Mathematics for Foundation Phase 1 Study Guide PDF

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2023

Caryn Naudé

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This study guide provides an introduction to the learning outcomes, study schedule, and various assessment methods for the module Mathematics for Foundation Phase 1. It covers different study sessions and lessons. It explains the importance of the study material for successfully teaching mathematics in the Foundation Phase. The guide includes details on how to effectively engage with the course content, and self-assess progress.

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Mathematics for Foundation Phase 1 STUDY GUIDE MFP201-DL-SG-E4 *MFP201-DL-SG-E4* Contents Introduction 1 How to use this Study Guide 3 Learning Outcomes and Assessment Standards...

Mathematics for Foundation Phase 1 STUDY GUIDE MFP201-DL-SG-E4 *MFP201-DL-SG-E4* Contents Introduction 1 How to use this Study Guide 3 Learning Outcomes and Assessment Standards 7 Your Study Schedule 8 Study Session 1 11 Study Session 2 71 References 154 STUDY GUIDE / MFP201-DL-SG-E4 / Page (i) Written by Caryn Naudé First edition January 2020 Second edition June 2020 Third edition November 2020 Fourth edition November 2023  STADIO (PTY) LTD 75 Silverton Road, Musgrave, Durban, 4001 This document contains proprietary information that is protected by copyright. All rights are reserved. No part of this document may be photocopied, reproduced, electronically stored or transmitted, or translated without the written permission of STADIO (PTY) LTD. STUDY GUIDE / MFP201-DL-SG-E4 / Page (ii) Introduction Welcome to the module Mathematics for Foundation Phase 1 (MFP201)! This module is designed to equip you with the necessary knowledge and skills enabling you to teach mathematics successfully in the Foundation Phase. Mathematics in Foundation Phase 1 will focus on the teaching and learning of selected mathematics content areas to Foundation Phase learners. This module will form the foundation of Mathematics for Foundation Phase 2 (MFP301) and Mathematics for Foundation Phase 3 (MFP402). Each module will incorporate appropriate learning theories, pedagogical principles and content areas to develop effective mathematics teachers in and will follow a logical progression and also extend the students' knowledge into the Intermediate Phase. Each module will endeavour to empower students to teach in a way that will make Foundation Phase learners realise mathematics makes sense and has value in their everyday lives. Plan your time carefully, work diligently, and study throughout the course, not only for your examination, then you will be sure to succeed! Prescribed Readings This module requires you to read the following prescribed texts:  McDermott, L. and Rakgokong, L. 1996. Excell in teaching Mathematics: Teacher's Manual. Cape Town: Kagiso Education.  Van de Walle, J.A., Karp, K.S. and Bay-Williams, J.M. 2023. Elementary and Middle School Mathematics: Teaching Developmentally. 11th edn. Essex: Pearson Education. The above prescribed texts are the primary resources in this module. You'll be told when to read various sections of it as you work through this study guide. You'll also need to read parts of the following documents as part of your studies for this module:  Department of Basic Education. South Africa. 2011. Curriculum Assessment Policy Statement (CAPS): Foundation Phase. Mathematics Grades R-3. Pretoria: Government Printing Works. We often refer to the above document simply as the Mathematics Grades R-3 CAPS document in this study guide. Finally, you'll have to read the document indicated below, which sets out the referencing techniques that we use, which are based on the Harvard system. You'll need to use these referencing conventions when writing your assignment.  STADIO (PTY) LTD. 2022. STADIO Referencing Guide. 2nd edn. Durban: STADIO (PTY) LTD. Study Guide  STADIO MFP201-DL-SG-E4 Page 1 Please also note that your prescribed textbooks and this study guide will be sent to you in hardcopy, but you'll have to download the Mathematics Grades R-3 CAPS and Government documents and the STADIO Referencing Guide from the Learning Management System (LMS), which is our online student portal. Terms used in this Module Please take note of the following terms and their definitions or abbreviations, as you'll encounter them frequently in this module. CAPS The acronym CAPS refers to the Curriculum and Assessment Policy Statement, and specifically in this guide the Mathematics Grades R-3 CAPS will be used. It is important that you download this document and explore this to get familiar with how it is laid out. Manipulatives A manipulative in mathematics is a resource that is used to help demonstrate abstract mathematics concepts to young learners. There are many examples of manipulatives such as ten frames, hundred charts, counters, base ten blocks, measuring jugs etc. There is a substantial list of examples of suitable manipulatives for Foundation Phase in Lesson 2 in this study guide. Assessment of this Module Everything you need to know about the formal assessment of this module can be found in the Mathematics for Foundation Phase 1: Assessment Guide. This assessment booklet can be found on the LMS on the module page. Page 2 MFP201-DL-SG-E4 Study Guide  STADIO How to use this Study Guide We compiled this study guide to help you work through the prescribed study material for this module in a logical and manageable way. The study guide also gives you extra theory and explanations where necessary, and offers many opportunities for self-reflection and self-assessment. We suggest that you flip through and skim the entire guide to get an overview of the module's structure and content. Your Study Schedule Your study schedule, which comes a little further on in this study guide, is a summary of your module content, and clearly indicates the following:  study sessions;  lessons;  prescribed readings;  learning outcomes per lesson; and  suggested lesson timeframes. Study Sessions and Lessons in this Module You'll find two study sessions in your study schedule, as shown below. Study Session 1 The first study session contains the following two lessons:  Lesson 1: Becoming an effective Mathematics Teacher; and  Lesson 2: Understanding the Mathematics Curriculum. Study Session 2 The second study session is the longest, and contains the following three lessons:  Lesson 3: 'Numbers, Operations and Relationships';  Lesson 4: 'Space and Shape' (Geometry); and  Lesson 5: 'Patterns, Functions and Algebra'. Study Guide  STADIO MFP201-DL-SG-E4 Page 3 Learning Outcomes and Assessment Standards for this Module We list learning outcomes and assessment standards for the module as a whole just before your study schedule. We also list learning outcomes for each study session at the beginning of that study session. The learning outcomes tell you exactly what you need to be able to do at the end of the module or study session, while the assessment standards tell you how you can check that you've achieved the learning outcomes. Self-assessment during this Module Since you are studying via distance learning, it's important for you to take charge and monitor your own progress. To help you do this, we have incorporated various modes of self-assessment into the module which you can use to gauge your knowledge and understanding of the material. Specifically, we include the following types of self-assessment in this study guide:  self-reflections;  writing activities;  online activities;  practical activities;  self-assessment tests; and  competence checklists. We explain each type of self-assessment below. Self-reflections Self-reflections are activities that ask you to think about important topics or questions. They do not require you to do anything other than think carefully about something. Self-reflections are your opportunities to work through new concepts, identify areas in your life or work that need change, and solve problems. Writing activities Writing activities require you to write down answers to questions. Space is provided after each activity to allow you to write your answers in this study guide. Writing activities are your opportunities to demonstrate your subject knowledge and apply theory in practice. They are invaluable practice, as they help you to clearly assess your ability to answer similar questions in an exam or other formal assessment setting. Depending on the nature of the writing activity, we sometimes provide answers below which you can use to assess the accuracy of your own answers. We also sometimes provide model answers to open-ended questions, usually to showcase the manner in which such questions should be answered. A model answer is therefore not the only legitimate answer to a question, but rather is an example of the sort of answer that would be acceptable to an examiner. Page 4 MFP201-DL-SG-E4 Study Guide  STADIO Online activities Online activities usually ask you to go online to read an article or post, or watch a video. Note that when you are told to engage with online materials, you should consider the content of those materials to form part of the module content. Practical activities Practical activities usually ask you to physically do something. These practical activities allow you to apply your theoretical knowledge to various situations in the real world. We encourage you to use practical activities as a way to evaluate yourself and find the weak spots in your understanding of the module content. Self-assessment tests You'll find self-assessment questions and answers at the end of every study session in this module. The self-assessment tests are similar to writing activities, only they come at the end of a study session, and so require you to respond to questions based on all of the content in that study session, and often in a more integrated fashion. There is no space in the study guide to write down your answers – you should do this in a notebook of your own. We encourage you to actually write down your answers, not just think them through, as you need to practise articulating your thoughts, reasoning (or working out), and understanding effectively in words. Once you've written out your answers to all the questions, only then should you compare your answers with the model answers provided, as this exercise will give you a good indication of how well you've understood the content of that study session and are able to apply the theory. Competence checklists We have also included a competence checklist at the end of each study session to help you confirm that you can perform the listed learning outcomes. Each competence checklist has a list of the learning outcomes for that session, with a little checkbox next to each for you to tick. Essentially, the learning outcomes are a summary of your goals for working through a given module, and you want to know that you can confidently tick each box before undertaking the final exam. Good Luck with your Studies! Remember that this is a distance-learning module. Since you don't have a lecturer standing next to you while you study, and you've no classmates to check yourself against, you need to apply self-discipline. Nobody will check to see if you've completed any of the self- assessments in this study guide – as you know, they do not form part of your formal assessment for this module. That said, we once again encourage you to work through them all, as they form an important part of this module and your learning experience. Study Guide  STADIO MFP201-DL-SG-E4 Page 5 The study skills you'll develop by undertaking a distance-learning module include self- direction and responsibility. Embrace the challenge, as self-direction and responsibility are important life skills that will help you to succeed in all areas of your life. Page 6 MFP201-DL-SG-E4 Study Guide  STADIO Learning Outcomes and Assessment Standards LEARNING OUTCOMES ASSESSMENT STANDARDS Upon successful completion of this module, We know that you have met the learning you must be able to do the following: outcomes when you can do the following: LO 1: Explain the manner in which AS 1: Use relevant theories to analyse children learn and understand how children learn and understand mathematics. mathematics and explain how this will inform the design of LO 2: Compare different learning theories mathematics activities, providing and unpack their implications for relevant examples. classroom practice. AS 2: Compare and contrast different LO 3: Create a learning environment that learning theories and how they that promotes the development of have changed to accommodate 21st century skills in learners. each generation of learners. LO 4: Apply ideas when planning lessons AS 3: Identify, explain and should be able that depict the inclusion of content to implement measures to promote knowledge and skills as set out in 21st century skills in learners in the the Foundation Phase mathematics classroom. Curriculum and Assessment Policy Statement (CAPS). AS 4: Apply knowledge from the Curriculum and Assessment Policy LO 5: Develop age-appropriate Statement (CAPS) in Foundation mathematics activities based on the Phase mathematics practically CAPS. during lesson presentations. LO 6: Establish a sound foundation of AS 5: Design developmentally- number concept in the early years. appropriate activities/lessons based on the content areas that are LO 7: Teach the four basic operations in- stipulated in the CAPS for context including the problem- Mathematics Grades R-3 solving approach. document. AS 6: Explain number concept and the four basic operations and how these should be taught to Foundation Phase learners at their developmental level using practical activities. AS 7: Implement the problem-solving approach when designing mathematics activities/ planning lessons. Study Guide  STADIO MFP201-DL-SG-E4 Page 7 Your Study Schedule Page 8 STUDY SESSION 1 LESSON PRESCRIBED READING LEARNING OUTCOMES TIMEFRAME LESSON 1: Becoming an effective  McDermott, L. and Rakgokong, L. 1996. Excell in  Establish an understanding and appreciation of 2 weeks Mathematics Teacher Teaching Mathematics: Teacher's Manual. Cape Town: how young learners construct their knowledge. Kagiso Education. Section One: Mathematics Learning  Recognise and implement strategies to help and Teaching: reduce mathematics anxiety in teachers.  Unit 2: The Theory of Constructivism (pages 5 to 9).  Analyse the constructivists and behaviourist  Unit 4: Learning (pages 31 to 35). learning theories.  Understand the implication of the theories in the  Van de Walle, J.A., Karp, K.S. and Bay-Williams, J.M. mathematics classroom. 2023. Elementary and Middle School Mathematics:  Create a learning environment that promotes the Teaching Developmentally. 11th edn. Essex: development of 21st century skills in Foundation Pearson Education. Part I: Teaching Mathematics Phase mathematics. Developmentally: Big Ideas and Research-Based Practices: MFP201-DL-SG-E4  Chapter 2: Exploring What It Means to Know and Do Mathematics (pages 40 to 55).  Chapter 3: Teaching Problem-Based Mathematics (pages 57 to 81). LESSON 2: Understanding the  Department of Basic Education. South Africa. 2011.  Gain an in-depth understanding of the 2 weeks Mathematics Curriculum Curriculum Assessment Policy Statement (CAPS): mathematics curriculum as set out in the CAPS. Foundation Phase. Mathematics Grades R-3. Pretoria:  Apply ideas when planning lessons that depict Government Printing Works. the inclusion of content knowledge and skills as set out in the CAPS.  Van de Walle, J.A., Karp, K.S. and Bay-Williams, J.M.  Develop age-appropriate mathematics activities 2023. Elementary and Middle School Mathematics: for counting and mental mathematics activities Teaching Developmentally. 11th edn. Essex: for Foundation Phase learners. Pearson Education. Part I: Teaching Mathematics  Assess learners informally. Developmentally: Big Ideas and Research-Based  Develop appropriate checklists, and rubrics. Practices:Chapter 4: Planning in the Problem-based Classroom (pages 82 to 107).  Chapter 5: Creating Assessments for Learning (pages 109 to 128). Study Guide  STADIO STUDY SESSION 2 LESSON PRESCRIBED READING LEARNING OUTCOMES TIMEFRAME LESSON 3: 'Numbers, Operations  McDermott, L. and Rakgokong, L. 1996. Excell in  Demonstrate an understanding of early number 3 weeks and Relationships' Teaching Mathematics: Teacher's Manual. Cape Town: concept and number sense development in Kagiso Education: young learners.  Section One: Mathematics Learning and Teaching –  Understand computation and the integration Study Guide  STADIO Unit 3: Problem Solving (pages 22 to 30). with the basic operations.  Section Two: Number, Measurement and Geometry  Analyse the four basic operations, the – Unit 5: Number Concept Development and relationship among them and how to introduce Counting (pages 46 to 66). these in a lesson.  Section Two: Number, Measurement and Geometry  Establish an understanding of fractions. – Unit 6: Computation (pages 70 to 85).  Gain insight into the problem-solving approach.  Section Two: Number, Measurement and Geometry  Understand the different kinds of problems. – Unit 8: Fractions (pages 100 to 109).  Implement the problem-solving approach including different problem types when planning  Van de Walle, J.A., Karp, K.S. and Bay-Williams, J.M. lessons. 2023. Elementary and Middle School Mathematics: Teaching Developmentally. 11th edn. Essex: Pearson Education:  Part I: Teaching Mathematics Developmentally: Big Ideas and Research-Based Practices:Chapter 4: Planning in the Problem-based Classroom (pages 82 to 107).  Part II: Teaching Mathematics Developmentally: MFP201-DL-SG-E4 Concepts and Procedures in Pre-K – 8 ­ Chapter 7: Developing Early Number Concepts and Number Sense (pages 157 to 184). ­ Chapter 8: Developing Meanings for the Operations (pages 185 to 213). ­ Chapter 9: Developing Basic Fact Fluency (pages 215 to 242). Page 9 STUDY SESSION 2 (continued) LESSON PRESCRIBED READING LEARNING OUTCOMES TIMEFRAME Page 10 LESSON 4: 'Space and Shape'  McDermott, L. and Rakgokong, L. 1996. Excell in  Demonstrate an understanding of the geometric 2 weeks (Geometry) Teaching Mathematics: Teacher's Manual. Cape Town: concepts. Kagiso Education:  Use the correct terminology when designing  Section Two: Number, Measurement and Geometry planning and teaching 'Space and Shape' – Unit 9: Geometry and Spatial Sense concepts. (pages 110 to 127). Van de Walle, J.A., Karp, K.S. and Bay-Williams, J.M. 2023. Elementary and Middle School Mathematics: Teaching Developmentally. 11th edn. Essex: Pearson Education. Part II: Teaching Mathematics Developmentally: Concepts and Procedures in Pre-K – 8:  Chapter 19: Developing Geometric Thinking and Geometric Concepts (pages 539 to 577). LESSON 5: 'Patterns, Functions and  Van de Walle, J.A., Karp, K.S. and Bay-Williams, J.M.  Demonstrate understanding of geometric and 1 week Algebra' 2023. Elementary and Middle School Mathematics: number pattern development in young learners. Teaching Developmentally. 11th edn. Essex:  Apply knowledge of 'Patterns, Functions and MFP201-DL-SG-E4 Pearson Education. Part II: Teaching Mathematics Algebra' when planning lessons for learners as Developmentally: Concepts and Procedures in Pre-K – outlined in the CAPS. 8:  Chapter 13: Algebraic Thinking (pages 332 to 366). Study Guide  STADIO Study Session 1 Welcome to the first of your two study sessions for this module. Let's start by looking at the learning outcomes for this particular study session. Learning Outcomes After you've completed Study Session 1, you should be able to do the following:  establish an understanding and appreciation of how young learners construct their knowledge;  recognise and implement strategies to help reduce mathematics anxiety in teachers;  analyse the constructivists and behaviourist learning theories;  understand the implication of the theories in the mathematics classroom;  create a learning environment that that promotes the development of 21st century skills in Foundation Phase Mathematics;  gain an in-depth understanding of the mathematics curriculum as set out in the CAPS;  apply ideas when planning lessons that depict the inclusion of content knowledge and skills as set out in the CAPS;  develop age-appropriate mathematics activities for counting and mental mathematics activities for Foundation Phase learners;  assess learners informally; and  develop appropriate checklists, and rubrics. We now move on to give you an overview of the two lessons in this study session. Overview of Study Session 1 Welcome to the start of your journey of becoming an effective Foundation Phase mathematics teacher! You may be a little nervous about mathematics because of your own personal experiences (either positive or negative) with the subject in your school career. Fear not! Often the best mathematics teachers are those who struggled with the subject themselves. Being successful in a subject and having the knowledge also does not automatically mean that you can teach the content. That is what the purpose of this module is – to prepare you to teach mathematics in the Foundation Phase. To do this effectively, we need to look at the various aspects of becoming an effective mathematics teacher and also, we need to understand the Foundation Phase mathematics curriculum (this will be CAPS). These are the topics we will be working through in Study Session 1. In Lesson 1, Becoming an effective Mathematics Teacher, we will start by contextualising mathematics in South Africa. Next, we will explore how students learn and understand mathematics. Thereafter, we will discuss learning theories that inform the teaching and learning of mathematics. We will then explore the role of the teacher in a Foundation Phase Study Guide  STADIO MFP201-DL-SG-E4 Page 11 mathematics classroom and 21st century skills in Foundation Phase mathematics. Finally, we will then explore a teacher's perspective: mathematics anxiety. In Lesson 2, Understanding the Mathematics Curriculum, we will explore what mathematics looks like in a South African Foundation Phase classroom by exploring various aspects of the Mathematics Grades R-3 CAPS document. We will then explore how to plan a Foundation Phase mathematics lesson. Lastly, we will explore informal assessment in Foundation Phase mathematics. It's time now to start Lesson 1. Page 12 MFP201-DL-SG-E4 Study Guide  STADIO Lesson 1: Becoming an Effective Mathematics Teacher  Prescribed readings Your prescribed readings for Lesson 1 are as follows:  Pages 5 to 9 of Unit 2: 'The Theory of Constructivism' in Excell in Teaching Mathematics: Teacher's Manual by L. McDermott and L. Rakgokong.  Pages 31 to 35 of Unit 4: 'Learning' in Excell in Teaching Mathematics: Teacher's Manual by L. McDermott. and L. Rakgokong.  Pages 40 to 55 of Chapter 2: 'Exploring What it Means to Know and Do Mathematics' in Elementary and Middle School Mathematics: Teaching Developmentally by Van de Walle et al.  Pages 54 to 81 of Chapter 3: 'Teaching Problem-Based Mathematics' in Elementary and Middle School Mathematics: Teaching Developmentally by Van de Walle et al. Note that you'll be told exactly when to read each of the above texts as you work through the lesson. Introduction It may surprise you at that we do not immediately start with working out answers to problems or learning about shapes in this module. Instead, we begin learning how to teach mathematics effectively by understanding the context of mathematics in South Africa. Most importantly, we look at how children learn and understand mathematics and the implications of this knowledge when adapting teaching and learning strategies as well as the learning environment to maximise learning. Before we begin, please consider the following:  there is no need to feel anxious! MFP201 is not all about calculations and formulas;  it is about understanding the foundations of learning and doing mathematics;  this includes having a look at the psychology of how children learn and understand mathematics and how to create an environment that promotes this; and  this module focuses on the Foundation Phase. We start this lesson by contextualising mathematics in South Africa. We then discuss how students learn and understand mathematics, including a discussion about the learning theories that inform the teaching and learning of mathematics and the implications of this. We then unpack the idea of 21st century skills and look at how these skills can be integrated into Foundation Phase mathematics. Finally, we discuss mathematics anxiety from a teacher's perspective and consider whether or not this has an impact on the way you personally approach teaching and learning in mathematics. Study Guide  STADIO MFP201-DL-SG-E4 Page 13 Let's begin by looking at the status of mathematics in South Africa. Contextualising Mathematics in South Africa Mention the subject of mathematics in South Africa and there is a good chance that you will elicit a negative response from learners and adults alike. There seems to be a love – hate relationship with the subject and many, many people feel that they are bad at mathematics and that this attitude is acceptable. Often, these feelings and attitudes about mathematics are passed down to children and this limits their ability to be successful at mathematics themselves. As an introduction to this module, this section of the lesson will briefly look at the status of mathematics in South Africa. To do this, we discuss the following areas:  the importance of mathematics;  causes of bad mathematics results; and  the language of mathematics. The importance of mathematics It is important for us as future mathematics teachers to reflect on our own feelings and attitudes to mathematics. It's time now for you to work through your first self-reflection activity, namely Self-reflection 1. Self-reflection 1 1. In your opinion, do you think it is important to study mathematics at school? Can you provide reasons for your opinion? 2. Do you think that mathematics is important in everyday life? Why/why not? 3. If you decided in point 1 and 2 above that mathematics is important, why do we struggle with mathematics in South Africa? Now that you have thought about your viewpoint on the importance of mathematics, let's take a look at some of the research that is available. For Online Activity 1, you will find a list of recommended sources for you to read. These sources provide context about mathematics in South Africa and also present valid arguments about why the status of mathematics is this way. The main ideas of these articles are also discussed after Online Activity 1. Page 14 MFP201-DL-SG-E4 Study Guide  STADIO Now do Online Activity 1 below.  Online Activity 1  Read the following articles: The links to these articles are given below and can also be found on the LMS. 1. 'Being bad at Mathematics shouldn't be a badge of honour' (Doug 2019) www.tes.com/news/being-bad-maths-shouldnt-be-badge-honour 2. 'South African performance on the trends in international mathematics and science study' (Letaba 2017) https://bit.ly/37JgzrV 3. 'Matric really does start in Gr 1' (Spaull 2017) mg.co.za/article/2017-01-27-00-matric-really-does-start-in-grade-one. 4. 'Why half of SA's primary school kids still can't do basic Mathematics' (Robinson 2019) https://www.citizen.co.za/news/south-africa/education/why-half-of-sas- primary-school-kids-still-cant-do-basic-maths/ 5. 'Government celebrates South Africa's dismal Mathematics and science results' (BusinessTech 2016) https://bit.ly/2Tx7e1I 6. 'Matric Mathematics pass numbers don't add up' (Hlatshaneni 2019) https://www.citizen.co.za/news/matric-maths-pass-numbers-dont-add-up/ Let's explore the main ideas from these articles. In the article, 'Being bad at Mathematics shouldn't be a badge of honour' by Doug (2019), we learn that it is socially acceptable to struggle with Mathematics, and in many cases this struggle is worn as 'almost a badge of honour.' Doug (2019) asks the question of whether the current generation is weak in Mathematics because they inherited the inability to 'be good' at the subject or should we rather be questioning the approach to teaching and learning Mathematics and education in general. In the article, 'Matric really does start in Gr 1', Spaull (2017) speculates that the root of South Africa's ongoing under- and non-achievement culture in mathematics is not to be found in the high school, but instead much earlier, in the Foundation Phase or earlier. He came to the following significant conclusion: "We need to acknowledge that matric starts in Grade 1 (and even earlier), and that it really is possible to improve primary schooling if that is where we focus most of our time, energy and resources" (Spaull 2017). Study Guide  STADIO MFP201-DL-SG-E4 Page 15 Please work through Self-reflection 2. Self-reflection 2 What are your thoughts about the following extracts from the articles you have read? 1. "Half of all South African pupils who attended school for five years can't do basic calculations. This is according to a 2015 TIMMS report on mathematics achievements among Grade 5 learners in South Africa" (Robinson 2019). 2. "The TIMSS results show that South Africa was ranked second last out of 48 countries for Grade 4 mathematics, second last for Grade 8 mathematics and stone last for Grade 8 science out of 38 countries" (BusinessTech 2016). 3. Professor Anthony Essien (cited in Hlatshaneni, 2019), head of mathematics at Wits University, proposes that the ability to learn Mathematics was more of a function of language skills than just logical thinking. 4. "79% of South African Grade 6 mathematics teachers were classified as having content knowledge levels below the level at which they were teaching" (Robinson 2019). Let's now briefly look at some of the reasons for poor mathematics results in South Africa. Causes of bad mathematics results There is no one reason that can be identified as the cause of our learners' poor performance in Mathematics. Machaba (2013) has identified certain factors that could contribute to poor Mathematics performance. These include:  the teaching methods that are commonly used for mathematics (whole class teaching);  failure to use knowledge associated with mathematics (for everyday activities);  language barriers;  lack of flexibility;  quality of educator – child interaction;  motivation to do mathematics; and  teachers' mathematics knowledge (or lack of). Page 16 MFP201-DL-SG-E4 Study Guide  STADIO Please work through Self-reflection 3. Self-reflection 3 With reference to factors identified by Machaba (2013), consider the following: 1. Why do you think teachers insist on whole class teaching? How could this approach negatively impact mathematics performance? 2. Is it possible to make mathematics relevant to learners' daily life? 3. Do you think that teachers' who have good mathematics knowledge automatically will be good Mathematics teachers? We mentioned the topic of language barriers above, let's now move on to discuss the language of mathematics because it is a factor that influences a learner's performance in mathematics. The language of mathematics South Africa is a country of wonderful diversity and it is not uncommon for learners in a school to speak many different languages. However, this poses a challenge to teaching and learning mathematics. Mathematics can be described as a second language, (and sometimes even a third language depending on a learner's home language and the language of instruction at the school) for learners because it includes formal learning of concepts specific to the subject and it has its own unique set of vocabulary that is used to describe and interpret mathematics relationships, and should be taught as such. This idea of mathematics as a language is supported by the following definition of mathematics we can find in the Mathematics Grades R-3 CAPS document (Department of Basic Education 2011: 8) Mathematics is a language that makes use of symbols and notations for describing numerical, geometric and graphical relationships. It is a human activity that involves observing, representing and investigating patterns and qualitative relationships in physical and social phenomena and between mathematical objects themselves. It helps to develop mental processes that enhance logical and critical thinking, accuracy and problem-solving that will contribute to decision-making. Please work through Self-reflection 4. Self-reflection 4 How much do you think mathematics language influences the performance of learners in the subject? Study Guide  STADIO MFP201-DL-SG-E4 Page 17 Now that we have contextualised mathematics in South Africa, we now move on to discuss how young children learn and understand mathematics because knowing this will impact our approach to teaching and learning in Foundation Phase mathematics. How Young Children Learn and Understand Mathematics Teaching and learning mathematics is not only about having a good knowledge of the subject itself. As you may have noticed, MFP201 is not all about calculations and formulas, but rather it aims to provide insight into the foundations of learning and doing mathematics. This means we must examine relevant psychological aspects of how children learn and understand mathematics and also how to create an environment that promotes learning. To do this, we discuss the following points:  learning theories that inform the teaching and learning of mathematics;  active knowledge creation and learning;  21st century skills in Foundation Phase mathematics; and  a teacher's perspective of mathematics anxiety. It is also now time to start engaging with your prescribed readings. Please work through the following readings now and make notes to supplement what appears in this study guide:  Pages 5 to 9 of Unit 2: 'The theory of constructivism' in Excell in teaching Mathematics: Teacher's Manual by L. McDermott and L. Rakgokong.  Pages 31 to 35 of Unit 4: 'Learning' in Excell in Teaching Mathematics: Teacher's Manual by L. McDermott and L. Rakgokong.  Pages 40 to 55 of Chapter 2: 'Exploring What It Means to Know and Do Mathematics' in Elementary and Middle School Mathematics: Teaching Developmentally by Van de Walle et al. Learning theories that inform the teaching and learning of mathematics The process of learning is extremely complex and there are various theories that try to explain how learning actually happens. Before we start working through these theories, please work through Self-reflection 5. Self-reflection 5 1. What is your definition of 'learning'? 2. How does an individual learn? 3. Are learning and understanding different processes? As you would have discovered in Self-reflection 5, defining the concept learning is extremely challenging. There are also various perspectives in literature that define learning differently. Page 18 MFP201-DL-SG-E4 Study Guide  STADIO These different perspectives are influenced by different theories that have been developed over time, as we learn more about how the brain functions. Theories provide different interpretations of how children learn and understand mathematics. They can be thought of as lenses or tools for understanding how an individual learns. One theory is not more correct than the other and it is important to note that learning theories are not teaching strategies, instead they inform the way we teach. Figure 1 below shows us what learning theories we will be exploring in this module: Constructivism Behaviourism Learning theories Figure 1: Learning theories Study Guide  STADIO MFP201-DL-SG-E4 Page 19 Before we start looking at each one of these in detail, we need to have a broad understanding of what it means to 'do' mathematics? To do this, please complete Writing Activity 1 below.  Writing Activity 1  Read the pages 40 to 46 in Van De Walle et al. (2023) and summarise the main points of 'What is Mathematical Proficiency?" below. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ Model answer: There are various points you could have noted, including: (1) Mathematics is about generating strategies for solving a problem, applying the selected strategy, and then checking to see whether the strategy is correct and efficient. (2) Finding and exploring regularity and order, including patterns – and then trying to make sense of it, and relate it to the 'real' world. (3) Exploring generalisations about mathematics operations and learn about relationships between the operations. (4) Engaging in the science of pattern and order. As you would have noticed in your response to Writing Activity 1 above, learning and 'doing' mathematics consists of various, complex mental processes that should also carefully reflect mathematics in everyday life as well. Let's now explore this idea of learning theories and compare the various perspectives. See if you notice any features of each learning theory discussed below in your own experience with learning mathematics. Page 20 MFP201-DL-SG-E4 Study Guide  STADIO We will begin by discussing behaviourism. Behaviourism McDermott and Rakgokong (1996:9) describe in detail behaviourism, also called the 'traditional approach', as the transmission (passing on) of knowledge from the teacher to the learners in a very set way. You may have also heard this approach referred to as 'chalk-and- talk' or learning 'rote' (meaning learning off by heart often without understanding what you are learning). This approach recognises the main role of the teacher as the only source of knowledge for learners and learners are considered as being 'blank slates' with little to no consideration for their prior knowledge or experiences. Learners are passive in receiving information which they must learn (very often without understanding) and listen to teachers and only concentrate on giving the answers that the teachers wanted using the method that the teacher demonstrated. Figure 2 below is a good representation of a behaviourist approach to learning. Knowledge Figure 2: Pouring knowledge into an empty head Teachers who follow a behaviourist approach don't feel that explanations are important, and the focus is on the student's ability to provide the correct response. Other features of behaviourism include:  mathematics is taught in a specific order – firstly addition will be taught, then subtraction, then multiplication then division and only after this word problems are introduced;  few connections between the operations and other mathematics concepts are explicitly taught;  the belief is that learning is observable by the actions of the learners. in other words, changes in behaviour indicates that learning has taken place;  application of rules and algorithms (a step-by-step procedure for calculations) means that learners have learnt;  the 'why' is seldom explained, most often only the how; and  the focus is on 'rote' learning (we can also think of this as learning off by heart even if we don't understand what we are learning), drill (repeating over and over until it is learnt 'off by heart') and timed tests. Study Guide  STADIO MFP201-DL-SG-E4 Page 21 Please work through Self-reflection 6. Self-reflection 6 Take some time to think about your own beliefs about how mathematics should be taught. Consider the following questions: 1. Who do you think is responsible for learning in a classroom? 2. What is the teacher's and the learner's role in learning? 3. Do you think that the traditional approach has any advantages or disadvantages? To further investigate your beliefs and experiences about behaviourism complete Writing Activity 2 below.  Writing Activity 2  Test yourself. Solve the following problems in the space provided: 1. 68 + 28 = ? _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ 2. 73 – 17 = ? _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ Answers: (1) 96; (2) 56 Page 22 MFP201-DL-SG-E4 Study Guide  STADIO Let's take a moment to consider how you solved the problems outlined in Writing Activity 2 above. Possibly, in order to solve these problems, you followed the traditional 'soldier sum' approach as illustrated in Figure 3 below: Figure 3: Soldier sum approach If I asked you why you solved the problems using this specific strategy most of you would say that that is how you were taught. You were provided with a specific method (algorithm) including carrying over or borrowing which has been applied consistently throughout your life. From looking at the information about behaviourism above, it is likely that we can agree that as teachers of Foundation Phase mathematics, we don't want to follow this approach. Let's now look at the alternative approaches to learning, namely constructivism. Principles of constructivism The name, constructivism should already remind you of the word construct or to build. As you would have read in your prescribed readings at the beginning of this section, a constructivist approach recognises the following:  learners are not blank slates but rather create and construct their own learning;  learning is seen as an active process;  as learning happens brain networks are added to, rearranged, modified continually which allows the learner to give meaning to things using existing schemas (prior knowledge);  the schemas/networks are products of constructing knowledge and are tools used to construct new knowledge;  new knowledge is added through reflective thought and there is an important process of connecting new ideas to existing ideas. This process is explained in Figure 2.9 (Van de Walle et al. 2023: 50);  the social environment, including various stakeholders, play a vital role in learning; and  there are different theorists who fall into the broad category of constructivism, but for the purpose for this module we will only focus on Piaget and Vygotsky. Study Guide  STADIO MFP201-DL-SG-E4 Page 23 NOTE You are going to see the word schema often in learning about constructivism, and especially Piaget. A schema (also called a scheme) can be described as a mental template or framework where information is stored and new information fits into this template. It is an abstract concept to explain the way information is stored in the brain without getting too technical about the complex way in which the human brain functions. There is most likely new terminology mentioned in the points above. These concepts are mentioned in Piaget's theory of cognitive development and Vygotsky's sociocultural theory. We will look at both of these in detail now. Let's start by unpacking Piaget's theory of cognitive development. Piaget's theory of cognitive development Piaget explained that cognitive development happens through two processes, the organisation and adaptation of information. These two processes are displayed in Figure 4 below: Cognitive Development The ongoing process of Organisation arranging information and experiences into schemas Assimilation: Children mould new information into existing schemas The mind's ability to adapt to the Adaptation environment. It is achieved by: Accomodation: Children change their schemas to restore a state of equilibrium Figure 4: The process of cognitive development The process of adaptation can be challenging to understand. Let's look at the processes of assimilation and accommodation in a bit more detail. Assimilation and accommodation Assimilation refers to the process of learning a new concept that 'fits' in with your prior knowledge. In this case, the new information is interpreted and added to an existing schema (prior knowledge). In contrast, accommodation refers to the process of modifying and adapting an existing schema or creating a new schema because incoming information is in contradiction (the new concept does not fit the existing knowledge) to what is already known. Page 24 MFP201-DL-SG-E4 Study Guide  STADIO The information above may not really make sense until we apply it to an example in the classroom. Let's do this now by referring to the following examples: Assimilation Think of the concept of addition. When children are introduced to addition, two numbers are added together, and the answer always is bigger than each of the two numbers. This becomes a predictable pattern and each experience with numbers in addition is assimilated. Therefore, when learners are introduced to adding three or more numbers, this new knowledge is simply added to the children's existing knowledge of addition. Accommodation Refer to the example above about the predictability of addition. So far, children have learnt that the sum (answer to an addition problem) is bigger than each of the numbers in the problem. However, this is not always the case – what happens when learners are introduced to the concept of adding zero? Their schema of addition will have to be adapted and modified to include the concept of adding zero and how that affects the answer. Now, children have to understand that when adding zero, the answer to an addition problem will be the same as one of the numbers in the addition problem. Piaget's theory of cognitive development also explains how children's mode of learning and ability to learn new concepts and skills changes according to their age. This explanation has been categorised in the following four stages:  sensorimotor stage;  preoperational stage;  concrete operational stage; and  formal operational stage. Study Guide  STADIO MFP201-DL-SG-E4 Page 25 The main characteristics of each of these stages are evident in Figure 5 below: Preoperational (2 – 7 years) Sensorimotor (0 – 2 years) Can use symbols to represent objects The infant learns and explores their Can't reason logically yet environment using their senses Has not developed concepts of conservation Develops object permanance and reversability Stages of Cognitive Development Concrete operational (7 – 11 years) Formal operational (11 years – adulthood) Develops concepts of conservation and classification Abstract thinking Can think logically about concrete objects Logical reasoning can be applied to abstract Reasoning skills start developing contexts Figure 5: Features of each stage of cognitive development Now that you have been introduced to the main features of Piaget's theory of cognitive development you can complete Online Activity 2 below.  Online Activity 2  Watch the following videos to learn more about Piaget's theory of cognitive development. As you watch these videos, add to your notes and the information found in this study guide. The links to these videos are given below and can also be found on the LMS. 1. Piaget's Stages of Development (misssmith891) https://www.youtube.com/watch?v=TRF27F2bn-A 2. Schemas, assimilation, and accommodation | Processing the Environment | MCAT | Khan Academy (khanacademymedicine 2015) www.youtube.com/watch?v=xoAUMmZ0pzc&t=10s Now that you have quite a good understanding of one of Piaget's theory of cognitive development, we can move on the discuss Vygotsky's sociocultural theory. Sociocultural theory – Vygotsky You have already read about the sociocultural theory in your prescribed readings. Vygotsky's sociocultural theory share many ideas with other theorists within the broad category of constructivism. In addition, Vygotsky proposed that learning depends on new knowledge falling within the zone of proximal development (ZPD) and through social interactions Page 26 MFP201-DL-SG-E4 Study Guide  STADIO (influenced by tools of mediation). The unique characteristics of the sociocultural theory include:  recognition that the social environment is vital for learning and understanding;  information is internalised according to the individual's (ZPD) and learning needs scaffolding and support by more knowledgeable others (MKOS); and  semiotic mediation (language and other tools such as diagrams, symbols pictures and actions) that convey meaning and social knowledge or cultural practices (knowledge that is passed down). You may have already heard of the ZPD in other psychology modules you have completed, but let's quickly revisit what this construct means. The zone of proximal development (ZPD) The ZPD describes an abstract zone in an individual's brain where learning takes place. For learning to happen, learners need to be provided a problem that is within their ZPD and they require support (a term called 'scaffolding') from teachers, parents and competent peers (also referred to as more knowledgeable others). Scaffolding occurs when the teacher asks the learner questions and provides them with clues, prompts or hints to solve difficult problems and promote learning. Once the learner is able to solve similar problems on their own, the teacher withdraws (removing the scaffolding) allowing the learner to be more independent. Figure 6 below provides another way of looking at the ZPD. Figure 6: Zone of proximal development Study Guide  STADIO MFP201-DL-SG-E4 Page 27 Now complete Online Activity 3 below.  Online Activity 3  Watch the video Vygotsky's Zone of Proximal Development (bcb704 2012) to learn more about Vygotsky's sociocultural theory. As you watch it, add to your notes and the information found in this study guide. The link to this video is given below and can also be found on the LMS. youtube.com/watch?v=0BX2ynEqLL4&t=68s We have now finished working through the learning theories that we need to be familiar with as Foundation Phase mathematics teachers. Let's briefly look at the implications of these learning theories for teaching mathematics in the Foundation Phase. Implications for teaching mathematics Van de Walle et al. (2023: 50 – 55) provide insight into how learning theories influence teaching and learning mathematics in a Foundation Phase classroom. Various factors have been outlined below in Figure 7, but you will need to add further notes to these so that you are able to discuss them. Treat errors as Promote discussion. learning Scaffold new content. opportunities. Build new knowledge Engage students in Encourage multiple from prior knowledge. productive struggle. approaches. Provide opportunities Provide opportunities Reflective thought: to communicate Honour diversity. about mathematics. interconnected and interrelated ideas. Figure 7: Implications for teaching mathematics Page 28 MFP201-DL-SG-E4 Study Guide  STADIO Please work through Self-reflection 7. Self-reflection 7 Take some time to think about constructivism. Consider the following question: Is constructivism an approach to learning that you think is recommended for Foundation Phase mathematics? Let's move on to discuss the idea of active knowledge creation and learning, which is one of the fundamental principles of constructivism. Active Knowledge Creation and Learning In this section of the lesson we discuss active knowledge creation. We will explore this by discussing the following points:  active learning;  a learning-centred approach; and  manipulatives as tools to support learning. Let's begin by discussing active learning now. Active learning A brief discussion about active learning can be found in McDermott and Rakgokong (1996: 31 – 32). Active learning occurs when learners are involved in the learning process and are active in creating knowledge by exploring, discussing, experimenting, and engaging in solving problems, both individually and collaboratively. Study Guide  STADIO MFP201-DL-SG-E4 Page 29 Complete Writing Activity 3 below.  Writing Activity 3  Read about active learning on pages 31 to 38 in McDermott and Rakgokong (1996) and make a list of the characteristics of an active learner below. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ Model answer: Active learners can: start their own activities, take responsibility for their own learning, make decisions, solve problems, transfer learning from a familiar context to a new or different context, organise, share their understanding of numbers in multiple ways, evaluate their own abilities, feel positive or good about themselves as learners of mathematics. The Department of Basic Education (2011) supports the idea of active learning. The Mathematics Grades R-3 CAPS document (Department of Basic Education 2011: 12) proposes that mathematical experiences should include opportunities to "do, talk and record", which promotes an active approach to learning that focuses on the process of learning and not just the final product (the recording part or writing down the answer) that we see at the end of the learning process. When learners are given the opportunity to explore complex, abstract mathematics concepts and then talk about these explorations they are engaging in active knowledge creation. In order for this to be promoted, the classroom environment should be learning-centred and it should focus on the complete learning experience, including the role of the learner, the teacher, the environment and the manipulatives, resources and materials. Remember, we stated what a manipulative is under the terms for this study guide in the introduction of the module. Refer back to this definition if you need to remind yourself what a manipulative is. Page 30 MFP201-DL-SG-E4 Study Guide  STADIO Please work through Self-reflection 8. Self-reflection 8 With reference to the idea of active learning and your knowledge of learning theories, consider the following: 1. How does active learning relate to the constructivism learning theories we discussed earlier? 2. Are you better able to describe how do children learn and how new knowledge created? Can you explain the importance of activating prior knowledge and existing schemas to new knowledge? (Hint – think about how children organise and adapt to new information). After careful thinking in Self-reflection 8 above, we must also recognise the importance of ensuring that mathematics content taught in a Foundation Phase classroom is relevant to our learners' worlds. Mathematics should be connected to life and the world outside of school to build a secure foundation for the more abstract mathematics concepts that will come later in the school curriculum (think algebraic theories or trigonometry). Learners also need to see the value of mathematics in life and for their future career prospects; if they are not exposed to these, they will be limited in their career choices because of inadequate mathematics knowledge and experience. Study Guide  STADIO MFP201-DL-SG-E4 Page 31 Let's use our understanding of active learning and learning theories and do Writing Activity 4 below.  Writing Activity 4  Using your understanding of how children learn and understand mathematics, critically compare the following Grade 1 lesson plan extracts and answer the questions that follow: Extract 1 – Lesson topic: Addition 1 to 10 For the development of the new concept, Addition for numbers 1 to 10. Learners begin by playing a game, 'What number am I' where the teacher gives them clues such as 'I am one more that 7 and two less than 10'. The teacher will also have flash cards with 'magic numbers' such as 6, 8 and 10 that learners can draw from a bag. Learners will use their magic number and explore and experiment with a ten-frame chart and use manipulatives, such as bottle tops, to create different number combinations to make their magic number. These combinations can be compared and discussed with a partner. The teacher can ask questions and observe learners demonstrating their combinations. Extract 2 – Lesson topic: Addition 1 to 10 The teacher will tell learners that they are going to work with the number 10. The teacher explains to the learners that they must use the ten frame and bottle tops to make different combinations. The teacher starts by telling the learners to put three bottle caps onto their ten frame. The learners are then told to fill up the spaces to make 10. The teacher tells the learners to try again, this time using five bottle tops. Learners are then given a worksheet to complete on their own. The worksheet is then marked. Counter Counter Counter 1. Compare the two extracts above. What is different? What is the same? _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ Page 32 MFP201-DL-SG-E4 Study Guide  STADIO  Writing Activity 4  (continued) 2. Which lesson extract do you think is more effective? Why? _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ 3. What is the role of the learner and teacher in each? _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ 4. In which extract is active learning happening? Why do you say so? _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ Answers: Answers will vary. In Writing Activity 4 above, you should have noticed that Extract 1 demonstrated active learning, and learners were involved in the learning process. However, in Extract 2, the teacher was the main stakeholder and learners followed their instruction with very little to now room for exploring Mathematics on their own. Many teachers think that it is enough to Study Guide  STADIO MFP201-DL-SG-E4 Page 33 provide learners with manipulatives. Their thinking is that if learners use manipulatives, then active learning will happen. However, from what we have learnt so far, we know that this is not the case. To really understand what role manipulatives play let's briefly look at how manipulatives can support learning. Using manipulatives to support learning The role of manipulatives in mathematics is significant in the Foundation Phase. Young learners need to have active, hands-on experiences with manipulatives to help them understand abstract mathematical concepts. A manipulative is not a mathematical concept, but rather a physical representation of an abstract concept. They help to demonstrate the understanding process and provide learners with a way of making their thinking process visible. If we were to think about the idea of fractions – if we were asked to visualise the idea of ⅙, if we had never had the opportunity to work with and see examples of circles, squares that showed us what ⅙ looked like we would not be able to visualise it. This is what manipulatives do, they provide a concrete mental representation of a concept that would otherwise be very challenging, if not impossible to understand. Teachers also now have access to virtual manipulatives, and we will look at this topic in the next section of this study guide. Manipulatives can also include teacher-made resources and a lot of recycling material is also extremely useful, for example: bread tags and milk bottle lids make excellent counters. We now move on to discuss 21st century skills in Foundation Phase mathematics because as teachers of a very exciting, technologically advanced future, we need to prepare ourselves to teach learners of the future. 21st Century Skills in Foundation Phase Mathematics In this section of the lesson we discuss 21st century skills in Foundation Phase mathematics by discussing the following ideas:  21st century skills and the Fourth Industrial Revolution (4IR);  integrating 21st century thinking in Foundation Phase Mathematics; and  focus on problem-solving in the Foundation Phase classroom. Let's begin by discussing the idea of 21st century skills. Page 34 MFP201-DL-SG-E4 Study Guide  STADIO 21st century skills and the Fourth Industrial Revolution (4IR) The term 21st century skills has been a buzzword in education for a long time now. These are skills that have been identified as vital knowledge, skills, characteristics and attitudes that learners will need to develop in order to function in the future society. There is various research that looks at what this concept means, let's look at two now:  Wessling (2010) states that: "Twenty-first-century learning embodies an approach to teaching that marries content to skill".  Kolk (2011) proposes that: "A 21st century classroom must engage and energize and prepare all students to be active participants in our exciting global community". One of the biggest mistakes that teachers make is to think that the term refers only to the use of technology in the classroom. Teaching learners to use technology is of course important, but more importantly are the skills they develop using this technology. This means that 21st century skills can also be developed without the use of technology and the classroom does not have to be well-resourced with computers or tablets to be future-focused. The skills (you may also hear these referred to as the 4Cs) that are being referred to above include the following:  critical thinking;  creativity;  collaboration; and  communication. The list of these skills has grown over time to include many other aspects. Another current buzzword in education is the Fourth Industrial Revolution (known as 4IR). The Fourth Industrial Revolution (4IR) refers to a future dependent on technology. Many jobs as we know today will be automated in the future and the workforce is going to need a very particular skill set to be able to be productive in society. Drivers of the 4IR include the Organisation for Economic Cooperation and Development (OECD) and the World Economic Forum (WEF). Subsequently, the WEF has identified new list of skills that are required to succeed in the 4IR (Davos, 2016). These include:  complex problem-solving;  critical thinking;  creativity;  people management;  coordinating with other people;  emotional intelligence;  ethical judgements and decision making;  service orientation;  negotiation; and  cognitive flexibility. Study Guide  STADIO MFP201-DL-SG-E4 Page 35 If you look at the above skills, they are not ones that could be automated, but rather they are uniquely human. These skills do not link to 'knowing' things, but rather knowing what to do about things based on ethics, working with others and coming up with new, creative solutions to problems. Now complete Online Activity 4.  Online Activity 4  There are many interesting videos articles and blogs that provide more information about 21st century skills and 4IR. Watch the related videos on the LMS. Then take some time and research the following online: 1. What are 21st century skills? 2. What is the Fourth Industrial Revolution? 3. For this you may want to look at the two organisations mentioned above as well (OECD and WEF). 4. How will these phenomena affect education now and in the future? Once you have done this, log on to the LMS and create a discussion forum post where you discuss one or two aspects that you found especially interesting in your research. You do not only have to agree with what you read, rather think about how it all translates back to a South African Foundation Phase classroom. If we refer back to the list of skills mentioned above, we need to consider whether or not they could be developed in a Foundation Phase mathematics classroom. Let's do this now. Integrating 21st century thinking in Foundation Phase mathematics It may not always be recognised, but mathematics is the ideal subject to develop the skills mentioned above. All of these skills can be developed through problem-solving. Learners are required to think critically about a problem, think about different strategies and also evaluate their strategies (critical thinking and creativity). They are also required to communicate in various stages of problem solving and often even working with others to solve a problem (collaboration). For the reasons mentioned here, a really great mathematics teacher will teach through problem solving. Teaching through problem-solving As we mentioned above, problem-solving is critical in a Foundation Phase mathematics classroom and it allows the teacher to promote the development of a variety of skills. Please take the time to read the following:  Pages 57 to 81 of Chapter 3: 'Teaching Problem-Based Mathematics' in Elementary and Middle School Mathematics: Teaching Developmentally by van de Walle et al. Page 36 MFP201-DL-SG-E4 Study Guide  STADIO Now, complete Writing Activity 5.  Writing Activity 5  Create a mind-map in the space below that outlines all of the important features about teaching through problem-solving that you found in your prescribed reading outlined above. Don't forget to include the topic of 'worthwhile tasks'. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Model answer: The answer will vary. Let's now take some time to think about how technology can be integrated into Foundation Phase mathematics. Using technology in mathematics You may hear about Science, Technology, Engineering and Mathematics (STEM) subjects that are introduced early in a learner's school career (sometimes this is introduced as early as pre- school). You may have noticed in the media that the South African Department of Basic Study Guide  STADIO MFP201-DL-SG-E4 Page 37 Education has introduced a curriculum for coding and robotics (if this is new to you a quick internet search may be in order). Coding and robotics can highly promote the learning of mathematics. Please complete Online Activity 5 below.  Online Activity 5  Watch the following videos. The links to these videos are given below and can also be found on the LMS. 1. How to Teach Kids to Code (Payne 2018) www.youtube.com/watch?v=s6BVSs2I7ow 2. Learning to Code, Coding to Learn (Buchanan 2012) www.youtube.com/watch?v=8vXgjfBmzFs 3. Robotics is STEM (Science Technology Engineering Math) www.youtube.com/watch?v=Q6Cy3z5QunI 4. Teaching Coding Through Robotics (CBS News) www.youtube.com/watch?v=MOYDmWfdmSA There is also a plethora of websites, apps, videos, virtual manipulatives and technological devices that can support learning in mathematics. Some of the most widely recognised include:  Khan Academy;  GeoGebra;  Minecraft;  National library of Virtual Manipulatives (you do need JAVA for this);  DIDAX Virtual Manipulatives;  Math Playground;  Kahoot;  Symbaloo (I particularly like BrainPop, but there is a lot to explore);  YouTube; and  Mentimeter. Try and work through the above list when you are online, there is so much to learn and explore and it is the hope that you feel inspired to use some of these suggestions in your mathematics teaching. Your future classroom might include virtual reality, augmented reality and artificial intelligence! We can definitely agree that 21st century and 4IR skills can be integrated into Foundation Phase mathematics – if we can really be successful at this, we can provide real value to the present and future success of our learners. We now move on to discuss a teacher's perspective: mathematics anxiety, because becoming an effective mathematics teacher means that you need to have a positive attitude to mathematics. Page 38 MFP201-DL-SG-E4 Study Guide  STADIO A Teacher's Perspective: Mathematics Anxiety In this section of the lesson we discuss mathematics anxiety from a teacher's perspective. A teacher's attitude towards a subject will also influence their learners' feelings towards the subject. Let's briefly explore this idea. You would have read briefly about mathematics anxiety in McDermott and Rakgokong (1996: 32). You would have noticed that many people joke about mathematics anxiety, but it is a real condition with physiological and psychological manifestations. Mathematics anxiety can be as a result of a negative experience in a mathematics learning experience where you were made to feel unconfident, or bad about yourself. Perhaps you were left feeling humiliated for providing the wrong answer, or others laughed at your reasoning. An experience like this can leave you feeling fearful, nervous and withdrawn from mathematics. These negative feelings stay with us and now we are going to be faced with teaching a subject that we don't feel positive about. If this is the case, then we need to find a way to instil a love for mathematics in ourselves again so that we can do the same for our learners. Please work through Self-reflection 9. Self-reflection 9 1. What do you think you can do to develop a passion for mathematics (or strengthen this passion if you already are passionate about the subject)? 2. How will a teacher's attitude towards mathematics influence a learner's attitude? For now, it's time to end our discussion on Becoming an Effective Mathematics Teacher and move on to discuss Understanding the Mathematics Curriculum. We do this in Lesson 2. Only move on to Lesson 2 if you are confident that you understand the content in the present lesson. If anything is unclear to you, remember that you can email us to discuss the matter. All of the information that you need is in the Welcome Letter and on the LMS. Study Guide  STADIO MFP201-DL-SG-E4 Page 39 Lesson 2: Understanding the Mathematics Curriculum  Prescribed readings Your prescribed readings for Lesson 2 are as follows:  Mathematics Grades R-3 CAPS document.  Pages 82 to 107 of Chapter 4: 'Planning in the Problem-Based Classroom' in Elementary and Middle School Mathematics: Teaching Developmentally by J.A. van de Walle et al.  Pages 109 to 128 of Chapter 5: 'Creating Assessments for Learning' in Elementary and Middle School Mathematics: Teaching Developmentally by J.A. van de Walle et al. Note that you'll be told exactly when to read each of the above texts as you work through the lesson. Introduction In Lesson 1, we explored how children learn and understand mathematics. In Lesson 2, we'll look more specifically at how mathematics is covered in a South African Foundation Phase classroom. To do this, we are firstly going to explore various aspects of CAPS. Secondly, we are going to look at how to plan a Foundation Phase mathematics lesson. Lastly, we are going to discuss informal assessment in Foundation Phase mathematics. Let's start this lesson by unpacking CAPS. Curriculum and Assessment Policy Statement: CAPS CAPS is the policy statement or document for learning and teaching, from Grade R to Grade 12, in South African schools. The significant amount of information and the progressive nature of learning means that CAPS is separated into various phases, namely: Foundation Phase (Grades R to 3), Intermediate Phase (Grades 4 to 6), Senior Phase (Grades 7 to 9) and FET (Grades 10 to 12). As we know, this module only focuses on Foundation Phase mathematics; therefore, we will only be unpacking this specific phase. As mentioned in the introduction of the module, you need to be familiar with CAPS and that is why specific page numbers have not been mentioned here. It is expected that you explore the document and learn where to find the required information. Certain extracts are provided below, and then you will be provided with a specific page number. Page 40 MFP201-DL-SG-E4 Study Guide  STADIO The Mathematics Grades R-3 CAPS document provides Foundation Phase teachers with vital information including time allocations per phase, weighting allocations and content areas. The subject, mathematics, is further separated into five content areas. Let's start looking at these now. CAPS: time allocations It would be a challenge for teachers to plan how much time that is spent on each subject in each grade without having specific guidelines. Figure 8 below shows an extract from the Mathematics Grades R-3 CAPS document (Department of Basic Education 2011: 6) that shows how much time per week should be spent on certain subjects. Figure 8: CAPS time allocation – Foundation Phase As you can see, in the Foundation Phase, seven hours per week should be spent doing mathematics. If we know how long that we should spend doing mathematics it is also important for us to know which content areas, we should be focusing on in each grade. We can see this in the example that follows. Study Guide  STADIO MFP201-DL-SG-E4 Page 41 CAPS: Weighting – content areas In the extract from the Mathematics Grades R-3 CAPS document (Department of Basic Education (2011: 10) provided in Figure 9, guidelines are provided for each of the five content areas is weighted differently according to each grade. Figure 9: CAPS weighting of content areas Please take note of the very important explanation provided at the bottom of the extract. This explanation provides a rationale for spending more time on 'Numbers, Operations and Relationships' in comparison with the other content areas in Foundation Phase. Let's take some time and look at these content areas in more detail now. CAPS content areas The content areas are umbrella headings for the content taught over all the grades in mathematics. Each content area has many different topics. There are five main content areas for mathematics:  Numbers, Operations and Relationships;  Patterns, Functions and Algebra;  Space and Shape (geometry);  Measurement; and  Data Handling. Let's look at the topics covered in each content area briefly now. Page 42 MFP201-DL-SG-E4 Study Guide  STADIO Numbers, Operations and Relationships Figure 10 provides an overview of this content area. If you look at the various topics it is easy to see how understanding many of these provides a foundation for learning other, more complex mathematics concepts, and that is why most of the time is spent on this content area. Figure 10: 'Numbers, Operations and Relationships' topics It is important to also recognise how progression happens in this content area. Teachers in each grade need to understand this progression as we need to know what learners have experienced before (prior knowledge), and, also how the topic will be built upon in the future. The main progression in 'Numbers, Operations and Relationships' happens in three ways:  the number range increases;  different types of numbers are introduced; and  the calculating strategies change. It is also important to note the following:  as the number range for doing calculations increases up to Grade 3, learners should develop more efficient strategies for calculations; and  contextual problems should take account of the number range for the grade as well as the calculation competencies of the learners. Let's look at the second content area, 'Patterns, Functions and Algebra' now. Study Guide  STADIO MFP201-DL-SG-E4 Page 43 'Patterns, Functions and Algebra' This content area in Foundation Phase focuses mostly on learning about and exploring patterns. The topics are shown in Figure 11. Figure 11: 'Patterns, Functions and Algebra' topics Progression in 'Patterns, Functions and Algebra' happens in the following ways:  complete and extend patterns represented in different forms; and  identifying and describing patterns. Describing and exploring patterns lays the foundation for learners in the intermediate phase to describe rules for patterns. This in turn becomes more formalised in algebraic work in the senior phase in the form of what we commonly remember as algebra. Let's briefly look at the third content area, 'Space and Shape' (geometry) now. 'Space and Shape' (geometry) The content area, 'Space and Shape' comprises topics that we commonly refer to as geometry. The topics are shown in Figure 12. Figure 12: 'Space and Shape' topics Page 44 MFP201-DL-SG-E4 Study Guide  STADIO The main progression in 'Space and Shape' in the Foundation Phase includes:  focusing on new properties and features of shapes and objects in each grade; and  moving from learning the language of position and matching different views of the same objects to reading and following directions on informal maps. Let's briefly look at the fourth content area, measurement now. Measurement When most of us think about measurement we first think about units of measurement such as centimetres, kilogrammes and litres. Learners in the Foundation Phase will explore many topics when learning about measurement. These are outlined in Figure 13 below. Figure 13: Measurement topics The main progression in measurement across the Foundation Phase includes:  the introduction of: ­ new forms of measuring; ­ new measuring tools starting with informal tools and moving to formal measuring instruments in Grades 2 and 3; and ­ new measuring units, particularly in Grades 2 and 3.  calculations and problem-solving with measurement should include the number work that has already been covered (this means that this is also heavily dependent on 'Numbers, Operations and Relationships'). Let's briefly look at the final content area in the Mathematics Grades R-3 CAPS document, Data Handling. Study Guide  STADIO MFP201-DL-SG-E4 Page 45 'Data Handling' 'Data Handling' is the content area that has the lowest weighting of all the content areas in the Foundation Phase. 'Data Handling' includes the following topics shown in Figure 14: Figure 14: Data-handling topics The main progression in 'Data Handling' is outlined here:  moving from working with objects to working with data;  working with new forms of data representation; and  learners should work through the full data cycle at least once a year – this involves collecting and organising data, representing data, analysing, interpreting and reporting data. Now that we have had a broad overview of the topics covered by in CAPS, let's take a moment to think about the resources that are recommended in CAPS. Let's do this now. Recommended Resources for the Foundation Phase Mathematics Classroom Earlier in this study guide we discussed the importance of manipulatives with reference to how young children learn. The Mathematics Grades R-3 CAPS document (Department of Basic Education, 2011: 16) provides a list of recommendations about the types of manipulatives that should be available in a Foundation Phase mathematics classroom. Before we look at the list, let's see what you think. Page 46 MFP201-DL-SG-E4 Study Guide  STADIO Please complete Writing Activity 6.  Writing Activity 6  Make a list of 10 mathematics resources that you think would be important in a Foundation Phase classroom: 1. _____________________________________________________________________ 2. _____________________________________________________________________ 3. _____________________________________________________________________ 4. _____________________________________________________________________ 5. _____________________________________________________________________ 6. _____________________________________________________________________ 7. _____________________________________________________________________ 8. _____________________________________________________________________ 9. _____________________________________________________________________ 10. _____________________________________________________________________ Answer: Your list can include (among others) the following manipulatives: (1) Counters; (2) Large dice; (3) A big counting frame; (4) A height chart; (5) Big 1-100 and 101-200 number grid posters; (6) Different number lines (vertical and horizontal); (7) A set of flard cards (expanding cards); (8) Play money (remember, legally you can't photocopy real money); (9) A calendar for the current year; (10) A large analogue wall clock; (11) A balance scale; (12) Building blocks; (13) Modelling clay; (14) A variety of boxes (different shapes and sizes); (15) A variety of plastic bottles and containers to describe and compare capacities; (16) 3D shapes (sphere (ball), rectangular prism (box), cube, cone, pyramid, cylinder); (17) 2D shapes (different sized rectangles, circles, different sized and shaped triangles); (18) Mathematical games e.g. Ludo, Snakes and Ladders, Jigsaw puzzles, Dominoes, Tangrams etc. How many of these have you heard of? Research the ones that you don't know, and you may find them to be extremely useful in a particular lesson. You now have an idea of what to teach and when, and also what can be used in teaching, but let's move on to exploring what a mathematics lesson should look like. What does a Mathematics Lesson look like? There are various approaches, structures and lesson plan templates to teaching mathematics in the Foundation Phase. However, before we start looking at a structured mathematics lesson, let's consider how mathematics

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