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

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 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(2012) Title of the thesis or dissertation PhD (Chemistry)/ M.Sc (Physics)/ M.A (Philosophy)/M.Com (Finance) etc [Unpublished]: University of Johannesburg Retrieved from: I R O (Accessed: Date) NUMBER CONCEPT DEVELOPMENT IN GRADE 1: CHILDREN’S PERFORMANCE AND TEACHERS' PEDAGOGICAL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 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 i 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 ii all the pupils and the Grade 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 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 iii 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 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 year compared to the results at the end of their Grade year, after mathematical concepts have been taught by the Grade teacher I set out to explore teachers’ methods of instruction when teaching mathematics to Grade 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 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; iv 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 v 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 THE RESEARCH PROBLEM 1.1.1 Background: South African early grade classrooms 1.1.2 Number concept development: learning mathematics for teaching 1.2 RESEARCH QUESTION 1.3 THE AIM AND OBJECTIVES OF THE STUDY 1.3.1 Objectives 1.4 THEORETICAL BACKGROUND 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 vi 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 vii CHAPTER 4: RESEARCH RESULTS AND DISCUSSIONS OF FINDINGS ON GRADE TEACHERS’ PEDAGOGICAL CONTENT KNOWLEDGE AND THE TEACHING OF MATHEMATICS IN THE CLASSROOM 69 4.1 INTRODUCTION 69 4.2 DATA SETS AND – RESULTS OF THE MARKO-D TEST 72 4.2.1 4.3 Observations during the MARKO-D test 88 DATA SETS AND – TEACHER PEDAGOGY: QUALITATIVE DATA 95 4.3.1 Data set 3: Classroom observations 96 4.3.2 Data set 4: Interviews with the Grade 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 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 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 viii APPENDIX K: INTERVIEW QUESTIONS WITH GRADE TEACHERS Interview questions with Grade Teachers How you feel about mathematics in general? How you feel about teaching mathematics to Grade children? What challenges you experience? What are the strategies that you follow when teaching mathematics? How did the knowledge on the different Levels of Mathematical Development influence your way of teaching? What did you different? What information had the biggest impact on your way of teaching? How you evaluate yourself to determine whether your lesson/teaching were successful? Why you think children in general struggle with mathematics? 230 APPENDIX L: TRANSCRIPT OF INTERVIEW QUESTIONS WITH GRADE TEACHERS Interview with Grade Teacher (17 September 2015) Teacher A How you feel about mathematics in general? Feelings about mathematics Obviously, I feel maths is very important, especially as a Grade Teacher I think getting good ground in Early mathematics mathematics will serve the children well throughout their experience whole school careers And how did you feel about mathematics when you Teacher’s previous were at school? experience about I hated maths! I’m not a mathematical person at all I was mathematics Falling behind in ok in junior school and then I missed a section swopping mathematics schools in High School and I was lost and had to go for Positive experience about mathematics extra maths lessons So it’s important for me that maths is sort of easy for children So that it’s not intimidating later How you feel about teaching mathematics to Grade children? I find teaching maths to Grade children very interesting because, they are just learning the number concepts and to see them suddenly just click and get the concept that they try to get Yeah, and it’s interesting to see how they are learning What challenges you experience? Just concepts that are difficult to them, especially like place value We’ve done it over and over again and they find it very difficult Money is very difficult, I think it’s 231 because it’s a different medium that they have to work with Uhm! Yes that’s what they struggle with They tend to grasp the basic stuff more easily and when it becomes more abstract it becomes more difficult What are the strategies you follow when teaching mathematics? I try to make maths quite visual so that they can actually see the numbers We draw it on the board and count it out I use sticky dots in their books so that they can actually see and count what they are doing, or cross-out when they are subtracting Just try and make it as visual and concrete as possible because maths is so abstract later on So you keep it as concrete as possible? Yes as much as is possible How did the knowledge on the different Levels of Mathematical Development influence your way of teaching? What did you different? The information I got was mainly on how important the number line is and that the children should see that is smaller than and that made a big impact on me and that’s why I try to make maths so visual so that they can at least see that is bigger than So that’s what made quite a difference to me Yeah and to help me focus on the number line and where the numbers come And the part-part-whole concept? I never used it that much I know that XXXXXX uses it a lot, but I like the idea of it What information had the biggest impact on your way of teaching? Mainly the number line In what way? I think because I came from Grade 2, we’ve move past that, so going back to numbers to make them understand that is things and to physically see that it is bigger 232 Yes the cardinality! Yes that they get the number How you evaluate yourself to determine whether your lesson/teaching was successful? I just generally base it on the children How many children come and ask for help If there are too many children then they’ve missed it, then I have to re-assess my lesson or re-do it If they seem to have grasped it or are happily working on their own and as I walk around and see they got it, then I know, it was successful Why you think children in general struggle with mathematics? I think mainly because they haven’t grasped the basic idea of what numbers are So they can’t subtract, because they not know what they’re subtracting from what I think it can be intimidating, because there are these random symbols that they have supposed to use to solve problems and it’s meant to make sense to them and it often doesn’t So it’s definitely basics, getting that all in and then moving from there Do you think teachers in general, I don’t know for how long have you been teaching? Six years You’ve come across different foundation phase teachers, at different schools Do you think teachers’ knowledge is adequate to teach mathematics? To foundation phase children yes, generally I think higher up it becomes a problem and teaching it differently I just find that some people believe in different methods and drilling children is quite popular I feel that doesn’t really reach them concepts, but I think that they need to know sums off by heart So I think it’s difficult to say, but they need to know and understand why - = and why, because you take it away, rather than to just know it off by heart To know how it really works If you think back to your training, you think teachers who are studying are getting adequate training and information on how to teach mathematics? Oh no, definitely not! I studied through -(university) and we basically just got packs and had to sums ourselves and there was no training on this is the stage 233 and you have to take this step and the children has to grasp the step There was none of that Just can you the maths, yes you can maths, ok you can teach maths Without actually knowing That’s why I found this information so helpful, because I never really thought how to teach maths before They never showed me a way to it, especially because I was coming to Grade and that’s the introduction to everything Rather than okay you know your numbers, now we are going to times and times is grouping and we can that and it’s fine So of the very first basics, there was none of that at all Do you think if you had more knowledge on how children learn mathematics that would have made it easier to teach them? Oh yes definitely! Yes you would plan your lessons around what would be easiest for them to grasp, rather than you would just try to teach them 234 Interview questions with Grade Teachers – (Teacher B, 15 Sept 2015) Positive previous experience general? Previous experience Teacher: “Ehh, I love maths Why I love maths? influences the present How you feel about mathematics in Because I had a teacher who made time to help me understand maths So I love maths! I thoroughly loved it.” Teacher qualities If teaching methods fail How you feel about teaching mathematics to Grade children? Teacher: “It can be scary at times (giggle) But you Children’s feelings and state have to have a lot of patience and always of mind remember that maybe they don’t grasp it the first time, then a second time, a third time The moment children feel nervous, they stop learning So you’ve Children not understand Cardinality got to be patient and know what you And if a child doesn’t understand the way you teach him now, you’ve got to find another way to teach him.” 10 What challenges you experience? Strategies that teacher Teacher: “The children don’t have the concept, that follows they, they can count but they don’t have the Materials: concrete concept that is (showing her fingers).” Number line 11 What are the strategies you follow when teaching mathematics? Teacher: “Uhm! To start with something concrete and to stay with something concrete until a child feels comfortable to move on and does it on its own The number line, because that gives him a better understanding and … to actually it How 235 can I explain? Take children and take away 2, let them it.” Children has to Interviewer: “Something physical, so they can experience experience it Do you mean…I don’t want to put words in your mouth, but you mean they have to experience it?” Teacher: “Yes, that’s it! Experience it! Let them go pick up stones and count it” 12 How did the knowledge on the different Levels of Mathematical Development influence your way of Knowledge had an impact teaching? What did you different? Counting does not Teacher: “Yes it made a big, big difference, because you mean cardinality have to understand that even though a child can count they Part-part whole don’t know the concept, they don’t know the amount, how Number line: much it is And the part-part-whole is like a wonderful that opens a distance between numbers world for children, because they seem to understand and grasp much better when you give them the part-part whole sums and then For children to understand the distance Grouping and division between and is the same as 51 and 56 is exactly the same but grouping and division can be a nightmare for children, that can be an absolute nightmare So never teach Concrete materials both together, never ever.” Children have to Interviewer: “What did you find when you taught the Gr experience it children about grouping and division?” Teacher: “Grouping they seem to understand easier, but division you should be teaching with sweets You get a box of smarties and tell the child to divide that between him and his friends Because that is a physical thing they and they understand it and that’s the thing Grouping never works when you choose children because they choose their friends and others feel left out.” 236 The use of concrete materials Influence of knowledge on teaching Interviewer: “So it’s important to use the right concrete materials?” Teacher: “Absolutely!” Part-part whole 13 To experience the concept What information had the biggest impact on your way of teaching? Teacher: “I think the part-part-whole where you use a … I’ve got a chart where you count out the complete number …” Methods of teaching empowers the teacher Interviewer: “Whole number” Teacher: “Whole number and then put part-part in each block and that gives children a better understanding that you can break up something You can extend this by Evaluation of teaching giving them a whole number and then give them parts to fill up So they learn how to the number train sums If children don’t So that helped a lot So whoever thought of that, God understand bless him!” 14 How you evaluate yourself to determine whether your lesson/teaching was successful? If children don’t After a week you can either ask a child to explain to the understand class how to it or you can give them work and see if they can still it If they can still it, the foundation was there If they can’t, you’ve failed and you have to Try a different method start all over again They have to, have to Interviewer: “That’s my follow-up question: If you find that a child cannot it after a week, what would you do?” Teacher: “You just have to start right from the bottom and lay the foundation again and again Maybe your lesson wasn’t on their level so they didn’t understand it, you need to change that Firstly, not go back to the same lesson Prepare another one and see it in a different way 237 If it’s or children who cannot understand, sometimes it is better to get one of their friends to Children learn from explain it to them, because they seem to learn much children easier and better from each other.” 15 Why you think children in general struggle with mathematics? I tell you what! Children don’t share anymore, because mommy will buy a packet of sweets for each child, so Concrete materials they don’t learn to share That’s one of the reasons needed to grasp concepts why they cannot division, they can’t group things together They don’t have a lot of concrete work, so you got to implement a lot of concrete work Experience of Interviewer: You have a lot of experience How many teaching years have you been teaching? Teacher: “Eighteen years: Interviewer: “Now if you look back to those 18 years, and you think about the teachers Do you think in Teachers stuck in their way of teaching general those teachers’ knowledge about maths is adequate to teach foundation phase? Find different ways of Teacher: “No I don’t think so You see a lot of times as explaining a teacher when I listen to the teachers it is my more like Make it enjoyable for my way or no other way I it this way If they don’t children Positive understand it we’ll it the same and again, but it feeling never works Because if a child cannot grasp it the first negative about time you explain it, you need to find another way I mathematics Teacher feels often think as teachers, when it comes to maths, it’s a Teachers need more quick you know a quick subject You don’t put so much knowledge on effort into it You don’t make it fun; you have to make it teaching mathematics fun You make English and life skills fun, but not Maths I think there are a lot of teachers who not like maths And it is a difficult subject to explain to little children to 238 anyone, why this and why that I think a person should write a book on how to improve maths in your classroom 239 APPENDIX M: CODES TEACHERS’ INTERVIEWS – TEACHER A Teacher interview: Teacher A Previous experiences that teacher had Feelings about mathematics Laying a good maths foundation Early mathematics experience- leads to later success Teacher’s experience at school about mathematics: negative Falling behind in mathematics Previous experience about mathematics Strategies that teacher follows: Make mathematics easy Positive experience about mathematics not intimidating later Method of teaching: visual 2dimensional Materials: dots Method of teaching: visual & concrete Teaching method for subtracting: visual Visually experience and use 3dimensional objects Build one level on another Methods of teaching: knows off by heart, bonds Challenges teacher experience Difficult concepts: place value Repetition of difficult concepts Difficult concept: money Abstract concepts are difficult Methods of teaching mathematics are inadequate 240 How children learn? Learning process of grasping Basic concepts are grasped faster Abstract concepts are difficult Influence of knowledge Understanding how children think The use of a number line: sequence and intervals Position of a number on the number line Method of teaching: visual Teacher didn’t incorporate all the information given: part-part-whole Use of the number line to teach Importance of the cardinality of numbers Evaluation of teaching Evaluation of lesson: children’s actions Children who don’t understand Children who are able it = learnt Measure success: child is able to it Success: children grasped Difficulties children struggle with Cardinality of numbers Lack of basic cardinality leads to further difficulties Children can feel intimidated Symbols are abstract and have no meaning To understand the problem Teacher’s knowledge about mathematics alone is not enough to teach Methods of teaching mathematics are inadequate Different methods of teaching: drilling Teacher’s knowledge of mathematics sufficient for foundation phase 241 Adequate training for teachers Training of teachers on pedagogical content University: training teachers not adequate Mathematics knowledge but no pedagogical content knowledge PCK: No training of teachers Teaching mathematics to children: PCK Teaching basics first PCK helps to plan and teach more effectively More knowledge for teacher will assist children to learn easier Better planning = preparation 242 APPENDIX N: CODES TEACHERS’ INTERVIEWS – TEACHER B Teachers’ interviews Teachers’ interviews - Teacher B Previous mathematics experience that the teacher had Positive previous experience Previous experience influences the present Teacher’s influence results in love for mathematics Teacher qualities: Teacher qualities: patience If teaching methods fail, try again Children’s state of mind Children’s feelings and state of mind - nervous Children who not understand - use other methods Strategies that teacher follows Materials: concrete Use different things to explain same concept Number line gives better understanding Child has to participate Use objects (3dim) to explain The use of correct concrete materials Children have to experience it To experience the concept Materials to assist with explanations Start with basics and build upon it Children learn from children Concrete materials needed to grasp concepts Make it enjoyable for children Positive feeling = fun 243 Challenges teacher experiences in class Children not grasp Cardinality Do not understand concepts Influence of knowledge Knowledge had an impact on how children understand mathematics Teacher learnt new concepts: part-part-whole: Counting does not mean cardinality Number line: distance between numbers: rationality Decomposing of numbers Methods of teaching empower the teacher Difficulties children struggle with Grouping and division Quantity of work taught Decomposing of numbers Concrete materials needed to grasp concepts Evaluation of teaching If children don’t understand Repetition Try a different method Level of teaching on child’s level If child is able to it after time = success Explain concept again before moving on to the next Teacher’s knowledge about mathematics alone is not enough to teach Experience of teaching in years Proper Graduate training Teachers stuck in their way of teaching Find different ways of explaining Make it enjoyable for children Positive feeling = fun Teacher feels negative about mathematics Teachers need more knowledge on teaching mathematics 244 ... 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... xi CHAPTER 1: INTRODUCTION AND BACKGROUND 1. 1 THE RESEARCH PROBLEM 1. 1 .1 Background: South African early grade classrooms 1. 1.2 Number concept development: learning mathematics... 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