9944_Grade R teacher’s pedagogical content knowledge about the development of children’s numerical cognition

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How to cite this thesis
Surname, Initial(s). (2012). Title of the thesis or dissertation (Doctoral Thesis / Master’s
Dissertation). Johannesburg: University of Johannesburg. Available from:
http://hdl.handle.net/102000/0002 (Accessed: 22 August 2017).
i

University of Johannesburg
Faculty of Education

Grade R teacher’s pedagogical content knowledge about the development of
children’s numerical cognition

Lerato Bernice Ndabezitha
201008659

Dissertation submitted for the degree
Magister Educationis
in
Childhood Education
at the
University of Johannesburg

Supervisor: Professor Elizabeth Henning
Co-supervisors: Professor Nadine Petersen
: Dr Sonja Brink

Date: September 2018

ii
DECLARATION
I declare that this is my original research for the purpose of the dissertation, Grade R
teacher’s pedagogical content knowledge about the development of children’s
numerical cognition.
…………………………………………………..
Lerato Bernice Ndabezitha
11 September 2018
iii

SUMMARY

The study aimed to find out what grade R teachers know about children’s number
concept development, specifically as proposed by the model of Fritz et al. (2009,
2013). I was motivated to conduct the study by my observations of Grade R
classrooms in my work as teacher educator. I had noticed that teachers’ practice in
mathematics teaching by way of the four signs of number calculations and some
counting – much of which was not age-appropriate. This type of discourse is prevalent
in most classes in Soweto schools I have visited as a researcher and as a teacher in
the foundation phase. In a teacher development project for grade R teachers,
the Meerkat Maths programme, I had the opportunity to investigate how 15 of the
participating teachers had progressed during and after the project. I observed teaching
in their classrooms and also their engagement in the development project at the
university.
Based on a framework of teacher career development, I analysed data obtained from
interviews and observations. The data were analysed in conventional qualitative
content analysis mode. The main finding was that although teachers were able to
reflect on the model of number concept development of Fritz et al. (2009; 2013) which
they had encountered in the development project, they found it hard to infuse their
knowledge into a structured curriculum-in-practice. I concluded that they were not able
to rely on their intuitive pedagogy, coupled with the model of number concept
development that they had learned, likely due to the strict demands of the school
curriculum content with which they had to comply.

iv

ACKNOWLEDGEMENTS

I would like to acknowledge the following individuals and groups who have contributed
to the completion of this dissertation.
I would first like to thank my research supervisors, Professor Elizabeth Henning,
Professor Nadine Petersen and Dr. Sonja Brink. I have learnt so much from you as
my mentors and I cannot thank you enough for your invaluable guidance and insight.
You have allowed this study to be my own work, but steered me in the right direction
whenever I needed it.
To the teachers who allowed me to interview and also observe them in their classroom,
I would like to say thank you for contributing to my study.
To the almighty God who gave me strength to keep on keeping on. Thank you my God
for the spirit of knowledge, wisdom and understanding. I am nothing without you.
Thank you for using me as a vessel to write this dissertation so that you become
glorified through me because I am indeed a living sacrifice.

Finally, I want to thank my family, friends and loved ones who supported and
encouraged me through this journey. Thank you.

v

FUNDING ACKNOWLEDGEMENTS
I would like to acknowledge the financial assistance of the National Research
Foundation (NRF). A bursary for this study was awarded by the NRF-DST SARChI
Chair: Integrated Studies of Learning Language, Mathematics and Science in the
Primary School. Chair holder: Prof Elizabeth Henning

vi

TABLE OF CONTENTS

DECLARATION
……………………………………………………………………………………………..
ii
SUMMARY
…………………………………………………………………………………………………..
iii
ACKNOWLEDGEMENTS ………………………………………………………………………………
iv
FUNDING ACNOWLEDGEMENTS…………………………………………………………………. v
TABLE OF CONTENT
……………………………………………………………………………………
vi
LIST OF ADDENDA ……………………………………………………………………………………… x
LIST OF FIGURES
………………………………………………………………………………………..
xi
LIST OF TABLES
………………………………………………………………………………………..
xiii
CHAPTER 1
Background and overview
……………..Error! Bookmark not defined.
1.1 Background ………………………………………………………………………………………… 1
1.2 Framework of the study ………………………………………………………………………… 5
1.3 Research problem ……………………………………………………………………………….. 7
1.3.1 Aims and objectives of the study
………………………………………………………. 8
1.4 Terminology
………………………………………………………………………………………… 8
1.5 Research design
………………………………………………………………………………….. 9
1.6 Research ethics clearance ……………………………………………………………………. 9
1.7 The structure of the dissertation …………………………………………………………….. 9
1.8 Conclusion
………………………………………………………………………………………… 10
CHAPTER 2
Teacher knowledge and children’s early numeracy ………………….. 12
2.1 Introduction
……………………………………………………………………………………….. 12
2.2 Number concept development
……………………………………………………………… 15
2.2.1 Perspectives on counting ………………………………………………………………. 21
2.2.2 Innate number knowledge
……………………………………………………………… 25
vii

2.2.3 Explicit instruction of number concepts ……………………………………………. 26
2.2.4 Language as a tool in numeracy learning ………………………………………… 27
2.2.5 Learning difficulties and learner support ………………………………………….. 30
2.3 Teachers’ trajectory towards adaptive expertise
……………………………………… 32
2.4 Professional development after initial teacher education
………………………….. 35
2.5 Learning communities
…………………………………………………………………………. 37
2.6 Teacher’s subject knowledge of number concept development ………………… 39
2.7 The teacher as stable, adaptive expert practitioner …………………………………. 40
2.8 Conclusion: Teachers’ ‘toolkit’ and number concept development …………….. 42
CHAPTER 3
Research Design…………………………………………………………………. 43
3.1 Introduction: A case of grade R teachers’ knowledge
………………………………. 43
3.2 Sampling: An intact group of teachers …………………………………………………… 46
3.3.1 Observations ……………………………………………………………………………….. 48
3.3.2 Interviews
……………………………………………………………………………………. 48
3.3.3 Advantages and disadvantage of interviews …………………………………….. 50
3.3.4 Application of interviews in this research …………………………………………. 50
3.4 Interview questions …………………………………………………………………………….. 52
3.6 Validity and reliability ………………………………………………………………………….. 54
3.7 Ethical considerations
…………………………………………………………………………. 55
3.8 Conclusion
………………………………………………………………………………………… 56
CHAPTER 4
The data of the study …………………………………………………………… 57
4.1 Introduction
……………………………………………………………………………………….. 57
4.2 Narrative of the research process: From collection of data to analysis of data61
4.3 The outcome of the analysis
………………………………………………………………… 67
4.3.1 A stimulating mathematics environment is important for the teacher ……. 67
4.3.2 Grade R teacher qualification not yet recognized
………………………………. 68
viii

4.3.3 Teachers value concrete manipulatives for teaching learners number
concepts
……………………………………………………………………………………………… 71
4.3.4 Teachers recognize that grade R is play orientated …………………………… 73
4.3.5 Grade R maths curriculum is not inclusive ……………………………………….. 74
4.3.6 Too many worksheets for learners ………………………………………………….. 73
4.3.7 Teachers value professional development
……………………………………….. 74
4.3.8 Professional performance reviews diminish teachers’ inclusive pedagogy

………………………………………………………………………………………………………….. 74
4.3.9 Teachers know that learners struggle with number concepts ……………… 75
4.3.10 Teachers do not yet fully understand levels of numerical development . 75
4.3.11 Teachers need to know about number concept development
……………. 75
4.4 Conclusion
………………………………………………………………………………………… 79
CHAPTER 5
Discussion and conclusion
……………………………………………………. 80
5.1 Introduction: “Abafundisi bawazini ngezibalo zabantwana”? …………………….. 80
5.2 Research findings: The main categories
………………………………………………… 80
5.2.1 A stimulating maths environment is important to the teacher
………………. 81
5.2.2 Grade R teacher qualification is not yet recognized …………………………… 85
5.2.3 Teachers value concrete manipulatives for teaching learners number
concepts
……………………………………………………………………………………………… 91
5.2.4 Teachers recognize that grade R is play orientated …………………………… 95
5.2.5 Grade R maths curriculum is not inclusive ……………………………………….. 98
5.2.6 Too many worksheets for learners ………………………………………………… 102
5.2.7 Teachers value professional development
……………………………………… 104
5.2.8 Professional performance reviews diminish teachers’ inclusive pedagogy

………………………………………………………………………………………………………… 106
5.2.9 Teachers know that learners struggle with number concepts ……………. 107
5.2.10 Teachers don’t understand levels of numerical development
…………… 107
5.2.11 Teachers need to know about number concept development
………….. 107
ix

5.5 Conclusion
………………………………………………………………………………………. 110
Reference list …………………………………………………………………………………………… 111
ADDENDUM A …………………………………………………………………………………………. 125
ADDENDUM B …………………………………………………………………………………………. 126

x

LIST OF ADDENDA

ADDENDUM A
Ethics Clearance form………………………………………128
ADDENDUM B
Excerpt from Meerkat Maths Training Programme………129

xi

LIST OF FIGURES

Figure 1.1 A framework for understanding teaching and learning (from Darling
Hammond & Bransford, 2005: 11)
………………………………………………… 4
Figure 1.2
Dual theoretical lense
…………………………………………………………………. 7
Figure 2.1
A map to navigate the chapter by
……………………………………………….. 13
Figure 2.2
Signposting of sections in the chapter
…………………………………………. 16
Figure 2.3
A conceptual model for early numerical concepts (adapted from Fritz et
al., 2013) ………………………………………………………………………………… 17
Figure 2.4
The dimensions of adaptive expertise (from Darling-Hammond &
Bransford, 2005:48) …………………………………………………………………. 33
Figure 2.5
Types of knowledge for professional learning (from Snow et al.,
2005:210)
……………………………………………………………………………….. 35
Figure 3.1
Research process ……………………………………………………………………. 45
Figure 3.2 Content analysis (from Henning et al., 2004:104-106) ………………….. 54
Figure 4.1
Interview process (adapted from Henning, Van Rensburg & Smit,
2004; Cresswell, 2009) …………………………………………………………….. 57
Figure 4.2
Codes that derived from transcriptions
……………………………………….. 62
Figure 4.3
New codes after they had been reviewed by the critical reader
……… 63
Figure 4.4
New codes derived and colour coded
………………………………………… 64
Figure 4.5
Ten categories
………………………………………………………………………… 65
Figure 4.6
Eleven categories
…………………………………………………………………….. 65
Figure 4.7
Pattern of findings …………………………………………………………………… 66
xii

Figure 4.8
Tracking the analysis backwards to ‘raw’ data ……………………………… 67
Figure 4.9
Comparison of views among the five schools ………………………………. 70
Figure 4.10 Worksheet which required learners to say which is ‘more’ – meerkats or
lady bugs
………………………………………………………………………………… 77
Figure 4.11 Comparing quantity (of dots)
……………………………………………………… 78
Figure 4.12 Comparing quantity (of dots)
……………………………………………………… 78
Figure 5.1 Levels and types of knowledge for professional learning (from Snow et
al., 2005:210) ………………………………………………………………………….. 81
Figure 5.2 The reporting process
………………………………………………………………… 89
Figure 5.3 The ‘ama ali’ game ……………………………………………………………………. 94
Figure 5.4 The ‘Izingedo’ game…………………………………………………………………… 94
Figure 5.5 Hopscotch
………………………………………………………………………………… 94

xiii

LIST OF TABLES

Table 2.1
Results in mathematics in SACMEQ II and III (2007) ……………………… 14
Table 3.1
An overview of the analysis process (based on a model by Smith, 2010)

…………………………………………………………………………………54
Table 4.1
Interview questions for data collection
………………………………………….. 58
Table 4.2
Interview work plan
……………………………………………………………………. 59
Table 4.3
Observation work plan
……………………………………………………………….. 60
Table 4.4
Teacher demographics
………………………………………………………………. 60
Table 4.5
Excerpts from data sets
……………………………………………………………… 62
Table 4.6
The differences between initial and revised codes …………………………. 63
Table 5.1
Grade R daily lesson plan
…………………………………………………………. 101

1

CHAPTER 1
BACKGROUND AND OVERVIEW

1.1 Background
In teacher development interventions, ‘uptake’ is a perennial issue of debate and
discussion (Davis, Palinscar, Smith, Arias & Kademian, 2017). In South Africa, large
scale studies such as the Early Grades Reading Study (Taylor, Kotze & Mohohlane,
2016; 2017) have been conducted to find out which component of a three-tiered
intervention in the North West Province showed uptake by teachers and an effect on
learner performance. Another study, he Grade R Mathematics Study of the University
of Cape Town and the Western Cape Education Department (WCED) showed no
improvement in learner performance, although the uptake by teachers was shown to
be positive (JET Education Services, Kelello Consultancy, 2018). Henning (2012)
found that although teachers changed their discourse about practice, there was no
evidence of change to their practice. Similar findings come from a study by Radebe
(2018), in which four foundation phase teachers showed little change to their practice
during an in-depth intervention mentoring project.
Faced with this type of evidence, I embarked on a study to find how a group of grade
R teachers responded and adapted their practice during and after an intervention
programme aimed at enhancing their understanding of pre-numeracy development in
children. My reasoning was that teachers who have learned about children’s
conceptual development, and who have studied mathematical cognition literature in
professional development workshops, are likely to change their views, their discourse
and their practice, no matter how brief or intense the professional development project
may have been. I assumed that teachers would report that they change their practice
according to their newly acquired knowledge (Henning, 2012). According to this
author’s study, teachers in the early grades increasingly used conceptual development
terminology and used less of the discourse of the four mathematical operations and
counting, as is the observed discourse pattern of teachers talking about their work in
professional development workshops that I attend. In classroom observations I have
noted that teachers who have not been
2

exposed to developmental theory of mathematical cognition refer to their practice of
this subject in the foundation phase as the teaching of adding, subtracting, multiplying
or dividing. This type of discourse is prevalent in most classes in the Soweto schools
I have visited as a researcher and a teacher in the foundation phase.
Often, when grade R teachers engage with the Curriculum and Assessment Policy
Statement (CAPS) for grade R mathematics, they feel that they are confronted with a
document that challenges not only their mathematics knowledge, but also their
knowledge of how children learn.1 In the light of some of the available research on the
topic in South Africa (Department of Basic Education, 2011), and my experience with
teachers, I argue that teachers need to have knowledge of typical child number
concept development for them to teach number concepts effectively. In the WCED
study it was found that the teachers, despite an intense development programme, did
not manage to improve the mathematical knowledge of grade R learners of two
Western Cape districts over a year.2 I argue that teachers need to have a basic form
of pedagogical content knowledge (PCK), which, according to Shulman (1987)
includes knowledge of the learner. This is not only knowledge of who each individual
child in a grade R class is and what their background is, but also knowledge of what
cognitive developmental psychology tells us about normative number concept
development and learning progression (Clements & Sarama, 2011). I believe teachers
will have a stronger sense of how to teach and assess young learners if they know
more of the developmental aspects of learning numeracy. There is an abundance of
literature on this aspect of child development and in the second chapter of this
dissertation I will discuss some of this literature, with leading scholars, such as Carey
(2009), Spelke (2000) and Dehaene (2011) included. Without knowledge of
developmental aspects of early numeracy, I would argue that teachers’ PCK is
incomplete.
Shulman (1987; 2009) argues that teaching has a professional knowledge base, a part
of which is PCK. He argues that teaching requires knowledge of the content,
knowledge of pedagogy and knowledge of learners. His typology of teacher knowledge

1https://www.education.gov.za/Curriculum/CurriculumAssessmentPolicyStatements(CAPS)/CAPSFoundation/tabi
d/571/Default.aspx
2 JET Education Services (2018) and Kelello Consultancy (2018) conducted an evaluation in this intervention,
utilising the MARKO-D SA test (Henning, Fritz, Balzer, Herholdt, Ragpot, & Ehlert, 2018) in three South African
languages – isiXhosa, Afrikaans and English.
3

has been taken further by many scholars, with Darling-Hammond and Bransford
(2005) basing large parts of their handbook for teacher education and development
on his work. In my study, the aspect of Shulman’s model that will stand out is
pedagogical content knowledge (PCK). In the PCK toolkit of the average foundation
phase teacher, I argue, knowledge of mathematical cognition of grade R children is
largely absent.
At the university where I conducted research, the organising principle for foundation
phase teacher education is the developing and learning child (Ragpot, 2014). In the
case of mathematics learning, this would be the object of the PCK toolkit – the child
who is learning mathematics and developing concepts and procedures, along with the
child who learns to read, the child who learns to learn science; the child who learns to
learn history, and so forth. It is noteworthy that Darling-Hammond and Bransford
(2005:19), in their leading handbook, Preparing Teachers for a Changing World,
accentuate knowledge of the learner as a primary variable in the model of teaching
that they suggest as a conceptual framework for teacher education.
From this view I could further argue that, as much as the teacher needs to know
mathematics and the pedagogy of mathematics, she also needs to know how to gauge
a child’s learning and development. For that, teachers need to have a sound
understanding of young children’s mathematical cognitive development. This view is
expressed by Darling-Hammond and Bransford (2005:11) in their exposition on how
people learn (HPL) which is derived from earlier work by Bransford, Brown, and
Cocking (1999).

4

Figure 1.1
A framework for understanding teaching and learning (from Darling
Hammond & Bransford, 2005: 11)
In my study I enquired into the experiences and views of 15 grade R teachers (most
of whom were not ‘qualified’ and thus served as ‘practitioners’) from primary schools
in the same area of Johannesburg. I wanted to find out what the teachers understood
about children’s learning of numerical concepts. The teachers had attended a maths
training programme, known as the Meerkat Maths teacher development programme
which is based on the conceptual model of early numerical conceptual development
of Fritz, Ehlert and Balzer (2013).
The programme aimed to equip teachers with knowledge of the mathematical
knowledge that children in grade R already have and can build on, and also how to
deal with variations in early mathematics understanding among children. In doing so,
I posed the overall research question: What do grade R teachers know about the
development of early numerical cognition?
It was important for me to find this out because I argued that if grade R teachers know
how number concepts develop, they will be better able to prepare children to learn
number concepts. Towards this end, I designed and conducted the study from a dual
framework, comprising of the notion of dimensions of adaptive expertise as proposed
5

by Darling-Hammond and Bransford (2005)3 and types of knowledge for professional
learning as conceptualised by Snow, Griffin and Burns (2005). Figure 1.2 presents a
framework for understanding mathematical cognition for grade R teachers. In this
figure the framework of these theories are set out, followed by a brief discussion.
In addition to this frame, I also utilised a developmental model of early numeracy
development, which is based on different developmental theories of mathematical
understanding: Fritz, Ricken and Balzer (2009) and Fritz, Ehlert and Balzer (2013)
modelled a developmental sequence of early numerical development and confirmed
this in different samples with German children as well as in several studies in South
Africa (Herholdt, 2017; De Villiers, 2015). The first three levels of the Fritz et al. (2013)
model are relevant to grade R. The levels do not only follow each other chronologically,
but build upon each other. This means that a subsequent level can only be reached
when the preceding level had been accomplished.
1.2 Framework of the study
The research question required a conceptual ‘gauging’ of teacher knowledge. In order
to do this, I applied the dual framework that I have already mentioned. These two
models, which are discussed at greater length in the next chapter, are briefly set out
below and their application as conceptual lens in the study is depicted in Figure 1.2.
According to Darling-Hammond and Bransford (2005) ‘routine experts’ and ‘adaptive
experts’ are lifelong learners. Routine experts develop fundamental skills that they
apply throughout their teaching experience. Adaptive experts, on the other hand, do
not depend on fundamental skills only. They allow their skills to develop continually.
This makes their skills more flexible in the long run over the course of their teaching
career.
According to Snow et al. (2005), there are certain principles that guide teachers in
what can and should occur during professional training development. There are five
types of knowledge for professional learning that guide this, namely: ‘declarative’,
‘situated’, ‘stable’, ‘expert’ and ‘reflective knowledge’.

3 The data analysis leaned to the use of the Snow et al. (2005) model more than the one of Darling-
Hammond & Bransford (2005)
6

Types of
knowledge for
professional
learning

Snow et al. (2005)

Dimensions of
adaptive expertise

Darling-Hammond and
Bransford (2005)
gaze at
Teacher knowledge of conceptual model of early numerical development
(Fritz et al., 2013)
Dual theoretical lens
7

Figure 1.2
Dual theoretical lens
1.3 Research problem
The problem that I formulated for my research project comes from my experience of
foundation phase teachers whom I have encountered and observed. I noticed that
teachers mainly teach mathematics by focusing on procedural knowledge (what steps
to follow when doing maths) and pay scant attention to the conceptual knowledge that
underlie procedures such as, for instance, double digit addition or subtraction and
overall additive relations (Roberts, 2016) and multiplicative relations (Long, 2015). My
supposition is that this gap comes about because teachers approach mathematics
from divergent perspectives. They teach concepts in a haphazard way instead of
looking at it from a convergent perspective of concepts which a child builds, one after
the other. Many teachers, I believe, think when they have taught a concept, the
learners in their class have learned it and all that is needed is some homework and a
bit of repetition and a few assessments that can be scored. De Villiers (2017) argues
in a similar vein.
Grade R teachers are doubly disadvantaged – not only are most of them ‘unqualified’
(in the sense of not having a formal qualification) – but they are also regarded as ‘lower
level’ teachers who are paid less than qualified teachers. South African research on
grade R has shown that currently, the children who benefit most from attending grade
R programmes which are connected to the primary school curriculum are privileged
children (Van der Berg, Girdwood, Sheperd, Van Wyk, Kruger, Viljoen, Ezoebi &
Ntaka, 2013). For many less privileged children in grade R however, the school day is
often merely a child-minding day, with unqualified teachers who are remunerated very
poorly (in comparison to their grade one counterparts, who receive a full salary with
benefits (Van der Vyver, 2012). There is no question that more expertise is needed in
grade R classrooms.
Ultimately, the study is driven by an understanding that the content knowledge of
grade R teachers comprises knowledge of mathematics, literacy, language, science,
social science, art and culture, sport and so on. Teachers have to know some of all of
these learning areas, but, ultimately, they have to know the children who are learning
about the world from their teachers. In my view, teachers also must know some models
of learning which could guide them in their understanding of how children learn.
8

Finding out how teachers think and talk during, and specifically after an intervention to
strengthen their knowledge of children’s learning is only one step in the direction of
finding out how teachers build (more) knowledge of numerical cognition of children in
the early childhood years.
1.3.1 Aims and objectives of the study
The overall aim of the study was to find out what the teachers know about the
development early numerical cognition of children. The specific objectives were to:
 Observe teachers in their classrooms when they teach grade R mathematics
after having attended the intervention and using these observations, not only
as background data, but also to inform the design of an interview schedule;
 Observe the teachers during the workshops;
 Explore teachers’ knowledge by way of semi-structured interviews about the
topic;
 Analyse the data for content, identifying what teachers may have learned in the
teacher development programme and how they have applied this knowledge.
1.4 Terminology
The main terms I have used in this dissertation relate to the unit of analysis of the
study, which Trochim (2006:1) describes as follows: “The analysis you do in your study
determines what the unit is”. In Chapter 4 the analysis of the data is set out and it
consists of what the teachers know about a specific topic and what they do in their
classrooms concerning this topic.
Grade R teaching: This is the reception year, which is nearly universal in South African
public schools.
Number concept development: The term relates to a specific view of number concept
development, which comes largely from the field of developmental cognitive
psychology.
Teacher professional development: This refers to in-service teacher education
programmes.
Teacher knowledge: The term is borrowed from the work of Shulman (1987) referring
to a specific typology of the knowledge and skills of a teacher.
9

1.5 Research design
I approached this research as a descriptive case study (Yin, 2013; Stake, 2005), and
utilised interviews and observation notes as data collection tools. Teachers were
interviewed in IsiZulu or/and English. The interviews were transcribed and also
translated, where needed. The analysis of the transcribed texts was conducted largely
inductively, synthesising the units of meaning as they were identified by the primary
researcher (Henning, Van Rensburg & Smit, 2004; Strauss & Corbin, 1999). However,
when elements of the conceptual model of early numerical development (Fritz et al.,
2013) were identified, deductive coding occurred, but that was only minimally, because
it turned out that the teachers had not internalised enough of the programme content
to operationalise their knowledge. After the coding, the data were scrutinised and the
codes were categorised to form categories and to construct a set of findings with a
main pattern across all of them.
1.6 Research ethics clearance
I was aware of the ethical measures I had to consider when doing my research. This
study was with human participants and I took cognisance of their rights. The specific
measures I took included: a) applying for overall ethics approval via faculty processes
(see addendum A); b) requesting informed consent from participants to be part of the
research; c) as part of informed consent, the ethics procedure was explained to
participants prior to the interviews and they were informed that the data would only be
used for the purpose of this study and will be reflected in a research report which may
be viewed by others and d) that the findings will not expose the identities of the
participants’ and the name of the schools they are teaching at. I also treated
participants’ responses with a strong measure of confidentiality and information
obtained was only discussed with my research supervisors who also are cognisant of
ethical measures.
1.7 The structure of the dissertation
The dissertation is organized in five chapters. The chronology presented below
represents a summary of the layout of the dissertation report.
10

The first chapter comprises the background and motivation of the study. The chapter
further presents the theoretical framework which underpinned the study, with
objectives clearly stated.
In the second chapter, I present a specific literature study related to teachers’
knowledge for teaching, number concepts for grade R learners’ acquisition and I also
looked at the development of early number concepts.
The third chapter outlines the research methodology of this case study and gives
details of the case when viewed through the theoretical lens. The data collection
methods and the process of data analysis also form part of this chapter. The chapter
ends with a discussion of the ethical issues that were considered.
The fourth chapter offers the analytic description of how data collected from the grade
R teachers were analysed, citing relevant examples from the raw data and showing
the analytic process.
In the last chapter, I present the research findings and discuss the findings, make
suggestions for practitioners, researchers and policy makers in the department of
education.
An outline of the chapter titles is presented here.
Chapter 1: Orientation and background
Chapter 2: Teacher knowledge and children’s early numeracy
Chapter 3: Research design
Chapter 4: The data of the study
Chapter 5: Discussion and conclusion
1.8 Conclusion
The chapter introduced the study by describing the research problem along with the
motivation to conduct the study. At tertiary level, there are programmes to train
teachers for grade R teaching. At the level of government, there are support systems
within the school hierarchy to support in-service teachers. However, there seems to
11

be a lack of expertise of teacher competencies in number concept development
teaching because most grade R teachers are not properly qualified. The main
argument of the study is that teachers in this transition time of children’s education are
expected to prepare them for the demanding grade one mathematics curriculum. The
teachers had been participants in a development programme, which focused on
seriation and classification as concepts as well as theory of number concept
development (Fritz et al., 2013). The study was thus conducted to find out what they
may have learned from this intervention and how they applied their knowledge in the
classroom.