© 2020 University of South Africa All rights reserved Printed and published by

© 2020 University of South Africa All rights reserved Printed and published by the University of South Africa Muckleneuk, Pretoria LADMMM6/1/2021 10001107 InDesign, Florida Campus HSY_Style LADMMM6/1/2021 (iii) CONTENTS  Page OVERVIEW OF CONTENTS OF MODULE ix Study unit 1: EXPLORING WHAT IT MEANS TO ‘DO’ MATHEMATICS 1 WELCOME 1 UNIT OUTCOMES 2 AN INTRODUCTION TO MATHEMATICS EDUCATION 2 The history of mathematics education 2 Mathematics education in South Africa 3 Why is educational change needed in South Africa? 4 What is mathematics? People’s views 5 How does CAPS define mathematics? 7 WHAT DOES IT MEAN TO ‘DO’ MATHEMATICS? 8 Contrasting perceptions of teaching school mathematics 10 Mathematics as a science of pattern and order 11 The verbs of doing mathematics 15 What is basic mathematics? 17 An environment for doing mathematics 19 EXAMPLES OF DOING MATHEMATICS 21 The five content areas 21 EXPLORING PATTERNS IN MATHEMATICS 27 Repeating patterns 28 Growing patterns 29 UNIT SUMMARY 36 SELF-ASSESSMENT 36 REFERENCES 37 Study unit 2: DEVELOPING UNDERSTANDING IN MATHEMATICS 38 WELCOME 38 UNIT OUTCOMES 40 INTRODUCTION 40 A CONSTRUCTIVIST VIEW OF LEARNING 41 The construction of ideas 42 Implications for teaching 45 Examples of constructed learning 45 Construction in rote learning 46 Understanding 47 Examples of understanding 49 Benefits of relational understanding 51 (iv) CONTENTS TYPES OF MATHEMATICAL KNOWLEDGE 56 Conceptual understanding of mathematics 58 Procedural knowledge of mathematics 61 Procedural knowledge and doing mathematics 63 A CONSTRUCTIVIST APPROACH TO TEACHING THE FOUR OPERATIONS 64 Classroom exercises on the basic operations 64 THE ROLE OF MODELS IN DEVELOPING UNDERSTANDING 68 Using models in the teaching of place value 71 Using an abacus to explain grouping in tens 72 Models and constructing mathematics 74 Explaining the idea of a model 76 Using models in the classroom 77 Models in your classroom 77 STRATEGIES FOR EFFECTIVE TEACHING 78 UNIT SUMMARY 79 SELF-ASSESSMENT 80 REFERENCES 81 Study unit 3: TEACHING THROUGH PROBLEM-SOLVING 82 WELCOME 82 INTRODUCING PROBLEM-SOLVING 83 Discussion of case studies 84 What then is problem-solving? 85 DEVELOPING PROBLEM-SOLVING TASKS 85 Problem-solving activities for space and shape 85 Starting where the learners are 88 Developing different kinds of mathematical ideas 92 Concepts and relationships constructed by connecting ideas 93 Developing procedures and processes 95 Good problems have multiple entry points 97 Designing and selecting effective tasks 98 A THREE-PART LESSON FORMAT 99 Before, during and after 100 Teacher’s actions in the before phase 101 Things to think about 102 Teacher’s actions in the during phase 102 Teacher’s actions in the after phase 103 Working towards problem-solving goals 104 Different levels of cognitive demands in tasks 104 THE VALUE OF TEACHING USING A PROBLEM-BASED APPROACH 107 UNIT SUMMARY 108 SELF-ASSESSMENT 109 REFERENCES 110 LADMMM6/1 (v) Contents Study unit 4: PLANNING IN THE PROBLEM-BASED CLASSROOM 111 WELCOME 111 UNIT OUTCOMES 112 INTRODUCTION 113 PLANNING A PROBLEM-BASED LESSON 113 Variations of the three-part lesson 115 Workstations and games as problem-based activities 115 TEACHING AND LEARNING IN SMALL GROUPS 116 Why ‘group work’? 116 What is group work? 118 What makes co-operative learning work? 120 Individual and group accountability 120 Reflecting on the work of the groups 122 Arranging learners into groups 123 ASSESSING GROUP WORK 127 Self-assessment in group participation 128 Peer assessment in group participation 128 Teacher assessment of group interaction 129 DEALING WITH DIVERSITY 130 Plan for multiple entry points 131 Listen carefully to learners 131 DRILL AND PRACTICE 132 What drill provides 132 What practice provides 133 LESSON PLANNING 133 UNIT SUMMARY 138 SELF-ASSESSMENT 139 REFERENCES 140 Study unit 5: BUILDING ASSESSMENT INTO TEACHING AND LEARNING 141 WELCOME 141 UNIT OUTCOMES 143 INTRODUCTION 143 WHY ASSESS? 145 Programme of assessment 145 The purposes of assessment 146 Continuous assessment 147 WHEN TO ASSESS? 148 An example of assessment before teaching and learning 149 WHAT TO ASSESS? 153 Assessing for reasoning, understanding and problem-solving 154 Reflecting on what to assess through the teaching of data handling 156 (vi) CONTENTS HOW TO ASSESS 158 The phases of assessment 158 Assessment methods, tools and techniques 159 Methods of assessment 162 Self-assessment 162 Peer or group assessment 163 Observation 164 Performance-based tasks 165 Tools for assessment 166 Portfolios 166 Rubrics 169 HOW TO INTERPRET THE RESULTS OF ASSESSMENT? 174 Three points of reference 175 HOW TO REPORT 177 SELF-ASSESSMENT 179 REFERENCES 180 Study unit 6: EXERCISES ON TEACHING DATA HANDLING 181 OVERVIEW 181 Tables, lists and tallies 181 REPRESENTING DATA 183 Pictograms 184 Bar graphs 185 Pie charts 188 Line graphs and broken line graphs 189 INTERPRETING DATA 192 Interpreting pictograms 193 Misleading pictograms 193 Interpreting bar graphs 194 Misleading bar graphs 195 Interpreting pie charts 195 Interpreting line graphs 196 Misleading line graphs 197 LADMMM6/1 (vii) Contents Study unit 7: EXERCISES ON TRANSFORMATION GEOMETRY 199 TRANSFORMATION GEOMETRY 199 Pre-knowledge 199 ISOMETRIC TRANSFORMATIONS 201 Translation 201 Reflection 202 Rotation 206 SYMMETRY 210 Line symmetry 210 Line symmetry of a pair of points 211 Rotational symmetry 213 Point symmetry 214 ENLARGEMENTS (DILATIONS) 215 The centre of enlargement 216 The factor of enlargement 217 TESSELLATIONS 217 (viii) LADMMM6/1 (ix) 1OVERVIEW OF CONTENTS OF MODULE Dear Student We would like to welcome you all as students studying the module: Teaching Mathematics (SP Subject Didactics). This module focuses on the approach to the teaching of mathematics and how the learning of mathematics occurs. This learning area requires a strong emphasis on basic and applied mathematical principles. This module will empower you with mathematics teaching skills and methods; and will equip you with techniques that allow you as the mathematics teacher and facilitator to expose the learners to the basics of mathematics, and to the application and use of mathematics in the real world outside the classroom. The module Teaching Mathematics (SP Subject Didactics) is intended as a guide to teaching mathematics for in-service teachers in primary schools. In order to teach mathematics in this twenty first century, teachers need to understand what it means to do mathematics; where will the journey of mathematics education take our learners and how we can help them adapt in the current knowledge generation. Furthermore, teachers need awareness that the twenty first century includes an era of the fourth industrial revolution, where technology knowledge and usage is critical. To help learners, we need to answer a few important questions: • What is mathematics? What is mathematics learning and teaching in South Africa about today? • How does mathematical learning take place? • How can we teach mathematics effectively, particularly in diverse classrooms? • What is ‘basic’ in mathematics? What is the fundamental mathematical knowledge that all learners need, irrespective of the level of mathematics learning they will ultimately achieve? • How do we assess mathematics learning most effectively? These questions are important for all learning and teaching, but particularly for learning and teaching mathematics in diverse classrooms. In terms of the policy on inclusive education, all learners – whatever their barriers to learning or particular circumstances in life – must learn mathematics. The module is divided into six units, each of which addresses the above questions from a different perspective. Although the units can be studied separately, they should be read together to provide comprehensive guidance in answering the above questions. (x) OVERVIEW OF CONTENTS OF MODULE UNIT 1:  EXPLORING WHAT IT MEANS TO ‘DO’ MATHEMATICS This unit gives a historical background to mathematics education in South Africa. The traditional approach to teaching mathematics is then contrasted with an approach to teaching mathematics that focuses on ‘doing’ mathematics, and mathematics as a science of pattern and order in which learners actively explore mathematical ideas in a conducive classroom environment. UNIT 2: DEVELOPING UNDERSTANDING IN MATHEMATICS In this unit, the theoretical basis for teaching mathematics – constructivism – is explored. Varieties of teaching strategies based on constructivist understandings of how learning best takes place are described. UNIT 3: TEACHING THROUGH PROBLEM-SOLVING In this unit, the shift from the rule-based, teaching-by-telling approach to a problem- solving approach to mathematics teaching is explained and illustrated with numerous mathematics examples. UNIT 4: PLANNING IN THE PROBLEM-BASED CLASSROOM In addition to outlining a step-by-step approach for a problem-based lesson, this unit looks at the role of group work and co-operative learning in the mathematics class, as well as the role of practice in problem-based mathematics classes. UNIT 5:  BUILDING ASSESSMENT INTO TEACHING AND LEARNING This unit explores the following questions in terms of assessment in mathematics classrooms: Why assessment? (the purposes of assessment); What to assess? (achievement of learning, but also understanding, reasoning and problem-solving ability); How to assess? (methods, tools and techniques); How to interpret the results of assessment? (the importance of criteria and rubrics for assessment in mathematics); and How to report on assessment? (developing meaningful report cards). UNIT 6: EXERCISES ON TEACHING DATA HANDLING UNIT 7: EXERCISES ON TRANSFORMATION GEOMETRY LADMMM6/1 (xi) Overview of contents of module THE STRUCTURE OF THE UNITS Each unit consists of the following: • Welcome to the unit – from the three teachers who discuss their challenges and discoveries about mathematics teaching uploads/Ingenierie_Lourd/ ladmmm6-study-guide.pdf

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