beliefs about classroom assessment: a qualitative study … · beliefs about classroom assessment:...
TRANSCRIPT
BELIEFS ABOUT CLASSROOM ASSESSMENT: A QUALITATIVE
STUDY OF PRE-UNIVERSITY MATHEMATICS TEACHERS
Suzieleez Syrene Abdul Rahim
BScEd(Hons) MEd(Maths Ed)
This thesis is presented for the degree of
Doctor of Philosophy
at
The University of Western Australia
Graduate School of Education
2012
ii
iii
ABSTRACT
Assessment reform movements globally and in neighbouring Asian countries have
brought about a proposed change to the assessment system in Malaysia, from a highly-
centralised examination system to a more holistic system. Within this context, the aim
of the study reported in this thesis was to develop understandings about Malaysian pre-
university mathematics teachers‟ beliefs about classroom assessment. The overarching
aim consisted of three more focused and subsidiary aims; (a) to investigate teachers‟
beliefs about the role and purpose of classroom assessment; (b) to investigate teachers‟
classroom practice with regard to mathematics assessment and the relationship between
their espoused beliefs and what they actually do; and (c) to investigate teachers‟ beliefs
about change to their classroom assessment practices.
To achieve these aims, a qualitative, multiple-case study approach was adopted within
an interpretive framework. Case studies were conducted of five mathematics teachers of
a pre-university programme at a private college in Malaysia. Data collection techniques
comprised in-depth, semi-structured interviews, non-participant observations, document
collection, and reflective journal entries.
The data were coded and analysed in two stages. The first stage involved within-case
analysis for each of the five case studies. The second stage involved a cross-case
analysis which generated key themes for each of the three subsidiary aims. Appropriate
measures such as triangulation of data, thick description, and member checking were
taken to ensure trustworthiness of the data.
The key findings reflected the teachers‟ beliefs that (a) classroom assessment provides
feedback for instructional decisions and understanding students as learners, (b)
iv
classroom assessment assists in the development of the „whole‟ individual, (c) fairness
and authenticity are main issues of concern for alternative assessment and (d) peer
support, professional development and awareness of changes are factors that facilitate
change in teachers‟ assessment practices.
The findings of this study contribute to the knowledge base of mathematics teachers‟
beliefs with regard to assessment. At the same time, the findings can inform pre-service
and in-service teacher training programmes for mathematics teachers in Malaysia so
that they will be consistent with the proposed assessment reform.
v
ACKNOWLEDGEMENTS
First and foremost I would like to thank both my supervisors, Winthrop Professor Grady
Venville and Winthrop Professor Anne Chapman for their patience in guiding me
through this challenging journey. Without their patience, guidance, advice and wisdom I
would not have been able to complete this thesis. Thank you for believing in me. I also
thank them both from the bottom of my heart for being my friend when I faced difficult
situations and challenges during this journey. I have learnt so much from the both of
you and I am so blessed to have such dedicated supervisors. To the both of you, I
dedicate this thesis.
This thesis is also dedicated to my family who have been with me throughout this
journey. This thesis was a family „project‟. I thank my wonderful, supportive parents,
Abdul Rahim Mohd Nor and Haleena Lai with all my heart for instilling in me the love
for knowledge and for always believing in me. To my dear husband, Tajul, thank you
for all the wonderful support and understanding. To both my lovely children, Hazem
and Afiqah, thank you for being so patient with mummy and thank you for all the joy
and laughter. To my sister, Hazeleen, thank you for all your prayers and good wishes.
My sincere thanks to the postgraduate students at the GSE for the wonderful
companionship throughout this journey. Special thanks to my „study buddies‟ Michele
Toner, Elaine Lopes, Jasmine McDonald, Lesley Williams, Mignon Shardlow, Michelle
Striepe, Janty and Sharifah Fatimah Syed Ahmad (Bon), thank you all for sharing and
caring. To Umnea, Carol, Sharon, Thanh, Param, Amy, Zarin, Carmen and Adrian,
thank you for the conversations and friendship. To the GSE staff, thank you for
everything.
vi
DECLARATION
I declare that this thesis is my own work and does not contain work that I have
published, nor work under review for publication. I certify that this dissertation does not
incorporate without acknowledgement any material previously submitted for a degree or
diploma in any university. I also certify that to the best of my knowledge this
dissertation does not contain any material previously written or published by another
person without due reference being made to this fact in the text.
Suzieleez Syrene Abdul Rahim
vii
TABLE OF CONTENTS
ABSTRACT iii
ACKNOWLEDGEMENTS v
DECLARATION vi
TABLE OF CONTENTS vii
LIST OF TABLES xi
LIST OF FIGURES xii
LIST OF APPENDICES xiii
LIST OF ABBREVIATIONS xiv
CHAPTER ONE: INTRODUCTION
Introduction 1
Background of the Researcher 1
Background and Context
Assessment
Global Assessment Reform
Assessment Reform Movement in Asian Countries
The Malaysian Context
3
3
6
9
10
Significance of the Research 13
Overview of the Methodology 14
Overview of the Findings 15
Conclusion 16
CHAPTER TWO: LITERATURE REVIEW
Introduction 17
An Exploration of Assessment in the Literature 18
viii
Definitions of Assessment in the Literature
Definitions of Classroom Assessment in the Literature
Definitions of Alternative Assessment in the Literature
Assessment and Learning
Changing Trends in Assessment
Assessment Reform in Asian Countries
Assessment Reform in Malaysia
Assessment in Mathematics
Mathematics Teachers‟ Assessment Practices
19
23
24
26
28
32
38
40
43
Teachers‟ Beliefs
Definition and Influence of Beliefs in the Literature
Teachers‟ Beliefs and Mathematics
Teachers‟ Beliefs and Mathematics Assessment
46
46
48
54
Conclusion 59
CHAPTER THREE: METHODOLOGY
Introduction
Research Approach
Research Design
Context and Participants of the Research
Data Collection
Data Analysis
Trustworthiness
Ethical Considertaions
Conclusion
61
62
65
70
75
85
89
95
96
ix
CHAPTER FOUR: CASE STUDIES
Introduction
Case Study One: Lulu
Case Study Two: Kate
Case Study Three: Anakin
Case Study Four: Jane
Case Study Five: Wendy
Conclusion
97
100
122
142
158
176
190
CHAPTER FIVE: CROSS-CASE ANALYSIS AND DISCUSSION
Introduction
Purpose and Role of Classroom Assessment
Classroom Assessment Practices
Reasons for Change to Classroom Assessment Practices
Issues and Concerns Regarding Change in Assessment Practices
Suggestions to Facilitate Change in Assessment Practices
Conclusion
191
192
202
208
214
220
223
CHAPTER SIX: CONCLUSION
Introduction
Overview of the Research
Summary of the Key Findings
Contribution to the Literature Review
Implications for Professional Practice
Limitations of the Research
Personal Postscript
224
224
226
236
237
239
240
REFERENCES 242
x
Appendix A
Appendix B
Appendix C
Appendix D
Appendix E
Appendix F
Appendix G
Appendix H
Appendix I
Appendix J
Appendix K
268
269
270
271
272
275
276
277
278
279
280
xi
LIST OF TABLES
Table 3.1 Characteristics of Participants 74
Table 3.2 Summary of Data Collection for Each Participant 67
Table 4.1 A Comparison of the Eight Research Participants 99
and Subset of Five Case Studies Presented in this
Thesis Based on Different Aspects of their Background
Table 5.1 Traditional and Alternative Assessment Techniques 204
Employed by Participants
xii
LIST OF FIGURES
Figure 3.1 The Research Design showing the Data Collection and
Analysis Process 69
Figure 3.2 Mind Map of Categories Generated from Cross-case
Analysis 70
Figure 5.1 Themes for Teachers‟ Beliefs about the Role and Purpose
Of Classroom Assessment 192
Figure 5.2 Categories of Teachers‟ Assessment Practices 203
Figure 5.3 Themes for Teachers‟ Beliefs about Reasons for Changes to
be made to Classroom Assessment Practices 209
Figure 5.4 Themes for Teachers‟ Beliefs about Issues and Concerns
Regarding Change in Classroom Assessment Practices 215
Figure 5.5 Themes for Teachers‟ Beliefs about Suggestions to Facilitate
Change in Assessment Practices 220
Figure 6.1 Overview of the Research Findings 227
xiii
LIST OF APPENDICES
Appendix A Information Sheet (for the institution) 268
Appendix B Consent Form for Research Participant (for the institution) 269
Appendix C Information Sheet (for the participant) 270
Appendix D Consent Form for Research Participant (for the participant) 271
Appendix E Framework of Guiding Interview Questions 272
Appendix F Reflective Account 275
Appendix G Excerpt of Summary of Category for Lulu 276
Appendix H Example of Lulu‟s Mind Map 277
Appendix I Working Summary Sheet of Individual Case study 278
Appendix J Theme Summary Sheet 279
Appendix K Observation Summary Sheet 280
xiv
LIST OF ABBREVIATIONS
MES Malaysian Examinations Syndicate
MLD Mathematics Learning Difficulties
NCTM National Council of Teachers of Mathematics
NRC National Research Council
PISA Programme for International Student Assessment
PMR Penilaian Menengah Rendah
SPM Sijil Pelajaran Malaysia
STPM Sijil Tinggi Pelajaran Malaysia
TIMSS Trends in International Mathematics and Science Study
UPSR Ujian Penilaian Sekolah Rendah
1
CHAPTER ONE
INTRODUCTION
Introduction
The aim of the research reported in this thesis was to develop understandings of
Malaysian pre-university mathematics teachers‟ beliefs about classroom assessment.
This aim consisted of three subsidiary and more focused aims. The first subsidiary aim
was to explore teachers‟ beliefs about the purpose of classroom assessment. The second
subsidiary aim was to investigate teachers‟ classroom practice with regard to
mathematics assessment and the relationship between their espoused beliefs and what
they actually do.The third subsidiary aim was to investigate teachers‟ beliefs about
change to their assessment practices, including their beliefs about reasons for change,
barriers to change and factors that may facilitate change.
Background of the Researcher
As a child, my parents put great emphasis on the importance of gaining a strong
foundation in mathematics. Their belief was that learning mathematics enhanced
problem-solving skills which are essential to everyday life. This early awareness of the
importance of mathematics shaped my love for the subject.
I was very fortunate to have very dedicated mathematics teachers throughout my
schooling years. They showed me the beauty of learning mathematics and that
mathematics was not just a subject about numbers.Their teaching approaches made
learning mathematics enjoyable. The teacher who left the greatest impression on me
was my upper secondary mathematics teacher. Her teaching approach focused mainly
on the process rather than the product. I remember how she made my classmates and I
2
tear out all the answer pages at the back of our mathematics textbook and throw them
into the wastepaper basket. Her reason was simple; she did notwant us to focus on the
answers. She believed that the focus of mathematics learning should be on solving
problems and making decisions about the most suitable method to solve a particular
problem, not just on getting the answer. Her approach helped shape us into more
confident students as far as mathematics was concerned.
At university, the undergraduate course that I took was a combined course between the
Faculty of Science and the Faculty of Education. I spent four years training to be a
teacher, specialising in mathematics. My lectures at the science faculty provided me
with the content-based knowledge. Mathematics was reduced to learning about
mathematical theorems, proofs and laws. I felt that I had stepped into a complex and
unknown zone. My university examinations consisted of questions that required me to
write lines and lines of mathematical statements that did not have much meaning for
me. These written examinations were beginning to „shake‟ my confidence in
mathematics. At that time, I felt that learning mathematics involved a lot of memorising
and that took the joy out of learning mathematics.
My teacher training was overseen by the education faculty. It was during my time at the
education faculty that I was introduced to pedagogy; the „art of teaching‟. I learnt about
the diversity in student-learning, different teaching approaches and various assessment
techniques. It was the lessons in my Mathematics Teaching Methods course during my
third and final year that kept my love for mathematics „alive‟. This course provided me
with the opportunities to learn and explore the world of education that would prepare
me to take on my role as a mathematics teacher for the next 13 years.
3
My career as a teacher saw me mostly dealing with students who were deemed low-
achievers by the system. Throughout my experience of teaching these students, I
realised that,based on the grades that they obtained through written tests, they perceived
themselves as „hopeless‟ students in mathematics. But as a teacher, I saw promising
potential in some of them. That was the start of my interest in mathematics assessment.
In 2006, the Malaysian Examinations Syndicate (MES) announced that the national
educational assessment system would be reviewed. Malaysia was officially taking its
first step of moving away from the pervading examination-oriented culture. When I was
offered a government scholarship to further my studies, I chose classroom assessment to
be the focus of my research, specifically mathematics teachers‟ beliefs about classroom
assessment. I wanted to contribute to the knowledge-base of this reform.
Background and Context
Assessment
Assessment is an important component in the teaching and learning process as it
provides teachers with information that is important for decision-making in the
classroom. Good quality instruction exists when teachers have a sound understanding of
the state and nature of students‟ learning. Stiggins (1994) explains that good classroom
assessment of students‟ achievement generates information which forms the basis of
teachers‟ understanding of students‟ strengths and weaknesses in their learning. In
addition, information that is collected through the use of various classroom assessment
techniques provides a guide for teachers in making educational decisions (Brookhart &
Nitko, 2008; Nitko & Brookhart, 2007). These decisions concern students‟ learning and
development, as well as the suitability and effectiveness of classroom instruction (Linn
& Miller, 2005).
4
Research indicates that about one third to one half of teachers‟ time is spent on
assessment-related activities (Stiggins & Conklin, 1992). Assessment acts as a tool for
teachers to gain insight into the meanings constructed or assigned by students to ideas
or concepts taught in the classroom. Webb (1994) explains that this aspect of
assessment allows the teacher to gauge whether the idea or concept taught was
conveyed successfully to the students. Furthermore, assessment provides feedback to
teachers about the appropriateness of the use and selection of instructional materials as
well as the effectiveness of instructional methods deployed in the classroom (Linn &
Miller, 2005; Nitko & Brookhart, 2007). Hence, assessment informs teachers of the
effectiveness of their own teaching and planning of instructional activities.
Assessment is also important to students. Nitko and Brookhart (2007) state that students
view assessment as a way of informing them of their progress and also to identify the
learning areas that need improvement. Feedback from assessment reinforces successful
student learning (Linn & Miller, 2005). Nitko and Brookhart (2007) further emphasise
that students who receive regular feedback through assessment are better motivated to
learn as they feel more involved in their own learning. The feedback received allows
students to feel that they are being given another chance to do better in their learning.
Therefore, feedback from assessment guides improvement of student learning as well as
contributing to student motivation.
There are various types of assessment, but the main four types are summative,
formative, diagnostic and evaluative assessment (Hackling, 2004; Hornby, 2003; Linn
& Miller, 2005). Hackling (2004) claims that teachers are only directly involved in
formative, diagnostic and summative assessment. Formative assessment is carried out
during the teaching and learning process. The purpose of formative assessment is to
5
provide continuous feedback to both teachers and students of learning difficulties and/or
successes (Linn & Miller, 2005). The continuous nature of feedback during formative
assessment supports on-going improvement of the teaching and learning processes.
The use of formative assessment has helped in raising the level of achievement among
students, especially the low-attainers (Black & Wiliams, 1998). Black and Wiliams
(1998) explain that continuous feedback provides students with specific guidance on
their strengths and weaknesses in their learning. Thus, students are given a chance to
improve their learning to enable them to raise their level of academic achievement.
Diagnostic assessment provides information about students‟ prior knowledge as well as
identifying learning difficulties. Summative assessment is conducted to assess
achievement at the end of instruction, whereas formative assessment is for monitoring
and informing learning progress during instruction. Previously, summative assessment
was the main focus of students‟ assessment, but due to recent education reform
movements, formative and diagnostic assessment have gained more significance
internationally as important aspects of assessment of students‟ learning.
Assessment can also been categorised as formal or informal. Formal assessment is more
structured compared to informal assessment and usually involves a standardised test or
examination. For formal assessment, there is a fixed schedule, specific instructions for
the implementation of the assessment and a fixed amount of time to complete the
assessment (Wragg, 2001). Informal assessment is less structured and carried out during
lessons. Asking the class a question to determine whether the students are able to follow
the lesson is a simple example of informal assessment. This form of assessment
provides immediate feedback to teachers. Wragg (2001) explained that walking around
6
the class and monitoring students while they work on mathematics questions is another
example of informal assessment. This activity helps the teacher to identify students who
are facing difficulties with the lesson but are too shy to put their hand up to ask for
assistance.
There are different techniques of assessment that can be employed in the classroom to
collect information about students‟ achievement (Angelo & Cross, 1993; Nitko &
Brookhart, 2007). Examples of assessment techniques considered to be more traditional
include tests, exams, quizzes and textbook exercises. More contemporary and
innovative assessment techniques include concept maps, group work, portfolios,
journals and presentations. An assessment technique can be used for more than one
purpose, for example, a portfolio can be used for formative and summative purposes
(Bell, 2007).
Global Assessment Reform
In the early 1980s, issues concerning assessment focused on large-scale standardised
tests. These standardised tests were used extensively as the main measure of student
learning and the extent to which educational objectives were being achieved. According
to Stiggins and Conklin (1992), the focus of most research on assessment was on the
development and improvement of standardised tests because standardised test scores
were the accepted indicators of the effectiveness of schools during that period.
Trends in International Mathematics and Science Study (TIMSS) and Programme for
International Student Assessment (PISA) are two international assessment programmes
which allow participating countries to compare students‟ educational achievements
across member countries. TIMSS focuses on the assessment of mathematics and science
7
whereas PISA focuses on reading, mathematics and science. Both TIMSS and PISA
consist of traditional paper-and-pencil tests and questionnaires. The results from TIMSS
and PISA serve as indicators of the effectiveness of schools in comparison with other
participating countries. The assessment policies of these participating countries put
emphasis on standardised testing.
In contrast, standardised tests are now deemed by some commentators as insufficient as
measures of achievement as students are assessed on a limited range of instructional
objectives (Shepard, 2000). This perception of standardised tests is further supported by
Linn and Miller (2005) who state that, globally, the educational assessment scenario is
moving away from an examination and testing culture towards a more flexible
assessment culture as a wider range of assessment techniques is being implemented in
the classroom.
Popham (2008) points out that, traditionally, the main reasons teachers assessed
students were to monitor their progress, to diagnose their strengths and weaknesses, to
assign grades, to rank students and to assess instructional effectiveness. He further
explains that in today‟s educational milieu, students‟ academic performances have
become part of the teacher evaluation process, test results are used by the public as
indicators of educational effectiveness, and the use of a variety of assessment
techniques can improve instructional quality. Traditionally, the main purpose of
assessment was for the teacher to monitor the quality of teaching and learning (Popham,
2008). The purpose of assessment in today‟s educational scenario has expanded, not
only to monitor students‟ performances, but the teachers‟ performances as well. The
public‟s emphasis on test results as indicators of educational effectiveness has also
caused tension between high-stakes public examinations and teachers‟ freedom to use a
8
variety of assessment techniques. Emphasis on tests as the main assessment technique in
the classroom has the effect of compelling teachers to teach to the test through the use
of more teacher-centred pedagogies as well as narrowing the content knowledge taught
(Au, 2007).
The current philosophy and practices of assessment evolved as a result of an
international educational reform movement which began in the late 1980s. This reform
has brought about change to the purpose of assessing student learning as well as the
methods of assessment being used by teachers (Shepard, 2000). The basis of this reform
was the introduction of the constructivist theory of learning which acknowledges
students as active knowledge builders. Von Glasersfeld (1995) states that the two
underlying principles of constructivism are: knowledge is constructed by the individual
through interaction with his or her surroundings; and, reality is subjective as it is
constructed by the individual based on his or her experiences. Constructivists view
students as active learners, not as learners who passively accept knowledge being taught
to them (von Glasersfeld, 1995; Van de Walle, 2007).
During the educational reform, constructivists, like von Glasersfeld (1987), came to
view mathematical knowledge to be less about students‟ ability to recall facts and more
about students‟ ability to generate solutions or new ideas. Mathematics educators are
now more inclined to view mathematics as a subject which involves communication,
problem solving and logical reasoning (Van de Walle, 2007). Mathematics teaching is
moving towards creating a classroom environment that helps students construct their
own knowledge through engagement with realistic mathematical problems (Van de
Walle, 2007).
9
The effects of this reform have caused a shift in thinking about mathematics learning
and teaching, which, in turn, is prompting changes in mathematics assessment. In an
ongoing cyclical process, reform in mathematics assessment will influence the
development of mathematics instruction and curriculum that supports students‟ learning
(Nisbet & Warren, 2000).
Assessment Reform Movement in Asian Countries
The global education reform movement made its initial impact in the Asia region in the
form of national assessment reforms in countries such as Hong Kong and Singapore.
The year 2001 brought about reform in the Hong Kong education system. The focus of
the reform was to concentrate on acquiring generic skills in addition to subject content
knowledge (Chan, Kennedy, Yu &Fok, 2006). This automatically brought about reform
in the national assessment system which now aims to find a balance between summative
and formative assessment.
Singapore embarked on a reform in education around the turn of the 21st Century and
focused on a curriculum that emphasised thinking skills, creativity and communication
skills (Sellan, Chong & Tay, 2006). This new education system in Singapore
encouraged the nation to be actively engaged in lifelong learning. The national
assessment moved in tandem with this reform and focused more on coursework and oral
communication. The underpinning principle of the education reform in Singapore was
to prepare the nation to meet challenges in today‟s globalised world.
With these recent reform movements in neighbouring Asian countries, Malaysia also
decided to revamp its national educational assessment system to meet the demands of an
increasingly globalised world. Hence, preparations are under way to develop a new
10
national assessment system for schools that was initiated in the year 2006. The focus is
to move away from the highly centralised examination system to a system that is a
combination of centralised examination and school-based assessment (Tuah, 2007).
The Malaysian Context
In Malaysia, educational assessment is currently dominated by high stake tests.
Students‟ achievement is measured by standardised tests at the school level and national
level. There are three major national examinations in Malaysia which are taken by all
students in government schools. The results from these three examinations are used for
certification and selection. The Malaysian Examination Syndicate (MES) is the
governing body in charge of the national examinations. The first national examination is
taken at the end of the sixth year of primary school. This examination is known as the
Ujian Penilaian Sekolah Rendah (UPSR) or Primary School Assessment Test and
signals completion of six years of teaching and learning at the primary level.
The remaining two national examinations are taken at the secondary level. The standard
number of years in a Malaysian secondary school is five years, three years of lower
secondary and two years of upper secondary schooling. The second national
examination, called the Penilaian Menengah Rendah (PMR) or the Lower Secondary
Assessment, is taken at the end of the third year of secondary school. The third
examination is known as the Sijil Pelajaran Malaysia (SPM) or Malaysian Education
Certificate. This examination is the most important of the three as the results are used to
determine entry into higher education. The SPM is taken at the end of the second year
of upper secondary school.
11
Upon completion of the SPM, Malaysian students have the option of enrolling in post-
secondary programmes conducted in private or government institutions. An example is
the Sijil Tinggi Persekolahan Malaysia (STPM) or Malaysian Higher Education
Certificate pre-university programme conducted by the Ministry of Education at
selected government secondary schools. The duration of this programme is 18 months
and is known as „Form Six‟ in the secondary schools. The Malaysian Examinations
Council is the governing body in charge of the STPM examination taken at the end of
the programme. The STPM results are used for admission into Malaysian public
universities as well as certain internationally recognised universities. Besides the
STPM, students may apply for admission into local or overseas universities with other
pre-university qualifications such as A-Level, diploma or matriculation results. The A-
Level programme is usually conducted by private colleges whereas diploma and
matriculation courses may be conducted at both private and government institutions.
The general populace of Malaysia associate successful and effective learning in school
with the number of A grades obtained in these national examinations. It has been the
norm in Malaysia to publicly highlight the number of A grades obtained by schools and
students in all the major examinations. Too much focus on the results of standardised
tests has created a classroom environment where teachers teach to the test and students
learn for the test rather than focusing on more general and sustained cognitive
development and knowledge construction.
Recently in Malaysia, there was a call for reform to the national education assessment
system. The people within the education system, such as teachers, principals,
curriculum developers and students, wanted a better balance between assessment of
learning and assessment for learning (Tuah, 2007). Parties such as parents and
12
politicians requested that a more holistic approach to national examination and
assessment conducted by the schools should be taken as part of the monitoring of
students‟ learning and development. The call for reform was an initial step towards
paving the way for a less examination-oriented education system.This turn of events
with regard to assessment is consistent with the Malaysian National Education
Philosophy which emphasises the development of individuals in a holistic and
integrated manner so as to realise their full potential.
In a response to the call for assessment reform, the then Director of the Malaysian
Examinations Syndicate, Datuk Dr Adi Badiozaman Tuah, proposed changes in the
Malaysian assessment system including a balance between summative and formative
assessment, reduction in over reliance on scores obtained through the national
examinations, due recognition of school assessment, assessment of all aspects of the
curriculum and an increase in the use of various methods to gather information about
students‟ learning (Tuah, 2007). Datuk Dr Tuah further expressed the intention of
empowering schools and teachers to conduct quality assessment of and for the learning
needs of the students.
The proposal included five types of assessment that, in combination, would be
introduced as a step towards a more holistic assessment system. These included central
assessment, school assessment, central examinations, psychometric tests and physical
activity assessment (Tuah, 2007). School assessment refers to any form of assessment,
summative or formative in nature, which is developed and implemented by teachers in
schools. Classroom assessment, as a subset of school assessment, refers to any form of
formative assessment designed and implemented in the classroom by the teachers.
13
Feedback from formative classroom assessment should be used to inform the process of
student learning and not to judge their learning (Brookhart & Nitko, 2008).
Significance of the Research
The focus of this research is pre-university mathematics teachers‟ beliefs about
classroom assessment at a college where the institution‟s assessment policy had
undergone reform from traditional paper-pencil techniques to a mixture of traditional
and alternative assessment techniques. Research about the beliefs of the mathematics
teachers at this college provides an understanding of the beliefs of teachers having to
undergo reform to their classroom assessment practices.
As a consequence of the proposed changes in assessment of mathematics in Malaysia,
there was a need for research to be conducted to gain insights into Malaysian
mathematics teachers‟ current beliefs about classroom assessment and how those beliefs
impact on classroom practice. The findings of this study create a knowledge base and
the findings serve as an initial contribution to the assessment reform movement in
Malaysia.
Thompson (1984) explained that if mathematics teachers‟ classroom behaviour is
influenced by their views, beliefs and preferences, then steps need to be taken to
identify and understand these beliefs or views and how they are related to teachers‟
instructional practice. Only then would any attempts to improve the quality of
mathematics instruction be fruitful. Hence, to better inform the mathematics assessment
reform in Malaysia, the research is essential for providing information about
mathematics teachers‟ beliefs of classroom assessment.
14
Due to the relatively small amount of research that has been conducted regarding
mathematics teachers‟ beliefs about assessment (Adams & Hsu, 1998; Delanshere &
Jones, 1999; Nisbet & Warren, 2000), the findings of this research will contribute to the
broader body of knowledge on mathematics assessment. Through an examination of
Malaysian pre-university mathematics teachers‟ beliefs about classroom assessment,
this research contributes to the body of knowledge within Malaysia and also
internationally.
The findings of the research serve as a source of information to the Malaysian
government in their on-going effort to improve pre-service and in-service teacher
training programmes for mathematics teachers so they will be consistent with the
proposed assessment reform in Malaysia.
Overview of the Methodology
The aim of this research was to gain understandings into mathematics teachers‟ beliefs
about classroom assessment. The study consisted of six research questions:
1. What are the teachers‟ beliefs about the purpose of classroom assessment in
the teaching and learning process?
2. What classroom assessment techniques do teachers currently use?
3. How do teachers‟ beliefs influence their selection of classroom assessment
techniques?
4. What do teachers believe are the reasons for change to their classroom
assessment practices?
5. What do teachers believe are barriers to change in their classroom
assessment practices?
15
6. What do teachers believe are factors that may facilitate change in their
classroom assessment practices?
To achieve the aim, a qualitative case study within an interpretive framework was
adopted. The participants of the study were selected through purposive sampling. Eight
pre-university mathematics teachers at one college were initially involved in this study.
The research process was conducted in two parts; the pilot study and the main study.
The main sources of data were interviews, non-participant observations, document
collection and reflective accounts. Two levels of data analysis were conducted; within-
case analysis and cross-case analysis. Five individual case studies were generated and
themes emerged from the cross-case analysis.
Overview of the Findings
The findings of this study provided an insight into the teachers‟ beliefs aboutthe role
and purpose of classroom assessment, their assessment practices and their beliefs about
change to their assessment practices.
Five themes emerged about the teachers‟ beliefs about the role and purpose of
classroom assessment including(a) feedback to inform instructional decisions, (b)
feedback about students as learners, (c) communication of students‟ progress and
achievement, (d) motivating students in their learning, and, (e) preparation platform for
external examinations.
The case study teachers were categorised into three groups with regard to their
classroom assessment practices; (a) the innovative assessor; (b) the „sitting on the fence‟
assessor; and (c) the traditional assessor. The basis of these categories was the extent to
16
which the teachers used traditional and alternative assessment techniques in their
classrooms.
Four themes emerged from the analysis describing the teachers‟ beliefs about reasons
for change. The themes were: (a) „teaching to the test, learning for the test‟; (b)
developing the „whole‟ individual; (c) enjoying the learning process; and, (d)
informative and fair assessment. Issues and concerns that the teachers had about change
that arose from the findings were related to: (a) resistance to change; (b) reluctance to
share; (c) authenticity; (d) students‟ response; and (e) issues with assessment
techniques.
Finally, the teachers provided suggestions to facilitate change to assessment practices.
The suggestions related to: (a) professional development; (b) peer collaboration and
support; (c) implementation; and (d) awareness.
Conclusion
This chapter has provided an overview of the background and context of the research,
the significance of the research, the research methodology employed and the research
findings. Chapter Two discusses the theoretical and empirical literature that underpins
this research. Chapter Three outlines the methodology adopted for data collection and
analysis. Next, Chapter Four presents the findings in the form of individual case studies.
Chapter Five discusses the themes that emerged from the cross-case analysis. Chapter
Six concludes the research with an overview of the study its findings and a discussion of
the implications for professional practice.
17
CHAPTER TWO
LITERATURE REVIEW
Introduction
To reiterate, the central aim of the study reported in this thesis was to gain
understandings of Malaysian pre-university mathematics teachers‟ beliefs about
classroom assessment. Specifically, the research sought to gain insights into the
teachers‟ beliefs about classroom assessment after having undergone changes in their
assessment practices.
This chapter presents the conceptual framework which guided the research. The
conceptual framework was informed by theoretical and empirical literature about
assessment and the perspective that teachers‟ beliefs influence their classroom practice
(Nespor, 1987; Pajares, 1992). The literature review was ongoing throughout all phases
of the research, from the data collection to the writing of this thesis. As the focus of the
research was on assessment and teachers‟ beliefs, this chapter is divided into two main
sections, each analysing the literature on one of these constructs. The first section
examines the various definitions of classroom assessment from the existing literature
followed by a discussion on the relationship between assessment and learning and
changing views of teaching and learning. The discussion then focuses on assessment
reform in the Asia Pacific region, narrowing down to the Malaysian context, and
continues with a consideration of the literature that examines assessment specifically in
mathematics education. This section ends with an overview of research on mathematics
teachers‟ assessment practices. The second section begins with an exploration of the
literature that provides definitions of beliefs and a brief analysis of the literature that
investigated the influence of teachers‟ beliefs on their classroom teaching practices. The
18
discussion continues with an overview of research on teachers‟ beliefs, particularly
within the subject of mathematics. This is followed by an overview of research on
teachers‟ beliefs and mathematics assessment. The chapter then concludes with a
summary of issues that arose from the discussion presented in the chapter.
An Exploration of Assessment in the Literature
Instruction, learning and assessment are closely interrelated and this relationship is
important for the success of education (Biggs, 1999; Broadfoot, 1998; Sebatane, 1998;
Wiliam, 2007). Assessment is crucial to the teaching and learning process because
information about students‟ learning is important to teachers. As elegantly expressed by
Kulm (1994), assessment permits teachers to “find out what we know, do not know, and
might like to know about our students” (p. 11). Most teachers face the challenge of
creating and implementing sound assessment that provides relevant and useful feedback
about students‟ learning (Butler & McMunn, 2006).
Assessment is not a „stand-alone‟ component in education; it has a significant impact on
both curriculum and instruction (National Research Council [NRC], 2001). According
to the National Research Council [NRC] (2001), “what is measured and how
information is used depend to a great extent on the curriculum that is taught and the
instructional methods used” (p. 51). The curriculum outlines the competencies that need
to be taught and learnt, whereas instruction encompasses the teaching approaches and
learning activities to achieve the curricular goals. The Council describes assessment as a
process that evaluates the extent of student achievement of the goals set within the
curriculum. These three components, curriculum, instruction and assessment, need to be
aligned for assessment to support learning (NRC, 2001).
19
Research has revealed that teachers spend a significant amount of their time, about one-
third or one-half, on assessment activities (Crooks, 1988; Stiggins & Conklin, 1992).
Stiggins, Arter, Chappius and Chappius (2006) accentuate the potential benefits of
effective classroom assessment through their claim that it “can motivate the
unmotivated, restore the desire to learn, and encourage students to keep on learning, and
it can actually create, not simply measure, increased achievement” (p. 3). Accordingly,
they make the strong assertion that the ability to use assessment well can be seen as a
characteristic of good teaching. As a consequence, it is critical for teachers to be
assessment literate and to be effective assessors. According to Stiggins (2000), teachers
who are assessment literate understand the importance of selecting appropriate
assessment methods “to gather dependable information about student achievement” (p.
5) and are able to use this information to provide students with learning opportunities
that can raise their achievement. In the current educational milieu, assessment is
expected to play a supportive as well as evaluative role in the teaching and learning
process.
Definitions of Assessment in the Literature
This subsection presents a discussion about the definitions and descriptions of
assessment that exist in the literature. There are various definitions and descriptions of
assessment, most of which are closely associated with the process of decision-making.
Every day teachers are constantly making decisions in their respective classrooms;
decisions such as which group of students needs additional assistance, whether a lesson
plan for a particular class is practical, or whether the teaching strategy employed is
effective. These decisions require teachers to have access to relevant and reliable
information to guide the decision-making process so that good decisions are made.
20
Assessment is seen as a process which involves two main activities; gathering of
relevant information, followed by the interpretation of this information with the
intention of enlightening teachers about the extent to which their students have learnt
something (Harlen, 1994). Information gathered in assessment is a “purposeful
collection of data, both qualitative and quantitative” (Gallagher, 1998, p. 12) to assist
teachers in making decisions about their students. In an elaboration of Gallagher‟s
(1998) definition, Miller, Linn and Gronlund (2009) explained that for educational
assessment, quantitative data are numerical expressions of student achievement,
whereas qualitative data are simply cases of non-numerical descriptions of student
learning. An example of quantitative data is obtaining 35 correct answers of 50
questions in a multiple-choice test. A comment such as “Laura‟s work was neat”
(Miller, Linn & Gronlund, 2009, p. 29) is an example of a qualitative description.
Airasian (2001) included the further stage of synthesis, describing assessment as “a
process of collecting, synthesizing and interpreting information to aid in decision-
making” (p. 8). Anderson (2003) provided a more general definition of assessment
compared with that of Airasian (2001), whereby assessment was defined as a process of
“gathering information about students ... to aid teachers in the decision-making process”
(p. 4). According to Gronlund (2003), assessment consists of a variety of techniques
that are used with the intention of determining the level of students‟ achievement with
regard to the learning objectives. White (2007) also emphasised assessment as a data-
gathering process. He described assessment as “a process for gathering evidence and
making judgements about students‟ needs, strengths, abilities and achievements” (p.
44).
21
In broader educational contexts, assessment is seen as providing documented
information needed for decision-making at the institutional, programme and classroom
levels (Evans, 2002). This view is supported by Brookhart and Nitko (2007), who refer
to assessment as a process of “obtaining information that is used for making decisions
about students, curricula, programs, and schools” (p. 4). These two definitions of
assessment extend the use of information gathered from assessment from the classroom
level to the programme and institutional levels.
The definitions and descriptions presented above highlight assessment as a process with
two main characteristics. The first characteristic is that assessment generates
information, and the second characteristic is that assessment involves making decisions
and judgements based on the gathered information.
Wiliam (2007), Berry (2008) and Miller, Linn and Gronlund (2009) provide more
detailed descriptions of assessment than the descriptions presented in the previous
paragraphs. Wiliam (2007) viewed assessment as a process that is interactive in nature
and functions “as a bridge between teaching and learning, helping teachers collect
evidence about student achievement in order to adjust instruction to better meet student
learning needs, in real time” (p. 1054). According to Wiliam (2007), assessment is
interactive because teachers can act upon the information generated about the teaching
and learning process that goes on in the classroom. He stressed that assessment can
support learning in many ways as well as measure learning. Berry (2008) described
assessment activities as “conscious and systematic activities used by teachers and
students” (p. 6) to generate information which are then analysed, interpreted and
inferred upon to assist in making decisions that enhance teaching and learning in the
classroom.
22
Miller, Linn and Gronlund (2009) depicted assessment as a broad term that
encompasses all procedures that generate information about the state of student learning
and also involves “the formation of value judgments concerning learning progress” (p.
28). An example of a value judgement formed in assessment might be that students have
made good learning progress. Accordingly, assessment can play an important role in
efforts to enhance effective teaching. It is a systematic process which “begins with the
identification of learning goals, monitors the progress students make toward those goals,
and ends with a judgment concerning the extent to which those goals have been
attained” (Miller, Linn and Gronlund, 2009, p. 26). In brief, Wiliam (2007), Berry
(2008) and Miller, Linn and Gronlund (2009) portrayed assessment as a comprehensive
process of educational decision-making that assists the enhancement of teaching and
learning.
The key similarity between the descriptions of assessment presented so far in this
chapter is the focus on assessment as a data-gathering process that assists decision-
making. Their descriptions by Wiliam (2007), Berry (2008) and Miller, Linn and
Gronlund (2009) differed from the previous definitions because of the emphasis on and
elaboration of the relationship between assessment, teaching and learning. Wiliam
(2007) regarded assessment as an interactive process that links teaching and learning,
whereas Berry (2008) and Miller, Linn and Gronlund (2009) referred to assessment as a
systematic process that enhances teaching and learning. Assessment-related activities
serve as important sources of information for both teachers and learners. For teachers,
decisions with regard to, for example, types of teaching materials, learning activities,
teaching approaches and types of feedback to be given are guided by information
gathered through assessment (Wiliam, 2007). Students, on the other hand, use
assessment-generated information to improve their learning, for example, by identifying
23
their areas of strength and weakness, understanding the reasons for the mistakes they
make and to be aware of the learning goals that need to be achieved (Berry, 2008).
Therefore, assessment is important for regulating the quality and effectiveness of the
teaching and learning process.
In summary, all the definitions and descriptions presented in the above discussion
potray assessment as a process, the main product of which is a collection of information
that is crucial in guiding the educational decision-making process to enhance teaching
and learning.
Definitions of Classroom Assessment in the Literature
The research reported in this thesis was focused specifically on classroom assessment.
This subsection presents a brief discussion of the definitions of classroom assessment in
the literature and ends with the definition of classroom assessment in this research.
Classroom assessment is the assessment process that is carried out within the classroom
(Buhagiar, 2007). It is a process involving the collection of information and evidence of
student learning to support classroom instruction, as well as to provide useful feedback
to students to enhance learning (Kulm, 1994; Shepard, 1989). It provides information
that notifies teachers about student learning, to appraise and guide classroom instruction
and to offer students feedback that may be used to improve their learning (Ercikan,
2006).
More recently, McMillan (2007) defined classroom assessment as “the collection,
evaluation, and use of information to help teachers make decisions that improve student
learning” (p. 8). He further elaborated that information generated from classroom
24
assessment guides teachers‟ instructional decisions to create a conducive learning
environment as well as inform students about the extent of their learning. Nolan and
Taylor (2008) provided a more elaborate description of classroom assessment. They
described classroom assessment as “a set of tools and processes that support all
students‟ learning” (p. 9) and assist teachers in making sensible decisions in the
classroom. According to Brookhart (2009), through classroom assessment, teachers are
able to monitor students‟ learning growth and grasp of concepts in relation to the
learning objectives within the classroom context. The literature indicates that the main
focus of classroom assessment is to improve instruction and learning at the classroom
level.
Drawing from the definitions presented in the above review, classroom assessment in
this research refers to the assessment process that encompasses assessment techniques
employed to gather information, interpretation and use of the acquired information for
decision-making within the classroom context with the intent of enhancing teaching and
learning.
Definitions of Alternative Assessment in the Literature
There are two categories of assessment: traditional assessment and alternative
assessment. Traditional assessment usually involves the use of conventional paper and
pencil tests which may consist of multiple-choice or true or false questions (Wu & An,
2007). The focus of traditional assessment is on recall or memorisation of knowledge.
The term „alternative assessment‟ is widely used in current literature on assessment,
especially in the literature related to aspects of assessment reform. Alternative
assessment refers to a form of assessment that is non-standardised and non-traditional
(Huerta-Macias, 1995; Wu & An, 2007), for example, projects, portfolios and journals.
25
For example, Gipps and Stobart (2003) refer to alternative assessment as the use of non-
traditional forms of assessment such as portfolios and projects, and “an alternative use
of assessment as part of the learning process” (p. 173).
Alternative assessment, according to Janisch, Liu and Akrofi (2007), is classroom-based
and consists of a variety of methods other than formal, standardised testing for the
purpose of determining student learning. While the context of the assessment is not a
feature in the other definitions of alternative assessment mentioned here, Janisch, Liu
and Akrofi (2007) view alternative assessment as assessment “situated in the
classroom” or “classroom-based assessment” (p. 222). Alternative assessment gives
emphasis to both the process and product of learning (Janisch, Liu & Akrofi, 2007;
Raymond, 1994). This differs from traditional assessment which mainly focuses on the
product of learning.
Berry (2008) elaborated that alternative assessment is “an ongoing process during which
teachers diagnose learning problems, monitor student learning and help them through
giving quality feedback, as well as adjusting teaching with reference to learning
outcomes” (p. 82). Such characteristics portray alternative assessment as part of the
teaching and learning process and not as a separate entity. She emphasised that
alternative assessment provides a more comprehensive „picture‟ of students‟ learning
performance in the various aspects and domains of learning. Also, alternative
assessment provides opportunities for students‟ involvement in the learning process as
well as continuous feedback with regard to their learning. Hence, Berry‟s (2008) view
of alternative assessment is in agreement with the views of Janisch, Liu and Akrofi
(2007) and Raymond (1994) which is that alternative assessment focuses on the process
as well as the product of learning.
26
Based on the above discussion of the various definitions of alternative assessment in the
literature, alternative assessment in the context of this research refers to various forms
of assessment that are different from traditional assessment forms, which support and
enhance teaching and learning, and focus on both the process and product of learning.
Assessment and Learning
The relationship between learning and assessment is an important foundation in
assessment because “every assessment is grounded in a conception or theory about how
people learn, what they know, and how knowledge and understanding progress over
time” (NRC, 2001, p. 20). An understanding of what constitutes knowledge and
learning influence the assessment design and selection of assessment items that are
considered appropriate to assess what is deemed important knowledge and skills from
students.
A factor that contributed to the call for reform in educational assessment was the
emergence of broader views of learning, especially from constructivist perspectives
(Gipps, 1994, 1999). Behaviourists viewed learning as the accumulation of knowledge
(Shepard, 2000). The purpose of assessment from the behaviourist perspective focused
on the students‟ abilities to „reproduce‟ knowledge taught in the classroom, thus,
verifying whether “the student either has or has not learned the content” (Delandshere &
Jones, 1999, p. 219).
Learning from a constructivist perspective, however, involves students constructing
knowledge (von Glasersfeld, 1987) and not simply the transfer of knowledge from
teacher to students as viewed by behaviourists. From this perspective, students are not
viewed as passive learners, but are considered active constructors of knowledge. As
27
learning is seen as “a process of personal knowledge construction and meaning making”
(Gipps, 1999), each student would have a unique „construction‟ of knowledge. As a
result, a constructivist perspective of knowledge means that assessment needs to
encompass diverse techniques to capture the varied forms of knowledge constructed by
students.
From a constructivist perspective, diversity in students‟ learning is acknowledged as
knowledge is individually constructed and constantly being developed. Delandshere and
Jones (1999) argue that if learning is seen as “a process of constant development…then
assessment is more likely to be seen as providing documentation and feedback” (p.
219). Therefore, assessment reform from a behaviourist to constructivist perspective has
altered the role of classroom assessment to generating information and feedback to
encourage students to take charge of their learning as well as guide teachers in preparing
classroom activities “relevant for the diverse individual students in their class”
(Delandshere & Jones, 1999, p. 219).
The focus of assessment in the teaching and learning process has, therefore, broadened
to include improvement of individual learning as well as increasing students‟ and
teachers‟ involvement during assessment (Gipps, 1999). With regard to the classroom
context, Stiggins (1994) emphasised that, “the greatest potential value of classroom
assessment is its ability to make students full partners in the assessment process” (p.
15). This „redefined‟ role of assessment permits the use of a wider range of assessment
techniques in addition to the usual paper-pencil testing techniques. When using
alternative assessment techniques, teachers are able to collect more diverse information
about their students‟ learning, thus adding more details to the teachers‟ overall
understanding of their students. For example, students might keep a reflective journal
28
that demonstrates to the teacher the knowledge they started with, the techniques they
used to overcome any misunderstandings they had and the knowledge with which they
completed a particular module of work. This approach would allow the teacher to
understand the improvements that the student has made from start to finish rather than a
single snapshot of the knowledge they have at the end of a module that a traditional
approach to assessment would provide.
In conclusion, a change in the way learning and teaching is conceived and understood
has resulted in a need to reform the way students are assessed. Although the
behaviourist view of assessment is still present in most assessment practices, especially
end-of-topic and end-of-semester tests (Delandshere, 2002), it is important to note that
the literature in this section brings to light not a call to „do away‟ with the traditional
techniques, but supports the implementation of broader approaches to assessing
students.
Changing Trends in Assessment
The literature shows that assessment in education has undergone significant
developments in recent decades. Traditionally, the purpose of assessment was strongly
connected with the processes of selection, certification, accountability and reporting
(Gipps & Stobart, 2003; Worthen, 1993). The focus of assessment for most of the 20th
Century was on summative data, generated mainly from traditional standardised tests
and examinations (Broadfoot, 2009; Earl, 2005). These traditional forms of assessment
were previously seen as a gateway that allowed students to progress from one level of
education to the next as well as a „vehicle‟ to social mobility (Earl, 2005). What
followed in concert with this view was an emphasis on the role of assessment in
education to sort and classify students based on their individual achievement; that is,
29
assessment was carried out mainly for summative purposes to address issues such as
accountability, certification and selection. Hence, high-stakes standardised testing, made
up of traditional paper-pencil tests, was dominating the educational assessment
scenario.
The 1980s and 1990s presented an era where educators and researchers came to realise
the limitations of traditional modes of assessment in providing information about
students‟ learning (Bell & Cowie, 2001; Earl, 2005; Stiggins, 2001). Traditional paper-
pencil tests were gradually viewed as providing information that lacked detail and was
not specific enough about students‟ achievement to assist teachers in their decision-
making process. Nor did they enable teachers to better understand students‟ learning.
At the same time, assessment seemed to be considered a separate entity from the
teaching and learning process (Birenbaum, 2003). This was probably due to the fact that
examinations and tests were mostly carried out at the end of a learning unit or course
(Lambert & Lines, 2000). Supporters of alternative assessment called for a shift in focus
for assessment in schools “from large-scale assessment for accountability to classroom
assessment for teaching and learning” (Earl, 2005, p. 6). Sound educational assessment
lies in the combination of traditional standardised testing and classroom assessment and
striking a balance between the two (Stiggins, 1994). Advocates of alternative
assessment held the view that classroom assessment carried out by teachers needed to
be given more value as it is an important facet to help enhance and develop student
learning
The issue brought to light by advocates of alternative assessment was not about a move
to replace the traditional assessment modes of examinations and standardised tests, but a
30
call for the development and use of alternative forms of assessment that better support
student learning (Gipps & Stobart, 2003). Stiggins (2001) emphasised that for change to
happen in the educational assessment scenario, important changes needed to be made to
the way assessment is viewed. These changes included recognition: (a) that assessment
is important for instructional purposes as well as for accountability; (b) that classroom
assessment generates important information for classroom-based decision-making; and,
(c) that assessment consists of various modes that can fulfil different assessment
purposes. What can be gathered from this discussion so far, is that standardised testing
and classroom assessment as well as traditional and alternative assessment approaches
each generate useful information to provide a better and more comprehensive „picture‟
of students‟ state of learning and achievement.
The start of the 21st Century saw the expansion of the role of assessment from that of
„measuring‟ student achievement to that of supporting and improving student learning.
According to Broadfoot (2009), awareness of change in assessment grew at the turn of
this century, due to an “increasingly broad recognition among assessment professionals,
teachers, researchers and even policy makers that there is a need for a substantial
change of approach to assessment” (p. viii). Stiggins (1994) explained that in the new
century there would be a greater need for “information managers” than “information
memorizers” (p. 31); that is, people who possess the skills to seek and use information
to solve problems would be more highly valued by society. Therefore, contemporary
educational assessment is seen to focus more on higher-level educational outcomes such
as problem-solving skills and higher-order thinking. These are far more complex skills
compared with low-level cognitive skills such as memorising skills, which often are the
focus of standardised testing (Birenbaum, 2003).
31
The traditional assessment system, which focuses on grading and positioning students
(Berlak, 1992) is insufficient for addressing the increasing role and purpose of
assessment in today‟s classrooms. The most apparent trend in the literature is towards
assessment for multiple purposes (Bell & Cowie, 2001). This trend suggests that
assessment should fulfil a broader array of purposes, for example, assessment should
“support teaching and learning, provide information about pupils, teachers and schools,
act as a selection and certification device, as an accountability procedure, and drive
curriculum and teaching” (Gipps, 1994, p. 1).
The current literature on assessment draws attention to three different purposes of
assessment which are formative, summative and evaluative (Wiliam, 2007). Of the three
purposes, formative assessment is assessment that supports learning, hence the term
„assessment for learning‟. Summative assessment or assessment of learning serves the
purpose of “certifying the achievements or potential of individuals” and evaluative
assessment is for “evaluating the quality of educational programs or institutions”
(Wiliam, 2007, p. 1056). Assessment for learning is considered to support learning by
providing teachers with information to alter instructional activities that can create a
more conducive learning environment and offer learning opportunities to maximise
student learning. In contrast, assessment of learning is assessment that provides sound
and dependable information of students‟ competence in relation to the standards set
within the curriculum and their standing compared with their peers (Earl, 2005).
Gipps and Stobart (2003) explain that the underpinning notions of learning and the
conceptions of the relationship between assessment and learning are what differentiate
traditional and alternative assessment approaches. They further elaborate that the beliefs
32
held by teachers about learning would influence the assessment approach adopted in the
classroom.
Assessment Reform in Asian Countries
Examinations have long been a dominant characteristic of the educational scenario in
Asian countries. Asian countries have been known to rely heavily on examinations as
the main form of assessment to provide information to educators to assist in making
decisions and judgements about students‟ performance (Berry, 2011). According to
Kennedy and Lee (2008), public examinations are a topic of “community interest and
concern in many Asian countries” (p.73) because these examinations are said to be “part
of the social structure of many Asian societies” (p. 74). Public examinations are seen as
„social vehicles‟ that „transport‟ individuals from one level to the next within the social
structure of a community (Earl, 2005; Kennedy & Lee, 2008). In addition to elevating
the „social status‟ of an individual, public examinations are also perceived as a route to
enter institutions of higher learning. This is because public examinations are seen “to
function as a selection mechanism screening out some students and endorsing others for
further education” (Kennedy & Lee, 2008, p. 74). The emphasis on examinations in
Asian countries has resulted in a culture of teaching and learning to the test, hence
producing individuals who are efficient in memorising but „poor‟ in managing
information (Stiggins, 1994).
The global assessment reform resulted in a wave of assessment reform which began at
the end of the last century in some Asian countries such as China, Hong Kong and
Singapore (Berry, 2011; Chan, Kennedy, Yu & Fok, 2006; Sellan, Chong & Tay, 2006;
Tan, 2011; Yu, Kennedy, Fok & Chan, 2006). Assessment reform was closely
associated with curriculum reforms in these Asian countries which, in turn, resulted
33
from redefined perspectives of teaching and learning as discussed in the previous
section about assessment and learning. The presence of assessment reform in the Asia
region signals a reduced reliance on examinations as the main form of gathering
information about student learning and an increase in awareness of the role of
assessment in the improvement and development of the teaching and learning process
(Berry, 2011).
In the next few paragraphs, assessment reform in three Asian countries including China,
Hong Kong, and Singapore are examined in more depth. These countries have more
geographic and cultural similarities with Malaysia compared with Western countries
and thus are more likely to provide insight into the possible nature and outcomes of
assessment reform in Malaysia.
Chinese education is influenced by Confucian philosophy and in accordance with this
philosophy the focus of education is to instil high moral values and knowledge within
individuals to enable them to „function harmoniously‟ in society (Hui, 2005).
Examinations have played a significant role in China‟s education system. China has a
long history of using examinations as “the main instrument for making decisions on
educational opportunities and government official selection” (Berry, 2011, pp. 49-50).
According to Confucian philosophy, government officials should be individuals who
demonstrate “knowledge, wisdom and virtue-qualities that were viewed as essential for
the harmonious administration of state affairs at all levels” (Berry & Adamson, 2011, p.
7). It is inappropriate, therefore, for these individuals to be selected on the basis of their
family standing in society (Hui, 2005). Hence, examinations in China potentially
provide individuals with the „gift‟ of social mobility, enabling them to move from one
level of social standing to a higher level in society.
34
Through the centuries, the importance put on examinations in China has resulted in a
passive learning environment for students. Teachers concentrate on knowledge content
and skills that are the focus in examinations, drill students with numerous questions that
are similar to those in the actual examinations, and encourage memorising (Han Min &
Yang Xiumen, 2001). As a result, students tend to perceive that the objective of learning
is to pass examinations.
Assessment reform in China began in the 1990s but was given more serious attention
and focus at the turn of the 21st century. According to Berry (2011), the year 2001
marked “the beginning of China‟s curriculum reform for basic education” (p. 52). With
this reform in motion, assessment is now seen to play a more crucial role in the teaching
and learning process, and not just for the purpose of selection and accountability. As
Berry (2011) explains, “assessment reform in basic education was closer to the heart of
assessment for learning” (p. 52).
Although China has taken proactive steps to reform its examination-oriented assessment
system to one that provides more support to students‟ learning, Wang (2008 as cited in
Berry, 2011), mentioned that there are two major challenges that need to be overcomed
for the success of this reform. The first challenge is the teachers‟ limited understanding
of assessment for learning and the second challenge is the teachers‟ lack of ability to
incorporate assessment into teaching and learning. For example, a case study of three
primary school teachers‟ assessment practices in Guangzhou by Berry and Gao (2009)
found that, firstly teachers lacked the required understanding and knowledge of the
concepts of assessment for learning. Secondly, due to this lack of understanding and
knowledge, teachers in this study had not implemented relevant assessment techniques
35
effectively in their classrooms. This situation may possibly be due to the examination
culture mentality that still exists within the Chinese community (Berry, 2011).
Like China, Hong Kong‟s education system is influenced by Confucian philosophy,
whereby examinations remain at the heart of the system. In the 1990s, Hong Kong
responded to the global assessment reform through the introduction of the Target-
Oriented Curriculum, a reform initiative that “attempts to link assessment and learning”
(Berry, 2011, p. 55) with the introduction of formative assessment. However, teachers
found this form of assessment “very difficult to handle and too time-consuming to
handle” (Berry, 2011, p. 55). In 2000, another initiative was introduced in a continuous
effort to reform Hong Kong‟s assessment system, with a special emphasis on
assessment for learning. According to Berry (2011), this time around, two new forms of
assessment were introduced, the Basic Competency Assessment for primary education
and junior secondary, and School-based Assessment for secondary education.
According to Hill and Wan (2006), school-based assessment was introduced because
written examinations are not able to assess certain aspects of the curriculum‟s learning
outcomes. For example, outcomes that involve „practical work‟, require particular
equipment and resources, such as laboratories and workshops, for example, or more
time to complete tasks such as research projects, constructing portfolios or composing
or rmaking something” (Hill & Wan, 2006.)
Hong Kong too faced challenges in its effort to bring about change to the assessment
system. For example, Carless (2005) carried out a study in relation to the
implementation of the Target-Oriented Curriculum and discovered that parents were
resistant to the change and teachers did not receive adequate support from their
36
colleagues and school management. According to Berry (2010), Hong Kong is still
struggling to implement the assessment reform policies as teachers‟ assessment
practices in schools are still „traditional‟ in nature.
Singapore, which is Malaysia‟s closest geographical „next-door neighbour‟ compared to
China and Hong Kong, embarked on a journey of systematic educational reform at the
turn of the 21st century. According to Tan (2011), Singapore‟s Ministry of Education
introduced a new vision for education in 1997. The vision of “Thinking Schools
Learning Nation” was to guide the reform of Singapore‟s education system to “an
ability-driven system” (Tan, 2011, p. 76). Singapore‟s revised national school
curriculum put more emphasis on thinking skills, problem-solving skills and students‟
ability to be independent learners (Sellan, Chong & Tay, 2006). Learning activities
encouraged students to use higher-order thinking skills and be creative when solving
problems. Assessment practices required revision to accommodate the changing focus
of the curriculum. New approaches to learning required a more appropriate approach to
assessment that was different from traditional paper-pencil methods. Accordingly,
Sellan, Chong and Tan (2006) described a shift in Singapore‟s assessment system that
focused on coursework and assessment techniques that foster thinking skills and
creativity.
Although Singapore has taken steps in the direction of reforming the nation‟s
assessment system, Singapore faces similar challenges to those faced by China in their
effort to move away from an assessment culture which centres on examinations. Song
and Koh (2010) commented that a learning culture that “comes with an emphasis on
high-stakes examinations over the past four decades” (p. 2) is going to need time to „go
away‟ despite the initiative by the Singapore government to bring change to the
37
“educational assessment landscape in schools” (p. 2). Research carried out between
2004 and 2005 showed that “teachers‟ assessment practices focused on the format of
drill and practice of basic knowledge and skills, of factual and procedural knowledge”
(Koh & Luke, 2009, p. 313).
In 2009, in an on-going effort towards educational assessment reform, Singapore‟s
Ministry of Education made a decision to replace examinations in primary grades 1 and
2 with school-based assessment that would better support learning. This is an on-going
initiative by the ministry to lessen the emphasis on examinations, especially for students
in the early years (Tan, 2011). Koh, Lim and Habib (2010) argued that the Primary
Education Review and Implementation committee proposed that assessment in schools
should support the holistic development of the pupil. Therefore, year-end examinations
were replaced with assessment techniques that provide regular and continuous feedback
on students‟ learning. Tan (2011) claims that although there is considerable emphasis on
the idea of holistic learning, “less thought is devoted to the forms of assessment practice
that prompts students to learn holistically” (p. 100). Tan (2011) explains that a holistic
approach to assessment should encompass assessment practices that encourage students
“to demonstrate how well knowledge is connected with each other, rather than how
much knowledge is understood” (p. 100). He argues that in such an environment,
assessment will enhance learning and motivate students to learn beyond focusing on
examinations.
In summary, China, Hong Kong and Singapore have assessment systems that adhere to
Confucian philosophy. Within this philosophy, examinations are seen as the main tool
for determining an individual‟s „future‟. However, each of these countries has initiated
steps to incorporate strategies consistent with the growing global assessment reform that
38
promotes the view that assessment is not only a tool for selection and certification, but
has a more significant role in the learning and teaching process. Research suggests,
however, that there are challenges along the way, resulting in slowness of change.
Change from an examination-oriented system to a holistic assessment system in these
contexts has had varying degrees of success in terms of the initial goals of the proposed
reforms.
Assessment Reform in Malaysia
Due to the presence of assessment reform in neighbouring countries within the Asia
region, Malaysia has taken steps to revamp its educational assessment system.
According to Ong (2010), since 1997 the Malaysian Examination Syndicate [MES] has
implemented a certain degree of continuous school-based assessment, for example, “the
science practical and oral tests for Malay and English languages” (p. 99) which are
carried out by school teachers with the purpose of monitoring students‟ learning growth
and progress. The MES has also initiated workshops and seminars with the purpose “to
receive feedback and opinions from educators and the public” (Ong, 2010, p. 99) with
regard to necessary changes to the assessment system which previously had put great
emphasis on performance in national examinations.
In 2007, the Ministry of Education [MOE] introduced the National Educational
Blueprint of which one of its aims was to produce well-balanced individuals who are
confident as well as creative (MOE, 2007). To achieve this aim, one recommendation
was to expand “school-based assessment and alternative assessment to provide more
holistic and accurate evaluation than the current one-off examinations” (Ong, 2010, p.
100). As a result, the MES made public the proposal for a new National Eduational
Assessment Sytem [NEAS] which would reduce the emphasis on examinations (Tuah,
39
2007), focusing on „humanising‟ assessment. In addressing the call in the National
Education Blueprint for an assessment system that focuses on human capital
development, the NEAS needed to “move towards criterion-referenced, task-based and
holistic evaluation of knowledge, skills and competence” (Raja Musa, 2008, p. 383). In
Malaysia, the overemphasis on examination results as an important product of schooling
has encouraged teachers to „teach to the test‟. The implementation of the NEAS has the
potential to change the view of schools as institutions that centres on preparing students
to perform well in examinations to one that views schools as institutions where the
focus is “more on educating and guiding them [students] to acquire knowledge and
skills (MES, 2007).
As mentioned in the previous chapter, five forms of assessment were introduced in the
NEAS: school assessment; central assessment; central examination; psychometric tests;
and physical activity assessment. School assessment entails any form of assessment that
is “planned, developed, conducted, examined and reported by teachers in school‟ (Tuah,
2007, p. 6). As for central assessment, the MES determines the standards, instruments,
data analyses and guidelines for this form of assessment. Schools are responsible for
administering, marking and reporting central assessment. Central examination measures
the content of the national syllabus and is fully conducted by the MES. Central
examination involves certification of students‟ performance by the MES. Psychometric
tests are implemented with the purpose of measuring students‟ innate abilities whereas
physical activity assessment is for measuring and evaluating students‟ performance and
involvement in extra-curricular activities (Tuah, 2007).
The introduction of these five forms of assessment in the NEAS would enable the
collection of various data about students‟ development, performance and achievement
40
(Kaur, 2007). Furthermore, the objective of the NEAS was “to reduce over-reliance and
over-dependence on data obtained through centralised public examinations for
information about students” (Kaur, 2007, p. 2). Therefore, the NEAS is a more holistic
assessment system that is seen as an effort by the MES to fulfil the Malaysian
government initiative to improve the quality of human capital development (Tuah,
2007, p. 8).
Assessment in Mathematics
From a behaviourist perspective, the learning of mathematics is the transfer of
knowledge or the “transmission of rules and facts” (Delandshere & Jones, 1999, p. 218)
from the teacher to students, emphasing the notion that mathematics teachers „pass on‟
knowledge to students. However, the emergence of the theory of constructivism brought
into question this notion of learning. From a constructivist perspective, learning in
mathematics involves the construction of mathematical knowledge and meanings
assigned by students to these constructions. In tandem with this shift in the way learning
is viewed, the National Council of Teachers of Mathematics [NCTM] (1989) in the
United States emphasised that the learning of mathematics was more than mastering
computational skills. The NCTM advocated the idea that the focus of mathematics
learning should also involve communication skills and higher-order thinking skills such
as critical, analytical and problem-solving skills. Consequently, what was considered
important in the learning of mathematics moved from mastery of computational skills to
understanding of mathematics.As a result of this shift of focus to understanding, there
was a rise in interest in alternative assessment in mathematics (NCTM, 1995).
The Assessment Principle documented in the Principles and Standards for School
Mathematics (NCTM, 2000) states that “assessment should support the learning of
41
important mathematics and furnish useful information to both teachers and students” (p.
22). This principle focuses attention on the idea that assessment needs to be an essential
component of instruction to guide teachers in making instructional decisions with the
intent of improving and supporting student learning of mathematics.
According to Webb (1993), mathematics assessment needs to be a “continuous and
dynamic process, rather than static testing at specific points in time” (p. 2) to enable
teachers to understand the meanings that have been assigned to mathematical ideas and
concepts. He further clarifies that assessment in mathematics is also an important tool to
monitor students‟ progress towards mathematical competence. According to Clements
and Ellerton (1995), “students who give correct answers to paper-and-pencil
mathematics items sometimes have little or no understanding of the mathematical
concepts and relationships which the tests were designed to measure” (p. 184.). As
mathematics is a complex domain, teachers require multiple sources of information to
determine the development of students‟ mathematical thinking. The use of alternative
assessment techniques is an appropriate avenue for providing teachers with this type of
information.
Traditional forms of mathematics assessment, usually standardised, written tests, focus
on “the recording of the amount of correct or incorrect answers” (Thompson, Thompson
& Else, 2000, p. 29). Formal, standardised, written tests provide limited opportunities
for students to communicate their mathematical thinking and understandings.
Alternative mathematics assessments present much greater opportunities for students to
apply their mathematical knowledge to show what they can do and produce and not just
what they can remember or reproduce. At the same time, teachers are able to “gain
insight into students‟ broad knowledge and understanding of mathematics, not just skills
42
and procedures” (Kulm, 1994, p. 3). For example, when students present their solutions
to mathematical problems on the board, teachers are provided with a „view‟ of students‟
mathematical thoughts and reasoning approaches.
Through the use of alternative assessment, mathematics teachers are able to create
detailed profiles of their students‟ mathematical learning and/or personal background,
for example, their interests, attitudes and experiences with regard to mathematics
(Thompson, Thompson & Else, 2000). Kulm (1994) outlines how alternative
assessment provides teachers with a comprehensive understanding of, and better insight
into, their students‟ mathematical thinking as well as personal background, which, in
turn, assists them in becoming better mathematics teachers. Teachers are then able to
decide on suitable approaches to “developing conceptual understanding and the
relationships between procedures and concepts” (Kulm, 1994, p. 3-4). The use of
alternative assessment in the mathematics classroom presents teachers with an array of
assessment techniques such as portfolios, journals, presentations and projects. These
techniques can be carried out at the individual or group level. Techniques such as
projects and presentations can be implemented at both individual and group level,
whereas reflective journals are better suited as an individual activity because entries into
the journals are personal in nature.
Traditional forms of testing in mathematics seem to put importance on finding the
correct answer. This emphasis encourages students to strive to be an expert “test taker”
(Kulm, 1994), thus creating test anxiety among them. According to Kulm (1994), “test
anxiety is perhaps the greatest factor in producing poor attitudes towards mathematics”
(p. 5). As mentioned earlier, the use of alternative assessments allow students to exhibit
their mathematical ability through a variety of techniques. Students are presented with a
43
wide range of opportunities to produce good results. Better performances in
mathematics help increase students‟ self-esteem and lessen test anxiety. Therefore,
students may develop better attitudes towards mathematics.
The use of alternative assessment in mathematics signals a change from an emphasis on
producing correct answers to enhancing mathematical thinking, reasoning and problem
solving skills among students. For example, a scoring rubric provides “focused
feedback to the students about the solution process. It emphasises processes,
procedures, and extended thought rather than only the correct answer” (Kulm, 1995, p.
6). A scoring rubric may be constructed to award students with marks for presentation
of a solution even if the final answer is incorrect. These marks are a form of feedback
on the problem solving strategies that students choose to use.
Mathematics Teachers’ Assessment Practices
This subsection presents an overview of research conducted on mathematics teachers‟
assessment practices. There is limited research on mathematics teachers‟ assessment
practices in the published literature, however, one such research is a qualitative case
study of five secondary school mathematics teachers in Ontario, Canada (Suurtamm,
2004). This research investigated the participating teachers‟ experiences in
implementing authentic assessment in their respective classrooms. Through interviews,
the teachers emphasised that although it was not compulsory to adopt authentic
assessment in their classrooms, they chose to do so because they were aware of the
importance of applying mathematics in real-life contexts. By doing so, these teachers
felt that students are able to develop a better understanding of mathematics as well as
appreciate and value mathematics in their everyday life.
44
The findings indicated that the teachers faced numerous barriers in their quest to
implement authentic assessment in their mathematics classrooms, including “balancing
traditional expectations in the curriculum with authentic assessment activities, matching
new assessment techniques with traditional reporting methods, and dealing with feelings
of isolation and alienation from colleagues” (Suurtamm, 2004, p. 503). Collegial and
administrative support was considered crucial by the five case study teachers for the
success of efforts to introduce authentic assessment techniques.The findings suggested
that adequate resources, such as exemplars of problems and assessment tools as well as
professional development and cooperation among teachers, are needed to support the
use of authentic assessment.
Watt (2005) conducted a survey to investigate the assessment approaches employed by
60 mathematics teachers, ranging from Years 7 to 12, from 11 secondary schools
situated in Sydney‟s metropolitan area. In addition, the researcher investigated the
attitudes of these 60 teachers towards the use of alternative assessment techniques. The
findings showed that traditional assessment techniques, namely mathematics tests, were
the most favoured among the teachers. The preferred alternative assessment methods
that the teachers used included observations, assignments, oral and practical tasks. As
for their attitudes towards alternative assessment, all teachers were in agreement that
written mathematics tests provided a better measure of students‟ mathematical abilities
as students advanced through the school years. The reason for this view was the
subjective nature of alternative assessment techniques in comparison with traditional
techniques. Another finding was that teachers with the least teaching experience seemed
to be more receptive to alternative methods than very experienced teachers.
45
Buhagiar and Murphy (2008) conducted a study of the classroom assessment practices
of 12 mathematics teachers in Malta who were teaching in a sixth form college. A
qualitative case study approach was adopted and data collection techniques included
interviews, observations, and document collection. Most of the teachers adopt a teacher-
centred approach in their classrooms. Their main concern was to prepare students for
examinations and finishing the syllabus. The findings showed that the teachers‟
assessment beliefs and practices were “well aligned with their still very traditional
classroom approaches” (Buhagiar & Murphy, 2008, p. 174). The main focus of the
teachers‟ assessment practices was on formal assessment such as class tests. The study‟s
findings suggest that this focus on traditional assessment exists because of the high-
stakes examination environment that exists within the school system. A
recommendation that emerged from this study is that to improve teachers‟ classroom
assessment practices, teachers need to work in “an educational system that truly values
„learning‟ and „everyone‟ in all its endeavours” (Buhagiar & Murphy, 2008, p. 180).
One other research focusing on mathematics teachers‟ assessment practices was
conducted by Suurtamm, Koch and Arden (2010). This research was part of a bigger
study investigating how Grades 7 to 10 Ontario teachers understand the mathematics
curriculum. The findings showed that the teachers‟ assessment practices were consistent
with assessment reform. The teachers employed a wide range of assessment techniques
such as mathematics journals, assignments and performance tasks to gain a better
insight into their students‟ understanding. Assessment activities were observed to be
integrated with instructional activities, aligning with the view that assessment needs to
support learning.
46
In summary, this discussion on research carried out on mathematics teachers‟
assessment practices provide an insight into the types of assessment techniques
employed by mathematics teachers, the constraints that they faced when implementing
these techniques and steps taken to overcome barriers.
Teachers’ Beliefs
Definition and Influence of Beliefs in the Literature
Defining the construct „belief‟ is a complex matter; as McLeod and McLeod (2002)
argue, “Yet so far at least, there is no real agreement on a single definition of the term
[belief] in the literature…” (p. 118). There exist various definitions of beliefs (Leder &
Forgasz, 2002; McLeod & McLeod, 2002) because of the varied interpretations and
connotations assigned to it in the literature. According to Rokeach (1968), belief is “any
simple proposition, conscious or unconscious, inferred from what a person says or
does” (p. 113) and further elaborated that “all beliefs are predispositions to action” (p.
113).
Beliefs are assumed to have a substantial influence on behaviour (Pajares, 1992; Ajzen
& Fishbein, 1980). Ajzen and Fishbein (1980) stated that “beliefs are the basic
determinants of any behaviour” (p. 223). They further stressed that a change in an
individual‟s beliefs will bring about change in his or her behaviour. Teachers‟ beliefs
according to Kagan (1990) refer to “highly personal ways in which a teacher
understands classrooms, students, the nature of learning, the teacher‟s role in the
classroom and the goals of education” (p. 423). Schoenfeld (1992) later describes
mathematical beliefs as “an individual‟s understandings and feelings that shape the
ways that the individual conceptualizes and engages in mathematical behaviour…” (p.
358), whereas Ponte (1994) refers to beliefs as undeniable personal truths held by an
47
individual. According to Swan (2006), “beliefs underpin personal thought and
behaviour” (p. 59).
Research has indicated the influence of teachers‟ beliefs on their instructional practices
(Elbaz, 1981; Isenburg, 1990; Shavelson & Stern, 1981; Shulman, 1987; Thompson,
1992). Hofer and Pintrich (1997), for example, viewed teachers‟ belief systems as a
reflection of the teachers‟ personal views about knowledge, which, in turn, influence
their instructional decisions in the classroom. These beliefs have a substantial effect on
what teachers view as appropriate classroom behaviour and instructional decisions.
Isenberg (1990) explained that research on teachers‟ beliefs systems helps educators
understand the variations in instructional practices of teachers in the classroom. He
argued that teachers need to understand that their beliefs have an influence on their
instructional practices. According to Beswick (2005), teachers‟ beliefs are “susceptible
to change when they are made explicit” (p. 143). She further mentioned that teachers‟
beliefs and practices develop and change together.
Fullan (1993) explained that the low rate of success in the introduction of educational
changes caused researchers to shift their attention to investigating teachers‟ beliefs as a
significant component in implementing reform. Furthermore, teachers are seen as key
agents of change, who are crucial in determining the success or failure of any
educational reform. As a consequence, it is important to consider teachers‟ beliefs when
designing reform so that meaningful changes will be successfully carried out in the
classroom (Eisenhart, Shrum, Harding & Cuthbert, 1988; Isenburg, 1990).
Drawing from the definitions and discussion presented in the above review, the term
beliefs in this research refer to an individual‟s personal understandings, views and
48
feelings that are regarded as true to the individual and guide their behaviour in and
understanding of the world around them.
Teachers’ Beliefs and Mathematics
This subsection presents a discussion on research conducted on mathematics teachers‟
beliefs and their instructional practices. There has been considerable research conducted
in this area and such studies have generated findings that indicate mathematics teachers‟
beliefs have a significant influence on their instructional practices (Anderson, Sullivan
& White, 2005; Aguirre & Speer, 2000; Barkatsas & Malone, 2005a; Beswick, 2005,
2007; 2008; Calderhead, 1996; Ernest, 1989; Grootenboer, 2008; Wilkins, 2008; Yates,
2006). Due to the complexity of the concept of belief, research on mathematics
teachers‟ beliefs and their classroom practices have generated findings that show
inconsistent findings; consistency between teachers‟ beliefs and their instructional
practices and inconsistency between teachers‟ beliefs and their instructional practices.
Following is a discussion of several studies conducted with a focus on mathematics
teachers‟ beliefs as well as the research findings that demonstrates the effect of
teachers‟ beliefs on their classroom practices.
Considerable research has been conducted on mathematics teachers‟ beliefs about the
nature of mathematics, mathematics teaching and learning and the influence of these
beliefs on their classroom practices. The findings of these studies indicate that there is a
relationship between teachers‟ beliefs and the instructional practices that they adopt in
their classroom. Examples of such research were studies conducted by Aguirre and
Speer (2000), Beswick (2007; 2008), Wilkins (2008) and Cross (2009).
49
One such study was a qualitative case study of two mathematics teachers in the United
States, carried out by Aguirre and Speer (2000) who argued that a teacher‟s beliefs were
central to his/ her “selection and prioritization of goals and actions” (p. 327) in the
classroom. The findings of the study identified that the combination of the teachers‟
beliefs about the nature of mathematics and teaching and learning influenced the
teachers‟ instructional goals and classroom practice. For example, one of the case study
teachers held the belief that mathematics consisted primarily of formulas and
procedures and that the teaching of mathemathics involved teachers providing clear and
relevant mathematical examples to help students understand. As a consequence, her
main focus in classroom discussions was to provide students with appropriate and clear
examples to ensure students were aware of the formulas and procedures that they were
expected to know.
A study conducted by Beswick (2007/2008) identified differences between the beliefs
of a group of 22 K-8 mathematics teachers with regard to students with mathematics
learning difficulties (MLD) and students in general. These teachers had been exposured
to an array of pedagogical approaches for teaching students with mathematical learning
difficulties through their participation in a professional learning programme. The
findings indicated that the teachers‟ beliefs guided their preference for particular
teaching approaches and teaching materials. For example, the teachers held beliefs that
for MLD students, „surviving‟ the mathematics classroom was more important than
gaining conceptual understanding, and a greater use of concrete materials was needed
when teaching MLD students.
Wilkins (2008) conducted a study that investigated the relationship among four teacher-
related variables: mathematical content knowledge; attitudes toward mathematics; use
50
of inquiry-based instruction; and their beliefs about the effectiveness of this form of
instruction, involving 481 in-service elementary teachers from two school districts in
the South Eastern United States. The findings of this study suggested that “teachers‟
beliefs partially mediated the effects of content knowledge and attitude on instructional
practices” (Wilkins, 2008, p.156). The findings implied that teachers with higher
mathematical content knowledge held the belief that success in learning mathematics
was due to traditional approaches, hence, their tendency to subscribe to more traditional
methods of instruction. In contrast, teachers who experienced less success in learning
mathematics tend to hold beliefs that non-traditional instructional approaches are more
effective in promoting student mathematical understanding. Therefore, their preference
was more towards alternative modes of instruction.
Another study that explored the relationship between the mathematics teachers‟ beliefs
and their classroom practices was a qualitative, collective case study of five, ninth-grade
mathematics teachers from two suburban county schools in the southeastern United
States conducted by Cross (2009). The findings of this study indicated a strong
alignment between the participating mathematics teachers‟ beliefs and their classroom
practices, suggesting that the teachers‟ beliefs are “a fairly reliable predictor” (Cross,
2009, p. 341) of the types of instructional approaches employed in the classroom. For
example, three of the five case study teachers viewed mathematics as a collective set of
concepts, rules, skills and procedures. These teachers held beliefs that the teacher was a
„provider‟ of knowledge and that mathematics learning involved application of correct
procedures to achieve computational accuracy. Hence, these three teachers provided
students with classroom activities that enhance learning through repetitive practice of
mathematical procedures.
51
The four studies discussed above show that the teachers‟ beliefs were consistent with
their instructional practices. The findings of these studies corroborate the view that
teachers‟ beliefs have a strong effect on their instructional practices. The implications of
these studies emphasise the importance of understanding teachers‟ beliefs to better
understand their instructional practices. The findings provide a clearer picture of the
different influences of teachers‟ beliefs on their instructional practice, for example,
guiding the selection of teaching approaches and teaching materials.
Unlike the research discussed above which showed consistency between teachers‟
beliefs and instructional practices, the following discussion focuses on research that
generated findings which showed inconsistency between teachers‟ beliefs and
instructional practices. Raymond (1997) conducted a study involving six novice
elementary school teachers and identified that one of the teachers portrayed
inconsistency between her beliefs and instructional practices. This teacher‟s beliefs
about mathematics were categorised as traditional whereas her beliefs about
mathematics teaching and learning were categorised as non-traditional. Raymond
(1997) observed that this particular teacher‟s classroom practices were traditional in
nature, consistent with her beliefs about mathematics but inconsistent with her beliefs
about mathematics teaching and learning. The findings of this study implied that this
inconsistency arose from the teacher‟s concern with issues such as time constraints,
insufficient resources, standardised tests and classroom management.
Barkatsas and Malone (2005a) conducted a study involving 465 Greek secondary
mathematics teachers, ranging from teachers of Grade 7 to teachers of Grade 12. This
research was conducted in two parts; a survey using a 34-item questionnaire and a case
study of an experienced teacher. Through the case study of Ann, an experienced teacher,
52
Barkatsas and Malone (2005a) reported that Ann‟s beliefs were not always consistent
with her classroom practices. Based on the case study findings, they suggested that
classroom environment and external expectations contributed to these inconsistencies.
For example, teachers were expected to focus on preparing their students for university
entrance examinations and guide the students to achieve the highest score possible.
Therefore, although Ann was a firm believer of collaborative approaches and active
participation in teaching and learning, her classroom practices were not in line with her
espoused beliefs.
The findings in both studies (Barkatsas & Malone, 2005a; Raymond, 1997) suggest that
inconsistencies between teachers‟ beliefs and their practices do not exist because of the
teachers themselves, but due to certain aspects of the context they teach in, for example,
an examination-oriented environment. Such a situation could pose a challenge for
teachers with non-traditional views of teaching and learning to implement approaches
such as group work or presentations.
Research on the relationship between teachers‟ beliefs and instructional practices has
also generated different categories of beliefs. Two such studies were conducted by
Barkatsas and Malone (2005a) and Beswick (2005). Barkatsas and Malone (2005a)
identified two categories that characterised the Greek teachers‟ beliefs about
mathematics and the teaching and learning of mathematics, which were contemporary-
constructivist orientation and traditional-transmission-information processing
orientation. Teachers in the contemporary-constructivist category held beliefs that
mathematics is a way of thinking, learning mathematics enhances problem-solving
skills, and teaching and learning activities should be student-centred to allow self-
reflection on learning and also encourage collaborative learning. Teachers in the
53
traditional-transmission-information processing category viewed mathematics as a
collection of facts and information, teaching and learning activities as teacher-centred
and rote learning is important.
Beswick (2005) investigated 25 secondary mathematics teachers‟ beliefs about the
nature of mathematics, mathematics teaching and learning and the relationship between
their beliefs and the teachers‟ classroom practices from six rural secondary schools in
Tasmania, Australia. Three teacher belief clusters emerged from the findings of this
study: (a) Content and clarity cluster, (b) Relaxed problem solvers cluster, and (c)
Content and understanding cluster. Teachers in the content and clarity cluster held the
belief that computational skills were important for mathematics and that the teaching of
mathematics focussed on providing answers as well as clear explanations of
mathematical content to students. Teachers in the relaxed problem solvers cluster held a
problem-solving view of mathematics and their views about mathematics teaching were
in total contrast with that of the beliefs held by content and clarity teachers. The main
concern of teachers in the content and understanding group was students‟
understanding. They strongly believed that sequencing of mathematical topics is critical
in the teaching of mathematics. These three clusters offered a specific insight into the
characteristics of the different types of teacher beliefs about the nature of mathematics,
and mathematics teaching and learning.
In summary, this subsection has presented insights into the relationship between
teachers‟ beliefs and their instructional practices through research. Some studies have
shown consistencies between beliefs and practices whereas others have shown
inconsistencies. Categories of teachers‟ beliefs that have been produced from research
54
provide an awareness of the types of beliefs that possibly exist and the characteristics of
practices in relation to those beliefs.
Teachers’ Beliefs and Mathematics Assessment
As previously mentioned, considerable research has been conducted on the relationships
between classroom teachers‟ beliefs about mathematics, their teaching and learning of
mathematics and their classroom practices in general, but comparatively little research
has been reported on mathematics teachers‟ beliefs specifically about assessment such
as their beliefs about the purpose of assessment and their use of assessment data (Nisbet
& Warren, 2000). Adams and Hsu (1998) argued that research in relation to
mathematics assessment is important because assessment should be used to establish the
mathematical content and approaches to enhance students‟ mathematical learning.
Moreover, such research provides information about the level of compatibility between
the teachers‟ conceptions and proposed mathematics assessment practices. The
following discussion provides an overview of key research conducted to date in relation
to mathematics teachers‟ beliefs and assessment.
Adams and Hsu (1998) conducted a survey in public schools in a southeastern state in
the the United States, to investigate mathematics teachers‟ conceptions of assessment
and their assessment practices in the classroom. The findings indicated that the
mathematics teachers‟ conceptions of assessment covered a wide range of assessment
techniques and approaches. Importantly, it was found that these conceptions structure
the selection of assessment techniques that the teachers employed in the classroom and
the frequency of usage. These conceptions indicate that teachers have their own beliefs
about which assessment technique is relevant or suitable to be used in their respective
classrooms. For example, teachers in Grades 3 and 4 felt that homework was important
55
to assess students‟ understanding, compared with teachers of first and second graders
who felt that homework was less important.
Delanshere and Jones (1999) conducted a study of three elementary mathematics
teachers from two schools in the United States undergoing curriculum reform. By
examining the relationship between assessment, teaching and learning, factors that
characterised the teachers‟ beliefs about assessment in mathematics emerged. The
researchers identified three main factors that shaped the teachers‟ beliefs; (a) the
external or relevant policy definition of the purposes and functions of assessment, (b)
teachers‟ views of the official curriculum and their position with regard to the subject
matter, and (c) teachers‟ understanding of students and their learning. The teachers in
the study tended to view the learning and teaching of mathematics as two non-related
components and limited their assessment practices to making judgements of the
students‟ mastery of the subject matter. Hence, the classroom assessment practice of
these elementary teachers was focused on evaluating what students could or could not
do in mathematics.
Nisbet and Warren (2000) investigated the nature of Australian primary school teachers‟
beliefs, focusing on beliefs about mathematics teaching and assessment. These
researchers also examined the relationship between teacher characteristics such as
gender, grade level, qualifications and years of experience with their beliefs. The
findings suggested that the teachers believed that the main purpose of assessment is to
inform teachers and students of the learning in the classroom. Of the two, using
assessment to inform the teacher was rated the highest priority. Hence, the assessment
techniques adopted in the classroom were more structured to generate information on
their performance as a teacher. Other findings from the study by Nisbet and Warren
56
(2000) showed that female teachers tended to use assessment to inform teachers and
learners more than male teachers, and the use of assessment as a diagnostic tool
decreased as age level of students increased.
Research conducted in New Zealand by Pfannkuch (2001) investigated primary and
secondary mathematics teachers‟, ranging from teachers of Year 3 to Year 13, beliefs
and assessment practices. The dominant belief among the primary teachers was that the
purpose of assessment was to generate information about students‟ progress in their
learning. The primary teachers used a variety of assessment techniques, whereas the
secondary teachers tended to restrict use of alternative assessment techniques to non-
examination mathematics classes. The secondary teachers believed that the purpose of
assessment was to provide information about the students‟ potential to succeed in
national examinations. Further, the secondary teachers believed that it is important for
students to be informed of the level of their mathematical ability and assessment
provided them with such information. Another finding of this study was that both
primary and secondary teachers shared the belief that alternative assessment methods
allowed all students to feel a sense of achievement, which differed from the effect of
traditional written examinations that tend to label non-performing students as „failures‟.
The study findings also revealed that the primary teachers held a strong belief that
“assessment must show that each child has progressed or achieved success” (Pfannkuch,
2001, p. 198).
The findings in the three studies (Delanshere & Jones, 1999; Nisbet & Warren, 2000,
Pfannkuch, 2001) suggest that teachers view the main purpose of assessment is to
generate information to assist them in making judgements about students‟ progress in
their learning. Another purpose of assessment indicated by the research findings is to
57
inform teachers about their performance as a teacher. Guided by these beliefs, teachers
select assessment techniques that they deem suitable to generate the required
information.
Research also has been conducted with the specific aim to investigate the typology of
mathematics teachers‟ beliefs with regard to assessment. One such study was the survey
conducted by Barkatsas and Malone (2005b), mentioned earlier in this chapter, which
involved 465 Greek secondary mathematics teachers comprising Grade 7 to Grade 12
mathematics teachers. The aim of this research was to identify the types of beliefs that
existed amongst the Greek mathematics teachers in relation to assessment and teacher
characteristics such as gender, professional development, years of teaching experience,
position held and educational qualification. Data collection was carried out through a
survey using a 19-item questionnaire. The study findings identified three categories of
orientation for the beliefs of these teachers in relation to mathematics assessment: (a)
socio-constructivist orientation; (b) problem-solving orientation; and (c) accountability
orientation.
Teachers in the socio-constructivist orientation category believe that students should be
provided opportunities to experience challenging, problematic mathematical situations
to enhance their mathematical learning, emphasising that the construction of
mathematical knowledge is through their experiences (Barkatsas & Malone, 2005b).
Teachers who adopt a problem solving orientation to mathematics assessment believe
that teachers should present students with problems relating to real-life situations that
cultivate investigative skills as well as a variety of problem-solving techniques
(Barkatsas & Malone, 2005b). Teachers in the accountability orientation group believe
that mathematics assessment is to provide information for accountability purposes such
58
as feedback on students‟ progress and to determine students‟ progression to the next
grade level (Barkatsas& Malone, 2005b).
The findings with regard to the influence of teacher characteristics on beliefs about
mathematics assessment in the research by Barkatsas and Malone (2005b) among others
suggested that: (a) gender did not influence the Greek mathematics teachers‟ beliefs
about assessment; (b) the more experienced teachers subscribed to the problem solving
orientation to mathematics assessment; and, (c) the social constructivist view of
mathematics assessment was more common among teachers than among principals and
vice-principals, especially teachers with Masters and first degrees in mathematics. In
conclusion, Barkatsas and Malone (2005b) contend that an understanding of the types of
beliefs about mathematics assessment that exists among mathematics teachers provides
a platform for a better understanding of why teachers do what they do in their respective
classrooms.
In response to recent changes in the mathematics syllabus in Australia, Panizzon and
Pegg (2008) carried out research to explore 25 rural secondary science and mathematics
teachers‟ changing assessment practices and the implications it had on their classroom
practice. This two-year study involved participating teachers in professional
development sessions. The study established that there was a change in the teachers‟
perception of learning and this change was mirrored in their instruction and assessment
practices. The teachers‟ perceptions of assessment had shifted from a process of
gathering students‟ marks to generating information for diagnostic purposes to inform
teaching practices that improve students‟ learning. The findings of this study suggest
that exposure to appropriate professional development sessions regarding assessment
practices may influence change to teachers‟ beliefs about assessment.
59
The discussion above emphasises the importance of understanding and identifying
mathematics teachers‟ beliefs in relation to assessment, as these beliefs have an
influence on the selection and implementation of assessment techniques in the
classroom.
Conclusion
An important point ascertained from the literature is that mathematic teachers‟ beliefs
about curriculum reform influence the pace and level of success of the reform (Handal&
Herrington, 2003; Gooya, 2007). Handal and Herrington (2003) emphasise that the
current global reform of assessment of learning in mathematics education will only be
successful if teachers‟ beliefs about this reform are taken into consideration. If
mathematics teachers‟ beliefs match the principles that underpin mathematics
assessment reform, then it is more likely they will implement the change with
confidence and the degree of success of implementation is likely to be promoted.
Alternatively, if mathematics teachers‟ beliefs are not consistent with the underpinning
principles of mathematics education reform, then this mismatch is likely to negatively
impact on the success of the reform (Battista, 1994; Handal & Herrington, 2003).
Moreover, it is likely that teachers will not be willing or cooperative in implementing
changes in their classroom practices. Teachers will continue with their individual
agenda in the classroom and the main objectives of mathematics assessment reform are
unlikely to be achieved.
This chapter provided an overview of the literature about assessment and teachers‟
beliefs. Relatively few studies have been conducted in the area of mathematics teachers‟
beliefs about assessment as most research to date has focused on mathematics teachers‟
beliefs and the teaching and learning of mathematics. This study will, therefore, make a
60
contribution to the literature about mathematics teachers‟ beliefs about assessment in an
era when assessment reform is gaining momentum. The next chapter discusses the
methodology adopted for this study.
61
CHAPTER THREE
METHODOLOGY
Introduction
As outlined in Chapter One, the overarching aim of this study was to develop
understandings of Malaysian pre-university mathematics teachers‟ beliefs about
classroom assessment. This overarching aim was divided into three subsidiary aims.
The first subsidiary aim was to explore the teachers‟ beliefs about the purpose of
classroom assessment. The second subsidiary aim was to investigate the teachers‟
classroom practices with regard to mathematics assessment and the relationship between
their espoused beliefs and what they actually do. The third subsidiary aim of this study
was to investigate the teachers‟ beliefs about change to their assessment practices; their
beliefs about reasons for change, barriers to change and factors that may facilitate
change.
The purpose of this chapter is to provide a description of the methodological
framework, and the methods of data collection and analysis that were used to meet these
aims. The chapter is presented in seven sections. The first section provides the
justification of the research approach taken, that of a qualitative approach within the
interpretive paradigm. A justification and outline of the case study research design is
presented in the second section. The third section provides a description of the research
site and the people who participated in the study. The fourth section presents an
explanation of the data collection procedures and the use of qualitative data collection
techniques including interviews, observations, reflective accounts and document
collection.The data analysis process is outlined in the fifth section. The sixth and
seventh sections address the trustworthiness and ethical considerations for the study
62
respectively. Finally, the chapter ends with a brief summary that leads to the findings
chapter.
Research Approach
Interpretive Paradigm
This study was conducted within the interpretive research paradigm. The central
principle of interpretivism is that human beings are constantly interpreting and making
sense of the world in which they live (Williamson, 2002). From this perspective, human
activity is viewed as a social experience which involves „making meaning‟ of these
experiences (Schwandt, 2000). Interpretivists rebuff the idea of an objective reality that
is said to exist „out there‟ and waiting to be discovered. Reality in the interpretive
paradigm is viewed as relative in nature and encompasses personal meanings that are
developed in association with the individual‟s experiences. These constructed meanings
function as guidelines to the individuals in their effort to understand the world they live
in (Creswell, 2007; Radnor, 2001; Willis, 2007). Because of these varied versions of
reality which are individually and socially constructed, the idea of multiple realities is
said to exist in this paradigm.
Within the interpretive paradigm, knowledge is said to be acquired through the
interpretation of social reality. Interpretivists consider social reality as a collection of
personal and context-specific meanings which is constructed and negotiated by the
individuals involved in the context (Gall, Gall & Borg, 1999). Hence, the building
blocks of knowledge are interpretations of experiences and meanings assigned to those
experiences (Radnor, 2001). Interpretivists view knowledge as a construction of
negotiated meanings between the researcher and the participant (O'Donoghue, 2007).
These negotiated meanings are exclusive to the research context.
63
According to Schwandt (2000), conducting research in the interpretive paradigm is an
effort to comprehend the complexity of the experienced world through the „eyes‟ of the
individuals who live it. Interpretive research explores people and their interpretations,
meanings and understandings (Mason, 2002). This viewpoint is further supported by
Williamson (2002) who emphasised that interpretivist researchers are concerned with
“the beliefs, feelings and interpretations of participants” (p.31).
Furthermore, interpretive research focuses on revealing and understanding the
constructed meanings of an event for those involved. The interpretivist researcher tends
to construct this understanding based on the individuals‟ views of the specific social
context being investigated (Creswell, 2007). Therefore, researchers who adopt this
paradigm are interested in understanding social reality through the meanings that
humans make of their experiences (Eisenhart, 1988; Crotty, 1998; O‟Donoghue, 2007).
In summary, interpretive research is concerned with understanding human meanings of
the world they live in.
The interpretive paradigm was selected to underpin this study because the focus of the
research was to develop understandings of Malaysian pre-university mathematics
teachers‟ beliefs about classroom assessment. This study was concerned with
uncovering the multiple realities of the teachers, their experiences and personal
constructions of their understandings of classroom assessment in their mathematics
classroom. This is important as research has indicated that teachers‟ beliefs influence
what they do in the classroom (Anderson, Sullivan & White, 2005; Barkatsas &
Malone, 2005b; Beswick, 2007; Pajaras, 1992; Yates, 2006). The research reported in
this thesis, therefore, presents an interpretivist account of the beliefs of Malaysian pre-
university mathematics teachers about classroom assessment.
64
Qualitative approach
As the study was located within the interpretive paradigm and the emphasis was on
developing an understanding of the participants‟ beliefs, a qualitative approach to data
collection was considered the most appropriate. According to Merriam (2009),
“qualitative researchers are interested in understanding how people interpret their
experiences, how they construct their worlds, and what meaning they attribute to their
experiences” (p. 5). This is further supported by Lichtman (2006) who states, “the main
purpose of qualitative research is to provide an in-depth description and understanding
of the human experience” (p. 8). Qualitative research, therefore, aims to capture and
provide a thorough description of the social experience of individuals and meaning
given to those experiences (Gay & Airasian, 2003; Lichtman, 2006; Merriam, 2009;
Sarantakos, 2005). Through these thorough descriptions, qualitative researchers aim to
provide a „window‟ into the world of the participants as seen and lived by them.
Creswell (2007) explains that a qualitative approach is the preferred form of inquiry
when the focus of the research is to provide a detailed understanding of a particular
issue. Gay and Airasian (2003) are of the opinion that “qualitative, interpretive research
is useful for describing and answering questions about participants and context” (p.
163). This form of research allows the researcher to seek an understanding of a group or
phenomenon, as well as the participants‟ view towards events, beliefs or practices. This,
in turn, is consistent with the interpretive framework which is guided by the search for
understanding through acts of interpretation (Willis, 2007). In the research reported in
this thesis, the group of interest was pre-university mathematics teachers and the
phenomenon of interest was classroom assessment. Further, the focus was to seek
understanding of participating teachers‟ beliefs about classroom assessment.
65
In the field of education, qualitative researchers gather evidence with the purpose of
disclosing the multiple realities held by the participants of specific educational contexts
(Burns, 1994). Hence, the qualitative researcher is only interested in the „truth‟ from the
participants‟ perspective. It is through the use of qualitative methods such as interviews
and observations that the researcher is able to have access to the individual‟s
constructed meanings in a specific educational setting. According to Burns (1994),
“qualitative methods attempt to capture and understand individual definitions,
descriptions, and meanings of events” (p. 238), hence the selection of a qualitative
approach for the research reported in this thesis.
Research Design
A case study research design was considered to be appropriate for this research because
case studies explore events “within the parameters of openness, communicativity,
naturalism, and interpretativity, as informed by the interpretive paradigm” (Sarantakos,
2005). According to Creswell (2007), case study research is a “study of an issue
explored through one or more cases within a bounded system, such as a context, a
system (i.e. a setting, a context)” (p. 73).
An advantage of case study research is that it involves an in-depth investigation of a
phenomenon with the aim of better understanding the phenomenon of interest (Punch,
2009; Stake, 2005; Yin, 2009). Another advantage is that case study research employs
the use of multiple data collection methods and multiple sources of data, generating a
variety of information that allows the researcher to provide a rich description of the
phenomenon under study (Punch, 2009). A common concern about case study research
is to do with the issue of generalisability (Punch, 2009; Yin, 2009). The concern is
about whether a case study can produce generalisable results from a sample size that is
66
deemed not representative of the broader population. However, in qualitative case study
research, the focus is on the transferability, and not the generalisability of the findings
(Merriam, 2009; Mertens, 2010). The notion of transferability allows the readers to
decide on the extent the case study findings are transferable to their own research
situations. The issue of transferability is further discussed later in this chapter.
In order to address the research questions, an in-depth understanding of the
phenomenon of interest, that is, teachers‟ beliefs about classroom assessment, in the
specific context of Malaysian pre-university mathematics classrooms was necessary
(Merriam, 2009; Yin, 2009; Punch, 2005; Creswell, 2007). The research questions
were:
1. What are the teachers‟ beliefs about the purpose of classroom assessment in the
teaching and learning process?
2. What classroom assessment techniques do teachers currently use?
3. How do teachers‟ beliefs influence their selection of classroom assessment
techniques?
4. What do teachers believe are the reasons for change to their classroom
assessment practices?
5. What do teachers believe are barriers to change in their classroom assessment
practices?
6. What do teachers believe are factors that may facilitate change in their
classroom assessment practices?
A multiple case study design was considered the most appropriate research design
(Bogdan & Biklen, 2007; Punch, 2009; Yin, 2009), as it provided a specific context and
detailed focus on a significant number of pre-university mathematics teachers with a
67
wide distribution of attributes such as age, gender, years of teaching experience and
whether they had received teacher-training. Moreover, multiple case studies present a
variety of outlooks on the issue being investigated. In line with the interpretive
paradigm that underpinned this study, this qualitative case study design allowed the
researcher to gain an in-depth understanding of a specific situation and meanings for
those in that situation (Merriam, 2009), while “aiming to preserve and understand the
wholeness and unity of the case” (Punch, 2009, p. 119). This was an enquiry into a real-
life context. The case study was thus naturalistic and aimed to provide a thick
description using comprehensive data collected from the investigated context (Willis,
2007).
Case study research provides a rich and detailed account of the phenomenon being
studied (Merriam, 2009; Punch, 2009; Yin, 2009). This account may offer new or
further enriched insights that can form the basis of future research. According to
Merriam (2009), case study has an important role in extending and advancing a field‟s
knowledge base. Therefore, the findings of this study will contribute to the knowledge
base of mathematics teachers‟ beliefs in the context of assessment.
The unit of analysis or „case‟ in this study was the individual teacher and the context of
the study was a private college in Malaysia. The multiple case study design adopted in
this study allowed the researcher to understand the investigated issue from different
individuals‟ perspectives (Yin, 2009). It involved the same data collection procedures
for each mathematics teacher. To develop an in-depth understanding of the teachers‟
beliefs about classroom assessment, data were collected from multiple sources
(Creswell, 2007; Yin, 2009) including interviews, non-participant classroom
68
observations, reflective accounts and document collection, as discussed later in this
chapter.
Figure 3.1 depicts the framework for the data collection and data analysis process,
showing the different phases from the pilot study to the generation of themes in the
main study.
69
Figure 3.1 The Research Design showing the Data Collection and Analysis Process
MODIFICATIONS OF
INTERVIEW QUESTIONS
EIGHT CASE STUDY TEACHERS
(Inclusive of Teacher 1 & Teacher 2 from the pilot study)
INTERVIEWS CLASSROOM
OBSERVATION
DOCUMENT
COLLECTION
REFLECTIVE
ACCOUNT
PILOT STUDY
TEACHER 1
(Interview)
TEACHER 2
(Interview)
ANALYSIS
(Interview data)
MAIN STUDY
DATA ANALYSIS
WITHIN-CASE
ANALYSIS
CROSS-CASE ANALYSIS
INDIVIDUAL CASE
STUDY REPORTS
(Chapter 4)
EMERGING THEMES
CROSS-CASE ANALYSIS
AND DISCUSSION
(Chapter 5)
70
Context and Location of the Study
This research was conducted at a single site, one private college located in the state of
Selangor, Malaysia, specifically in the context of a pre-university programme offered by
the college. This private college offers three different pre-university programmes, each
one following an external and internationally recognised prescribed curriculum. These
pre-university programmes provide enrolled students, who have finished their
secondary education, with a foundation for admission into tertiary programmes at local
and international educational institutions. The medium of instruction in all three pre-
university programmes is English. Each of these three pre-university programmes
assesses their students differently. One programme‟s assessment encompasses 100%
external examinations while another programme includes 50% continuous assessment
and 50% external examination. The other programme‟s assessment consists of 70%
continuous assessment and 30% external examinations.
The pre-university programme which was the specific context for this research is an 18-
month programme which is divided into three semesters. The students in this
programme are assessed through two external examinations prepared and marked by the
governing body that provides the curriculum. The examinations are carried out in two
stages; each stage contributes 50 percent of the overall marks. The first stage of the
examination is conducted after the first 12 months of the programme, followed by the
second stage of the examination which is conducted at the end of the remaining six
months. Although this pre-university programme subscribes to an internationally
recognized curriculum, the teaching staff consists of local Malaysian teachers.
The reason for the selection of this college as the site of the research was that the
mathematics teachers of this pre-university programme were required to employ a
71
mixture of formative and summative assessment techniques. The purpose of the
formative assessment is for monitoring the academic progress of the students, even
though the programme is totally examination-oriented. This situation provided a unique
and helpful context for the research because the use of more formative assessment is a
goal of the assessment reform in Malaysia. These mathematics teachers at this college
provided relevant participants for the researcher to gain insight into what it meant to
incorporate this mixture of assessment techniques, while being accustomed to a totally
examination-oriented situation. This situation is similar to what other teachers at
Malaysian schools will face when the assessment reform is gradually implemented into
the current education scenario.
Participants
In qualitative research, the aim is to develop an in-depth understanding of an event or
phenomenon of interest (Creswell, 2012). According to Creswell (2012), it is necessary
for a qualitative researcher to “purposefully or intentionally” (p. 206) select participants
or sites. This form of „deliberate‟ or „purposive‟ sampling is required as there is a
necessity to select participants who are able to provide the researcher with information
to best understand the phenomenon being studied (Creswell, 2007; Creswell, 2012;
Merriam, 2009; Punch, 2005). In purposive sampling, the basis for the selection of
participants is that they are “information rich cases” (Patton, 2002, p. 46). As Patton
(2002) emphasised, “information-rich cases are those from which one can learn a great
deal about issues of central importance to the purpose of the research” (p. 46).
According to Bryman (2008), “in purposive sampling, the researcher samples with
certain research goals in mind” (p. 415). The researcher‟s intent is to select participants
who are able to provide information that is relevant “to the research questions that are
72
being posed” (Bryman, 2008, p. 415). Merriam (2009) emphasised that for purposive
sampling, the researcher determines the selection criteria for choosing the participants
of the study. She also points out that the selection criteria decided by the researcher
“directly reflect the purpose of the study and guide in the identification of information-
rich cases” (p. 78).
As mentioned in the previous section, the selection of the college as the site of the
research and selection of the pre-university programme as the specific context of the
research was purposeful to meet the criterion of a context where a mixture of formative
and summative assessment techniques were employed in an examination-oriented
situation. The research participants were selected from the total population of 17
mathematics teachers who were teaching in the college‟s pre-university programme.
The 17 teachers comprised 13 female and four male teachers. An important ethical
aspect in qualitative research is voluntary participation by the participants in the
research (Silverman, 2010). Of the 17 mathematics teachers, eight were voluntarily
willing to take part when approached by the researcher; therefore they became the
participants of this study.The programme in which these teachers taught provided a
relevant context for this study. Although external examinations were the main form of
student assessment, the implementation of alternative assessment techniques was also a
requirement by the management of the college to gauge the students‟ learning progress.
These teachers were appropriate participants as they have the experience of using
various techniques of assessment in an environment that subscribes only to
examinations as the main form of student assessment. Hence, the experiences of these
eight teachers after having undergone change from a totally traditional paper-pencil
assessment culture to a mixture of traditional and alternative assessment culture
73
provided the researcher with a window into their beliefs about change to their
assessment practices, which was a focus of this study.
The eight participants were a mixture of senior and junior mathematics teachers, with
teaching experience ranging from two years to 30 years. Within the group, there were
various teacher-training backgrounds; some had received formal teacher-training
whereas others had not. However, those with no teacher-training had learnt the art of
teaching through the various teaching positions that they had held throughout the years.
The characteristics of the eight participant teachers are listed in Table 3.1 on the
following page.
74
Table 3.1
Characteristics of Participants
Participant
pseudonym
Gender
Age
Qualification
Major
Teacher
Training
Teaching
Experience
(years)
Jane Female Late
50s
Bachelor of
Science
Diploma of
Education
Mathematics
Mathematics
Yes 30+
Kate Female Late
50s
Bachelor of
Science
Diploma of
Education
Mathematics
Mathematics
Yes 30+
Kay Female Early
30s
Bachelor of
Science with
Education
Masters
Mathematics
Statistics
Yes 7
Wendy Female Late
20s
Bachelor of
Science with
Education
Masters
Mathematics
Statistics
Yes 4
Lulu Female Mid
40s
Bachelor of
Science
Mathematics No 20+
Duke Male Late
30s
Bachelor of
Science
Masters
Mathematics
Statistics
No 18
Cindy Female Mid
30s
Bachelor of
Science
Masters
Mathematics
Statistics
No 8
Anakin Male Late
20s
Bachelor of
Science
Masters
Mathematics
Statistics
No 4
75
Data Collection
Data collection was carried out in two stages; first, the pilot study and second, the main
research, which was carried out using the revised procedures from the pilot study. The
pilot study was carried out from mid-March 2008 to mid-April 2008. Sampson (2004)
and Yin (2009) recommend the use of a pilot study to assist in improvement and
modifications of the data collection process, for example, refinement of interview
questions or the suitability of the data collection techniques selected. During the pilot
study, the researcher had arranged meetings with the college administration to discuss
access to the participants as well as establishing rapport with the administration
personnel and potential participants. This opportunity was used to identify potential
participants for the study and allowed the researcher to explain the purpose and
procedures of the study to these potential participants.
During the pilot study, the researcher was able to recruit two participants who were
willing to be interviewed at short notice. The purpose of these interviews was to trial the
initial interview questions and to identify any modifications that needed to be made to
the questions. All these activities were carried out after the application for ethics
approval was approved by Human Research Ethics Committee at the The University of
Western Australia.
As a result of the pilot study, the researcher identified some interview questions which
needed modifications. The main reason for the modifications was that some of the
questions were vague because of the way they were worded and participants did not
really understand the focus of the question. An example of such a question is,“How do
you define classroom assessment in your mathematics class?” The pilot study teachers
indicated problems with the word “define”. For example, they answered this interview
76
question by giving reasons for carrying out assessment. This particular interview
question was later modified and reworded as, “Can you explain to me what classroom
assessment means to you?” Another example is the interview question, “What are the
factors that you consider when you choose to use a particular assessment technique?”
The teachers struggled to provide an answer for this question. When the researcher
asked the teachers why they found it difficult to comprehend what the question
required, the feedback was the question was too wordy and the word “factors” was
considered rather intimidating. Hence, the question was modified to, “Why do you
choose to use these assessment techniques?”
The researcher maintained constant communication with the potential participants
identified during the pilot study via emails and telephone conversations in order to build
rapport. Through this communication, the researcher was able to secure confirmation of
participation from eight teachers for the main stage of data collection. These eight
teachers were inclusive of the two teachers who were interviewed during the pilot study.
The rapport established between the researcher and the eight participants was
maintained throughout the study through regular contact, again via emails and telephone
conversations.
The main stage of data collection was carried out from mid-August 2008 to mid-
October 2008. This was deemed a suitable period after consulting the potential
participants and college management during the pilot study. There are two main student
intakes in a year for the selected pre-university programme which was the specific
context for this study. The first intake is in January and the second intake is in July.
During the period this research was conducted, the participating teachers were involved
in the teaching of three exisiting intakes, the July 2007 intake from the previous year,
77
the January 2008 intake and the July 2008 intake in the current year. It is usual practice
in this programme that students in the July intake from the previous year and the
January intake of the current year sit for the internal trial examination during mid-
August to early September of the current year. These students are then required to sit for
the relevant external examination in the months of October and November in the same
year.
The researcher asked the eight participating teachers about the suitability of the period
from mid-August 2008 to mid-October 2008 for the main data collection and they all
agreed that during this period they had more free time to be interviewed as their
students would have finished sitting their internal trial examination. All eight
participants were provided with an information sheet explaining the study and a consent
form for them to sign as a formal confirmation of their agreement to participate in the
study (see Appendix A to Appendix D for samples of the information sheets and
consent forms for the institution and participants).
The main stage of the data collection was carried out in three phases. Phase one
involved an in-depth and detailed, individual interview with each case study teacher.
Phase two consisted of classroom observations of each case study teacher delivering
two lessons that included assessment as much as possible. Five of the eight case study
teachers also participated in reflective writing during this second phase. Only five of the
teacher participants were agreeable and comfortable with writing reflective accounts
about classroom assessment. Phase three consisted of extensive document collection
and a follow-up, individual interview with each teacher. Some examples of documents
collected were assessment handouts, marking schemes, and students‟ work. These
multiple sources of data allowed the researcher to build an in-depth understanding of the
78
teachers‟ beliefs, as well as enable triangulation to contribute to the trustworthiness of
the findings. The data collection techniques employed in this study, including
interviews, non-participant observation, document collection and reflective accounts,
are further elaborated in the following sub-sections.
The qualitative research process is said to be emergent (Creswell, 2007; Merriam,
2009), whereby the data collection procedures may be changed or modified as the
researcher is doing fieldwork. The researcher kept a journal documenting thoughts and
issues that arose during the fieldwork. This allowed the researcher to reflect on the data
collection procedures and enabled appropriate modifications to be made to ensure that
good quality and relevant data were collected to achieve the study‟s aim without
compromising ethical standards. An example of a modification that took place during
the data collection process occured when the researcher sought permission from the
participating teachers to video tape classroom activities during the observation. The
teachers were very reluctant to agree to video taping of them teaching. The researcher
documented the difficulties faced when trying to get the teachers‟ permission to
videotape classroom activities. When reflecting on the relevant entries in the journal
which documented the responses to the issue of videotaping, the researcher noticed that
all participating teachers were very apprehensive and self-conscious about being
videotaped. As a consequence, the researcher abandoned the plan to videotape
classroom activities and decided to rely on detailed observational fieldnotes to be taken
during the observation process. This modification resulted from the researcher being
aware of and responsive to the feelings and wishes of the participants, and an alternative
method was developed that would enable adequate data to be collected in order to
answer the research questions.
79
Table 3.2 below provides a summary of the data collected from the eight participating
teachers.
Table 3.2
Summary of Data Collection for Each Participant
Participant
Pseudonym
Interviews
Classroom
observations
Reflective
accounts
Documents
Lulu
3 2 6
Kate
2 2 2
Anakin
3 2 2
Jane
2 2 7
Wendy
3 2 1
Duke
3 2 0
Kay
3 2 0
Cindy
2 2 0
Interviews
The main method of data collection for this study was interviews. An interview is a
good way of probing perceptions, beliefs, definitions and meanings constructed by an
individual (Beswick, 2007; Luft & Roehrig, 2007; Punch, 2005). According to Yin
(2009), interviews are “one of the most important sources of case study information” (p.
106). Interviewing is essential when the focus of the study is on the participants‟
thoughts and beliefs; qualities that cannot be observed (Merriam, 2009; Patton, 2002).
This is possible because interviews allow the researcher to “enter into the other person‟s
perspective” (Patton, 2002, p. 341). Interviews vary in terms of the structure and the
depth of the interview (Punch, 2009); therefore, the type of interview selected should
80
correspond with the research aims (Fontana & Fey, 1994). Semi-structured interviews
were used in this study as a way of probing the case study teachers‟ beliefs about
classroom assessment.
Qualitative interviewing is more open-ended and allows flexible questioning of
participants. The semi-structured interview is less structured and “all of the questions
are more flexibly worded or the interview is a mix of more and less structured
questions” (Merriam, 2009, p. 90). For this study, the interview technique employed
consisted of general questions as a guide (see Appendix E) to allow a smooth flow
throughout the interview, to ensure appropriate data to answer the research questions
were captured and to facilitate data analysis. Merriam (2009) explained that the semi-
structured interview allows “the researcher to respond to the situation at hand, to the
emerging worldview of the respondent, and to new ideas on the topic” (p. 90).
According to Punch (2005), there must be a degree of flexibility as the interview
progresses. This manner of flexibility allowed the researcher in this study to probe
further if interesting issues arose during the interview and follow-up questions were
presented to elicit more information from the case study teachers (Creswell, 2005;
Punch, 2005; Richards & Morse, 2007). The probes or follow-up questions were
developed during the course of the interviews, evolving with the context of the
interviews. The participants‟ responses to the follow-up questions contributed to the
richness of the qualitative data.
A challenge of conducting interviews is when the participant is not able to effectively
articulate his or her views during the interview. In such a situation, the richness of the
interview data may be compromised. Therefore, it is crucial for the researcher to „dig‟
81
further into the responses of such participants with the use of probes or follow-up
questions (Punch, 2010). Time management is another challenge of conducting
interviews as they can be time-consuming (Mertens, 2010). The researcher needs to
consciously keep time during the interview. For example, if the participant is providing
responses that are not relevant or beyond the scope of the interview question, the
researcher needs to bring the focus of the conversation back to the issue that is being
probed.
Prior to the interviews conducted with the participants in the main stage of the data
collection process, the interview protocol was pilot tested to ensure the suitability of the
questions. The two pilot interviews were transcribed and the transcripts analysed. Draft
case studies were constructed for the pilot study participants. Guided by initial findings
from the pilot study, some revision of the interview questions was implemented to
enhance their suitability to illicit rich information of the teachers‟ beliefs in the second
stage of the study.
Each of the participants was interviewed for about one hour prior to the classroom
observations and then for about one hour after each of the two classroom observations.
For the two teachers who were previously involved in the pilot study, their individual
interview prior to the classroom observation during the main stage of the data collection
was a continuation of the pilot interview, but with the purpose of further probing their
previous responses. Conducting three rounds of interviews allowed the researcher to
further explore teachers‟ beliefs and follow-up on the issues highlighted in the
classroom observations. However, three teachers had two interviews because of time
constraints due to their individual teaching time-table. They were interviewed after both
rounds of classroom observations. All interviews were tape-recorded and fully
82
transcribed for analysis, with the consent of the participants. The combination of
interviews and classroom observations strengthened the research rigour by improving
triangulation in the data collection process. According to Mertens (2010), “triangulation
involves checking information that has been collected from different sources or methods
for consistency of evidence across sources of data” (p. 258). The notion of triangulation
is further discussed in the section on trustworthiness later in this chapter.
Non-participant observation
Observation in qualitative research tends to be more unstructured and naturalistic
(Punch, 2009). Some researchers view observation as a technique that provides them
with an opportunity to understand the intricacy of human behaviour (Lichtman, 2010),
uncovering the complex relationships that exist in a natural, social setting such as the
classroom or institution (Marshall &Rossman, 2006) and documenting the incidents as
they occur (Merriam, 2009). There are several reasons for conducting observations in
qualitative research. Firstly, an observer may detect incidents or issues that may assist in
understanding the context under investigation (Merriam, 2009). Secondly, observation
is a way of triangulating other data, that is, it provides a “cross-check on data obtained
in interviews” (Simons, 2009, p. 55). Thirdly, there may be incidents or issues from the
observations “that can be used as reference points for subsequent interviews” (Merriam,
2009, p. 119). Last, but not least, observations allow the researcher to encapsulate “the
experience of those who are less articulate” (Simons, 2009, p. 55) in the research
context. These reasons justified the use of observation as an additional data collection
technique in this study.
The technique of non-participant observation was used in order to “refrain from
interventions in the field” (Flick, 2009, p. 223). A challenge of doing non-participant
83
observation is that the presence of the researcher may cause a change in behaviour of
the participant during observation (Burns, 1994; Creswell, 2009). This was addressed
by establishing good rapport with the participants to help them feel less self-conscious
during the observation.
The non-participant observation of the assessment-related teaching practices of the eight
participants was carried out in the main stage of the data collection. The observations
involved two visits to each mathematics teachers‟ classroom, with their permission, to
observe the activities in the classroom. The researcher approached all the participants
and requested a copy of their individual teaching time-table. Then, arrangements were
made with the individual teachers on the suitable times for the researcher to conduct the
classroom observations. Most of the classroom observations were carried out during a
single teaching period, but there were some observations conducted during a double
teaching period. The selection of time slots for observations depended on the
availability of both the researcher and teacher participant. The classes selected for the
observations included the use of a range of assessment techniques.
The researcher took descriptive field notes as a record of what was observed during the
course of the observation (Creswell, 2005). These descriptive field notes contribute to
the „thick‟ description of the teachers‟ beliefs about classroom assessment. An
observation protocol was prepared to guide the researcher in taking field notes. The
field notes described the events that unfolded in the classroom, such as the interactions
between the teacher and students, the interactions between students and students, the
various kinds of assessment tasks, comments or responses made by the teacher and
students, tasks that teachers had mentioned in the interviews and also tasks that had not
been mentioned.
84
Documents
Another valuable source of data for the case studies was documents (Punch, 2005).
During the fieldwork, the researcher approached the teachers for samples of documents.
These documents were categorised into two types; teaching related documents and
personal documents. Two of the six research questions in this study focussed on the
assessment practices of the participating teachers. The two research questions are “What
classroom assessment techniques do teachers currently use?” and “How do teachers‟
beliefs influence their selection of classroom assessment techniques?” Teaching related
documents provided information about the different assessment techniques that teachers
employed their classroom as well as relevant samples of documents related to these
techniques. Documents such as students‟ work, assessment handouts, marking schemes
and marking rubrics prepared by teachers were considered teaching-related documents.
Entries in teaching portfolios and curriculum vitae were examples of documents deemed
as personal documents. The remaining four research questions sought to identify the
teachers‟ beliefs in relation to the purpose of assessment and change with regard to
assessment practices. Background information of these teachers such as educational
qualification, work experience and positions held were obtained from their respective
curriculum vitae. Most of the teaching portfolios collected from the teachers in this
study contained entries that documented their respective teaching philosophy, views
about the role of a teacher, views about learning, views about assessment, their
achievements and responsibilities as well as future plans for professional development.
Entries regarding assessment in the teaching portfolio also provided information about
assessment techniques that were relevant to the research questions about teachers‟
assessment practices.
85
All these documents were collected and analysed with a view to understanding the
participating teachers‟ beliefs about classroom assessment. Documents were provided
by the teachers either in paper or electronic form.
Reflective Accounts
Another form of data collected for this study was reflective accounts of assessment-
related practices. The teachers were asked to complete a reflective account (see
Appendix F) in electronic form to document their thoughts and feelings about
assessment, their assessment practices and assessment activities carried out in their
respective classrooms. The researcher requested that the teachers send two reflective
accounts each month for six months through email to the researcher. The reason for the
documentation of teachers‟ reflective thoughts was that some teachers might express
their thoughts and feelings better through written correspondence compared with verbal
communication during interviews. In addition, the reflective accounts provided
additional and deeper insights into the teachers‟ beliefs. Only five of the eight teachers
agreed to complete the requested reflective accounts. The other three teachers felt that
this activity was too time-consuming.
Data Analysis
Qualitative data analysis is a “process of making sense out of the data” (Merriam, 2009,
p. 175) and this process is “inductive and iterative” (Lichtman, 2010, p. 188). Data
analysis involved processing the data and drawing out interpretations to address the
research questions. The data analysis process for this study was guided by the
qualitative data analysis framework of Miles and Huberman (1994). This framework
encompasses three components: data reduction, data display and drawing conclusions
and verification. The analysis of all the data sources in the study, which consisted of
86
interview transcriptions, field notes, reflective accounts and documents, was guided by
these three components.
Creswell (2009) suggests that, before the data analysis process, the researcher needs to
“organize and prepare the data for analysis” (p. 185). For example, all interviews were
transcribed by the researcher, and field notes and other sources of data were sorted and
arranged into meaningful clusters. Then, the researcher read and re-read all the data
collected to get an overall impression of the information generated as well as the general
meaning of the data for each participant. Through this activity, the researcher became
more familiar with all the data collected (Marshall & Rossman, 2006).
The first component in the data analysis framework (Miles & Huberman, 1994) is data
reduction. Data reduction refers to the processes of selecting, simplifying, abstracting
and transforming data into themes by coding (Creswell, 2007; Miles & Huberman,
1994). Coding in this research involved assigning labels, tags or names to segments of
data from all sources. Transcripts of interviews, field notes, documents collected and
reflective accounts were read many times to gain a general „feel‟ of the data. During this
stage of data analysis, sentences or paragraphs from interview transcripts, field notes,
reflective accounts and documents collected from participants were segmented. These
text segments were then assigned code words or phrases that best reflect the meaning
portrayed by these text segments (Creswell, 2012). Some codes were actual words of
the participants. According to Creswell (2012), this type of code is referred to as an in-
vivo code. All the data were re-read and re-coded several times with initial codes, also
known as descriptive codes (Miles &Huberman, 1994), being refined and modified
during the course of the data analysis process.
87
The next level of coding in this framework was more analytical. It involved
interpretating and establishing the relationship between elements of the coded data.
These coded data were then grouped and integrated into more meaningful units or
categories, as illustrated by the example in Appendix G. Next, themes were generated
by identifying the relationships between the categories.
The next stage was data display. Data display is the process of organising and
constructing the coded data into suitable graphical representations such as summaries,
tables and mind maps. The purpose of data display is to help the researcher understand
what is going on with the data through visual representation of the data collected (Miles
& Huberman, 1994). In this study, the coded data from the transcripts of each teacher
were displayed in the form of a mind map. These mind maps helped the researcher get a
general insight into the overall view of the teachers‟ beliefs (refer to Appendix H for an
example of a mind map of one of the case study teachers).
The final stage in the data analysis process was drawing conclusions and verification
from the data. The purpose of this stage was “to integrate what has been done into a
meaningful and coherent picture of the data” (Punch, 2009, p. 175). Individual case
studies were generated for five of the eight case study teachers. These five teachers
were selected because they were representative of all the participating teachers and they
had provided data from all four sources including interviews, non-participant
observation, documents, and reflective accounts.
Merriam (2009) mentions that there are two levels of analysis: within-case and cross-
case analysis. In this study, conclusions about the teachers‟ beliefs were obtained by
performing single-case analyses for each case, followed by cross-case analysis of all
88
five cases (Maxwell, 2005; Merriam, 2009). Appendix I presents an example of a
working summary sheet of findings for each individual case study. This summary sheet
was constantly updated as the within-case analysis progressed for each participating
teacher. The purpose of this summary sheet was to provide a guide for the researcher
when writing up the case studies.
For each case, the categories drawn from the data reduction process were the main focus
in performing the next level of analysis which is the cross-case. A cross-case analysis
was then conducted using the categories that emerged from individual case studies as
the basis for this analysis. For example, a category that emerged from the single-case
studies was teachers‟ understandings of assessment as generating information for
identifying high and low achievers. This category then became one aspect of the cross-
case analysis. Figure 3.2 on the next page shows an example of a mind map of the
categories generated from the cross-case analysis of the individual case studies in
relation to the role and purpose of classroom assessment. This form of data display
provided the researcher with an overview of the categories involved for the next level of
analysis which was identifying the emerging themes. Appendix J displays an example
of a summary sheet for categories obtained from the cross-case analysis and collapsed
to generate themes for the role and purpose of assessment.
89
Figure 3.2 Mind Map of Categories Generated from Cross-case Analysis
Trustworthiness
The term “trustworthiness” refers to a set of criteria put forward by Lincoln and Guba
(1989) with regard to the issue of quality in qualitative research. There are four aspects
of trustworthiness including: credibility; transferability; dependability; and
confirmability. The techniques employed in this study to ensure the trustworthiness of
the research are outlined with reference to these four criteria.
Credibility
The notion of credibility refers to the extent to which the data collected has been
„accurately‟ interpreted to correspond with the investigated phenomenon. The
responsibility of the researcher in this study was to ensure that the interpretation of the
90
data reflected the meaning and understanding held by the teacher participants about the
issue being investigated, in this case, their beliefs about classroom assessment.
Triangulation was one strategy adopted by the researcher to ensure credibility of the
data collected. A form of triangulation was the use of various methods of data collection
(Merriam, 2009). There were four different data collection techniques used in this study;
interviews, non-participant observation, documents, and reflective accounts. Another
strategy for enhancing credibility was the use of multiple sources of data. This involved
“comparing and cross-checking” (Merriam, 2009, p. 216) data obtained from classroom
observations as well as interviews conducted with the case study teachers, “with
different perspectives or from follow-up interviews” (Merriam, 2009, p. 216) with the
same case study teachers. All this provided the researcher with more credible, consistent
and complete data from which to develop findings for this study (Creswell, 2007;
Maxwell, 2005; Merriam, 2009).
Another technique for ensuring credibility was to obtain feedback from the participants
regarding the interpretations of their individual data arrived at by the researcher. This
process is known as member-checking (Merriam, 2009). Member-checking is an
activity carried out by the researcher with the intention of „getting the green light‟ from
the participants that the researcher‟s findings and interpretations of the research data is
on the „right track‟ (Maxwell, 2005). All participants were invited to take part in the
member-checking process. Initially, each teacher was sent, via email, an individual
written summary of the researcher‟s findings based on the analysis of the data collected
from each of them. The participants were asked to provide comments and feedback on
these findings with the purpose of “seeking confirmation that the researcher‟s findings
91
and impressions are congruent with the views of those on whom the research was
conducted” (Bryman, 2008, p. 378).
Three participanting teachers provided verbal feedback through discussion via
telephone conversations with the researcher. Two of the participating teachers
commented that they felt overwhelmed by the quantity of the information provided. To
address this concern, the researcher offered to send them, via email, their individual
mind maps as a summary of the researcher‟s findings to facilitate the member-checking
process. Six mind maps were sent to the two teachers; each addressing the research
questions of the study. An example of these mind maps is shown in Appendix H. The
final three participating teachers chose not to participate in the member-checking
exercise, citing heavy workloads and tight work schedules. However, these teachers
were happy for their data to be used for the purpose of the research despite not
participating in member checking. The teachers who participated in the member-
checking process were agreeable to the findings presented to them and were interested
in the data and interpretation of the data by the researcher.
Transferability
Transferability is a term that is associated with the aspect of applicability, that is, the
degree to which the research findings are transferable to other, similar settings (Mayan,
2009; Merriam, 2009). Although the reconstructed meanings and findings in a
qualitative study are specific to that particular research context, there is a possibility that
these findings and meanings “may be common to a larger group” (Baxter & Eyles,
1997, p. 515). Stake (1994) suggests that a particular qualitative study can be viewed as
an exisiting example within a larger group. Hence, the possibility of the transferability
92
of the research findings to similar contexts outside the study is something that can be
considered.
According to Baxter and Eyles (1997), it is the qualitative researcher‟s responsibility to
“provide data which allow for transferability, rather than necessarily demonstrating its
existence, by providing the database on which such judgements may be made by others”
(p. 516). To address the issue of transferability, the researcher needs to provide a
sufficient, detailed, thick description of the research context that allows the reader to
have a comprehensive understanding of this context (Lincoln & Guba, 1985; Shenton,
2004). Shenton (2004) elaborates that background information of the study helps
“establish the context of the study” (p. 73) and thick description of the phenomenon
being investigated assists the reader to “determine how far they can be confident in
transferring to other situations the results and conclusions presented” (p. 70).
To ensure a „high‟ degree of transferability of the findings on teachers‟ beliefs about
classroom assessment in the study reported in this thesis, the researcher provided a thick
description of the study in terms of the research context, participants, processes and
findings. This thick description serves as a guide to help the reader make a judgement
and decide on the transferability of the findings on the participating teachers‟ beliefs to
situations that are similar to that of this reported study (Lichtman, 2010).
Dependability
According to Merriam (2009), the idea of dependability in qualitative research refers to
the issue of “whether the results are consistent with the data collected” (p. 221). The
focus of the issue of dependability is on the “researcher-as-instrument and the degree to
which interpretation is made in a consistent manner” (Baxter & Eyles, 1997, p. 517).
93
Baxter and Eyles (1997) assert that the notion of dependability is “largely concerned
with documenting the research context” (p. 516). Hence, dependability of the research
findings relies upon “the researcher‟s ability to show convincingly how they got there,
and how they built confidence that this was the best account possible” (Richards, 2009,
p. 152).
Lincoln and Guba (1985) suggest that a strategy to ensure the dependability of the
research findings is to develop an audit trail. Merriam (2009) elaborates that “an audit
trail in a qualitative study describes in detail how data were collected, how categories
were derived, and how decisions were made throughout the inquiry” (p. 223). As such,
a direct manner of addressing the issue of dependability in qualitative research is to
provide a detailed report of the steps and processes involved in carrying out the research
(Baxter & Elyes, 1997; Shenton, 2004). In this study, the researcher created an audit
trail by providing detailed descriptions of the data collection process, data collection
techniques, data analysis procedures as well as data analysis techniques used in drawing
out categories from the data. Further, codes are provided in the case studies chapter that
make this audit trail explicit to the reader. Documents and records related to this study,
for example field notes, interview transcripts, reflective accounts, data analysis
summary sheets and mind maps, were collected and kept in a safe, but easily accessible
place (Bryman, 2008).
Confirmabillity
The notion of confirmability, according to Lincoln and Guba (1985), refers to the
“extent to which the data and interpretations of the study are grounded in events rather
than the inquirer‟s personal constructions” (p. 324). According to Baxter and Elyes
(1997), confirmability focuses on “both the investigator and the interpretations” (p.
94
517), unlike dependability which focuses more on the consistency of the research
process. Addressing the issue of confirmability involves taking steps to make certain
that the research findings are “the result of the experiences and ideas of the informants,
rather than the characteristics and preferences of the researcher” (Shenton, 2004, p. 72).
One approach that is used to meet the confirmability criterion is by means of an audit
trail. As mentioned earlier in the discussion on dependability, an audit trail involves
documenting “the researcher‟s decisions, choices, and insights” (Morse & Field, 1995,
p. 144) during the research process.
Although an audit trail is an approach used to address both dependability and
confirmability, the purpose of the audit trail is viewed differently in each of these
criteria. For confirmability, the audit trail is a platform for the reader to examine “the
integrity of research results” (Shenton, 2004, p. 73) by providing the reader means of
tracing and confirming the researcher‟s interpretations back to the relevant source of
data. Shenton (2004) further explains that for dependability, audit trail “allows the
reader to assess the extent to which proper research practices have been followed” (p.
71), focusing more on “how the research was done” (Baxton & Elyes, 1997, p. 517).
In this study, an audit trail was provided to address the confirmability criterion. The
audit trail provided by the researcher allows the reader to understand and track the
research process and analysis of data from the raw data to the point of the research
findings. Each teacher and each source of data relevant to the teachers were given
specific codes for referencing purposes. Therefore, the audit trail makes it possible for
the reader to determine the extent of acceptance of the data that emerged and the
researcher‟s interpretation of these data (Shenton, 2004).
95
Ethical Considerations
The procedure and methods used for this research were submitted and approved by the
Human Research Ethics Committee at The University of Western Australia prior to any
contact with participants. Once ethics approval had been granted, all the participants
were briefed on the purpose of the study and procedures involved during data collection.
They also were provided with an information sheet which contained written information
about the nature of the study and were required to sign a consent form to establish their
willingness to take part in the study. A copy of the signed consent form was given to the
participants and they understood that they could withdraw at anytime during the course
of the study.
The researcher had obtained permission from the Academic Director of the college and
the Programme Director of the pre-university programme to gain access to the teachers
and all relevant resources at the study site. A consent form also was signed by the
management and it was only upon obtaining this consent that the researcher identified
potential participants.
With regard to the relationship between the researcher and participants of this study,
there was no conflict of interest between the two parties. The researcher did not work at
the college and did not have any authority over the teachers. Therefore, a power
relationship was non-existent between the researcher and participants.
To maintain the anonymity of all participating teachers, pseudonyms were used in the
production of this thesis. The teachers themselves determined the pseudonym of their
choice. The five case study teachers reported in the next chapter were each assigned a
set of codes for reference throughout the findings discussion. The first set of letters
96
identifies the data source, the second set of letters depicts the teacher‟s pseudonym, and
the number at the end represents the number of the data source. The set of letters that
represents each of the data sources are: „Int‟ for interview, „Obs‟ for observation field
notes, „Doc‟ for document, and „Ref‟ for reflective account. As for the pseudonym of
the individual case study teacher; LL represents Lulu, KT for Kate, AK for Anakin, JN
for Jane, and WN for Wendy. For example, IntKT1 depicts Kate‟s (KT) first (1)
interview (Int). These codes also facilitated the data trail mentioned previously in the
„Trustworthiness‟ section of this chapter.
Last, but not least, all interview transcripts, raw interview data, field notes, documents
and reflective accounts were stored either in hard copy or electronic forms. All of the
data pertaining to the study were kept under lock and key in a secure office and the
researcher was the only person to have access to them.
Conclusion
This chapter has summarised the methodology that was employed in this study. The
interpretive paradigm that underpinned this study and the reason for a qualitative
approach were discussed at the beginning of this chapter. This was followed by the
rationale for selecting a case study design. An explanation was provided for the
selection of the research site and participants. Further, the chapter outlined the data
collection procedures of the study and elaborated the data collection techniques
employed. The data analysis framework was then discussed followed by issues
pertaining to trustworthiness of the data. Finally, a discussion of ethical issues was
presented. The next chapter presents the findings of the study in the form of five
individual case studies.
97
CHAPTER FOUR
CASE STUDIES
Introduction
This chapter presents the findings of the case study research on the beliefs and practices
of five pre-university mathematics teachers with regard to assessment. The overall aim
of this study was to understand Malaysian pre-university mathematics teachers‟ beliefs
in one college about classroom assessment in a context where the participants were
required by the management of the college to incorporate alternative assessment
techniques in their classroom practices. Each case study describes the participant‟s
beliefs about classroom assessment after they had changed the way they assess their
students from a culture of 100% paper-pencil testing to a mixture of examinations and
alternative methods of assessment.
The five teachers presented as case studies in this chapter were selected because they
represented the diverse backgrounds of the entire cohort of participating teachers. An
examination of Table 4.1 shows that the five selected teachers reported in this chapter
were representative of the eight participating teachers in the aspects of gender, age,
academic qualifications, teacher training and years of teaching experience. Of the five
teachers, one was male teacher and four were female and the age of these five case
studies range from their 20s to 50s. The academic qualifications of the five teachers
included bachelors degrees, diplomas of education and masters degrees. Three of the
five selected teachers had teacher training whereas the other two did not. The teaching
experience of the five teachers ranged from less than five years to more than 30 years.
98
Another reason for the selection of these five teachers was that they were considered the
most information-rich cases because all four types of data were collected from them. All
five were interviewed, observed, provided reflective accounts and relevant documents
whereas the other three teachers decided not to provide reflective accounts as they
perceived this activity as too time-consuming. When the data collected from all eight
teachers were analysed and a case study was constructed for each of the eight teachers,
the case studies of the five selected teachers provided „rich‟ information for the
researcher to address the study‟s aims compared with the information from the other
three teachers‟ case studies.
Each case study begins with a brief background of the teacher participant and continues
with a discussion of the findings which is structured around nine aspects: experiences
related to assessment; beliefs about teaching, learning and assessment; purpose of
classroom assessment; types of classroom assessment; classroom assessment
techniques; beliefs about the need for change in assessment practices; problems and
concerns regarding change in assessment practices; and, suggestions to facilitate
change. The first two aspects presented in each case study emerged from the data and
provide information about the participant‟s experiences and beliefs about teaching,
learning and assessment, allowing the reader to gain an initial understanding of the
background of each participant in their role as a teacher. The remaining seven aspects in
each case study were derived from the key themes of the research questions, thus
providing a systematic discussion of the findings for each research question.
99
Table 4.1
A Comparison of the Eight Research Participants and Subset of Five Case Studies
Presented in this Thesis Based on Different Aspects of their Background
Background Aspect
8 Research Participants 5 Case Studies
Gender
Female
6 5
Male
2 1
Age Range
26-36
4 2
37-47
2 1
48-58
2 2
Qualification
Bachelor
8 5
Diploma
2 2
Master
5 2
Teacher Training
With
4 3
Without
4 2
Teaching Experience
Less than 10 years
4 2
10≤Years<25
2 1
25≤Years<40
2 2
100
Case Study One: Lulu
Background
Lulu was a mathematics teacher in her mid-40s with about 20 years of teaching
experience. She has a Bachelor‟s degree but no teacher-training background. Lulu was
teaching in a secondary school before moving to her current position as a pre-university
mathematics teacher in the private college where the research was conducted. When she
first started teaching at the college, she was teaching in a programme that made
considerable use of alternative and innovative assessment techniques. Lulu was in that
programme for 13 years before being transferred to teach the current programme to
which she was attached to. At the time of the data collection, she had been teaching in
this programme for about 4 years. Hence, she has been teaching in the college for 17
years. Some of her previous significant roles in the institution involved overseeing the
running of mathematics revision workshops and examination seminars, and the training
of peer tutors. She also was the chairperson of the student academic support committee.
Experience Related to Assessment
When asked about her previous experience with assessment as a student, Lulu recalled
that “whenever there is an assessment, to me means I have to study. To me that is the
only understanding” (IntLL1). Lulu‟s statement illustrates that as a student, she viewed
classroom assessment as a tool used by the teachers to „force‟ the students to study.
Lulu further elaborated that her teacher believed that giving class tests was a way to
ensure that students revised topics that had been taught because “I think my teacher
gave me a classroom test or whatever because my teacher wanted me to study that
particular topic.” (IntLL1).
101
As a teacher, Lulu had both similar and different beliefs about classroom assessment
compared with when she was a student. Like her teacher, Lulu also uses classroom
assessment to “make my students study” (IntLL1). Lulu explained that class tests are
given to encourage the students to study because if “there‟s no test they would not
study” (IntLL1). In her opinion, the implementation of classroom assessment helps
create an environment where students will have a reason to constantly study and revise
their lessons. Although Lulu shared the belief that assessment is a tool that can be used
to „force‟ students to study,but more importantly, she strongly believed that classroom
assessment also informed her of students‟ understanding of the lessons carried out in the
classroom. According to Lulu, information about students‟mathematical understanding
is important for teachers to guide the planning of teaching and learning.
Lulu had previous experience teaching a programme which had an assessment system
that was a mixture of examinations and alternative assessment. This assessment system
comprised 50 percent internal assessment and 50 percent external examination.
According to Lulu, the internal assessment was administered by the teachers and usually
encompassed projects or directed investigation activities. This programme provided
Lulu with experience using assessment techniques other than the traditional paper-
pencil tests and examinations.
Teaching, Learning and Assessment
Lulu held a progressive view about the purpose of education. For Lulu, “getting an
education is to prepare them [students] for the real world” (IntLL2). She elaborated that
learning was more than „equipping‟ oneself with the ability to obtain good academic
results. Lulu was of the opinion that developing individual personality is a part of the
learning process. She mentioned that it is important for students to develop their
102
personality rather than just concentrate on grades. Lulu explained that “A‟s don‟t really
count anymore. Everyone can get an A nowadays.” (IntLL2). According to Lulu, for
students to function effectively in the working world, it is essential for them to
recognise their own strengths, weaknesses and fears, factors that make up part of their
personality. Then, students need to learn to improve their weaknesses, overcome their
fears and maximise their strengths because, Lulu argued, an individual is guided by his
or her personality.
When Lulu first started her teaching career, her philosophy was “students‟ success is
my achievement” (DocLL1). To Lulu, her success as a teacher depended on “the
students‟ success at obtaining good results and gaining admissions into good
universities” (DocLL1). She expressed the belief that the main role of the teacher was to
guide students to attain high academic achievement in examinations. However, recently
she had reflected on her teaching practice and had come to realise that teaching is not
just about preparing students to achieve good grades in examinations. As a result, Lulu
experienced a change in the way she viewed teaching and learning. She began to focus
more on understanding, encouraged better interaction among students as well as with
the teacher, and enhanced the use of higher-order thinking skills. The following excerpt
from her teaching portfolio document illustrates Lulu‟s change in view with regard to
teaching and learning.
I started to prepare students for in-depth understanding rather than
superficial awareness of the subject matter, make them participate in higher
order thinking. I make them explore and discover the concepts and theories
rather than adopting them. I ask more than tell in class, and I practise
collaborative learning. I also try to ... build better rapport with them.
Students are finding me more approachable ... they are more interactive in
classes now. (DocLL1)
103
Lulu stated that the role of a teacher in the classroom is to facilitate learning. For
instance, teachers should not just provide answers to questions. Lulu elaborated that
“when a student asks a question, I redirect the question to the rest of the class. If nobody
is able to give me an answer, I would guide them to arrive at the solutions” (DocLL1).
This practice was noted in lessons conducted during both classroom observations. When
there were questions from students with regard to the mathematical problem in
discussion during the lesson or from the previous day‟s homework, Lulu attended to
these questions by illiciting possible steps or methods of arriving at the solution from
the rest of the class (ObsLL1, ObsLL2). Next, she listed all the suggested steps or
methods for discussion on the board. Lulu also offered relevant hints whenever
necessary to assist the students. It is evident in these observations that from Lulu‟s
opinion, finding a solution to a mathematical problem should be a joint effort between
the teacher and students, and not the sole responsibility of the teacher.
According to Lulu, if teachers act as facilitators of learning then students need to be pro-
active in their learning. In most cases, Lulu seemed to try as best as possible to adopt a
student-centred approach to teaching but for classes of weaker students, “my lessons
tend to be more teacher-centred” (DocLL1). The following excerpt illustrates some of
Lulu‟s approaches to provide students with opportunities to play a pro-active role in
their learning.
Certain days and for certain classes, I teach by giving them bits of
information and let them formulate the concepts and ideas on their own. I
encourage them to investigate, probe and to consolidate information and
build on it. (DocLL1)
As a consequence of her belief that education prepares students to „function‟ in the real
world, Lulu held the view that exposure to alternative assessment techniques helps
prepare students to face working life. Lulu pointed out that “when they [students] come
104
out and work, they don‟t work as an individual. They work as a team” (IntLL1). For
Lulu, traditional assessment techniques such as paper-pencil test focus on individual
skills and abilities. Through alternative assessment, however, students learn about team
work, cooperative skills as well as interpersonal skills, all of which Lulu considers
important in the work place. For Lulu, experience with alternative assessment can help
in the development of students into individuals capable of facing the real world.
Purpose of Classroom Assessment
The purpose of classroom assessment for teachers
Lulu was of the opinion that, for teachers, the purpose of classroom assessment is to
provide information to help them select the appropriate teaching pathways and adjust
the pace of their classroom teaching to suit the students. Results from classroom
assessment indicate whether students have difficulty understanding a particular concept
that has been taught. Lulu explained that this information helps the teacher gauge
whether to proceed with teaching the next concept or to repeat a particular lesson, hence
determining the pathway for each sequence of lessons. Furthermore, with reference to
assessment results, Lulu said that decisions about the pace of the lessons can be made,
and teachers are able to determine how fast they should teach so as to allow the students
to follow the lessons comfortably. This is shown in the following quote:
But more on the fact that… I want to test their understanding. I want to
know their level and also because of that I can use that [information] to
adjust my teaching speed. (IntLL1)
Lulu pointed out that classroom assessment is important to teachers as it is a way of
informing them about their classroom teaching. By carrying out classroom assessment,
a teacher is informed of how well the students have understood his/her teaching because
“if there‟s no classroom assessment, I really don‟t know how much they understand my
105
teaching” (IntLL1). According to Lulu, this form of feedback can serve as an indicator
of the quality of the teacher‟s classroom teaching.
Lulu expressed the opinion that classroom assessment can be considered to be an
„identification tool‟. There were two aspects to Lulu‟s view about the role of classroom
assessment as an „identification tool‟. The first was that assessment is a tool to identify
high-achieving and low-achieving students, and the second was it enables immediate
identification of students‟ understanding. According to Lulu, teachers are able to
identify “who are the weaker students, who are the better ones” (IntLL1) from the
compilation of classroom assessment results; hence, the identification of students based
on their level of achievement. Lulu expressed the view that with this information in
hand, teachers are able to attend to the low-achieving students and help them with their
weaknesses and problems in learning mathematics because teachers “have to give
special attention to the weaker ones” (IntLL1).
With regard to students‟ understanding, Lulu commented that it is important for
teachers to be informed of students‟ mathematical understanding and expressed the
view that classroom assessment provides teachers with the opportunity “to assess how
much they [students] know about the mathematical concepts that we have gone
through” (IntLL1). According to Lulu, “it‟s through assessment that we can identify
students‟ weaknesses in which area and where are their strengths” (IntLL1) thus
providing the teacher with a better comprehension of students‟ mathematical
understanding. Furthermore, with this understanding, teachers can “focus on their
[students‟] weaknesses and improve them from there” (IntLL1).
106
Lulu further elaborated that when a teacher asks the question “Do you understand?”
during lessons (ObsLL1), all the students usually nod their heads indicating they
understand. According to Lulu, this may not be true for all students because the students
who do not understand may nod as well. There are also students who do not understand,
but choose to keep quiet. Lulu commented that in situations such as these, without
classroom assessment she would “not know who understand my teaching” (IntLL1).
According to Lulu, the implementation of classroom assessment is important because,
students do not necessarily respond to the teacher‟s questions in the classroom ways that
reflect their true understandings. Lulu commented that in some cases “some students
when you ask them orally they‟re very scared. Even if they know, they might not be
able to answer your question well.” (IntLL1). What this statement suggests is that there
are students who may not be confident enough to respond to the teacher‟s questions.
Further elaboration by Lulu clarifies that she understands that a characteristic of Asian
students is to be fearful of giving their opinions freely, especially in the presence of
someone who is deemed to be of a higher standing, such as a teacher. The following
interview excerpt illustrates this point.
I think that is typical of Asian students. They‟re very afraid to speak in
public and they‟re very afraid to speak to someone of a higher status like
their teachers. If they are talking among their friends, they will not have
any problem. (IntLL1)
In addition, Lulu stated that classroom assessment generates information that can be
used for reporting purposes. Teachers need to notify parents about the students‟
progress because “parents want to know how well their kid is coping in school”
(IntLL1). Lulu also mentioned that information from classroom assessment was useful
in instances where she was required to give forecast results and also write testimonials
for her students, mainly for university applications. When asked how classroom
107
assessment helped in the preparation of forecast results and testimonials, Lulu explained
that information from classroom assessment “serves as a guide to judge a student‟s
ability and knowledge” (RefLL7).
Lulu also expressed the opinion that classroom assessment presents her with many
opportunities to understand and „get to know‟ her students better as individuals. She
pointed out that it was also necessary for her to have knowledge of “the inner side of
them [students], rather than just their A‟s, their grades” (IntLL2). According to Lulu,
teachers “need to know them [students], their opinions, their points of view about maths
and also about life” (IntLL2). She used what she calls non-conventional assessment
techniques, such as journals and reflective logs, to provide opportunities for students to
express themselves. Lulu emphasised that to be able to teach well, a teacher needs to
know her students personally, as shown in the following interview excerpt:
If I don‟t know my students how am I going to teach them? If I treat all of
them as robots, treat all of them as the same individual, then I cannot do a
good job if they‟re treated all the same. (IntLL2)
I feel that by understanding them better, I can approach them, I can teach
them using the approach that is suitable. (IntLL2)
The purpose of classroom assessment for students
As mentioned in the previous section, classroom assessment assesses how much
students know about the mathematical concepts taught in the classroom. As pointed out
by Lulu, information from classroom assessment of students to identify their areas of
strengths and weaknesses. She elaborated that, through classroom assessment, a student
will know the extent to which he or she is able to manage his or her learning of
mathematics, “if there is no assessment ... you actually do not know how well you can
cope in a particular topic” (IntLL1). Students also are able to identify their weaknesses
and concentrate on addressing them in order to contribute to self-improvement.
108
Moreover, through classroom assessment, students are able to identify specific mistakes
they have made. Lulu explained that it is important for students to be aware of the
mistakes they made in previous assessments. This will enable them to learn from these
mistakes so as not to repeat them in future assessments.
The students that Lulu was teaching at the time of the data collection had to sit for two
major examinations during the course of the programme. Lulu emphasised that
classroom assessment serves as a form of preparation for the students in facing their
final examination. For this purpose, the forms of classroom assessment that Lulu was
referring to were traditional paper-pencil tests. The written tests provided students with
the experience of answering and dealing with questions that are similar to those in the
actual examinations. In this context, Lulu views classroom assessment as a platform to
prepare students for examinations.
Types of Classroom Assessment
When Lulu was asked about instances when she carried out classroom assessment, she
explained that she tends to carry out classroom assessment at the beginning of a new
topic. Lulu further explained that this form of assessment helps her identify students‟
prior knowledge or „background‟ knowledge of a particular topic before she starts
teaching. She pointed out that such information is useful to teachers when making
instructional decisions, for example teaching pace, because “if they [students] have a
good background, then you can go through faster” (IntLL1). For example, at the
beginning of the introductory lesson for the concept of optimisation which involves the
application of differentiation in finding maximum and minimum values to solve
mathematical problems, Lulu asked the students what they understood by the term
“optimisation” (ObsLL1). Students raised their hand and provided various explanations
109
of optimisation from their perspective. This activity provided Lulu with a quick
overview of the students‟ understanding of optimisation before continuing with the rest
of the lesson.
Lulu also administered assessment at the end of a topic, or a few topics, for the purpose
of assessing how much the students have learnt and understood the mathematical
concepts. For this purpose, she explained that she “gives them [students] monthly class
tests” (DocLL1). She mentioned that the monthly tests are usually given after the
completion of two or three mathematical topics. These tests provided Lulu with an idea
of the students‟ general understanding of each of these mathematical topics.
Lulu‟s students also are continuously assessed during lessons as well as given work at
different stages of their learning. Lulu actively involves students in lessons through the
use of oral questioning through which she assesses the students‟ understanding
(ObsLL1, ObsLL2). Some examples of Lulu‟s questions are: “What conclusion can you
make from the graphs?”, “What is the purpose of doing this?”, “What do you need to do
next?”, “What relationship do you see between the two graphs?” and “When is the
function defined?” (ObsLL1, ObsLL2). In addition to oral questions, Lulu also assessed
students‟ understanding through the class work such as questions from the textbook or
handouts which she prepares. While students are doing class work, Lulu walks around
the class to check on individual student‟s work (ObsLL1, ObsLL2) because she wants
“to make sure everyone is on the right track” (DocLL1).
Lulu also mentioned that she carries out informal assessment in the classroom. When
asked to explain her view about the difference between formal assessment and informal
assessment, Lulu made a clear distinction between the two. Formal assessment involves
110
recording of marks or grades, whereas there is no record taking in informal assessment.
Lulu further elaborated that informal assessment is carried out just to “see how well
they understand the topic rather than giving them a grade for the assessment” (IntLL2).
Lulu said an example of formal assessment is the monthly class test she gives, while
oral questions and short quizzes are examples of informal assessment.
Classroom Assessment Techniques
There are a variety of assessment techniques that can be adopted by teachers in their
respective classrooms. Lulu mentioned that she has been using an array of assessment
techniques in her classroom for the past three years. During the interview, Lulu talked
about 12 different assessment techniques that she employs in her classroom. These
included oral questions, multiple-choice questions, mathematics journal, feedback
forms, projects, quizzes, reflective logs, peer assessment, written class tests,
presentations, directed investigation and mathematics games.
As stated earlier, Lulu regularly used oral questions during lessons to gauge students‟
understanding of the mathematical concept or topic being taught. Examples of the
questions that Lulu uses in the classroom were listed in the previous section and these
questions focus on higher-order thinking. Students‟ responses provide Lulu with an
overview of the level of their understanding in relation to the concept or topic that they
are learning. In addition to oral questions, written class tests which are given on a
monthly basis were also mentioned. These tests usually assessed students‟ overall
mathematical knowledge in relation to two or three topics. Lulu also used quizzes or
multiple choice questions to test simple mathematical concepts as well as to get
immediate feedback on students‟ comprehension of these simple concepts.
111
In one of the interviews, Lulu mentioned that she had introduced mathematics games as
an alternative form of assessment (IntLL1). She pointed out that mathematics games can
help strengthen and assess students‟ understanding of mathematical concepts. She
further elaborated that she had designed a mathematics game for the topic of
differentiation. Lulu remarked that when students solve the questions given, she can
gauge their comprehension of differentiation. She further explained that “the students
will enjoy the game but at the same time, it will reinforce their concept of
differentiation. They have to do it very quickly because they are competing against one
another in the game” (IntLL1). Mathematical games create an atmosphere of
competitiveness among the students in their attempt to complete the game as quickly as
possible. Lulu said students usually come to class well-prepared to compete and during
the game, they also learn from their friends.
A unique assessment activity that Lulu had introduced in her mathematics classroom
was the directed investigation activity. This technique consists of structured questions
which go beyond the syllabus. Students are required to answer questions which will
finally help them come up with a conclusion or formula for the given situation. When
asked why she had introduced a form of assessment that went beyond the required
syllabus, Lulu explained that she wanted the students to experience something different
which allowed them to “go beyond the level that a student is expected to know”
(IntLL1). As a result, students would be able “to think globally” (IntLL1) and are not
just confined to familiar situations within the syllabus.In Lulu‟s opinion, the
examination questions just require the students to “vomit out” (IntLL1) what they learnt
in the classroom. She is more concerned about giving her students different learning
experiences through a variety of assessment techniques that might prepare them for
112
higher education, “I told them that I am preparing you for university ... so far they have
been doing well” (IntLL1).
Another assessment technique that Lulu had introduced was the reflective log. The
reflective log (DocLL2) is a document that students use to monitor their progress in the
examinations set by the college. Lulu explained that the students fill in the reflective log
after every examination and set targets for themselves in all forthcoming examinations.
They make comments about their achievement in relation to the targets set. Lulu‟s
students also made suggestions about how to improve their achievement for the next
examination. Lulu‟s role was to provide feedback on all comments and suggestions
made. She is able to identify her students‟ problems and this helps her understand them
better; “they get to tell me what is the problem ... what they need to do to improve”
(IntLL2). According to Lulu, the reflective log scaffolds students in the process of self-
evaluation.
In addition to reflective logs, Lulu has asked her students to keep a journal. One reason
Lulu introduced journal writing as an assessment activity was because as mentioned
earlier, she was required to write testimonials for her students. In order to provide „fair‟
testimonials, she explained, there are “a lot of things that I need to know about them
personally” (IntLL2). Moreover, Lulu pointed out that she lacks the opportunity to
become more acquainted with her students when she is in the classroom. This opinion is
illustrated in the following statement.
And also ... usually when we go to class we don‟t have the time to know
them well. So it‟s through this maths journal writing that I get to know them
better. What is their fear, the affective part of them, not just about their
maths content part. (IntLL2)
113
Lulu provides her students with a question and they are expected to jot down their
thoughts or feelings regarding issues raised by the question. There are two kinds of
questions that Lulu gives for the journal writing; the first is based on the mathematical
content being taught and the second, as Lulu puts it, is “the affective part” (IntLL2). The
students are given one week for a question that is categorised as being affective, and
two weeks for a question related to mathematical content. These short, regular entries
provide Lulu with a better „picture‟ of each student. Lulu gives feedback to the students
by writing her comments next to their journal entry. The following accounts illustrate an
example of each type of question.
For example, in one of the question I ask them to describe their feeling
when they come to the board to do work. I want to know whether they have
fear, whether they have confidence and so on. ...some students said “Oh, I
don‟t fear anything at all”. Some students said, “I‟m very worried because
I‟m so scared of making mistakes, you know.” ...I will write down the
comment do not worry about making mistakes, you know. It‟s better that
you learn from your mistakes now than later. (IntLL2)
For example, the first topic is on quadratic. ...What is their favourite method
and why do they particularly like this method? For GP [geometric
progression] and AP [arithmetic progression], I ask them to give an example
...in their daily life, rather than just questions that are so theoretical that they
see in the textbook. (IntLL2)
Projects are another form of alternative assessment that Lulu has introduced to her
students. According to Lulu, she used projects to assess the students‟ understanding of
the topic on statistics and that the statistics project was a group project. Each group was
given the freedom to determine their own topic of interest with the condition that was
suitable for data collection. Lulu provided each group an instruction sheet with specific
guidelines (DocLL3) and briefed them on the procedures.
First of all, I give them instruction what they need to do. Like you know,
they have to collect data ... what they need to do throughout the procedure,
and then I give them an activity plan, so that I give them a rough guideline
for each week. What are the activities involved for each week? So that they
can get their work done on time. They can complete their project on time.
(IntLL3)
114
Lulu explained that the group project offers students the opportunity learn about and
experience “the importance of collaborative learning and team work” (RefLL1), skills
which she feels are crucial for students to have in order to be effective employees in the
workforce. Lulu further explained that besides data collection, the project also involved
processes such as data analysis, data interpretation and forming a conclusion for the area
of inquiry that they had selected to study. Each group was required to prepare a report
of the project and this report had to be submitted to the teacher. Lulu noticed from her
observations in the classroom that being involved in the statistics project helped
encourage students to be creative.
Well, some of them can come up with very interesting interpretation. So,
they never fail to amaze you. Sometime they are better than us when it
comes to creativity, ... not all of us are very creative. Some students are
extremely creative. They can come up with ideas that are really creative and
innovative. They can come up with ideas that shock you. (IntLL3)
Presentation and peer assessment are two other forms of alternative assessment that
were employed by Lulu in the classroom by incorporating both forms into the group
statistics project. Besides handing in a compilation of written work for marking, each
group also was required to do an oral presentation of their respective projects and the
rest of the class was given the opportunity to assess their classmates‟ presentations. This
was Lulu‟s initial step to introducing peer assessment to her students. The peer
assessment is conducted as a group task and marks are based on a rubric (DocLL4)
provided by Lulu. The following excerpt from a reflective entry by Lulu describes how
the oral presentations were carried out during one of her lessons.
Students did their presentation today. There were seven teams. Each team
was given 15-20 minutes. Their presentations were so impressive with all
sorts of animations. Most of them inject humour into their project too. I
made them sit with their team members in class while the presentations
went on. The teams started off with the team leaders introduced the
members, followed by giving an introduction to their survey such as the
objective and significance of the study, location of the study and sampling
techniques used. The second members would present the data in various
115
forms (stem-leaf, cumulative frequency, box-whisker, histogram) the third
members do the comparison between two sets of data (female and male)
using calculations of measure of locations, standard deviations etc. The last
members present the summary of findings and conclusions. (RefLL6)
The Need for Change in Assessment Practices
Lulu expressed the opinion that there was a need for change as the current assessment
system in the pre-university programme she is teaching has too much emphasis on rote
learning and “the current way of assessing the student is just testing them on their
memory” (IntKT1). According to Lulu, the overall assessment of students should also
include some coursework to allow teachers to assess the students‟ progress
continuously. Lulu stated that a single examination is not a fair measure of students‟
real mathematical ability. She explained that coursework, comprising alternative
assessment activities, will allow skills such as students‟ communication skills to be
assessed in mathematics, which is not possible through traditional paper-pencil
techniques. To elaborate, students should be made to do presentations to allow them to
interpret and draw conclusions from mathematical results, and then communicate this
information to others. The following responses from Lulu illustrate her beliefs about the
need for change.
One exam is not enough to determine a student‟s success or failure. It‟s only
through continuous assessment that you can test a student‟s real ability.
(IntLL1)
Well, you have to assess them continuously on every topic through different
forms of assessments like project, quiz, presentation, rather than just written
exam at the end of the year, at the end of the course. (IntLL1)
In Lulu‟s opinion, mathematics assessment should assess students‟ analytical and
interpretive skills. She reiterated that learning mathematics or mathematical problem-
solving is not just about getting the correct answers. She mentioned that mathematical
algorithms and getting mathematical results can be done with the use of today‟s
116
technology; “With all the mathematical software, any question can be answered using
software, for example graph sketching. Even if you want the student to integrate, even a
simple calculator can do the job” (IntLL1).
Another point that Lulu made was that with the current assessment system, students
learning mathematics are only interested in the solutions. The important thing for the
students when they face an assessment is getting the correct answers. In Lulu‟s own
words, “they are not willing to think on their own” (IntLL1). Lulu pointed out that
change is necessary to get students to go beyond focussing only on the correct solution
and to better encourage them to think about the process of mathematical thinking. She
further elaborated that by enhancing students‟ ability to think “we actually prepare them
for the real world because in the real world you don‟t just solve mathematical problems,
get the answer and that‟s it. They have to look at things from different angles
[perspectives]” (IntLL1). For Lulu, there is a need to check their understanding by
means of triangulating different assessment techniques.
Problems and Concerns about Change in Assessment Practices
Prior to the research being conducted, Lulu taught in a secondary school where the main
form of assessment was written class tests. When she moved into a programme which
required her to adopt a mixture of traditional and alternative assessment techniques,
there were hurdles which she had to overcome. She is of the opinion that if the current
assessment system is reformed, teachers will face similar problems to those she
experienced during this transition.
The main barrier Lulu envisioned was the teachers‟ fear of change because they would
have to learn something new and unfamiliar. She felt teachers will be worried about
117
whether they will be able to grasp the different unfamiliar assessment techniques as they
are already „comfortable‟ with the traditional written class tests. This could create an
atmosphere of „unwillingness‟ among teachers to learn different assessment techniques.
Most of the teachers are not exposed to the different kind of assessment.
Most of them are more used to the written test. So if they do not know, they
definitely cannot impose it on the students. In the first place, the teacher has
to be trained on the different kinds of assessment before they can do it with
the students. If a teacher is not willing to learn, then of course the teacher
will not be happy with the change. (IntLL1)
Lulu expressed the opinion that some teachers hold a strong belief that, for
mathematics, the written test is still the best method of assessment. She explained that
most of these teachers are very comfortable with the traditional paper-pencil class tests
as this form of assessment is easy to carry out and marking is simple. Also, teachers do
not need to spend too much time on preparing written tests. Therefore, “if they are not
willing to accept new changes, then of course it will give problems” (IntLL1).
Lulu also explained that to set questions for some assessment techniques, for example,
the directed investigation activity is “extremely, extremely difficult” (IntLL1). For
activities such as projects, Lulu explained that the teachers need to determine and
organize the activities to be carried out throughout the project. Teachers need to spend
extra time to carry out appropriate preparation and, according to Lulu, “most teachers
are not willing to do so” (IntLL1).
Lulu also pointed out that there is the problem of the „reliability‟ of students‟ work.
Teachers sometimes feel that they do not really know whether the students actually
completed homework tasks through their own effort. Therefore, teachers sometimes feel
that the marks given to students for work such as projects and group work might not be
reflective of the actual effort put in. For example, for group work, some members may
118
not have contributed to the work done. This is another barrier that Lulu mentioned that
needed to be addressed before successful assessment reform. The following excerpt
from Lulu‟s reflective account illustrates this point.
A group of students came up to tell me that one of the team members is not
contributing. He is in charge of drawing the box and whisker plot but has
done it wrongly. When the rest asked him to redo it he again did it wrong. I
told them I would talk to the boy. (RefLL5)
Suggestions to Facilitate Change in Assessment Practices
Although articulate about these barriers, Lulu was of the opinion that they can be
overcome. She emphasised that change has to be gradual; “we have to do it [change]
step by step” (IntLL2). According to Lulu, the first time something new is implemented
it is always difficult, but as time goes by, the implementation will be easier. As Lulu put
it, “we have to”.
But the first round, the first step is very difficult. But once you‟ve overcome
the first step, it would be very easy. I think for most people the fear is to
cross the first hurdle. But once you can cross the first hurdle it‟s alright.
(IntLL1)
Lulu suggested that pre-service teachers must be given early exposure to and hands-on
experience with various assessment techniques. She reiterated that if teachers
themselves are not familiar or „comfortable‟ with the various assessment techniques,
they will find it a challenge to implement these techniques in the classroom because “if
you do not know something, you cannot practice it” (IntLL1). As for in-service
teachers, Lulu suggested that they must be sent for relevant, adequate and context-
specific training on the different forms of assessment techniques. This would enhance
their confidence when implementing these techniques in the classroom.
119
Another suggestion provided by Lulu was that school management should promote
teacher peer-sharing sessions where teachers who are well-versed with the different
assessment techniques can offer advice and support to „novice‟ teachers in this regard.
Lulu explained that she learnt a lot about different assessment techniques from
colleagues who were familiar or had been exposed to these different techniques. For
example, Lulu came to know about mathematics journals from a colleague who gave
her a sample of an entry written by a student.
In addressing the „reliability‟ of students‟ work, Lulu suggested that certain parts of
assessment tasks, such as projects and group work, should be done in the classroom.
This would allow teacher supervision and provide a comparison for aspects of
individual work. Lulu also proposed that members in each group assess each other
because “when they work as a group they know best who is working and who is not”
(IntLL2).
Lulu also stressed that if the examination requirement was changed to include assessing
students in various ways, then the students must be encouraged to accept other
assessment techniques beside the usual written class tests. Parents‟ support is equally
important to teachers when implementing a new assessment system. Teachers will face
fewer problems from the public if parents are made aware of the advantages in assessing
students with a variety of assessment techniques.
Of course parents also have to give support. If the parents do not believe in
the new way of assessment, then the teachers will have a problem too. To
make sure they do not complain about the different ways of assessing the
students, we have to make it clear to them that this will bring benefit to the
students. (IntLL2)
120
Summary of Lulu’s Beliefs
As a student, Lulu viewed assessment as a tool used by her teachers to make sure
students revised their work. At the beginning of her teaching career, she associated her
„success‟ with the grades her students achieved in their examinations. Slowly, this view
changed and Lulu began to emphasise understanding and connection to the „real world‟
as the focus of an education. Hence, she now feels that students must not be assessed
solely through the traditional paper-pencil testing.
Lulu used a wide range of non-traditional assessment techniques with her current
students. Lulu expressed the importance of developing learning relationships with her
students through understanding their learning needs. These classroom assessments
provide Lulu with information that helps her plan more effective lessons for students, as
well as informing her of the quality of her teaching. Lulu also uses assessment as a way
of identifying the academic ability of students; whether they are high-achievers or low-
achievers. This, in turn, makes it possible for her to determine which students need
further attention and assistance with their lessons. Alternative assessments, according to
Lulu, give her an opportunity to „know‟ her students better, academically and
personally. She mentioned that students can use information from classroom assessment
to guide their studies, to monitor how well they are doing.
Lulu was supportive of change in assessment practices for a number of reasons, but
mainly because the current system focusses too much on assessing the students‟
“memory work”. According to Lulu, the main issue that will underpin problems that
may arise when changing assessment practices is the fear of changing from something
familiar to the unknown for teachers. Accordingly, the main suggestion Lulu made to
121
address this concern was to provide teachers with adequate and relevant professional
development on different assessment techniques.
122
Case Study Two: Kate
Background
Kate was a mathematics teacher in her late 50s. She has a bachelor degree in
mathematics and also holds a Diploma in Education, with a major in mathematics and a
minor in physics. Kate has about 30 years of experience teaching mathematics. She had
been in her current position as a pre-university mathematics teacher for about six years
at the time of data collection. Kate had the opportunity to hold the position of
Chairperson of the Mathematics Nucleus Group which involved planning activities for
all the mathematics teachers in the pre-university programmes in the college. Her
previous teaching experience included teaching in a government school, a private school
and an international school, thus exposing her to various educational programmes.
According to Kate, however, her past teaching experience mostly involved examination-
oriented programmes. Therefore, she had very limited experience with alternative forms
of assessment.
Experiences Related to Assessment
As a student during the 1950s and 1960s, Kate experienced an education system that
was very examination-oriented. One of her mathematics teachers likened the then
education system to a pyramid, where the good students were the ones who “got
through the system” (IntKT1). Selection to educational programmes and courses was
based solely on performance in examinations. Due to the strong emphasis on
examinations, Kate described the assessment process as a „filtering process‟ because
“assessment, especially examinations was like a filter. By the time you reach the top,
there are few [students] left.” (IntKT1). Students who did not pass the examinations at
different stages of the education system dropped out of school.
123
Life was very competitive when Kate was a school student as there were limited
resources as well as a limited number of colleges and universities for further studies. A
point of interest brought up by Kate was that education was taken very seriously
because it was a “way of social mobility” (IntKT1). This meant that an individual who
could attain a good education was able to “step into a different lifestyle” (IntKT1).
Obtaining a good education, therefore, meant getting good results in examinations. It is
evident that the environment that Kate grew up in may have impacted on her beliefs
about the importance of assessment which she associates very strongly with traditional
paper-pencil class tests. In her opinion, her students are still experiencing a similar
examination-focused environment to those she experienced as a student. Kate
mentioned, “out of 30 years, it‟s the same system I myself was in, and now my students
are in” (IntKT1). Examinations and class tests are still the main forms of assessment for
students in the current education scenario in Malaysia. Achievement in examinations
was still the main criterion for selection to an array of educational opportunities.
Teaching, Learning and Assessment
Kate said that she considered mathematics to be an an important subject because
students who learn mathematics learn “how to be analytical” (IntKT1). Kate deemed
analytical skills to be essential for students when studying at university. She explained
that, in mathematics, students need to learn how to analyse and apply formulae and not
just memorise them. Mathematics learning engages students in determining which
method is most appropriate for solving particular kinds of problems. Kate expressed her
belief that students not only gain good analytical skills through learning mathematics
but also enhance their ability to cope with different situations. The following comment
illustrates this viewpoint.
124
What they should learn is the analytical part of it rather than memorizing the
formula. The application part they should learn… They should be able to
analyse what is the correct way of doing it…I feel that if a person is
analytical, no matter which field they are in, they should be able to cope
with it. (IntKT1)
For Kate, communication was central to teaching and learning mathematics. Kate
pointed out that the ability to communicate mathematically involves solving problems
through the use of formulae and the skill to infer. Students must be able to identify and
apply related concepts and information to solve mathematical problems. Kate explained
that mathematical communication is not only through “using words and writing”
(IntKT1). The following interview excerpt further illustrates Kate‟s point.
…this is two points. Find the equation of the plane that passes through the
midpoint and perpendicular to this line….able to communicate to me means
they know how to get the normal, how to get the point on the plane.…I take
it as communication. (IntKT1)
Kate also expressed her belief that teachers also need to possess good communication
skills. She is of the opinion that “for effective teaching, a teacher should have a good
knowledge of the content of the course we teach and possess the ability to communicate
well” (DocKT1). With good communication skills, teachers are able to “help students
understand the fundamental concepts” (DocKT1).
Kate mentioned that the role of a teacher is “to prepare students for examinations and to
obtain excellent results in examinations” (DocKT1). She further elaborated that “one of
my main objectives in teaching is to ensure students learn the fundamental content of
the course we teach and complete the course successfully and be ready to pursue higher
education” (DocKT1). Therefore, from Kate‟s point of view, her responsibility as a
teacher was to ensure students succeed in their examinations which, in turn, would
provide them with the opportunity to progress to higher education.
125
As mentioned previously, Kate felt that students today have a similar assessment
experience to that which she experienced when she was a student. When asked to
elaborate on her views of mathematics pedagogy today compared with the earlier years
in her career, however, she explained that although the content is still the same, there is
a difference in the learning and teaching of the content. To her, mathematics was
grounded in its basics and the learning of mathematics involves progressing to more
sophisticated levels.
Without level by level build up, you can‟t learn maths… Last time, you
teach them three plus three using finger counts, but maybe now there are
other ways of teaching. But eventually, the things you learn are the same.
(IntKT2)
Kate said she was not as concerned about the way the students learn, as with what they
learned. What was important for her was the results of their learning. “How you learn it
I don‟t really care, but at the end, the result has to be there… give me the correct
answer” (IntKT1). To ensure the desired results are produced, Kate focuses on teaching
students the formulae and guides them through “the steps of doing it” (IntKT1). An
example of this was observed when she was discussing questions from a past
examination paper with her class (ObsKT1). Kate explained to the students what was
required by the question and proceeded to write the solution on the board. This was
followed by her thorough explanation of every step in the solution, which she said to the
students were important to write down and she also indicated where the marks were
allocated (ObsKT1). The observation indicated that teaching of mathematics emphasises
familiarity with the appropriate routine of solving a particular problem.
Kate expressed her strong belief that examinations and class tests are very important to
establish whether students have done well or not. Traditional paper-pencil assessments
126
are her main source of information about her students‟ learning. The following
interview excerpt illustrates this point.
For me, I feel examination is very important to assess the student, you
know? Otherwise, how do we distinguish whether the student, this student,
has done very well, this student has not. … I take class tests very seriously
all the time. (IntKT1)
When asked to compare mathematics assessment and assessment in other subjects, Kate
expressed her opinion that mathematics assessment and science assessment are “almost
the same” (IntKT1). Both subjects are logical and involve the explanation of concepts.
Mathematics and science are subjects which deal with facts. To Kate, there is a clear
distinction between right and wrong when answering mathematics questions. In that
sense, answers are seen as „absolute‟. She stated emphatically:
Right means right and wrong means wrong…clear cut between right and
wrong in answering questions…the answer is fixed in that sense. (IntKT1)
At the time of the data collection, Kate taught in a programme that followed a fixed
syllabus. She explained that she adheres to the syllabus “very, very closely” (IntKT1)
and “does not go beyond” (IntKT1) the syllabus. The mathematics assessment that Kate
implements in her class consists of an array of questions that are varied in terms of
levels of difficulty. She also stressed that these questions should always be within the
scope of the syllabus. Past years‟ test questions serve as a source of reference in
determining suitable items in the mathematics assessment.
Purpose of Classroom Assessment
The purpose of classroom assessment for teachers
According to Kate, classroom assessment provides teachers with information about their
students‟ understanding of the mathematical concepts taught. To emphasise this point,
Kate mentioned that there were times she assumed the students understood certain
127
concepts taught in a lesson. When the students were given class tests, however, the test
results indicated that they had not “exactly grasped” or “exactly understood” (IntKT1)
the concepts she had explained. Classroom assessment provided her with a clearer
„picture‟ of the actual situation of the students‟ understanding of these concepts, and
this made her realise that she should not make assumptions that students always learn
what is being taught:
One time my student, when he saw something like cos B over cos A, he
cancelled the two cos. Left B over A only. This is not right. How can cos
cancel cos? That‟s why we think they understand but they go and do wrong
things like this. (IntKT1)
Another matter of concern for Kate was the extent to which students were able to retain
their understanding of taught concepts after a certain period of time. In her words,
“…after a certain period of time, you do not know whether they can retain, whether they
exactly understood what we‟ve taught and whether they retain that or not” (IntKT1).
She further explained that class tests, given after a certain period of time after students
learn a particular concept, provide teachers with information about their ability to retain
understanding. This information added to the „picture‟ she developed of her students‟
understanding of the lessons.
Classroom assessment also gave Kate a better understanding of the “standard of the
class” (IntKT2), by which she meant the students‟ level of mathematical understanding
as a whole class. This information was essential to her with regard to new classes she
was teaching. Students are admitted into the programme Kate was teaching at the time
of the interview based on their forecast results. Although she had records of their
previous achievements and forecast, she pointed out that it was crucial for her to be
aware of the current standard of the class:
128
Especially new classes. We want to know. Ok, we can look at their past year
grade, I mean their past exam forecast and everything from other school.
But we still have to know the standard of the class. Where they stand, you
see. (IntKT1)
Kate emphasised that classroom assessment also serves as a source of information to
guide adjustments or improvements in the teacher‟s instructional plans and decisions
that would be more beneficial to students. She explained that this role of classroom
assessment has three aspects. Firstly, teachers are able to adjust their teaching pace to
suit the students. For example, for more able students, the teacher can teach at a faster
pace. Secondly, teachers can decide on a more appropriate selection of mathematical
examples used in lessons with „good‟ students. The teacher can give harder examples
and discuss “more challenging questions” (IntKT1). Kate was of the opinion that this
would be particularly beneficial to students with higher mathematical ability. Thirdly,
teachers are able to appropriately modify the presentation of lessons; for example, the
sequence in which the mathematical examples or mathematical concepts were to be
taught for a particular topic; as well as revise the teaching approaches adopted in the
classroom. To Kate, these instructional restructurings are important for the enhancement
of students‟ understandings of lessons. The following comment illustrates these points.
... maybe I shouldn‟t have put the example this way. Maybe I should have
started with a simpler example. Maybe I should have explained it from a
different view point. (IntKT1)
Kate claimed that a further purpose of classroom assessment is to aid teachers in
identifying students‟ areas of strength and weakness. According to Kate, it is important
for teachers to recognise in which topics or fields of mathematics the students are “good
or weak” (IntKT1). For example, she was able to identify, through classroom
assessment, that one of her students was proficient in pure mathematics but had
difficulties in statistics. Kate expressed the view that when teachers are able to
129
recognise students‟ areas of strength and weakness, they are able to “guide them
accordingly and not blindly” (IntKT1). Teachers can determine which areas students
need further assistance with and what other suitable exercises should be made available
to the students to address their weaknesses.
Information from classroom assessment also assisted Kate in identifying the high-
achieving and low-achieving students. Teachers are able to identify “who are in the
better category, who are in the weaker category” (IntKT1). Therefore, teachers are able
to identify and group students based on their level of achievement as well as identify the
extent of their individual academic achievement.
Kate further explained that sometimes during her lessons she requests the help of high-
achieving students to assist her with the low-achieving students. This helps her address
the issue of trying to attend to every student‟s enquiries during lessons. Indirectly then,
classroom assessment helps Kate with her classroom teaching.
You‟ll know who to ask to help another student…you know A is good and
B is weak. Sometimes when you‟re explaining to another student,…ask A
to help B. (IntKT2)
Further, Kate uses information from classroom assessment for reporting and forecasting
purposes. Parents should be able to “see the progress of their kids” (IntKT1). Kate felt
that information such as this was crucial for parents to have as “some may think that
their kids are performing very well” (IntKT1) when in fact they are not. This would
avoid any misunderstanding between the teacher and parents regarding their child‟s
achievement. In the programme that Kate was teaching, the external examination body
required teachers to give a forecast for each student which serves as a guide in the
marking process of the final examination. For Kate, information from classroom
130
assessment guides her decision when providing these forecast results because “we
cannot simply forecast” (IntKT2). The forecasts are confidential and sent back to the
external examination body about a month before the final examination.
Even though Kate explained her view that mathematics assessment involved evaluating
whether the answers are correct or not, she also expressed the belief that, to a certain
extent, mathematics assessment should also draw attention to students‟ “way of getting”
(IntKT1) these answers. According to Kate, the correct answers are the final products in
mathematics assessment, but the “the way of getting it, the steps of getting it, the
method of getting it” (IntKT1) all need to be given attention. The display of techniques
and approaches employed by the students in mathematics assessment represents the
processes of their thoughts in getting the answers.
Kate further elaborated that the presentation of solutions in a mathematics assessment
provides a visual display of the students‟ “state of mind” (IntKT1). If a student is able to
present a systematic set of steps to the answer, this implies that the student has a “clear
logical mind” (IntKT1). On the other hand, when students display a mixture of right and
wrong steps, lots of cancellation or writing steps that are not in line with the context of
the question, then this shows that these students are in a “very confused state” (IntKT1).
According to Kate, this may be due to a poor foundation in mathematics or lack of
understanding. She can then attend to this issue in subsequent lessons.
Kate expressed her belief that information derived from her students‟ performance in
classroom assessment can guide improvements and adjustments for future assessments.
If the students had performed very well in a given test, then teachers can “give a harder
test next time” (IntKT1). Teachers can rate the level of difficulty of the class tests they
131
administer in class by referring to the students‟ results of these class tests. This
information would enable the teachers to modify the level of difficulty of the
assessment items in future assessments, so that these items match the mathematical
ability of the students in the class.
Finally, Kate stated that she views classroom assessment as a way of preparing her
students for their external examinations. The students Kate was teaching at the time of
the interview were required to sit two major examinations during the course of their
study. As mentioned previously, Kate expressed a strong opinion that it was important
for her to prepare her students for examinations. She stated this explicitly: “My main
intention is to train my students for exams” (IntKT1).
The purpose of classroom assessment for students
According to Kate, it was important for the students to be informed about their
understanding of the mathematical concepts they had learnt in class. Classroom
assessment provided them with such information. Kate mentioned that some students
assumed they possessed good mathematical knowledge and, therefore, underestimated
the level of difficulty of the course. Hence, these students did not put sufficient effort
into their studies. This led to under-achievement in class tests. Classroom assessment
for Kate, therefore, served to inform students of their „actual‟ understanding of the
lessons.
Because some students they thought they‟re very good. So they
underestimate the course. They don‟t put in effort. Then they don‟t perform
well in the test. (IntKT1)
Kate was of the opinion that classroom assessment also provides students with
information about “where they stand compared to their classmates” (IntKT1). When
132
students are aware of their ranking in relation to their classmates, they are able to
determine their level of performance in comparison with the class or cohort. Based on
this information, the students can judge whether they have put „enough effort‟ into their
studies. Kate explained, “If they did not put enough effort, then they should do
something about it” (IntKT1).
Kate further elaborated that classroom assessment guides students in their effort to
improve their understanding of the subject. Students would be able to decide “what to
do and how much time they need to spend” (IntKT1) as well as “know what to
adjust”(IntKT1) with regard to their learning. According to Kate, a student‟s results
from classroom assessment may imply the degree of suitability of the learning approach
employed by the student. If the results are not good, this might imply that the student
would need to change their approach to learning. For example, if a student was
attempting to learn by simply attempting a number of mathematics exercises and they
were getting poor results, then this would implicate the „ineffectiveness‟ of the method
of learning employed.
And if they really find the subject hard, ...maybe their method of learning is
wrong. …because they have not had the correct method of learning. From
the assessment they should try to change. (IntKT1)
Kate said that she valued the notion that classroom assessment plays a part in
encouraging communication among peers. Students‟ assessment results indicate to the
class who the high-achieving students are. Kate tried to help as many students as
possible if they come across any difficulties. However, sometimes there are too many
queries and not enough time. In instances like these, some of the weaker students
approach the better students for help, thus, they are able to learn from their peers.
133
Types of Classroom Assessment
Kate implemented classroom assessment mainly for summative purposes. She
emphasised that the main focus of classroom assessment is to collect information about
students‟ understanding. As Kate mentioned, “my main idea is I want to know whether
they‟ve understood or not” (IntKT1). She usually gives class tests after teaching
between two to three topics, depending on the length of the topics. Kate commented that
assessment that is based on a combination of more than two or three topics does not
allow teachers to assess all aspects of one particular topic in sufficient depth. Kate
mentioned that students should be assessed as frequently as possible by giving class
tests “maybe one month once” or “preferably after every topic” (IntKT1).
Occasionally, assessments were carried out at the commencement of a topic. Kate only
used this „pre-assessment‟ for topics which students have learnt previously, never for
completely new and unfamiliar topics. The purpose of Kate‟s pre-assessment exercise
was to identify students in the class “who can do and who cannot do” (IntKT1) the
topic. What this means is that Kate was able to identify students who have prior
knowledge of the mathematical concepts that will be used in the forthcoming topic.
Simple assessment techniques are also used by Kate during lessons to continuously
assess students as well as to provide immediate feedback to them. These techniques are
described in the following section.
According to Kate, formal assessment refers to a test where the students are assigned
fixed seating arrangements, answer a question paper which is collected and marked “out
of 100” (IntKT1). She gave examinations and class tests as examples of formal
assessments. Kate also mentioned that, formal assessment is a “very standardised”
(IntKT1) form of assessment where all the students “do the same thing” (IntKT1). On
134
the other hand, she associated informal assessment with activities such as classroom
questioning, asking students to present their work on the whiteboard and undertaking
projects.
Classroom Assessment Techniques
Assessment techniques that Kate employed in her classroom were dominated by
traditional paper and pencil class tests. Examinations and class tests are her main forms
of formal assessment. The structure of her classroom test is from direct to more
challenging questions as the test items progress. She explained that this structure
benefits the students because it enables them to “score some points” (IntKT1) so as not
to feel discouraged. Kate stressed that it was important for her to provide her students
with opportunities to feel confident.
Kate pointed out that she does not believe in giving her students „surprise‟ class tests
but prefers that they prepare for any test given. Kate explained that she gives students
enough time to prepare for class tests. Her rationale for not giving surprise class tests
was that she wants “to see after preparation how much they [students] know” (IntKT1).
According to Kate, some students are able to comprehend a lesson quicker and better
than others, whereas some may need “to go home and digest” (IntKT2) the concepts
that have been taught. For this reason, surprise class tests are not fair for students who
need longer time to comprehend what they have learnt.
Kate also uses two assessment techniques which she described as informal assessment.
One such technique is to ask students to come out to the front of the class and present
their work or solve questions on the whiteboard. Kate usually asks for volunteers from
the class to participate in this assessment activity. This is then followed by Kate‟s
135
explanation or comment on the student‟s work. If there are no volunteers, Kate takes it
upon herself to write the solutions on the board and explain. Students are able to view
the “different types of mistakes” (IntKT1) made. According to Kate, presenting work on
the board followed by an explanation is an activity that provides immediate feedback to
students as well as to inform the teacher instantly of the students‟ understanding.
During classroom observation, it was noted that the other method of informal
assessment used frequently by Kate during lessons was verbal questioning. Every so
often, after explaining something, she asked her students, “So, understand?” a phrase
which means “Do you understand?” during the two observed lessons (ObsKT1,
ObsKT2). Most of the time, the students would respond by nodding their heads, as if
saying, “Yes”. Sometimes students respond as a class and there were times when there
was no response. In the latter case, Kate usually just turned around and continued with
the lesson, assuming the students‟ silence as a sign that they have understood what she
had taught. There were no follow-up questions to further elicit responses from the
students. Kate also mentioned that in class, she seldom uses open-ended questions like
“how do you do it?” or “why did you do it like this?” The classroom observation
(ObsKT1) corroborated this view as no open-ended questions were observed.
When asked about the kinds of non-traditional assessment carried out in class, Kate
pointed out that, up until now, she had only used mind maps and guided assignments.
She described an example where students were instructed to draw a mind map to
“present everything you know about integration to me”. The second type of non-
traditional assessment used by Kate, was a guided assignment on the concept
differentiation from first principles. This concept was not in the syllabus for the topic of
differentiation but students constantly asked her about this concept because it appeared
136
in the beginning of their textbook. She prepared a guide for the assignment and students
were required to follow the instructions closely. Due to the nature of this assignment,
Kate said that “practically they can do very well” (IntKT2). Kate also mentioned that
she intends to explore the use of group work, but has not done so as yet.
For Kate, the principle of assessment focuses on “what I give them and what they come
back with” (IntKT2). She rarely bothered about “what their brains are thinking”
(IntKT2), hardly ever paying attention to the students‟ thought processes. However,
Kate‟s experiences using some alternative assessment techniques have opened her mind
to the idea that these techniques may help her understand her students better, especially
in situations where she could not comprehend why some students have difficulty
understanding simple concepts. Kate explained further that she expects certain students
to understand the lessons better than others, but there were times she was taken by
surprise that these students faced difficulties. According to Kate, alternative assessment
techniques may inform teachers of “what is in the brain” (IntKT2), referring to students‟
thinking. Through these techniques she feels she may be able to tap into her students‟
thinking and try to grasp the reason her students might have trouble understanding some
of her explanations during lessons.
The Need for Change in Assessment Practices
In Kate‟s view, one reason for change towards more alternative assessment practices
was “that it‟s a trend and we have to move to continuous assessment” (IntKT2). She
seemed apprehensive about moving away from traditional testing to more non-
traditional assessment. Kate emphasised that “at the moment, I still feel that written
examination is still the best way to assess” (RefKT2). She acknowledged that some
students face anxiety problems whenever they sit for a test or examination and they
137
would be disadvantaged if their performance was based solely on traditional class tests.
Some of these students may be good at mathematics, but do not perform well during
examinations or class tests. Non-traditional assessment may be one way of overcoming
the problem.
According to Kate, she did not reject new ideas but she felt confident with her current
assessment practices because it has a track record of producing results. She was worried
that if she were to take on new and different assessment approaches, the students‟
results might be compromised. Kate explained that “if a student should get an A and if
under my guidance don‟t get an A, I will feel bad you know” (IntKT2). For Kate,
preparing her students for examinations is her main concern because selection of
students into institutes of higher learning is based on their results.
Problems and Concerns about Change in Assessment Practices
The major concern Kate expressed about non-traditional assessment practice was the
issue of fairness. Any form of assessment must be a fair assessment because “everyone
must stand an equal chance” (IntKT2). Kate pointed out that traditional assessment such
as written tests were “fair in a sense it‟s standardised. It‟s the same question paper.
Everybody sit with the same duration of time, everybody has the same amount of time
to prepare” (IntKT2). In her opinion, fairness is ensured through standardisation of
classroom assessment. This concern about standardisation encompasses teachers‟
marking and grading, selection of the content to be assessed and the level of difficulty
of the tasks assigned. The following comments illustrate Kate‟s concerns regarding
fairness in assessment.
My grading and your grading might be different. That‟s not fair. (IntKT1)
It cannot be you‟re A is better than my A. (IntKT1)
138
Kate pointed out that even within the same school, teachers teaching the same subject
might set a different number of topics or items for their assessment. Some teachers may
even prepare tasks that are simpler or easier compared with another teacher‟s tasks. In
this situation, students who are given the easier assessment will perform better. Then
this would be unfair for other students. Kate explained, “…education has to be fair to
kids…if unfairness comes in, then it is no good” (IntKT1).
Another concern expressed by Kate about non-traditional assessment was the
implementation of projects as a form of assessment. This concern is again related to the
issue of fairness. She doubted that some students put sufficient effort into doing the
assigned project. According to Kate, some may receive assistance from family members
in completing the projects. Furthermore, a student may have a parent or parents who are
mathematics teachers or who have a good foundation in mathematics. These students
may receive “good guidance” (IntKT1) for their project. She was of the opinion that this
may be inequitable for others without such guidance.
Kate also was concerned about the use of the internet in completing tasks for non-
traditional assessments. There is fear of plagiarism among students as they have access
to so much information on the internet. Students may “copy, cut and paste” (IntKT2)
the information required for their assessment and submit it as their own work.
In addition, Kate expressed the belief that teachers also would initially experience some
uneasiness “moving away from a very familiar method of assessment” (IntKT2). When
asked further about this feeling of uneasiness among teachers, Kate elaborated that this
feeling was not due to “whether we know how to do it or not” (IntKT2), but more to do
139
with the fact that some teachers are doubtful about the techniques used to assess the
students.
Time was a factor that is of utmost importance for Kate. She explained that she did not
mind trying different classroom assessment techniques, as long as they fitted into her
teaching schedule. What is crucial for Kate was to ensure that she completes teaching
the syllabus within the stipulated time frame; “I cannot be like taking up so much of my
time, until I cannot complete what I want to teach them” (IntKT2).
Suggestions to Facilitate Change in Assessment Practices
One suggestion made by Kate to facilitate change in the way teachers assess students in
the classroom was to organise workshops where the focus is to provide teachers with
hands-on activities on various alternative assessment techniques (RefKT2). According
to Kate, most of the workshops that she had attended focused on the theoretical aspects
of assessment, which she found provided no useful input for the teachers to try out the
techniques on their own.
Teach us how to do it…Don‟t tell us the theory on how to do it. At the end,
we still don‟t know how to do it. (IntKT2)
Another suggestion was to allocate only a small percentage of students‟ overall
performance to non-traditional assessment. This would reduce the effects of plagiarism
and the disadvantage for students who have minimal access to resources and guidance
in completing the non-traditional assessments assigned to them. In this way, Kate
suggested a student‟s achievement is better represented overall.
According to Kate, if class tests are part of the classroom assessment, then they must be
standardized and all students must sit for these class tests simultaneously. The reason
140
for this is to avoid „leakage‟ of the test questions. Kate also insisted that these class tests
make up a larger percentage of the classroom assessment compared with other tasks,
such as projects, because class tests are more important and fair.
Kate also expressed her view that students must be briefed about what is expected from
them with regard to the tasks they must complete. She further emphasised that proper
guidelines must be accessible to students so that they are aware of the criteria for
judgement by the teachers for the work they submit. In Kate‟s words, “You must tell
them what you expect, what you want.” (IntKT2). She suggested that in order to be fair,
students must be given correct guidelines with regard to assessment criteria and
standards. Another point that Kate expressed strongly was that it was very important
that teachers must themselves be familiar with these guidelines so that they are able to
provide students with the proper guidance.
Summary of Kate’s Beliefs
Kate‟s experience as a student in an education system that was very examination-
oriented seems to have had a significant influence on her views of classroom
assessment. She feels that traditional paper-pencil tests are the best form of assessment
for mathematics.
Kate said that the teaching and learning of mathematics involves enhancing students‟
analytical skills, ability to communicate mathematically, problem-solving skills in
various situations and familiarity with routine. Although to Kate the correct answers are
the focus and final products of mathematics assessment, she also believes that the
presentation of the “steps of getting it [answers]” (IntKT2) is important for feedback to
the teacher and students.
141
For Kate, the main purpose of classroom assessment is to prepare her students for their
external examinations. Other less prominent purposes of classroom assessment include
the identification of students‟ understanding, to direct the path and pace of teaching, for
reporting students‟ academic achievement, the identification of high-achieving and low-
achieving students, as well as students‟ strengths and weaknesses and to inform
students‟ of their ranking among their peers.
Kate showed a strong preference for examinations and class tests, which she categorised
as formal assessment. Informal assessment used by Kate included classroom
questioning, requesting students to present their work on the board and undertaking
projects. She prefers to assess students as frequently as possible and occasionally carries
out “pre-assessment” activities at the start of a new topic. Kate does not like “surprise
tests” as she feels students should always be given enough time to prepare.
Change in her approach to mathematics assessment would be the result of a general
trend rather than any conceived notion by Kate of a need for change. Concerns that Kate
explained with regard to change in assessment practices focused on the issue of fairness,
but also included the time needed for alternative assessment.
To address these concerns, Kate suggested that teachers must be given opportunities to
attend workshops that incorporate hands-on activities on alternative assessment
techniques. The results of alternative assessments should be a small percentage of
students‟ overall performance. Students should be given suitable briefing and guidelines
on what is expected from them with regard to the alternative assessments given by
teachers. Last, standardisation of items being assessed, guidelines and marking must be
given appropriate attention.
142
Case Study Three: Anakin
Background
Anakin was a young mathematics teacher in his late 20s. He holds a bachelor‟s degree
in mathematics and, at the time of the interviews, he was pursuing a Master of Science
degree, majoring in applied statistics. Anakin had no teacher-training background and
ventured into teaching on a part-time basis during his third year as an undergraduate
student. His teaching experience, which was about three years, has seen him as a part-
time tutor at the university where he was an undergraduate and also as a temporary
substitute teacher at a primary school. He began his current position in 2006. Like Kate,
Anakin also had the opportunity of taking on the role of Chairperson of the
Mathematics Nucleus Group whose members consist of all the mathematics teachers
teaching in the pre-university programmes in the college.
Experiences Related to Assessment
Recalling his secondary school days, Anakin remembered that only school
examinations, and not class tests were included as formal assessment. He also
mentioned that the only form of classroom assessment that his mathematics teacher
used to assess them was homework. He could not recall any implementation of
discussion activities and the classroom environment was very formal. According to
Anakin, this context made mathematics “quite boring for me” because he did not
understand what he was doing. He emphasised that now, as a teacher, he does not want
to establish this kind of classroom environment for his students.
We understand the concept,… but sometimes, we are not sure why we study
topics like differentiation, integration. What I do now, for certain topics, I
tell them why we learn it. (IntAK1)
Anakin described an interesting experience with assessment that he had during his final
year at university. One of his lecturers assessed the students by giving them homework
143
and they were required to do class presentations. Anakin explained that he felt this
lecturer was “special” due to the one hour examination that he administered at the end
of the course. One section of the examination, which was worth 40 out of 50 marks,
required students to write down three questions pertaining to the topics learnt during the
course. In Anakin‟s view, this was an exceptional examination because instead of
answering questions, the students were expected to create questions.
Anakin expressed his opinion that this was a unique way of assessing students‟
understanding. He explained that the lecturer could gauge students‟ understanding
through the questions they presented. Anakin added that this form of assessment
encouraged the students to take their studies seriously, and to try to understand the
concepts as best as possible, to enable them to come up with good quality questions
during the examination. The following excerpt from the interview further illustrates this
point:
Of course you really have to go and study seriously about the topic. .... Of
course if you ask stupid questions, you cannot expect you get high marks.
He can see if you really study the topics or not. For sure I‟ll ask not good
quality questions if I don‟t study properly. (IntAK1)
Make sure we really know what the important concept that we don‟t really
understand. Then only we ask questions about it to the lecturer in the exam.
So, when he sees our questions he knows, oh, that part or this part we don‟t
understand. (IntAK1)
Teaching, Learning and Assessment
Anakin expressed a strong belief that mathematics is a subject that helps enhance
students‟ problem-solving skills. During his undergraduate days, Anakin noticed that
“the maths students can do very well compared to other students. I strongly believe that
maths can help us improve our problem-solving skills” (IntAK1). Anakin suggested the
reason for this is that the mathematics students engage in learning activities such as
144
computer programming tasks, which requires them to come up with programmes that
would do specific tasks, thus enhancing their problem-solving skills.
Anakin pointed out that apart from developing understandings of mathematical concepts
and problem-solving skills, it is important that students understand why they are
learning certain mathematical topics and their applications. He explained that during his
school days, he found mathematics to be a boring subject because he could not
appreciate the application of the topics he was learning. As a mathematics teacher, he
stressed that he always made sure that he explained to his students the application of the
topics learnt. For example:
Let‟s say, if I‟m doing trigo [trigonometry], I‟ll tell them about what is
oscillation, period and how we use this idea in physics. Like differentiation,
maximum and minimum problem, I‟ll tell them…why when the water fall
it‟s in a sphere and not other shape…because they want minimum surface
area. All this and what I know I‟ll tell them. (IntAK1)
When asked to explain what he understood by student-centred and teacher-centred
approaches, Anakin emphasised that because he did not receive any teacher-training, he
felt that his understanding of these two terms was vague. To him, a teacher-centred
approach involves activities that are passive and boring because “teachers just talk and
you just sit down there listening” (IntAK2). An interesting comment Anakin made was
that he viewed tests as having „teacher-centred‟ characteristics, as the teacher was still
the focus of this assessment activity. The following excerpt exemplifies this view:
If I give them a test, before that I have to teach what‟s in the test. Still is
teacher-centred. I have to teach them what they need to know. Seems like I
am the one who feeds them. (IntAK3)
For Anakin, a student-centred approach allows students to “learn more from
themselves” (IntAK3) and encompass any “work they do or any activities that they do
during the class” (IntAK3). He emphasised that engagement in student-centred activities
145
helps mould students to become independent learners. Activities of this nature provide
students with the opportunity to learn “what‟s in the lesson by themselves and more
than what they gain from me [teacher]” (IntAK3), by doing the work themselves.
Anakin favoured the student-centred approach as he felt that “if every time I‟m the one
teaching in front, I find that it‟s quite boring” (IntAK1). Based on the classroom
observation, however, Anakin‟s teaching appeared to be more inclined towards the
teacher-centred approach. An example was observed during a lesson on sketching
modulus function graphs (ObsAK1). He explained to the class that the graph of the
function y=|2x+1| was shaped like the letter „V‟ and is a combination of two linear
graphs, y = 2x +1 and y = –2x –1. Anakin instructed the students to sketch the graph of
y=|1–x| as he walked around the class to check on their graphs. This was followed by
him sketching the correct graph on the whiteboard and commenting on the mistakes
made by some of the students in their sketches. This approach was observed throughout
the lesson for the sketching of other modulus function graphs. The lesson was
dominated by the teacher‟s explanation and students were seen to be rather quiet.
According to Anakin, mathematics is a subject that is mastered through constant
involvement in practice and exercises because it focuses to a great extent on
calculations. As Anakin put it, “maths is a subject where practice makes perfect”
(IntAK1). Therefore, mathematics assessment should encompass activities that are
mainly centred around calculations. This is in stark contrast with Anakin‟s anecdote
about his university lecturer.
In comparison to mathematics assessment, Anakin is of the opinion that assessment for
subjects such as literature and biology requires students to possess good writing skills.
146
These skills allow them to write with clarity and express themselves well in the
assessment activities. Therefore, in Anakin‟s opinion, assessment activities such as
presentations are seen to be more relevant to these subjects compared with mathematics.
For other subjects like literature, biology, all these they need very clear
explanations. So for me, these kinds of subjects, they need to write more to
show they understand more. These kinds of subjects, maybe they need to
do more presentation. (IntAK1)
Purpose of Classroom Assessment
The purpose of classroom assessment for teachers
According to Anakin, one purpose of assessment is to provide teachers with information
about their students‟ understanding of the topic taught. In Anakin‟s words, “by doing
assessment, then only I know how far they understand the topic” (IntAK1). For Anakin,
it is through assessment that teachers are able to identify how much of the content
taught in a lesson was understood by the students.
Anakin emphasised that classroom assessment informs teachers of the suitability of the
teaching methods employed in the classroom. This information guides teachers in
deciding whether certain teaching techniques adopted are still suitable in helping
students understand better. Anakin mentioned that if there is a need for change in the
way students are taught, assessment will indicate this need to teachers.
If I always use method one, is it still okay to use? If not good to use when
teaching, so maybe I need to try method two. When I see the results are
good, then I know, oh, my teaching is okay. But if students‟ marks no good,
maybe I have to teach another way. (IntAK1)
For Anakin, assessment in the form of students‟ presentations serves as a tool to
encourage his students to „self-study‟. He uses this form of assessment for the
mechanics component in the mathematics syllabus for students in the science stream.
This assessment activity requires the students to study the topics in mechanics on their
147
own as a form of preparation before coming to class. These topics have not been taught
by Anakin prior to the presentations, but according to him, the students have learnt all
the mechanics topics in their secondary school physics class. The class presentations are
carried out to allow the students to communicate their understanding of the topics to
their peers.
When asked about what he did with the classroom assessment results, Anakin explained
that they are recorded and kept by the teacher. This compilation of students‟ results
allows the teacher to monitor their students‟ learning and their progress. Anakin
mentioned that up-to-date results are compared with previous results to gauge any
improvement made by the students.
The purpose of classroom assessment for students
According to Anakin, the final mark or grade from an assessment is most important for
the students. Results are used by the students who are submitting an application for
further studies at the university level. Hence, the product of assessment is seen to fulfil a
selection criterion for university applications.
Secondly, Anakin expressed the view that assessment “can help the student understand
the concept and improve their problem-solving skills” (IntAK2). He suggested that
students enhance their understanding of mathematical concepts by engaging in
assessment activities, as well as improve their problem-solving skills. For example,
Anakin assigned his students lots of homework, “every day for sure they have
homework” (IntAK1). He stated that by doing lots of homework, his students will have
lots of practice in applying their mathematical skills to solve the various mathematical
148
problems. This is consistent with his earlier statement that mathematics is mastered
through constant practice.
Types of Classroom Assessment
Anakin stated that he is not familiar with the terms formal and informal assessment,
however, he was happy to explore what these terms immediately conveyed to him. He
said that formal assessment probably involves activities that must be carried out and
students will “for sure get something from the assessment” (IntAK1). Anakin
mentioned that for formal assessment students are not allowed to carry out discussions
or have access to references. He described examples of formal assessment as
examinations, class tests and homework. Anakin considered homework as a form of
formal assessment because it “must be done” (IntAK1), it is straightforward and if
students do homework, they will gain understanding of what was taught in class.
Anakin described informal assessment as activities that permit discussions amongst
students and the use of references (RefAK3). He mentioned group work and
presentations as examples of informal assessment.
Anakin occasionally carried out assessment at the start of a new topic. This was only
done for new topics which are related to topics that students have learnt previously, not
for unfamiliar topics. Anakin explained that the pre-assessment activity is usually a
short quiz of about 10 questions that need short answers only.
Assessment activities are sometimes carried out by Anakin at the end of a lesson, when
time permits. Anakin explained that sometimes about ten minutes before the end of a
lesson, he gives the students a question to answer. The students solve the question and
he collects the work and marks it. An example was observed during a lesson on
149
sketching modulus function graphs (ObsAK1). During the last 15 minutes of the lesson,
Anakin instructed the students to answer two questions from the textbook. At the end of
the lesson, Anakin collected the students‟ work to be checked and marked. This activity
reflected Anakin‟s concern about and dedication to his students‟ learning.
Classroom Assessment Techniques
In Anakin‟s opinion, alternative assessment techniques are not very useful for assisting
students to improve their learning of mathematics. Rather, he felt that these alternative
techniques play a better role in helping the students enhance their personality. Although
Anakin is relatively new to the teaching profession, he claimed that he employed a
variety of classroom assessment techniques. These assessment techniques include
homework, class tests, presentations, oral questions, quizzes and peer assessment.
Anakin‟s favourite assessment technique is homework. He expressed the belief that
homework is the best form of assessment as it is an effective way of helping students
improve their mathematics learning. An example Anakin gave to emphasise this belief
was when he had to take over a class with low-achieving students. He „forced‟
homework every day onto the students and, later, the class tests results showed that
there was improvement in students‟ achievement. He assigned homework every day,
from the textbook or other mathematics reference books (ObsAK1). Anakin emphasised
that he is very strict about students doing their own homework. He further explained
that he makes it clear to his students that sharing of homework with other students is not
tolerated in his class. A warning is repeatedly issued to all his students that they would
be in trouble if caught. The following interview excerpt illustrates this point.
I‟m very strict. I warn the students. I tell them in the class, if you borrow
your work to your friend to copy I don‟t think it will help him. Of course if I
150
don‟t know, then you are lucky. If one day I see somebody copy homework,
then both of you are finished! (IntAK1)
Class tests are carried out after three topics or about once every 3 weeks. Anakin
explained that if the topics are lengthy and students need to do a lot of studying, then
there would be a test after two topics instead of three. Although there are teachers who
prefer to “give every topic one test” (IntAK1), Anakin claimed that this practice is “a bit
useless” (IntAK1). He expressed the belief that when students prepare for class tests
after every topic, there‟s very little studying done as students only need to revise a few
concepts. This situation may deceptively enable them to do well in the class tests.
According to Anakin, these „good‟ results could mislead the students into thinking that
they are performing well in their tests, when it is not a true representation of their actual
academic ability. Anakin explained his reasoning that examinations cover a number of
topics that have been taught over an extended period of time, at one sitting; hence,
students may not be able to achieve as well as expected, because they have more
concepts to remember and approaches to select from when solving problems.
In addition to tests, Anakin assessed his students through presentations. He used two
types of presentations; Power Point presentations and presentations on the board. The
Power Point presentation was carried out for all four topics in the mechanics section of
the mathematics syllabus. The class was divided into four groups and each group is
assigned a topic. According to Anakin, the students are given ample preparation time,
“if including holidays they have about ten weeks” (IntAK1). The Power Point
presentation addresses the content of the topic, useful examples related to the topic and
good quality questions for discussion. Each group presentation was a combination of
individual presentations, as each group member is responsible for different aspects of
the topic assigned to them.
151
Anakin explained that the second type of presentation involves the students first
discussing an assigned problem in groups and then coming out to the board to present
their answers to the rest of the class. This type of activity was carried out “once in two
or three weeks” (IntAK2). The group discussions usually last about 20 to 30 minutes of
the one-hour lesson. Anakin explained that some of the problems are taken from the
miscellaneous exercises in the textbook or from past years‟ examination papers.
Oral questions were used by Anakin during lessons and power point presentations. The
classroom observations indicated that the questions Anakin asked during lessons were
typically, “Understand?”, “Okay or not?” and “Any questions?” (ObsAK1; ObsAK2).
Questions to provoke the students‟ thinking about the concept being taught were not
evident during the observation. The questions were more for Anakin to reaffirm that he
still had the students‟ attention. Another instance where Anakin claimed that oral
questions are used by Anakin is during the group power point presentations. He said he
allows the students to have a question and answer session between the presenters and
the observers. According to Anakin, this questioning process helps him identify how
well the presenters prepared and understand the topic. In Anakin‟s words,
When they present, I can see did they really prepare during the past few
weeks. Did they really study? Do they really understand the concept?
Because if you don‟t understand the concept, definitely you cannot give a
good presentation. During the presentation, I allow the audience to ask
questions, to argue if they don‟t agree. So from this process, I can see if the
presenter was really fully prepared or not. (IntAK2)
Anakin explained that peer assessment is a technique that he recently implemented in
Anakin‟s classes during presentations. Each student was given a list of the presenters‟
names and was required to rate each of them out of 10 marks. Anakin said that he
insisted students write comments to justify why they awarded those marks to the
presenters. By insisting on written comments, he could make the students “pay attention
152
to the person who‟s presenting” (IntAK2). The following comments further illustrate
Anakin‟s approach.
Other than rating, I want them to write comments also. I told them I don‟t
want to see something like „very good‟ only. If good, which part is good.
You have to explain more. I want them to do all this. If they give six marks,
they have to let me know why the six marks. (IntAK2)
So the presentation sometimes is actually quite boring. ... Even though
boring but I want them to listen. Because by paying attention, even though
it‟s boring, only then they can write out the comments. (IntAK2)
The Need for Change in Assessment Practices
Anakin articulated his opinion that the current assessment system influences teachers to
teach to the test. The teachers‟ main focus in the current system is the examinations and
they feel their job is to prepare students for these examinations only. This emphasis on
examinations hinders the teachers from engaging the students in activities that will help
them understand and enjoy learning mathematical concepts taught in class. According
to Anakin, the change will enable the teachers to pay less attention to the examinations
and to change their teaching approaches so as to allow students to take pleasure in and
benefit from the process of learning.
Another thing is,…they concentrate only on the exams. This will make the
main job of the teacher to prepare the students for the exam. They are not
teaching the students to understand and enjoy the subject. It‟s just 100% I
prepare you for the exam. …They are preparing students for exams only.
Not to enjoy the process of learning. (IntAK1)
Anakin mentioned that the introduction of alternative assessment techniques allows
students to grow into more balanced and capable individuals (RefAK3). The students
learn and enhance „soft‟ skills such as organising a group, managing group members,
discussion skills, how to do a presentation and team work from taking part in these
alternative assessment activities. Anakin elaborated that some students have a tendency
to be “really quiet in class....like to work by themselves. ...don‟t like to mix with other
153
people” (IntAK1). Therefore, the group presentations can “help improve their
relationships with other people” (IntAK1). Moreover, Anakin expressed his belief that
through alternative assessment techniques, students are more likely to become
independent learners, and rely less on the teacher to “feed them” (IntAK1). As
mentioned earlier, these skills are important to what Anakin referred to as the growth of
the students‟.personality. The following excerpts illustrate Anakin‟s ideas about the role
of alternative assessment in the growth of one student‟s personality:
We really need to learn how to work together with other people. We need
group work, no matter what kind of group work it is. As long as they can
work together. (IntAK1)
...assess students in many areas like critical thinking and level of reasoning,
the ability to cope with stress, the performance of the ability to
communicate knowledge and interpersonal skills and by doing so, we train
them to be even more capable. (RefAK3)
Anakin explained his view that any change in the assessment system would be
welcomed by students; that they will enjoy learning more as there will be fewer class
tests to sit. Students will not be under constant pressure as they are now, to prepare for
class tests and examinations. Anakin commented that in an environment which is
dominated by high-stakes testing, students tend to solve mathematical problems without
proper understanding.
...they can learn somewhat in a more enjoyable way because tests and
exams will bring pressure to students. (RefAK3)
Students will only know how to solve the numerical problems by using
formulas without better understanding of the whole concept. (RefAK2)
Problems and Concerns about Change to Assessment Practices
One of the issues that Anakin suggested will arise from the introduction of alternative
assessment techniques is teachers‟ need to change their instructional practices.
Instructional techniques that cater for “teaching to the test” (IntAK1) will be deemed
154
unsuitable in this new context. Having said that, Anakin stressed that although this is an
issue, it is just a small issue, because problems will arise only at the beginning.
I think they will face a problem but I don‟t think it‟s a very big problem.
Maybe the teachers have to change the way they teach. Since the system
will change, the way they teach will also have to change. So maybe at the
beginning there are some problems. (IntAK1)
Anakin expressed his concern that there will be an increase in teachers‟ workload. For
example, when he instructed the students to prepare Power Point presentations, he had
to go through each group‟s set of slides to make sure that the content was suitable for
the topic. For activities such as projects, Anakin commented that a lot of time is taken
up with preparation.
For me as a teacher I find that my work is a bit more, because for example,
the presentation I have to go through their slides first. …When they submit,
I have to go through all the slides, all the examples they choose. I find that I
have to go through the solutions all this you know. I find that my workload
is quite a lot. I told you about the project that I want to do for stats. I am
now still preparing. (IntAK2)
Another point made by Anakin was that the challenge of group work is to be able to
give clear instructions to the students about what is required from them. He stressed that
if the teacher does not give clear instructions, students are not able to start on their
group work efficiently. Another issue is that the first time an assessment system is
implemented there are no exemplars of previous group work to show the students as a
guide.
Suggestions to Facilitate Change to Assessment Practices
Anakin suggested that teachers need to be trained in the various alternative assessment
techniques. The training should encompass preparing the relevant activities, how to
prepare a fair marking guide that has “clear scales to follow” (IntAK1). At present, no
clear guidelines exist.
155
Another suggestion to facilitate this change that Anakin mentioned was to create good
networking among lecturers. Teachers need to meet up regularly, maybe “once in about
three or four months” (IntAK3) and share their experiences of using different
assessment techniques. In Anakin‟s opinion, he would benefit more from listening to
other lecturers‟ experiences with assessment techniques, “...experience is more
important compared to reading from a book, ...it‟s better to ask other senior lecturers”
(IntAK3).
Anakin also emphasised the importance of resources for teachers to use as points of
reference. It would help if teachers were provided with easy access to websites on the
internet that subscribe to alternative assessment techniques. This would enable teachers
to have access to a rich source of ideas and examples of different kinds of activities that
are suitable for and manageable with their students.
Summary of Anakin’s Beliefs
Anakin expressed the belief that mathematics is a subject that is mastered through
consistent practice and solving of problems. This practice will enhance students‟
problem-solving skills and enrich their understanding of mathematical concepts. He
emphasised that teachers need to provide students with opportunities for meaningful
learning. Students should be told the reason for learning a particular topic or concept so
that they are able to apply those concepts more effectively.
For Anakin, the main purpose of assessment is to provide teachers with information
about their students‟ understanding of the concepts as well as the suitability of the
teaching approaches adopted in the classroom. In addition, he mentioned that
assessment could serve as a „tool‟ to encourage students to do some preparation before
156
coming to class. In Anakin‟s opinion, assessment results are vital for students‟
application to university. Students also are able to gauge their progress in their studies
by referring to their assessment results.
Anakin‟s favourite assessment technique was homework. He emphasised that if students
consistently completed their homework, their understanding of mathematics will be
better. He does not have a clear understanding of informal and formal assessment, due
to his non-teacher training background. Other assessment techniques that Anakin claims
to have employed are tests, presentations, oral questions and peer assessment.
For Anakin, a change in the way students are assessed would allow students to enjoy the
learning process better, as examinations and tests are the cause of stress for students and
teachers alike. Implementation of alternative assessment techniques also will assist in
students‟ growth as capable and independent learners. Anakin is of the opinion that
students‟ involvement with various assessment activities will hone their social and
communication skills.
A concern that Anakin mentioned with regard to a change in assessment practices was
that it will require teachers to reassess the way they teach in the classroom, as they
cannot employ approaches that „teach to the test‟. He was concerned that the
introduction of various assessment techniques will increase teachers‟ workload as
preparation time needs to be increased.
To address these concerns, Anakin suggested that teachers be given appropriate training
in the various assessment techniques. He feels that teachers learn best from their peers‟
experiences with different assessment techniques. Regular meetings should therefore be
157
scheduled to allow teachers to create useful networks as well as a sharing culture.
Anakin also proposed that teachers be provided with access to sufficient resources, such
as the internet, to find suitable assessment activities for their respective students.
158
Case Study Four: Jane
Background
Jane was an experienced mathematics teacher in her late 50s with more than 30 years of
teaching experience. She graduated with a Bachelor of Science and received her
Diploma in Education after attending a one-year teacher-training course. Most of her
teaching experience has been at the secondary school level, focused on the teaching of
mathematics. According to Jane, she is very passionate about teaching, and was inspired
by one of her secondary mathematics teachers, of whom she has a very high regard and
is still in contact with till this day. She mentioned that if she has any mathematics
related problems, she still seeks this particular teacher, who is in his 70s, to discuss
ways to solve the problem. During her service in secondary schools, Jane held the
position of Head of Mathematics until her retirement. After opting for an early
retirement, Jane worked for about six months as a mathematics consultant for a
technology-based company, reviewing courseware related to mathematics. This was
followed by her current employment as a pre-university mathematics teacher. Although
Jane had limited experience with non-traditional assessment techniques, she said she is
willing to try new alternative assessment techniques.
Experience Related to Assessment
Jane grew up in an era which was very much dominated by examinations. As a student,
she held the view that assessment was about getting good marks. According to Jane,
during that era, a student who did well in examinations and achieved high marks was
considered “a good student” (IntJA1). Jane described an incident that happened when
she was in Standard Five (the equivalent to Year Five), that showed how important
marks were to her as a student. When one of her teachers made a mistake of giving her
fewer marks in a test paper than she deserved, she showed how upset she was with this
159
particular teacher. The teacher reacted by scolding her and insisted that she should not
behave in that manner because of a few marks. This incident made her realise that
marks were not everything.
Throughout her career as a teacher in the national schools, Jane was involved in the
marking of the national mathematics examination paper for secondary students.
Students in the national schools sat this mathematics paper in their final year of
secondary school. Jane had high respect for examiners during the early years of her
teaching career. They took their role as examiners very seriously. According to Jane,
they were meticulous in their marking, keeping strictly to the marking scheme and made
sure their marking was not flawed by mistakes.
It was exams only. ... But I would say I salute to them, the markers. The
attitude of marking papers was really very serious. They mark so slowly. ...
All the details and little things they jot down. A mistake shouldn‟t happen
even for one mark. …You shouldn‟t run away too much from the scheme. It
shouldn‟t happen for maths. …They are very careful. (IntJA2)
Jane commented that over the years as a school teacher, she had noticed that the types
of questions asked in examinations have changed quite significantly. An example she
gave was a change in questions for the additional mathematics paper in the national
examination, which students sit at the end of secondary school. Previously the questions
“were all subjective” (IntJA1), but now the additional mathematics paper includes
“structured type of questions” (IntJA1). She stresses that these structured questions are
“very much easier” (IntJA1) and they give the impression of guiding the students.
Teaching, Learning and Assessment
Jane expressed a rather holistic view of education. She mentioned that “education is not
only getting knowledge” (IntJA2) but also involves “moulding the whole personality of
160
the person” (IntJA2). Therefore, from Jane‟s point of view, education is important for
cognitive development and social development. She conveyed her view that there is a
need for students to learn to socialize so they can also gain knowledge from their
surroundings. She claimed that if you have “no socialising skills then you totally don‟t
know what‟s happening in the outside world” (IntJA2).
As for teaching, Jane stated that a teacher‟s job is “just not imparting knowledge”, but
also, as mentioned earlier on, “moulding the personality of the person” (IntJA1). She is
of the opinion that teachers have some influence over their students‟ behaviour because
the “teacher is one companion that they [students] can confide in” (IntJA1). Jane
pointed out that the incident that happened to her when she was in Standard Five left her
feeling embarrassed because she was a “model student” (IntJA1). According to Jane,
she felt thankful that her teacher scolded her as this episode taught her that she must
apologise for any of her misbehaviour:
I think she gave me less marks. I was upset. The teacher scolded me. …first
time I get a good scolding from a teacher. …practice and practice how to
say I‟m sorry. I‟d never say I‟m sorry to a stranger. (IntJA1)
According to Jane, learning is a process where students not only acquire knowledge
because they “learn something”, but most importantly, they learn how to apply the
knowledge that they have gained. To Jane, mathematics learning is about learning to
solve problems. She elaborated that learning mathematics provided students with an
environment that enabled them to develop their minds to be “analytical, systematic,
logical” (IntJA1). These skills, in Jane‟s opinion, are essential in our daily lives. An
example she mentioned to emphasise this viewpoint referred to Polya‟s problem-solving
model. The following interview excerpt illustrates this example.
You see, the Polya‟s four steps are literally teaching you how to understand
a situation, how to analyse and then plan and run it, then check. So, you
161
apply to other things when you step out to society and work. Unconsciously,
it‟s from your training in maths. (IntJA2)
Jane emphasised that assessment should be carried out continuously. She disagrees with
the notion that students‟ performance be based on a single assessment, usually given at
the end of the year. Jane prefers to “see the consistency of the interest or improvement
or deterioration of students…over a period of time” (IntJA1). She feels that assessment
given on a regular basis provides teachers with a better understanding of their students.
As Jane put it, “consistent assessment would be more useful” (IntJA1).
Jane considered mathematics assessment “much easier to do” (IntJA1) compared with
assessment in other subjects because the answers in mathematics are „absolute‟. She
elaborated that this feature of mathematics answers made it easier for the teacher to
detect “if something has gone wrong” (IntJA1). This was further justified by the
statement “if for one plus one someone says is three, then you know something‟s wrong
somewhere…I can see straight away” (IntJA1). For other subjects, Jane explained that
the answers are more varied and diversified; to the extent “sometimes all are
acceptable”. She believed that in such situations, an answer “depends on the teacher,
how much he or she can accept as correct, the so called correct answer” (IntJA1).
Purpose of Classroom Assessment
The purpose of classroom assessment for teachers
Jane regards classroom assessment as a form of feedback about students‟ learning.
Assessment provides her with information about how much her students have
„absorbed‟ from the lessons. In Jane‟s view, if she did not “get back anything good”
(IntJA1) from the assessment, then she would take it as a “failure on my part” (IntJA1).
To her, this meant that she did not do a good job of teaching the students.
162
For Jane, assessment informs her of the quality of her teaching. She explained that it
was important to her to know “if I am teaching them properly” (IntJA1). Jane uses
assessment as a means for students to enlighten her about any dissatisfaction they have
regarding her teaching. In Jane‟s opinion, one reason why students may have difficulty
understanding the lesson could be due to the fact that they “don‟t like my teaching”
(IntJA1). She explained that if there are any adjustments to be made to improve her
teaching, she needed feedback from the students as a guide.
Jane strongly expressed the belief that classroom assessment does not just involve
assessing the learning material, but also involves assessing the students‟ personality. For
example, she assesses a student‟s personality through their handwriting or the way they
speak. If the handwriting is small or the student is too soft-spoken, she takes it that this
student has poor self-confidence. Jane addressed situations like these by taking it upon
herself to counsel students and “show them they can have confidence in themselves”
(IntJA2). She stressed that by assessing the student‟s personality, teachers are able to
“see his attitude towards learning, whether he‟s always positive” (IntJA2). The teacher
was able to identify whether students are trying their best, or whether they are just
content with their current performance.
According to Jane, she kept the assessment results such as marks from tests,
examinations, or assignments, for reporting purposes. These results are sent to the
parents to inform them of their children‟s academic performance because “parents sure
want to know whether their kids are studying properly. They spend so much,...they hope
their child do well” (IntJA1). Jane also receives requests from students for a forecast
grade as this is a usual requirement for their university applications. Hence, the forecast
that Jane provides is based on the assessment results that she recordes and keeps.
163
The purpose of classroom assessment for students
Jane mentioned that apart from teachers, assessment also informs students about their
own learning. Assessment provides students with information on “how much they
understand the lessons” (IntJA1). Jane believes good marks indicate that students have
understood the lesson well, whereas low marks suggest the need “to do something to get
better results next time” (IntJA2). Jane explained that “do something” here could mean
that these students need to study harder or practice doing maths more to improve their
achievement scores.
In Jane‟s opinion, students can be motivated to perform better through assessment. She
noticed that students are excited whenever they knew they were getting their marks
back. Upon receiving their marks, Jane has observed an array of facial expressions
among the students, portraying their individual feelings about the marks received. She
also said she sometimes notices an atmosphere of competitiveness among her students.
This atmosphere, according to Jane, motivates students to do better. Students with high
marks feel a boost in their self-confidence, whereas the ones who do not perform as well
are driven to work harder to improve their results.
You can see some happy, some not satisfied. They compare and ask each
other, what did you get? …feel good if they higher marks than their friends.
…got low marks maybe get motivated to study harder. (IntJA1)
Types of Classroom Assessment
When asked about formal assessment, Jane explained that she views formal assessment
as „traditional‟ in nature, stating that tests and examinations are examples of formal
assessment approaches. Furthermore, she mentioned that students are placed in an
examination venue and required to answer the examination questions within a „fixed‟
time frame. Marks from formal assessments are meant for the parents and the college‟s
164
records. To Jane, formal assessment involves “rigid memorisation of facts and
reproduction of what has been taught” (RefJA7). In her opinion, therefore, formal
assessment is rigid and requires students to regurgitate everything that they have learnt,
emphasising memory work.
Jane regarded informal assessment as more „casual‟ compared with formal assessment,
because students “need not worry about the outcome” (IntJA1). “Outcome” here refers
to marks or grades. She elaborated that informal assessment offers teachers a variety of
methods to assess students because they have “a bigger range to judge the ability and
capability” of students. As for the marks given for any informal assessment activity,
Jane stressed that these marks are not for the students, but for her own reference. The
following interview extract describes an example of a particular informal assessment
activity carried out by Jane.
For example at the end of the lesson, like that day I had ten minutes left, ...
So I said, “... Look at question number ten. Why not try question number
ten now”. Ok they will try. After five minutes I said, “Please pass up your
work before you walk off”. ... So it‟s not a formal thing. ... they pass up. I
just see from there I know whether they understand or not. But I record the
mark not to grade them but for myself. (IntJA1)
Jane mentioned that her normal practice before starting a new topic is to question the
students orally. This activity allows her to gain some insight into the students‟ prior
knowledge, to “see how much they know before I start” (IntJA1). This information is
useful to Jane in helping her to decide on a suitable approach to the topic. Below is an
example that illustrates this point made by Jane.
Like vector. They know it‟s just magnitude and direction. They just
memorize from the textbook; magnitude and direction. They can‟t tell you
much. So, you really need to introduce first to them. Some may have
misconceptions. (IntJA1)
165
During an interview, Jane also described a particular activity that she had carried out at
the beginning of the first lesson for the topic of statistics. She had projected the word
„statistics‟ on the white board and asked the students to state what crossed their minds
when they looked at that word (RefJA1). Jane explained that she was surprised at the
outstanding responses that she received from the class members. The students had
mentioned words and phrases that were significantly related to statistics. Examples of
these words and phrases included data, graphs, pictures, calculation of means, drawing
histogram, probability distribution and getting information. According to Jane, this
simple activity helped put the students “into the right frame of mind to participate in
what we are going to cover in the following six weeks” (RefJA1) for the topics of
statistics.
Classroom Assessment Techniques
Jane expressed the opinion that teachers should adopt a mixture of traditional and
alternative assessment techniques in the classroom. She employs an array of assessment
techniques such as tests, feedback forms, quizzes, multiple-choice questions, oral
questions, observations, presentations, peer assessment and peer feedback.
Jane had introduced the use of a feedback form the previous year, following her attempt
to create a classroom environment that was more student-centred. The feedback form
functioned as an outlet for students to write their comments or suggestions regarding
their learning, as well as the teaching approaches carried out in the classroom. Jane
received considerable feedback from her students about how they felt about the way she
taught and also how she could help them learn more effectively. The following
interview excerpt elaborates on this point.
166
I ask them in particular what do you think of student-centred learning. They
tell me I still like the teacher teach first, then we do. ... What I did is the
other way. I let them do first, then I patch up the holes. Some [students] say,
I prefer you guide us first, then we patch up. That‟s something I didn‟t
notice. I thought it‟s nice to let them explore, discover first. (IntJA1)
Another assessment technique that Jane used was to present the students with a past
examination question in relation to the topic taught (RefJA2). This assessment activity,
which was usually carried out at the end of a topic that she had finished teaching,
informed Jane of the quality of her teaching. She stated that she felt very happy that one
of her students commented that he enjoyed this form of assessment. Another positive
point that Jane realised about this assessment activity was that it gave the students an
opportunity to “clarify any doubts concerning that chapter” (RefJA2), and this was
followed by the effect that students were becoming “more certain of what they have
learnt because it has served as a summary of the topic too” (RefJA2). The following
excerpt describes how this activity was carried out in the classroom.
At the end of each topic, I put up an exam question, from any exam board,
as a class assessment. I then walk round to check their answers, praising
those who can do and giving guide or hint to those who are not certain of
the solution. By this way I can have immediate feedback to check the
effectiveness of my teaching. (RefJA2)
According to Jane, multiple-choice questions present students with a variety of possible
answers to work with. This makes it more interesting for students. Jane prefers to give
questions that emphasise application, because she feels that the current trend pays a lot
of attention to students‟ ability to apply what they have learnt. She views this type of
question to be more interesting to answer.
In addition to usual monthly tests, quizzes and multiple-choice questions, another
technique used by Jane to assess her students was oral questioning. Jane used this form
of assessment for two purposes. The first was to identify students‟ prior knowledge at
167
the beginning of a new topic. This information helps her make decisions about suitable
teaching approaches for the topic. In Jane‟s words, “At least I know what they know.
Then I know what to do” (IntJA1). The second purpose was to get feedback from the
students regarding their learning. Based on this feedback, Jane is able to identify among
her students “which one is paying attention, which one is not” (IntJA1), as well as
which of the students need more attention.
In Jane‟s class, students also are required to do presentations of two types. The first
type involves students coming out to the whiteboard and presenting their solution to a
question that Jane has given them in class. If a student cannot complete the solution,
that particular student selects another student to assist in solving the question (ObsJA1).
When asked the reason for doing this, Jane explained that this encourages more student-
involvement and “make everyone on their toes” (IntJA2), because the other students do
not know who will be called out next. Furthermore, students do not feel embarrassed if
they cannot solve the problem because there is another student there to help them with
it. According to Jane, this activity creates a learning atmosphere that is fun and
promotes teamwork, because the students and their friends must work together to get
the solution.
…they find it interesting because they‟re laughing and joking. Sometimes
they may call their friend or they may call their enemy. To make the
atmosphere not so rigid. Make it more lively. (IntJA2)
The second type of presentation used by Jane was a group Power Point presentation that
is usually carried out after Jane has completed teaching all the topics in the mechanics
syllabus. She used this group presentation as one of the assessments for the second
semester, as well as a revision for mechanics. There was an element of peer assessment
in this activity because Jane gives the groups who make up the audience, an opportunity
168
to assess the group that is presenting. She explained that each group‟s presentation is
assessed on four aspects; content, clarity, presentation and creativity. The following
extract describes how Jane carried out this activity for one of her classes.
Class was divided into six groups with specific chapters assigned to each of
them. Each group was given 15 minutes for presentation. The observing
groups were given a chance to assess their peers‟ performance with scale
sheets running from 1 (weak) to 5 (excellent) on content, clarity,
presentation and creativity. (RefJA3)
As mentioned earlier, Jane pointed out that there are times when students learn more
effectively from their peers. Therefore, she paired up the good students with the weak
students by arranging them as seating partners in the classroom. Her idea is that the
better students are able to help explain mathematical concepts that the weaker students
have difficulty understanding. Jane assessed the weaker students‟ learning progress by
getting verbal feedback from the better students. She did not keep any formal records of
their progress, just feedback by means of conversation with the better students on
whether the weaker students understand the concepts taught. According to Jane, “most
of the feedback I get is very positive” (IntJA2).
The Need for Change in Assessment Practices
Jane stated that the use of alternative assessment methods would enable teachers to
understand their students better. Teachers would be able to determine whether students
were consistent in their studies, because “if a student is consistent, he should be given
credit” (IntJA1). She explained that through alternative assessment methods, teachers
could see their students‟ “attitude towards learning, whether always trying for the best
or just contented with their percentage” (IntJA1).
169
Jane emphasised that too much emphasis on grades creates a perception among students
that “if I don‟t do well, I‟m not good. I won‟t have a future.” (IntJA2). She claimed that
alternative assessment enhances students‟ confidence in their learning, especially the
weaker students. To Jane, assessment techniques other than the traditional pencil-paper
tests, provided students with opportunities to feel good about themselves as well as
show that they are capable of some form of achievement. The comment below
illustrates this point.
They don‟t feel humiliated. They don‟t feel they are inferior group. They
feel they can do something. …At least they can produce something. We
give them confidence in life. They don‟t feel they are drop-outs. The child
may be good in some other thing, at least it‟s appreciated. (IntJA2).
Another point brought up by Jane was that, “There are many sources of intelligence.
Different people have different kinds of intelligence.” (IntJA1). She asserted that this
difference among individuals should be recognized. As Jane put it, “I would like the
assessment system to look at the intelligences” (IntJA1). The recognition of students‟
different intelligences gives them “a chance to excel, to do well” (IntJA1).
According to Jane, it is not a fair judgement if the students‟ progress was based solely
on a test or examination that was administered after a long period. She elaborated that a
student could have been doing well in his or her studies, but due to some unforeseen
circumstances on the day of the examination, this student may not perform well. Jane
mentioned that this situation is a disadvantage to the student. Two examples that Jane
referred to as unforeseen circumstances were “poor health or family problems”
(IntJA1).
According to Jane, a mixture of examinations and alternative assessments would suit the
students better. She explained that such a change to the assessment system would
170
reduce the emphasis on examinations, which, in turn, would deter students from
studying for the purpose of examinations only. To Jane, this approach to assessment
would emphasise that “understanding is really important” (IntJA1) to learning, and that
it‟s not about “swallowing it [knowledge] and reproducing it” (IntJA1). Students would
not look at assessment as a test on memory, where “If you have a good memory, you
score.” (IntJA1).
Problems and Concerns about Change in Assessment Practices
Jane mentioned that a problem she faced when assessing students through group work
was that “you cannot differentiate in group work” (IntJA1). She finds it difficult to
determine “who has done how much” (IntJA1) among the group members. She
explained that it is unfair when some group members do more than others, because “the
lazy fellow doesn‟t do anything, but the good ones try very hard because they know the
marks matter” (IntJA1). In Jane‟s opinion, this is a problem most teachers need to be
aware of when engaging in group work as a form of assessment.
Another problem that Jane encountered when trying to incorporate more alternative
assessment techniques into her teaching was the lack of communication among teachers
regarding the different assessment techniques they were using in their classrooms.
According to Jane, some teachers are just working individually and are “still reluctant to
share” (IntJA1). Jane mentioned that she did not know what the other teachers were
doing and neither were they aware of what she was doing. She elaborated that she tried
assessing her students with mind maps after having talked to another teacher and found
out that this teacher had used mind-maps as a form of assessment. Jane confided that
“you don‟t expect me to talk to everybody and ask what they are doing. People say, you
are so nosy. Why do you bother about what I do” (IntJA1). When Jane was a school
171
teacher, “you wouldn‟t dare to share. People [teachers] would have thought you‟re so
arrogant. Who do you think you are?” (IntJA1). So, she felt that this apprehension of
being thought of as a nosy person and a show off could be one of the reasons for
teachers not to converse among themselves about their assessment practices.
Jane also expressed concern with the issue of authenticity. She elaborated that when
using alternative assessment techniques, teachers do not have any control over the
authenticity of students‟ work. This, in her opinion, was different from examinations
where students do not receive any “external assistance, so it‟s really his or her
performance” (IntJA2). To Jane, the internet was a contributor to the problem of
authenticity. As she put it:
The thing you cannot check is, you don‟t know which one copy from
internet. ...I cannot tell and it‟s impossible to check. So whether it‟s a
genuine product of the child or the group, I cannot tell. (IntJA2)
Jane emphasised that if teachers are not able to establish whether the work is done by
the students themselves, then this situation will be disadvantagous to the better students.
According to her, the better students “cannot excel because the not so good students
copy from internet. That thing [work] may turn out to be even better than this good
fella‟s effort” (IntJA2).
For Jane, implementing alternative assessment in the classroom may cause students to
become lazy in their studies. She explained that in a system where examinations are the
main form of assessment, students are constantly under pressure to perform; whereas
the use of alternative assessment creates an environment that is less stressful for
students. As Jane put it, with less pressure “the students can enjoy life” (IntJA2).
172
Another problem that Jane mentioned was that senior teachers may resist change to their
assessment practices because they associated change with more work for teachers. On
top of that, she argued that senior teachers are “so comfortable with their own methods.
They say, why should I change?” (IntJA2). Jane agreed that alternative assessment
would increase teachers‟ workload as these alternative techniques involve students
being assessed on a more regular basis than paper-pencil techniques. The following
excerpt illustrates her point.
I believe to be more work. You are now looking at the child continuously.
...you‟re seeing the child all year round. You‟re paying more attention.
...Definitely more work. It takes up more of your attention on the child.
(IntJA2)
Suggestions to Facilitate Change in Assessment Practices
Jane gave four suggestions that could help make the change to more alternative
assessment a smooth transition. The first of these suggestions is to have uniform
guidelines for the teachers to refer to. Jane suggested that teachers needed to meet
regularly to discuss issues related to assessment; such as the type of assessments that are
suitable to carry out and “the minimum standard we should expect” (IntJA1) from the
students regarding the work they submit. She feels that teachers need to agree on “what
you call good and what you call excellent [work]” (IntJA1). Meetings such as these
would provide opportunities for teachers to discuss and come to a consensus on what is
seen as good quality work and what is not.
The second suggestion that Jane made was to create “a common bank where everybody
contribute” (IntJA1). This „common bank‟ that Jane referred to was a folder created on
the public network that all teachers have access to. Jane‟s idea was to provide a venue
for teachers to share with other teachers the assessment activities that they have tried out
in their respective classrooms, as well as sharing the positive and negative points of
173
using these activities. She believes that teachers tend to “learn a lot from other teachers”
(IntJA1). The following excerpt from the interview further illustrates Jane‟s point.
I think learning from experience is easier than learning from books.
Different people, different approaches. We gather all the points and we see
which one is suitable for us. (IntJA1)
Jane spoke about an incident during a mathematics department meeting last year where
she proposed that “the weightage of the written examination should be heavier”
(IntJA1) than the alternative assessment. Although Jane mentioned that alternative
assessment provides opportunities that “help the weaker ones gain more confidence in
life” (IntJA2), she feels that teachers still needed to clearly discriminate the
achievement levels of the students. One way of doing so, according to Jane, is to
maintain the written examination as this method of assessment would “give them a
chance to show that they‟re better students” (IntJA2).
The last of Jane‟s suggestions was to provide teachers with proper training about
assessment. Jane explained that pre-service teachers at the teacher-training level, as well
as in-service teachers, need to be convinced that alternative assessments techniques are
another legitimate way of assessing their students. If teachers have doubts about these
techniques, they will not be able to implement them „whole-heartedly‟. Jane suggested
that teachers, whether they are pre-service or in-service teachers, need to be given high
quality training on how to properly assess students. The trainers must be people who are
experts in the field of assessment.
Summary of Jane’s Beliefs
Like Kate, Jane‟s early experience with assessment was with examinations and tests. As
a student, her understanding of a good student was someone who achieved high scores
174
in examinations. For Jane, teaching was not just about imparting knowledge, but also
involves teachers in guiding students to become better people with positive
personalities. Jane views learning as not just about knowledge acquisition, but also
about knowing how to use the acquired knowledge. Assessment is something that Jane
believes has to be done continuously. She explained that it is unfair to judge a student‟s
ability based on a single examination.
Classroom assessment provided Jane with information about her students‟ learning as
well as their attitudes towards learning and mathematics. Assessment also informed
Jane‟s students about how well they are doing in their studies. This information may
help in motivating students to work harder to better their mathematical abilities.
Jane associates formal assessment with traditional paper-pencil testing techniques and
views informal assessment as more casual approaches, where students do not have to
worry about the marks or grades. She feels that students should be assessed with a
mixture of traditional and alternative assessment techniques. Some of the different
assessment techniques that Jane employs in the classroom are tests, feedback forms,
quizzes, multiple-choice questions, oral questions, observations, presentations, peer-
assessment and peer feedback.
With the use of varied assessment techniques, Jane is able to gather a variety of
information about her students as they have more opportunities to show their
mathematical ability. This allows her to understand her students better, including their
various intelligences. Issues Jane mentioned regarding the use of alternative assessment
techniques included the difficulty of determining individual contribution in group work
and the authenticity of students‟ work. Further, Jane felt that alternative assessment
175
exerted less pressure on students compared with examinations, and students may take
their studies less seriously. Jane believed that senior teachers might put up some
resistance to the idea of change because they are comfortable with the traditional paper-
pencil testing.
To overcome these barriers, Jane suggested that teachers need to have guidelines that
they could refer to when in doubt. One suggestion was to create a „common bank‟ of
different assessment techniques contributed by other teachers. Jane also mentioned that
appropriate training about assessment was essential for teachers‟ knowledge and
confidence in implementing unfamiliar assessment techniques.
176
Case Study Five: Wendy
Background
Wendy was a young mathematics teacher in her late 20s who has only been teaching for
four years. She is a trained teacher who graduated with a Bachelor of Science with
Education. At the time of the interviews, Wendy was in her final semester of a Master
of Statistics degree. Wendy‟s only experience teaching in a secondary school was when
she was placed for a three-month teaching practicum. Wendy‟s four years of teaching
experience were in private institutions where she taught in pre-university programmes.
Wendy had just moved from another private institution to her current position prior to
the interviews. At her former employment, she was teaching in a programme where
alternative assessment techniques were used extensively to assess the students. In this
regard, Wendy is quite different from the other teachers because she comes from a
background with extensive experience of alternative assessment techniques.
Experience with Assessment
Wendy recalled that during her school days, learning mathematics involved substantial
memory work; “a lot of memorising when it comes to maths” (IntWN1). When she was
a student, Wendy felt that “as long as you‟re doing the past years‟ questions, and you
keep doing the same type of questions, you will be able to score” (IntWN1). She
explained that, as a student, she did not understand why she needed to learn
mathematics. To her, mathematics was a subject that she had to take rather than wanted
to take. Later, as a teacher, she realised that mathematics was a form of knowledge that
she could “apply it in my everyday life” (IntWN1). Hence, Wendy now had come to
appreciate mathematics in a different way because she recognised its application to
everyday life.
177
Wendy described an incident when she was in secondary school where she was required
to prepare a portfolio for a subject that was not mathematics. The subject teacher
instructed the students about what to put into the portfolio, to the extent that “I will just
have to copy everything the teacher tells me” (IntWN1). Another incident happened
when she was in sixth form, one of her science subject teachers provided results for the
experiments during a laboratory practical, “we were not allowed to experiment”
(IntWN1). Therefore, the reports, which were one of the assessments items, were all the
same. Wendy claimed this experience “hindered my interest in science” (IntWN1). She
further claimed that these two incidents hindered her interest in alternative assessment.
At her previous workplace, Wendy was involved in a programme where the teachers, by
referring to guidelines of the syllabus to be covered, were the ones who determined the
amount of content to be assessed as well as how the students were to be assessed. She
explained that the students were assessed with different assessment techniques over and
above the usual paper-pencil tests. According to Wendy, in a semester of 14 weeks of
this programme, “every two or three weeks, we gave them [students] a short
assessment. It could be a written test,...quizzes, assignments, presentations...”
(IntWN1).
Teaching, Learning and Assessment
Wendy described education as a process that involves “drawing out the best from
someone” (IntWN2). She elaborated that an individual goes to school to explore and
find out what their strengths are. Wendy said that education guides the individual to
realise the ability or abilities they possess which enable them to do things.
178
To Wendy, being a teacher meant being a person who could help students realise their
potential. During her first year of teaching, she was „drilling‟ her students for
examinations, and her main concern was for them to obtain an „A‟, not caring about the
way they went about it. After a while, Wendy “realised that it cannot be done because
not everyone is gifted academically” (IntWN1). The following excerpt illustrates an
incident that happened in her first year of teaching that helped her come to this
realisation about students‟ potentials and assessment.
So then,...I think it was during orientation day, I was walking down the
stairs, and this whole group of students who couldn‟t do maths, they could
sing and they could perform. They actually have a band and they perform
...nightly. And like, they earn tonnes of money performing as a band. ...
after a while, I realized, what the heck, so what if you get an “A” for maths.
...I realized education is basically telling someone that, you know, you have
potential to do something. That helped me as a teacher. How does that
influence me in my teaching? Come to assessment, I don‟t assess them
anymore 100 percent examination-based. So a lot of it is these alternative
assessment things. (IntWN2)
When preparing her lessons, Wendy made sure that she incorporated the “real world
kind of things” (IntWN2). Wendy did this so that the lessons were “meaningful to them
[students], they‟ll be able to retain the knowledge longer” (IntWN2). To Wendy,
learning mathematics helps train students “to be obedient”. By this she meant that, “if
this is the way to get the answer, you just have to go through it, accept it” (IntWN3).
The following interview excerpt further illustrates this view.
...when I taught integration, differentiation to students who have never done
additional maths...they will have to learn to accept the formula, the concept,
the product rule and stuff like that, before they can actually get the answer.
So it‟s just a way of thinking, a way of behaving when it comes to maths.
(IntWN3)
According to Wendy, mathematics assessment is different from assessment in other
subjects. To elaborate further, she gave an example of comparing assessment in
mathematics and a subject she had taught previously, thinking skills. According to
179
Wendy, for mathematics, marks are usually awarded when you show the steps that were
followed before arriving at the answer, whereas for thinking skills, marks are given as
long as the student is able to give opinions about an issue and back it up with certain
reasons. For Wendy, mathematics assessment is not just about getting correct answers,
as the following interview excerpt shows.
If a student showed me an answer only ..., I will probably not award him
any mark. If he shows me his reasoning skills and why he actually does
certain steps, I will give him more marks as compared to just the correct
answer. (IntWN1)
As previously mentioned, Wendy stressed that it is important for students to be taught in
a way that results in them experiencing „meaningful‟ learning. Therefore, when it comes
to assessment, Wendy said her assessments are “more on what they can use in their
everyday life. ...not so much on exam topics” (IntWN1). Below is an example she gave
to further explain her point.
...we talk a little bit about compound interest. ...my assessment in that topic
was, I got these students to go out to look for types of interest. One is
simple interest and the other is compound interest. Under simple interest I
actually started off with telling them the number of bankrupts in Malaysia
comes from the age group 24-30. So then I got them to look at credit cards
and what were their interest rate like ... As for compound interest, like when
they‟re going to buy houses, I got them to calculate the interest. So these are
things that you know will be meaningful to them. (IntWN1)
Wendy elaborated further that she did not believe in giving students „surprise‟ quizzes
because she held on to the principle that students must be given time to prepare to be
assessed. She stressed that her tests would only cover topics which she had taught and
would incorporate different levels of skills such as recalling formulas and application.
Another factor that Wendy felt was important when planning an assessment was the
language used. According to Wendy, the language in any assessment item has to be
simple so that students can understand easily what is required of them.
180
Purpose of Classroom Assessment
Wendy had limited views about the purpose of classroom assessment. She did, however,
emphasise that classroom assessment is more important for students than teachers. The
following section presents the purpose of classroom assessment from Wendy‟s point of
view.
The purpose of classroom assessment for teachers
Classroom assessment informed Wendy about her students‟ learning, “whether they‟ve
learnt something that they can apply in their life” (IntWN1). This information helps her
in planning her lessons. For example, from class tests, Wendy could identify the
concepts with which students need further assistance. This, in turn, assisted her in
deciding appropriate activities in future lessons that would enhance the students‟
learning, as well as be in line with her view that students‟ learning must be meaningful.
Wendy used the information obtained from classroom assessments to guide her in
determining forecast grades for her students. According to Wendy, she was required to
give forecasts as it was a requirement when students put in their applications to
university. She takes “the average of what they have done, like all the tests and
assignments” (IntWN1) to provide a forecast for the students.
Moreover, Wendy explained that she had carried out a form of assessment to inform her
about her teaching. She mentioned that when students were not doing well in the class
tests, she would ask them to individually write on a piece of paper their perceived
reasons for their inadequate performance in the tests. An example of a response that she
received from the students was that she was giving them too much homework. They felt
that the amount of homework given was a burden to the extent that “they don‟t have
181
time to study” (IntWN1). Wendy‟s response to this comment was to „cut down‟ on the
amount of homework she was giving them.
The purpose of classroom assessment for students
In Wendy‟s opinion, classroom assessment was important for students because the
results inform students about their learning. From the assessment results, students are
able to ascertain how well they are doing in their studies. According to Wendy, if the
results indicated that they lack certain knowledge, then the students would “need to
study more” (IntWN1). Wendy reiterated that when students do classroom assessment
they are indirectly being trained to comply with instructions. As she put it; “It trains
you to be obedient. Following the steps. These are the steps you need and require to
solve a problem” (IntWN1).
Wendy mentioned that students needed results from classroom assessment for
university applications. The results indicate “how much they already know, the
background they already know to actually get into university” (IntWN1).
Types of Classroom Assessment
When asked how often she gives class tests, Wendy mentioned that she usually gives
her students a test after two or three topics. She also added that “if I‟ve already ended
the chapter, then I‟ll give them a quiz” (IntWN1). It was noted during classroom
observations, she frequently asked her students questions before starting the lesson
(ObsWN1, ObsWN2). Her questions would mostly be about the homework given the
previous day or any previous work assigned.
182
Before starting a new topic, Wendy said she had a question and answer session. She
explained that she did this to “ensure that they [students] understand what had happened
before, so that they can continue on with today‟s class. To refresh their memory.”
(IntWN1). Wendy emphasised that this question and answer session was for new topics
that had some relation to the previous topic she had taught.
Wendy‟s view of formal and informal assessment is based on the requirement of
assigning grades “in black and white” (IntWN1). Wendy views formal assessment as a
form of assessment that requires teachers to assign grades to students. Informal
assessment to Wendy includes any activities, such as students explaining to the teacher
“orally or in written form” (IntWN1) what they have learnt from the lesson, and does
not involve grades being assigned to students.
Classroom Assessment Techniques
The different classroom assessment techniques Wendy said she has used in her
mathematics classroom included class tests, verbal questions, reports, presentations and
projects. She also mentioned that there are three other assessment techniques that she
has recently explored; journals, portfolios and peer feedback. The following discussion
provides Wendy‟s description of the different assessment activities she carried out in
class.
Wendy required her students to do a project, based upon 100 marks, for statistics. This
project was carried out as a group project. According to Wendy, the students were
required to form groups of three, and she allowed the students to choose their own
group members. The reason she gave for doing this was so that “you [students] don‟t
come to me and say so and so didn‟t do any work” (IntWN1). Then, Wendy instructed
183
each group to select a topic that would allow them to collect data. An example of a topic
that Wendy‟s students came up with was the number of calories in a burger. Then,
Wendy explained that she distributed guidelines about how the 100 marks would be
distributed. The guidelines, Wendy explained, informed students of the aspects that
would be judged and the distribution of marks.
In addition to projects, Wendy‟s students were also exposed to presentations as a form
of classroom assessment. There were two instances where Wendy used presentations to
assess her students. The first was during her teaching, where it was observed (ObsWN1,
ObsWN2) that Wendy would call out students to present their solutions on the board for
the other students to see. When asked the reason for doing this, Wendy explained that
when she stood at the back of the class when students present their work on the board,
she is able to experience what the students face when she teaches at the front. Some of
the things that she gathered from observing her students were that it was important to
“list out every single step on the board” (IntWN1), not to “think out loud” too much
because some students may not be able to follow her trail of thought, to make sure her
hand writing on the board was legible, and “I‟ve learnt how to divide the board and
write on it nicely” (IntWN1).
The other situation when presentations are required by Wendy is after the students have
completed their data collection for the statistics project. According to Wendy, each
group is expected to do “a ten-minute presentation on their findings” (IntWN1). The
presentations address how they collected the data, representation of the data, the
analysis of the data and their interpretation of the data. Wendy added that a written
report was to follow the presentation and must be submitted to her for marking and
feedback.
184
The next assessment technique that Wendy explained was verbal questions to her
students, especially when they go to the board and present their solutions (ObsWN1).
Wendy elaborated that by doing so, she could determine “whether they actually did the
question by themselves” (IntWN1), and she could also probe into their thinking and
„see‟ the concept they used. It was also observed that when a student asked Wendy a
question during a lesson, she would first get other students in the class to give their
answer before explaining her answer. There were a lot of questions „thrown‟ to the
students when Wendy was teaching (ObsWN1).
The Need for Change in Assessment Practices
Wendy argued that even if students do not “do too well in their exams, it doesn‟t mean
that they haven‟t acquired any maths knowledge” (IntWN1). When she used alternative
assessment techniques, she became aware that some of the low-achieving students had
other abilities which allowed them to do well in certain ways. At the same time, Wendy
mentioned that through the different assessment techniques, some quiet students showed
that they were able to produce interesting things. The following interview excerpt
illustrates this.
A lot of students who sit quietly at the back of the class, they came out with
wonderful stuff. They could compare if you were to eat the burger from
McDonalds, Burger King and Wendy, which one will make you grow fat
faster. That kind of thing. It was amazing. (IntWN1)
Wendy further added that when students were asked the reason for learning the
mathematical concepts that were taught to them, most of them had no idea. The students
were not interested with the „why‟ of learning these mathematical concepts, but instead
were more concerned that they were doing enough questions that would get them an
„A‟. Wendy mentioned that if students were having such thoughts, then it was
185
„unhealthy‟ thinking. Students needed to learn “because you want to and not because of
the mark” (IntWN1). The interview excerpt below further illustrates this point.
When I asked them why is it that you need to learn exponential function and
log, nobody could tell me why they want to learn it. And I asked them don‟t
you want to know why? They said no. We just want to do the questions and
get a good grade. At the end of the day, if we know all this can we get an A?
(IntWN1)
Another point made by Wendy was that learning should be assessed continuously. She
asserted that students‟ learning should not be judged based on “a one day test, or a one
and a half hour test” (IntWN1). To Wendy, it is unfair judgement if learning was based
solely on paper and pencil testing, “it should not be based on exam only” (IntWN1).
The reason being that the results could disadvantage students if, on the day of testing,
they were facing situations where “could be they have fallen sick, or something
happened in their family” (IntWN1). The students would not have given their best
because situations as such would interfere with their concentration and thinking. Hence,
the results would not be reflective of the students‟ actual academic ability.
Wendy stressed that alternative assessment techniques provide students opportunities to
“learn something that you could never learn from an exam setting” (IntWN1). She
mentioned presentation skills, communication skills and team work as examples of
skills that can be learnt through the students‟ involvement in assessment activities such
as projects and presentations. Wendy also said that it is important that students are able
to communicate the knowledge they possess; as she put it “you have all the knowledge
in your head only it‟s no point. It‟s useless. You must be able to communicate it out.”
(IntWN2).
186
Problems and Concerns about Change to Assessment Practice
Wendy pointed out that there will be teachers who will resist the move to change
teachers‟ assessment practices. She elaborated that there will be teachers who feel that
what they have been doing all this while worked well and there is no necessity to bring
in new ideas or change. “You can‟t control what people feel and what people think”
(IntWN2). She stated that teachers who believe in the need for change should just “keep
on doing, keep on believing, keep on doing” (IntWN2).
Another issue of concern Wendy spoke of during the interview was that teachers find it
difficult to determine whether the work submitted was actually done by the students
themselves. She mentioned that if a student „misuses‟ the opportunity to learn from
doing the work assigned to him or her, then this student will face difficulties when
studying in university. Wendy explained, “maybe when he goes to university, he
doesn‟t know how to do research, involve in plagiarism, you know” (IntWN2).
Two other issues mentioned by Wendy were that doing alternative assessment is time
consuming and also requires a lot of work. She elaborated that activities like projects
and presentations take up quite a lot of classroom time which was limited. This would
be a problem for teachers when the main focus was to complete teaching the syllabus. In
addition, the preparation needed for planning alternative assessment activities require
many things to be done, for example, preparing guidelines and looking up resources and
materials.
A problem that Wendy faced when trying to implement alternative assessment activities
in the classroom was the response from the students. She further explained that for
students who were more examination-oriented they felt that these activities were “a
187
waste of time” (IntWN1), whereas on a more positive note, the students who could not
cope with examinations only, really enjoyed the activities. According to Wendy, after
watching their friends having a good time with the assessment activities, the more
examination-oriented students learnt to join in the fun.
Suggestions to Facilitate Change in Assessment Practices
One of the suggestions made by Wendy was that teachers needed time to adjust to
change, “there always must be a grace period to adjust” (IntWN3). She said that
explanations must be given to teachers about the reasons for the change, because the
teachers need to be convinced that change is necessary. When teachers are convinced
and given time to adapt, then “they [teachers] will be okay” (IntWN3).
According to Wendy, when the teachers are convinced, they need support to move in
the direction of the required change. She elaborated further that support was crucial at
this stage because if the teachers faced difficulties in the course of implementing the
change, they would need “to be able to find support that will respond very quickly”
(IntWN3). An example that Wendy gave was, “probably like an online person that I
could communicate with” (IntWN3).
For assessment activities such as presentations, Wendy suggested that teachers could be
invited into another teacher‟s class to observe the students‟ presentations. Then, the
score for each presentation would be the average of the marks given by all the teachers
present. As for projects, Wendy mentioned that a second marker could be appointed, in
addition to the class teacher, to mark the students‟ projects.
188
Summary of Wendy’s Beliefs
Wendy is a young teacher with only 4 years teaching experience. Before joining her
current employer, she had experience teaching a programme that included alternative
assessment techniques in the classroom. Therefore, she has some experience with these
different methods of assessment.
Wendy remembered that, as a school student, she never understood why she had to
learn mathematics. She realised that during those days, mathematics learning involved a
lot of memory work. Preparation for examinations was all about doing past years‟
questions. Now as a teacher, she has learnt to appreciate mathematics more because of
an awareness of its usefulness in our everyday life.
Wendy perceived a teacher as someone who helps students realise the potential they
have and how to use their respective abilities to function in the world. Education in
Wendy‟s eyes is a process of drawing out the best potential in an individual. As far as
mathematics learning is concerned, Wendy felt that learning mathematics helps students
to be more disciplined because in mathematics, there are steps to be followed when
solving questions.
From Wendy‟s point of view, classroom assessment provides her with information
about whether her students have gained knowledge that they can apply in their everyday
lives. Wendy bases her students‟ university application forecasts on the accumulated
assessment results of the students. Information from classroom assessment also guides
her in planning lessons.
189
In Wendy‟s opinion, classroom assessment informs students about how well they are
coping with their studies. The results guide the students in identifying their strengths
and weaknesses. Students also become aware of how much knowledge they possess
from assessment results and this supports their preparation for study at university.
For Wendy, the difference between formal and informal assessment is that formal
assessment involves assigning grades, whereas informal assessment does not. Before
starting a new topic, Wendy usually has a question and answer session to help students
recall the concepts learnt previously. Other assessment techniques employed by Wendy
in her classroom are class tests, verbal questions, reports, presentations and projects.
Journals and portfolios are two other techniques that she would like to try in the future.
Wendy believed that alternative assessment provides different opportunities for students
to show their academic abilities. Further, alternative assessment activities can „open up‟
the students‟ minds to the use of the mathematical concepts that they learn in their
everyday lives. According to Wendy, the use of alternative assessment provides a fairer
judgement of students‟ academic progress as it is usually carried out in a continuous
manner.
Some of the issues that were of concern to Wendy are the authenticity of the students‟
work that is submitted and the resistance that some teachers may have towards the call
for change in the way they assess students. Wendy also mentioned increase in workload
and time constraints as other issues to be looked into when trying to incorporate these
alternative assessment techniques.
190
A suggestion given by Wendy to overcome these obstacles was to talk to the teachers
and inform them of the rationale for incorporating alternative assessment techniques in
the classroom. Further, she suggested that once teachers are convinced, some form of
support must be provided as a point of reference when they get into any difficulties.
Conclusion
This chapter described the five case study teachers‟ beliefs about assessment. Although
they came from various backgrounds, they held a mixture of similar and contrasting
beliefs about assessment. The descriptive individual case studies provided a „window‟
into these teachers‟ views, thoughts and feelings about assessment as well as their
practices. This allowed the researcher to better comprehend participating teachers‟
beliefs about assessment and their classroom assessment practices. The next chapter
provides a cross-case analysis of the five case studies. The discussion is structured so
that it directly addresses the research questions.
191
CHAPTER FIVE
CROSS-CASE ANALYSIS AND DISCUSSION
Introduction
This chapter has two aims. The first aim is to present a discussion of the key themes that
emerged from the cross-case analysis of the five individual case studies that were
presented in the previous chapter. The second aim is to address the research questions
through this discussion. The overarching aim of this study was to develop
understandings of Malaysian pre-university mathematics teachers‟ beliefs about
classroom assessment. To address this overarching aim, six research questions were
formulated. The six research questions are as follows:
1. What are the teachers‟ beliefs about the purpose of classroom assessment in
the teaching and learning process?
2. What classroom assessment techniques do teachers currently use?
3. How do teachers‟ beliefs influence their selection of classroom assessment
techniques?
4. What do teachers believe are the reasons for change to their classroom
assessment practices?
5. What do teachers believe are barriers to change in their classroom
assessment practices?
6. What do teachers believe are factors that may facilitate change in their
classroom assessment practices?
This chapter is organised into five sections, each of which relates to the research
questions of the study: purpose and role of classroom assessment; teachers‟ classroom
assessment practices; reasons for change in classroom assessment practices; issues and
192
concerns regarding change in classroom assessment practices; and suggestions to
facilitate change in classroom assessment practices. The second and third research
questions are both addressed in the second section, which discusses teachers‟ classroom
assessment practices. For each of these five sections, the discussion is interwoven with
relevant literature to provide a richer insight into the key themes that emerged across all
five case studies. This chapter ends with a brief summary before proceeding to the
concluding chapter.
Purpose and Role of Classroom Assessment
Five key themes emerged from the cross-case analysis of the five teachers‟ beliefs about
the role and purpose of classroom assessment as illustrated in Figure 5.1 below. These
five themes were: (a) feedback to inform instructional decisions; (b) feedback about
students as learners; (c) communication of students‟ progress and achievement; (d)
motivating students in their learning; and (e) fulfilling external requirements.
Figure 5.1 Themes for Teachers‟ Beliefs about the Role and Purpose of Classroom
Assessment.
BELIEFS ABOUT THE
ROLE AND PURPOSE OF
CLASSROOM
ASSESSMENT
Feedback to inform
instructional decisions Motivating students
in their learning
Preparation platform for
external examinations
Feedback about
students as
learners
Communication about students‟
progress & achievement
193
Feedback to Inform Instructional Decisions
All five teachers regarded classroom assessment as a source of information that guides
their instructional decisions with the intention of creating a better teaching and learning
environment. This shared belief is consistent with research findings that teachers gather
evidence and information from classroom activities as well as classroom assessments to
help them make instructional decisions that present students with opportunities to get
the most out of their learning (Stiggins, Arter, Chappius & Chappius, 2006). The
findings of the present study showed that classroom assessment provides all five case
study teachers valuable feedback that directs them in making suitable modifications to
their instructional plans. The literature also indicates with the prevailing view that good
teachers make appropriate alterations to their instruction by gauging their students‟
learning progress through the use of classroom assessment information on a regular
basis (Stiggins, Arter, Chappius & Chappius, 2006; McMillan, 2007). All participating
case study teachers expressed great interest and concern for the well being of their
students as learners and that it is important for them to create a conducive learning
environment in their classroom.
Classroom assessment generates useful information that allows teachers to deduce the
effectiveness of their teaching (Silver & Kenney, 1995; Philippou & Christou, 1997;
Nitko & Brookhart, 2007; Popham, 2008). This was the general feeling among the five
study participants. These teachers said that feedback from classroom assessment allows
them to improve aspects of their teaching that they believe would enable more effective
classroom learning. Teaching pace, teaching approaches, and the pathway of teaching
are examples of aspects of teaching that four of the teachers said were important in
ensuring teaching effectiveness. Lulu and Kate explained that they made adjustments to
their teaching pace to allow students to follow lessons comfortably, at a pace
194
commensurate with their level of mathematical ability. For example, they can teach at a
faster pace if the students are „good‟ students, but they need to teach at a slower pace if
the students are struggling to understand mathematical concepts and processes.
The suitability of teaching approaches adopted in the classroom also was seen by the
participating teachers as important in assisting students to understand mathematical
concepts taught. This idea also is supported by the literature (Panizzon & Pegg, 2008).
The shared view of the teachers was that they need to be aware of the effectiveness of
the teaching approaches that they adopt so that timely adjustments can be made. An
example given was the selection of mathematical examples and questions that teachers
use in their teaching. Kate took the view that teachers of students with high-
mathematical ability can provide difficult and more complex mathematical examples as
well as establish discussions of questions that challenge these students beyond their
current mathematical understanding. Thus classroom assessment assists the teachers in
regulating their use of teaching materials and techniques with the aim of enhancing
students‟ learning.
Three of the five teachers, Lulu, Kate and Wendy, mentioned that they regularly made
decisions about the pathway of their teaching. Classroom assessment presented these
teachers with feedback that is used as a guide for making decisions pertaining to the
direction of their lessons, such as whether to proceed with teaching the next concept or
whether to repeat a lesson. According to Nitko and Brookhart (2007), teachers should
re-teach mathematical concepts or topics that students struggle or fail to comprehend
before moving on to the next concept or topic. Alternatively, if students have
successfully grasped the intended mathematical concept or topic, then the teacher can
choose to further develop the students‟ knowledge of these concepts or topics or
195
advance to a different concept or topic (Brookhart & Nitko, 2008). It seems from the
findings of the present study that some teachers do regularly use this approach as a
means to organise and guide their instructional design and curriculum delivery.
The participating teachers also said they used feedback to inform instructional decisions
about teaching sequence, a construct closely related to the pathway of teaching. The
literature also indicates that information drawn from classroom assessment assists
teachers in making decisions about the need to revamp their planned teaching sequence
(Popham, 2008). Lulu, Kate and Wendy said that a teacher‟s decision about the order in
which mathematical examples are to be presented for a particular lesson or the sequence
in which mathematical concepts of a specific topic are to be taught, would have an
impact on students‟ understanding. They argued that quality presentation and logical
progression of lessons are critical to ensure coherence as well as to enhance students‟
understanding of mathematical concepts. Hence, it seems that for these teachers,
teaching sequence and assessment practices are closely interrelated and inform their
curriculum planning and delivery.
One teacher, Kate, pointed out that information obtained from classroom assessment is
also useful for the planning of future assessments. In Kate‟s case, the emphasis was on
class tests where her students‟ performance was used as an indication of the level of
difficulty of the assessment. According to Kate, teachers can modify and adjust the level
of difficulty and suitability of future assessmentsbased on information from previous
assessments. This gives the impression that, for Kate, information from classroom
assessment does not just inform alterations to the planning of classroom instruction and
delivery, but also guides the creation of future assessments as well as modification of
current assessments that are to be further implemented. This concurs with the view that
196
assessment provides information for future actions of teachers (White, 2007), in this
case, the planning of future assessments.
Feedback about Students as Learners
All participating teachers conveyed that a most significant purpose of classroom
assessment is to provide feedback about students as learners. The discussion in this
section touches on the different aspects of information generated by classroom
assessment that can help teachers create „profiles‟ of their students as learners. Some
examples of the aspects of such profiles are level of mathematical achievement, learning
areas of concern and ranking. Through these profiles, teachers have a broader
understanding of their students as learnersin relation to both academic content and non-
academic content. Here, academic content refers to the content of the subscribed
mathematics curriculum and non-academic content refers to features such as attitude,
communication skills and personal skills. These profiles also can serve as guidelines to
assist teachers in creating appropriate opportunities for learning that best suit their
students‟ learning needs and, thereby, maximise their learning.
Three of the case study teachers placed great emphasis on the role of classroom
assessment in assisting them to identify the range of students‟ levels of mathematical
achievement. Lulu, Kate and Jane believe that the students‟ level of mathematical
achievement, as indicated by the results of classroom assessment, also indicates their
levels of understanding of what has been taught. In addition, these three participating
teachers agreed that when teachers are able to distinguish between high-achieving and
low-achieving students, they are able to identify which students face difficulties with
their learning. They articulated a shared view that, with this information in hand,
teachers are able to plan timely and appropriate remedial actions to assist their students
197
in overcoming any learning issues they face (Linn & Miller, 2005). This implies that it
is crucial for teachers to seek information about their students‟ level of mathematical
achievement as this information becomes the guiding factor for planning and
fordeciding appropriate remedial-based instruction. In addition, remedial learning
activities need to be carried out to allow students sufficient time to address their
learning issues before the next round of assessment.
One of these three teachers, Kate, held an interesting view about another use of
information with regard to high and low achievers. Kate uses information about high-
achievers and low-achievers in her class to promote collaborative learning among her
students. She encourages the low-achieving students to approach the high-achieving
students for help; at the same time, requesting the better students to render assistance to
the weaker students. Kate proposed that assessment activities are potential social
activities that can strengthen student-student relationships as well as encourage students
to be independent learners.
In addition to identifying students‟ mathematics achievement, three of the five teachers
said that classroom assessment is a means of identifying their students‟ strengths and
weaknesses. This coincides with a finding of a study conducted by Phillippou and
Christou (1997) which highlighted that teachers considered that one of the main roles of
assessment was to diagnose their students‟ learning. Lulu, Kate and Wendy were of the
opinion that such information assists teachers in planning learning activities that further
enhance students‟ understandings in areas where they already have good comprehension
and improve on learning areas of concern. Students can be strategically guided by this
input about their mathematical achievement to help them address their problematic
learning areas. The findings of this study suggest that when planning their lessons,
198
teachers need to be well-informed about their students‟ strengths and weaknesses. In
addition to guiding suitable selection of curriculum delivery approaches, teachers need
to be mindful when assigning assessment tasks to students. An unnecessary emphasis on
a particular type of assessment activity can disadvantage some students (Panizzon &
Pegg, 2008). These students may unfairly and consistently achieve low marks if a
narrow set of assessment tasks is set by the teacher.
An interesting view that both Kate and Wendy expressed in relation to identifying
students‟ areas of strength and weakness was that classroom assessment provided them
with opportunities to „see‟ their students‟ flow of thought. The students‟ presentations
of solutions in solving mathematical problems are „visuals‟ of their state of
mathematical understanding. With some insight into how their students think, both Kate
and Wendy expressed beliefs that, as teachers, they had a better understanding of why
some of these students faced difficulties in certain learning areas. Teachers who make
good decisions which bring about effective classroom instruction are those who
“understand what students know and how they think” (Davis &McGowen, 2007, p. 19).
Both these case study participants seemed to suggest that teachers need to have „visual
access‟ to their students‟ thinking because it allows teachers to clearly „see‟ their
students‟ shortcomings in their mathematical understandings, hence making better
instructional decisions.
The findings indicated that results from classroom assessment bring to the teachers‟
attention the students‟ ranking among their classmates or among their peers in the
cohort. Lulu and Kate explained that information about students‟ ranking provided them
with a „picture‟ of their students‟ level of mathematical ability and knowledge in
comparison with their peers. As for the students, ranking indicates to them their level of
199
performance as well as their ability to cope with their studies in comparison with their
peers. Both Lulu and Kate expressed views that with this information, the students are
able to judge whether they need to adopt better learning approaches or decide on
necessary actions to be taken to enable them to perform as well as or better than their
peers. This finding suggests that assessment motivates students to work harder and
increase their effort to learn.
Two of the five participating teachers expressed the belief that classroom assessment
provided opportunities for teachers to become better acquainted with their students
personally as individuals. Lulu and Jane were the only teachers in this study who said
that classroom assessment was not just about informing teachers about their students‟
academic ability or learning progress. They claimed that classroom assessment also was
useful for obtaining other information about students such as their attitudes towards
learning, views about mathematics (Lowery, 2003), and their level of self-confidence.
Lulu and Jane argued that if teachers had a more comprehensive understanding of their
students, they would be better informed and be able to take appropriate measures to
assist them in their mathematics learning.
Communication about Students’ Progress and Achievement
All five participating teachers were in agreement with the view that classroom
assessment generates information that serves as evidence of students‟ learning progress
and achievement. In addition to teachers and students, the teachers explained, parents
also have an interest in classroom assessment. Assessment information is shared with
parents to keep them informed about their child‟s level of achievement and current
progress (Nitko & Broadfoot, 2007; Stiggins, 2008). Kate stressed that this form of
communication was crucial for parents and students alike; because access to such
200
information prevents misleading assumptions about „how well‟ the students were doing
in their studies.
The teachers in this study explained that they drew on the results from classroom
assessment to guide them in giving forecast grades to students as well as writing
individual student testimonials. The necessity for these forecast grades and testimonials
arose because they are part of the requirements for university applications made by their
students. In instances where formal assessment results are unavailable for use, teachers
can refer to the classroom assessment information as a guide to communicate students‟
progress and achievement to the required party.
Consequently, assessment results are seen as the main source of information for gauging
students‟ academic performance for the purpose of communicating with various
stakeholders. Another use of assessment information by the teachers in this study was to
assist in predicting their students‟ actual examination results. This coincides with
research (Buhagiar& Murphy, 2008) that reported teachers claimed that with „sufficient
knowledge‟ of students from classroom assessments, they are able to forecast the
students‟ actual examination results.
Motivating Students in their Learning
Classroom assessment was viewed by two of the participants of this study, Jane and
Wendy, as a source of motivation for students, however, they differed in their opinions
about the way classroom assessment motivates students in their learning.
Jane conveyed the belief that results from classroom assessment were a driving force
that motivates students in their learning. For the better students, receiving good results
201
meant that they had performed well in the given assessment. The feeling of achievement
made them feel good about themselves; indirectly adding to their self-confidence with
regard to learning mathematics. Jane felt that when low-achieving students witness the
joy of getting good results among their peers, they may be motivated to study harder to
improve their performance. This finding is consistent with a finding of a study by
Pedersen, Williams and Bulu (2004) which highlighted that teachers generally believe
grades can motivate students.
Wendy, on the other hand, expressed her belief that when students have experienced
success in solving mathematical problems, they will show more interest in learning
mathematics. Success in solving mathematical problems is seen by the students as a
sign that they are capable of doing mathematics. Research has shown that students
involved in problem-solving activities develop self-confidence because they need to
have “confidence to take necessary risks in problem-solving” (Suurtamm, 2004, p.503).
As a result, students show more confidence in their ability to learn mathematics.
Jane and Wendy emphasised that, in addition to gauging students‟ learning in the
classroom, assessment activities have the potential to create confident and independent
mathematics learners.
Preparation Platform for External Examinations
Two participants, Kate and Lulu, thought of classroom assessment as a platform to
prepare students to face external examinations. These case study teachers stated that
classroom assessment, specifically class tests, prepares their students for the final
external examination at the end of the course. Both participating teachers explained that
exposure to class tests was a way of familiarising students with the process and format
202
of the formal examination. Of the two teachers, Kate articulately argued that an
important purpose of assessment is a preparation platform and linked this with her value
of preparing her students for examinations. It seems that for teachers like Kate, whose
students face high-stakes examinations, one of their priorities in the classroom is to
prepare their students to pass (Buhagiar& Murphy, 2008). This is often achieved by
duplicating the actual assessment items and the examination environment through
classroom assessment.
Classroom Assessment Practices
This section on classroom assessment practices is divided into two subsections. The
first subsection presents a discussion of the nature of the assessment practices of the
five case study teachers. The second subsection presents an overviewof the different
assessment techniques employed by the case study teachers.
Nature of Assessment Practices
In this subsection, the case study teachers are grouped into three categories based on the
nature of their classroom assessment practices. The basis for these categories is the
extent of these teachers‟ use of traditional and alternative assessment techniques to
assess their students. The three categories are the (a) innovative assessor, (b) „sitting on
the fence‟ assessor, and (c) traditional assessor. The discussion in this subsection
includes the characteristics of each category and the relation of these characteristics to
the teachers‟ beliefs about classroom assessment. Figure 5.2 is a diagrammatic
representation of these three categories and Table 5.1 is a summary of the different
assessment techniques employed by the participants.
203
Figure 5.2 Categories of Teachers‟ Assessment Practices
CATEGORIES
OF TEACHER
ASSESSMENT
PRACTICES
INNOVATIVE ASSESSOR
Uses alternative assessment
techniques more frequently
than traditional assessment
techniques
‘SITTING ON THE FENCE’
ASSESSOR
Attempts to achieve balance
between the useof
alternative and traditional
assessment techniques
TRADITIONAL ASSESSOR
Uses traditional assessment
techniques more frequently
than alternative assessment
techniques
204
Table 5.1 Traditional and Alternative Assessment Techniques Employed by
Participants
Participant Traditional Assessment
Techniques
Alternative Assessment
Techniques
Lulu Class test
Multiple-choice questions
Oral questions
Mathematics journal
Feedback form
Project
Quiz
Reflective log
Peer feedback
Presentation
Directed investigation
Mathematics game
Kate Class test Presentation
Oral question
Mind map
Assignment
Jane Class test
Multiple-choice questions
Feedback form
Oral questions
Observation
Presentation
Quiz
Peer feedback
Anakin Class test
Homework
Presentation
Oral question
Quiz
Peer feedback
Wendy Class test Oral questions
Project
Report
Presentation
Peer feedback
Journal
Portfolio
205
The innovative assessor employs alternative assessment techniques more frequently
than traditional assessment techniques. Some of the alternative assessment modes used
included projects, journal writing, reflective forms, peer feedback and presentations.
Lulu and Wendy were the two teachers included in this category and each used more
than six alternative assessment techniques. Lulu was deemed to be „highly‟ innovative
because she used alternative assessment modes extensively. Wendy was considered an
innovative assessor even though she used few alternative assessment techniques in
comparison with Lulu. The findings in the previous chapter suggest that both teachers
believe that alternative assessment was significantly important because the focus of
learning at present is to prepare students to function in the real world and is not only
about their ability to get high grades.
Both Lulu‟s and Wendy‟s expressed beliefs about assessment emphasised the
importance of providing continuous feedback to students to guide them in their learning.
Students need to be informed constantly about their progress to allow them to decide on
steps to improve gradually; thereby, encouraging students to take responsibility for their
own learning. They expressed the view that education is about preparing students with
skills that will enable them to function effectively in the real world. Alternative
assessment modes provide exposure to such skills, for example teamwork,
communication and management skills. Guided by these beliefs, Lulu and Wendy‟s
selection of assessment techniques encompass a wide range of alternative assessment
techniques compared with the other teachers in the study.
The „sitting on the fence‟ assessor tries to maintain a reasonable balance between the
use of alternative and traditional assessment modes in the classroom. In this category,
the participants used four to six alternative assessment techniques. Jane and Anakin
206
expressed the belief that alternative assessment techniques are just as useful as
traditional assessment techniques, therefore, these teachers juggled the two. Their
beliefs incorporate the idea that traditional assessment provides „reliable‟ information
about students‟ learning, whereas alternative assessment offers students of various
calibre an opportunity to „feel good‟ about themselves and for teachers to „know‟ their
students better. Although trying to maintain some balance between alternative and
traditional assessment, they both hold the belief that traditional assessment is slightly
more „superior‟ to alternative assessment because, to them, most importantly,
assessment must be reliable and fair. Their view is that the results from alternative
assessment are just „add-ons‟ to help understand students better. These results are not
„suitable‟ to be used as the main indicator of students‟ learning because of their
questionable level of „reliability‟. As a consequence, Jane and Anakin incorporate more
traditional assessment modes in their respective classrooms.
Last but not least, the traditional assessor is a strong believer in the value of traditional
paper-pencil assessment. Written tests are the main form of classroom assessment
employed by the traditional assessor. Only one teacher in this study fell into this
category. Kate is result-oriented as her main concern is preparing her students for
examinations and she only used four alternative assedssment techniques. Kate clearly
explained her belief that traditional assessment is a fair and standardised form of
assessment. She was not against the idea of using alternative assessment in the
classroom, but she holds the belief that such assessment approaches simply provide
weaker students, or students with examination anxiety, with a different venue to show
their learning capabilities. She expressed the conviction, however, that students should
always „officially‟ be assessed by means of paper-and-pencil testing because, in Kate‟s
opinion, alternative assessment lacks rigour and reliability.
207
Classroom Assessment Techniques
The findings reported in the previous chapter potrayed a variety of assessment
techniques employed by all the case study teachers. These ranged from traditional
paper-pencil tests to alternative techniques such as projects, presentations and journals.
This subsection provides an overview of these various assessment techniques.
All case study teachers „embraced‟ the new component in the college‟s assessment
policy, which evolved from only traditional paper-pencil tests to a mixture of traditional
and alternative techniques, with diverse levels of „commitment‟. Of the five
participating teachers, Lulu showed a „high‟ dedication towards the implementation of
alternative assessment. Many of the classroom assessment techniques Lulu used were
unfamiliar to the majority of teachers in the pre-university programme in which she was
involved. Some of these techniques are mathematics journals, reflective logs, directed
investigation and peer assessment. Kate, on the other hand, could be considered the
opposite of Lulu. Kate demonstrated a lack of interest in alternative assessment
techniques. Her classroom assessment practice was dominated by traditional tests and
examinations.
Peer feedback was a technique that was of interest to four of the case study teachers;
Lulu, Anakin, Wendy and Jane. Although they said they did not have much experience
with this technique, all four teachers had implemented it in their classes. There were
times when peer assessment was carried out successfully in their classes and at times,
they failed. The findings showed that these four teachers were keen to make relevant
adjustments, improvements and learn more about peer assessment to ensure better
implementation in future.
208
Presentation was an assessment technique that was popular among four of the case
study teachers, with the exception of Kate. These presentations varied from individual
presentations to group presentations or coming up to the board and presenting solutions.
Oral questioning was a technique that was used by all five participants. During
classroom observations, it was noticed that oral questioning and presentations were two
frequently used assessment techniques.
An interesting finding from this study was Anakin‟s „devotion‟ to homework as a form
of assessment. Although considered by most teachers to be a common activity, Anakin
provided a strong argument that homework is a dominant assessment activity in his
classes. This finding provides a „fresh‟ perspective on the role and purpose of
homework in the classroom.
Although the five case study mathematics teachers were from the same pre-university
programme, the findings of this study revealed an array of assessment techniques they
employed in their classes. This finding suggests that individual teachers have individual
preferences with regard to assessment techniques, guided by their beliefs. Therefore,
this characteristic of the teachers need to be acknowledged when introducing new
assessment practices.
Reasons for Change to Classroom Assessment Practices
This section synthesises the reasons that the case study teachers believe warrant a
change to assessment practices in the classroom. All teachers, with the exception of
Kate, expressed the belief that there was a need for change to be made to their present
classroom assessment practices. Kate was the only teacher who was not really
convinced that her classroom assessment practices required change. She made it very
209
clear that if she had to adopt alternative assessment practices, the reason for doing so
would be because it was a trend. Four themes emerged from the cross-case analysis that
looked at the perceived reasons for change to be made to current assessment practices
across the five case studies. They are (a) „teaching to the test, learning for the test‟, (b)
developing the „whole‟ individual, (c) enjoying the learning process, and (d)
informative and fair assessment. These themes are presented in Figure 5.3.
Figure 5.3 Themes for Teachers‟ Beliefs about Reasons for Changes to be made to
Classroom Assessment Practices.
‘Teaching to the test, learning for the test’
All case study teachers, with the exception of Kate, were of the opinion that one reason
why teachers need to change their assessment practices was that they tend to „teach to
the test‟. The shared belief was that the „culprit‟ behind this issue was the current
external assessment system, prescribed by the external curriculum provider, is result-
oriented and emphasises memory work. Research indicated that there is a growing trend
that advocates multiple approaches to assessment because traditional paper-pencil
BELIEFS ABOUT REASONS
FOR CHANGE TO
CLASSROOM ASSESSMENT
PRACTICES
Informative and fair
assessment Enjoying the
learning process
Development of the
„whole‟ individual
„Teaching to the test, learning
for the test‟
210
methods are not capable of addressing every learners‟ needs (Lowery, 2003) although it
is reported that one-third to one-half of teachers‟ time is spent on assessment-related
activities (Stiggins & Conklin, 1992). The four participating teachers felt that such
practice creates a learning environment where teachers channel most of their energy into
preparing students for examinations. The effect of this practice on students is that
students „learn for the test‟ and perceive learning as merely familiarising themselves
with mathematical content and items that are similar to those in the actual examination.
As for teachers, the effect of „teaching to the test, learning for the test‟ may see them
planning learning activities around particular mathematical topics or concepts that are
deemed likely to appear in the examinations; as well as constructing assessment items
that are as similar as possible to those in the actual examination. The case study teachers
expressed concern that such practices and activities may create a learning culture that
focuses on memorizing rather than understanding, and stresses „reproduction‟ of
mathematical knowledge and skills (Buhagiar& Murphy, 2008). In short, such
circumstances may highlight the focus of mathematics assessment on getting the
expected „product‟ or the „right answer‟; a situation not supported by Lulu, Jane,
Anakin and Wendy.
Further, Jane and Wendy highlighted that the continuous emphasis on examinations,
results and grades brings about negative conceptions about achievement and learning
among their students. They noticed that students tend to label themselves „failures‟ if
their assessment results, especially results from traditional paper-pencil tests, are not
good. Wendy pointed out that some students may interpret their inability to achieve
good results as an indication that they have poor knowledge and poor memory. She
viewed this perception as unjust and misleading for the affected students.Although it
211
was discussed earlier that classroom assessment can help motivate students, it was also
noted that if teachers put a negative emphasis on the role of assessment, it can distort
the true value of assessment as a motivator for learning.
Developing the ‘Whole’ Individual
In this section, the „whole‟ individual refers to a broader and holistic vision of the
individual, encompassing the intellectual domain, affective domain and personal skills.
In an environment where the emphasis is on examinations and results, little attention is
given to the development of the „whole‟ individual. Four of the five participants, Lulu,
Jane, Anakin and Wendy were of the opinion that the implementation of alternative
assessment techniques, such as presentations and projects, in the classroom provides an
environment that assists in the enhancement of the students‟ affective domain and
personal skills. The affective domain includes factors such as student motivation,
attitudes and values; whereas personal skills includes communication skills, ability to
build rapport, being tolerant towards others, leadership skills, organisational skills and
working in a team. As for the intellectual domain, alternative assessment approaches
encouraged students to hone their thinking skills to raise their general levels of
cognition.
These four case study teachers discovered that, for students who struggled to do well in
the traditional paper-pencil approach to assessment, the use of alternative assessment
techniques in the classroom made it possible for them to showcase their strengths in
their learning that could not be captured by paper-and-pencil testing. This, in turn,
helped enhance the students‟ self-confidence because they felt they were able to
„produce or do something‟ as evidence of their individual learning abilities.Suurtamm
(2004) emphasised that it was important to develop students‟ self-confidence for the
212
enhancement of mathematics learning, because mathematics was about solving
problems.
According to Lulu and Jane, students need to develop into „whole‟ individuals for their
survival in the workforce and the real world. They shared the view that teachers
focussing on examinations alone are not providing their students with opportunities to
adequately develop survival skills that are much needed by the students to function
outside the classroom. These case study teachers agreed that it is important for teachers
to be aware of this aspect of assessment and create as many assessment opportunities as
possible that would assist in their students‟ development into a „whole‟ individual and
take on their role effectively in society. The findings of this study suggest that
classroom activities, whether learning or assessment activities, should address the needs
of students seeking to prepare themselves with the necessary skills to survive in the
workforce as well as in their daily lives.
Enjoying the Learning Process
Two participants, Anakin and Wendy, stated that the stress and anxiety, faced by most
students due to the emphasis on examinations and grades, deters students from enjoying
the process of learning mathematics. They take the view that, as a result of emphasis on
grades, students do not show any interest in knowing the reason for learning a particular
mathematical concept or topic. Neither do students bother to understand what they are
learning, as they are constantly paying attention to rote learning to prepare for
examinations. Both case study teachers argued that the view that mathematics
assessment is only about getting correct answers has to change as this has, more often
than not, been over-emphasised in the classroom.
213
Both these teachers believe that if there was a change in the way students were assessed
and there was a reduced emphasis on examinations, then students would be more likely
to enjoy the learning process. When emphasis is on learning rather than grades,
“students are more likely to be intrinsically motivated and encouraged to take risks that
challenge and expand their learning” (Davies, 2007, p.34). Teachers will thus be able to
engage students in assessment activities that could help enhance their understanding of
the mathematical concepts as well as enjoy learning in a meaningful way. Hence, a
better learning environment would exist in the classroom.
Informative and Fair Assessment
Traditional paper-and-pencil assessment techniques such as examinations and tests are
commonly used to assess students. Although that is the case, Lulu, Jane, Anakin and
Wendy shared the belief that it is risky and unfair to assess students‟ learning through
the use of tests and examinations only. Jane stressed that some students have
examination anxiety where they feel fearful of sitting examinations. Furthermore,
Wendy and Jane expressed concern that a student could face an unexpected event on the
day of the examination. Situations such as receiving news of a death in the family or
feeling ill, for example, might impact negatively on a student‟s performance in the
examination. Surprisingly, although Kate felt strongly that there was no necessity for
change to her classroom assessment practices, she shared Jane‟s view about the negative
impact of examination anxiety. In this one instance, Kate conceded that alternative
assessment techniques may be one way of addressing this situation.
An additional but related and common view among the teachers in this study was that,
in order for them to better understand students‟ learning they need to be assessed
continuously. With more information from assessment of their students‟ learning,
214
teachers would have a better „picture‟ of the state of their students‟ learning progress.
This, in turn, would provide the teachers and students with better information with
which to address any shortcomings in the learning process.
One of the case study teachers, Jane, emphasised that different students have different
ways of learning and, because of this, she strongly believes that teachers need to create
a fair assessment environment that caters for this difference among students.
Assessment tasks that take into account the diversity of students provide broader and
better opportunities for students to display their varied capabilities (Gipps & Stobart,
2009). Hence, students will not be at a disadvantage because of any one type of
assessment that may be emphasised in the classroom.
Issues and Concerns Regarding Change in Assessment Practices
Whenever there is a call for change, there is always a sense of apprehension from the
stakeholders. All participants, including Kate, expressed concerns and mentioned issues
in relation to change in teachers‟ classroom assessment practices. Kate expressesd the
understanding that the current assessment practices were sufficient, but if change in
assessment practices was to take place, then she had some concerns. From the cross-
case analysis, five themes emerged: (a) resistance to change, (b) reluctance to share, (c)
authenticity, (d) students‟ response, and (e) issues with assessment techniques. These
themes are represented in Figure 5.4.
215
Figure 5.4 Themes for Teachers‟ Beliefs about Issues and Concerns Regarding Change
in Classroom Assessment Practices
Resistance to Change
There were several factors that contributed to teachers‟ resistance to change in their
classroom assessment practices. Fear of change among teachers, especially experienced
teachers, was a common view shared by four of the five participating teachers. This fear
was seen as one of the contributing factors to resistance to change. Lulu, Kate, Jane and
Wendy felt that this fear of change was grounded in the lack of experience in moving
away from familiar assessment techniques to unfamiliar ones. Moreover, case study
teachers said there was likely to be strong concern when teachers feel the current
assessment techniques have „worked well‟ in terms of student assessment. Experienced
teachers may express reservations about change to their classroom assessment practices.
They may feel doubtful about the effectiveness of unfamiliar techniques and Lulu, Kate
and Jane said there were teachers who still strongly maintain that for mathematics, „test
is the best‟. In addition, these teachers also need time to learn new assessment
techniques; something many would be unwilling to do as they felt that it was
BELIEFS ABOUT ISSUES
ANDCONCERNSREGARDING
CHANGE TO CLASSROOM
ASSESSMENT PRACTICES
Resistance to change
Reluctance to share
Authenticity
Students‟
response
Issues with assessment
techniques
216
cumbersome and an additional burden. This suggests that when change or reform is
proposed, careful attention must be given to the reaction of experienced teachers,
especially those who are very comfortable with the current assessment system which
they believe have always produced high-achieving students.
Another contributing factor was the belief that the incorporation of alternative
assessment techniques would increase teachers‟ workload. Lulu, Jane and Wendy
explained that in their experience, alternative assessment techniques typically require
students to be assessed more often. On top of that, preparation and marking time for
assessment activities like projects and presentations is increased, as they often involve
creating guidelines or rubrics.
Time is essential to teachers as they constantly struggle to complete teaching the
required syllabus within the given timeline. Lulu, Kate and Wendy agreed that
conducting alternative assessment techniques is often time-consuming. For example,
group presentations can take up a lot of lesson time as each group is required to do their
presentation in front of their peers. The three teachers expressed concern that
assessment activities such as these interfere with their normal teaching schedule and
hinder them from finishing the syllabus. For Kate, completing the syllabus was the
highest priority for her as a teacher. The case study teachers, therefore, seemed to
propose that careful instructional planning involving allocation of time for teaching and
assessment activities is crucial for the successful implementation of alternative
assessment techniques.
217
Reluctance to Share
Jane was the only participant who expressed concern about the lack of sharing among
teachers with regard to classroom assessment. She explained her belief that if teachers
were pro-active in sharing with other teachers their experiences in using different
assessment techniques, this can assist a smooth transition from a dominant traditional
paper-pencil assessment culture to using a mixture of traditional and alternative
assessment techniques.
In addition, Jane explained that if teachers were aware that some of their colleagues are
using alternative assessment techniques in their respective classrooms, then they may be
compelled to try these new assessment techniques. The reason being they will not feel
that it is a „one man‟s journey into the unknown‟ as they know of others who are doing
the same. A sense of isolation is always a concern among teachers who are willing to
venture into something new (Suurtamm, 2004). If this issue can be addressed as early as
possible, then more teachers are likely to be keen on adopting alternative assessment
techniques in their classrooms.
Authenticity
Authenticity of students‟ work was a major concern raised by all teachers in this study,
with the exceptionof Anakin. Lulu, Jane, Kate and Wendy were worried that some
students hand in work that may not have been done by them. They stated that some
students may have had extensive assistance in completing assessment tasks; the worst
case scenario suggested was that some students engaged another person, such as a
parent, to do their work for them.
218
Another issue of concern for these four teachers in relation to the authenticity of
students‟ work is plagiarism among students. They commented that it is difficult to
ascertain whether work handed in is original or whether plagiarism had taken place.
Kate, one of the four teachers, highlighted and questioned the issue of fairness for those
students who might have done the work on their own, without any assistance. She felt
that the internet is the main source of plagiarism among students in completing
assessment tasks such as projects and presentations. This issue creates a disadvantage
for students who do the work without any assistance. Hence, teachers expressed concern
that good marks obtained for assessment activities such as projects may not be a true
reflection of the student‟s ability.
Students’ Response
Any implementation of new assessment techniques in the classroom will affect students
as well as teachers. An issue that Lulu and Wendy had to face when trying to implement
alternative assessment techniques was the negative response from some students,
especially those who were performing well in the current assessment system. These
students felt that it was a waste of their time to be involved in assessment activities that
had no bearing on their final results. Lulu and Wendy both felt that students with this
thinking lost out on the benefits of being involved in such assessment activities.
Jane drew attention to the issue that with the introduction of alternative assessment in
the classroom, students may begin to be less diligent with their studies. This was
because students may feel less pressure with alternative assessments compared with
examinations. Therefore, Jane emphasised that it is important for students to understand
that alternative assessment activities are just as important, are not „easier‟ than
examinations, but simply a different approach.
219
Issues with Assessment Techniques
All five case study teachers had, to some extent, attempted the use of alternative
assessment techniques in their respective classrooms. In the process of implementing
these techniques, they encountered some obstacles. For example, they faced difficulties
in identifying the level of individual commitment for group assignments. They were
aware of some students putting in more effort than others, but trying to determine „how
much‟ effort was put in by each group member was a challenge.
Other issues that the case study teachers had to deal with were the lack of exposure of
their students to certain assessment techniques like peer assessment, for example; as
well as preparations of assessment guidelines, rubrics and materials. For example, Lulu
encountered problems with her students, especially the weaker ones, when she tried to
implement peer assessment. Her students had no exposure to this form of assessment
and did not show a level of seriousness when asked to assess their peers. The task of
developing peer assessment skills is challenging (Bryant & Carless, 2010) and requires
consistent practice to be competent (Sadler, 1989). Another example came from Anakin.
He struggled with the task of producing „easy-to-follow‟ guidelines and rubrics for
marking. They all admitted that they struggled to come up with comprehensive
guidelines for students to follow and rubrics for marking. Designing a rubric can be a
challenging and daunting task, especially for those who are novice (Butler & McMunn,
2006).
Another issue that some of the teachers had with the implementation of alternative
assessment was related to time. Kate, Wendy and Anakin felt that the preparation and
„execution‟ of alternative assessment modes in the classroom was rather time-
consuming. For example, Anakin explained that he had to spend a lot of time going
220
through the Power Point slides of each group with the intent to provide feedback on the
contents of the slides before the presentation. Kate mentioned that she would not mind
trying out any alternative assessment modes, provided they do not take up her teaching
time or disrupt the teaching schedule.
Suggestions to Facilitate Change in Assessment Practices
This final section presents the suggestions made by the case study teachers to help
facilitate change in their assessment practices. Four themes emerged from the cross-case
analysis. They are:(a) professional development, (b) peer collaboration and support, (c)
implementation, and (d) awareness, as illustrated in Figure 5.5.
Figure 5.5 Themes for Teachers‟ Beliefs about Suggestions to Facilitate Change in
Assessment Practices
Professional Development
All participating teachers, except Wendy, stated that teachers should attend workshops
that provide them with practical knowledge about alternative assessment as well as
hands-on activities on how to implement these techniques. They felt that they would not
BELIEFS ABOUT SUGGESTIONS
TO FACILITATE CHANGE IN
CLASSROOM ASSESSMENT
PRACTICES
Professional
development Peer collaboration
and support
Awareness
Implementation
221
benefit from workshops or training that put emphasis on the theoretical rather than
practical aspects of alternative assessment.
Another suggestion was for resources, such as assessment books, to be made available
to teachers for reference at any time. Teachers would be able to further expand their
knowledge on the basics of assessment and also obtain ideas on how to create
alternative assessment activities for their classroom.
Peer Collaboration and Support
A suggestion made by Jane, Anakin and Wendy emphasised the role of peers in an
environment of change. They asserted that peer collaboration and support from their
colleagues was important to persuade teachers to embrace change. Peer sharing
sessions were deemed a good way of exchanging ideas about assessment techniques that
had been carried out and improved by other teachers. This activity allows teachers to
engage in active conversation with their peers about the issues that they face when they
implement certain assessment activities or get advice on how best to use a particular
assessment technique for their respective classes. This drive towards comradeship
among teachers can help build their confidence and overcome their fear of exploring the
use of alternative assessment techniques.
Another suggestion was to create an electronic „common bank‟ of assessment
techniques where any teacher could share their materials for the assessment activities
they had used successfully in their classrooms. The common view was that this would
be a valuable resource for teachers searching for ideas to vary their assessment
activities. Furthermore, when a teacher has adapted another teacher‟s assessment
activities, he or she could add on an improved version of the assessment activity. This
222
way, there will always be a variety of high quality assessment activities for teachers to
use.
Implementation
As discussed earlier, authenticity of students‟ work was an on-going issue for the
teachers with regard to alternative assessment. One suggestion to overcome this issue
was to make sure that certain aspects of the assessment activity are completed in class
under the teacher‟s supervision. This will increase the authenticity of students‟ work to
a certain extent.
Teachers from the case studies agreed that students must be briefed on what is expected
of them in the assessment activity assigned to them. This will create a sense of fairness
for the students, because it would be unfair for the teachers to grade students‟ work if
the students are not aware of the marking criteria. Therefore for effective assessment
and learning to take place teachers need to make explicit to students the assessment
criteria and students understand these criteria clearly (Elwood & Klenowski, 2002).
As Wendy put it, implementation of any change must be given a „grace period‟ to allow
all involved in the change to familiarise themselves with what is new to them.
According to Wendy, if teachers are given sufficient time to address changes to their
assessment practices, they will be able to accommodate the change better.
Awareness
Lulu and Jane suggested that teachers need to be made aware of any changes that are to
take place well before the change is implemented. This is to avoid unnecessary
misunderstanding about the purpose of the change. The teachers need to be briefed on
223
the changes to be made, and convinced that these changes are in their interest as well
the students‟ interest. Only when teachers are convinced about the reason for change,
can the change be implemented with a high degree of success (Battista, 1994; Handal&
Herrington, 2003).
It was also suggested by Lulu that parents should be briefed about changes that are to
take effect, as this concerns the education of their children. She elaborated that it was
important for parents to have this awareness, as their complaints can sometimes hinder
educational change.
Conclusion
This chapter discussed the themes that emerged from the cross-case analysis of the five
participants and was structured to address the research questions. Although the teachers
came from varied backgrounds, there were more commonalities than differences in
terms of their beliefs in each section. The next chapter concludes the thesis.
224
CHAPTER SIX
CONCLUSION
Introduction
The purpose of this chapter is to summarise and conclude the research reported in this
thesis. The chapter is divided into four sections. The first section provides an overview
of the research, briefly recounting the aims and background of the study, and the
methodology adopted for data collection and the approach to data analysis. The second
section presents a summary of the results, highlighting the key findings of the study.
The third section provides an argument about the implications of the research for
professional practice, particularly in Malaysia, but also for broader contexts of
mathematics assessment in other countries. The fourth section outlines the limitations of
this study and finally, the chapter concludes with a reflective postscript.
Overview of the Research
The study reported in this thesis set out with the overall aim to develop understandings
of Malaysian pre-university mathematics teachers‟ beliefs about classroom assessment.
This overall aim was divided into three subsidiary aims: to develop an understanding of
teachers‟ beliefs about the purpose of classroom assessment; to develop an
understanding of teachers‟ classroom practice with regard to mathematics assessment;
and to develop an understanding of teachers‟ beliefs about change to their assessment
practices.
Recently, assessment has been thought to have an increased significance in the teaching
and learning process. Assessment is an important part of teachers‟ classroom practice
that is of high current interest and critical scrutiny in education (Cumming & Wyatt-
225
Smith, 2009). Changing views about the teaching and learning of mathematics has put
more emphasis on communication, problem-solving and logical reasoning (Van de
Walle, 2007) and has indirectly transformed the role of assessment into a more active
one. Current assessment trends have moved schools and their teachers towards the use
of more alternative forms of assessment, focusing on increasing the role of assessment
in supporting students‟ learning rather than simply documenting what they know for
grading and reporting (Nisbet & Warren, 2000).
This global shift in the focus of assessment from „measuring‟ students‟ achievement to
the additional and important role of supporting students‟ learning has influenced the
national assessment reform movements in the Asian region, for example, in countries
such as Hong Kong and Singapore. Assessment in the Asian region has, for a long time,
been rooted in a traditional testing culture where examinations and tests are the main,
and often only forms of student assessment. Reform movements in countries within this
region have focused on finding a balance between formative and summative assessment.
At the time of the genesis of the doctoral research presented in this thesis, in 2007, the
Malaysian government made public its intention to revamp the national assessment
system from a narrow focus on high-stakes testing to a more holistic assessment system
with more alternative forms of assessment to be conducted in the classroom (Tuah,
2007).
Research has indicated that for any educational change to be introduced and
implemented with some degree of success, it is critical for participating teachers‟ beliefs
to be taken into consideration (Isenburg, 1990; Fullan, 1993). Teachers are the agents of
educational change. When teachers‟ beliefs are in line with the principles that underpin
the reform, they are more likely to implement the changes with confidence.
226
To achieve the aims of this study, a qualitative multiple case study was selected as the
most suitable approach to gain insight into the participants‟ „world‟ through their eyes.
The participants of the study, selected through purposive sampling, comprised pre-
university mathematics teachers from one college in Malaysia. The teachers provided a
unique, informative and informed sample due to the progressive assessment policy
adopted in the college prior to widespread reform in other schools in the Malaysian
context. Semi-structured in-depth interviews, non-participant classroom observation,
document collection and reflective accounts were used to illicit detailed and
multifaceted data from each of the participants. Data analysis was carried out in two
stages. The first stage, within-case analysis, involved analysing the data for each
individual teacher through the use of thematic mind maps and narrative construction.
The second stage, cross-case analysis, involved searching and analysing the cases for
emerging themes and evidence that confirmed and disconfirmed the themes.
The findings of the sresearch were reported in Chapter Four in the form of detailed and
rich case studies of five participating teachers with the intent of providing the reader
with an opportunity to „get to know‟ and understand them as unique individuals and to
gain insight into their prior experiences, beliefs and practices with regard to
mathematics assessment. Chapter Five provided a description and critical discussion of
themes that emerged across the five cases.
Summary of Key Findings
This section presents the key findings of the research reported in this thesis. The
discussion of these findings is organised into three sections which relate to the three
subsidiary aims of the study: (a) teachers‟ beliefs about the role and purpose of
classroom assessment; (b) teachers‟ assessment practices; and (c) teachers‟ beliefs
227
regarding change to assessment practices. The findings of the research highlighted
important relationships between these three aspects of the research that are represented
as a concept map in Figure 6.1.
Figure 6.1 Overview of the Research Findings
The three oval boxes in Figure 6.1 represent the three central components of the
research reported in this thesis: 1. beliefs about the role and purpose of classroom
assessment; 2. classroom assessment practices; and 3. beliefs about change to
assessment practices. The rectangular boxes represent five additional factors that
emerged from the data that appear to influence the three central components of the
Factors to facilitate
change
Influence
change to
Influence
Influence
Support
change to
Influence
change to
Influence
Influence
BELIEFS ABOUT
THE ROLE &
PURPOSE OF
CLASSROOM
ASSESSMENT
CLASSROOM
ASSESSMENT
PRACTICES
BELIEFS
ABOUT
CHANGE TO
ASSESSMENT
PRACTICES
Beliefs about
teaching &
learning
Experiences with
assessment
Influence
Influence
Beliefs
about
barriers to
change
Beliefs about
reasons for
change
228
research. The relationships between the central components (in the ovals) and the
additional components (in the rectangles) are represented by the arrows. The
direction of the arrow indicates the direction of the relationship and the word or
phrase on each arrow represents the nature of the relationship. For example, the
findings of the research showed that teachers‟ experiences with assessment and
their beliefs about teaching and learning influenced their beliefs about the role and
purpose of classroom assessment. These relationships are represented on the left
side of Figure 6.1. The teachers‟ classroom assessment practices were influenced by
their experiences with assessment and beliefs about the role and purpose of
classroom assessment. These relationships are presented at the bottom of Figure
6.1. Beliefs about reasons for change and barriers to change impacted on the
teachers‟ beliefs about change to assessment practices, which, in turn, influenced
their beliefs about the role and purpose of classroom assessment and classroom
assessment practices. Finally, the teachers argued that for them to initiate change in
classroom assessment practices, factors that facilitate such change need to be
present.
The significance of Figure 6.1 is that it presents an overview of the complex
interrelationships between the three central components of the research and the five
additional factors that were found to have an influence on the central components.
The concept map shows how a network exists among the various components and
that change to any aspect of this network is likely to have a ripple effect, resulting
in a number of changes. A more in-depth discussion of each of the central
components and its relationship with the addition factors is presented in the
following sections.
229
Teachers’ Beliefs about the Role and Purpose of Assessment
Although the five case study teachers came from one school site, the findings reported
in Chapter Four disclosed a wide spectrum of beliefs about the role and purpose of
classroom assessment within which commonalities and variations existed. Teachers‟
beliefs about the role and purpose of classroom assessment are represented as one of the
three central components in Figure 6.1. Four key findings emerged with regard to the
teachers‟ beliefs about the role and purpose of classroom assessment. The first key
finding was that teachers‟ expressed a strong and universal belief that classroom
assessment provides essential feedback to guide teachers in their decision-making
processes with regard to classroom instruction. According to the case study teachers,
feedback from classroom assessment assisted them in making sound decisions about
instructional design and curriculum delivery. Teaching pace, teaching sequence and
teaching approaches were examples of aspects of teaching mentioned by the participants
where decisions were guided by classroom assessment results. These decisions included
modifications and improvements that could enhance the effectiveness of their teaching,
therefore, providing students with more and better opportunities to take full advantage
of their learning environment (Stiggins, Arter, Chappius & Chappius, 2006). Thus,
classroom assessment functions as a „guiding framework‟ in making sound decisions
about instruction and curriculum delivery.
The second key finding was that participating teachers conveyed the belief that
classroom assessment presents crucial information about students as learners. All case
study teachers felt that such information was valuable in helping them become better
and effective teachers. The participating teachers expressed the belief that information
generated from classroom assessment assisted in „profiling‟ their students as learners.
Information such as students‟ areas of strength and weakness; students‟ level of
230
mathematical ability; and students‟ level of achievement in comparison with their peers
provided the teachers with a clear comprehension of their students‟ learning.
Interestingly, two case study teachers, Lulu and Jane, communicated the belief that non-
academic information, such as students‟ attitudes towards learning and views about
mathematics learning, were just as important. Both these teachers were of the opinion
that it is crucial to obtain both academic and non-academic information from assessment
to better understand their students as individual learners. This finding suggests that
teachers need to have an indepth and dynamic approach to the comprehension of the
diversity in their students‟ learning abilities and learning preferences so that the
planning and implementation of suitable learning activities and timely remedial actions
can be completed effectively. Hence, the role and purpose of classroom assessment
according to this group of teachers was extended beyond „measuring‟ students‟
achievements, to support their learning.
The third key finding of this study was the belief explained by the participating teachers
that classroom assessment is essential for producing information for a variety of
communication purposes. The data indicated that the participating teachers held the
view that it is important to provide documentation of students‟ learning progress to
parents. The case study teachers, guided by results from classroom assessment, prepared
forecasts and testimonials for communicating students‟ on-going progress to university
authorities to their university applications.
The fourth key finding of this study was the influence of formal assessment
requirements set by a particular body of authority, for example, the management of the
institution or the curriculum provider, on the teachers‟ beliefs about the purpose of
classroom assessment. In this study, the external examination requirements of the pre-
231
university programme in which the participating teachers were involved had a strong
and pervasive influence on two case study teachers‟ beliefs about the purpose of
classroom assessment. Both Kate and Lulu expressed the belief that one important
purpose of classroom assessment is to prepare students for the final, external
examination.
Teachers’ Assessment Practices
This study revealed a diversity of assessment practices that were used by the case study
teachers. The various assessment techniques employed by the teachers ranged from
alternative techniques such as projects, presentations and peer assessment to the
traditional paper-pencil testing technique. Teachers‟ classroom assessment practices are
represented as a central component in Figure 6.1. The participating teachers were
divided into three categories based on the extent to which they used traditional and
alternative assessment techniques. The three categories were (a) the innovative assessor;
(b) the „sitting on the fence‟ assessor; and (c) the traditional assessor.
The innovative assessors were characterised by their use of more alternative assessment
techniques than traditional assessment techniques. The innovative assessors explained
their belief that classroom assessment provides continuous feedback to support
students‟ learning and also to present opportunities for students to enhance skills, such
as management skills, teamwork and communication skills that would enable them to
function effectively in the real world.
The „sitting on the fence‟ assessors tended to make an effort to balance the use of
alternative and traditional assessment techniques. Although they attempted to maintain
such a „balance‟ the data indicated they seemed to favour traditional assessment
232
techniques. The sitting on the fence assessors justified this behaviour through an
expressed belief that traditional assessment techniques provide reliable and fair
information about students‟ learning, whereas information from alternative techniques
only supplements the information obtained from the traditional paper-and-pencil tests
and examinations.
The assessment practice of the traditional assessor was dominated by the traditional
paper-and-pencil techniques. Only one case study teacher belonged to this category.
Kate expressed her firm opinion that traditional assessment techniques are fairer and
more standardised compared with alternative assessment techniques and that alternative
assessment of students‟ learning is deficient in rigour and reliability.
Research has indicated that prior experiences have an influence on teachers‟ beliefs
(Barkatsas & Malone, 2005a). Therefore, experience is a factor that has an influence
over teachers‟ assessment practices (Keady, 2007) and is represented as a key
component in Figure 6.1. The teachers in this study all had experienced an examination
culture as a student and later, as a teacher. Teachers such as Kate and Jane, who had the
most experience of an examination-oriented assessment culture, tended to more ardently
argue their beliefs about the high level of fairness and reliability of traditional
techniques over alternative techniques compared with the other teachers. An interesting
finding, however, was that Kate‟s assessment practices were strongly dominated by
traditional assessment techniques which was consistent with her seemingly stronger
belief about the advantages of traditional assessment techniques. In comparison, the
data from Jane‟s case study indicated her belief in this regard was somewhat ambivalent
and it was evident that she tended to use more alternative assessment techniques. As a
233
consequence, Kate was considered to be a traditional assessor, whereas Jane was
considered to be a „sitting on the fence‟ assessor.
Furthermore, this study found that teachers‟ experiences with certain assessment
techniques influenced their preference for assessment techniques that they employed in
the classroom (Figure 6.1). If teachers had successfully implemented a particular
assessment technique, they were more willing to continue using that technique.
Assessment techniques that had given rise to complications and issues during
implementation were usually abandoned or rarely used again by the teacher. For
example, Lulu faced difficultites in using peer assessment. Her reaction to the future use
of such a technique was to simply use peer assessment with the intention of providing
exposure for her students, but she did not use it as a serious means of assessing their
understanding.
The findings of this research also suggested that teachers‟ experience with assessment
techniques that required longer preparation and implementation also influenced their
assessment practices. Techniques requiring longer preparation and implementation time
were deemed to be suitable only if there was a strong justification to use them as well as
sufficient time to implement them in the classroom.
Teachers’ Beliefs about Change to Assessment Practices
This section describes the key findings with regard to the case study teachers‟ beliefs
about change to their assessment practices. These key findings summarise the
discussion in Chapter Five in relation to reasons for change, barriers to change and
factors to facilitate change, all of which are represented in Figure 6.1.
234
Holistic learning
All participating teachers, except Kate, agreed that there was a need to adopt more
alternative assessment techniques in the classroom to create diversity in the ways
students are assessed. They generally were of the opinion that too much emphasis on
marks and grades has resulted in a culture in Malaysia where teacher „teach to the test‟.
The four case study teachers agreed that this culture of assessment has stifled students‟
academic and personal development. They were concerned that students were not
experiencing a learning environment which supports their development into „whole‟
individuals and students were not enjoying the learning process. According to the
participating teachers, in such an environment, learning is seen as a cumbersome „chore‟
by students. An argument presented by Anakin and Wendy was that if students enjoy
learning, they will learn more, and learn better because they are more likely to pay
attention and be engaged in classroom activities. Davis (2007) concurred that, if
teachers emphasise and value learning above grades and competition, then students
would be more motivated to learn. Moreover, the findings reported in Chapter Four
highlighted the negative effects of an emphasis on grades and marks on students,
especially the academically challenged students. The focus on marks and grades may
demotivate students and work against the establishment of encouraging good
relationships between students and teachers in the classroom (Harlen & Deakin-Crick,
2003).
Assessment, especially the use of alternative techniques, was seen by the case study
teachers to support students‟ learning by providing various opportunities for students to
portray various abilities. Students are able to “represent their learning in ways that best
suit them” (Berry, 2008, p. 83). Alternative assessment techniques also were seen by the
participating teachers to offer students an environment that promotes holistic individual
235
development. Through the use of alternative assessment, such as journals and portfolios,
students are able to “reflect on their own learning and make plans for their learning
needs” (Berry, 2008, pp. 82-83).
Fairness and Authenticity in Alternative Assessment
There were two important issues that were of great concern to the case study teachers
with regard to the use of alternative assessment techniques; fairness and authenticity. In
this study, the issue of fairness encompassed students‟ work and and teachers‟ grading.
When students handed in work for assessment, for example, projects, the participating
teachers, with the exception of Anakin, explained that there often was a question in their
minds about who had actually done the work. Teachers said that they serious challenges
in determining the authenticity of students‟ work, and that it would be unfair if students
received good grades for work that they did not do themselves. An example was given
by Kate who mentioned that plagiarism from the internet was one contributor to the
issue of fariness and authenticity. Furthermore, the teachers explained that if one of
their students had access to resources that other students did not have access to, this also
would be an issue of fairness.
Importance of Professional Support in Adopting Changes
The findings presented in this study described the importance of factors such as peer
support, resources and professional development in encouraging and enabling the move
to change in teachers‟ assessment practices (Figure 6.1). To change their practices, the
teachers in this study argued that peer support was a crucial factor. If there was strong
support from peers with regard to implementing new assessment techniques, the case
study teachers said they would be willing to try to embrace change. The key reason they
expressed was they would not feel „alone‟ in venturing into the „unknown‟.
236
Another factor that participating teachers made clear would contribute to facilitating
change to their assessment practices was the availability of opportunities for
professional development. The findings of this research showed that if changes were
required from the teachers, they must first be given appropriate training to enable and
support them to carry out different assessment techniques. Some techniques would
clearly be relatively new to them, and lack of familiarity would result in lack of
confidence in using them in their classroom. Sustained, supportive and targeted
professional development has the potential to address these concerns about lack of
familiarity and confidence (Keady, 2007).
Contribution to the Research Literature
An important contribution the findings of this research contribute to the literature is the
knowledge of the diversity of mathematics teachers‟ beliefs about classroom
assessment, even at one single site. The findings reported in this thesis add to the
knowledge-base of mathematics teachers‟ beliefs in general (Beswick, 2006,
2007/2008; Speer, 2008; White, Way, Perry & Southwell, 2005;2006) and specifically
with regard to assessment (Adams & Hsu, 1998; Delandshere & Jones, 1999; Nisbet &
Warren, 2000). Moreover, the findings contribute to current research on assessment
within the Asian context, importantly in Malaysia, a country that is in the process of
reforming its assessment system. It is important to understand teachers‟ beliefs because
this study has shown that the participating teachers‟ beliefs about the role and purpose
of assessment influence their preference for particular assessment approaches and
techniques (Figure 6.1). The findings also acknowledged the role that the teachers‟
individual experiences with assessment played on their beliefs about assessment and
practices (Figure 6.1). For example, Kate had limited experience with alternative
237
assessment, thus, she expressed a strong belief in traditional assessment techniques as
the most reliable form of mathematics assessment.
Another finding of this study was the importance of considering teachers‟ beliefs when
implementing change (Figure 6.1). This finding provides strong, additional support for
previous research that has indicated the importance of aligning teachers‟ beliefs with the
principles that underpin any reform movement (Battista, 1994; Gooya, 2007; Handal&
Herrington, 2003). As stated earlier, if there is consistency between teachers‟ beliefs
and the reform principles, then the chances of a successful reform is increased. In the
research reported in this thesis, the change in the assessment policy by the management
of the college was aligned with a constructivist perspective of assessment. The results
reported in Chapter Four indicated, however, that a behaviourist perspective of
assessment underlined the beliefs of two of the five case study teachers, Kate and Jane.
The behaviourist perspective was predominant in Kate‟s expressed beliefs compared
with Jane‟s which was consistent with the finding that Kate‟s assessment practices had a
greater emphasis on traditional paper-pencil techniques. In summary, if the majority of
teachers‟ beliefs are not aligned with the underlying principles of any reform, the
probability of a successful reform will be reduced.
Implications for Professional Practice
An important implication of the findings of this research is the need for professional
support to enable changes to teachers‟ assessment practices. Teachers need to be
provided with on-going professional development that is practical and focussed on
classroom behaviours, teaching resources and, the familiarisation with and learning of
the new assessment techniques that are to be implemented. In addition, research
(Goodrum, Hackling & Rennie, 2001; Keady, 2007) has shown that peer support plays a
238
significant role in „enticing‟ teachers to try new approaches. The institution that was the
focus of this research is likely to benefit should opportunities be provided to teachers to
build peer networks and sharing of knowledge.
Another implication of this research is with regard to the training programme of pre-
service teachers. Pre-service teachers need to be provided with knowledge of, and
hands-on experience with alternative assessment techniques. Besides acquiring
knowledge of alternative assessment, an interesting suggestion from Lulu was that the
assessment of pre-service teachers should include alternative assessment. The rationale
for this was to allow pre-service teachers to experience the implementation of
alternative assessment from their students‟ perspective. This exposure would provide
them with a better understanding of the advantages and disadvantages of different
assessment techniques. Research (e.g. Keys, 2003) suggests teachers‟ beliefs that are
influenced by their personal experience have an effect on their choice of instructional
strategies, in this case their preference for particular assessment techniques (Figure 6.1).
Professional development, for in-service teachers and training programmes for pre-
service teachers, both need to be planned carefully and strategically. The purpose of
such programmes should not only be to enhance the teachers‟ theoretical and practical
knowledge with regard to alternative assessment, but also to align their current beliefs
about assessment with the underlying principles of the assessment reform. The
implication of this study‟s findings in this aspect is relevant to the Malaysian context as
the country is currently in the process of implementing change to the assessment
system.
239
Limitations of the Study
There are several limitations to the findings of this study. The first limitation relates to
sampling. The participants in this research were obtained through purposive sampling
and were limited to mathematics teachers involved in a pre-university programme that
followed a prescribed external curriculum. In contrast, the focus of the assessment
reform in Malaysia is on the national assessment system in schools. The context of the
research in the pre-university programme thus may seem at odds with the intended
context of reform in Malaysian schools. However, the aim of the research required
information be provided by Malaysian mathematics teachers who had undergone the
transformation from traditional assessment practices to a mixture of traditional and
alternative assessment practices. Therefore, it was important to conduct the research in a
context where this transition had recently taken place.
Further, the number of participants involved in this research was eight and five of these
eight participants were selected for the reporting of the research in this thesis. As a
result, the findings of this study are limited in terms of generalisability. As discussed in
Chapter Three, however, the findings should be considered in terms of transferability
(Merriam, 2009; Stake, 1994), a term consistent with the qualitative paradigm in which
the research was conducted. Thick description of the findings (Lichtman, 2012; Lincoln
& Guba, 1985) in the form of case studies and detailed discussion of the methodology
employed also provide information for other researchers and practitioners to accurately
consider the transferability of the findings to their own context.
The third limitation was the perceptions of the participating teachers and the
management of the college. Although the researcher had established good rapport with
the case study teachers before data collection, there were a few participating teachers
240
who were nervous about the potential for the college management to access the data,
especially the interview data. Several participants mentioned to the researcher that they
preferred that the management of the college had minimal access to the research data.
There were instances during the interviews where the teachers were a little hesitant to
give their views. The researcher addressed this situation by approaching and negotiating
with the college‟s management that any request for access to individual teacher‟s data
would need to have the approval of the teacher. Regardless of this negotiated position,
the findings may have been limited by cautious responses that were due to some
teachers‟ concerns about the potential lack of anonymity.
Personal Postscript
Although I was aware of the significance of the influence of teachers‟ beliefs on their
instructional practice through my readings of the research literature, my personal
journey through this study provided me with the opportunity to observe and experience
first hand the consequence of the case study teachers‟ beliefs about classroom
assessment on their assessment practices. This research on teachers‟ beliefs within the
classroom context has made me realise how important it is to have a comprehensive
understanding of teachers. By understanding teachers and their beliefs better,
implementing changes is likely to be a less challenging task. Because policy-makers and
educational leaders are better informed from the findings of this study about the nature
of any possible „resistance‟ from teachers, appropriate steps can be taken to increase the
success of change. An important awareness that I gathered from communicating with
the participating teachers throughout the data collection process was that teachers feel a
sense of empowerment and self-confidence when approached for their views or
opinions about change. One of the case study teachers commented that she felt
appreciated because her views were sought and considered important enough for me to
241
report in this thesis. At the same time, by being aware of the issue of change in
assessment practices through this study, this teacher had actually made a conscious
effort to think more deeply about this move to reform assessment, which she claimed
she would not normally have done. The research reported in this thesis has not only
contributed to the research literature on assessment but has also made a personal impact
on the way I view the importance of research about beliefs as a whole.
242
REFERENCES
Adams, T. L., & Hsu, J.-W.Y. (1998). Classroom assessment: Teachers' conceptions
and practices in mathematics. School Science & Mathematics, 98(4), 174-180.
Aguirre, J. & Speer, N. (2000). Examining the relationship between beliefs and goals in
teacher practice. Journal of Mathematical Behaviour, 18(3), 327-356.
Airasian, P. W. (2001). Classroom assessment: Concepts and applications (4th ed.).
New York: McGraw Hill.
Anderson, J., Sullivan, P., & White, P. (2005).Using schematic model to represent
influences on, and relationships between, teachers‟ problem-solving beliefs and
practices. Mathematics Education Research Journal, 17(2), 9-38.
Anderson, L. W. (2003). Enhancing the quality of teacher decision making. Mahwah,
NJ: Lawrence Erlbaum.
Angelo, T.A. & Cross, K.P. (1993). Classroom assessment techniques: A handbook for
college teachers (2nd ed.). San Francisco, CA: Jossey-Bass.
Au, W. (2007). High-stakes testing and curricular control: A qualitative metasynthesis.
Educational Researcher, 36(5), 258-267.
Ajzen, I & Fishbein, M., (1980). Understanding attitudes and predicting social
behaviour. Englewood Cliffs, NJ: Prentice Hall.
243
Barkatsas, A., & Malone, J. (2005a). A typology of mathematics teachers‟ beliefs about
teaching and learning mathematics and instructional practices. Mathematics
Education Research Journal, 17(2), 69-90.
Barkatsas, A., & Malone, J. A. (2005b). Secondary mathematics teachers' beliefs about
mathematics assessment and components that influence these beliefs. Nordic
Studies in Mathematics Education, 2, 35-59.
Battista, M.T. (1994). Teacher beliefs and the reform movement in mathematics
education. Phi Delta Kappan, 75(6), 462-467.
Baxter, J. & Eyles, J. (1997). Evaluating qualitative research in social geography:
Establishing „Rigour‟ in interview analysis. Transactions of the Institute of
British Geographers, New Series, 22(4), 505-525.
Bell, B. (2007). Classroom assessment of science learning. In S. K. Abell & N. G.
Lederman (Eds.), Handbook of Research on Science Education (pp. 965-1006).
Mahwah, N.J.: Lawrence Erlbaum Associates.
Bell, B. & Cowie, B. (2001). The characteristics of formative assessment in science
education. Science Education, 85, 536-553.
Berlak, H. (1992). The need for a new science of assessment. In H. Berlak, F. M.
Newmann, E. Adams, D. A. Archbald, T. Burgess, J. Raven, & T. A Romberg
(Eds.), Towards a new science of educational testing and assessment (pp. 1-22).
Albany: State University of New York Press.
244
Berry, R. (2008). Assessment for learning. Aberdeen, HK: Hong Kong University Press.
Berry, R. (2011). Educational assessment in Mainland China, Hong Kong and Taiwan.
In R. Berry & B. Adamson (Eds.), Assessment reform in education: Policy and
Practice (pp. 49-62). Dordrecht: Springer.
Berry, R. & Adamson, B. (2011). Assessment reform past, present and future. In R.
Berry & B. Adamson (Eds.), Assessment reform in education: Policy and
Practice (pp. 3-14). Dordrecht: Springer.
Berry, R. & Gao, L. (2009). Teachers‟ classroom assessment practice in China. Paper
presented at the International Conference on Primary Education. Hong Kong
Institute of Education, Hong Kong.
Beswick, K. (2005). The beliefs/practice connection in broadly defined contexts.
Mathematics Education Research Journal, 17(2), 39–68.
Beswick, K. (2006). The importance of mathematics teachers' beliefs. Australian
Mathematics Teacher, 62(4), 17-22.
Beswick, K. (2007). Teachers‟ beliefs that matter in secondary mathematics classrooms.
Educational Studies in Mathematics, 65(1), 95-120.
Beswick, K. (2007/2008). Influencing teachers‟ beliefs about teaching mathematics for
numeracy to students with mathematics learning difficulties. Mathematics
Teacher Education and Development, 9, 3-20.
245
Biggs, J. (1999). Teaching for quality learning at university. Buckingham: Open
University Press.
Birenbaum, M. (2003). New insights into learning and teaching and their implications
for assessment. In M. Segers, F. Dochy, & E. Cascallar (Eds.), Optimising new
modes of assessment: In search of qualities and standards (pp. 13-36). The
Netherlands: Kluwer Academic.
Black, P. & Wiliam, D. (1998).Inside the Black Box. Phi Delta Kappan, 80(2), 139-
147.
Bogdan, R. C., & Biklen, S. K. (2007).Qualitative research for education: An
introduction to theories and methods (5th ed.). Boston, MA: Pearson/ Allyn&
Bacon.
Broadfoot, P. (1998). Records of achievement and the learning society: A tale of two
discourses. Assessment in Education, 5(3), 447-477.
Broadfoot, P. (2009). Foreword. In C. Wyatt-Smith & J. Cumming (Eds.), Educational
assessment in the 21st century (pp.v-xi). Dordrecht: Springer.
Brookhart, S. M. (1997). A theoretical framework for the role of classroom assessment
in motivating student effort and achievement. Applied Measurement in
Education, 10(2), 161-180.
246
Brookhart, S. M. (2009). Assessment and examinations.In L. J. Saha and A. G. Dworkin
(Eds.), International handbook of research on teachers and teaching (pp. 723-
738). New York: Springer.
Brookhart, S. M., & Nitko, A. J. (2008). Assessment and grading in classrooms. Upper
Saddle River, NJ: Pearson Merrill Prentice Hall.
Bryant, D. A. & Carless, D. R. (2010). Peer assessment in a test-dominated setting:
Empowering, boring or facilitating examination preparation? Educational
Research for Policy and Practice, 9, 3-15.
Bryman, A. (2008). Social research methods (3rd ed.). New York: Oxford University
Press.
Buhagiar, M. A. (2007). Classroom assessment within the alternative assessment
paradigm: revisiting the territory. Curriculum Journal, 18(1): 39 - 56.
Buhagiar, M. A., & Murphy, R. (2008). Teachers' assessments of students' learning of
mathematics. Assessment in Education: Principles, Policy & Practice, 15(2),
169 - 182.
Burns, R. B. (1994). Introduction to research methods (2nd ed.). Melbourne: Longman.
Butler, S. M., & McMunn, N. D. (2006). A teacher's guide to classroom assessment:
Understanding and using assessment to improve student learning. San
Francisco, CA: Jossey -Bass.
247
Calderhead, J. (1996). Teacher‟s: beliefs and knowledge. In D. C. Berliner & R. C.
Calfee (Eds.), Handbook of educational psychology (pp. 709-725). New York:
Macmillan.
Chan, J. K., Kennedy, K. J., Yu, F.W. & Fok, P. (2006). Assessment policy in Hong
Kong: Implementation issues for new forms of assessment. Paper presented at
the 32nd International Association of Educational Assessment (IAEA)
Conference, Singapore.
Clarke, D. J. (1992). Activating assessment alternatives in mathematics.Arithmetic
Teacher, 39(6), 24-29.
Clements, M. A. & Ellerton, N. F. (1995). Assessing the effectiveness of pencil-and-
paper tests for school mathematics. In B. Atweh & S. Flavel (Eds.), MERGA 18
Galtha: Proceedings of the eighteenth annual conference of mathematics
education research group of Australasia (pp. 184-188). Darwin: Northern
Territory University.
Creswell, J. W. (2005). Educational research : Planning, conducting, and evaluating
quantitative and qualitative research (2nd ed.). Upper Saddle River, NJ: Merrill.
Creswell, J. W. (2007). Qualitative inquiry and research design: Choosing among five
approaches (2nd ed.). Thousand Oaks, CA: Sage
Creswell, J. W. (2009). Research design: Qualitative, quantitative, and mixed methods
approaches (3rd ed.). Thousand Oaks, CA: Sage.
248
Creswell, J. W. (2012). Educational research: Planning, conducting, and evaluating
quantitative and qualitative research (4th ed.). Boston, MA: Pearson.
Crooks, T. J. (1988). The impact of classroom evaluation practices on students. Review
of Educational Research, 58(4), 438-481.
Cross, D. I. (2009). Alignment, cohesion, and change: Examining mathematics
teachers‟ belief structures and their influence on instructional practices. Journal
of Mathematics Teacher Education, 12(5), 325-346.
Crotty, M. (1998). The foundations of social research: Meaning and perspective in the
research process. Crows Nest, NSW: Allen & Unwin.
Cumming, J. J., & Wyatt-Smith, C. (2009). Framing assessment today for the future:
Issues and challenges. In C. Wyatt-Smith & J. Cumming (Eds.), Educational
assessment in the 21st century (pp. 1-17). Dordrecht: Springer.
Davis, G. E., & McGowen, M. A. (2007). Formative feedback and the mindful teaching
of mathematics. Australian Senior Mathematics Journal, 21(1), 19-29.
Delanshere, G. G. (2002). Assessment as inquiry. Teachers College Record, 104(7),
1461-1484.
Delanshere, G. G. & Jones, J. H. J. (1999). Elementary teachers‟ beliefs about
assessment in mathematics: A case of assessment paralysis. Journal of
Curriculum and Supervision, 14(3), 216-240.
249
Earl, L. M. (2005). Thinking about purpose in classroom assessment: Assessment for, as
and of learning. Deakin West, ACT: Australian Curriculum Studies Association.
Eisenhart, M. A. (1988). The ethnographic research tradition and mathematics education
research. Journal for Research in Mathematics Education, 19(2), 99-114.
Elbaz, F. (1981). The teacher‟s “practical knowledge”: A report of a case study.
Curriculum Inquiry, 11, 43-71.
Elwood, J. & Klenowski, V. (2002). Creating communities of shared practice: The
challenges of assessment use in learning and teaching. Assessment & Evaluation
in Higher Education, 27(3), 243-256.
Ercikan, K. (2006). Developments in assessment of student learning. In P A. Alexander
& P. H. Winne (Eds.), Handbook of educational psychology (2nd. ed.) (pp. 929-
951). Mahwah, NJ: Lawrence Erlbaum.
Ernest, P. (1989). The knowledge, beliefs and attitudes of the mathematics teacher: A
model. Journal of Education for Teaching, 15(1), 13-33.
Evans, C. D. (2002). Understanding assessment. Delta Pi Epsilon, 44(1), 3-12.
Flick, U. (2009).An introduction to qualitative research (4th ed.). London: Sage.
250
Fogarty, R. & Pete, B. M. (2010). The Singapore vision: Teach less, learn more. In
James Bellanca & Ron Brandt (Eds.), 21st century skills: Rethinking how
students learn (pp. 97-116). Bloomington: Solution Tree Press.
Fontana, A., & Frey, J. H. (1994). Interviewing : The art of science. In N. K. Denzin&
Y. S. Lincoln (Eds.), Handbook of qualitative research (pp. 361-376). Thousand
Oaks, CA: Sage.
Fullan, M. (1993). Change forces: Probing the depths of educational reform. London,
U.K.: Falmer Press.
Gall, J. P., Gall, M. D., & Borg, W. R. (1999). Applying educational research: A
practical guide (4th ed.). New York: Longman.
Gallagher, Jo D. (1998). Classroom assessment for teachers. Upper Saddle River, NJ:
Prentice Hall
Gardner, H. (1992). Assessment in context: The alternative to standardized assessment.
In B. R. Gifford & M.C O'Connor (Eds.), Changing assessment: Alternative
views of aptitude,achievement, and instruction(pp. 77-119). Boston: Kluwer.
Gay, L. R., & Airasian, P. (2003). Educational research: Competencies for analysis and
applications (7th ed.). Upper Saddle River, NJ: Merrill / Prentice Hall.
Gipps, C. V. (1994). Beyond testing: Towards a theory of educational assessment.
London: The Falmer Press.
251
Gipps, C. V. (1999). Socio-cultural aspects of assessment. Review of Research in
Education, 24, 355-392.
Gipps, C., & Stobart, G. (2003). Alternative assessment. In T. Kellaghan & D.L.
Stufflebeam (Eds.), International handbook of educational evaluation (pp. 549-
575). Dordrecht: Kluwer Academic.
Gipps, C., & Stobart, G. (2009). Fairness in assessment. In C. Wyatt-Smith & J.
Cumming (Eds.), Educational assessment in the 21st century: Connecting theory
and practice (pp. 105-118). Dordrecht: Springer.
Goodrum, D., Hackling, M., & Rennie, L. (2001). The status and quality of teaching of
science in Australian schools. Canberra: Department of Education, Training and
Youth Affairs.
Gooya, Z. (2007). Mathematics teachers‟ beliefs about a new reform in high school
geometry in Iran. Educational Studies in Mathematics, 65, 331-347.
Gronlund, N. E. (2003). Assessment of student achievement (7th ed.). Boston, MA:
Pearson Education
Grootenboer, P. (2008). Mathematical belief change in prospective primary teachers.
Journal of Mathematics Teacher Education, 11, 479-497.
Hackling, M.W. (2004). Assessment in science. In G.Venville & V. Dawson (Eds.), The
Art of Teaching Science. NSW: Allen & Unwin.
252
Handal, B. & Herrington, A. (2003). Mathematics teachers‟ beliefs and curriculum
reform. Mathematics Education Research Journal, 15(1), 59-69.
Harlen, W. (1994). Issues and approaches to quality assurance and quality control in
assessment. In W. Harlen (Ed.), Enhancing quality in assessment (pp. 11-25).
London: Paul Chapman.
Hofer, B. K. &Pintrich, P. R. (1997). The development of epistemological theories:
Beliefs about knowledge and knowing and their relation to learning. Review of
Educational Research, 67(1), 88-140.
Hornby, W. (2003). Assessing using grade-related criteria: A single currency for
universities? Assessment & Evaluation in Higher Education, 28(4), 435-454.
Hui, L. (2005). Chinese cultural schema of Education: Implications for communication
between Chinese students and Australian educators. Issues in Educational
Research, 15(1), 17-36.
Huerta-Macias, A. (1995). Alternative assessment: Responses to commonly asked
questions. TESOL Journal, 5(1), 8-11.
Isenburg, J. P. (1990). Teachers‟ thinking and beliefs and classroom practice. Childhood
Education, 66, 322-327.
Janisch, C., Liu, X. & Akrofi, A. (2007). Implementing alternative assessment
opportunities and obstacles. The Educational Forum,71(3), 221-230.
253
Kagan, D. M. (1990). Ways of evaluating teacher sognition: Inferences concerning the
Goldilocks principle. Review of Educational Research, 60(3), 419-469.
Kaur, S. (2007, May 9). Assessment to replace UPSR and PMR. The Star. (retrieved
from http://www.thestar.com.my)
Keady, W. (2007).The impact of professional learning on science teachers’ beliefs
about assessment and their assessment practice (Unpublished doctoral thesis).
Edith Cowan University, Perth, Australia.
Kellaghan, T., &Greaney, V. (2001). The globalisation of assessment in the 20th
Century. Assessment in Education: Principles, Policy & Practice, 8(1), 87-102.
doi: 10.1080/09695940120033270
Kennedy, J. K. (2007). Barriers to innovative school practice: A socio-cultural
framework for understanding assessment practices in Asia. Paper presented at
the Redesigning Pedagogy Culture, Understanding and Practice Conference,
Singapore.
Kennedy, J. K. & Lee, J. C.-K (2008). The changing role of schools in Asian societies:
Schools for the knowledge society. Oxon: Routledge.
Keys, M. P. (2003). Primary and secondary teachers shaping the science curriculum:
The influence of teacher knowledge (Unpublished doctoral thesis). Queensland
University of Technology, Queensland.
254
Koh, K., & Luke, A. (2009). The authentic and conventional assessment in Singapore
schools: An empirical study of teacher assignments and student work.
Assessment in Education: Principles, Policy & Practice, 16(3), 291-318.
Kulm, G. (1994). Mathematics assessment: What works in the classroom. San
Francisco: Jossey-Bassey.
Lambert, D. & Lines, D. (2000). Understanding assessment: Purposes, perceptions,
practice. London: RoutledgeFalmer.
Leder, G. C. & Forgasz, H. J. (2002). Measuring mathematical beliefs and their impact
on the learning of mathematics. In G. Leder, E. Pehkonen& G. Torner (Eds.),
Beliefs: A hidden variable in mathematics education. (pp. 95-113). Dordrecht,
The Netherlands: Kluwer Academic Publishers.
Lichtman, M. (2006). Qualitative research in education: A user's guide. Thousand
Oaks, CA: Sage.
Lichtman, M. (2010). Qualitative research in education: A user's guide (2nd ed.).
Thousand Oaks, CA: Sage.
Lincoln, Y. S., & Guba, E. G. (1985). Naturalistic inquiry. Thousand Oaks, CA: Sage.
Linn, R.L. & Miller, M. D. (2005). Measurement and Assessment in Teaching (9th
ed.).
Upper Saddle River, N.J.: Prentice Hall
255
Lowery, N. V. (2003). Assessment insights from the classroom.The Mathematics
Educator, 13(1), 15-21.
Luft, J. A., & Roehrig, G. H. (2007). Capturing science teachers‟ epistemological
beliefs: The development of the Teacher Beliefs Interview. Electronic Journal of
Science Education, 11(2), 38-63.
Malaysia Examinations Syndicate (2007). Introduction to National Educational
Assessment System (NEAS) Paper presented at the International Forum On
Educational Assessment System: Looking Ahead, Petaling Jaya, Malaysia.
Marshall, C. & Rossman, G. B. (2006).Designing qualitative research (4th ed.).
Thousand Oaks, CA: Sage
Mason, J. (2002). Qualitative researching (2nd ed.). London: Sage.
Maxwell, J. A. (2005). Qualitative research design: An interactive approach (2nd ed.).
Thousand Oaks, CA: Sage.
Mayan, M. J. (2009). Essentials of qualitative inquiry. Walnut Creek, CA: Left Coast
Press.
McLeod, D. B., & McLeod, S. H. (2002). Synthesis- beliefs and mathematic education:
Implications for learning, teaching and research. In G. Leder, E. Pehkonen& G.
Torner (Eds.), Beliefs: A hidden variable in mathematics education? (pp. 115-
123). Dordrecht, The Netherlands: Kluwer Academic Publishers.
256
McMillan, J. H. (2007). Classroom assessment: Principles and practice for effective
standards-based instruction (4th ed.). Boston, MA: Pearson / Allyn & Bacon.
Merriam, S. B. (2009). Qualitative research: A guide to design and implementaion. San
Francisco, CA: Jossey-Bass.
Mertens, D. M. (2010). Research and evaluation in education and psychology:
Integrating diversity with quantitative, qualitative and mixed methods (3rd ed.).
Thousand Oaks, CA: Sage.
Miles, M. B., & Huberman, A. M. (1994). Qualitative data analysis (2nd ed.).
Thousand Oaks, CA: Sage.
Miller, M. D., Linn, R. L., & Gronlund, N. E. (2009). Measurement and assessment in
teaching (10th ed.). Upper Saddle River, NJ: Pearson Education.
Ministry of Education (2007). Pelan Induk Pembangunan Pendidikan 2006-2010.
Putrajaya: Bahagian Perancangan dan Penyelidikan Dasar Pendidikan.
Morse, J. M., & Field, P. A. (1995). Qualitative research methods for health
professionals (2nd ed.). Thousand Oaks, CA: Sage.
National Council of Teachers of Mathematics (1989). Curriculum and evaluation
standards for school mathematics. Reston, VA: National Council of Teachers of
Mathematics.
257
National Council of Teachers of Mathematics (1995). Assessment standards for school
mathematics. Reston, VA: National Council of Teachers of Mathematics.
National Council of Teachers of Mathematics (2000). Principles and Standards for
school mathematics. Reston, VA: National Council of Teachers of Mathematics.
National Research Council (2001). Knowing what student know: The science and design
of educational assessment. Washington, D.C.: National Academy Press.
Nespor, J. (1987). The role of beliefs in the practice of teaching. Journal of Curriculum
Studies, 19(4), 317-328.
Nisbet, S., & Warren, E. (2000). Primary school teachers‟ beliefs relating to
mathematics, teaching and assessing mathematics and factors that influence
these beliefs. Mathematics Teacher Education and Development, 2, 34-47.
Nitko, A. J., & Brookhart, S. M. (2007). Educational assessment of students (5th ed.).
Upper Saddle River, NJ: Pearson Merril Prentice Hall.
O'Donoghue, T. (2007). Planning your qualitative research project: An introduction to
interpretivist research in education. Abingdon, OX: Routledge.
Ong, S. L. (2010). Assessment profile of Malaysia: High-stakes examinations dominate.
Assessment in Educaion: Principles, Policy & Practice, 17(1), 91-103.
258
Patton, M. Q. (2002). Qualitative research and evaluation methods (3rd ed.). Thousand
Oaks, CA: Sage.
Pajares, M. F. (1992). Teachers‟ beliefs and educational research: Cleaning up a messy
construct. Review of Educational Research, 62(3), 307-332.
Panizzon, D., & Pegg, J. (2008). Assessment practices: Empowering mathematics and
science teachers in rural secondary schools to enhance student learning.
International Journal of Science and Mathematics Education, 6, 417-436.
Pedersen, S., Williams, D., & Bulu, S. (2004). Teachers‟ beliefs underlying their
assessment practices in a problem-based learning activity. In L. Cantoni& C.
McLoughlin (Eds.), Proceedings of World Conference on Educational
Multimedia, Hypermedia and Telecommunications (pp. 3701-3706).
Chesapeake, VA: Association for the Advancement of Computing in Education.
Phillipou, G., & Christou, C. (1997). Cypriot and Greek primary teachers' conceptions
about mathematical assessment. Educational Research and Evaluation, 3(2),
140-159. Doi: 10.1080/1380361970030203
Pfannkuch, M. (2001). Assessment of school mathematics: Teachers' perceptions and
practices. Mathematics Education Research Journal, 13(3), 185-203.
Ponte, J. P. (1994). Knowledge, beliefs and conceptions in mathematics teaching and
learning. Paper presented at the 5th International Conference on Systematic
Cooperation between Theory and Practice in Mathematics Education, Pavia.
259
Popham, W. J. (2008). Classroom assessment: What teachers need to know (5th ed.).
Boston, MA: Pearson/Allyn& Bacon.
Punch, K. F. (2005). Introduction to social research: Quantitative and qualitative
approaches (2nd ed.). London: Sage.
Punch, K. F. (2009). Introduction to research methods in education. London: Sage.
Radnor, H. A. (2001). Researching your professional practice: Doing interpretive
research. Buckingham: Open University Press.
Raja Musa, Raja Mohamed Fauzi (2008). Testing and assessment in Malaysia. In I. A.
Bajunid (Ed.), Malaysia, from traditional to smart schools: The Malaysian
educational odyssey (pp. 358-384). Shah Alam: Oxford Fajar.
Raymond, A. M. (1994). Assessment in mathematics education: What are some of the
alternatives in alternative assessment? Contemporary Education, 66(1), 13-17.
Raymond, A. M. (1997). Inconsistency between a beginning elementary school
teacher's mathematics beliefs and teaching practice. Journal for Research in
Mathematics Educatio, 28(5), 550-576.
Richards, L. (2009). Handling qualitative data (2nd ed.). London: Sage Publications.
Richards, L., & Morse, J. M. (2007).Readme first for a user's guide to qualitative
methods (2nd ed.). Thousand Oaks, CA: Sage.
260
Rokeach, M. (1968). Beliefs, attitudes and values: A theory of organization and change.
San Francisco: Jossey-Bass.
Sadler, D. R. (1989). Formative assessment and the design of instructional systems.
Instructional Science, 18(2), 119-144.
Sampson, H. (2004). Navigating the waves: The usefullness of a pilot in qualitative
research. Qualitative Research, 4(3), 383-402.
Sarantakos, S. (2005). Social research (3rd ed.). New York, NY: Palgrave Macmillan.
Saxe, G. B., Gearhart, M., Franke, M. L., Howard, S., & Crockett, M. (1999). Teachers'
shifting assessment practices in the context of educational reform in
mathematics. Teaching and Teacher Education, 15, 85-105.
Schoenfeld, A. H. (1992). Learning to think mathematically: Problem-solving,
metacognition and sense-making in mathematics.In D. A. Grouws (Ed.),
Handbook of research on mathematics learning and teaching (pp. 334-370).
New York: MacMillan.
Schwandt, T. A. (2000). Three epistemological stances for qualitative inquiry:
Interpretivism, hermeneutics, and social constructionism. In N. K. Denzin& Y.
S. Lincoln (Eds.), Handbook of qualitative research (2nd ed., pp. 189-213).
Thousand Oaks, CA: Sage.
261
Sebatane, E. M. (1998). Assessment and classroom learning: A response to Black &
William. Assessment in Education: Principles, Policy & Practice, 5(1), 123-130.
Sellan, R., Chong, K. & Tay, C. (2006). Assessment shifts in the Singapore education
system. Paper presented at the 32nd International Association of Educational
Assessment (IAEA) Conference, Singapore.
Shavelson, R. J., & Stern, P. (1981).Research on teachers‟ pedagogical thoughts,
judgments, decisions and behavior. Review of Educational Research, 51, 455-
498.
Shenton, A. K. (2004). Strategies for ensuring trustworthiness in qualitative research
projects. Education for Information, 22, 63-75.
Shepard, L. A. (1989). Why we need better assessments. Educational Leadership,
46(7), 4-9.
Shepard, L.A. (2000). The role of assessment in a learning culture. Educational
Researcher, 29(7), 4-14.
Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform.
Educational Review, 57, 1-22.
Silver, E. A. & Kenney, P.A. (1995).Sources of assessment information for instructional
guidance in mathematics. In T. A. Romberg (Ed.) Reform in school mathematics
262
and authentic assessment (pp. 38-68), Albany, State University of New York
Press.
Silverman, D. (2010). Doing qualitative research (3rd ed). London: Sage Publications.
Simons, H. (2009). Case study research in practice. London: Sage.
Song, E. & Koh, K (2010). Assessment for learning: Understanding teachers’beliefs
and practices. Paper presented at the 36th International Association of
Educational Assessment Annual Conference, Bangkok, Thailand.
Speer, N. M. (2008). Connecting beliefs and practices: A fine-grained analysis of a
college mathematics teacher's collections of beliefs and their relationship to his
instructional practices. Cognition and Instruction, 26(2), 218-267.
Stiggins, R. J. (1994). Student-centered classroom assessment. New York: Macmillan
College Publishing.
Stiggins, R. J. (2000). Learning teams for assessment literacy. Orbit,30(4), 5-7.
Stiggins, R. J. (2001). Student-involved classroom assessment (3rd ed.). Upper Saddle
River, NJ: Merill-Prentice Hall.
Stiggins, R.J. (2008). An introduction to student-involved assessment for learning (5th
ed.). Upper Saddle River, NJ: Pearson Merrill Prentice Hall.
263
Stiggins, R. J., Arter, J., Chappius, J., & Chappius, S. (2006). Classroom assessment for
student learning: Doing it right using it well. Portland, OR: Educational Testing
Service.
Stiggins, R. J., & Conklin, N. F. (1992). In teachers’ hands: Investigating the practices
of classroom assessment. Albany, NY: State University of New York.
Suurtamm, C. A. (2004). Developing authentic assessment: Case studies of secondary
school mathematics teachers' experiences. Canadian Journal of Science,
Mathematics and Technology Education, 4(4), 497-513.
Suurtamm, C., Koch, M. & Arden, A. (2010). Teachers‟ assessment practices in
mathematics: Classroom in the context of reform. Assessment in Education:
Principles, Policy & Practice, 17(4), 399-417.
Swan, M. (2006). Designing and using research instruments to describe the beliefs and
practices of mathematics teachers. Research in Education, 75, 58-70.
Tan, K. (2011). Assessmentfor learning reform in Singapore-Quality, sustainable or
threshold? In R. Berry & B. Adamson (Eds.), Assessment reform in education:
Policy and Practice (pp. 75-88). Dordrecht: Springer.
Taylor, C. S. & Nolan, S. B. (2008).Classroom assessment: Supporting teaching and
learning in real classrooms (2nd ed.). Upper Saddle River, NJ: Pearson.
264
Thompson, A. G. (1984). The relationship of teachers' conceptions of mathematics and
mathematics teaching to instructional practice. Educational Studies in
Mathematics, 15(2), 105-127.
Thompson, A. G. (1992). Teachers‟ beliefs and conceptions: A synthesis of the
research. In D. A. Grouws (Ed.), Handbook of research on mathematics
teaching and learning (pp. 127–146). New York: Macmillan.
Thompson, D., Thompson, K. & Else, N. (2000). Alternative assessment .Australian
Primary Mathematics Classroom, 5(4), 29-32.
Tuah, A. B. (2007). National Education Assessment System: A proposal towards a more
holistic education assessment system in Malaysia. Paper presented at the
International Forum On Educational Assessment System: Looking Ahead,
Petaling Jaya, Malaysia.
Van de Walle, J. A. (2007). Elementary and middle school mathematics: Teaching
developmentally (6th ed.). Boston, MA: Pearson.
Von Glasersfeld, E. (1987). Learning as a constructivist activity.In C.Janvier (Ed.),
Problems of representation in the teaching and learning of
mathematics.Hillsdale, N.J.: Lawrence Erlbaum Associates.
Von Glasersfeld, E. (1995). Radical constructivism:A way of knowing and learning.
London, U.K.: The Falmer Press.
265
Watt, H. M. G. (2005). Attitudes to the use of alternative assessment methods in
mathematics: A study with secondary mathematics teachers in Sydney,
Australia. Educational Studies in Mathematics, 58(1), 21-44.
Webb, N.L. (1993). Assessment for the mathematics classroom. In N.L. Webb & A.F.
Coxford (Eds.), Assessment in the Mathematics Classroom (1993 Yearbook)(pp.
1-6).Reston, VA: National Council of Teachers of Mathematics.
White, A. (2007). Assessment in school mathematics.In N. Idris (Ed.), Classroom
assessment in mathematics education (pp. 43-60). Selangor: McGraw-Hill
White, A. L., Way, J., Perry, B., &Southwell, B. (2005/2006).Mathematical attitudes,
beliefs and achievement in primary pre-service mathematics teacher education.
Mathematics Teacher Education and Development, 7, 33-52.
Wiliam, D. (2007). Keeping learning on track: Classroom assessment and regulation of
learning. In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics
teaching and learning (pp.1053-1098).
Wilkins, J. L. M. (2008). The relationship among elementary teachers‟ content
knowledge, attitudes, beliefs, and practices. Journal of Mathematics Teacher
Education, 11(2), 139–164.
Williamson, K. (2002). Research methods for students, academics and professionals:
Information management and systems (2nd ed.). WaggaWagga, NSW: Centre
for Information Studies, Charles Stuart University.
266
Willis, J. W. (2007). Foundations of Qualitative Research: Interpretive and Critical
Approaches. Thousand Oaks, CA: Sage.
Wilson, M. S., & Cooney, T. (2002). Mathematics teacher change and development. In
G. Leder, E. Pehkonen& G. Torner (Eds.), Beliefs : A hidden variable in
mathematics education (pp. 127-147). Dordrecht, The Netherlands: Kluwer
Academic Publishers.
Worthen, B. R. (1993). Critical issues that will determine the future of alternative
assessment. Phi Delta Kappan, 74, 444-456.
Wragg, E. C. (2001). Assessment and learning in the secondary school. London:
Routledge.
Wu, Z. & An, S. (2007). Using alternative assessment to assess students‟ learning.In N.
Idris (Ed.), Classroom assessment in mathematics education (pp. 19-31).
Selangor: McGraw-Hill
Yates, S. (2006). Elementary teachers‟ mathematics beliefs and teaching practices after
a curriculum reform.In J. Novotna, H. Moraova, M. Kratka & N. Stehlikova
(Eds.), Proceedings of the 30th Conference of the International Group for the
Psychology of Mathematics Education, Vol.5 (pp. 433-440). Prague: PME.
Yin, R. K. (2009).Case study research: Design and methods (4th ed.). Thousand Oaks,
CA: Sage.
267
Yu, F. W, Kennedy, K. J., Fok, P. , & Chan, J. K., (2006). Assessment reform in basic
education in Hong Kong: The emergence of assessment for learning. Paper
presented at the 32nd International Association of Educational Assessment
(IAEA) Annual Conference, Singapore.
268
Appendix A
INFORMATION SHEET (for the institution)
Research Title: Post-secondary Mathematics Teachers‟ Beliefs About Classroom
Assessment: The Malaysian Context
Dear Sir/Madam,
SuzieleezSyrene Abdul Rahim is undertaking this research as a Doctor of Philosophy
student of the Graduate School of Education, The University of Western Australia.
With the assistance of your institution, we are hoping that data will be gathered from
mathematics teachers to assist in developing a better understanding of Malaysian post-
secondary mathematics teachers‟ beliefs about classroom assessment.
The data collection techniques that will be employed in this research are individual
interviews, classroom observations, reflective journal writing and document collection.
There will be 2 individual interviews to be conducted with every participant and the
interviews will be audio recorded. The interviews will be conducted at a place and time
of the participants‟ convenience. Classroom observations will be conducted with the
consent of the participants. All information provided by the participants will be treated
as strictly confidential and will not be made available to anyone else.
If your institution is willing to participate in this study, you will be required to sign the
Consent Form. A copy of the signed Consent Form will be given to you for your
record. Your participation in this study does not prejudice any right to compensation,
which you may have under statute or common law. Should you decide to withdraw from
the study, you may do so at any time without prejudice.
If you wish to know more about this study, please email me at
[email protected] or call (618) 64883811. You may also contact Associate
Professor Anne Chapman at [email protected] or call (618)64882387 or
contact SuzieleezSyrene at [email protected] or call (618)64888026 for
further details regarding this study. Your cooperation is greatly appreciated.
Yours sincerely
Professor Grady Venville
The Human Research Ethics Committee at the University of Western Australia requires
that all participants are informed that, if they have any complaint regarding the manner,
in which a research project is conducted, it may be given to the researcher or,
alternatively to the Secretary, Human Research Ethics Committee, Registrar‟s Office,
University of Western Australia, 35 Stirling Highway, Crawley, WA 6009 (telephone
number 6488-3703). All study participants will be provided with a copy of the
Information Sheet and Consent Form for their personal records.
269
Appendix B
CONSENT FORM FOR RESEARCH PARTICIPATION (for the institution)
Research Title: Post-secondary Mathematics Teachers‟ Beliefs About Classroom
Assessment: The Malaysian Context
I ( the representative of the participating institution) have read the information provided
and any questions I have asked have been answered to my satisfaction. I agree to
participate in this activity, realising that I may withdraw at any time without reason and
without prejudice. I understand that my participation is voluntary.
I understand that all information provided is treated as strictly confidential and will not
be released by the investigator. The only exception to this principle of confidentiality is
if a court subpoenas documentation. I have been advised as to what data is being
collected, what the purpose is, and what will be done with the data upon completion of
the research.
I agree that research data gathered for the study may be published provided my name or
other identifying information is not used.
____________________ _________
Name of institution Date
______________________________________
Name and position of person granting permission
____________________ _____________________
Signature Email address
The Human Research Ethics Committee at the University of Western Australia requires
that all participants are informed that, if they have any complaint regarding the manner,
in which a research project is conducted, it may be given to the researcher or,
alternatively to the Secretary, Human Research Ethics Committee, Registrar‟s Office,
University of Western Australia, 35 Stirling Highway, Crawley, WA 6009 (telephone
number 6488-3703). All study participants will be provided with a copy of the
Information Sheet and Consent Form for their personal records.
270
Appendix C
INFORMATION SHEET (for the participant)
Research Title: Post-secondary Mathematics Teachers‟ Beliefs About Classroom
Assessment: The Malaysian Context
Dear Sir/Madam,
SuzieleezSyrene Abdul Rahim is undertaking this research as a Doctor of Philosophy
student of the Graduate School of Education, The University of Western Australia.
With your assistance, we are hoping that data will be gathered from mathematics
teachers to assist in developing a better understanding of Malaysian post-secondary
mathematics teachers‟ beliefs about classroom assessment.
The data collection techniques that will be employed in this research are individual
interviews, classroom observations, reflective journal writing and document collection.
There will be 2 individual interviews to be conducted with every participant and the
interviews will be audio recorded. The interviews will be conducted at a place and time
of the participants‟ convenience. Classroom observations will be conducted with the
consent of the participants. All information provided by the participants will be treated
as strictly confidential and will not be made available to anyone else.
If you are willing to participate in this study, you will be required to sign the Consent
Form. A copy of the signed Consent Form will be given to you for your record. Your
participation in this study does not prejudice any right to compensation, which you may
have under statute or common law. Should you decide to withdraw from the study, you
may do so at any time without prejudice.
If you wish to know more about this study, please email me at
[email protected] or call (618) 64883811. You may also contact Associate
Professor Anne Chapman at [email protected] or call (618)64882387 or
contact SuzieleezSyrene at [email protected] or call (618)64888026 for
further details regarding this study. Your cooperation is greatly appreciated.
Yours sincerely
Professor Grady Venville
The Human Research Ethics Committee at the University of Western Australia requires
that all participants are informed that, if they have any complaint regarding the manner,
in which a research project is conducted, it may be given to the researcher or,
alternatively to the Secretary, Human Research Ethics Committee, Registrar‟s Office,
University of Western Australia, 35 Stirling Highway, Crawley, WA 6009 (telephone
number 6488-3703). All study participants will be provided with a copy of the
Information Sheet and Consent Form for their personal records.
271
Appendix D
CONSENT FORM FOR RESEARCH PARTICIPATION
(for the participant)
Research Title: Post-secondary Mathematics Teachers‟ Beliefs About Classroom
Assessment: The Malaysian Context
I ( the participant) have read the information provided and any questions I have asked
have been answered to my satisfaction. I agree to participate in this activity, realising
that I may withdraw at any time without reason and without prejudice. I understand that
my participation is voluntary.
I understand that all information provided is treated as strictly confidential and will not
be released by the investigator. The only exception to this principle of confidentiality is
if a court subpoenas documentation. I have been advised as to what data is being
collected, what the purpose is, and what will be done with the data upon completion of
the research.
I agree that research data gathered for the study may be published provided my name or
other identifying information is not used.
____________________ ____________
Participant‟s Name Date
____________________ _____________________
Participant‟s signature Email address
The Human Research Ethics Committee at the University of Western Australia requires
that all participants are informed that, if they have any complaint regarding the manner,
in which a research project is conducted, it may be given to the researcher or,
alternatively to the Secretary, Human Research Ethics Committee, Registrar‟s Office,
University of Western Australia, 35 Stirling Highway, Crawley, WA 6009 (telephone
number 6488-3703). All study participants will be provided with a copy of the
Information Sheet and Consent Form for their personal records.
272
Appendix E
FRAMEWORK OF GUIDING INTERVIEW QUESTIONS
PART A (General beliefs about classroom assessment)
Q1. Can you explain to me what classroom assessment means to you?
Q2. In your opinion, what is the purpose of carrying out classroom assessment?
Q3. How do you think mathematics assessment is different from assessment in other
subjects?
Q4. For your students, at the post-secondary level, what important aspects of
mathematics do you think should be focused on in a classroom assessment?
Why are they important?
Q5. How do you view mathematics classroom assessment now compared to the time
when you were a student?
Q6. What would you say if someone made a comment that mathematics assessment is
all about getting correct answers?
Q7. Throughout your career as a mathematics teacher, how have the assessment
approaches changed in Malaysia?
PART B (Classroom assessment in teaching and learning)
Q1. Why is classroom assessment important to you as a mathematics teacher?
Q2. Do you think classroom assessment is important to the students? Why?
Q3. Do you think classroom assessment is more important to the teacher than to the
students? Why?
Q4. What do you do with the results of your classroom assessment?
How are the results useful to you as a teacher?
Q5. How are the assessment results useful to the students?
PART C (Types of classroom assessment)
Q1. When do you usually carry out classroom assessment in your mathematics
teaching?
273
Q2. Why do you carry out classroom assessment in these instances?
Q3. Could you give me examples of other instances or situations where you also carry
out classroom assessment in your mathematics lessons?
Q4. What does formal / informal assessment mean to you?
Q5. Could you give me some examples of formal/informal assessment?
PART D (Classroom assessment techniques)
Q1. What are the different assessment techniques that you choose to use in your
mathematics classroom?
Q2. Why do you choose to use these assessment techniques?
Q3. What are the factors that you consider when you choose to use a particular
assessment technique in your mathematics classroom?
Why are these factors important?
Q4. Is the classroom environment different when you use different assessment
techniques?
Could you give me an example?
Q5. Were there any instances where you tried a particular assessment technique and it
was not successfully carried out?
Why did this happen?
Q6. Are there any classroom assessment techniques that you feel are not suitable to be
used in mathematics? Why?
PART E (Changes to classroom assessment practices)
Q1. What is your opinion of any proposal to change the current assessment system?
Q2. Do you think there is a need to reform the current assessment system?
Q3. What are the changes that you feel are necessary to be made to the way you
currently assess the students?
Why are these changes necessary?
Q4. How will this change effect you as a teacher?
Q5. How will this change effect the students?
274
Q6. What are the problems that teachers would face when this change is implemented?
Q7. What are the steps that can be taken to overcome these problems?
Q8. Who else will face difficulties if the current assessment system is changed?
275
Appendix F
REFLECTIVE ACCOUNT
Name :
Date :
Lesson Topic:
Type of assessment carried out:
Please feel free to write down your thoughts or feelings with regards to the assessment
activity that you‟ve recently carried out. The following questions are just guidelines.
Q1. Were you satisfied with how the assessment went?
Q2. Which part of the assessment did you feel was interesting? Why?
Q3. Are there any parts of the lesson where you felt needed to be modified?
Q4. What do you think the students gained from the assessment?
Q5. Were there any problems that you faced during the assessment?
Teacher’s written response:
276
Appendix G
Excerpt of Summary of Category for Lulu
Research question: Role and purpose of assessment
CODE CATEGORY
Understand teacher‟s teaching Quality of teaching
Proceed with teaching Direction of teaching
Move on with teaching Direction of teaching
How much students understand Inform students‟ understanding
Evaluate students‟ understanding Inform students‟ understanding
Standing in class Ranking
Adjust teaching speed Teaching pace
Prepare for final exam Preparation for external exams
Suit students‟ pace Teaching pace
Coping with studies Coping ability
Report to parents Reporting
Learn from mistakes Guide improvement
277
Appendix H
Example of Lulu’s Mind map
Context-specific training for in-
service teachers
Explain the advantages of
alternative assessment to
students
Change exam requirements
Early exposure for pre-service
teachers
LULU’S
SUGGESTIONS FOR
FACILITATING
REFORM
Include teachers’ peer-sharing sessions
Address public awareness (parents &
society)
Some aspects of classroom assessment to be done in class so monitoring can occur
278
Appendix I
Working Summary Sheet of Individual Case Study
Participant :Anakin
1. Experience related to
assessment
1.Experienced examination-oriented system as a student in school
2. Homework was the only assessment activity given in school
3. „Interesting‟ assessment experience in final year at university
(come up with questions for the examiner)
2.Beliefs about teaching,
learning & assessment
1. Maths enhance students‟ problem-solving skills
2. Important for students to have meaningful learning (e.g
understand why they learn a particular mathematical concept)
3. Has a vague understanding of student-centred and teacher-
centred approach.
4. Preference for student-centred approach
5. Mastery of maths is through constant practice
6. Presentation activities is more relevant for subjects like biology
and literature
3.Purpose of assessment
(for teachers)
1. Inform students‟ understanding
2. Informs suitability of teaching methods
3. Initiate “self-study” among students, e.g presentations that are
carried out in his classroom
4. Monitor progress of the students
4. Purpose of assessment
(for students)
1. Fulfill selection criteria for university application
2. Enhance their understanding (e.g homework as an assessment
activity)
5. Types of classroom
assessment
1. Formal assessment (must be carried out; get something out of it;
examinations, homework, class tests)
2. Informal assessment (discussion and references permitted)
3. Diagnostic assessment (before starting a new topic)
4. Formative assessment (give a short question at the end of a
lesson)
6. Classroom assessment
techniques
1. Homework
2. Class tests
3. Presentations
4. Oral questions
5. Quizzes
6. Peer assessment
7. Beliefs about change
in assessment practices
1. Current system influences teachers to teach to the test (focus on
exams)
2.Allow students to enjoy the process of learning
3. Alternative assessment allows students to develop into more
balanced and capable individuals
4. Less stressful for students
8. Concerns about
change in assessment
practices
1. Teachers need to change their instructional practices
2. Increase in workload
3. Ability to provide clear instructions to students on what is
expected from them
4. No samples from previous students (this is the first time
implementing this)
9. Suggestions to
facilitate change in
assessment practices
1. Training in various assessment techniques
2. Good networking among teachers for sharing purposes
3. Easy access to resources
279
Appendix J
Theme Smmary Sheet
Research question: Role and purpose of assessment
CATEGORY
THEME
1.Directs pathway of teaching
Feedback for instructional
decisions
2.Teaching pace
3.Teaching effectiveness
4. Adjustments for future assessments
5.Reporting, forecasting, testimonials
Communication of students’
progress and achievement
6.Identify high-achieving and low-achieving
students
Feedback regarding students
7.Informs students‟ learning
8.Ranking
9.Informs coping ability
10.Identify students who can follow the lessons
11. Identify areas of strengths and weaknesses
12.Understand students better as individuals
13. Monitor students‟ progress
14. Encourage revision
Motivating students in their
learning
15. Encourage students to study
16. Motivation to perform better
17. Prepare for external exams
Preparation platform for external
examination
280
APPENDIX K
OBSERVATION GUIDE SHEET
Name of Teacher :
Date :
Lesson Topic :
WHAT TO OBSERVE
OBSERVATION NOTES
1. Type of assessment
2. Instructions given
3. Materials used
4. Procedures carried out
5. Teacher‟s responses
6. Students‟ responses
7. Classroom environment