my precious family:
My beloved Ayah and Ibu
Siblings
Nieces and Nephews
&
My friends;
Embong bin Salleh, Kasnawati bt Md Yassin, Mohd Hairudin Embong, Farahdilla
Mohd Noor, Hartini Embong, Muhamad Fadhil Long, Zuraidah Embong, Mohd
Ghadafi Embong, Nabila Farahin Osman, Kartina Embong, Mohamad Fahmi Sofi,
Haleesha, Riyadh, Faris, Firash, Iqbal, Erika, Yusuf & Anis
Who gives me strength,
Who provide unswerving supports and doa’
Your heartiness, love and understanding are irreplaceable.
Al-fatihah to my late Father, Embong bin Salleh.
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ACKNOWLEDGEMENTS
In the name of Allah of the Most Gracious, the Most Merciful, May His Peace and
Mercy be on His Holy Prophet Muhammad SAW, the Messenger of Allah. First of all,
I would like to express my humble gratitude and gratefulness to Allah who in His Mercy
has given me good health, patience and commitment to complete my dissertation.
In conducting this study, I would like to express my gratitude and appreciation
to my supervisor, Asst. Prof. Dr. Madihah Khalid for her invaluable guidance, advice
and helpful suggestions throughout the completion of this study.
I am also grateful to Prof. Dr. Rosnani Hashim and Assoc. Prof. Dr. Joharry
Othman, and all committed faculty of Kulliyyah of Education who have in one way or
another contributed to the completion of this dissertation. In addition, my special thanks
are extended to the all principals who involved in this research (SMK Padang Midin,
SMK Seri Berang, SMK Sultanah Asma, SMK Syed Ibrahim, SMK Meru, SMK
Seksyen 7, SMK Indahpura 1, SMK Dato Jaafar and International Islamic School) who
gave me permission to conduct the research in the stated school.
Finally, I would like to express my special appreciation to my beloved mother,
Kasnawati binti Md. Yassin, my late father Embong Salleh, siblings, nieces and
nephews for their love, patience and words of encouragement.
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TABLE OF CONTENTS
Abstract ...................................................................................................................... ii Abstract in Arabic ...................................................................................................... iii
Approval Page ............................................................................................................ iv Declaration ................................................................................................................. v Copyright ................................................................................................................... vi Dedication .................................................................................................................. vi Acknowledgements .................................................................................................... viii
List of Tables ............................................................................................................. xii List of Figures ............................................................................................................ xiv List of Abbreviations ................................................................................................. xv
CHAPTER ONE: INTRODUCTION .................................................................... 1 1.1 Background of the Study ......................................................................... 1
1.1.1 Errors in Mathematics ................................................................... 3
1.1.2 Algebraic Tiles .............................................................................. 5 1.2 Statement of the Problem ........................................................................ 7
1.3 Objectives of the Study ........................................................................... 10 1.4 Research Questions ................................................................................. 10
1.5 Theoretical Framework ........................................................................... 10 1.6 Conceptual Framework ........................................................................... 13
1.7 Significance of the Study ........................................................................ 17 1.8 Delimitation of the Study ........................................................................ 19
1.9 Definition of Terms ................................................................................. 20 1.10 Summary ................................................................................................ 22
CHAPTER TWO: LITERATURE REVIEW ....................................................... 23 2.1 Introduction ............................................................................................. 23
2.2 What are Integers?................................................................................... 23 2.3 Integers in the Malaysian Mathematics Curriculum ............................... 24
2.4 Difficulties in Learning Integers ............................................................. 26
2.5 Errors in the Operation of Integers.......................................................... 27 2.6 Some Common Errors of Integers........................................................... 31 2.7 What Are Errpors? .................................................................................. 32
2.8 Causes of Errors in Integers .................................................................... 33 2.9 Strategies for Teaching Integers.............................................................. 36 2.10 Constructivism in the Practice ............................................................... 36
2.10.1 Constructivism and Piaget ......................................................... 40 2.10.2 Constructivism and Skemp ........................................................ 43
2.10.3 Representation ........................................................................... 46 2.10.4 Modelling in Mathematics ......................................................... 53 2.10.5 Algebraic Tiles ........................................................................... 54
2.10.5.1 How algebraic tiles help in integers ............................ 67 2.11 Summary ................................................................................................ 69
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CHAPTER THREE: RESEARCH METHODOLOGY ...................................... 70 3.1 Introduction ............................................................................................. 70
3.2 Research Design ...................................................................................... 70 3.3 Population and Sample ............................................................................ 74
3.3.1 Population ..................................................................................... 74 3.3.2 The Schools and Classes ............................................................... 74 3.3.3 Sample .......................................................................................... 75
3.4 Instrumentations ...................................................................................... 76 3.4.1 The Instruments ............................................................................ 76 3.4.2 Interview Protocols ....................................................................... 79 3.4.3 Observation Checklist ................................................................... 79
3.5 Ethical Procedure .................................................................................... 80 3.5.1 Lesson Plans for Intervention ....................................................... 81
3.6 Pilot Study ............................................................................................... 82
3.7 Data Collection and Analysis .................................................................. 84 3.8 Summary ................................................................................................. 86
CHAPTER FOUR: DATA ANALYSIS AND RESULTS .................................... 88
4.1 Introduction ............................................................................................. 88 4.2 Research Findings ................................................................................... 88
4.2.1 Part I: Respondents’ Demography ................................................ 90 4.2.2 Part II: Research Question 1 ......................................................... 91
4.2.2.1 Item Polarity .................................................................... 91
4.2.2.2 Item Fit ............................................................................ 93
4.2.2.3 Construct Validity ........................................................... 94 4.2.2.4 Consistency Of Result With Purpose Of Measurement .. 96
4.2.3 Part III: Research Question 2 ........................................................ 98
4.2.4 Part IV: Research Question 3 ....................................................... 116 4.2.4.1 Observation Data ............................................................. 116
4.2.4.2 Teachers’ Interview ......................................................... 119 4.2.5 Part V: Research Question 4 ......................................................... 132 4.2.6 Promotes Problem Solving ........................................................... 140
4.2.6.1 Fun to Use........................................................................ 142 4.2.6.2 Motivating ....................................................................... 144
4.3 Summary ................................................................................................. 145
CHAPTER FIVE: DISCUSSION, CONCLUSION AND RECOMMENDATION ...... 146 5.1 Introduction ............................................................................................. 146 5.2 Discussion ............................................................................................... 146
5.2.1 Research Question 1: Is the EIIT for the Malaysian Form 1
pupils valid? .................................................................................. 146 5.2.2 Research Question 2: What are the types of errors and
misconception in the operation of integers? ................................. 148 5.2.3 Research Question 3: What are the causes of errors in
solving problems in operation of integers? ................................... 150
5.2.4 Research Question 4: Does teaching using algebraic tiles
give any effect on the pupils’ performance in the operation
of integers? .................................................................................... 153 5.3 Limitations of the Study .......................................................................... 154
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5.4 Conclusion .............................................................................................. 155 5.5 Recommendation for Further Research .................................................. 155
REFERENCES ......................................................................................................... 158
APPENDIX A: TEST OF THE FOUR OPERATIONS OF INTEGERS ......... 168 APPENDIX B: INTERVIEW GUIDE FOR TEACHERS ................................. 175
APPENDIX C: INTERVIEW GUIDE FOR PUPILS ......................................... 176 APPENDIX D: LESSON PLANS ......................................................................... 177 APPENDIX E: OBSERVATION CHECKLIST ................................................. 194 APPENDIX F: PERMISSION FROM THE MINISTRY OF
EDUCATION ............................................................................... 195 APPENDIX G: PERMISSION FROM THE STATE EDUCATION
DEPARTMENT ........................................................................... 198
APPENDIX H: RELIABILITY RESULT FOR PILOT STUDY ...................... 206 APPENDIX I: VALIDITY RESULT FOR PILOT STUDY ............................. 207 APPENDIX J: USED OBSERVATION CHECKLIST ..................................... 208 APPENDIX K: CLASSROOM OBSERVATION REPORT ............................. 209
APPENDIX L: TEST ON THE FOUR OPERATIONS OF INTEGERS
(RECOMMENDED VERSION) ................................................ 211
xii
LIST OF TABLES
Table No. Page No.
3.1a Addition of Integers 77
3.1b Subtraction of Integers 78
3.1c Multiplication of Integers 78
3.1d Division of Integers 78
3.1e Problem Solving on Integers (Word Problem) 78
3.2 Data Collection and Analysis 86
4.1 Number of Respondents for Each School 90
4.2 Profile of the Respondents 90
4.3 Profile of Respondents Interviewed 91
4.4 Item Polarity Statistics: Correlation Order (Operations in Integers) 92
4.5 Item Statistics: Misfit Order 94
4.6 Reliability of Item Difficulty Estimates 97
4.7 Reliability of Person Ability Estimates 98
4.8 Percentage of Correct/Wrong Answers 98
4.9 The Mean, Median, Standard Deviation and Interquatile Range of
the Data 99
4.10 The Percentage Pupils Choosing Options 101
4.11 Test Items and Pupils’ Justification for Answer Given 109
4.12 Observation of Themes 117
4.13 Demographic Details of Interviewed Teachers 120
4.14 Themes and Subthemes of the Interview 120
4.15 Profile of Respondents for the Intervention 133
4.16 Mean Score for Control Group 134
4.17 Paired-Sample Test for Control Group 134
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4.18 Mean Scores for Treatment Group 135
4.19 Paired-Sample Test Treatment Group 135
4.20 ANOVA Homogeneity of Variance 137
4.21 Homogeneity of Regression 138
4.22 Levene’s Test of Equality of Error Variances 139
4.23 Tests of Between-Subjects Effects 140
4.24 Algebraic Tiles in Problem Solving 142
5.1 Types of Errors and Errors 148
xiv
LIST OF FIGURES
Figure No. Page No.
1.1 Conceptual Framework of the Research 14
2.1 Number Line of Integers 23
2.2 Visual and Algebraic: A Comparison between a Visual Image and
Algebraic Arguments 42
2.3 Lesh’s Model of Representations 50
2.4 A Set of Algebra Tiles 54
2.5 The Yellow and Red Squares 55
3.1 Methodology of the Study 73
4.1 Writing Map for Operations of Integers 96
4.2 Standardized Residual for Posttest-Score 138
xv
LIST OF ABBREVIATIONS
PISA Programme in International Student Assessment
TIMSS Trends in Mathematics and Science Study
1
CHAPTER ONE
INTRODUCTION
1.1 BACKGROUND OF THE STUDY
Mathematics is known as an abstract subject which constantly develops and changes
from time to time (McEwan, 2000). Despite being one of the most important subjects,
many pupils enter high school with severe gaps in their understanding of basic concepts
and skills in mathematics. These weaknesses make it difficult for them to understand
higher-level mathematics. One of the basic concepts which functions as a precondition
to the higher levels of mathematical concepts and skills involves a specific part of the
number system which is the integer. Integers are positive and negative numbers and the
numbers must not be in the form of fractions or decimals. They can be even or odd. For
instance, -10, 500, and 0 are all integers, while one-half (1
2), 4.3 and pi are not integers.
The important skill required in learning integers is performing basic operations on it,
which involves signs of the numbers and the signs of the required operation.
Basic operations of integers seem simple, yet, according to Alsina and Nelson
(2006), pupils tend to get confused and they struggle when asked to solve simple
mathematical problems. It is difficult for the pupils because they have been taught to
follow rules and procedures in a very abstract manner without going through models
for better conceptual understanding. Hence, it is desirable that the pupils should grasp
the fundamentals of mathematics so that they are able to learn the advanced
mathematical processes easily. Furthermore, having good mathematical skills will
ultimately save the pupils’ time during examination and reduce the need for tutoring or
remediation. Moreover, since each process builds upon prior knowledge and successful
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application of these skills, it is extremely important that the fundamentals are solid for
every school pupil.
Another important element in building a strong fundamental in mathematics is
the teaching methods used in the classroom. Since every public school in Malaysia is
using the same syllabus, the only difference among them is the teachers’ methods of
teaching. Each teacher has his/her own ways of teaching in order to encourage pupils’
learning and their participation in acquiring knowledge. Teachers play a vital role in
ensuring that pupils understand the mathematical concepts systematically and
comprehensively. Teachers are strongly encouraged to be flexible and creative
throughout the teaching and learning process to make teaching mathematics effective.
Besides that, teachers should know the nature of pupils’ learning styles, strengths and
weaknesses so that an effective teaching and learning environment can be designed.
Recognising pupils’ errors in solving mathematical problems, for instance, will assist
teachers in improving pupils’ achievement in mathematics.
This study aims to identify pupils’ errors in the operations of integers, following
the development and validation of an instrument which is used to identify the types of
errors in integers. Another part of the study uses the quasi-experimental design to
examine a teaching method which is believed to potentially help pupils in understanding
the concepts of integers. As part of the study, the pupils’ progress is also monitored and
their particular weaknesses identified and targeted. An instructional intervention using
algebraic tiles is applied to explain for the errors in solving problems involving the
operations of integers so that this issue can be minimised, if not eliminated.
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1.1.1 Errors in Mathematics
According to Drews, Dudgeon, Lawton and Surtees (2014), errors can be divided into
three categories: careless errors, computational errors, and conceptual errors.
Meanwhile, Graeber and Johnson (1991) believe that the characteristics of errors are:
(i) self-evident, in which the person does not feel the need to prove them; (ii) coercive,
in which the person is compelled to use them in an initial response; and (iii) widespread,
in which it happens among both naive learners and more academically-able pupils.
From the definition of errors, it shows that misconceptions are conceptual errors.
Errors are wrong answers due to poor planning. A planning must be systematic
so that pupils are able to apply the right ideas in certain situations. According to
Roselizawati and Masitah (2014) and Radatz (1980), errors are the symptoms of the
fundamental conceptual structures that become the cause of errors. The underlying
beliefs and principles in the cognitive structure that become the cause of systematic
conceptual errors are known as misconceptions. Therefore, when teachers explain about
pupils’ errors, they have to look at the current pupils’ schema and how they interact
with each other with instructions and also experience.
Making errors in mathematics is one of the significant learning barriers among
pupils. It is however also one of the best ways to learn, in essence by making mistakes.
It leads to a deepening of the pupils’ knowledge and a challenge to the pupils’ thinking.
However, errors must be dealt in decent ways. Most pupils’ errors are not of an
accidental character, but are attributable to individual problem-solving strategies and
rules from previous experience in the mathematics classroom that are incompatible with
the teachers’ instructions or techniques, or pupils’ observed patterns and inferences
during instruction. Receiving incorrect or false information may lead to pupils’ failure
to relate to their existing knowledge hence ruining their schemas due to the
4
misapprehension. There is therefore, a risk of pupils learning the inaccurate procedure
if they link patterns with the wrong understanding.
In an attempt to understand new information, according to Ashlock (2002), rules
are overgeneralised and overspecialised, thus leading to fallacies and erroneous
procedures. In case teachers do not step in or pedagogical solutions are not discovered,
some of the errors will last for long periods of time. Moreover, some pupils feel that
there is no need for them to learn from the mistakes that they have made. This is one of
those beliefs that deter them from realising their mistakes and learning from them. A
staunchly fossilised idea that there is no relationship between correct and incorrect
methods of solving mathematical problems leads them to the start of a question
disregarding the errors they made in its solution. Besides that, pupils are of the notion
that mathematics includes rules and procedures which are discrete. Consequently,
believing firmly in such ideas leads them to consider mathematics to be a ridiculous
subject.
In addition, pupil errors are unique and reflect their understanding of a concept,
problem or procedure. Analysing pupil errors may reveal the erroneous problem-
solving process and thus, provide information on the understanding of and the attitudes
towards mathematical problems. Upon analysing diagnostic on performance tests in
solving text problems, erroneous patterns demonstrated by pupils which are due to other
difficulties such as inadequate understanding of texts or incorrect number manipulation
can be determined. Pupil errors are usually persistent unless the teacher intervenes
pedagogically. By examining each of their written work diagnostically, teachers would
be able to look for patterns and hence find possible causes for errors. Subsequently,
teachers will develop strategies which can be used to encourage pupils to reflect on their
understanding. According to Skemp (1976), concepts and schemata are stable once they
5
are formed and are held to be resistant to change. Thus, good examples of concepts are
required in order for proper concepts to be established. However, pupils are not always
successful in acquiring or developing correct conceptual structures which result in
errors. Errors must not be seen as obstacles or ‘dead ends,’ but must be regarded as an
opportunity to reflect and learn. Teachers should recognise these errors and then
prescribe them in an appropriate instructional strategy to be more diagnostically-
oriented in order to avoid any subsequent major conceptual problems. The diagnosis
itself should be continuous throughout the instruction.
1.1.2 Algebraic Tiles
Having identified pupils’ errors, the question then becomes how to deal with them. In
this study, the use of algebraic tiles is incorporated in the teaching method as an added
representation to make teaching and learning more meaningful. According to Cakir
(2008), Longfield (2009) and Savion (2009), adopting a student-centred pedagogy is
the best way to address errors. In contrast to traditional, teacher-centred methods which
position the teacher at the literal and figurative centre of the room, student-centred
methods aim to place pupils at the centre of their learning process and to empower them
as agents of their own learning. Using a range of problem-solving activities is a good
place to start since teachers can use some shorter activities and some extended activities
depending on pupils’ necessities. Open-ended tasks are easy to implement because they
provide all pupils the opportunity to achieve success, together with critical thinking and
creativity. This is the essence of this research.
Pupils’ understanding can be enhanced using multiple representations. The
algebraic tiles mentioned above are a representation of the concrete form that is used to
enhance pupils’ understanding. Dealing with multiple representations and their
6
connections plays a key role for learners to build up conceptual knowledge in the
mathematics classroom. According to Duval (2006) and Goldin and Shteingold (2001),
representations play a special role in mathematics. As mathematical concepts can only
be accessed through representations, they are crucial for the construction processes of
the learners’ conceptual understanding. Multiple representations are ways to symbolise,
describe and refer to the same mathematical object. They are used to understand,
develop, and communicate different mathematical features of the same object or
operation, as well as connections between different properties. Multiple representations
include graphs and diagrams, tables and grids, formulas, symbols, words, gestures,
software codes, videos, concrete models, physical and virtual manipulatives, pictures,
and sounds. Therefore, representations are the thinking tools for doing mathematics.
What these methods have in common is that, in placing pupils at the centre of the
learning process, they engage them in an authentic process of discovery. It shows that
when pupils are presented with compelling and authentic learning problems, they
become more motivated and engaged. Activity-based methods also heighten the
likelihood that pupils will challenge each other, or their own errors, which is thought to
have a more transformative effect compared to having one’s ideas challenged by the
teacher (Goldsmith, 2006). The prominent representation that will be the focus of this
study is the “algebraic tiles” which is considered as concrete. Similar to other
representations such as verbal, real world and pictorial, the algebraic tiles is assumed to
assist pupils’ comprehension of the symbolic stage which will at the same time enhance
their conceptual understanding.
7
1.2 STATEMENT OF THE PROBLEM
Malaysian pupils’ performance in the “Trends in Mathematics and Science Study”
(TIMSS) and “Programme in International Student Assessment” (PISA) has resulted in
a great worry that it would undermine the nation’s aspiration for the Vision 2020
(Ideasorgmy, 2014). Much has been talked and reported on Malaysian pupils’
achievements in these two international tests and the major concern pertains to the
teaching and learning of mathematics in our school system. Essentially, many
Malaysian pupils seem to depend on rote memorisation in learning mathematics.
Teachers seem to teach pupils using rules and procedures in order to get the correct
answers and, hence, neglect their conceptual understanding (Lim, 2011). Lim (2011)
expresses concern that many teachers teach pupils for the sake of passing examinations
instead of emphasising on the understanding of concepts. She believes that this situation
occurs due to the challenging nature of teaching conceptual understanding which
requires extensive preparations and good content knowledge from the teachers. Lim
also views that the teaching of mathematics in many schools in Malaysia can still be
characterised as teacher-centred.
On the other hand, the Ministry of Education recommends a focus on five
elements in the teaching and learning of mathematics which include problem solving,
communication, reasoning, mathematical connections, and application of technology
(MOE, 2003). However, in the case of operations of integers, teachers prefer to provide
the pupils with rules to be memorised. This is followed with drilling them with enough
practice to make them stick to the rules. This practice may lead to a poor understanding
and misapplication of the rules since the chances for pupils to get confused are high
with so many rules to remember. For example, those who answer 6 + (–2) = –8 may
argue that 2 added to 6 is 8, yet there is a minus sign which makes the answer a negative.
8
The fact that the rules are only applied to the multiplication of a positive and a negative
integer and not for the addition of integers is lost without proper understanding.
However, it is important to explore the possible reasons as to why pupils answer
the question in such a way. It is more interesting to explore the errors from a certain
pattern that can explain the pupils’ thinking or their conceptual understanding as in this
case. In many situations, pupils tend to use their previous knowledge and to apply
strategies they have used to whole numbers in addition and subtraction when dealing
with integers. This makes the teachers’ approaches in teaching integers an important
investigation to understand how teachers think when teaching this subject and to
determine their level of knowledge on this topic. By conducting such investigation, a
proper solution could be identified on how to overcome problems with regards to pupils’
errors of integers.
This study is also a part of a diagnostic exercise to identify gaps in teachers’
content knowledge and pedagogical skills as promoted in the Malaysian Education
Blueprint (2012). Teachers are expected to understand their pupils’ thinking processes
and should be able to correct them at the earlier stage so that the problems shall not
persist as they grow up into adults. Sadler (2012) makes evident that a significant
proportion (38%) of adult pupils in between 18 to 25 years of age provide wrong
answers to routine problems on the operations of integer due to many different reasons
which can be resolved if certain measures are taken to improve the situation earlier.
In addition, teachers’ lack of instruments can be used to diagnose the types of
errors that pupils do in solving problems involving operations of integers. Some studies
related to this topic seem to rely on self-constructed instruments that have yet to be
verified or validated (Egadowatte, 2011; Sadler, 2012; Schindler & Hubmann, 2013;
Rubin, Marcelino, Mortel & Lapinid, 2014). Therefore, this research produces a
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validated Errors Identification Integer Test (EIIT) instrument that can be used to
identify pupils’ errors in solving problems involving addition, subtraction,
multiplication and division of integers, together with a full guideline or manual on how
to use the instrument. The instrument was developed based on the existing literature
(see Bny Rosmah, 2006) and also this specific research. Teachers may use the
instrument and the suggestions on how to teach and counter pupils’ errors by
emphasising on their conceptual understanding.
Another phase of this research involves the instructional intervention. A quasi-
experimental method was employed to test the method that would develop ways to help
pupils’ improvement in the operations of integers. This research aims at a particular
weakness which resulted from the findings using EIIT and interviews. The intervention
took a maximum of three weeks of normal teaching and learning process. In addition,
the intervention was conducted in a formal and specific class. This should help the
teachers involved to monitor their pupils’ progress along with the intervention process.
Besides that, the intervention for this research employed one of the strategies in the
teaching and learning of integers, namely the algebraic tiles.
Briefly, this study aims to address the above-mentioned problems by focusing
on the operations of integers. In achieving this aim, the types of errors that pupils
perform can be identified, the causes of errors examined, and a teaching model is
proposed. Additionally, the instructional intervention process assists the teachers in
identifying the paramount method in teaching the operations of integers. The instrument
was also validated to ensure that the pupils and teachers would be able to distinguish
the errors in solving the mathematical problems.