COPYRIGHT AND CITATION CONSIDERATIONS FOR THIS THESIS/ DISSERTATION
o
Attribution — You must give appropriate credit, provide a link to the license, and indicate if
changes were made. You may do so in any reasonable manner, but not in any way that
suggests the licensor endorses you or your use.
o
NonCommercial — You may not use the material for commercial purposes.
o
ShareAlike — If you remix, transform, or build upon the material, you must distribute your
contributions under the same license as the original.
How to cite this thesis
Surname, Initial(s). (2012) Title of the thesis or dissertation. PhD. (Chemistry)/ M.Sc. (Physics)/
M.A. (Philosophy)/M.Com. (Finance) etc. [Unpublished]: University of Johannesburg. Retrieved
from: https://ujcontent.uj.ac.za/vital/access/manager/Index?site_name=Research%20Output (Accessed:
Date).
NUMBER CONCEPT DEVELOPMENT IN GRADE 1: CHILDREN’S
PERFORMANCE AND TEACHERS’ PEDAGOGICAL SKILLS
By
ADELE PETRO VAN DER BERGH
DISSERTATION
Submitted in accordance with the requirements for the degree of
MASTER OF EDUCATION
in
CHILDHOOD EDUCATION
In
FACULTY OF EDUCATION
at the
UNIVERSITY OF JOHANNESBURG
SUPERVISOR: PROFESSOR L RAGPOT
CO-SUPERVISOR: PROFESSOR E HENNING
August 2018
i
DECLARATION
Student Number: 201145656
I hereby declare that the dissertation, Number Concept Development in Grade 1:
Children’s Performance and Teachers’ Pedagogical Skills, is my own work and
that all sources I have used or quoted have been indicated and acknowledged by
means of complete references.
___________________________
August 2018
Adele Petro van der Bergh
Date
ii
ACKNOWLEDGEMENTS
There are so many people who are a part of my life, and especially along this
journey, whom I would like to thank for their love and support. To everyone who
encouraged me and who had an input in my life during the duration of this study, I am
truly grateful.
My Lord and Heavenly Father, who has given me the strength and insight to start and
complete this journey. Thank you!
My husband and lifelong friend, Herman, thank you for all you love and support and
always believing in me. Thank you for allowing me to use our precious family time to
complete this task. To my daughters, Bronwyn and Tammy, who had to sacrifice
family time so that their mom could complete this task. My family, you mean the world
to me! Thank you!
My friend, Ingrid Reyneke, who became my study partner and close friend. Thank
you for all the many hours spent working together and the encouragement during
these three years to keep going! It made this journey one of immense learning and
deep friendship.
Professor Lara Ragpot, who undertook to be my supervisor despite her many other
academic and professional commitments. Thank you for the countless hours of
support and your wisdom!
Thank you, Professor Elbie Henning, for being my co-supervisor and for believing in
me. Your wisdom, support and expertise encouraged me to strive to higher academic
standards!
I would like to thank the principal, Mrs Liz van Tonder, and Orban School, who
granted me the opportunity to conduct my study at my own workplace. Thank you to
iii
all the pupils and the Grade 1 teachers who participated so willingly and eagerly in
this journey. Without you, this study would not have been a success.1
Thank you, Monica Botha, for the time to proofread and edit my work. Your countless
words of support and encouragement gave me strength for the final stretch.
1 Orban School gave permission to disclose its name and the principal’s name. The teachers and the
children remain anonymous and are treated with confidentiality.
iv
ABSTRACT
South Africa is challenged by a serious deficit in early mathematics learning in our
schools. This study argues that teachers need a deeper understanding of how
children learn and develop concepts in early mathematics. As a practitioner, I
realized that we do not know enough about how young children learn mathematics,
and that we teach, based on our own intuitive theories of learning, focusing on
memory and facts, as well as methods and procedures.
The aim of the study was to investigate how children perform on a diagnostic
numeracy competence test at the beginning of their Grade 1 year compared to the
results at the end of their Grade 1 year, after mathematical concepts have been
taught by the Grade 1 teacher. I set out to explore teachers’ methods of instruction
when teaching mathematics to Grade 1 children, specifically number concepts.
The literature study includes discussions about the theories on cognitive
development and investigations made by neuroscientists and developmental
psychologists on children’s development of number concepts. At the same time, I
investigated Shulman’s (1987) notion of pedagogical content knowledge (PCK) and
the role of the teacher in making the subject matter accessible to the child, claiming
that knowledge of child development is part of a teacher’s PCK toolkit. The overview
of literature concludes with a discussion on dyscalculia, which is prevalent in five
percent of children.
The qualitative design was based on data of the performances of the pupils from the
two Grade 1 classes in the sample. Data were collected by way of observations and
interviews.
The findings show that:
i. The clear usage of language to teach and explain mathematics throughout
schooling is essential for learning;
v
ii. Concepts at different levels of mathematical cognitive development are taught
throughout schooling and some specifically in Grade 1;
iii. Children’s approach to mathematics highlight the difficulties they encounter;
iv. Well-trained teachers use different strategies and evaluate procedures, to
ensure maximum learning in mathematics lessons; and
v. The use of concrete materials fulfils an important role in early grades
mathematics learning.
The study proposes that if knowledge of how children learn mathematics influences
the well-trained teacher to teach better, and leads to his/her pedagogical content
knowledge improving, he/she should be able to assist children to build on their
previous mathematical knowledge. This happens through active engagement and
participation, the use of concrete materials and exploration, and learning new
concepts.
Key words: Numeracy cognition, pedagogical content knowledge, conceptual
development, diagnostic test, MARKO-D test
vi
TABLE OF CONTENTS
DECLARATION ……………………………………………………………………………………………. i
ACKNOWLEDGEMENTS ……………………………………………………………………………… ii
ABSTRACT
………………………………………………………………………………………………… iv
TABLE OF CONTENTS
……………………………………………………………………………….. vi
LIST OF FIGURES
……………………………………………………………………………………….. x
LIST OF TABLES ……………………………………………………………………………………….. xi
CHAPTER 1: INTRODUCTION AND BACKGROUND
………………………………………. 1
1.1
THE RESEARCH PROBLEM
……………………………………………………………… 1
1.1.1
Background: South African early grade classrooms …………………………. 2
1.1.2
Number concept development: learning mathematics for teaching……… 5
1.2
RESEARCH QUESTION
……………………………………………………………………. 7
1.3
THE AIM AND OBJECTIVES OF THE STUDY
……………………………………… 8
1.3.1
Objectives ………………………………………………………………………………….. 8
1.4
THEORETICAL BACKGROUND
…………………………………………………………. 9
1.5
METHODS AND DESIGN ………………………………………………………………… 10
1.5.1
Research design ……………………………………………………………………….. 10
1.5.2
Sampling
………………………………………………………………………………….. 11
1.5.3
Data collection ………………………………………………………………………….. 11
1.5.4
Data analysis ……………………………………………………………………………. 12
1.6
RESEARCH ETHICS ………………………………………………………………………. 13
1.7
TRUSTWORTHINESS …………………………………………………………………….. 14
1.8
STRUCTURE OF THE STUDY …………………………………………………………. 15
1.9
SUMMARY …………………………………………………………………………………….. 16
CHAPTER 2: ASPECTS OF YOUNG CHILDREN’S EARLY MATHEMATICAL
CONCEPT DEVELOPMENT
………………………………………………………………………… 17
vii
2.1
INTRODUCTION
…………………………………………………………………………….. 17
2.2. INITIAL THEORIES OF CHILDHOOD COGNITIVE DEVELOPMENT
…….. 20
2.3
CORE COGNITION OF NUMBER …………………………………………………….. 24
2.3.1
Core system 1: the core representation of numerical magnitude –
approximate number system (ANS) ………………………………………………………… 25
2.3.2
Core system 2: the precise representation of distinct entities or small
quantities – object tracking system (OTS)
…………………………………………………. 26
2.3.3
Symbolic learning of number……………………………………………………….. 27
2.4
A CONCEPTUAL MODEL OF NUMBER DEVELOPMENT …………………… 30
2.5
TEACHING MATHEMATICAL CONCEPTS AND PEDAGOGY ……………… 42
2.6
LANGUAGE AND THE TEACHING OF MATHEMATICS ……………………… 47
2.7
CONCLUSION ……………………………………………………………………………….. 50
CHAPTER 3: RESEARCH DESIGN AND METHODOLOGY ……………………………. 52
3.1
INTRODUCTION
…………………………………………………………………………….. 52
3.2
RESEARCH DESIGN TYPE …………………………………………………………….. 52
3.2.1
The MARKO-D test ……………………………………………………………………. 53
3.2.2
Interviews and observations ……………………………………………………….. 54
3.3
METHOD: DATA COLLECTION PROCESS ……………………………………….. 54
3.4
METHOD: SAMPLING …………………………………………………………………….. 57
3.5
METHOD: COLLECTION OF DATA
…………………………………………………… 59
3.5.1
MARKO-D test ………………………………………………………………………….. 59
3.5.2
Observations: MARKO-D test (a) and Classroom (b) ……………………… 62
3.5.3
Interviews
…………………………………………………………………………………. 64
3.6
DATA ANALYSIS ……………………………………………………………………………. 65
3.7
ETHICAL CONSIDERATIONS ………………………………………………………….. 66
3.8
CHAPTER SUMMARY …………………………………………………………………….. 67
viii
CHAPTER 4: RESEARCH RESULTS AND DISCUSSIONS OF FINDINGS ON
GRADE 1 TEACHERS’ PEDAGOGICAL CONTENT KNOWLEDGE AND THE
TEACHING OF MATHEMATICS IN THE CLASSROOM …………………………………. 69
4.1
INTRODUCTION
…………………………………………………………………………….. 69
4.2
DATA SETS 1 AND 2 – RESULTS OF THE MARKO-D TEST ………………. 72
4.2.1
Observations during the MARKO-D test ……………………………………….. 88
4.3
DATA SETS 3 AND 4 – TEACHER PEDAGOGY: QUALITATIVE DATA
…. 95
4.3.1
Data set 3: Classroom observations …………………………………………….. 96
4.3.2
Data set 4: Interviews with the Grade 1 teachers
………………………….. 103
4.4
ANALYSIS OF THE RAW DATA FROM QUANTITATIVE AND
QUALITATIVE METHODS
………………………………………………………………………. 117
4.5
CATEGORIES THAT WERE DERIVED FROM THE DATA
…………………. 117
4.6
FINAL THEMES ABSTRACTED FROM THE RAW DATA
…………………… 120
4.6
CONCLUSION ……………………………………………………………………………… 121
CHAPTER 5: DISCUSSION OF FINDINGS, RECOMMENDATIONS AND
CONCLUSION …………………………………………………………………………………………. 122
5.1
INTRODUCTION
…………………………………………………………………………… 122
5.2
DISCUSSION OF FINDINGS ………………………………………………………….. 123
5.2.1
The clear usage of language to teach and explain mathematics
throughout Grade 1 is essential for learning
……………………………………………. 123
5.2.2
Well-trained teachers use different strategies and evaluation
procedures, to ensure maximum learning in mathematics lessons …………….. 127
5.2.3
Concepts at different levels of mathematical cognitive development
are taught throughout schooling and some specifically in Grade 1 …………….. 133
5.2.4
Children’s approach to mathematics activities highlight the difficulties
they encounter
……………………………………………………………………………………. 137
5.2.5
The use of concrete materials fulfils an important role in early grade
mathematics learning ………………………………………………………………………….. 140
5.3
THE LIMITATIONS OF THIS STUDY
……………………………………………….. 141
ix
5.4
RECOMMENDATIONS ………………………………………………………………….. 142
5.5
SUMMARY …………………………………………………………………………………… 143
REFERENCES …………………………………………………………………………………………. 145
APPENDIX A: ETHICS CLEARANCE
…………………………………………………………. 160
APPENDIX B: PARENT/GUARDIAN CONSENT LETTER (PRE-TEST) ………….. 166
APPENDIX C: PARENT/GUARDIAN CONSENT LETTER (POST-TEST)
………… 167
APPENDIX D: MARKO-D RAW SCORE PRE-TEST RESULTS
……………………… 168
APPENDIX E: MARKO-D RAW SCORE POST-TEST RESULTS …………………… 176
APPENDIX F: PRE-TEST RESULTS – LEVELS …………………………………………… 198
APPENDIX G: POST-TEST RESULTS – LEVELS ………………………………………… 204
APPENDIX H: CLASS A – PRE- AND POST-TEST LEVEL PERCENTAGE
…….. 212
APPENDIX I: CLASS B – PRE- AND POST-TEST LEVEL PERCENTAGE………. 217
APPENDIX J: SCRIPTS FOR MATHEMATICS LESSONS
…………………………….. 222
APPENDIX K: INTERVIEW QUESTIONS WITH GRADE 1 TEACHERS
………….. 230
APPENDIX L: TRANSCRIPT OF INTERVIEW QUESTIONS WITH GRADE 1
TEACHERS
……………………………………………………………………………………………… 231
APPENDIX M: CODES TEACHERS’ INTERVIEWS – TEACHER A
………………… 240
APPENDIX N: CODES TEACHERS’ INTERVIEWS – TEACHER B ………………… 243
x
LIST OF FIGURES
Figure 2.1: Model of mathematical conceptual development
…………………………….. 32
Figure 3.1: Data collection procedure
……………………………………………………………. 54
Figure 3.2: Illustration of the four data sets used in the study
……………………………. 55
Figure 4.1: Illustration of the four data sets used in the study
……………………………. 70
Figure 4.2: A diagram that illustrates the data analysis process
………………………… 72
Figure 4.3: Class A: MARKO-D pre-test and MARKO-D post-test results …………… 74
Figure 4.4: Class B MARKO-D pre-test and MARKO-D post-test results ……………. 75
Figure 4.5: Pre- and post-test results from Classes A and B
…………………………….. 76
Figure 4.6: Combined average scores for Classes A and B on the pre- and
post-tests ………………………………………………………………………………………… 77
Figure 4.7: Scores on the pre- and post-test on Level I – MARKO-D test …………… 78
Figure 4.8: Scores on the pre- and post-test on Level II – MARKO-D test ………….. 80
Figure 4.9: Scores on the pre- and post-test on Level III – MARKO-D test …………. 81
Figure 4.10: Scores on the pre- and post-test on Level IV – MARKO-D test
……….. 83
Figure 4.11: Scores on the pre- and post-test on Level V – MARKO-D test
………… 85
xi
LIST OF TABLES
Table 4.1: MARKO-D test results and extracted codes ……………………………………. 86
Table 4.2: MARKO-D observation codes
……………………………………………………….. 90
Table 4.3: Combined codes for data sets 1 and 2 to abstract categories
for the MARKO-D test
……………………………………………………………………….. 93
Table 4.4: Data set 3: Teacher A (1) Classroom Observation: Observation
text to codes ……………………………………………………………………………………. 99
Table 4.5: Data set 4: Teacher A (2) Classroom Observation: Codes
to categories ………………………………………………………………………………….. 100
Table 4.6: Teacher B Classroom Observations: Codes to categories ………………. 102
Table 4.7: Teacher A Interview: Codes to categories …………………………………….. 107
Table 4.8: Teacher B Interview: Codes to categories …………………………………….. 113
Table 4.9: Abstracted categories from the combined interviews with
Teachers A and B …………………………………………………………………………… 115
Table 4.10: Categories from the data sets 1 to 4
…………………………………………… 117
Table 4.11: Themes extracted from the categories from all 4 data sets ……………. 119
1
CHAPTER 1: INTRODUCTION AND BACKGROUND
1.1
THE RESEARCH PROBLEM
In my day-to-day practice as a teacher in the foundation phase2, and as a part-time
lecturer in pedagogy for Grade R3 for education students, I have come to realize that
teachers do not always know enough about how young children learn mathematics.
Furthermore, I have realized that we teach, based on our own intuitive theories of
learning, focusing much on memory and facts, as well as on methods and
procedures. We therefore teach children ways of ‘doing’ mathematics and ways to
remember the processes, and we do that reasonably well. However, I have come to
the conclusion that we do not pay enough attention to helping children to build
concepts, for example, of numbers. I realized that I would need to know more about
mathematics concept development and, like De Villiers (2016), I decided to study
mathematical cognition of children in order to use the knowledge I may encounter to
change my own practice. I thus wanted to become a teacher who works as a
practitioner in the knowledge-rich profession of teaching, and have what Gitomer and
Zisk (2015:1) refer to as “deep knowledge”.
Shulman (1987) theorized, as early as 1987, that teacher knowledge includes
knowledge of the child, and of learning. Gitomer and Zisk (2015) argue that the
knowledge base for teachers has increased and that pedagogical content knowledge
(PCK) is not purely about teaching or content, but also about developing a better
understanding of how to create new and improved teaching materials. They argue,
furthermore, that teachers have to know more to practise better, and part of that
knowing is to gain knowledge on how the young child learns mathematics (Gitomer &
Zisk, 2015). In the same way, professional development is aimed at a better
understanding of what it takes to be an effective teacher (Ball, Thames & Phelps,
2008). The knowledge base that will be considered in this dissertation is that of some
aspects of children’s learning of mathematics at the very start of their ‘institutional’
2 Foundation phase in the South African context refers to the phase from Kindergarten to Grade 3.
This is very similar to the Elementary Phase in the USA.
3 Grade R in the South African context refers to the year before Grade 1, the same as Kindergarten in
the USA.
2
learning. It is at this point of their institutional education where the foundations for
learning can be laid solidly, or, as in many instances in South Africa, can form an
unstable base from which to learn the abstract world of mathematics. This study
focuses on the foundations of mathematical learning, when children make the
transition from core knowledge, or innate knowledge, to the world of symbolic
knowledge; when language and other symbols enter their mathematical learning
(Butterworth, 1999; Carey, 2009; Dehaene, 2011; Henning & Ragpot, 2015; Spelke,
2010).
I will argue that it is at this intersect, where the child moves from innate knowledge to
symbolic knowledge, that the obstacles in the path of conceptual development of
mathematics, specifically numeracy, may be lodged. It is also here that the teacher
plays a crucial role in introducing and teaching children the mathematical concepts
that will form the foundation of their further learning. Cognitive neuroscientist
Stanislas Dehaene, claims that “all humans start on the same rung of the arithmetic
‘ladder’”, but adds that we do not all climb at the same rate to the same level and that
“progress on the conceptual scale of arithmetic depends on the mastery of a toolkit of
mathematical inventions” (Dehaene, 2011:263). I would argue that these ‘inventions’
are the children’s own and that teachers need to understand children’s psychology,
as observed in their actions and by what they say. In a school situation, I believe it is
the teacher who is responsible for providing the child with suitable tools, both
concrete and abstract. These tools will help the child to build the psychological
technologies (Rose & Abi-Rached, 2013) needed to work in what becomes a world of
highly abstract thinking in the knowledge field of mathematics.
1.1.1
Background: South African early grade classrooms
In South African schools, children are introduced to the symbolic world of
mathematics (and of literacy) in Grade R and in Grade 1. Here they enter a world of
full symbolic learning. We, the teachers, need to acquire more knowledge about the
level of conceptual development of children when they enter this world of symbolic
learning. My view is that such knowledge could help the teacher to introduce children
to usable tools and to guide them to develop sound numerical concepts.
3
Until recently, the Department of Basic Education (DBE) performed annual national
assessments (ANA) on specific subjects to establish the academic performance of
children nationally in that subject (DBE, 2012). According to the diagnostic report on
the performance of children in the foundation phase in the ANA for mathematics of
2012, there is a disturbing division between children in schools that perform well and
those in schools where the children perform poorly (DBE, 2013). Henning (2013b)
assumes that these test results are evidence that children who know more will learn
more.
The 2014 Annual National Assessments Results, released by the Minister of Basic
Education, show a slight improvement in the overall performance in ANA tests, with
average percentage scores increasing by a maximum of eight percent in
mathematics in Grade 1 (DBE, 2015)4. These scores also increased in all other
grades, except Grade 9. The Minister pointed to the problem specific to mathematics
teaching in South African schools, and said that the low scores were due to the fact
that children have a very poor understanding of mathematics, and that teachers did
not have the knowledge necessary to teach mathematics at Grade 9 level. The
Minister called for drastic intervention to address this problem (DBE, 2014).
The ‘poor grasp of mathematical concepts’ to which the Minister of Basic Education
(DBE, 2014:3) refers, needs to be addressed in South African schools, and it is one
of the considerations that will be kept in mind during this study.
Conditions in South African education are somewhat challenging, due to our cultural
and language diversities, as well as our history. Some children are taught in their
home languages from Grade R to Grade 3, and then switch to English. This is clearly
a disadvantage for children who have to make this switch, compared to children
whose education starts in their home language and then continues in that same
language. Literature is available on the evidence that the variance in language, when
describing mathematical concepts, plays a significant role (Bowerman & Choi, 2003;
Bowerman & Levinson, 2001; Henning, 2013a; Levine & Baillargeon, 2016; Levinson,
2003; Spaull & Kotze, 2015; Spelke, 2003). In a study on the assessment of Grade 1
4 The Department of Basic Education released the 2014 report on the ANA tests in 2015, which
accounts for the difference in reference dates.
4
in Gauteng, Fritz, Ehlert and Klüsener (2014) found that children in English schools
showed great variance when the home language is taken into consideration. These
authors also found that, of the Grade 1 children in English schools in the Gauteng
Province, only fifteen percent used English as home language.
According to Henning (2013a:143), the past curriculum reforms were based on “an
as yet unclear diagnostic image of what the vastly diverse children of the country are
capable of at the same age in, for example, Grade R and Grade 1”. She mentions
that for the past 30 years, there has been no new standardized test in South Africa
that can diagnose children’s competence in mathematics in this age group, and that
the ANA tests are merely descriptions of the child’s performance as a reflection of a
school’s curriculum implementation and coverage, and that it is a systemic
evaluation. This raises a great concern and, although the ANA tests cannot be
described as truly diagnostic, the level of performance in South African schools is
disturbing and cannot be ignored.
To address this disturbingly low level of performance in mathematics in South Africa,
there is a need for a test that is valid and reliable to test the number concept
development of a child. A reliable diagnostic test of number concept development
(regarding additive relations of number) should assess the child’s competence as an
individual. In this study, I make use of the Mathematical and Arithmetical
Competence Diagnostic (MARKO-D)5 test that tests the individual developmental
level of number concept of a child, and that will be explained in detail in the next
paragraph. The MARKO-D test is different from the ANA, as Henning (2013a) argues
that the ANA assesses mostly some curriculum knowledge; it measures how far the
teacher has progressed with the curriculum and how the children have been able to
keep up with the teacher. The discussion of the MARKO-D test follows.
5 The MARKO-D (Mathematical and Arithmetical Competence Diagnostic) test from Germany, that
measures the level of numeracy knowledge, has been standardized in four languages for South
African children. The languages for which norms have been developed are isiZulu, Sesotho,
Afrikaans, and English.
5
1.1.2
Number concept development: learning mathematics for teaching
In recent research in the Gauteng Province, a German-originated diagnostic
interview test [Mathematical and Arithmetical Competence Diagnostic test (MARKO-
D)] was translated into four South African languages, standardized and norms were
developed locally. The MARKO-D test was used in this study as part of a project to
expand the use of this test. This test was originally developed by a group of
researchers in Germany in 2009, (Fritz, Ricken & Balzer, 2009). Through the
collaboration of a research team from a university in Johannesburg in South Africa,
this test was adapted to investigate the early numerical competence of children in
South Africa. Teixeira (2013), as cited in Henning, 2013a:145), refers to the
MARKO-D test, as a measurement of mathematical and arithmetical cognitive
development which was designed to assist teachers and psychologists as a
diagnostic test of individual children to enable professionals to take remedial action if
necessary. Teixeira says that the test, if used on a larger scale, could help address
the strengths and challenges within the national curriculum and teaching practices.
In this study, I will include a component on the foundation phase (elementary phase)
teacher, and specifically how a Grade 1 teacher’s knowledge of mathematics
learning may influence children’s learning. The MARKO-D research team mentioned
previously found, in a small research sample, that teachers do not give much thought
to children’s mathematical cognition, but that they “teach from a notion of their
instructional methods as origin of children’s understanding” (Henning, 2013a:153).
The team also found a direct relationship between the curriculum, the teacher’s
teaching, and children’s performance. In this view, the teacher is an instrument of the
curriculum, whereas I would argue that the curriculum should be a tool in the hands
of the teacher. Henning (2013a) explains that, although teachers’ discourse is full of
child-centeredness terminology, their practice is curriculum and pedagogy driven.
Henning (2013a) found that teachers begin to change their discourse about teaching
when they encounter knowledge about mathematical cognition and conceptual
development.
In this study I investigated whether there is a relationship between a teacher’s
teaching (with an understanding of a child’s learning) and children’s performance.
6
The critical questions are: Where does the Grade 1 teacher start to teach
mathematics concepts and what existing knowledge of the children does the teacher
build on, when teaching mathematics? What is the mathematical background that the
child has when entering the formal mathematical world of school? How do children
make “their world mathematical”? (Henning & Ragpot, 2015:75).
From what we have learnt from recent literature, children use their innate knowledge
of both object discrimination and approximation, and what they have learnt in their
environment through experience. Children enter the world with the ability to see
approximate different quantities (Approximate Number System, or ANS), and to
distinguish between and name objects up to quantity three (Object Tracking System,
or OTS) (Henning & Ragpot, 2015:77). According to Henning (2013a), children make
their world mathematical with the tools that we, the teachers, families and other
caregivers, use to explain and present mathematics to them.
Young school children’s ability to make sense of this mathematical world, depends
on the type of ‘tools’ that they encounter along this path of conceptual development;
and whether they have the ability to use them functionally. Tools are also conceptual
(mind) tools and language tools. I argue that once a child is in school, it is the
responsibility of educators and the school system to refine the ‘mastery’ of the child’s
emerging conceptual toolkit and skills (concepts and procedures). Dehaene
(2011:271) refers to the quantity code (codes for quantities and the development of
number cognition) and states that “it is not only rendered accessible by education; it
must be extremely refined”. This extreme refinement applies to every area of
mathematics development and it is the responsibility of the teacher to provide
opportunities for children to learn the code and to learn its meaning and its use,
ultimately its use in abstract form.
When children start school, they already have a history of mathematical knowledge
and learning, based on the innate core knowledge systems and their experience of
their environment. In several studies, this mathematical history has been shown to be
a key factor in the development of mathematical knowledge and competence at
school age (Fritz, Ehlert & Balzer, 2013). Children who have had positive
experiences with number (and space) concepts, have a good chance of achieving
7
success at school, and children with a lack of good knowledge might experience
difficulties in learning mathematics (Aunola, Leskinen, Lerkkanen & Nurmi, 2004;
Krajewski, 2003; Landerl & Kaufmann, 2008).
This indicates that children in a class at a school will all have different mathematical
histories and that learning might not take place at the same pace for each child. In
any case, development is individual, and norms indicate a distribution of children
from different backgrounds, socially and economically, and most probably not with
the same exposure to mathematical experiences. This is one of the many reasons
why the MARKO-D test is reliable, as it tests the individual child instead of a group
average.
According to Henning (2013a), research findings have indicated that teachers teach
from a notion of their instructional methods as the origin of children’s understanding.
They do not give much, if any, thought to the children’s mathematical cognition or
their developmental teaching and the children’s conceptual learning, as they do with
facts and procedures. Although their discourse abounds with child-centeredness
terminology, their practice is driven by the curriculum and pedagogy of procedures
and facts.
As part of my research, while employed at this school, I investigated the instructional
methods of the two Grade 1 teachers. I also investigated the Grade 1 children who
participated and completed the MARKO-D (number concept) Test. I wanted to inquire
if there would be a difference in the outcome of this test after one year of learning in
the teacher’s class. The results of these tests, the children’s understanding of the
mathematical/ numerical concepts and the type of methods of instruction after one
year are discussed as the data of the study.
1.2
RESEARCH QUESTION
The above discussion of the identified problem has led me to the formulation of my
research question:
8
How do children perform on a diagnostic numeracy competence test at the
beginning of their Grade 1 year compared to the results at the end of their Grade 1
year, after mathematical concepts have been taught by the Grade 1 teacher?
The sub-questions are:
Which level of mathematical (number) concept development showed the most
improvement in results, after being taught for a year by the Grade 1 teachers?
What methods of instruction and teaching did the Grade 1 teachers use to
transfer the mathematical knowledge of concepts to the children?
1.3
THE AIM AND OBJECTIVES OF THE STUDY
The study was conducted with the aim of addressing the main research question,
thus to find out how children perform on this diagnostic numeracy competence test
(the MARKO-D) at the beginning of their Grade 1 year compared to the results at the
end of their Grade 1 year, after mathematical concepts have been taught by the
Grade 1 teacher.
The overall aim of the study was to find out how children perform on this diagnostic
test (the MARKO-D test) and to relate the assumed change in the teachers’ practice
and knowledge of mathematics cognition to the performance of two classes at two
observation times. The first test was administered at the beginning of the Grade 1
year, and the second set of tests took place after a year of teaching. The aim was
therefore to conduct a field experiment, but without a control group. The sampling
was also not randomised, as the classes were intact groups.
1.3.1
Objectives
The study’s objectives were to obtain information about:
1. The performance of Grade 1 children in two suburban South African
classrooms from the same school on a numeracy competence test (the
MARKO-D test) at the beginning of their Grade 1 year and again at the end of
the same year;
9
2. The level of concept development from the MARKO-D test that showed the
highest increase in test results, when the pre-test results were compared to
the post-test results at the end of the year; and
3. The methods of teaching that were utilised and how the Grade 1 teachers’
knowledge of developmental mathematical cognition might have featured in
their teaching.
1.4
THEORETICAL BACKGROUND
The theoretical model of mathematics concept development, as propounded and
validated by Fritz et al. (2013), forms the theoretical frame of the study. This model
claims that early number concept development happens in phases and that these
can be mapped onto a unidimensional Rasch model (Bond & Fox, 2012), that can
identify the child’s level of number development. The design and background of the
model will be discussed in depth in Chapter 2. The following is a brief summary as
introduction to the model:
Level I: Counting – On this level, the child begins to understand that the counting
rhyme has meaning and that one object could be represented by one number in
“one-to-one-correspondence,” using the count nouns – the numerals they learn from
their environment (Gelman & Gallistel, 1978).
Level II: Ordinality – Due to a number’s position in a number string, or number line,
the child develops an ordinal representation of numbers (Fritz et al., 2013). The child
learns that numbers increase in size further along the number line, which is a mental
representation of preceding and subsequent numbers.
Level III: Cardinality – This refers to cardinal understanding that each number-word
(numeral) refers to the number of elements in a set. The child recognizes that there is
a relationship between the number name and the number of objects the name
represents, which is always the same.
Level IV: Decomposability of numbers – Each number can be partitioned into
smaller numbers, but in turn also forms part of a bigger number. This is also known
as the part-part-whole concept and is often practised at school as number bonds.
Level V: Relationality – As the child has mastered the concepts of ordinality and
cardinality of numbers, the relation one number has to another can now be practised
10
(Fritz et al., 2013). Thus five is one bigger than six, but also three smaller than eight;
also five is the difference between 20 and 15.
Coupled with this model, the study will also explore what Shulman (1987) has termed
“pedagogical content knowledge”, meaning that apart from the content of a subject, a
teacher not only has to know methods of instruction and other tools, but also has to
know the child. In the case of this study, the aspect of the child that is emphasized is
the forming of mathematical concepts, specifically arithmetical and number concepts.
The study therefore has a developmental slant.
1.5
METHODS AND DESIGN
1.5.1
Research design
In this case study, one is able to find detailed data about the phenomenon that has
been studied (Henning et al., 2004). One of the characteristics of this case study
research was the use of multiple data sources, a strategy which also ensured the
credibility of the data, as described by Patton (1990) and Yin (2003), with the
classrooms as the “bounded system” (Stake, 1988:255). The study is a field
experiment, which means that it is not a true experiment with a randomized sample
and a control group. It was conducted in situ with classes at the same school and
there was no random assignment and also not a true experimental and control group.
This ‘design-type’ allows for description of the teachers’ inherent style, which I would
argue, would continue to play a notable role in their teaching, which is not
controllable. The two classes can also not be fully comparable, although they
comprise children from similar backgrounds. Nevertheless, the study is, arguably, an
interventionist inquiry that includes observations of classrooms, since patterns of
mathematical cognition will be noted in classroom teacher discourse. The assumed
shift in teacher talk will be argued as a shift in pedagogical content knowledge (PCK),
in other words, that the teacher will have become aware of the developmental issues
in numeracy.
11
1.5.2
Sampling
The sampling for this study consisted of 37 participants, comprising 35 Grade 1
children from two different classes in a dual-medium private school in Johannesburg,
and the two Grade 1 teachers who taught these classes. The Grade 1 children
participated in the MARKO-D test at two different times and these test results were
then analysed.
1.5.3
Data collection
Data for the study were collected by means of two data sources, the MARKO-D test,
and teacher observations and interviews (for a more in-depth discussion on
progression of data collection, see paragraph 3.5 and Figure 3.1 Data Collection
Procedure). The MARKO-D data consisted of two different data sets, namely the test
results of the MARKO-D test gathered from two testing opportunities, at the
beginning of the children’s Grade 1 year and a second time at the end of their Grade
1 year, and the qualitative observations made during the MARKO-D test by the
facilitators. The remainder of the data was sourced from classroom observations and
teacher interviews. The data were organized into four different data sets, from data
gathered from the four data collection methods (see Figure 3.2: Illustration of the two
data sources and four data sets used in the study):
i. MARKO-D test results;
ii. Observations by test administrators;
iii. Classroom observations; and
iv. Teachers’ interviews.
i.
MARKO-D quantitative test results:
Phase 1: During February of the children’s Grade 1 year, data were gathered by
using the MARKO-D test; and
Phase 2: During November of the same year, the same group of Grade 1 children
was assessed again, using the same test instrument.
Repeating the test with the same group of children showed how individual children,
as well as the entire group of children, progressed in their number conceptual
12
development during their Grade 1 year. This “design type” (Henning, Van Rensburg
& Smit, 2004) of the study can thus be described as follows: It comprised of two
phases of an ex post facto naturalistic (field) experiment. In an ex post facto study, or
after-the-fact research, the investigation starts after the fact (Grade 1 teaching) has
occurred without interference from the researcher. De Villiers (2016) notes that
although ex post facto design is not a true experiment (and not a design experiment),
it has some of the basic logic of inquiry of experimentation (Simon & Goes, 2013).
ii.
MARKO-D observations
During the completion of the MARKO-D test by the Grade 1 children, the test
facilitators made observational notes on the children’s behaviour during the test.
iii.
Classroom observations
Apart from the test instrument, data were further gathered during classroom
observations, where I observed the two Grade 1 teachers’ classes (four lessons
each) and identified the discourse markers from the categories as developed by
Henning et al. (2004).
iv.
Teacher interviews
Further data were gathered during interviews with both the Grade 1 teachers. The
interviews were done individually after the MARKO-D test was administered and after
a year of teaching those Grade 1 children (see paragraph 3.5.3 for more detail
regarding these interviews). These structured interviews would provide a teacher’s
perspective on the methods of teaching and her knowledge of mathematical concept
development, while the observations indicated how the Grade 1 children learnt in
class.
1.5.4
Data analysis
Test data:
The test scores were analysed to establish whether the level of competence of the
children in the two classes differed. I looked at overall test scores of each level of the
MARKO-D test which are based on the conceptual model with levels ranging from
Level I to Level V (see paragraph 1.4) and focused on the increase in levels. These
