Add number concept development in grade 1 children’s performance and teachers pedagogical skills

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 G

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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(2012) Title of the thesis or dissertation PhD (Chemistry)/ M.Sc (Physics)/ M.A (Philosophy)/M.Com (Finance) etc [Unpublished]: University of Johannesburg Retrieved

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NUMBER CONCEPT DEVELOPMENT IN GRADE 1: CHI LDREN’S PERFORMANCE AND TEACHERS' PEDAGOGICAL SKILLS

DECLARATION

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

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

all the pupils and the Grade 1 teachers who participated so willingly and eagerly in

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

Orban School gave permission to disclose its name and the principal’s name The teachers and the

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

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

the role of the teacher in making the subject matter accessible to the child, claiming

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;

ii Concepts at different levels of mathematical cognitive development are taught throughout schooling and some specifically in Grade 1;

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

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

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

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

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

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

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

CHAPTER 1: INTRODUCTION AND BACKGROUND

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

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

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

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

‘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

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

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

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

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)

(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

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

as yet unclear diagnostic image of what the vastly diverse children of the country are

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

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

individual In this study, I make use of the Mathematical and Arithmetical

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

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)

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)

curriculum, whereas I would argue that the curriculum should be a tool in the hands

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

teaching (with an understanding of a child’s learning) and children’s performance

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

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

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

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

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

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

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

The above discussion of the identified problem has led me to the formulation of my research question:

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?

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

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

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

knowledge of developmental mathematical cognition might have featured in their teaching

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

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

(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

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

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

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

i MARKO-D quantitative test results:

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

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

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

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

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

test scores also showed what level of concept development had the biggest increase

in the pre-test compare to the post-test results The observations done by the test facilitators were coded and categorized according to units of meaning, and themes were extracted

Teacher data:

discourse markers of teachers’ pedagogical language, using relevant utterances to construct a pattern of discourse, as explained by Fairclough (2003), and used by Henning (2013a) and Lamberti (2013)

With regard to the classroom discourse, lessons were observed at four points during the year, noting the discourse markers for pedagogy that show a leaning towards teaching concepts that are suited to the level of conceptual development of the children The teacher discourse was identified by way of observations (of mathematics lessons) and interviews with the teachers (which were recorded and transcribed) The data were then analysed, using a schedule as suggested by Henning et al (2004:106) In the comparison of discourse patterns, the researcher did not only capture the teachers’ style, but specifically the possible shift in discourse markers

The process of capturing, displaying and analysing the data was recorded for replication purposes, as suggested by qualitative research specialists such as Denzin and Lincoln (2001, 2002), Merriam (1998), and Miles and Huberman (1994) All of these specialists’ text on analysis was the primary source

In the last phase of the analysis, the results of the data sets were compared to see whether there was a relationship between the discourse patterns of the teachers and the performance of the children

To ensure that this study was done ethically, I abided by the University of Johannesburg’s ethics protocol Firstly, ethical consent to conduct the study was

obtained from the University of Johannesburg’s Faculty of Education’s ethics committee I also submitted a letter to the governing body of the private school to obtain permission to conduct my study at that particular school A letter of consent was also given to the parents/guardians of the Grade 1 children who participated in the study, to give permission for their children to participate, as they were under the age of 18 years and their parents or guardians had to give written consent The consent was given for the children to undertake the MARKO-D test and to be observed in their classrooms during mathematics lessons during their Grade 1 year These letters also ensured the parents/guardians and the school that the identity of each individual child and the information would be treated with confidentiality and that they were free to withdraw from this study at any given time The teachers were

would remain anonymous through all aspects of the study The two teachers in the study gave written consent that they could be interviewed and voice-recorded

Feedback on the findings will be communicated to the parents/guardians after the study had been examined during a parent feedback meeting During this meeting, I will give the parents/guardians general results, without singling out any children or teachers The feedback will be discussed with the teachers and the school management before the parent/guardian meeting

ability to measure what it is intended to measure accurately (De Vos, Strydom, Fouche & Delport, 2006) The instrument used, namely the MARKO-D test, had recently been validated and norms had been developed for the South African population in four South African languages and it is now a reliable instrument to use

in the South African context (Henning, Fritz-Stratmann, Balzer, Herholdt, Ragpot & Ehlert, in press)

The data gathered from the four different data sets, namely the (i) MARKO-D test results and (ii) observations during the administration of the MARKO-D test, (iii) the teacher classroom observations and (iv) the teacher interviews, were analysed and

results were triangulated, as advocated by Crotty (1998) for optimal reliability of findings Further trustworthiness was assured by means of triangulation of data collection methods, as well as through the use of member checks of transcribed data (Lincoln & Guba, 2000) As the participants in this study consisted of the Grade 1 children and teachers, the data gathered were therefore reliable as it measured the performance of Grade 1 children and their teachers during their Grade 1 year This adheres to the stipulations of trustworthiness as outlined by De Vos et al (2006), where it is mentioned that reliability in a study is heightened when the study tested what it set out to test This ensures that the study is trustworthy

Chapter 1 provides an overview of the study It identifies the background to the research problem, focuses on the motivation for the study and includes the problem statement and research question as well as the objectives of the research The research design and data generating methods are also discussed briefly

In Chapter 2, I discuss the theories that underpinned the study and focus on these theories in relation to the research question The chapter includes studies of early grade mathematics as well

Chapter 3 focuses on the research design I will explain the qualitative and quantitative methods that were used to collect data from the four data sets These were gathered from the MARKO-D test results and observations, and from the observations in the two Grade 1 classes and the interviews with the teachers

In Chapter 4, I report on the analysis of the data that I have collected The data are grouped in four data sets The results of the two tests using the quantitative method, namely the MARKO-D test, are analysed and compared The qualitative methods of observation and interviewing of the teachers are analysed and themes that were constructed from the analysis are presented

Chapter 5 is the concluding chapter, where I discuss the findings in terms of the research and use these findings to show how the study responded to the research

question Recommendations to guide other teachers to improve their understanding

of the mathematical cognitive development of children follow with suggestions for further study

This chapter served as an introduction to the study, focusing on the research problem, giving the background and the motivation for the study, while referring to some of the literature that has contributed to my understanding of mathematical cognition of young children upon entering school I also highlighted the dual theoretical framework of the study that looks at a specific model of number concept development, paired with the pedagogical content knowledge that the teacher utilizes

to teach young schoolchildren The design was a field experiment done with two classes at a school and was conducted in two phases at two different intervals during the same year The sampling consisted of 37 participants from the school and an ethical procedure was followed to protect the participants at all times The data were collected by means of four data sets and then carefully analysed to get to the final findings This chapter is thus an overview of the investigation on the performance of Grade 1 children in the diagnostic test (the MARKO-D test) Furthermore, it relates and argues the assumed change in the teachers' classroom practice and knowledge

of mathematics cognition to the performance of children from two classes at two observation times

CHAPTER 2: ASPECTS OF YOUNG CHILDREN’S EARLY MATHEMATICAL CONCEPT DEVELOPMENT

development, specifically their mathematical conceptual development I particularly want to investigate how general ideas about children’s mathematical conceptual development may shed light on children’s mathematics performance in the early grades in South African schools Current low performance of South African children

on global mathematics assessments (DBE, 2015) and the low mathematics performance of Grade 12 children (DBE, 2015), have raised many concerns regarding the possible causes and/or origins of this stunted achievement This chapter will thus focus on some of the literature at the intersect of: 1) global trends and ideas on children’s early mathematics development and the performance and the support thereof; and 2) the relation of these ideas to the learning and teaching of mathematics in the South African foundation phase classroom

Using insights, I have gained from the literature that I have studied, I argue that before one could address the serious deficit in early mathematics learning in South Africa (Spaull & Kotze, 2015), one should first have an understanding of current

specifically, an investigation into current theories of the children’s numerical cognition will afford greater insight into what the typical obstacles are that children experience

in the learning of numbers and the understanding of arithmetic Based on an understanding of current literature on the topic, I suggest approaches for overcoming the identified learning problems and propose an operationalized set of constructs in the research tasks to be discussed in Chapter 3

I have already mentioned in Chapter 1 (paragraph 1.1.1) that this study is motivated

by the concern I share with others, about the lack of progress of learning mathematics in the crucial early years of schooling The results of the last annual national assessments (ANA) in 2015, and diagnostic reports of the last few years on the mathematical performance of children in the foundation phase in education in

South Africa, are of concern (DBE, 2013; 2015) According to Ms Angie Motshekga, Minister of Basic Education (DBE, 2015), there is a poor grasp of mathematical concepts among the children in South African schools and teachers seem to lack the ability to teach them In her report on the ANA tests results (DBE, 2015), she indicates that there is a need for a deeper understanding of how children learn and develop cognition of mathematics and how such an understanding can possibly be of use to teachers to adapt their instruction

Annually, there are numerous workshops and seminars held by provincial education

classroom strategies and children’s procedural knowledge in order to improve

mathematics results An example of such workshops or seminars is the Sediba

Project (2013) that was launched at the North-West University, which aimed to

deepen the content knowledge of in-service, historically disadvantaged teachers in

example is Save the Children South Africa (SCSA, 2015), aimed at managing the

mentoring programme of 100 community-based student teachers to become qualified and highly committed foundation phase teachers This programme promotes

experience and theoretical knowledge Despite these efforts, there does not seem to

be an improvement in the results, as is clear from the above-mentioned results from the ANA 2015

Teachers often emphasize procedural knowledge and mathematics fact recall to the detriment of sufficient conceptual knowledge Somehow, they do not maintain a balance between these two crucial epistemological distinctions From studies by Venkat (2013) and Graven (2015), it is evident that teacher talk is controlled by methods of the teaching of operations, thus procedures, more than the concepts that are required to perform the four main mathematical operations themselves To grasp early conceptual development of number and operations, theoretical knowledge is required This chapter aims to discuss some of the ideas that one needs to understand the learning difficulties and successes that young children encounter when they start formal education in mathematics better I will look at their cognitive

Starting from a Piagetian point of view on childhood cognitive development, I will add ideas of Vygotsky (1978) I will then discuss ideas from contemporary psychologists such as Susan Carey (2009), Elizabeth Spelke (2000) and Karen Wynn (1990, 1992), and focusing on the contemporary notions of childhood conceptual change Lastly, I will look at how developments in neuroscience have given us an alternative picture to the Piagetian-focused insights on the development of mathematical concepts (Dehaene, 2014; Dillon, Pires, Hyde & Spelke, 2015) I will look at Dehaene’s (1997, 2011) scientific findings about arithmetic and the human mind, as this will shed light on the development of mathematical cognition of the Grade 1 child The reason why these researchers are prominent in my discussions is that their ideas focus on childhood mathematical development The understanding brought forward by these modern social scientists, about the way in which children develop mathematical concepts, could assist me in my study on developmental conceptual basis of learning (and its subsequent influence on teaching) of mathematics in the Grade 1 classroom

I will discuss mathematical conceptual development, keeping in mind that certain numerical concepts may be in place, innately, long before the child starts school (Dehaene, 2011), while others need to be developed in the classroom and in other environments (Dehaene, 2011) This discussion on innate number systems will include the approximate number system (ANS) and the object tracking system (OTS), as discussed by Feigenson, Dehaene and Spelke (2004) They describe ANS

as the innate ability to distinguish between different quantities, therefore between more or less objects in a set The OTS is described by Carey (2009) as a system that files individual objects neurologically and allows for the recognition or tracking of one, two or three objects As my study focuses on the Grade 1 child, these innate systems should be in place and able to assist the child when formal mathematics training starts in Grade 1

I will furthermore argue that together with the OTS and the ANS innate systems, certain mathematical concepts need to be in place in early numeracy development to understand mathematics As this study investigates the mathematical development of Grade 1 children, I will look at a model of mathematical cognition that establishes the

level of mathematical development The MARKO-D test was created by Fritz et al (2009), based on concerns about early numeracy development This model focuses

on the development of children aged four to eight, and the Grade 1 children in this study fall into that age group I will therefore argue why this is a relevant conceptual basis and measurement to use, according to which to investigate the results

Added to these two areas of discussion (psychology and neuroscience), in this chapter I will also look at contemporary ideas about pedagogy Schulman (1987) theorized, as early as 1987, that teacher knowledge includes knowledge of the child and of learning, also known as pedagogical content knowledge (PCK), and of the role

of the teacher in making the subject matter (such as mathematical concepts) accessible to the children (Fonseca, 2011), so that they can learn the foundational concepts needed as building blocks for their future learning I will investigate PCK and the impact it has on how children learn mathematics This will assist my investigation on the way in which Grade 1 children are taught and should be taught

As language plays such a central role in the understanding and development of mathematical concepts, I will argue the role of language in the learning of mathematics, as it develops through formal education I will look at the work of Carey (2009), Dehaene (2011), Fritz et al (2013), and Henning (2013a), and more specifically at their explanation of the use of linguistic terms for mathematical concepts and the meaning and impact of using of this mathematical language when learning mathematics Language is the tool that introduces the child to the world of mathematics The relevance of this argument will become clear in my discussion, as the child is exposed to language every day when he/she is taught by his/her teacher

For years, the field of childhood cognitive development has been dominated by the views of Jean Piaget, and although some aspects of his theory on childhood cognitive development are still relevant, many researchers have now developed ideas on children’s conceptual which go beyond Piagetian viewpoints (Ragpot,

2017) I do feel, however, that since Piaget occupies such a seminal position in the field, I will start my discussion on childhood development in general from a Piagetian basis and, in doing so, mention some of his ideas on the development of

work, as it is not relevant to this study

Piaget’s main proposal was that children learn best through experiences and by being actively involved in the process More than 60 years ago, Piaget (1952) based his theory of stages of cognitive development on how individuals experience and construct an understanding of their world and how this shapes their behaviour and personality (Berger, 1998) This study looks at the conceptual change that takes place while children learn mathematics in Grade 1 These children are able to learn symbolically during this phase of development

In clinical conversations with children, Piaget (1952) wanted to illustrate that abstract logic and logical thinking, occur by itself and that learning is not involved in this process Piaget asked questions that were far beyond children’s capabilities just to illustrate the pure thinking of children without the influence of previous experience or knowledge He started to believe that how children think, is more important than what they know (Berger, 1998), and yet the paradox is that they use what they know in their thinking

Piaget (1952) had several views on the development of mathematical concepts; he believed that early number, like other abstract concepts, must be constructed during sensorimotor interactions with the environment He believed that children are born without any preconceived ideas about arithmetic and that it takes years of observation in their environment before they really understand what a number is (Dehaene, 2011:31) Piaget (1952) collected proof that young children are unable to understand arithmetic In his experiment, where he hid a toy under a cloth and 10-month old babies failed to reach for it, he believed that the toy ceased to exist for the babies when it was out of sight His early observations have now been proved to have been a mistake (Wynn, 1990; 1992)

Through observations made by Piaget (1952, 1960), he came to the conclusion that children cannot seem to understand the concept of number before the ages of four to

(1952) found that children could not anticipate that the number of glasses and bottles stays the same if they are moved around or spaced differently Therefore, he said, the children could not conserve number He did not hold any theory of innate number concepts such as ANS or OTS (Carey, 2009)

development, where logical and mathematical abilities are progressively constructed

by observing the external world According to Dehaene (2011), these findings of Piaget had an impact on our education system and it was believed that children are not ready for arithmetic before the ages of six or seven Dehaene (2011) also explains that, according to the Piagetian theory, it is best to start by teaching logic and the ordering of sets, because these notions are prerequisites to the acquisition of number This is why “children spend much of their day piling up cubes of decreasing sizes, long before they learn to count” (Dehaene, 2011:32) We know that Piaget’s theory of stage-general cognitive development is in dispute (Ragpot, 2017); that the tests he used did not show what children were really capable of and that these were inadequate to determine when a child begins to understand the concept of number (Dehaene, 2011)

Lev Vygotsky is another early scholar of learning and development, and a contemporary of Piaget, who theorized about the conceptual development of cognitive competencies He argued that the relationship between learning and development in school-aged children cannot be separated (Vygotsky, 1978) He discusses three theoretical positions on this relationship The first one is based on the assumption that the maturational processes of child development are independent of learning, that learning is an external process and that it is not actively involved in the development process Even the work of Binet (1981), as discussed by

learning and that if a child’s mental functions (intellectual operations) have not matured to the extent that he is capable of learning a particular subject, then no

physical maturation normally takes place before learning, but is not the result of learning (Gauvain & Cole, 1997)

The second theoretical position that Vygotsky (1978) argued for, is that learning itself can also be regarded as (cognitive) development The two processes therefore occur simultaneously and learning and development coincide at all times

(1924), as described by Gauvain and Cole (1997), is an example of this relationship,

learning and development are related, that maturation depends on the development

of the nervous system and that learning itself is a developmental process Thus, the process of maturation makes the process of learning possible and the learning process stimulates the maturation process The problem with this theory is that learning in one area does not mean overall development in all areas

Wertsch & Tulviste, 1992) on learning and development, namely the notion of the

zone of proximal development (Gauvain & Cole, 1997) Gauvain and Cole (1997)

state that learning and development are interrelated from the very first day of a child’s life Once the child starts school, something new gets introduced to the child and his or her development, and learning takes place Gauvain and Cole (1997) explain that learning awakens internal developmental processes when the child interacts with people in his or her environment The child achieves independent development when these processes are internalized This process is best advanced within the zone of proximal development, which could be defined as:

… the distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance or in collaboration with more capable peers It defines those functions that have not yet matured but are in the process of maturation, functions that will mature tomorrow but are currently in an embryonic state (Gauvain & Cole, 1997:33)

From Gauvain and Cole’s (1997) point of view, learning is not antonymous with development, but organized learning results in mental development and sets in motion developmental processes that would be impossible if it were not for learning The whole process of learning takes place when the child is engaged with a person, such as his or her teacher, or even with more capable peers

mathematical cognitive development, but more recent ideas from contemporary psychologists, such as Carey (2009), Spelke (2000) and Wynn (1990, 1992) independently reveal more detail on when conceptual change in the domain of

innate systems with which children are born, that assist them in the development and learning of mathematical concepts and affords more awareness of when children start to develop numerical abilities

If one accepts that learning and development are interrelated (Gauvain & Cole, 1997), the question of the onset of both arises: How early does development take place in an infant or a child and when do they develop numerical abilities, or are they born with such knowledge? I will discuss the work of Wynn (1990, 1992) and Starkey,

conceptual change and advancement in mathematical cognition, as well as the work

of Carey (2009), who has conducted ground-breaking research on early number concepts and the innate number-related mental representations

Cognitive psychologist, Susan Carey (2009), proposes that a child is born with an explicit knowledge system How rich that innate knowledge system is, will determine how well those devices will be able to function Thus, the child with a rich knowledge system will have a better ability to learn than a child with a poor knowledge system Carey (2009) states that knowledge acquisition in the core domain is supported by innate domain-specific learning devices, which are universal

From birth, the brain is equipped with specific core knowledge (number) systems (Spelke, 2000) There are two core or innate systems: (i) core system 1, is the approximate representation of numerical magnitude This means that the child can represent, estimate or "guess" the size of small quantities comparatively; and (ii) core system 2, is the precise representation of distinct entities or small quantities up to three individual objects Feigenson et al (2004), explain that both these two systems are universally present in humans (and some birds and mammals), and do not come about through learning or cultural transfer These two systems will be discussed in more detail in the next section

approximate number system (ANS)

The approximate number system (ANS) is the approximate representation of

the size of different/varying small quantities The child is thus able to distinguish between a little/few and a lot These non-symbolic, approximate number representations are central to human knowledge of mathematics and form the basis for all subsequent numerical concepts (Feigenson et al., 2004)

Feigenson et al (2004) and Xu (2003) found that children have remarkable arithmetical abilities before any schooling even starts Children are able to solve approximate arithmetic problems, using basic operations They cannot get exact results, but they do get an approximate number to solve the problem (ANS) Uitzinger and Ragpot (under review) refer to the work of many leading researchers who have argued that the ANS is a mental system of quantification that has shown to be present in infants, children and adults (Dehaene, 2011) The system yields a representation of approximate number that captures the inter-relations between different quantities The approximate representation core system could be tested on the child’s ability to compare two given numbers or sets These two sets are compared according to the ratio of the magnitude of the two given entities It is easier

if the two entities display a big difference in ratio than if the ratio between them is not

so significant (Feigenson et al., 2004)

2.3.2 Core system 2: the precise representation of distinct entities or small quantities - object tracking system (OTS)

The object tracking system (OTS) equips the child to track and distinguish distinct individual elements (up to three) (Starkey et al., 1990) Children as young as six months are able to comprehend additive or subtractive changes of quantities within this number range (Koechlin, Dehaene & Mehler, 1997; Wynn, 1992) Core system 2, serves as a building block for the development of new numerical cognitive skills (Spelke, 2000:1233)

Cognitive psychologist Karen Wynn, based much of her work in the 1990s on

and development (Wynn, 1990, 1992, 1996) Wynn (1996) mentions that there is evidence that children are born with the ability to recognize and even mentally

cannot conserve number was not entirely accurate In her experiment (Dehaene, 2011:105) with two-and-a-half year old toddlers, Wynn (1990), showed that they can discriminate a number of actions, for example count the number of jumps that the

‘Big Bird’ (the character from Sesame Street) performs

In her well-known article about the study of addition and subtraction, Wynn (1992) showed that an infant was surprised when a hidden object did not appear once the screen had been dropped When one toy was shown and hidden and another one added and hidden, the child would spend a longer time at studying the one object than when two objects were revealed These results proved to Wynn that infants know that one plus one makes neither one nor three, but exactly two (Dehaene, 1997) Dehaene (1997:51) suggests that the knowledge that the babies displayed

become manifest as soon as the ability to memorize the presence of the objects behind the screen emerges, at around four months of age” This, however, applies only to quantities up to three

Spelke, in collaboration with fellow psychologists, Starkey and Gelman (1990), found

in their multimedia experiment with six-, seven-, and eight-month old infants that they are able to recognize three objects This is discussed by Dehaene (2011):

The baby looks longer at the slide whose numerosity matches the sequence of sounds that it is hearing It consistently looks longer at three objects when hearing three drumbeats, but now prefers to watch two objects when hearing two drumbeats It therefore seems likely that the baby can identify the number

of sounds … and is capable to compare it to the number of objects before its eyes (Dehaene, 2011:40)

Spelke (2000) found that the acquisition of intricate cognitive skills, such as reading, writing and calculating, depends on a number of systems that build on one another Fritz et al (2013) refer to numerous studies on the brain which show that the brain is equipped from birth with domain-specific core knowledge systems and that those systems allow for initial representations and reasoning about particular kinds of events and entities, such as objects, persons, places and numerosities (Spelke, 2000) Fritz et al (2013) further explain that, based on the efficiency of the core knowledge systems for numerosities, children acquire their first solid mathematical concepts when they start school, by which time they already have significant mathematical knowledge and experience of mathematical learning These authors point to evidence from longitudinal studies that the extent and complexity of this knowledge are important factors in the further development of mathematical knowledge and competences at school age

When language and other symbols enter children’s mathematical learning, they make

a transition from core or innate knowledge to a world of symbolic knowledge Scholars of mathematical cognition agree on this Henning and Ragpot (2015), describe this intersect of core knowledge with language symbols, where children map words onto concepts They explain that Levinson (2003), referred to this as plainly

"using language with understanding"(Henning & Ragpot, 2015:76) To use language successfully and to be able to communicate in the child’s own culture, means that the