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© 2021 University of South Africa All rights reserved Printed and published by the University of South Africa, Muckleneuk, Pretoria PDM4801/2022 10029478 Shutterstock.com images used Editor and Styler MNB_Style CONTENTS ORIENTATION AND WELCOME (vii) LEARNING UNIT 1 M AT H E M AT I C S C U R R I C U L U M I N P R O C E S S 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 INTRODUCTION CURRICULUM: A PLAN FOR LEARNING ACROSS MANY LEVELS 1.2.1 The term curriculum: extending the concept 1.2.2 Curriculum levels, institutions and products SOME CURRICULUM MODELS 1.3.1 The product model 1.3.2 The process model 1.3.3 The subject-centred model 1.3.4 The learner-centred model 1.3.5 The problem-centred model THE CURRICULUM DEVELOPMENT PROCESS APPROACHES TO CURRICULUM DESIGN 1.5.1 The instrumental approach 1.5.2 The communicative approach 1.5.3 The artistic approach 1.5.4 The pragmatic approach THE INTENDED, IMPLEMENTED AND ATTAINED CURRICULUM CONCLUSION ADDITIONAL LEARNING OPPORTUNITIES 1 2 2 3 5 7 8 8 8 9 9 10 13 14 15 15 16 17 22 22 LEARNING UNIT 2 S E L E C T E D M AT H E M AT I C S C U R R I C U L A A C R O S S T H E G L O B E 2.1 2.2 2.3 2.4 2.5 2.6 DIFFERENT COUNTRIES: DIFFERENT STRUCTURES DIFFERENT COUNTRIES: DIFFERENT BALANCE OF SUPPORT AND AUTONOMY DIFFERENT COUNTRIES: DIFFERENT PHILOSOPHICAL UNDERPINNINGS, CONTENT AND PEDAGOGY DIFFERENT COUNTRIES: DIFFERENT MATHEMATICS CURRICULA DIFFERENT CULTURES, DIFFERENT LEARNERS: SAME CURRICULUM? CONCLUSION PDM4801/2022 24 26 32 33 35 40 41 (iii) CONTENTS LEARNING UNIT 3 I N T E R N AT I O N A L L A R G E - S C A L E A S S E S S M E N T 3.1 3.2 3.3 3.4 TIMSS CURRICULUM AND ASSESSMENT FRAMEWORKS 3.1.1 An overview of the IEA studies 3.1.2 Assessment frameworks: content and cognitive domains 3.1.3 Assessment design, population and sampling SOUTHERN AND EASTERN AFRICAN CONSORTIUM FOR THE MONITORING OF EDUCATIONAL QUALITY 3.2.1 An overview of SACMEQ 3.2.2 The SACMEQ Project 3.2.3 Curriculum and assessment frameworks 3.2.4 Some results COMPARISON OF RESEARCH DESIGNS: INTERNATIONAL AND REGIONAL STUDIES CONCLUSION 43 44 45 46 54 56 56 57 59 60 61 62 LEARNING UNIT 4 C R O S S - E X A M I N I N G T H E C U R R I C U L U M A N D P O L I C Y S TAT E M E N T F O R M AT H E M AT I C S G R A D E R - 1 2 4.1 4.2 4.3 4.4 SOME THEORETICAL PERSPECTIVES A SUBJECT-CENTRED PERSPECTIVE ON THE CAPS 4.2.1 Why a theoretical and conceptual perspective? 4.2.2 Number, operations and relationships and the transitions 4.2.3 Patterns, functions and algebra and the transitions 4.2.4 Space, shape, geometry and measurement and the transitions 4.2.5 Data handling and probability and the transitions A PERSPECTIVE ON THE CAPS: A LEARNER-CENTRED APPROACH 4.3.1 Constructivist theory of learning 4.3.2 Realistic mathematics education A PERSPECTIVE: MATHEMATICS AS PROBLEM-SOLVING 65 66 69 70 71 74 76 80 81 82 83 85 LEARNING UNIT 5 D E S I G N I N G M AT H E M AT I C S C U R R I C U L A I N C O N T E X T 5.1 5.2 (iv) A COMMUNITY AND ITS CONTEXT THE SPIDERWEB OF CURRICULA COMPONENTS 5.2.1 Aims and objectives 5.2.2 Content and learning activities 5.2.3 Teacher’s role, pedagogical principles, materials and resources 5.2.4 Location and time 5.2.5 Assessment 87 88 89 90 91 92 93 93 Contents 5.3 5.4 PDM4801 AN INTEGRATED VIEW OF A MATHEMATICS CURRICULUM 5.3.1 Vision for a mathematics classroom 5.3.2 The learner and a productive disposition CONCLUSION 94 94 95 96 (v) ORIENTATION AND WELCOME i. Introduction Welcome to the module PDM4801: Curriculum Studies in Mathematics. This module is the fourth of five modules in the Post-Graduate Diploma in Mathematics Education. You are in the privileged position of already having obtained at least one degree at tertiary level. Your influential position in our country’s education may be strengthened even further through this post-graduate study in Mathematics Education. What you are about to experience, is deeper and broader than what you did at undergraduate level. Your view of mathematics education will be expanded beyond the local confines. Curriculum Studies is fundamental to a deeper understanding of the imperatives and purpose of mathematics education. Although the focus is on the South African curriculum, the module expands your view to the curricula of other countries across the globe, first-world through to third-world countries. Various components of these curricula are discussed for you to reflect on their impact on the development of the countries. The perspectives you gain in this module will enhance your awareness and appreciation of a contextual curriculum, which serves the needs of that region and cannot be applied as is, in the context of another region. We shall compare some of these curricula and contextualise our South African curriculum. ii. The course content This module forms part of the course: Post-Graduate Diploma in Mathematics, which is at NQF level 8. The course comprises the following modules: 1. Philosophical and Historical Perspectives in Mathematics Education 2. Theoretical Issues in the Teaching and Learning of Mathematics 3. Teaching and Learning in Mathematics Education 4. Curriculum Studies in Mathematics Education 5. Using Research in Mathematics Education Whereas you received the basic and fundamental equipment as a mathematics teacher during your undergraduate studies, this qualification raises your knowledge, understanding and insight of mathematics education to the next level. iii. Purpose and outcomes of this module The purpose of this module, Curriculum Studies in Mathematics Education, is to provide you with a variety of the components of the Mathematics curricula from different countries to analyse and compare these curricula. Furthermore, the module enables you to reflect on the curricula’s impact on the development of these countries. The study of different Mathematics curricula in selected countries endeavours to enhance awareness of a contextual curriculum. PDM4801 (vii) ORIENTATION AND WELCOME iv. Specific outcomes While working through the five learning units of this study guide, you will do the following: ● Develop an understanding of the role players of curriculum implementing in South Africa. ● Compare and critically review the Mathematics curricula in selected countries. ● Reflect on the analysis of international comparisons of mathematics achievement. ● Interrogate topics in the Mathematics curriculum to evaluate the trajectory of learning mathematics through all grades. ● Design mathematics curricula for a particular context. v. Structure of this study guide In each of the five learning units of this study guide, you will find the following: ● A table of contents specific to the learning unit to give you a overview of the learning unit’s structure. ● Outcomes of the learning unit to tell you what you need to know and understand once you have worked through the learning unit’s content. ● Activities follow almost every section. These are meant for you to apply the knowledge and understanding that you have gained in that section. It helps you to engage actively with the study contents. vi. How should you go about studying this module? It is not easy to study at a distance and you should not underestimate the time and effort involved. Once you have received your study material, please plan how you will approach and complete this module. Your work on each learning unit should involve the following: ● Skim through the learning unit and draw a basic mind map of its content. Then expand this map as your knowledge and understanding of the unit increases. If you have internet access, you can learn more about making mind maps at http://www. wikihow.com/Make-a-Mind-Map and http://www.mind-mapping.co.uk/make-mindmap.htm. ● Summarise every learning unit. ● Do a reflection exercise at the end of every learning unit. Each learning unit contains a checklist that guides your reflection on your learning. As you work, you must build up your study and exam preparation file. This study file will not be assessed, but it will be an extremely valuable tool for you in completing your assignments and revising for the examination. A study file is a folder or file in which you gather and compile additional and/or summarised information during the year as you work through the learning material. Such a file may contain the following: ● Answers to each activity in each learning unit ● A mind map/summary of each learning unit ● Your marked assignments (or a copy you made before submitting your assignment) ● Your reflections on each learning unit (viii) Orientation and welcome ● Where relevant, any extra reading material taken from the internet, additional books, medical and/or scientific journals ● A new vocabulary of words or a glossary of new terms in your own words vii. Overview of what is expected of you This orientation gives you an overview of and some general information about this module, including how you can study in this module, how to use myUnisa and about the assessment of the module. The study guide and tutorial letters contain everything you need to complete this module. However, you may also benefit from using the module website on myUnisa. You can do the following through myUnisa: ● Submit assignments (we recommend that you submit your assignments online as this will ensure that you receive rapid feedback and comments) ● Access the official study material ● Access the Unisa Library functions ● “Chat‟ to your lecturer or e-tutor and fellow students and participate in online discussion forums ● Access a variety of learning resources Check the site regularly for updates, posted announcements from your lecturer and additional resources uploaded throughout the semester. Please note that your lecturer may create a discussion forum for students to discuss issues among themselves. Please use this opportunity to the full. viii. Tutor site Depending on the number of students in a course, an e-tutor may be appointed. An etutor is there to support students’ learning and answer their questions. If the number of students does not warrant the appointment of an e-tutor, your lecturer will fulfil this role. Whether with your fellow students in the discussion forum or with an appointed e-tutor, please participate, as this will go a long way to enhance your learning. ix. The study material for this module The study material for this module includes: ● This Tutorial Letter 501 (the study guide) ● Tutorial Letter 101 ● Any other tutorial letters you may receive through the year ● Additional information provided by your e-tutor, on the myUnisa module site or any electronic communication; for example announcements or e-mails Tutorial Letter 101 will be part of your study pack or will be posted to you. You can also access it on myUnisa by clicking on ‘Official Study Material’ in the menu on the left of the module portal. PDM4801 (ix) ORIENTATION AND WELCOME Tutorial Letter 101 is just one of the tutorial letters that you will receive during the year. You must read this tutorial letter carefully. You may also receive follow-up tutorial letters during the year. x. Assessment in this module Activities in the study guide We would like to meet you and talk to you, but we realise that this is unlikely since you are a distance education student. Most of our communication will, therefore, be written communication. If we were in a classroom situation, we would ask questions to which you would respond immediately, but since we are not in a face-to-face classroom situation, we have set questions that you must answer in writing. These activities require you to give your opinion or link the content in the study guide with your everyday life experiences and prior knowledge. You have the opportunity to be creative to do practical work, to offer an opinion and to say when you do or do not agree. Testing yourself You must test yourself regularly through the self-assessment activities. Although each learning unit concludes with an exercise that expects you to reflect on what you have learned and whether you have achieved the outcomes, You must test yourself by constantly asking yourself “what do I know now, that I didn’t know before?” Also test yourself by using mind maps of each learning unit (as explained in the previous section). Assignments and the exam Your work in this module will be assessed as explained in Tutorial Letter 101. Familiarise yourself with the following information in Tutorial Letter 101: ● The calculation of assignment and examination marks ● The due dates and unique numbers of your assignments ● The submission of your assignments ● Examination periods, admission and marks Tutorial letter 101 also contains the actual assignment questions. While Tutorial Letter 101 will be sent to you, you can also access an electronic version on myUnisa under ‘Official Study Material’. xi. Orientation to using myUnisa You should be able to use the various menu options on the myUnisa site, which will enable you to participate actively in the learning process. These options include: myUnisa menu option What you will find here Official Study Material The study guide and tutorial letters will be stored under this option, as well as past examination papers. Announcements From time to time, the lecturer or your e-tutor will use this tool to give you important information about this (x) Orientation and welcome myUnisa menu option What you will find here module. You should receive an e-mail notification of new announcements placed on myUnisa. Calendar This tool shows important dates, such as examination dates and deadlines for assignments. This information will help you to manage your time and plan your schedule. Additional resources The lecturer may use this to post additional learning support material that might help you in your studies for this module. An announcement will be sent to students to inform you if anything has been added to this folder. Discussions This tool allows us to hold discussions as if we were in a contact setting. You can post any specific queries to the lecturer on the main module site. There will also be a forum for students where you can discuss issues among yourselves or support one another. Assessment info This tool allows you to submit your assignments electronically and to monitor your results. If you can, please submit your assignments via myUnisa. If you do not know how to do this, consult Tutorial Letter 101. In online interaction, always remember to be mindful of and respectful towards your fellow students and lecturers. The rules of polite behaviour on the internet are referred to as netiquette – a term that means “online manners”. To learn more about netiquette, visit: ● http://networketiquette.net/ ● http://www.studygs.net/netiquette.htm ● http://www.carnegiecyberacademy.com/facultyPages/communication/netiquette. html Please observe the rules of netiquette during your everyday online communications with colleagues, lecturers and friends. In particular, remember to be courteous towards your fellow students when using the Discussions tool. xii. Preview of this specific module This module consists of five learning units. The overview at the beginning of each learning unit prepares you for what is to come in that learning unit. For the five learning units in this module, the overviews are: 1. Designing and implementing a curriculum involve multiple considerations, the first of which is to define the curriculum. Then there are the role players at different levels of implementation, who play a part in the multiple components. A specific approach is followed for the plan and the process of developing a curriculum. A simplified way of thinking about the curriculum is to separate the intended curriculum, the implemented curriculum and the attained curriculum. You will encounter these concepts in many discussions about curriculum in learning unit 1. PDM4801 (xi) ORIENTATION AND WELCOME 2. Because many region-specific contextual factors play a part in the original design and the process of developing a curriculum, there will be many different curricula across the world, though their core curriculum content may be similar. One of the key differences will be whether the curriculum is centralised or decentralised and to what extent. For example, South Africa has a centralised curriculum, while New Zealand has guidelines for curriculum. Although each institution makes its own decisions, there is quality control over decentralised decisions. Learning unit 2 describes curricula from four countries and ask you to compare them according to various components. 3. In learning unit 3, we engage with curriculum and assessment frameworks, exemplified by two large-scale assessments, namely Trends in International Mathematics and Science Study (TIMSS) and Southern and Eastern Africa Consortium for Monitoring Educational Quality (SACMEQ). While exploring the curriculum and assessment framework of macro assessments, participating countries must know what is in the curriculum and what is going to be tested; and that the public and education department are confident that the study design is scientific and fair. 4. Learning unit 4 covers the following important aspects of the curriculum: ● The Curriculum and Assessment Policy Statement (CAPS) was developed from an existing curriculum, designed in 1997, known as Curriculum 2005. This learning unit explores how the CAPS curriculum can be mapped against a framework and judged in terms of the components discussed in learning unit 1. We also look at how the topics in CAPS align with the principles set out in learning unit 1. ● Central to most curricula is a philosophical approach to mathematics itself and the teaching and learning of mathematics. For the most part, a constructivist approach to teaching and learning mathematics is proposed. The modern theory of teaching and learning, together with the enactment of constructivism, namely Realistic Mathematics Education, will be explored and illustrated with examples. ● Most people will agree that the ultimate goal of mathematics is about solving problems. A problem-solving approach, initiated in the 1980s, and kept alive in many classrooms, will be discussed and explored. 5. Finally, learning unit 5 builds on learning unit 1, where the macro elements are discussed and Curriculum Design is elaborated with a focus on the micro-elements. xiii. What is expected of you in this specific module? If you approach this module with an open mind and intent to actively engage with the study material, you will succeed. What can you do from your side? ● Orientate yourself with the module and systematically engage with each learning unit by going through the table of contents and the specific outcomes. ● Understand a learning unit and do the activities to apply your understanding. ● Take the time to read the sources referred to in the text. ● Do assignments on time and endure hard times. Do not give up! (xii) Orientation and welcome xiv. Conclusion Good luck and enjoy the course. Please do not hesitate to contact your lecturers if there is something you do not understand. PDM4801 (xiii) LEARNING UNIT 1 Mathematics curriculum in process LEARNINGUNIT1 Table of contents INTRODUCTION 1.1 CURRICULUM: A PLAN FOR LEARNING ACROSS MANY LEVELS Activity 1.1.1: Defining the curriculum Activity 1.1.2: Curriculum levels, institutions and products 1.2 SOME CURRICULUM MODELS 1.2.1 The product model 1.2.2 The process model 1.2.3 The subject model 1.2.4 The learner-centred model 1.2.5 The problem-centred model Activity 1.2: Themba’s curriculum 1.3 THE CURRICULUM DEVELOPMENT PROCESS Activity 1.3: Using the spiderweb to reason about curriculum 1.4 APPROACHES TO CURRICULUM DESIGN 1.4.1 The instrumental approach 1.4.2 The communicative approach 1.4.3 The artistic approach 1.4.4 The pragmatic approach Activity 1.4: Comparing approaches to curriculum design 1.5 THE INTENDED, IMPLEMENTED AND ATTAINED CURRICULUM Activity 1.5: The intended, implemented and attained curriculum 1.6 1.7 CONCLUSION ADDITIONAL LEARNING OPPORTUNITIES REFERENCES PDM4801 1 LEARNING UNIT 1 MATHEMATICS CURRICULUM IN PROCESS OUTCOMES OF THIS LEARNING UNIT In this learning unit, you will develop a sound understanding of the role players of the curriculum in South Africa. At the end of learning unit 1, you should be able to do the following: ● ● ● ● ● ● 1.1 Explain the concept of a curriculum from various perspectives. Identify various educational approaches to curriculum. Describe and reflect on the process of curriculum design and development. Critically discuss the roles of various institutions in designing curricula. Unpack a variety of models of curriculum design. Critically reflect on the intended, implemented and attained curriculum. INTRODUCTION This learning unit focuses on the curriculum as it undergoes its development process from planning, through design to implementation. This process involves multiple considerations, such as defining the curriculum and identifying the role players at the different levels of the process and in its multiple components. Much work from the Netherlands has been incorporated in international studies and is, therefore, regarded as a global guideline in curriculum studies. Professor T Plomp was the chairperson of the IEA and a leading figure in the Netherlands Institute of Curriculum Development (SLO) together with Van den Akker and Thijs. Subsequently, we draw mainly from the work of Akker (2003, 2006) and Thijs and van den Akker (2009) as a basis for our discussion of the following: ● The term curriculum ● Five levels of curriculum in which different curriculum products are designed ● Four approaches to curriculum design, namely the instrumental, communicative, artistic and pragmatic approaches ● The curriculum process as it applies to the Southern African and global contexts A fairly general way of thinking about the curriculum process is to separate it into three phases, namely the intended, the implemented and the attained curriculum. These terms are associated with the International Association for the Evaluation of Educational Achievement (IEA) studies of which the TIMSS is one such study. Here we will look at how these terms have been used and the points of intersection. We also show how the phenomenon of teaching-to-the-test interacts with the three identified phases of the curriculum. 1.2 CURRICULUM: A PLAN FOR LEARNING ACROSS MANY LEVELS The term curriculum is used in many different ways. To some it means simply the content that is taught in the classroom on a particular topic, formerly known as a syllabus; to others, the term encompasses everything about schooling, from the policy documents issued at government level to the textbooks used in the school classroom. 2 Mathematics curriculum in process LEARNING UNIT 1 1.2.1 The term curriculum: extending the concept Here are two definitions pointing to the origins of the term curriculum: The first definition that came up in a search on the internet: Curriculum /kə ˈrɪkjʊləm/ the subjects comprising a course of study in a school or college. “Course components of the school curriculum.” http://folders.harveygs.kent.sch.uk/about/curriculum The second definition was compiled by the national institute for research into curriculum development in the Netherlands (SLO). In much of this module, we will be filling in all the pieces that make up this plan for learning that involves government departments, schools and teachers and learners. Thijs and van den Akker (2009) in Curriculum-in-Development use the term Leerplan or Plan for Learning. The Latin meaning of the word currere means to run. This verb can be applied to a course or to a vehicle. This conceptualisation captures the school curriculum, and has been translated into the Dutch, leerplan, and the English, plan for learning. Thijs and van den Akker (2009) Another way of defining the complex concept curriculum is by using explanatory models that show curriculum processes and products. Modelling is probably a more fruitful way of extending the concept than merely stating a brief definition. Travers and Westbury (1989) came up with conceptualising the curriculum in three phases, which can be associated with three dimensions on the same curriculum: ● The intended curriculum: what government and society envisions (a policy perspective) ● The implemented curriculum: what educators teach in the classroom (an executive perspective) ● The attained curriculum: what results learners show in assessment (an outcomes perspective) Figure 1.1 (as adapted for IEA purposes) elaborates on these ideas. The IEA uses this model for planning as the organising body of large-scale studies of educational achievements, such as TIMSS (in figure 1.1). This model, especially as it links with the ideas of Travers and Westbury (1989) will be discussed in some detail in paragraph 1.5. PDM4801 3 LEARNING UNIT 1 MATHEMATICS CURRICULUM IN PROCESS Figure 1.1: TIMSS Model of Potential Educational Experience (Schmidt, McKnight, Valverde, Houang, Wiley and David (1997:188) ACTIVITY 1.1.1 DEFINING THE CURRICULUM 1. Study the two definitions and the two more complex models of the curriculum in paragraph 1.1.1. Write a description of the curriculum for the following: a. A newspaper (two lines) b. A parent (four lines) c. A task team that would like to evaluate whether the educational system is functioning properly (10 lines) a. b. c. 4 Mathematics curriculum in process LEARNING UNIT 1 FEEDBACK ON ACTIVITY 1.1.1 The definition that one uses has to apply to the context. The newspaper may want to invite a public debate about the curriculum. Parents may be satisfied with knowing what their children are learning. The educational researcher has to have an elaborate model where all aspects of the system are taken into account. 1.2.2 Curriculum levels, institutions and products In the classroom, education may simply be seen as the interaction between a teacher and a learner towards some learning goal. However, in the country as a whole, public education has to manage a national system applicable to and fit for the entire country. At different levels of the curriculum, different organisations are involved. Van den Akker (2003) has identified the following five levels, in an order from the learner to the government minister of education: At the nano level, the focus is on the individual learner. In some schools, home-schooling settings and perhaps special schools, the educator might have designed an individual course of learning for a particular learner. At the micro-level, the focus is on the classroom as a teaching unit in a school, with the teacher as an important role player. The curriculum products at this level are the plan for teaching over the term, or over the year, the instructional materials, the textbooks, and in the South African context, the CAPS document that outlines the precise contents that should be taught, as well as the assessment plan for evaluating learning outcomes. At the meso level, the school and institutions with a similar purpose fit in. The school programme, timetables and extra-mural programmes are determined at this level. In some countries, the schools are autonomous, and in others, the government, at system level, keeps tight control. At the macro level, the focus is on the system, which is managed at the national level. In South Africa, the Department of Basic Education (DBE) is the national authority. At this level, the core objectives of the education system are conceptualised and the educational outcomes for the country are determined. The systems in most countries are divided into primary and high schools. In some cases, there are four levels, the first three years of schooling, the next three, and so on. The exit levels and exit examinations are determined at this national level. Other institutions are also part of the system, such as Umalusi, who is responsible for quality assurance, and the South African Qualifications Authority (SAQA), which is responsible for overseeing the design of qualifications. Before 1994, education was decentralised to provinces and each province was responsible for their own educational system. Each education department made its own decisions, within policy guidelines, about the content of their syllabus, the dates of their holidays and the setting of matric PDM4801 5 LEARNING UNIT 1 MATHEMATICS CURRICULUM IN PROCESS examinations. Decentralisation could result in duplication of functions and slightly varying perspectives and standards within a single education system. At the supra level, a group of independent countries cluster together for a particular purpose. In Europe, the individual countries adhere to some guidelines given by the European Union; for example, there is the Common European Framework of References for Languages. In Southern Africa, we have an international body, the Southern African Development Community (SADC) that has 16 member states. However, up until now, each country has had autonomy regarding education. Example of a tightly controlled school Let us take a normal primary school in South Africa. At this time (the 2020s) there is close control of the schools and what takes place in the classroom. The curriculum at grade level stipulates what has to be taught (content) when it should be taught (term 1) and how much time should be spent teaching it (1 week). The school-based assessment is monitored at a cluster level. Example of an autonomous school At schools that follow the educational reformer Rudolph Steiner, the Waldorf schools, the teacher has much more autonomy for deciding what he/she will teach (content), when to teach this topic and how much time is needed. As with all education systems, there are systems in place for accountability and support. At these schools, there is what is called a “College of Teachers”, made up of experienced teachers who guide less experienced teachers. If you can research education reformers such as Maria Montessori and Rudolph Steiner, you will see that many “reform” ideas have been incorporated into public education. ACTIVITY 1.1.2 CURRICULUM LEVELS, INSTITUTIONS AND PRODUCTS 1. Complete the table by identifying applicable examples of institutions at each level in the South African context. TABLE 1.1 Curriculum levels and curriculum products (Van den Akker (2003; 2006) in Thijs & Van den Akker 2009) Level Description SUPRA ● Common European Regional, Framework of international References for Languages MACRO System, national 6 Examples from Europe ● Core objectives, attainment levels Examples in the South African context Mathematics curriculum in process Level Description LEARNING UNIT 1 Examples from Europe Examples in the South African context ● Examination programmes MESO School, institute ● School programme ● Educational programme MICRO Classroom, teacher ● Teaching plan, instructional materials ● Module, course ● Textbooks NANO Pupil, individual ● Personal learning plan ● Individual course of learning 2. Choose one South African institution at the meso level and write 10 lines on their function, the institutions involved and the products delivered at this level. FEEDBACK ON ACTIVITY 1.1.2 1. For some of these blocks, you could answer from the top of your head, based on your experience as a teacher; for others, you would need to do a bit of research. It would be worthwhile doing that to start understanding why our education system differs from those of other countries – the reason being that our context is different, which should reflect in the curriculum. We cannot just take over another country’s curriculum as designed for their context and apply it in South Africa. 2. Keep in mind that we have public and private schools in South Africa and think about where the so-called “model C” schools fit. 1.3 SOME CURRICULUM MODELS Curriculum models try to explain and order processes and elements of the complex idea of the curriculum in a way that structures the concept in a comprehensible format. PDM4801 7 LEARNING UNIT 1 MATHEMATICS CURRICULUM IN PROCESS A curriculum model originates from the principles and criteria held by the designers of a curriculum. It is a theoretical framework, often in diagram form, which reflects their philosophical paradigm of knowledge acquisition and instruction. Furthermore, it reflects their approach to the role(s) of the teacher, the subject and the learner. See the following examples of various models, some currently in use, and some that have been in use in the previous century. 1.3.1 The product model In 1980, the Further Education Curriculum Review and Development Unit (FEU) in London made a broad distinction between the product and the process models (http://worldcat. org/identities/lccn-nb2007010602/). The category, product model, focuses mainly on the planning and the intentions of the curriculum that aims for some kind of desirable curriculum end-product resulting from the learning experience. They identify the following four products: ● The deficiency model begins with the assumption that learners have some kind of deficiency that has to be corrected, be it in literacy, numeracy, self-esteem, unrecognised special needs or some other area of their lives. ● The competency model focuses on the skills or the doing/acting part of learning and aims to develop these abilities. ● The information-based model is concerned with the acquisition of the knowledge underlying all learning – knowledge being the most important and prominent product of the curriculum. ● The socialisation model is concerned with integrating the learner into the social milieu within which learning may best take place, according to the approach to which the curriculum designers adhere. 1.3.2 The process model The emphasis of the process model is on activities and the effects of the curriculum through experiential learning in real life and exposure to this world. The quality of learning is prominent in the process models, for example: ● The reflective model is a process model where learners are taught how to view experiences and their learning critically and in detail, so that they can make links, discover relationships and see matters from different and alternative perspectives. The reflective model is not only concerned with knowledge acquisition, but also in part with how one feels about the learning. ● The counselling model is mainly concerned about the feeling aspect of learning. In this model, learners are taught to be in touch with how they feel and to understand those feelings. This model is especially applicable in cases where there are emotional barriers to learning, like anxiety. The curriculum would then design a helping relationship between the educator and the learner. 1.3.3 The subject-centred model The subject-centred model resembles the traditional curriculum where the focus is on a particular subject like mathematics or a discipline like languages. The learner is not in the centre of the design, but the instructional matter becomes the core of instruction. It demarcates what content should be taught and suggests how it should be learned. 8 Mathematics curriculum in process LEARNING UNIT 1 An example of a subject-centred curriculum is the so-called core curriculum, which is standardised across schools and provinces. Teachers are provided with a list of the content topics that they should teach, with examples of how they should teach the topics. A point of critique on this model is that it is not learner-centred and does not specifically take the learning styles and learning needs of learners into account, which may result in some learners falling behind and losing motivation. 1.3.4 The learner-centred model In the learner-centred curriculum, each learner’s individual needs and development goals matter. The point of departure is that learners are not the same, are at different points in their development and, therefore, have individual sets of needs. Learners are free to make choices within the curriculum because it is not a matter of one-size-fits-all. There is room for differentiation and learners have options as far as their activities, assignments and learning experiences are concerned. It is generally accepted that this approach motivates learners and keeps them engaged; however, from a teacher perspective, this is a labour-intensive curriculum and hardly attainable in overcrowded classrooms, such as is often the case in the South African context. It also requires highly skilled and experienced teachers to meet all learners’ needs at their developmental levels within the same class group. The individual learner’s choices and their needs should be maintained within a fine balance with the desired outcomes. 1.3.5 The problem-centred model The problem-centred curriculum is closely related to the learner-centred curriculum. The goal is to teach learners how to look at a real-life problem and come up with solutions, which prepares them for life and the problems that everyday living poses. This model makes the curriculum relevant and teaches them to be innovative and creative in solving problems; however, learners at lower levels of problem solving may be left behind, unless special attention is given to differentiation, as you will be required to do in Activity 1.2. ACTIVITY 1.2 A CURRICULUM FOR THEMBA You are an experienced retired teacher. Themba’s parents asked you to come up with an individual learning pathway for Themba’s future education until he can enter the labour market. Select a useful element from each of the mentioned curriculum models that may be worth considering when you do the design. For each element selected, motivate why you have chosen that element. Here is Themba’s story: Themba is 13 years old and he started struggling in the mainstream school system in Grade 4 when he was 10 years old. That was the time when he started developing epileptic seizures and he was given medication to keep the seizures under control. He takes one tablet at 6:00 and two at 18:00. The side-effects of this medication are that Themba becomes extremely tired about three hours after he has taken it and wants to fall asleep. His short-term memory is impaired and he cannot remember the detail of what he has been taught the previous day. The PDM4801 9 LEARNING UNIT 1 MATHEMATICS CURRICULUM IN PROCESS neurosurgeon identified brain damage and the prognosis is that Themba’s condition may slightly deteriorate to the extent that he will only be able to do routine tasks such as working in the production line of a factory. Themba hurts himself during seizures, he sometimes has temper outbursts and other children became afraid of him. He has repeated Grade 4 and was promoted to Grades 5 and 6 without having mastered all concepts at Grade 4 level. It became clear that he would not cope in the mainstream education system anymore. Themba’s parents decided to ask an experienced retired teacher to design a curriculum for him, by which he may acquire the basic life skills needed to function autonomously as an adult in society, given his medical condition and limitations. 1.4 THE CURRICULUM DEVELOPMENT PROCESS Before the curriculum development process begins, it is important to determine the purpose of education. Various education theorists have proposed educational purposes. Gert Biesta (2009), an educational philosopher, proposed three purposes: ● A qualification purpose: We might say that the purpose of teaching is so that the learners can obtain a school-leaving certificate, that is, a qualification. This purpose is important, as the qualification gives a signal to the employer, and the world out there, of the competence of the person. ● A socialisation purpose: Another purpose may be for the learners to socialise and learn the social mores and manners of their culture. In the Netherlands recently, it was found that learners knew very little about their Dutch culture; this aspect was brought into the curriculum. ● An individuation purpose: The purpose of individuation has to do with the development of individual talents and competencies and to ensure autonomy and independent thinking. The rationale behind the ordering of many aspects of education into a curriculum forms the basis of the curriculum development process. No singular or simple justification for a curriculum can be provided and we need to discuss curriculum from multiple perspectives. Figure 1.2 shows the interconnection of all the different parts of the curriculum that impact on the learner and the learning situation. 10 Mathematics curriculum in process LEARNING UNIT 1 Figure 1.2: The curricular spiderweb (Van den Akker (2003) in Thijs & van den Akker (2009:11)) The spiderweb connects the parts of the curriculum, which in turn raises questions to be answered while justifying the decisions made in developing the curriculum. ● What are the aims and objectives of the curriculum? ● Which learning content should be included? ● What learning activities will best enable the learning of concepts? ● What is the role of the teacher? Is the teacher the imparter of knowledge or the facilitator of reasoning, or sometimes one and sometimes the other? ● What materials and resources should be available? Do teachers choose these or are they provided by the DoE, who commissions the writing of the textbooks? ● Which pedagogy does the teacher follow that he or she thinks will support learning? Here a decision may need to be made regarding grouping and the working together of learners, in groups, or is all learning to occur individually? ● How can the available instructional time be managed? Is the time stipulated from outside the school or is there local control of time in each school? ● What is the location of learning? Does all learning happen inside the classroom or are excursions to the libraries, museums and parks encouraged? ● How, when and on which content will assessment take place? In Activity 1.3, you will start thinking about the rationale for a mathematics curriculum fit for learners in the same situation as Themba. Let us call this curriculum, the Themba Mathematics Curriculum (TMC). You will need to take some curriculum decisions and justify those decisions. It would be easier to answer these questions for a specific content area or topic even. The overarching curriculum decisions are, however, at another level: let them apply across the five content areas that learners will complete before they enter the labour market. Do not force content or be bound by phases or grades as we know them – rather focus on mathematics skills and knowledge that you think would or judge to be beneficial for this special curriculum. ACTIVITY 1.3 USING THE SPIDERWEB TO REASON ABOUT CURRICULUM 1. After each question, write down your answer, that is your own curriculum decision that you think will contribute best to the TMC PDM4801 Question Your decision What are the aims and objectives of the TMC? Aim: Objective 1: ____________________ 11 LEARNING UNIT 1 MATHEMATICS CURRICULUM IN PROCESS Question Your decision Objective 2: ___________________ Objective 3: ___________________ What learning content would you include? (Or how would you decide what content is in and what is out?) What type of learner activities would you include or advise teachers to use in this curriculum? What would you advise TMC teachers about their role in learners’ mathematics learning? What are the very basic materials and resources that you would advise in this curriculum? How would you incorporate learner grouping in the TMC? What will be the instructional time(s) that you would include in the curriculum? 12 Mathematics curriculum in process Question LEARNING UNIT 1 Your decision What is/are the location(s) where instruction (i.e. teaching and learning) should take place? What would be your approach to assessment in this curriculum? 2. Consider the parts of the spiderweb and write eight lines on how the content and the learning activities are related in the TMC. FEEDBACK ON ACTIVITY 1.3 Through this activity, we became aware that curriculum decisions need thorough consideration. We had to think deeply about our approach towards instruction before we could write an appropriate response. When we read the CAPS document again, we should be aware that there are educational paradigms, theories, epistemologies and ontologies underpinning curriculum decisions. They do not exist outside of a specific context and in all curricula, situational, political, historical and societal influences have their part to play in shaping the curriculum. We are now ready to discuss some approaches to curriculum and curriculum design that give direction to the decision makers and creators of curricula. 1.5 APPROACHES TO CURRICULUM DESIGN The models of curriculum describe different conceptualisations of the curriculum. This section discusses the following four quite distinct approaches to curriculum design: ● The instrumental approach, where the focus is on a systematic design process ● The communicative approach, where the focus is on engaging the many role players, building relationships and encouraging broad participation ● The artistic approach, where the design of the curriculum is akin to creating an artwork – the curriculum is seen as the product of a creative designer ● The pragmatic approach that focusses on practical usability PDM4801 13 LEARNING UNIT 1 MATHEMATICS CURRICULUM IN PROCESS Approaches to the design of a curriculum have much in common; however, they may differ in the following three areas: ● Sequence of the activities ● Characterisation of the activities ● The view of what is a good curriculum? The following excerpts describe to some extent the various approaches to curriculum design and discuss the implications of the approach for curriculum design and development. Each of the approaches has a leading educational theorist, or theorists, who have promoted the particular approach. These excerpts were taken from the Curriculum-in-development by Thijs and Van den Akker (2009:16–18). ● The instrumental approach, which has been promoted mainly by Ralph Tyler (1949). ● The communicative approach, which was advocated by Decker Walker (1990). ● The artistic approach, which is endorsed by Elliot Eisner (1979). ● The pragmatic approach is associated with design research, which is a modern approach to curriculum design, originating from the institute for curriculum development in the Netherlands, the SLO. 1.5.1 The instrumental approach Curriculum-in-development (Thijs & Van den Akker 2009:16) 14 Mathematics curriculum in process LEARNING UNIT 1 1.5.2 The communicative approach Curriculum-in-development (Thijs & Van den Akker 2009:16–17) 1.5.3 The artistic approach Curriculum-in-development (Thijs & Van den Akker 2009:17) PDM4801 15 LEARNING UNIT 1 MATHEMATICS CURRICULUM IN PROCESS 1.5.4 The pragmatic approach Curriculum-in-development (Thijs & Van den Akker 2009:18) In Activity 1.4, you will be comparing these four approaches to curriculum development in terms of sequencing and characterisation of activities. You will also decide what makes for a “good” approach. ACTIVITY 1.4 COMPARING APPROACHES TO CURRICULUM DESIGN 1. Complete the table. Draw your answers from the given excerpts. Instrumental approach Communicative approach Artistic approach Pragmatic approach Sequence of activities Characterisation of activities A good curriculum 2. Suppose that you were given the responsibility of redesigning the mathematics curriculum for the primary school from Grades 1 to 7. Which 16 Mathematics curriculum in process LEARNING UNIT 1 approach would you follow? Describe three aspects of the approach that are meaningful to you and explain why you find this approach meaningful. a. b. c. 3. A colleague argues that the pragmatic approach is the only approach that is feasible for a country with a limited budget. Make an argument for why elements of the other three approaches could also be incorporated and give examples of the elements to which you refer. FEEDBACK ON ACTIVITY 1.4 To check whether you were on the right track in responding to this activity, find the useful summary table of the four approaches (Thijs & Van den Akker 2009:16). Please ensure that you understand and can elaborate the cryptic descriptors under each column heading, indicating the various approaches: 1.6 THE INTENDED, IMPLEMENTED AND ATTAINED CURRICULUM The conceptualisation of curriculum phases, or strands, as given in the heading of this paragraph, was first developed by Travers and Westbury (1989, in Martin & Kelly 1996:1–4) for the IEA’s Second International Mathematics and Science Study (1995), a forerunner of what we now know as TIMSS. ● The intended curriculum refers to “what society would like to see taught”. ● The implemented refers to “what is taught in the classroom”. ● The attained curriculum refers to “what the students learn”. PDM4801 17 LEARNING UNIT 1 MATHEMATICS CURRICULUM IN PROCESS The IEA studies are based on this conceptualisation of the curriculum. This idea was elaborated by Schmidt et al (1997), who constructed a model against which to evaluate education systems around the world. Figure 1.1 contains this model (Schmidt et al 1997:188) and in the duplication of figure 1.1, we refer to the model, with an explanation of its elements alongside a reduced image. The phases of the curriculum are arranged horizontally: Column 1: the intended curriculum – what are students expected to learn? Column 2: the implemented curriculum – who provides the instruction? Column 3: an expansion of the implemented curriculum – how is instruction organised? Column 4: the attained curriculum – what have students learned? Down the vertical are the levels at which the curriculum is organised, that is system level, school or institution level, classroom level, and individual level. Duplicate figure 1.1: TIMSS Model of Potential Educational Experience (Schmidt et al 1997:188) These ideas are elaborated in the next excerpt. Martin, MO. 1996. Third International Mathematics and Science Study: An Overview in MO Martin and DL Kelly (eds.), Third International Mathematics and Science Study (TIMSS) Technical Report, Volume I: Design and Development. Chestnut Hill, MA: Boston College. Available online at https://timss.bc.edu/timss1995i/TIMSSPDF/TRall.pdf#page=101 1.4 THE CONCEPTUAL FRAMEWORK FOR TIMSS IEA studies have as a central aim the measurement of student achievement in school subjects, with a view to learning more about the nature and extent of student achievement and the context in which it occurs. The ultimate goal is to isolate the factors directly relating to student learning that can be manipulated through policy changes in, for example, curricular emphasis, allocation of resources, or instructional practices. 18 Mathematics curriculum in process LEARNING UNIT 1 Clearly, an adequate understanding of the influences on student learning can come only from careful study of the nature of student achievement, and the characteristics of the learners themselves, the curriculum they follow, the teaching methods of their teachers, and the resources in their classrooms and their schools. Such school and classroom features are of course embedded in the community and the educational system, which in turn are aspects of society in general. The designers of TIMSS chose to focus on curriculum as a broad explanatory factor underlying student achievement (Robitaille and Garden, 1996). From that perspective, curriculum was considered to have three manifestations: what society would like to see taught (the intended curriculum), what is actually taught in the classroom (the implemented curriculum), and what the students learn (the attained curriculum). This conceptualization was first developed for the IEA’s Second International Mathematics Study (Travers and Westbury, 1989). The three aspects of the curriculum bring together three major influences on student achievement. The intended curriculum states society’s goals for teaching and learning. These expectations reflect the ideals and traditions of the greater society, and are constrained by the resources of the educational system. The implemented curriculum is what is taught in the classroom. Although presumably inspired by the intended curriculum, the actual classroom events are usually determined in large part by the classroom teacher, whose behavior may be greatly influenced by his or her own education, training, and experience, by the nature and organizational structure of the school, by interaction with teaching colleagues, and by the composition of the student body. The attained curriculum is what the students actually learn. Student achievement depends partly on the implemented curriculum and its social and educational context, and to a large extent on the characteristics of individual students, including ability, attitude, interests, and effort. While the three-strand model of curriculum draws attention to three different aspects of the teaching and learning enterprise, it does have a unifying theme: the provision of educational opportunities to students. The curriculum, both as intended and as implemented, provides and delimits learning opportunities for students– a necessary though not sufficient condition for student learning. Considering the curriculum in all its aspects as a channel through which learning opportunities are offered to students leads to a number of general questions that can be used to organize inquiry about that process. In TIMSS, four general research questions helped to guide the development of the study: ● What are students expected to learn? ● Who provides the instruction? ● How is instruction organized? ● What have students learned? PDM4801 19 LEARNING UNIT 1 MATHEMATICS CURRICULUM IN PROCESS The first of these questions concerns the intended curriculum, and is addressed in TIMSS by an extensive comparative analysis of curricular documents and textbooks from each participating country. The second and third questions address major aspects of the implemented curriculum: what are the characteristics of the teaching force in each country (education, experience, attitudes and opinions), and how do teachers go about instructing their students (what teaching approaches do they use, and what curricular areas do they emphasize)? The final question deals with the attained curriculum: what have students learned, how does student achievement vary from country to country, and what factors are associated with student learning? The study of the intended curriculum was a major part of the initial phase of the project. The TIMSS curriculum analysis consisted of an ambitious content analysis of curriculum guides, textbooks, and questionnaires completed by curriculum experts and educationalists. Its aim was a detailed rendering of the curricular intentions of the participating countries. Data for the study of the implemented curriculum were collected as part of a largescale international survey of student achievement. Questionnaires completed by the mathematics and science teachers of the students in the survey, and by the principals of their schools, provided information about the topics in mathematics and science that were taught, the instructional methods adopted in the classroom, the organizational structures that supported teaching, and the factors that were seen to facilitate or inhibit teaching and learning. The student achievement survey provides data for the study of the attained curriculum. The wide-ranging mathematics and science tests that were administered to nationally representative samples of students at three levels of the educational system provide not only a sound basis for international comparisons of student achievement, but a rich resource for the study of the attained curriculum in each country. Information about students’ characteristics, and about their attitudes, beliefs, and experiences, comes from a questionnaire completed by each participating student. This information will help to identify the student characteristics associated with learning and provide a context for the study of the attained curriculum. The interaction of the intended, implemented and attained curriculum is necessarily complex. Depending on the approach that an education department or a curriculum developer takes on the development of a mathematics curriculum, one or other of these three phases could take on greater importance. It seems logical that the starting point is the intended curriculum, which then gets taken up in the implemented curriculum. The outcome is then the programme of assessment, which shows what has been attained. However, there may be a reverse of this direction. What is assessed might inform what is taught and what is taught may be a relatively small component of the intended curriculum. Jennings and Bearak (2014) studied the standards that form part of the state curriculum in three states of the USA, namely New York, Texas and Massachusetts, each presenting a somewhat different relationship between the intended, the implemented and the attained curriculum. 20 Mathematics curriculum in process LEARNING UNIT 1 Note the differences between the three states in the following article. Jennings, J & Bearak, J. 2014. Teaching to the Test in the NCLB Era: How test predictability affects our understanding of student performance. Educational Researcher 43(8): 381–389. The construct “teaching to the test” takes on different forms; for example, to limit the implemented curriculum to what is being tested, thereby influencing the attained curriculum. Another form is to adapt the school timetable to focus more on the subject or topics that are tested in the final examination, thereby giving undue emphasis to some parts of the intended curriculum, and less attention to others. This phenomenon is known as narrowing the curriculum. ACTIVITY 1.5 INTENDED, IMPLEMENTED AND ATTAINED CURRICULUM 1. Draw from Schmidt et al (1997) and Martin (1996) to explain each of these terms: intended, implemented and attained curriculum. 2. It is often said that assessment drives teaching and learning. If an aspect of the curriculum is not assessed, it will not be part of the implemented curriculum. The fact that the topic is in the intended curriculum has little authority. a) Argue for the notion that the attained (assessed or examined) curriculum is the aspect of most importance. Draw on Jennings and Bearak (2014) to support your argument. b) Present a counter argument that warns against this tendency of “teaching to the test” and make a case for keeping a broad and relevant curriculum rather than narrowing the curriculum. a) b) PDM4801 21 LEARNING UNIT 1 MATHEMATICS CURRICULUM IN PROCESS FEEDBACK ON ACTIVITY 1.5 In Activity 1.4 you have made your own voice heard as an informed academic scholar, who also knows the realities of teaching in South Africa. Despite all the challenges facing instruction and educators in this country, we trust and know that you have established through these new insights, your future approach towards “teaching to the test”. 1.7 CONCLUSION In learning unit 1, we have done the following: ● Described and reflected on the process of curriculation. ● Discussed the roles of various institutions in designing curricula. ● Unpacked some models of curriculum design. ● Reflected on the intended, implemented and attained curricula. Stemming from these discussions, some issues came to the fore, which will be discussed in the subsequent learning units. 1.8 ADDITIONAL LEARNING OPPORTUNITIES You might want to explore further and deeper the ideas that inform the design of the curriculum, see some of the following references. You might also want to explore some of the reform curricula such as Rudolf Steiner and Maria Montessori. Read the literature associated with TIMSS 2019. Check to what extent the same terms, intended, implemented and attained curriculum are used. Do they have the same meaning? Visit https://timssandpirls.bc.edu/ for more information. A modern curriculum reform has been Scotland’s Curriculum for Excellence (CfE), over the past decade (Priestley, Biesta & Robinson 2013). This reform has focused on school-based curriculum development, which has been democratic. Check yourself: Have you reached the goal and the specific Tick the box outcomes of learning unit 1? Did I develop a sound understanding of the role players of curriculum implementing? 22 1 Can I explain the concept curriculum from various perspectives? 2 Can I identify various educational approaches to the curriculum? 3 Can I describe and reflect on the process of curriculum design and development? 4 Can I critically discuss the roles of various institutions in designing curricula? Mathematics curriculum in process LEARNING UNIT 1 Check yourself: Have you reached the goal and the specific Tick the box outcomes of learning unit 1? 5 Can I unpack a variety of models of curriculum design? 6 Can I critically reflect on the intended, implemented and attained curriculum? References Akker, J. van den (2003). Curriculum perspectives: An introduction. In J. van den Akker, W. Kuiper & U. Hameyer (Eds.), Curriculum landscapes and trends (pp. 1–10). Dordrecht: Kluwer Academic Publishers. Akker, J. van den (2006). Curriculum development reinvented. In J. Letschert (Ed.), Curriculum development re-invented. Proceedings of the invitational conference on the occasion of 30 years SLO 1975–2005 (pp. 16–29). Enschede: SLO. Biesta, G. 2009. Good Education: What it is and why we need it. Inaugural lecture. Stirling, Scotland: The Stirling Institute of Education. Jennings, J & Bearak, J. 2014. Teaching to the Test in the NCLB Era: How test predictability affects our understanding of student performance. Educational Researcher. 43(8):381–389. Martin, MO. 1996. “Third International Mathematics and Science Study: An Overview” in MO Martin and DL Kelly (eds.), Third International Mathematics and Science Study (TIMSS) Technical Report, Volume I: Design and Development. Chestnut Hill, MA: Boston College. Available at https://timss.bc.edu/timss1995i/TIMSSPDF/TRall. pdf#page=101 (accessed on 27 April 2020). Priestley, M, Biesta, GJJ & Robinson, S. 2013. Teachers as agents of change: teacher agency and emerging models of curriculum, in M Priestley & GJJ Biesta (Eds.), Reinventing the curriculum: new trends in curriculum policy and practice, London: Bloomsbury. Schmidt, W, McKnight, H, Valverde, C, Houang, GA, Wiley, RT and David E. (1997). Many Visions, Many Aims: A cross-national investigation of curricular intentions in school mathematics. Dordrecht: Kluwer Academic Publishers Thijs, A & Van den Akker, J. 2009. Curriculum-in-Development. Enschede: SLO. Travers, KJ & Westbury, I. 1989. International studies in educational achievement, Vol. 1. The IEA study of mathematics I: Analysis of mathematics curricula. Pergamon Press. PDM4801 23 LEARNING UNIT 2 SELECTED MATHEMATICS CURRICULA ACROSS THE GLOBE LEARNINGUNIT2 Table of contents INTRODUCTION 2.1 DIFFERENT COUNTRIES: DIFFERENT STRUCTURES Activity 2.1: Comparing curriculum functions at different levels 2.2 DIFFERENT COUNTRIES: DIFFERENT BALANCE OF SUPPORT AND AUTONOMY Activity 2.2: Centralised and decentralised control 2.3 DIFFERENT COUNTRIES: DIFFERENT PHILOSOPHICAL UNDERPINNINGS, CONTENT AND PEDAGOGY DIFFERENT COUNTRIES: DIFFERENT MATHEMATICS CURRICULA Activity 2.4: Constructivist, Platonist, Formalist 2.4 2.5 DIFFERENT CULTURES, DIFFERENT LEARNERS: SAME CURRICULUM? Activity 2.5: Different cultures, different learners 2.6 2.7 CONCLUSION ADDITIONAL LEARNING OPPORTUNITIES REFERENCES OUTCOMES OF THIS LEARNING UNIT In this learning unit, you will compare and critically review the mathematics curricula in selected countries. At the end of learning unit 2, you should be able to do the following: ● Describe some similarities and differences between mathematics curricula in four countries, notably in terms of their curriculum management structures. ● Compare various national education ministries at the macro and meso levels in terms of centralisation or decentralisation of functions, the degree of autonomy of schools and the support they provide for education. ● Critique mathematics curricula by identifying and comparing the philosophical underpinnings, context and content. ● Discuss the alignment of curricula with cultural demands and alignment with learners’ profile. 24 SELECTED MATHEMATICS CURRICULA ACROSS THE GLOBE LEARNING UNIT 2 INTRODUCTION The many factors that impact on curriculum development affect countries differently. In learning unit 2, we present four country case studies, namely Singapore, the Netherlands, New Zealand and Kenya, providing some aspects of curriculum development and design in each of the countries that alert one to some of the important considerations when designing and implementing curricula. These countries represent an Eastern, European, African and island take on mathematics instruction. Figure 2.1: World map Source: Online Pictures: Creative Commons Almost all education ministries can be described with reference to the levels described in learning unit 1, namely the supra, macro, meso, micro and nano levels. However, how institutions are set up to carry out the functions at these levels differ from country to country. These ideas will be taken up in more detail in paragraph 2.1. A key difference between curricula in various countries will be whether the curriculum is centralised or decentralised and to what extent. The degree of centralisation or devolution to institutions depends on the levels of expertise in the schools and the confidence that those in authority have in the principals and heads of department to manage curricular functions. A related function is the support provided by the government and the accountability mechanisms that are in place. We will focus on these themes in paragraph 2.2. The curricula across the globe are mostly not explicit about the philosophy of mathematics underpinning the aims and outcomes of their individual curricula. The philosophical stance that the curriculum designers hold can at most be inferred from the way the curriculum is applied. In this learning unit, we will note three broad philosophical positions that can be applied to school curriculum: ● The Platonists, building on the ideas of Plato, hold a view of mathematics as existing distinctly apart from human beings’ conception of mathematics. There is a world of mathematics out there waiting to be discovered and one of their arguments to support this view is that there have been occurrences throughout the history of two mathematicians who had no connection with each other, making the same discovery at the same time. PDM4801 25 LEARNING UNIT 2 SELECTED MATHEMATICS CURRICULA ACROSS THE GLOBE ● The formalists understand mathematics as a formal system based on axioms and theorems, and as long as one adheres strictly to the rules, then what is being done mathematically is acceptable. In a mathematics class, one might have students working away at calculations and achieving the correct answer but may have little understanding of the meaning of the mathematics. The formalists would not see a problem here. ● The constructivists (formerly called intuitionists) challenge the field by insisting that mathematics must have a concrete material counterpart that is then represented by symbols. In a classroom, this group may insist that everything the student does has a connection with a concrete object. Elements of these branches of mathematical philosophy may be seen in various content and contexts found in mathematics curricula, as noted in the given examples. The ethno-mathematics movement reminded the mathematics community that mathematical ideas cannot be separated from the cultures within which they have been developed. Every culture across the globe has developed the ideas they need for progress and their children will come into contact with various mathematical concepts at various times. This topic will be discussed in relation to the curricula of the four selected countries (Singapore, the Netherlands, New Zealand and Kenya). 2.1 DIFFERENT COUNTRIES: DIFFERENT STRUCTURES The countries chosen for this learning unit represent an array of different structures at the macro level. Singapore is an interesting country as it has undergone some radical changes over the past 60 years and is now regarded as one of the more successful countries in terms of mathematics education. The Netherlands can be regarded as having an economy that has been relatively stable for several years and has a mathematics curriculum that is also regarded as strongly based on pedagogical theory. New Zealand has kept abreast of developments in mathematics education over the past 50 years and Kenya has recently undergone a curriculum reform. Some aspects of the relationship between the macro level, that is the national level, and the meso level, that of the institution or school, are highlighted in each of the curricula. In learning unit 1, we read about the levels that describe curriculum development and implementation, that is the supra (international), macro (national, government ministries), meso (school and institutions), micro (the classroom, the teacher), and the nano level (the learner). This section focuses on some aspects of the curricula of four countries that could be described according to these levels. Read the following four passages and identify the levels (see Thijs and Van den Akker (2009) and learning unit 1 of this module) and complete Activity 2.1.1. 26 SELECTED MATHEMATICS CURRICULA ACROSS THE GLOBE LEARNING UNIT 2 Singapore Singapore is a city-state at the southern end of the Malay Peninsula in Asia. 5.4 million people live on the island, of whom + 76% are Chinese. They speak Malay, Mandarin and Tamil, as well as English, which is taught as the first language at school and is the official language of the government. They are taught in English and their mother tongue. They go through six years compulsory primary school and complete an exit examination, which determines which secondary group they enter for the next four to five years: Special, Express, Normal (Academic) or Normal (Technical). Having completed the ordinary (O) level, they can do a diploma or certificate that raises them to A-level, which is tertiary. Figure 2.2: Map of Singapore Source: Online Pictures: Creative Commons The Singapore Ministry of Education’s curriculum aims to nurture its people, to develop a passion for learning and develop the capabilities to fulfil the potential and strengths for the good of the self, the society and the country. From the 1960s to the 1980s, the education system focussed on improving literacy and numeracy rates. The programme was driven from the Ministry of Education Head Quarters (MOE HQ). In 1997, a programme entitled “Thinking Schools, Learning Nation” (TSLN) was implemented. The implementation of TSLN marked a turning point in the focus of education to an education system characterised by flexibility, diversity and greater school autonomy. This intervention was created through changes at the levels of policy making by the educational governance, implementation across the education landscape, i. e. the schools, as well as the educational experience of learners themselves. The education system can be described as having a horizontal structure, in that there is a close relationship between the MOE HQ and the schools, with no intermediary layer. An interesting strategy is that there is the rotation of education officers and school leaders from the MOE HQ to the schools and back to MOE HQ, leading to a close alignment of policy and practice and encouraging a sense of common mission. The education system is centralised in some respects and decentralised in others. The MOE HQ is responsible for deciding on the national policies that relate to access to schooling for children, the criteria for admission, the funding rates for schools, and the school fees. They are also responsible for recruiting teachers, organising the training at the National Institute of Education (NIE) and deploying teachers to schools. The equitable distribution of resources to all schools is also their responsibility. The individual school is responsible for administrative matters and professional decisions, such as pedagogical approaches and special learning needs. The individual school also decides on the teacher’s function within the school; for example, which grade to teach and what school assignments to give in the respective grades. PDM4801 27 LEARNING UNIT 2 SELECTED MATHEMATICS CURRICULA ACROSS THE GLOBE Although there is an individual responsibility, the MOE HQ is available to assist wherever needed and as we said previously, there is a rotation of leader teachers, and ministry officials, meaning that the two levels of curriculum function intersect. In addition, there is a central oversight of the schools, which takes the form of a “holistic school evaluation and improvement” instrument for school self-evaluation. This instrument forms part of the School Excellence Model, which was developed in consultation with schools and key stakeholders. This internal evaluation is supported by an external validation every six years. This dual approach evaluation structure is not the only means of support for the schools: there is also a cluster system of the 12– 14 schools within geographical proximity, that forms a professional developmental network. The perspective on the curriculum in this section may be described as a sociopolitical perspective in that the government, representing society, has interacted to create this curriculum structure. We might also describe a view of this section as being from a technical-professional perspective, as the means of implementing the intended curriculum has been in focus. Source: http://timssandpirls.bc.edu/timss2015/encyclopedia/countries/singapore/ The Netherlands Netherland is a European country with territory in the Caribbean, together forming the United Kingdom of Netherland. The official language and language of tuition is Dutch, and English is a mandatory subject in secondary schools. There are several regional languages that enjoy official status, such as Limburgish, West Frisian and dialects of Dutch like Twents and Drents. Eight years of Primary school (5– 12 years old) and four years of Secondary school (13– 16 years old) are compulsory, after which a trade, a technical degree or an academic degree can be studied. The Dutch values are egalitarian and modern, and they have an aversion for anything that is non-essential. They put a high value on culture, art, architecture, literature and philosophy. Figure 2.3: Map of the Netherlands Source: Online Pictures: Creative Commons By contrast to other countries, Dutch schools have considerable autonomy. Their constitution includes a bill that entitles every individual to freedom of education. Both public and private schools have the autonomy to decide when and how to teach the core objectives of the Dutch curriculum based on their religious, philosophical or pedagogical views and principles. The Minister of Education, Culture and Science, at the national level, is responsible for the structure of the school system, school funding, school inspections, quality of the national examination and student support. 28 SELECTED MATHEMATICS CURRICULA ACROSS THE GLOBE LEARNING UNIT 2 Administration and management of schools are devolved to the individual school or the school board. The school boards of public schools usually have members from the municipality on the board. These local authorities are responsible for the implementation of the curriculum, the employment of teachers, administration of student matters and the administration of finances. Two thirds of schools in the Netherlands are privately run. These comprise Roman Catholic and Protestant schools; there are also schools designed on the principles and ideas of education reformers, such as Maria Montessori (Montessori schools), and Rudolf Steiner (Waldorf schools). All of the schools, both private and public are funded by the Ministry of Education. While the schools are essentially autonomous, they are also accountable. Every four years, Dutch inspectors visit the schools to ascertain whether they are performing to the standard expected by the Ministry. The schools that are struggling or not conforming to the core objectives of the Dutch curriculum will be visited more frequently than those who have good educational structures in place. Findings from these visits are reported to the school. Source: http://timssandpirls.bc.edu/timss2015/encyclopedia/countries/netherlands/ New Zealand New Zealand is a sovereign island country in the south-western Pacific Ocean with two main land masses, the North Island and the South Island. Internationally, the country ranks high, amongst others, in education and economic freedom. The Queen of England is their monarch. Apart from the national legislator (Parliament) and the executive (Cabinet), the country is organised in 11 regional councils and 67 territorial authorities for local government purposes. Primary and secondary school is compulsory from 6– 16 years. There are 13 school years and attending state/public schools is free. The country enjoys an OECD ranking of the seventh best education system in the world, with student performing exceptionally well in languages, mathematics and science. Over half of the population younger than 30 years old hold a tertiary qualification. In addition to private tertiary institutions, there are five state-owned types of tertiary institutions. Figure 2.4: Map of New Zealand Source: Online Pictures: Creative Commons New Zealand has a decentralised system with relative autonomy ceded to the school. There are four organisational bodies responsible for oversight of the schools. The Ministry of Education (MoE) has the overall authority to provide policy information to the New Zealand government to design and develop the national curriculum and to set out the operating guidelines for educational institutions. The MoE is also responsible for allocating funds and resources to educational institutions. PDM4801 29 LEARNING UNIT 2 SELECTED MATHEMATICS CURRICULA ACROSS THE GLOBE In an oversight role, the MoE collects, processes and reports on the educational statistics that arise from research studies, international large-scale studies and other surveys. They are also responsible for professional development programmes. The Evaluation Review Office (ERO) is responsible for evaluating the quality of educational programmes and compliance with regulatory provision and to monitor the effectiveness of the curriculum in promoting student learning. The New Zealand Qualification Authority (NZQA) is responsible for overseeing the coordination of the National Certificate of Educational Achievement (NCEA) The Educational Council of Aotearoa New Zealand (ECAZ) is the professional and regulating organisation for registering teachers. It seems that while there is nominal autonomy given to the schools there is centralised control of the curriculum, of evaluations, of teacher training, and of examinations. There is though in New Zealand no streaming of students into academic and vocational streams along their educational journey. Source: http://timssandpirls.bc.edu/timss2015/encyclopedia/countries/newZealand/ Kenya *A later paragraph will give some information on the Kenyan society and culture. The new Kenya Institute of Curriculum Development (KICD) aims to develop engaged, empowered and ethical citizens and to nurture every learner’s potential. The KICD, established in 2013, was the successor of the Kenya Institute of Education (KIE) (1976). The strategies set up are designed to be responsive to the needs of society, promote equity and access, and achieve the National goals of the country and society. The KICD was put in place to address needs observed in the country; for example, the government observed that there was a value and a behavioural crisis, especially among the youth. Another problem identified that the youth on leaving school were not equipped for finding jobs in the 21st century. The KCID was therefore tasked with preparing students for the current economies and equipping them with 21stcentury skills. The Basic Education Curriculum Framework Supporting the two primary objectives of developing engaged, empowered and ethical citizens and of nurturing every learner’s potential, are the following framework topics: Digital Literacy, Imagination and Creativity, Critical Thinking and Problem Solving, Learning to Learn, Citizenship and Self-efficacy and Communication and Collaboration. To address the need for values and general behaviour, the following values have been identified as critical for the society: responsibility, respect, excellence, care and compassion, understanding and tolerance, honesty and trustworthiness, trust and being ethical. In addition, there is a need for learners to be aware of the needs of the community and to contribute to the world. Another key aspect of the curriculum framework is to inculcate in learners an awareness of the environment: the core ideas to be promoted are that as inhabitants of the earth, we are the stewards, and therefore should be aware of the impact human beings are having on the earth. 30 SELECTED MATHEMATICS CURRICULA ACROSS THE GLOBE LEARNING UNIT 2 It is important to build relationships: the qualities of humility and open-mindedness are essential to engage with others; the skills of working in a team and communicating openly and honestly; and demonstrating respect, empathy and compassion for all people, all contribute to the building of society. Kenya has a diverse population and for the country as a whole to flourish, it is critical to value diversity. With these high ideals and determination to design a modern curriculum, many aspects of curriculum design have been addressed; for example, Instructional Design Theory based on principles of organising learning for deep conceptual understanding and focusing on learner-centred teaching rather than teachercentred. This curriculum also looks to constructivist theories based on the great thinkers of the past half-century, notably Piaget’s development theory, Vygotsky’s social constructivism, Dewey’s social constructivism, Garden’s multiple intelligences and Bruner’s cognitive development. While this focus on a modern curriculum is admirable, the critique by researchers is that a curriculum on its own without the supporting structures of smaller classrooms, the training of teachers to implement such a curriculum, and the necessary support structures in place. The monitoring, evaluation and reporting committee (MER) The implementation of this new curriculum, known as a competency-based curriculum (CBC), implemented in January 2019, was accompanied by close monitoring of the process by the MER. There is planned systematic tracking of activities and actions to assess progress. Critique of the system The main area of critique is the issue of teacher preparedness. Teachers who do not understand the change of theoretical underpinnings of a new curriculum may not be able to embrace a new pedagogy. A related critique is that the teaching materials are not available or adequate to address the new approach. “Many educators including those from Kenya, are now rejecting the externally driven approach to education reform. They propose instead an interactive and participatory approach which involves – and begins with- an evaluation by classroom teachers and district education personnel. This ensures that the views of the people closest to the process of teaching and learning are involved” (The Elephant Info 2019). Sources: https://kicd.ac.ke/curriculum-reform/basic-education-curriculum-framework/ https://www.aku.edu/news/Pages/News_Details.aspx?nid=NEWS-001985 https://www.theelephant.info/features/2019/10/17/will-the-new-competency-based-curriculumlead-to-declining-educational-standards-in-kenya/?print=pdf PDM4801 31 LEARNING UNIT 2 SELECTED MATHEMATICS CURRICULA ACROSS THE GLOBE ACTIVITY 2.1 COMPARING CURRICULUM STRUCTURES AT DIFFERENT LEVELS Complete the next table for the four countries under discussion, answering the five questions in the first column. Singapore 2.2 a. In what ways is the control in this curriculum centralised? b. At which levels is the institution autonomous (decentralised)? c. Describe the functions at the macro level. d. Describe the functions at the meso level. e. What information can you infer on the nano level? Netherlands New Zealand Kenya DIFFERENT COUNTRIES: DIFFERENT BALANCE OF SUPPORT AND AUTONOMY Degrees of centralisation and decentralisation at the national (macro) level and institutional (meso) level differ in the following ways: ● In Singapore, the curriculum from the 1960s to 1980s was focused and directed, as there was a need to improve literacy and numeracy across the country. Recently, however, the schools have been given greater autonomy in some respects. ● The Netherlands’ education system is differently structured. Their constitution allows for individuals to set up schools according to their philosophies, religion, or pedagogical views and principles; nevertheless, the core objectives of the Dutch curriculum are implemented by all schools. ● The New Zealand Ministry of Education grants the schools relative autonomy; however, there is a system of oversight to ensure quality education. ● Kenya has recently implemented curriculum reform. This reform has some interesting elements, in particular the determination to foreground the development of the potential of each learner and to make education central to the country’s economic growth. According to reports by the Kenyan Institute of Curriculum Development (KICD) and other observers, there are difficulties in the implementation phase of the new competency-based curriculum (CBC). Read the excerpts from each country and complete Activity 2.2. 32 SELECTED MATHEMATICS CURRICULA ACROSS THE GLOBE LEARNING UNIT 2 ACTIVITY 2.2 CENTRALISED AND DECENTRALISED CONTROL Read the short excerpts about the four countries in paragraph 2.1 and answer the questions. 1. In what ways might this curriculum be described as having (a) centralised control and in what ways is there evidence of the various levels being autonomous, and therefore described as (b) decentralised. Singapore Netherlands New Zealand Kenya (a) (b) 2. The relationship control-and-support is complex. Centralised control by ministries at the macro level is as necessary as providing support for schools at the meso level, for teachers in the classroom at the micro level, and for learners at the nano level. Discuss the interrelationship of control and support in two of the countries described in this unit. FEEDBACK ON ACTIVITY 2.2 Although we can certainly not go into every factor that may influence a country’s educational system, we cannot discount the political dispensation of countries in terms of their position on the democratic spectrum. There should be some correlation between the measure of control of systems, activities and citizens in general, and the political sentiments. Further research on curriculum development in one other country that you find interesting could be done. 2.3 DIFFERENT COUNTRIES: DIFFERENT PHILOSOPHICAL UNDERPINNINGS, CONTENT AND PEDAGOGY From the beginning of time, questions about what is true and reliable have been at the forefront of people’s thinking. What about life is dependable? These questions also applied to the development of counting and a number system, which was meant to support the activities of farmers and traders. The question was: how were people to believe that the numbers shown on a piece of papyrus was indeed a reflection of the PDM4801 33 LEARNING UNIT 2 SELECTED MATHEMATICS CURRICULA ACROSS THE GLOBE real happening that took place? For example, if the farmer exchanged five sheep for ten goats, do the symbols on the paper truthfully reflect this transaction? People thought philosophically about the questions of truth and reliability. Where is the truth? Is it in the mind of people and in what they perceive to be true? Is it in the external world and in what exists, independent of what people think? Or is it there where the mind meets the object and makes meaning from what is in the real world? Constructivism The close link from concrete and material reality to the mathematical symbols and representations is what interests the constructivist (or intuitionist) philosophers. They argue that the mathematical object or symbol can only be meaningful if it can be “constructed” from real objects (Davis & Hersh 1980). Platonism The Greeks made a distinction between the more ethereal (non-earthly) or theoretical phenomena like divine intervention and the practical day to day happenings and experiences. Plato (428/427 BCE) was a leading Greek philosopher who believed that mathematics existed apart from man’s thinking and could therefore be discovered. The Greeks took the position that “the whole of mathematics exists externally, independently of man, and the job of man is to discover (rather than create) these mathematical truths” (Davis & Hersh 1980:415). This thinking would be supported by the possibility that two mathematicians could make the same discovery at the same time – such an occurrence would support the Platonist philosophy. Formalism A third group of mathematicians believe that mathematics is something like a game, where one starts with a set of statements called axioms and build up theorems from those statements or axioms. In this way, one would progressively construct the body of mathematics according to some specific rules. In their thinking, there is little attention to the meaning of mathematical objects. What counts in mathematics is the rule, the axiom, the theorem and how the procedures are performed. This group of mathematical thinkers are known as the formalists. They regard the form of mathematics higher than its meaningful link(s) to the real world. For more detail, see Module 1 (Philosophical and Historical Perspectives) of this course. In summary, we can say that a curriculum that foregrounds the context of the learner and connects the mathematics to their everyday lives and experiences, may be drawing on a constructivist/intuitionist philosophical approach, whereas a curriculum that focuses mainly on following procedures and on obtaining the correct answer may be following a formalist approach. The Platonist view is more complex. According to Reuben Hersh (https://core.ac.uk/download/pdf/82047627.pdf): “The typical ‘working mathematician’ is a Platonist on weekdays and a formalist on Sundays. That is, when he is doing mathematics, he is convinced that he is dealing with an objective reality whose properties he is attempting to determine. But then, when challenged to give a philosophical account of this reality, he finds it easiest to pretend that he does not believe in it after all”. 34 SELECTED MATHEMATICS CURRICULA ACROSS THE GLOBE 2.4 LEARNING UNIT 2 DIFFERENT COUNTRIES: DIFFERENT MATHEMATICS CURRICULA We continue the case studies and now include excerpts describing the mathematics curricula of the four countries. These excerpts for three of the countries were taken from reports submitted to TIMSS 2015 Encyclopedia. Other information is available from the website of the countries’ ministries of education. For Kenya, information was obtained from the Kenya Institute of Curriculum Development (KICD). The Singapore Mathematics Curriculum Singapore’s curriculum, known as the Pentagon Model, has five important characteristics. Figure 2.5: Singapore mathematics curriculum framework Source: http://timssandpirls.bc.edu/timss2015/encyclopedia/ The five inter-related components support the development of problem-solving abilities. Each of these five components is elaborated in the curriculum documents and described for each grade. The focus on this section of the curriculum may be described as the substantive perspective, a focus on what is worth knowing. PDM4801 35 LEARNING UNIT 2 SELECTED MATHEMATICS CURRICULA ACROSS THE GLOBE The Singapore mathematics curriculum comprises a set of syllabi spanning 12 years, from primary to pre-university education. As mathematics is a hierarchical subject, higher concepts and skills are built upon foundational ones and must be learned in sequence. The curriculum is designed in a spiral manner where concepts and skills in each content strand (e. g., Numbers and Algebra, Geometry and Measurement) are revisited and built upon at each level to achieve greater depth and understanding. Teachers help their students to learn these concepts and skills by adopting age- and grade-appropriate pedagogical approaches. Central to these pedagogical approaches at the primary and lower secondary levels is the Concrete-PictorialAbstract (C-P-A) approach, whereby teachers lead students through activities that help build an understanding of abstract mathematical concepts from everyday experiences and meaningful contexts, using concrete and pictorial representations. Source: http://timssandpirls.bc.edu/timss2015/encyclopedia/ A central principle that we see in the Singaporean curriculum is that mathematical proficiency is developed through encountering situations or problems that are carefully designed for learning. These constructed situations serve two purposes, the first is to illustrate a concept by providing a context at the cognitive level of the child, and the second is to expand the existing conceptual structures (schemes) of the child through extending the complexity of the mathematical situation beyond the child’s current level of mastery. From the brief description of the Singapore mathematics curriculum, it is clear that the curriculum leaders have kept abreast of the research that has taken place over the past 40 years. The position taken in this curriculum is perfectly justifiable from a theoretical perspective: one of the challenges for mathematics education, noted by Vergnaud, a French mathematics didactician, 40 years ago, is that mathematical concepts are rooted in situations and problems (Vergnaud 1988:141– 142). From this perspective, we see that a single concept, for example, addition, may be applied to problems in many different contexts; and one problem context or situation may require many distinct concepts to solve one problem. Another aspect of learning mathematics in the Singapore curriculum is that a single concept, say subtraction, does not develop in isolation but develop in relationship with other concepts; for example addition, or counting backwards. A problem-solving curriculum allows learners to engage deeply with mathematics and develop a deep understanding of mathematics. The Netherlands Mathematics Curriculum Mathematics education in the Netherlands has been influenced by realistisch wiskundeonderwijs (realistic mathematics education). The content of the curriculum will be similar to other countries, though the pedagogical approach will differ in some ways. Here is an excerpt for the participation in the international study and can be found in the TIMSS 2015 Encyclopedia. 36 SELECTED MATHEMATICS CURRICULA ACROSS THE GLOBE LEARNING UNIT 2 The mathematics curriculum for primary school comprises 11 core objectives.8 These objectives describe the desired results of the learning process, but not the way in which they are to be achieved. In primary school, students should become familiar with mathematical basics, offered in a recognizable and meaningful context. Primary school students will gradually acquire familiarity with numbers, measurements, and two- and three-dimensional geometric shapes and solids, as well as the relationships and calculations that apply to them. Students will learn to use mathematical language while gaining mathematical literacy and calculation skills. By the end of primary school, students are taught how to: ● Use mathematical language ● Solve practical and formal mathematical problems and clearly demonstrate the process of finding a solution ● Identify different approaches for solving mathematical problems and learn to assess the reasonableness of solutions ● Understand the general structure and interrelationship of quantities, whole numbers, decimal numbers, percentages, and proportions, and use these to do arithmetic in practical situations ● Quickly carry out basic arithmetic calculations mentally, using whole numbers through 100, and learn the multiplication tables ● Count and calculate by estimation ● Add, subtract, multiply, and divide by taking advantage of number properties ● Add, subtract, multiply, and divide on paper ● Use a calculator with insight ● Solve simple geometrical problems ● Measure and calculate using units of time, money, length, area, volume, weight, speed, and temperature In 2010, so-called reference levels, or benchmarks for language (Dutch and English) and numeracy, were introduced to help raise student achievement in primary and secondary education. These levels describe the knowledge and skills students are expected to acquire at different stages in their school career. For numeracy in primary education, there are two important levels: the fundamental level (1F) and the advanced level (1S).9 The achievement level of at least 85 percent of students by the end of primary school should be at level 1F. Source: http://timssandpirls.bc.edu/timss2015/encyclopedia/ The expression “mathematics as a human activity” is attributed to Freudenthal (1971, 1973). This notion aligns with the idea that mathematics has its roots in everyday situations; it is then the generalisation of principles emerging from these situations that more abstract mathematics is developed. This approach, known as Realistic Mathematics Education (RME) is promoted from the Freudenthal Institute and has been a major influence on mathematics education in the Netherlands and other countries around the globe. The idea horizontal and vertical mathematisation in RME suggests that when learners encounter a mathematical situation in an everyday context, they use their existing knowledge to invent informal context-related solutions. When they create new knowledge while they reach solutions for the problem, it is said that they are mathematising horizontally. Following learners’ informal efforts to find solutions, the PDM4801 37 LEARNING UNIT 2 SELECTED MATHEMATICS CURRICULA ACROSS THE GLOBE teacher takes on a proactive role and starts guiding learners into formal mathematics. The teacher’s role is to provide learners with an opportunity to reinvent their informal solutions into formal mathematics (Freudenthal 1991). In guiding learners to make the turn from informal to formal mathematics, they extend mathematics vertically. A further RME principle, known as the intertwinement principle, aligns with the notion of conceptual fields. The idea is that mathematical concepts cannot be isolated to one domain – mathematical domains such as number, geometry, measurement and data handling are not considered as isolated curriculum chapters but as heavily integrated with one another. This horizontal integration provides connections across concepts. The New Zealand Mathematics Curriculum A general statement regarding Mathematics Education in New Zealand is the following: In Mathematics and Statistics, students explore relationships in quantities, space, and data, and learn to express themselves in ways that help them to make sense of the world around them (Ministry of Education. The New Zealand Curriculum 2015:16). Further in this curriculum document (p 26), the description of mathematics and statistics is elaborated: Mathematics “is the exploration and use of patterns and relationships in quantities, space, and time”, whereas statistics is the exploration and use of patterns and relationships in data”. They are related but have different ways of thinking and solving problems. They do, however, both “create models to represent both real-life and hypothetical situations”, and these situations are drawn from a wide range of contexts (Ministry of Education 2015:26). The next excerpt describes the focus of mathematics in the primary school, written for TIMSS 2015 Encyclopedia (Mullis et al 2016). Number and Algebra – Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals (simple), and percentages; know basic multiplication and division facts; know counting sequences for whole numbers; know how many ones, tens, hundreds, and thousands are in whole numbers; know fractions and percentages in everyday use; find fractions of sets, shapes, and quantities; record and interpret additive and simple multiplicative strategies using words, diagrams, and symbols, with an understanding of equality; generalize the properties of addition and subtraction with whole numbers; and connect members of sequential patterns with their ordinal position and use tables, graphs, and diagrams to find relationships between successive elements of number and spatial patterns. Source: http://timssandpirls.bc.edu/timss2015/encyclopedia/ Kenyan mathematics education In the new Competency-based Curriculum, introduced in Grades 1 and 4 in 2019, the focus on mathematics is described by the KICD as follows: The pre-primary mathematics activities should involve day-to-day life, they should focus on problem solving and should allow hands-on manipulation. 38 SELECTED MATHEMATICS CURRICULA ACROSS THE GLOBE LEARNING UNIT 2 The lower primary phase focuses on Numeracy and Number Work, the identification and value of placement in numbers (understanding of positionality and place value) and the basic operations, addition, subtraction, multiplication and division. The children further explore the world of mathematics in the higher primary phase by building on the skills learnt in earlier years, that is counting, adding, subtracting, multiplication and division. They develop an understanding of numbers and numerical operations used to develop strategies for mental mathematics, estimation and computational fluency. In the senior phase, the focus continues to be on problem-solving and computation, and numerical and physical measurements. In addition, there is a focus on developing logical reasoning to make rational decisions. ACTIVITY 2.4 PLATONIST, FORMALIST AND CONSTRUCTIVIST Read the short excerpts about the four countries and answer the following questions. 1. Which country’s curriculum do you think has elements of a constructivist/ intuitionist view of mathematics? Give a reason for your answer. 2. Which curriculum do you think has strong elements of a formalist view of mathematics? Why would you say so? 3. Describe in your own words, the Platonist view of mathematics. FEEDBACK ON ACTIVITY 2.4 Most of the curricula around the world follow a similar sequence of content and concepts. They would probably all aspire for fluency in computation and require some attention to problem-solving. In the higher grades, the mathematics almost certainly has elements of the major philosophies, constructivist, Platonist and formalist. PDM4801 39 LEARNING UNIT 2 2.5 SELECTED MATHEMATICS CURRICULA ACROSS THE GLOBE DIFFERENT CULTURES, DIFFERENT LEARNERS: SAME CURRICULUM? It is general practice in most countries that the intended curriculum is designed at the national (macro) level and then implemented at the school (meso) and classroom (micro) levels. When the curriculum is centrally controlled and tightly prescribed according to general national requirements, it may not be directly applicable in, for example, rural contexts. Some curricula design is flexible, which may enable teachers to use some discretion in how they teach a particular topic. Here are some of the interesting cultural facts about the different countries under discussion: Singapore, though small in size, has a variety of languages, religions and cultures. Former prime ministers have called Singapore a society-in-transition since it does not fit the description of a nation. Singapore’s culture has developed into its present modern culture that comprises a combination of Malay, European and Asian cultures. Singapore is known as the place where East meets West and as a Gateway to Asia. The so-called language Singlish has developed as part of the legacy of British colonial rule of the country. This language contains vocabulary and grammar that was influenced by a number of languages found across Asia, notably South and East Asia. People work hard and they party hard. Eating and drinking in social contexts are prominent in Singapore’s society. The fusion of cultures results in a rich variety of foods and drinks. Housing is mainly in blocks where people from different origins live in close proximity of each other and their interactions shape the identity of Singaporean society. Singapore holds education very highly because they regard it as the gateway to their future. Supplementary classes are attended from a very young age and education is draining and competitive because the view that a lack of education will hold a person back, is a strong motivation for complete and total engagement in education. The city-state of Singapore is not a concrete jungle but a green “garden city” where urban life meets nature. There are very strict societal rules, which in most cases are written into laws: chewing gum is prohibited in some places, as is smoking. Littering and throwing a wrapper anywhere meet with either a fine or community service to sweep the streets. Waste disposal is well regulated; therefore, Singapore is exceptionally clean. Retail shopping, street vendors and malls are plenty in Singapore. Queuing for long times is a regular occurrence as a result of its drive for organisation and order. The East African republic of Kenya borders the Indian Ocean and is situated between Somalia and Tanzania. It has a population estimated at 51+ million and the ethnic make-up is varied: Kikuyu 22%, Luhya 14%, Luo 13%, Kalenjin 12%, Kamba 11%, Kisii 6%, Meru 6%, other African 15%, non-African (Asian, European and Arab) 1%. Religions are Protestant 45%, Roman Catholic 33%, Muslim 10%, indigenous beliefs 10% and other 2%. Kenya is a multilingual country. Although Swahili and English are the official languages, more than 60 languages are spoken in the country. These mainly consist of tribal African languages as well as a minority of Middle Eastern and Asian languages spoken by descendants of foreign settlers (i.e. Arabic, Hindi, etc). The African languages come from three different language families – the “peoples” languages spoken in the centre and southeast, Nilotic languages (in the west) and Cushitic languages (in the northeast). ● Kenya is not a homogeneous country ethnicity wise. 40 SELECTED MATHEMATICS CURRICULA ACROSS THE GLOBE LEARNING UNIT 2 ● The make-up of Kenyans is that of 13 ethnic groups with an additional 27 smaller groups. ● The majority of Kenyans belong to tribes such as the Kikuyu, Luhya and Kamba. ● There are also minority tribes such as the Luo, Kalenjin, Maasai, Turkana, Rendille and Samburu. ● Around 13% of the population are of non-African descent, i.e. Indian, Arab and European. The Kenyan Constitution guarantees freedom of religion. Most Kenyans interweave native beliefs into traditional religion. Respect for ancestors is an important aspect of spiritual thinking in Kenya. They believe that a person only really dies once their relatives have forgotten them. Kenyans are group-orientated rather than individualistic. “Harambee” comes from a word meaning “to pull together” and defines the people’s approach to others in life. The concept is about mutual assistance, effort and responsibility. This principle has its roots in communal farming and herding. The extended family is the basis of the social structure. ACTIVITY 2.5 DIFFERENT CULTURES, DIFFERENT LEARNERS Which one of the curricula of Singapore and Kenya do you think allow the teachers the most flexibility to incorporate their local contexts and acknowledge cultural artefacts and history? Give reasons for your selection. FEEDBACK ON ACTIVITY 2.5 Most of the curricula around the world follow a similar sequence of content and concepts. They would all probably aspire to fluency in computation and require some attention to problem-solving. In the higher grades, the mathematics almost certainly has elements of the major philosophies – constructivist, Platonist and formalist. 2.6 CONCLUSION This learning unit has somewhat opened our horizons to compare and critically review the mathematics curricula in selected countries – we chose New Zealand, the Netherlands, Kenya and Singapore. The similarities and differences between PDM4801 41 LEARNING UNIT 2 SELECTED MATHEMATICS CURRICULA ACROSS THE GLOBE mathematics curricula in these four countries in terms of their curriculum management structures became clear. We realised that the various national education ministries follow different views to centralisation or decentralisation of powers and functions, the degree of autonomy of schools and the support they provide for education. We started to understand how the designers’ philosophical stance influence the characteristics of the curriculum. Lastly, we touched on the alignment of curricula with cultural demands and learners’ profile. The topics have not been exhausted and invite you to read further and engage in this most interesting topic. Check yourself: Have you reached the goal and the specific Tick the box outcomes of learning unit 2? Can you compare and critically review the mathematics curricula in selected countries? 1 Can I describe some similarities and differences between mathematics curricula in the four countries, notably in terms of their curriculum management structures? 2 Can I compare various mathematics curricula at the macro and meso levels in terms of centralisation or decentralisation of functions, the degree of autonomy of schools, and the support they provide for education? 3 Can I critique mathematics curricula according to their philosophical underpinnings, context and content? 4 Can I discuss the alignment of curricula with cultural demands and alignment with learners’ profile? References Davis, P. & Hersh, R. (1980). The Mathematical Experience. New York: Penguin Books. Vergnaud, G. (1988). Multiplicative Structures. In J. Hiebert & M. Behr (Eds.) Number concepts and operations in the middle grades. Hillsdale, New Jersey: National Council of Teachers of Mathematics. 42 LEARNING UNIT 3 INTERNATIONAL LARGE-SCALE ASSESSMENT LEARNINGUNIT3 Table of contents INTRODUCTION 3.1 TIMSS CURRICULUM AND ASSESSMENT FRAMEWORKS 3.1.2 Assessment frameworks: Content and cognitive domains Activity 3.1.2(a): Content domains Activity 3.1.2(b): Cognitive domains 3.2 SOUTHERN AFRICAN CONSORTIUM FOR THE MONITORING OF EDUCATIONAL QUALITY 3.2.1 An overview of SACMEQ 3.2.2 The SACMEQ project Activity 3.2.2: Aims and research design of SACMEQ 3.2.3 Curriculum and assessment frameworks 3.2.4 Some results Activity 3.2.4: SACMEQ – Some implications 3.3 COMPARISON OF RESEARCH DESIGNS: INTERNATIONAL AND REGIONAL STUDIES Activity 3.3: TIMSS and SACMEQ research design 3.4 CONCLUSION ADDITIONAL LEARNING OPPORTUNITIES REFERENCES OUTCOMES OF THIS LEARNING UNIT In this learning unit, you will reflect on the analysis of international and regional comparisons of mathematics achievement. At the end of learning unit 3, you should be able to do the following: ● Explain the rationale for the TIMSS curriculum frameworks and describe key components, the content and the cognitive domains. ● Present an overview of the TIMSS assessment framework that guides the design of test items and booklets and explain the rationale provided by the IEA. ● Describe the curriculum framework and key components of the SACMEQ study, under the auspices of UNESCO (United Nations Educational, Scientific and Cultural Organisation). PDM4801 43 LEARNING UNIT 3 INTERNATIONAL LARGE-SCALE ASSESSMENT ● Present the assessment framework guiding the design of test items and explain the rationale provided by SACMEQ. ● Compare the research designs, curriculum frameworks and the assessment frameworks of the international study and the regional study. INTRODUCTION Large-scale international studies presuppose that curricula across the globe follow a similar mathematics curriculum. In the 1990s, the International Association for the Evaluation of Educational Achievement (IEA) conducted studies of the curricula of the participating countries to establish a framework, which included both mathematical concepts, named the content domain and the cognitive processes required to solve the problems, known as the cognitive domain. A parallel study has been run under the auspices of UNESCO focused on Southern and East African countries, notably, the Southern (and Eastern) African Consortium for Monitoring Educational Quality (SACMEQ). In learning unit 3, we look at the curriculum and assessment frameworks of one of the large-scale studies driven and sponsored by the IEA and a second regional study under the auspices of UNESCO’s International Institute for Educational Planning (IIEP): ● The large-scale international assessment, Trends in International Mathematics and Science Study (TIMSS), initiated in the early 1980s. ● The regional Southern and Eastern African Consortium for the Monitoring of Educational Quality (SACMEQ or SEACMEQ) study, initiated in 1998. The participating countries need to know the structure of the curriculum framework used by the IEA and on what basis the tests are designed: What topics are in the curriculum frameworks? How are these topics going to be tested? Likewise, in the SACMEQ study, the participating countries need to know what is to be tested. A critical aspect of these studies is the confidence that the public and education departments place in the studies, which requires the research design and sampling strategies to be scientific and fair. 3.1 TIMSS CURRICULUM AND ASSESSMENT FRAMEWORKS In this part of the work, we give an overview of the IEA studies and discuss the framework of which they use for large-scale international mathematics assessment. In this part of the work, the excerpts in the blocks and the facts and ideas in the text were taken directly from the TIMSS Assessment Framework, referenced below: Mullis, IVS & Martin, MO. (Eds.). 2017. TIMSS 2019 Assessment Frameworks. Available at http://timssandpirls.bc.edu/timss2019/frameworks/. Martin, MO, Mullis, IVS & Hooper, M. (Eds.). 2016. Methods and Procedures in TIMSS 2015. Available at http://timssandpirls. bc. edu/publications/timss/2015-methods. html. 44 INTERNATIONAL LARGE-SCALE ASSESSMENT LEARNING UNIT 3 3.1.1 An overview of the IEA studies The IEA was formed in the late 20th Century. The first and second studies, which was conducted in a few countries in the 1980s, focused on science achievement. In 1995, the Third International Mathematics and Science Study (TIMSS) was administered. It has now become both a mathematics and science study and henceforth the acronym TIMSS stood for the Trends in International Mathematics and Science Study. Currently over 40 countries participate. In 2001, the IEA started conducting large-scale literacy studies called the Progress in International Reading Literacy Study (PIRLS). This learning unit focuses on TIMSS since it relates to mathematics. The theoretical framework for TIMSS: The focus of all these studies is an educational achievement; however, contextual information is collected from the participating countries to enable education departments and researchers to understand general, local and personal contexts more deeply. The overall theoretical framework of the IEA studies was described in learning unit 1; it makes explicit how the IEA understands the educational relationships and how each component affects the educational output. The IEA understands curriculum as comprising the intended curriculum within a general social and educational context, the implemented curriculum within a local, community and school context and the attained curriculum within the personal context or background of the learner. The assessment framework for TIMSS: For each of the IEA studies an assessment framework is designed to make explicit the content in the subject areas to be tested. For example, the assessment framework guiding the TIMSS serves to make explicit to participating countries the intended curriculum, from which the specific topics are drawn, and against which the items are conceptualised and designed. Grades 4 and 8 learners are targeted in most of the countries. For some countries, an exception is made and the Grades 5 and 9 learners are tested. The TIMSS is a longitudinal study that is administered every four years. To compare the results across the years, for example from 2003 to 2007, 2011, 2015 and 2019, the study design includes common items that are carried over across years. The TIMSS test design includes 14 booklets that are distributed across the learners in each of the participating countries. Some items are released for restricted use. Others are kept out of the public eye so that their use in subsequent years will not be jeopardised. The testing of every learner in every participating country would be too cumbersome and expensive. It is unnecessary, as it is the educational system, rather than the performance of individual learners, that is being assessed. In each country, 150 schools are randomly selected from a list of all possible schools across the country, meaning that each school has an equal chance of being selected. This selection is done by Statistics Canada. If, for example, one of the 150 schools cannot take part, perhaps the school has recently closed, the model identifies a replacement school. Within each school, one class is selected, again the class is selected at random. The teacher teaching that class then participates in the study. The same indicators in the contextual questionnaire are used every year, with some additional questions provided by the specific country. This questionnaire includes questions such as the qualifications of teachers. However, the information on teachers is not representative. Why? Because not every single Grade 4 or 5 teacher had the chance to be selected for the study, only the teachers of those classes selected for the PDM4801 45 LEARNING UNIT 3 INTERNATIONAL LARGE-SCALE ASSESSMENT study. Paragraph 3.1.3 gives more information about research design, population and sampling. 3.1.2 Assessment frameworks: content and cognitive domains The questions that will be discussed in this section are What is assessed in TIMSS? (content domain) and At what cognitive levels is this assessed? (cognitive domain). Content domains a) Content domain: number In the TIMSS research study, number is seen as providing the “foundation of mathematics” in the primary school. This domain comprises three topics that are differently weighted in terms of emphasis in the TIMSS test and accounts for 50% of the test. The test allocations within number are as follows: ● Whole numbers (25%) ● Expressions, simple equations and relationships (15%) ● Fractions and decimals (10%) Three aspects of number are regarded as essential in the TIMSS assessment framework, as in other frameworks. For example, the South African CAPS document lists the following categories: ● Understanding the place value underpinning whole numbers ● Being able to compute with whole numbers ● Using computation to solve problems Fractions and decimals are also an important component of CAPS. Usiskin (2004) considers the transition from working with numbers to working with variables as one of the most important transitions in number conceptualisation. It is therefore understood why TIMSS includes in its framework the understanding of prealgebra concepts, such as understanding the concept of variables (unknowns) in simple equations (for example, 105 ÷ = 15) and understanding the relationships between quantities. Usiskin also notes the transition from whole numbers to fractions and to rational numbers. In the process of making this transition, learners compare, add and subtract fractions and decimals and use this understanding to solve problems. The next textbox arranges the assessment framework from the TIMSS (Grade 4) for the number domain. For each subdomain, you will see descriptions of what is to be assessed. The main content topic is highlighted in bold. Whole numbers ● Demonstrate knowledge of place value (two-digit to six-digit numbers); represent whole numbers with words, diagrams, number lines or symbols; order numbers. ● Add and subtract (up to four-digit numbers), including computation in simple contextual problems. ● Multiply (up to three-digit by one-digit and two-digit by two-digit numbers) and divide (up to three-digit by one-digit numbers), including computation in simple contextual problems. 46 INTERNATIONAL LARGE-SCALE ASSESSMENT LEARNING UNIT 3 ● Solve problems involving odd and even numbers, multiples and factors of numbers, rounding numbers (up to the nearest ten thousand) and making estimates. ● Combine two or more properties of numbers or operations to solve problems in context. Expressions, simple equations and relationships ● Find the missing number or operation in a number sentence (e.g. 17 + w = 29). ● Identify or write expressions or number sentences to represent problem situations that may involve unknowns. ● Identify and use relationships in a well-defined pattern (e. g. describe the relationship between adjacent terms and generate pairs of whole numbers given a rule). Fractions and decimals ● Recognise fractions as parts of wholes or collections; represent fractions by using words, numbers or models; compare and order simple fractions; add and subtract simple fractions, including those set-in problem situations. (Fractions may have denominators of 2, 3, 4, 5, 6, 8, 10, 12 or 100.) ● Demonstrate knowledge of decimal place value, including representing decimals using words, numbers or models; compare, order and round decimals; add and subtract decimals, including those set-in problem situations. (Decimals may have one or two decimal places, allowing for computations with money. b) Content domain: measurement and geometry The content domains measurement and geometry take mathematics into the world around us. Geometry and measurement both have visual, quantitative and often, experiential features. Geometry enables us to understand the relationships between shapes, objects, sizes and dimensions not only visually, but even experientially; measurement enables us to quantify attributes of objects and phenomena in this world; for example, length, perimeter, area and volume, as well as mass and time. The two topic areas, measurement and geometry, are each allocated 15%, which is half of the 30% allocated to this content domain The TIMSS framework stipulates specific requirements for learners in Grade 4 when they engage in these two topics. As teachers, we could also consider taking an example from the TIMSS requirements for use in our teaching. Learners should do the following: ● Use a ruler to measure length. ● Solve problems involving length, mass, capacity and time. ● Calculate areas and perimeters of simple polygons. ● Use cubes to determine volumes. ● Identify the properties and characteristics of lines, angles and a variety of two- and three-dimensional shapes. Regarding spatial sense, students are required to do the following: ● Describe and draw a variety of geometric figures. ● Analyse geometric relationships and use these relationships to solve problems. PDM4801 47 LEARNING UNIT 3 INTERNATIONAL LARGE-SCALE ASSESSMENT Item writers from the participating countries contribute items that then go through a screening process. The 30% weighting for this section comes from the descriptions below. Please note the differences in spelling in British English and American English – for example, metre vs meter; litre vs liter. We use British English in South Africa. Here are the details of the assessment framework that guides the design of items on this topic. Measurement ● Measure and estimate lengths (millimetres, centimetres, metres, kilometres); solve problems involving lengths. ● Solve problems involving mass (gram and kilogram), volume (millilitre and litre), and time (minutes and hours); identify appropriate types and sizes of units and read scales. ● Solve problems involving perimeters of polygons, areas of rectangles, areas of shapes covered with squares or partial squares and volumes filled with cubes. Geometry ● Identify and draw parallel and perpendicular lines; identify and draw right angles and angles smaller or larger than a right angle; compare angles by size. ● Use elementary properties, including line and rotational symmetry, to describe, compare and create common two-dimensional shapes (circles, triangles, quadrilaterals, and other polygons). ● Use elementary properties to describe and compare three-dimensional shapes (cubes, rectangular solids, cones, cylinders and spheres) and relate these with their two-dimensional representations. c) Content domain: data Understanding quantitative information and the way it is represented in today’s society is critical for students in the 21st century. The many forms of media, including social media, all use charts, graphs and tables to represent data and to help organise information in such a way that it is generally comprehensible. In the TIMSS framework, this content domain comprises 20% at Grade 4 level. It is divided into two topic areas, namely reading, interpreting and representing data (15%) and using data to solve problems (5%). Below are the statements from the assessment framework guiding item writers when designing tests for each cycle. Reading, interpreting, and representing data ● Read and interpret data from tables, pictographs, bar graphs, line graphs and pie charts. ● Organise and represent data to help answer questions. Using data to solve problems ● Use data to answer questions that go beyond directly reading data displays (e.g. solve problems and perform computations using data, combine data from two or more sources, draw conclusions based on data). 48 INTERNATIONAL LARGE-SCALE ASSESSMENT LEARNING UNIT 3 In this subsection, we focused on Grade 4 content. All this information can be found on the TIMSS website at http://timssandpirls.bc.edu/timss2019/frameworks/ framework-chapters/mathematics-framework/mathematics-content-domains-eighthgrade/. ACTIVITY 3.1.2 (A) Content domains The designers of the TIMSS Grade 4 test incorporated the three content domains, namely number, measurement and geometry and data to construct a balanced assessment instrument. Each of the domains has two or more subdomains and within each subdomain, from two to five content areas. 1. Please complete the table with the information from the Grade 4 assessment frameworks. For the content areas, list only the key terms. Content domain 1. Number Subdomain 1.1 Content areas i. – ii. – iii. – iv. – 1.2 v. – i. – ii. – 2. Measurement and geometry 1.3 iii. – i. – 2.1 ii. – i. – ii. – 2.2 iii. – i. – ii. – 3. Data PDM4801 3.1 iii. – i. – 3.2 ii. – i. – 49 LEARNING UNIT 3 INTERNATIONAL LARGE-SCALE ASSESSMENT 2. The weighting across content domains differs. a) In the table, complete the percentage allocated for each content domain. b) Then discuss pros and cons of varying weightings of domains. Content domain Percentage Percentage for each subdomain Number 50% Whole numbers 25% Measurement and geometry Data c) You have to design a test for the Grade 4 class, containing 12 items to cover the entire curriculum. You decide to use the TIMSS assessment framework as a guide. Write down the contents of each item in short form. FEEDBACK ON ACTIVITY 3.1.2 (a) a) To complete this table, consult the various content domain information and CAPS. b) Weighting one topic less gives the idea that it is less important. c) Though answers may vary, all items should link back to the frameworks. Cognitive domains The TIMSS study centre has separated the content and cognitive domains in the framework, which is somewhat artificial because there are very few mathematics 50 INTERNATIONAL LARGE-SCALE ASSESSMENT LEARNING UNIT 3 questions that are purely content based. Most mathematics processing is dynamic and requires working with knowledge that has been reasoned by previous generations. Nevertheless, the cognitive domains sub-categories assist the test designer to write the questions and the education departments to understand the types of questions. Learners need to be familiar with the mathematics content as described in the previous section; however, the content questions require a range of cognitive skills that need to be clearly articulated. Therefore, according to the TIMSS assessment framework, the items have to cover the content domains but also three cognitive domains, namely: ● Knowing is the cognitive domain that covers “the facts, concepts and procedures” that students need to know. ● Applying is the cognitive domain that focuses on the application of knowledge and conceptual understanding to solve problems. ● Reasoning is the cognitive domain that takes the student beyond routine problems and where they are required to use their logical reasoning to engage with situations that are unfamiliar, contexts that are complex, and problems that require two or more steps. In the TIMSS frameworks, a learner is seen as mathematically competent when the content domains and the cognitive domains are integrated. The competencies include: ● Problem solving ● Providing a mathematical argument to support a strategy or solution ● Representing a situation mathematically (e.g. using symbols and graphs) ● Creating mathematical models of a problem situation ● Using tools such as a ruler or a calculator to help solve problems The three cognitive domains are addressed in both grades. The balance of items in each cognitive category differs in the two grades, where it is expected that the reasoning domain should have been more developed by Grade 8. The weighting of cognitive domains is shown in table 3.1. TABLE 3.1 Cognitive domains in TIMSS Percentages Cognitive Domains Knowing Grade 4 40% Grade 8 35% Applying 40% 40% Reasoning 20% 25% Knowing: The easy recall of basic facts, mathematical language, conventions, symbolic representations, measurement units and geometrical and spatial terms enable students to focus on the application or the problem to be solved. Included in knowing is knowing various procedures and steps to follow in solving the procedures. The language of mathematics and the properties of number are central to this cognitive domain. Table 3.2 shows the TIMSS descriptions of the categories of knowing. PDM4801 51 LEARNING UNIT 3 INTERNATIONAL LARGE-SCALE ASSESSMENT TABLE 3.2 Categories of knowing in TIMSS Recall Recall definitions, terminology, number properties, units of measurement, geometric properties and notation (e.g. a × b = ab, a + a + a = 3a). Recognize Recognise numbers, expressions, quantities and shapes. Recognise entities that are mathematically equivalent (e.g. equivalent familiar fractions, decimals and percentages; different orientations of simple geometric figures). Classify/Order Classify numbers, expressions, quantities and shapes by common properties. Compute Carry out algorithmic procedures for +, –, ×, ÷, or a combination of these with whole numbers, fractions, decimals and integers. Carry out straightforward algebraic procedures. Retrieve Retrieve information from graphs, tables, texts or other sources. Measure Use measuring instruments and choose the appropriate units of measurement. Applying: This domain involves applying what you know to solve mathematics problems in a range of contexts, either everyday contexts or more mathematical contexts. Also, in this domain is the ability to translate across multiple representations. Table 3.3 shows the TIMSS descriptions of the categories of applying. TABLE 3.3 Categories of applying in TIMSS Determine Determine efficient/appropriate operations, strategies and tools for solving problems for which there are commonly used methods of solution. Represent/Model Display data in tables or graphs; create equations, inequalities, geometric figures or diagrams that model problem situations and generate equivalent representations for a given mathematical entity or relationship. Implement Implement strategies and operations to solve problems involving familiar mathematical concepts and procedures. Reasoning: “Reasoning mathematically involves logical, systematic thinking” is the view of TIMSS, as well as other mathematics educators and researchers. In addition, it includes creative and intuitive thinking. Inductive reasoning applied to patterns can be used to solve problems, the solution of which is not immediately obvious. Deductive reasoning, based on assuming an axiom after which logical reasoning takes you to the answer, forms part of this domain. The categories of reasoning from the TIMSS framework for the cognitive domain are shown in table 3.4. 52 INTERNATIONAL LARGE-SCALE ASSESSMENT LEARNING UNIT 3 TABLE 3.4 Categories of reasoning Analyse Determine, describe or use relationships among numbers, expressions, quantities and shapes. Integrate/Synthesise Link different elements of knowledge, related representations and procedures to solve problems. Evaluate Evaluate alternative problem-solving strategies and solutions. Draw conclusions Make valid inferences based on information and evidence. Generalise Make statements that represent relationships in more general and more widely applicable terms. Justify Provide mathematical arguments to support a strategy or solution. ACTIVITY 3.1.2 (B) Cognitive domains a) In assessment, TIMSS distinguishes between content and cognitive domains. However, when it comes to designing items, the content and cognitive domains are integrated. The next table has two subdomains of number, whole number, and fractions and decimals in the top row. The left column contains the cognitive functions relating to the knowing domain. Look at the statements in the column under whole numbers, then write similar statements for the concepts of fractions and decimals for the same cognitive functions. Whole numbers Recall Recall the number facts Recognise Recognise odd numbers Classify/ order Order four-digit numbers from largest to smallest Compute Multiply a two-digit number by a one-digit number Fractions and decimals b) Choose one of the applying cognitive function descriptions and apply it to the content domain measurement. c) Choose one of the reasoning cognitive domain descriptions and use this description to reason about the content domain geometry. PDM4801 53 LEARNING UNIT 3 INTERNATIONAL LARGE-SCALE ASSESSMENT FEEDBACK ON ACTIVITY 3.1.2(b) At first glance, the intersecting domains seem to be complex, but once we start applying this integration, we realise that is can be a powerful tool in setting our assessments. It was worth the time engaging in this activity, you may agree. Please see Mullis and Martin (2017) for more information. 3.1.3 Assessment design, population and sampling The TIMSS study is about the measurement of student achievement that does justice to the broad range of content and skills in both mathematics and science. The monitoring of participating countries at four-year intervals, at both Grades 4 and 8, enables countries and their education departments to gauge trends. The same learners who are in Grade 4 during a test year (e.g. 2011), are in Grade 8 in the next TIMSS assessment year (2015). Every four years since 1995, the subsequent assessments have been linked to the one that preceded it. For example, the 2019 TIMSS study is linked to the 2015 study, with some common booklets. The change in education systems and the trend to go digital has resulted in half the countries in 2019 administering the assessment online. Grades 4 and 8 align with the International Standard Classification of Education (ISCED) levels where children in their fourth year of schooling have an average age of 9.5, while in their eighth year of schooling, the average age is 13,5. If we accept that our average Grade 1 learner in South Africa is 6.5 years old, our Grade 4 and Grade 8 learners’ ages correspond to this international standard classification of education. Because of an educational delay in our education system, perhaps caused by multilingualism that requires learners to focus on learning languages in these early years, a decision was made to test Grades 5 and 9. A sample of 150 schools, with one or two intact classes, will result in about 4 000 learners per country. This is the number of learners deemed to be necessary to provide an overview of the general mathematics ability over an entire country. In the assessment frameworks, the curriculum topics have been described together with their subtopics, concept clusters and individual concepts. The individual concepts must be represented fairly and therefore many assessment items are generated, evaluated and tested. A selection of these items is distributed across the 14 booklets. Each booklet at Grade 4 level contains about 12 mathematics assessment items (questions) and 12 science items (Grade 8 has more items per booklet). Each item appears in two booklets, which makes it possible to link the items when their results are analysed through a very sophisticated item response scaling process. Items out of the four content domains are fairly distributed across the booklets. Furthermore, the items, having been graded for difficulty level, are distributed across the booklets such that an equivalent number of difficult items occur in each booklet. For Grade 4, each booklet is estimated to take 18 minutes to complete the mathematics section and a further 18 minutes to complete the science section. Each learner completes two booklets, making their allowed time for completing the tests 72 54 INTERNATIONAL LARGE-SCALE ASSESSMENT LEARNING UNIT 3 minutes. For Grade 8, 22.5 minutes is allowed to complete each booklet, making the total time 90 minutes. From cycle to cycle, for example from 2015 to 2019, six booklets for each grade are kept secure so that these booklets may be used in the following cycle. Eight booklets are released for research or teaching purposes in each cycle. Thus, in each cycle, eight new booklets are generated. In Activity 3.1.3 you will engage further with the test design of TIMSS and get some exposure to the structure of this massive assessment study. ACTIVITY 3.1.3 TIMSS test design 1. The features of the TIMSS test design ensure that firstly there is comprehensive coverage of the entire content domain, and secondly that it includes a balance of the cognitive domain. Select three features of the test design that enable comprehensive coverage. Write a three-line explanation of the three features. 2. In the TIMSS study, longitudinal data must be obtained for countries to assess progress or lack of progress. How does the TIMSS test design ensure the security of the tests that are repeated from across the test cycles? (Write about eight lines.) 3. About how many schools and how many learners are tested in each country? Explain why this relatively small amount is enough for a country to gauge the effectiveness of the education system (write about eight lines). PDM4801 55 LEARNING UNIT 3 3.2 INTERNATIONAL LARGE-SCALE ASSESSMENT SOUTHERN AND EASTERN AFRICAN CONSORTIUM FOR THE MONITORING OF EDUCATIONAL QUALITY In this part of the work, the following sources were used to extract facts and ideas: UNESCO and IIEP information was based on http://www. iiep. unesco. org/en/ourexpertise/sacmeq. SACMEQ III information was based on Quality of Primary School Inputs in South Africa by Moloi and Chetty (2011) in SACMEQ III reports, Policy Brief, South Africa. Available at http://www. sacmeq. org/sites/default/files/sacmeq/reports/sacmeq-iii/ national-reports/s3_south_africa_final.pdf SACMEQ IV information was based on The SACMEQ IV Project in South Africa, by Department of Basic Education (2017) in SACMEQ IV reports, National Reports, South Africa. Available at http://www.sacmeq.org/sites/default/files/sacmeq/reports/ sacmeq-iv/national-reports/sacmeq_iv_project_in_south_africa_report.pdf. 3.2.1 An overview of SACMEQ The United Nations Educational, Scientific and Cultural Organization (UNESCO) is an agency of the United Nations (UN) aimed at contributing to the building of peace, the eradication of poverty, sustainable development and intercultural dialogue through education, the sciences, culture, communication and information. An arm of UNESCO (International Institute for Educational Planning, IIEP), created in 1963 in Paris, France, aims to develop the capacities of education departments to manage their education systems. It is through this arm that the Southern African Consortium for Monitoring Education Quality (SACMEQ) came into being. The IIEP conducted training programmes and offered technical assistance to education departments. In 1989, the Ministry of Education in Zimbabwe met with the Director of the International Institute for Educational Planning (IIEP) and together they planned a research study that involved a hands-on approach, named ‘learning by doing’. Out of this research came the Zimbabwean report From Educational Research to Education Policy: An example from Zimbabwe. In 1995, eight ministries of education located in Southern and Eastern Africa met to constitute the SACMEQ at the SACMEQ Coordinating Centre in Gabarone, Botswana. The SACMEQ projects, of which there have now been four, are large-scale research studies carried out in SACMEQ countries. The participating countries have increased to 16 (see table 3.5). They aim to assess the conditions of schooling and to assess the performance levels of learners and teachers in the areas of numeracy and literacy. Three large-scale studies have been completed, comprising of cross-national educational policy research projects at five to six-year intervals. 56 INTERNATIONAL LARGE-SCALE ASSESSMENT LEARNING UNIT 3 Table 3.5 indicates the countries in the region that took part in the SACMEQ project at different stages from 1995. TABLE 3.5 Participating countries in SACMEQ Project Years Countries SACMEQ I 1995–1999 Kenya, Malawi, Mauritius, Mozambique, Namibia, Zambia, Zanzibar and Zimbabwe SACMEQ II 2000–2004 Fourteen ministries SACMEQ III 2006–2011 Fifteen ministries SACMEQ IV 2012–2014 Angola, Botswana, Kenya, Lesotho, Malawi, Mauritius, Mozambique, Namibia, Seychelles, South Africa, Swaziland, Mainland Tanzania, Zanzibar, Uganda, Zambia, Zimbabwe. 3.2.2 The SACMEQ Project The SACMEQ project, made up of the ministries of education from African countries and officials from the IIEP, developed research instruments and collected useful information through up to date research methods. They aimed to monitor valid levels of achievement and to monitor changes in behaviour over time. The main aims of the study were to: ● measure performance across countries at single time points; and ● measure across different time points for a single country. To ensure consistency and reliability of test data, the tests and methodology are kept the same from cycle to cycle. The security of the test instruments is of utmost importance. This arrangement, while ensuring security, has the disadvantage of not allowing public scrutiny of the tests. Compare the arrangement that TIMSS has where some items are released after every cycle. Note that SACMEQ is conducted from within education departments; the resources, therefore, may be more constrained. Population and sampling Over each cycle, the desired target population of the Grade 6 learners in that country stays the same. The excluded target population is required to be less than 5%, which includes special schools and small schools with classes of less than 15 learners. The achieved sample size has to be 90% of the targeted sample size for schools and 80% for the learners. The sample schools are distributed proportionally across the provinces, with more schools sampled from the provinces with larger populations. The Educational Management Information System (EMIS) database established in 2013 is accessed by the SACMEQ Coordinating Centre (SCC) to conduct a two-way sampling design, first of schools and then of learners. The IIEP has a prescribed sampling process in two stages. In the first stage, the defined population is sampled with a probability proportional to the size of the school, so the probability of sampling a large school will be larger than sampling a small school. At PDM4801 57 LEARNING UNIT 3 INTERNATIONAL LARGE-SCALE ASSESSMENT the second stage, 25 learners from Grade 6 classes (not a whole class like in TIMSS) are randomly sampled from the sampled schools. In the SACMEQ study, teachers are also part of the study. They are tested and graded along with the learners. However, as with the TIMSS, the teachers are not a representative sample. In TIMSS, the intact class was the unit of analysis. In SACMEQ, it is the school as the first stage of sampling and the individual learner as the second stage. Therefore, any results found from testing teachers or information from the contextual questionnaire should be phrased as follows: The stated percentage of Grade 6 learners was in schools with teachers having that characteristic. For example, 35,4% of Grade 6 learners in the SACMEQ IV study were in schools that had teachers who achieved at Numeracy Level 8. Contextual questionnaires Questionnaires about the school context, classroom attributes, access to resources, home context and infrastructure and resources, were given to learners, teachers and school principals. The information on all these features of schooling was analysed by the National Research Coordinators and their research teams. The processes were directed from Gabarone, Botswana. ACTIVITY 3.2.2 Aims and research design of SACMEQ 1. Explain the distinction between the two main aims of SACMEQ by using an example of countries participating in the study. 2. What would be the population of SACMEQ from which sampling is done? 3. What is the unit of analysis in SACMEQ? 4. Give your opinion on the possible reasons behind excluding special needs learners and small schools with class sizes of less than 15 learners. 58 INTERNATIONAL LARGE-SCALE ASSESSMENT LEARNING UNIT 3 3.2.3 Curriculum and assessment frameworks An assessment framework is determined by the content that will be assessed, which includes concepts and procedures. It is important to have an explicit curriculum framework from which the assessment framework can be drawn. The assessment items can then be designed to align with these frameworks. This alignment is critical for fairness of the test across countries and for the validity of the test. In the case of SACMEQ, the mathematics curricula of all countries were not the same. The curricula were therefore analysed and compared. Those common sections of the various curricula formed the core of the test instrument. Other items were added and signposted as outside of the common curriculum. South African education officials and researchers can be assured that the SACMEQ test items and the overall test are curriculum based. The results can therefore inform the education ministry about the steps to be taken to improve the education system. Achievement levels The achievement levels serve as targets to be achieved by countries and are established as follows: the results of the tests are analysed and arranged on a continuum of proficiency from pre-numeracy to abstract problem-solving. The learners are also arranged along the continuum. The percentage of learners attaining each level are then reported. As can be seen from the descriptions in table 3.6, they refer to mathematical thinking, starting with foundational concepts and skills and moving up the levels to abstract problem-solving. Though the questions are based on the curriculum content, the cognitive demands are embedded in the questions. TABLE 3.6 SACMEQ mathematics competency levels and their descriptions Basic numeracy Advanced mathematical skills PDM4801 Level Competencies Basic maths skills 1 Pre-numeracy Applies single-step addition and subtraction. 2 Emergent numeracy Applies a two-step addition and subtraction involving carrying 3 Basic numeracy Translates verbal information into arithmetic operations. 4 Beginning numeracy Translates verbal or graphic information into simple arithmetic problems. 5 Competent numeracy Translates verbal, graphic or tabular information into an arithmetic form to solve a given problem. 6 Mathematically skilled Solves multiple-operation problems (using the correct order) involving fractions, ratios and decimals. 7 Concrete problem solving Extracts and converts information from tables, charts and other symbolic presentations to identify, and then solve multi-step problems. 59 LEARNING UNIT 3 INTERNATIONAL LARGE-SCALE ASSESSMENT Level Competencies Basic maths skills 8 Abstract problem solving Identifies the nature of an unstated mathematical problem embedded within verbal or graphic information and then translate this into symbolic, algebraic or equation form to solve a problem. 3.2.4 Some results In the SACMEQ III project, the middle point for Grade 6 learners was about level 3, with most learners at basic levels of numeracy, clustering around levels 2 to 3, and about 8% of learners showing high mathematics competency levels, at 6, 7 and 8. In the SACMEQ IV project, we see the middle point is edging towards level 4, with clustering around level 3 and 4, and with 15% achieving high mathematical competency. TABLE 3.7 Performance results of Grade 6 learners comparing SACMEQ III and IV Level SACMEC III SACMEQ IV 1 2 3 4 5 6 7 8 5.5 34.7 14.1 29.0 35.1 15.4 7.1 20.3 14.8 5.9 7.7 1.0 4.6 0.6 2.6 0.8 From the information in table 3.7, the education officials can concentrate their efforts at reform. Some teachers were also tested on the same test. Here we should be careful not to generalise, as the teacher sample is not representative of the teaching population. We are talking about the percentage of Grade 6 learners at schools with teachers having attained a particular level. TABLE 3.8 Performance results of Grade 6 teachers, comparing SACMEQ III and IV Level SACMEC III SACMEQ IV 1 2 3 4 5 6 7 8 0 0 0 0 0.2 0 3.2 1.4 9.8 7.2 21.8 23.4 37.2 32.4 27.8 35.4 From table 3.8 we see that in the SACMEQ III project, 88.8% of Grade 6 learners were taught in schools where teachers attained an advanced level of understanding, at least a level 6 of the curriculum as operationalised in the test. In the SACMEQ IV project, 91.2% of Grade 6 learners were taught in schools where teachers had an advanced understanding, at least a level 6, of the curriculum as operationalised in the test. 60 INTERNATIONAL LARGE-SCALE ASSESSMENT LEARNING UNIT 3 ACTIVITY 3.2.4 SACMEQ – some implications How would you explain a situation whereby a teacher was tested and attained mathematical knowledge and skills at level 6, yet some of her learners attained at level 7 and/or 8? How would you explain the difference in results from Grade 6 learners in SACMEQ III and IV? Would it be fair to say that SACMEQ is not only assessing the system but also developing the system? 3.3 COMPARISON OF RESEARCH DESIGNS: INTERNATIONAL AND REGIONAL STUDIES In this short section, we ask you to highlight the features of the TIMSS and the SACMEQ research design, and then note which features are common and which features are different. Why is the research design important? The research design of a study is important because the reliability and validity of the results depend on the aspects of the design that have been built into the research. The following features are worth noting: ● The aims of the study ● The research question ● The curriculum framework ● The assessment framework ● The sampling process ● The unit of analysis PDM4801 61 LEARNING UNIT 3 INTERNATIONAL LARGE-SCALE ASSESSMENT ACTIVITY 3.3 TIMSS and SACMEQ research design Compare the TIMSS and SACMEQ research designs in terms of the following: ● ● ● ● 3.4 The scope of the study How the curriculum and assessment frameworks are conceptualised The sampling process The unit of analysis CONCLUSION This learning unit presented some features of the TIMSS study run by the IEA, with NRCs in the various countries and investigated the curriculum and assessment frameworks, as well as the test design. We also presented some aspects of the regional SACMEQ studies, run by the ministries of education in the participating countries, under the auspices of UNESCO-IILB. We presented the achievement levels and looked at some results. What is presented here is a taste of what is available of the TIMSS and the SACMEQ websites. There are also articles written about both studies. Check yourself: Have you reached the goal and the specific Tick the box outcomes of learning unit 3? Can you reflect on the analysis of international and regional comparisons of mathematics achievement? 62 1 Can you explain the rationale for the curriculum frameworks and describe key components, the content and the cognitive domains? 2 Can you present an overview of the assessment framework guiding the design of test items and booklets and explain the rationale provided by the IEA? INTERNATIONAL LARGE-SCALE ASSESSMENT LEARNING UNIT 3 Check yourself: Have you reached the goal and the specific Tick the box outcomes of learning unit 3? 3 Can you describe the curriculum frameworks and key components of the SACMEQ study under the auspices of UNESCO? 4 Can you present the assessment framework that guides the design of test items and explain the rationale provided by SACMEQ? 5 Can you compare the research designs, curriculum frameworks and the assessment frameworks of the international study and the regional study? ADDITIONAL LEARNING EXPERIENCES TIMSS Reports on the South African participation in TIMSS can be accessed from the Human Sciences Research Council (HSRC). The following publications can be accessed at https://timssandpirls.bc.edu/. TIMSS 2019 & TIMSS 2015 Mullis, IVS & Martin, MO. (Eds.) 2017. Assessment Frameworks Mullis, IVS Martin, MO, Foy, P & Hooper, M. 2016. International Results in Mathematics November 2016. Mullis, IVS, Martin, MO, Goh, S & Cotter, K. (Eds.) 2016. Encyclopedia: Education Policy and Curriculum in Mathematics and Science October 2016. Mullis, IVS, Martin, MO & Loveless, T. 2016. 20 Years of TIMSS: International Trends in Mathematics and Science Achievement, Curriculum, and Instruction November 2016. Martin, MO, Mullis, IVS & Hooper, M. (Eds.). 2016. Methods and Procedures in TIMSS 2015 2016. Test yourselves 1. What do the acronyms IEA and TIMSS stand for? 2. What years has South Africa participated in the TIMSS study and the PIRLS study? 3. The TIMSS tests Grade 4 and 8. Why were Grades 5 and 9 tested in South Africa? 4. What South African organisation administered the TIMSS and PIRLS? 5. When do the next TIMSS and PIRLS studies take place in the Southern Hemisphere? Some more questions 1. The IEA: its history and its purpose. Has the IEA achieved its purpose? 2. The countries that participate in TIMSS and the countries that do not. Name the top-performing countries. Why do some countries choose not to participate? PDM4801 63 LEARNING UNIT 3 INTERNATIONAL LARGE-SCALE ASSESSMENT 3. Sampling and population. Describe these two terms in relation to TIMSS and PIRLS. How does the rigorous sampling process try to ensure comparability of the participating countries? 4. The TIMSS curriculum frameworks. How are these frameworks structured? Can there be a common curriculum across 40 countries? 5. The PIRLS curriculum frameworks. How are these frameworks structured? What components of literacy are in focus? 6. TIMSS test design. The test instrument has a matrix design. This means that not all items are answered by all students. Describe this design. What are the advantages of such a design? 7. Scaling the results. A Rasch type analysis is conducted on the data. The resulting analysis is presented on a scale. Explain the scale. To what does the 500-centre point refer? How are the results compared from cycle to cycle? 8. The dissemination of results. How are the results presented? What information can individual countries use to reflect on the current functioning of their education system? 9. Should South Africa use its educational resources to take part in international tests? How might these resources be spent differently? SACMEQ Some more readings on SACMEQ Moloi, MQ & Chetty, M. 2010. The SACMEQ III Project in South Africa: A Study of the Conditions of Schooling and the Quality of Education. Test yourselves 1. What do the following acronyms stand for: UNESCO, IIEP AND SACMEQ? 2. What years has South Africa participated in SACMEQ? 3. What grade was SACMEQ studying? 4. What South African organisation administered SACMEQ? 5. When do the next SACMEQ studies take place? Some more questions 1. Has the SACMEQ achieved its purpose? 2. Sampling and population. Describe these two terms in relation to SACMEQ. How does the rigorous sampling process try to ensure comparability of the participating countries? 3. How are the SACMEQ tests designed? Can there be a common curriculum across 16 countries? 4. Scaling the results. A Rasch type analysis is conducted on the data. The resulting analysis is presented on a scale. How are the points on the scale described? 5. The dissemination of results. How are the results presented? What information can individual countries use to reflect on the current functioning of their education system? http://www.sacmeq.org/sites/default/files/sacmeq/reports/sacmeq-iii/nationalreports/s3_south_africa_final.pdf http://www.iiep.unesco.org/en/our-expertise/sacmeq 64 LEARNING UNIT 4 CROSS-EXAMINING THE CURRICULUM AND POLICY STATEMENT FOR MATHEMATICS GRADE R-12 LEARNINGUNIT4 TABLE OF CONTENTS 4. CROSS-EXAMINING THE CURRICULUM AND POLICY STATEMENT FOR MATHEMATICS GRADE R-12 INTRODUCTION – ALIGNING EDUCATION TO THEORETICAL FRAMEWORKS 4.1 SOME THEORETICAL PERSPECTIVES 4.2 A SUBJECT-CENTRED APPROACH 4.2.1 Why a theoretical and conceptual perspective? Activity 4.2.1 – The critical years – A problem identified and a challenge 4.2.2 Number, Operations and Relations and the Transitions Activity 4.2.2 – Transitions 1, 6 and 7 4.2.3 Pattern, functions and algebra and the transitions Activity 4.2.3 – Transitions 2, and 4 4.2.4 Space and shape (geometry) and measurement Activity 4.2.4 – Transitions 3 and 4 4.2.5 Data handling and probability Activity 4.2.5 – Transition 5 4.3 A PERSPECTIVE: A LEARNER-CENTRED APPROACH 4.3.1 Constructivist theory of learning Activity 4.2.1 4.3.2 Realistic mathematics education Activity 4.3.2 4.4 A PERSPECTIVE: A PROBLEM-SOLVING APPROACH 4.4.1 A philosophic perspective on problem-solving 4.4.2 A pedagogical perspective CONCLUSION ADDITIONAL LEARNING EXPERIENCES PDM4801 65 LEARNING UNIT 4 CROSS-EXAMINING THE CURRICULUM AND POLICY STATEMENT FOR MATHEMATICS GRADE R-12 OUTCOMES OF THIS LEARNING UNIT In this learning unit, you will interrogate topics in the CAPS mathematics curriculum to evaluate the trajectory of learning of mathematics through all grades. At the end of learning unit 4, you should be able to do the following: ● Align topics in the mathematics curriculum with the theoretical frameworks. ● Apply topics in the mathematics curriculum with the process of mathematisation. ● Critically reflect on a mathematics curriculum based on problem-solving. INTRODUCTION In the first part of learning unit 4, we look at the CAPS curriculum from a subjectcentred approach. In the second part, we explore how a constructivist-aligned realistic mathematics education approach may be applied within the parameters of the CAPS. And finally, we reflect on how a problem-solving approach has been integrated. The Curriculum and Assessment Policy Statement (CAPS) was developed from a curriculum known as Curriculum 2005, designed in 1997. The CAPS curriculum can be mapped against various frameworks and judged in terms of the components discussed in learning unit 1. In learning unit 4, we look at how the progression of topics align with learners’ developmental processes that are considered essential for mastering mathematics. We want to see how the CAPS document develops mathematical concepts along with learners’ abilities to accommodate the progression cognitively. This means we superimpose or overlay developmental processes over the CAPS document. These developmental processes have been named transitions by an eminent mathematics educationist, Zalman Usiskin, from the University of Chicago. The mathematical concepts introduced in Grades R to 3 form the foundation for the intermediate phase, Grades 4 to 6, which then ensures that critical concepts are mastered so that learners entering Grades 7 to 9 are prepared. The final three years of schooling are a culmination of all the learning that has taken place in the 12 years, both in and out of the classroom. Central to most curricula is a philosophical approach to mathematics itself and the learning of mathematics. Some theoretical perspectives are discussed here. We align the current approaches to the models discussed in an earlier learning unit. For the most part, constructivism is the preferred approach to teaching and learning. This theory and an enactment of constructivism that is theoretically informed (Realistic Mathematics Education) will be explored. 4.1 SOME THEORETICAL PERSPECTIVES In learning unit 1, we explored four commonly known models of curriculum, which are the product model, the process model, the subject-centred model and the learnercentred model. An article by Robitaille and Dirks (1982) gives a general explanation of the development of mathematical models from conceptualising the nature of mathematics to developing a model to the curriculum development process to the decision about what to include in school mathematics and how to organise it. 66 CROSS-EXAMINING THE CURRICULUM AND POLICY STATEMENT FOR MATHEMATICS GRADE R-12 (Nature of) Mathematics Curriculum development process LEARNING UNIT 4 School Mathematics A curriculum model originates from the principles and criteria held by the designers of a curriculum. It is a theoretical framework, often in diagram form, which reflects their philosophical paradigm of knowledge acquisition and instruction. Furthermore, it reflects their approach to the role (s) of the teacher, the subject and the learner. For easy reference, we repeat here what has already been discussed about models. The subject-centred model The subject-centred model resembles the traditional curriculum that focuses on a particular subject like mathematics or a discipline like languages. The learner is not in the centre of the design, but the instructional matter lies to the core of instruction. It demarcates what content should be taught and suggests how it should be learned. An example of a subject-centred curriculum is the so-called core curriculum, which is standardised across schools and provinces. Teachers are provided with a list of the content topics that they should teach, with examples of how they should teach the topics. A point of critique on this model is that it does not specifically take the learning styles and learning needs of learners into account, which may result in some learners falling behind and losing motivation. The learner-centred model In the learner-centred curriculum, each learner’s individual needs and development goals matter. The point of departure is that learners are not the same, are at different points in their development and have individual sets of needs. Learners may make choices within the curriculum because it is not a matter of one-size-fits-all. There is room for differentiation and learners have options regarding activities, assignments and learning experiences. It is generally accepted that this approach motivates and keeps learners engaged; however, from a teacher perspective, this is labour intensive and hard to attain in overcrowded classrooms, such as is often the case in South Africa. It requires highly skilled and experienced teachers to meet all learners’ needs at their own developmental level within the same class group. Individual learner’s choices and their needs should be maintained within a fine balance with the desired outcomes. The product model vs the process model In 1980, the Further Education Curriculum Review and Development Unit (FEU) in London made a broad distinction between the product and the process models. The product model focuses mainly on the planning and the intentions of the curriculum that aims for some kind of desirable curriculum end-product resulting from the learning experience. The emphasis in the process model is on activities and the effects of the curriculum through experiential learning in real life and through exposure to this world. The product model emphasises learning outcomes, while the process model emphasises the way towards attaining the desired goals. PDM4801 67 LEARNING UNIT 4 CROSS-EXAMINING THE CURRICULUM AND POLICY STATEMENT FOR MATHEMATICS GRADE R-12 The problem-centred model The problem-centred curriculum is closely related to the learner-centred curriculum. The goal is to teach learners how to look at a real-life problem and come up with solutions to prepare them for life and the problems that everyday living poses. This model makes the curriculum relevant and teaches learners to be innovative and creative in solving problems; however, learners at lower levels of problem-solving may be left behind, unless special attention is given to differentiation. In summary, a curriculum is never unidimensional – it is always made up of a number of facets, though the emphasis can be stronger on one element or another. In most curricula, as we have seen in this module, there is a strong emphasis on the subject matter to be covered. There has been in the past five decades an emphasis on a learner-centred approach. In many curricula, there has also been some attention to problem-solving. The excerpt below is taken from the CAPS Foundation Phase (2011: 5–8). It gives the broad and specific aims of the curriculum that apply from Grade 1 to Grade 9. Read through these Curriculum Statements to identifying the types of influences that determined these statements. TABLE 4.1 CAPS Foundation Phase (2011:5–8): Broad and specific aims of the curriculum Grade 1–9 FP, IP and SP Mathematics documents The National Curriculum Statement Grades R - 12 aims to produce learners that are able to: identify and solve problems and make decisions using critical and creative thinking; work effectively as individuals and with others as members of a team; organise and manage themselves and their activities responsibly and effectively; collect, analyse, organise and critically evaluate information; communicate effectively using visual, symbolic and/or language skills in various modes; use science and technology effectively and critically showing responsibility towards the environment and the health of others; ● demonstrate an understanding of the world as a set of related systems by recognising that problem solving contexts do not exist in isolation. ● ● ● ● ● ● 2.2. What is Mathematics? 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. 2.3. Specific Aims The teaching and learning of Mathematics aims to develop the following in the learner: ● critical awareness of how mathematical relationships are used in social, environmental, cultural and economic relations; ● confidence and competence to deal with any mathematical situation without being hindered by a fear of Mathematics; ● a spirit of curiosity and a love of Mathematics; ● appreciation for the beauty and elegance of Mathematics; ● recognition that Mathematics is a creative part of human activity; ● deep conceptual understanding in order to make sense of Mathematics; and 68 CROSS-EXAMINING THE CURRICULUM AND POLICY STATEMENT FOR MATHEMATICS GRADE R-12 LEARNING UNIT 4 ● acquisition of specific knowledge and skills necessary for: – the application of Mathematics to physical, social and mathematical problems; – the study of related subject matter (e.g. other subjects); and – further study in Mathematics. 2.4. Specific Skills To develop essential mathematical skills the learner should: ● develop the correct use of the language of Mathematics; ● develop number vocabulary, number concept and calculation and application skills; ● learn to listen, communicate, think, reason logically and apply the mathematical knowledge gained; ● learn to investigate, analyse, represent and interpret information. ACTIVITY 4.1 THEORETICAL AND CONCEPTUAL PERSPECTIVES Read through the curriculum statements that provide the broad aims of the CAPS. Identify a statement that tends towards each of the following models: 1. Subject-centred model 2. Learner-centred model 3. Problem-solving model Explain why you think the statement fits the particular model. FEEDBACK ON ACTIVITY 4.1 When the aims of a curriculum are investigated, we can see the intentions that lean towards certain curriculum models. However, the aims do not speak the last word: What happens further down in the curriculum, how are these aims materialised within the document, and most of all, how are these aims implemented in classrooms? In line with the feedback on Activity 4.1, we consider that we are only exploring the intended curriculum, but we are anticipating how the curriculum might be implemented. 4.2 A SUBJECT-CENTRED PERSPECTIVE ON THE CAPS For many learners, mathematics presents great difficulty. For some it is an altogether logical experience to do mathematics and solve problems. Many articles and books PDM4801 69 LEARNING UNIT 4 CROSS-EXAMINING THE CURRICULUM AND POLICY STATEMENT FOR MATHEMATICS GRADE R-12 have been written on the subject and many teachers have struggled with what makes mathematics difficult. Here is one view on the topic: 4.2.1 Why a theoretical and conceptual perspective? Teachers and researchers who have worked in mathematics education have developed explanations for why mathematics might be difficult for some learners. Here we use the work of Usiskin who states “the particular developments in mathematical thinking that take place in Grades 7 to 10, or perhaps 6 to 9, or perhaps 6 to 10 – or should take place in those grades – constitute the most important set of developments in a person’s mathematical schooling” (2004:4). These developments comprise the following seven important transitions: 1. From whole number to real number 2. From number to variable 3. From properties of individual figures to general properties of classes of figures 4. From inductive arguments to deductive ones 5. From operations on two numbers to statistics with sets of numbers 6. From informal description to formal definition of mathematical ideas 7. From a view of mathematics as a set of memorised facts to seeing mathematics as interrelated ideas accessible through a variety of means Usiskin (2004:4) goes on to say that “teachers assume that students have made many of these transitions”, but if they have not made these transitions, then they are forced to memorise their way through algebra, geometry and functions. Usiskin (2004) has identified the grades from 6 to 10 as the transition years. We will focus on the transitions and then look across the CAPS content domains to identify how we could enable these transitions to take place. Read: Usiskin, Z. 2004. The importance of the transition years, Grades 7–10, in school mathematics. UCSMP Newsletter. ACTIVITY 4.2.1 THEORETICAL AND CONCEPTUAL PERSPECTIVES 1. From the description above, write in your own words what Usiskin believes is the problem and the challenge of the “transition years”. 2. What do you think are some of the difficult transitions that learners need to make when progressing in mathematics? 70 CROSS-EXAMINING THE CURRICULUM AND POLICY STATEMENT FOR MATHEMATICS GRADE R-12 LEARNING UNIT 4 FEEDBACK ON ACTIVITY 4.2.1 1. From a teacher and researcher perspective, Usiskin focuses on the developments in mathematical thinking that have to take place. 2. You might have some other ideas about why some learners experience mathematics as difficult. 4.2.2 Number, operations and relationships and the transitions This section explains each of the transitions and then maps the transitions onto sections of the CAPS where we think they can be integrated. We will look at the intended curriculum and take a subject-centred perspective by focusing on mathematical ideas. We elaborate Transition 1 and 6 here, as follows: Transition 1: From whole number to real number This transition has three major steps, from whole number to fraction, from fraction to rational number and from rational number to real number. It takes place predominantly in the number, operations and relationships domain. These steps begin in the foundation phase (FP) with understanding about counting, one-to-one correspondence, addition and subtraction, multiplication and sharing and grouping. Informal work with parts of wholes and parts of quantities leads to a formal understanding of fractions in the intermediate phase (IP). There are at least three different understandings of fraction, as follows: ● Fraction as a quotient (one number divided by another number) ● Fraction as a number between zero and one ● Fraction as a number that is not an integer A second step is the realisation that there is a new number system, the rational number system, incorporating fractions, for which there are interesting characteristics. For example, between any two points on a number line depicting rational numbers, there is another number. Also for each point on the number line, there is an infinite number of representations (Vamvakoussi & Vosniadou 2007). The understanding is that the same number can have different representations in the form of common fractions, decimal fractions and percentages. Grade 6 and 7 learners need to understand these representations and the characteristics of fractions, decimals and percentages. Read: Vamvakoussi, X & Vosniadou, S. 2007. How many numbers are there in a rational numbers interval? The third step, making the transition from rational to real numbers, takes place in the senior phase. The conceptual challenges here are greater and require a relinquishing of the idea that a number only represents an object. Transition 6 in Usiskin’s (2004) list of transitions, the transition from using informal descriptions for mathematical ideas, to using formal precise definitions, also needs attention within this content domain. The all-encompassing Transition 7, which takes learners “from a view of mathematics as a set of memorized facts to seeing mathematics as interrelated ideas accessible through a variety of means”, requires attention from the mathematics teacher at the intermediate and the senior phases. The mathematical idea that fractions, decimal fractions and percentages can be used interchangeably, PDM4801 71 LEARNING UNIT 4 CROSS-EXAMINING THE CURRICULUM AND POLICY STATEMENT FOR MATHEMATICS GRADE R-12 requires flexibility of thought, but also a deeper understanding of fraction and rational number. Please complete Activity 4.2.2, where you will engage further with the application of these transitions. ACTIVITY 4.2.2 NUMBER, OPERATIONS AND RELATIONS: IDENTIFY THE TRANSITIONS In the above section, we have touched on the transitions that apply to the number, operations and relations domain. Table 4.1 gives an overview of the main themes in this domain from the foundation phase to the senior phase. A deep understanding of concepts in the early years paves the way for the transitions. 1. Identify four concepts in the curriculum that are important in making the transition from whole number to real number. 2. List four activities that you might introduce to assist learners in making the transitions that you have identified in question 1. FEEDBACK ON ACTIVITY 4.2.2 When we start engaging with the practical facilitation of transitions in mathematics, we see that both pedagogical skill and subject content knowledge are required to mediate successful transitions. Furthermore, it seems that the right time for the transition is at the exit of a previous level, to make the entry to the next level a little smoother. Table 4.2 may assist with the basic information needed to complete this activity. 72 CROSS-EXAMINING THE CURRICULUM AND POLICY STATEMENT FOR MATHEMATICS GRADE R-12 LEARNING UNIT 4 TABLE 4.2 Curriculum overview of FP, IP and SP: Numbers, Operations and Relationships (CAPS 2011) General Content Focus FP Specific Focus IP Specific Focus SP Specific Focus Development of number sense that includes: The number range includes whole numbers to at least 1 000 and common fractions. ● the meaning of different kinds of numbers; ● the relationship between different kinds of numbers; ● the relative size of different numbers; ● representation of numbers in various ways; and ● the effect of operating with numbers. Number concept is developed through working with physical objects to count collections of objects, partition and combine quantities, skip count in various ways, solve contextual (word) problems, and build up and break down numbers. The range of numbers developed by the end of the Intermediate Phase is extended to at least 9-digit whole numbers, decimal fractions to at least 2 decimal places, common fractions and fractions written in percentage form. Representation of numbers in a variety of ways and moving flexibly between representations. Counting enables learners to develop number concept, mental mathematics, estimation, calculation skills and recognition of patterns. Number concept development helps learners to learn about properties of numbers and to develop strategies that can make calculations easier. Solving problems in context enables learners to communicate their own thinking orally and in writing through drawings and symbols. In this phase, the learner is expected to move from counting reliably to calculating fluently in all four operations. The learner should be encouraged to memorise with understanding, multiply fluently, and sharpen mental calculation skills. Recognising and using properties of operations with different number systems. Solving a variety of problems, using an increased range of numbers and the ability to perform multiple operations correctly and fluently. Attention needs to be focused in understanding the concept of place They build an understanding value so that the learner develops a of basic operations of sense of large addition, subtraction, numbers and multiplication and division. decimal fractions. Learners develop fraction The learner should concept through solving recognise and problems involving the describe properties sharing of physical of numbers and quantities and by using drawings. Problems include operations, including identity solutions that result in properties, factors, whole number remainders or fractions. Sharing involves multiples, and finding parts of wholes, and commutative, associative and also finding parts of distributive collections of objects. properties. PDM4801 73 LEARNING UNIT 4 CROSS-EXAMINING THE CURRICULUM AND POLICY STATEMENT FOR MATHEMATICS GRADE R-12 General Content Focus FP Specific Focus IP Specific Focus SP Specific Focus Learners are not expected to read or write fraction symbols. 4.2.3 Patterns, functions and algebra and the transitions Transition 2, “from number to variable”, aligns with the patterns, functions and algebra domain. There are different uses of a variable. One of these uses is the unknown. The unknown is often represented by a block or a picture in the early years, which requires the learner to know what is in that place, for example: 2 + [ ] = 10 or 2 + = 10 or 2 + __ = 10 Another use of a variable is as a pattern generaliser. The pattern for multiples of three, is 3m, where m stands for any integer. Both Transition 4, “from inductive arguments to deductive ones”, and Transition 7, “from a view of mathematics as a set of memorised facts to seeing mathematics as interrelated ideas accessible through a variety of means”, find a place in this content domain. Please revise the difference between inductive and deductive reasoning: Inductive reasoning is a method of reasoning in which the premises are viewed as supplying some evidence for the truth of the conclusion. It is also described as a method where one’s experiences and observations, including what is learned from others, are synthesised to come up with a general truth. Deductive reasoning is the process of reasoning from one or more statements to reach a logically certain conclusion. In the practice of teaching, the following may happen in terms of inductive and deductive reasoning: Inductive reasoning: Learners observe patterns of things that happen in mathematics; for example, they see that when you divide a fraction by a fraction, the answer is larger than the fractions. Now they arrive at a conclusion or a rule for themselves – they have discovered it themselves through inductive reasoning. Deductive reasoning: The teacher gives the facts, the rules, the formulae, the steps, the methods, the properties and the definitions and learners accept it and apply it. They can even apply it correctly without really understanding it or owning up to it. A mathematics example that contrasts the two types of reasoning is the following: When adding two odd numbers together one always gets an even number. Inductive reasoning would go like this, “I have added 100 pairs of odd numbers and I always get an even number, therefore this rule must be true”. This is inductive reasoning relying on observation. A deductive argument would go as follows: “Every odd number is made up of an even number plus one odd number. Two even numbers added together will result in an even number. The two extra odd numbers joined together makes an even number.” 74 CROSS-EXAMINING THE CURRICULUM AND POLICY STATEMENT FOR MATHEMATICS GRADE R-12 LEARNING UNIT 4 Or better still, we define an even number as a multiple of 2 and call it 2n. An odd number will then be 2n + 1. Another odd number we call 2m +1. When we add the two, we get 2n + 1 + 2m +1, which is equal to 2n + 2m + 2. We take out a common factor and get 2 (m + n + 1). This number is a multiple of two and therefore fits our definition of an even number. ACTIVITY 4.2.3 PATTERNS, FUNCTIONS AND ALGEBRA: IDENTIFY THE TRANSITIONS The above section touched on the transitions that apply to the patterns, functions and algebra domain. Table 4.3 gives an overview of the main themes in this domain from the foundation phase to the senior phase from CAS (2011). A deep understanding of concepts in the early years paves the way for the transitions. 1. Identify two concepts in the curriculum that are important in making the transition from a number to a variable. 2. List two activities that you might use to help learners in making the transitions. FEEDBACK ON ACTIVITY 4.2.3 The transition to variables may be one of the most important thresholds to overcome in guiding learners’ development towards abstract mathematical thinking. PDM4801 75 LEARNING UNIT 4 CROSS-EXAMINING THE CURRICULUM AND POLICY STATEMENT FOR MATHEMATICS GRADE R-12 TABLE 4.3 Curriculum overview of FP, IP and SP: Pattern, Functions and Algebra (CAPS 2011) General Content Focus FP Specific Focus IP Specific Focus SP Specific Focus Algebra is the language for investigating and communicating most of Mathematics and can be extended to the study of functions and other relationships between variables. A central part of this content area is for the learner to achieve efficient manipulative skills in the use of algebra. It also focuses on the: In this phase, learners work with both Numeric and geometric patterns are extended with a special focus on the relationships: – between terms in a sequence – between the number of the term (its place in the sequence) and the term itself. Investigation of numerical and geometric patterns to establish the relationships between variables. ● description of patterns and relationships through the use of symbolic expressions, graphs and tables; and ● identification and analysis of regularities and change in patterns, and relationships that enable learners to make predictions and solve problems. ● number patterns (e.g. skip counting); and ● geometric patterns (e.g. pictures). Learners should use physical objects, drawings and symbolic forms to copy, extend, describe and create patterns. Copying the pattern helps learners to see the logic of how the pattern is made. Extending the pattern helps learners to check that they have properly understood the logic of the pattern. Describing the pattern helps learners to develop their language skills. Focussing on the logic of patterns lays the basis for developing algebraic thinking skills. Number patterns support number concept development and operational sense built in Numbers, Operations and Relationships. The study of numeric and geometric patterns develops the concepts of variables, relationships and functions. The understanding of these relationships will enable learners to describe the rules generating the patterns. This phase has a particular focus on the use of different, yet equivalent, representations to describe problems or relationships by means of flow diagrams, tables, number sentences or verbally. Expressing rules governing patterns in algebraic language or symbols. Developing algebraic manipulative skills that recognize the equivalence between different representations of the same relationship. Analysis of situations in a variety of contexts in order to make sense of them. Representation and description of situations in algebraic language, formulae, expressions, equations and graphs. Geometric patterns include sequences of lines, shapes and objects but also patterns in the world. In geometric patterns learners apply their knowledge of space and shape. 4.2.4 Space, shape, geometry and measurement and the transitions Transition 3, the transition from identifying “properties of individual figures to general properties of classes of figures”, is a necessary transition in the space, shape and geometry domain. Transition 4, “from inductive arguments to deductive ones” is also a 76 CROSS-EXAMINING THE CURRICULUM AND POLICY STATEMENT FOR MATHEMATICS GRADE R-12 LEARNING UNIT 4 central construct in Euclidean geometry, where axioms and theorems are used as truths that do not need to be proved again, but that can be applied in proofs. Usiskin (2004) provides an example of the triangle and the sum of its angles. Taking an inductive approach, a teacher may ask the class to take a protractor, measure each of the angles and then add them. We might find some learners measuring accurately, some may get an answer of less than 180 and others an answer of more than 180. This route would be investigating the properties of individual figures, rather than investigating the properties of classes of figures. A deductive route would start with the fact of a revolution being 360 (a fact established by the Babylonians eons ago). The next step is to make a tessellation of a triangle as in Activity 4.2.4. Figure 4.1: Tessellation of a triangle The angles of the triangles are tessellated about the vertex, with each angle of the triangle appearing twice. Therefore 2 (Angle 1 + Angle 2 + Angle 3) = 360 Therefore Angle 1 + Angle 2 + Angle 3 = 180 ACTIVITY 4.2.4 SPACE, SHAPE, GEOMETRY, MEASUREMENT: IDENTIFY THE TRANSITIONS We have now touched on the transitions that apply to the space, shape, and geometry domain, which link to some extent to the measurement domain. Tables 4.4 and 4.5 give an overview of the main themes in this domain from FP to SP. 1. Identify three concepts in the curriculum that are important in making the transition from identifying “properties of individual figures to general properties of classes of figures”. 2. Identify three concepts that you might use to help learners making the transitions in measurement. PDM4801 77 LEARNING UNIT 4 CROSS-EXAMINING THE CURRICULUM AND POLICY STATEMENT FOR MATHEMATICS GRADE R-12 3. Identify transitions in space and shape (geometry) that are made from FP to IP and form IP to SP. You may want to look at Van Hiele’s levels of geometric thinking to inform your response. FEEDBACK ON ACTIVITY 4.2.4: Please revise Van Hiele’s levels of geometrical thought dealing with transitions: Level 0: Visualisation Level 1: Analysis Level 2: Abstraction Level 3: Deduction Level 4: Rigor Describes shapes based on their appearance Describes shapes based on their properties Recognises relationships between properties Proves theorems deductively through logical reasoning Establishes and analyses theorems TABLE 4.4 Curriculum overview of FP, IP and SP: Space and Shape (Geometry) (CAPS 2011) General Content Focus FP Specific Focus IP Specific Focus SP Specific Focus The study of Space and Shape improves understanding and appreciation of the pattern, precision, achievement and beauty in natural and cultural forms. It focuses on the In this phase learners focus on threedimensional (3-D) objects, two dimensional (2-D) shapes, position and directions. The learner’s experience of space and shape in this phase moves from recognition and simple description to classification and more detailed description of characteristics and properties of twodimensional shapes and three-dimensional objects. Drawing and constructing a wide range of geometric figures and solids using appropriate geometric instruments. ● properties, relationships; ● orientations, positions; and 78 Learners explore properties of 3-D objects and 2-D shapes by sorting, classifying, describing and naming them. Learners draw shapes and build with objects. Developing an appreciation for the use of constructions to investigate the properties of geometric figures and solids. CROSS-EXAMINING THE CURRICULUM AND POLICY STATEMENT FOR MATHEMATICS GRADE R-12 LEARNING UNIT 4 General Content Focus FP Specific Focus IP Specific Focus SP Specific Focus ● transformations of twodimensional shapes and threedimensional objects. Learners recognise and describe shapes and objects in their environment that resemble mathematical objects and shapes. Learners should be given opportunities to: – draw two-dimensional shapes and make models of threedimensional objects – describe location, transformations and symmetry. Developing clear and more precise descriptions and classification categories of geometric figures and solids. Learners describe the position of objects, themselves and others using the appropriate vocabulary. Learners follow and give directions. Solving a variety of geometric problems drawing on known properties of geometric figures and solids. TABLE 4.5 Curriculum overview of FP, IP and SP: Measurement (CAPS 2011) General Content Focus FP Specific Focus IP Specific Focus SP Specific Focus Measurement focuses on the selection and use of appropriate units, instruments and formulae to quantify characteristics of events, shapes, objects and the environment. It relates directly to the learner’s scientific, technological and economic worlds, enabling the learner to: In this phase the learners’ concept of measurement is developed by working practically with different concrete objects and shapes, learning the properties of length, capacity, mass, area and time. Learners should be exposed to a variety of measurement activities. Using formulae for measuring area, perimeter, surface area and volume of geometric figures and solids. ● make sensible estimates; and ● be alert to the reasonableness of measurements and results. PDM4801 Learners measure the properties of shapes and objects using informal units where appropriate, such as hands, paces, containers, etc. Learners compare different quantities by using comparative words such as taller/ shorter, heavier/lighter etc. Learners are introduced to standard units such as grams, kilograms; millilitres, litres; centimetres, metres. Learners should be introduced to the use of standardised units of measurement and appropriate instruments for measuring. They should be able to estimate and verify results through accurate measurement. Learners should be able to select and convert between appropriate units of measurement. ● Selecting and converting between appropriate units of measurement. ● Using the Theorem of Pythagoras to solve problems involving rightangled triangles. Measurement in this phase should also enable the learner to: – informally measure angles, area, perimeter and capacity/volume; – discuss and describe the historical development of measuring instruments and tools. 79 LEARNING UNIT 4 CROSS-EXAMINING THE CURRICULUM AND POLICY STATEMENT FOR MATHEMATICS GRADE R-12 General Content Focus FP Specific Focus IP Specific Focus Activities related to time should be structured with the awareness that learners’ understanding of the passing of time should be developed before they read about time. Measurement provides a context for learners to use common fractions and decimal fractions. SP Specific Focus 4.2.5 Data handling and probability and the transitions Transition 5, “from operations on two numbers to statistics with sets of numbers”, is central to data and probability. For ease of reference, one might find a report saying that the average South African family has 3,5 children. This does not mean that each family has three whole children and then a half child, but rather that when they surveyed 100 families, they found that there were 350 children. ACTIVITY 4.2.5 DATA HANDLING AND PROBABILITY: IDENTIFY THE TRANSITIONS In the above section, we have touched on the transitions that apply to the data handling and statistics domain. Table 4.6 gives an overview of the main themes in this domain from the foundation phase to the senior phase. 1. Identify three concepts in the curriculum that are important in making the transition from a number to a variable. 2. List three activities that you might introduce to assist learners in making the transitions. 80 CROSS-EXAMINING THE CURRICULUM AND POLICY STATEMENT FOR MATHEMATICS GRADE R-12 LEARNING UNIT 4 FEEDBACK ON ACTIVITY 4.2.5 Note the progression of this content domain across phases. TABLE 4.6 Curriculum overview of FP, IP and SP: Data handling and probability General Content Focus FP Specific Focus IP Specific Focus SP Specific Focus Data handling involves asking questions and finding answers in order to describe events and the social, technological and economic environment. Through the study of data handling, the learner develops the skills to collect, organize, represent, analyze, interpret and report data. Progression in Data Handling Learners should focus on all the skills that enable them to move from collecting data to reporting on data. Posing of questions for investigation. The main progression in Data Handling across the grades Learners should be is achieved by: exposed to: – a variety of contexts for − moving from collecting and working with interpreting data – a objects to range of questions that working with are posed and data; and answered related to − working with data. new forms of data Learners should begin representato analyse data critically The study of tion. through exposure to probability enables some factors that the learner to Learners should impact on data such as develop skills and work through from whom, when and techniques for the full data where data is collected. making informed cycle at least predictions, and once a year - this The focus of probability is to perform repeated describing involves events in order to list, randomness and collecting and uncertainty. organising data, count and predict outcomes. It develops awareness representing data, analysing, Learners are not that – different interpreting and expected to calculate situations have the probability of different probabilities reporting data. events occurring. of occurring – for Some of the many situations, above aspects of there are a finite data handling number of different can also be dealt possible outcomes. with as discrete activities. Collecting, summarizing, representing and critically analysing data in order to interpret, report and make predictions about situations. ● Probability of outcomes include both single and compound events and their relative frequency in simple experiments. (CAPS Grade R-3 p.34) 4.3 A PERSPECTIVE ON THE CAPS: A LEARNER-CENTRED APPROACH Many curricula advocate a learner-centred approach. Most notable is the curricula that lean heavily on the constructivist theory of learning; however, there is also realistic PDM4801 81 LEARNING UNIT 4 CROSS-EXAMINING THE CURRICULUM AND POLICY STATEMENT FOR MATHEMATICS GRADE R-12 mathematics education that aligns very well with the constructivist theory, though keeping an eye on the goal of learning formal mathematics. Both a constructivist theory and realistic mathematics education promote problem-solving as one of their central principles. In this section, we are going to first define constructivism, then describe the core ideas and then look at some of the CAPS statements to identify the match with a constructivist theory of learning. Secondly, we define realistic mathematics education, describe the core principles and identify which of these themes are present in the CAPS curriculum. 4.3.1 Constructivist theory of learning Read the following to enrich your insight into constructivism: McLeod, SA. 2019. Constructivism as a theory for teaching and learning. Simply Psychology. Available at https://www.simplypsychology.org/constructivism.html. Constructivism is “an approach to learning that holds that people actively construct or create their own knowledge and that reality is determined by the experiences of the learner” (Elliott et al 2000:256, in McLeod 2019). Further elaborating this notion, Arends (1998) notes that the guiding principle of this theory is that learners construct meaning through experience. This meaning is created by building on prior knowledge and new events. The principles of constructivism are as follows: ● Knowledge is constructed rather than innate. This principle ascribes learning to the educational environment, rather than to inherited so-called intelligence. ● Learning is an active process. The learner can, through actively engaging in an activity, construct meaning to knowledge. ● All knowledge is socially constructed. The community plays a guiding role. The term ‘zone of proximal development’ refers to that cognitive space in which a child can benefit from specific guidance from another person (Vygotsky 1978). ● Knowledge is personal. There is always a subjective interpretation of the knowledge that is transmitted and received. ACTIVITY 4.3.1 A CONSTRUCTIVIST PERSPECTIVE: IDENTIFY THE PRINCIPLES IN CAPS Read the statements in the excerpt from CAPS. Identify which constructivist principles align with these statements. For example, "All knowledge is socially constructed", aligns in some way with the second statement above. FP, IP and SP Mathematics documents (from CAPS) The National Curriculum Statement Grades R - 12 aims to produce learners that are able to: ● identify and solve problems and make decisions using critical and creative thinking (1); 82 CROSS-EXAMINING THE CURRICULUM AND POLICY STATEMENT FOR MATHEMATICS GRADE R-12 LEARNING UNIT 4 ● work effectively as individuals and with others as members of a team (2); ● organise and manage themselves and their activities responsibly and effectively (3); ● collect, analyse, organise and critically evaluate information (4); ● communicate effectively using visual, symbolic and/or language skills in various modes (5); ● use science and technology effectively and critically showing responsibility towards the environment and the health of others (6); and ● demonstrate an understanding of the world as a set of related systems by recognising that problem-solving contexts do not exist in isolation (7). FEEDBACK ON ACTIVITY 4.3.1 The paradigm or theoretical approach to which a curriculum adheres is seldom overtly observable. More often, we only become aware of that when we start understanding the implications of an educational paradigm or approach. 4.3.2 Realistic mathematics education Realistic mathematics education (RME) came from the Netherlands. In learning unit 2, we discussed some aspects of the Netherlands curriculum. Van den Heuvel- Panhuizen (2014:521) explains RME as follow: (Also available at https://www.icrme.net/uploads/1/0/9/8/109819470/rme_ encyclopaediamathed.pdf) “Realistic Mathematics Education – hereafter abbreviated as RME – is a domainspecific instruction theory for mathematics, which has been developed in the Netherlands. Characteristic of RME is that rich, ‘realistic’ situations are given a prominent position in the learning process. These situations serve as a source for initiating the development of mathematical concepts, tools, and procedures and as a context in which students can in a later stage apply their mathematical knowledge, which then gradually has become more formal and general and less context specific”. The core teaching principles guiding RME (Van den Heuvel-Panhuizen 2014:522–523) are the following: ● The activity principle. Active participation by learners is required. The learners must do mathematics and work with the concepts. Here mathematics is seen as “a human activity” (Freudenthal 1978). ● The reality principle. Learners must apply mathematics to solving “real-life” problems. The problem situations with which they engage should be meaningful to PDM4801 83 LEARNING UNIT 4 CROSS-EXAMINING THE CURRICULUM AND POLICY STATEMENT FOR MATHEMATICS GRADE R-12 the learner. The contexts should be rich and interesting to capture the interest and imagination of the learner. ● The level principle. The learner moves through levels of understanding. The first level involves “finding informal context-related solutions”. The next level involves creating various levels of short-cuts and schematisation – here the learner is beginning to experience the power of mathematical abstraction. ● The intertwinement principle. This level is when learners acquire insight into how concepts and strategies are related. The various content domains such as number, geometry, measurement and data handling are not regarded as separate and isolated sections, but rather as integrated. ● The interactivity principle. Although doing mathematics is an individual activity, it also is a social activity and learners benefit from interacting with others and reflecting on other strategies and ideas. ● The guidance principle. The term “guided reinvention” is used in RME to signify that learners are not inventing mathematics, some of which was discovered thousands of years ago, but rather they are developing their mathematical thinking. In the classroom, learners should often be having an ‘aha moment’, signifying that they have invented a new way of doing mathematics, that is new for the particular child. Two important concepts in RME are horizontal mathematisation and vertical mathematisation, a concept introduced by Treffers (1977). Horizontal mathematisation refers to the encountering of concepts in the everyday environment and learners informally applying their existing knowledge to solve these problems. In the classroom setting, the teacher can also purposefully present a situation that she planned to introduce a specific mathematical concept. During this informal way of engagement with mathematical ideas from the real world, learners may build their own mental, pictorial, symbolic or graphical models to express the way they understand the situation mathematically. Vertical mathematisation refers to the teacher following up on the learners’ informal explorations, by formalising the mathematical idea into a formula, a strategy, a theorem or a rule. She has to be alert to find opportunities to engage learners with higher, more formal mathematics. ACTIVITY 4.3.2 REALISTIC MATHEMATICS EDUCATION: REWRITING THE CURRICULUM In this section, we encountered the principles of RME. The CAPS states that mathematics is a “human activity”, thereby acknowledging these principles, but they are not explicit. Read the excerpt from CAPS below. In the right-hand column, rewrite what learners should be able to do using RME principles. FP, IP and SP Mathematics documents (from CAPS) The National Curriculum Statement Grades R - 12 aims to produce learners that are able to: ● identify and solve problems and make decisions using critical and creative thinking (1); 84 CROSS-EXAMINING THE CURRICULUM AND POLICY STATEMENT FOR MATHEMATICS GRADE R-12 LEARNING UNIT 4 ● work effectively as individuals and with others as members of a team (2); ● organise and manage themselves and their activities responsibly and effectively (3); ● collect, analyse, organise and critically evaluate information (4); ● communicate effectively using visual, symbolic and/or language skills in various modes (5); ● use science and technology effectively and critically showing responsibility towards the environment and the health of others (6); and ● demonstrate an understanding of the world as a set of related systems by recognising that problem-solving contexts do not exist in isolation (7). FEEDBACK ON ACTIVITY 4.3.2 Teachers may have many reasons why RME cannot work in the real classroom. One of the reasons would be that time is too limited to let learners engage in exploration before the teacher formalises the mathematics. At this stage, your thinking is already mature enough to build a counter argument why the benefits of RME outweigh the disadvantages. 4.4 A PERSPECTIVE: MATHEMATICS AS PROBLEM-SOLVING Mathematics is about solving problems, sometimes these problems are purely mathematical and sometimes the problem is rooted in a real-life context. Each theory in learning unit 4 has problem-solving as central to their philosophy. Usiskin’s (2004) transitions implicitly encourage learners and their teachers to develop more advanced mathematics. In RME, the goal is for learners to develop more formal mathematics through mathematisation. A constructivist approach ensures that learners have meaningful concepts that have been securely constructed through engagement with the concept. Check yourself: Have you reached the goal and the specific Tick the box outcomes of learning unit 4? Can you interrogate topics in the mathematics curriculum to evaluate the trajectory of learning of mathematics through all grades? ● Critically reflect on a mathematics curriculum based on problem solving. 1 PDM4801 Can I describe some similarities and differences between mathematics curricula in four countries, notably in terms of their curriculum management structures? 85 LEARNING UNIT 4 CROSS-EXAMINING THE CURRICULUM AND POLICY STATEMENT FOR MATHEMATICS GRADE R-12 2 Can I apply topics in the mathematics curriculum with the process of mathematisation? 3 Can I critically reflect on a mathematics curriculum based on problem-solving? Additional learning experiences Arends, RI. 1998. Resource handbook. Learning to teach (4th ed.). Boston, MA: McGrawHill. Dewey, J. 1938. Experience and Education. New York: Collier Books. Driscoll, M. 2000. Psychology of Learning for Instruction. Boston: Allyn & Bacon Elliott, SN, Kratochwill, TR, Littlefield Cook, J & Travers, J. 2000. Educational psychology: Effective teaching, effective learning (3rd ed.). Boston, MA: McGraw-Hill College. Freudenthal, H. 1983. Didactical phenomenology of mathematical structures. Reidel Publishing: Dordrecht. Freudenthal, H. 1991. Revisiting mathematics education. China lectures. Kluwer: Dordrecht Gravemeijer, KPE. 1994. Developing realistic mathematics education. CD-ß Press/ Freudenthal Institute: Utrecht. Schoenveld, AH & Arcavi, A. 1988. On the meaning of a variable. The Mathematics Teacher. 81(6): 420–427 Streefland L. 1991. Fractions in realistic mathematics education. A paradigm of developmental research. Kluwer: Dordrecht Treffers A. 1978. Wiskobas doelgericht [Wiskobas goaldirected]. IOWO: Utrecht Usiskin, A. (2004). A K-12 Mathematics curriculum with CAS: What is it and what would it take to get it? In W. Yang, S. Chu, T. de Alwis & K. Ang (Eds.), Proceedings of the 9th Asian technology conference in mathematics, 5–16. Blacksburg, VA: ATCM, Inc. Vamvakoussi, X & Vosniadou, S. (2007). How many numbers are there in a rational number interval? Constraints, synthetic models, and the effect of the number line. In S. Vosniadou, A Baltas & X. Vamvakousi (Eds.), Reframing the Conceptual Change Approach.in teaching and learning, 265–282. Oxford: Elsevier. Van den Heuvel-Panhuizen, M & Drijvers, P. 2014. Realistic Mathematics Education. In S Lerman (ed.), Encyclopedia of Mathematics Education, DOI 10.1007/978-94-007-49788, # Springer Science+Business Media Dordrecht 2014. Vygotsky, LS. 1978. Mind in Society: The development of higher psychological processes. Cambridge: Harvard 86 LEARNING UNIT 5 DESIGNING MATHEMATICS CURRICULA IN CONTEXT LEARNINGUNIT5 5. 5.1 5.2 DESIGNING MATHEMATICS CURRICULA IN CONTEXT INTRODUCTION A COMMUNITY AND ITS CONTEXT Activity 5.1: My Community Mathematics Education Curriculum – its Context THE SPIDERWEB OF CURRICULA COMPONENTS Activity 5.2.1: My CMEC – Aims and objectives Activity 5.2.2: My CMEC – Content and learning activities Activity 5.2.3: My CMEC – Teacher’s role and pedagogical principles Activity 5.2.4: My CMEC – Location and time 5.2 5.2.5 Assessment Activity 5.2.5: My CMEC – Assessment 5.3 AN INTEGRATED VIEW OF A MATHEMATICS CURRICULUM 5.3.1 Vision for a mathematics classroom Activity 5.3.1: Vision for a mathematics classroom 5.3.2 The learner and a productive disposition Activity 5.3.2: The learner and a productive disposition CONCLUSION ADDITIONAL LEARNING EXPERIENCES OUTCOMES OF THIS LEARNING UNIT In this learning unit, you will be exposed to designing a mathematics curriculum for a particular context. At the end of learning unit 5, you should be able to do the following: ● Analyse the structure of mathematics curricula in selected countries. ● Formulate an integrated view of a mathematics curriculum. PDM4801 87 LEARNING UNIT 5 DESIGNING MATHEMATICS CURRICULA IN CONTEXT INTRODUCTION The purpose statement of this entire module is to provide students with an insight into what the various components are that make up a mathematics curriculum. It is these components that vary from country to country and even from school to school. When one thinks of mathematics, one rarely thinks that there are many different perspectives about mathematics. Most often we think about numbers, algebra and geometry. In the first four learning units we have explored many different aspects of what would be considered when planning and developing a mathematics curriculum. For the education ministries of the many countries in the world, the planning for the mathematics curricula and the development of the curricula are connected with the needs of the country. An interesting story is that of Singapore, which focused from the 1960s to the 1980s on basic mathematics concepts and skills to give the whole country a sure foundation with adequate numeracy and literacy rates. From the 1980s; however, their approach has been more progressive with a focus on engaging with problem situations and developing problem-solving skills. Currently, Singapore has one of the highest achieving cohorts in international tests in mathematics. A country such as the United States of America is too diverse to demand a uniform curriculum across the various states. The central government, however, does stipulate standards for the entire country. By looking closely at the various countries’ curricula, we can hypothesise what components are significant and even think about incorporating some aspects of other curricula into one’s own. The critical notion, however, is that the country context is considered. What about South Africa, or one of the Southern Africa countries, or the East or North African countries? How might these curricula differ? And how might they be improved? Of course, the ministries of education can consider what is happening elsewhere in the world, but the danger might be that they adopt an external curriculum without adapting it to suit the particular country’s needs. 5.1 A COMMUNITY AND ITS CONTEXT In South Africa, we have a common curriculum for the entire country. If we encouraged region-specific curricula, how might they look? For example, how might you design a Community Mathematics Education Curriculum for the region where you live? Take a moment to think about the context of the community while you complete Activity 5.1. 88 DESIGNING MATHEMATICS CURRICULA IN CONTEXT LEARNING UNIT 5 ACTIVITY 5.1 My community mathematics education curriculum – its context Where is the community? Provide a rough map to pinpoint the community to which you refer. What is the composition of the people of this community? Describe the lives of the families that live in the community. What are the aspirations of the learners? What education programmes might ensure the progress and development of the community? What is your vision for the community in 10 years? 5.2 THE SPIDERWEB OF CURRICULA COMPONENTS In learning unit 4, we referenced an article by Robitaille and Dirks (1982) that argued a clear line from conceptualising the nature of mathematics to developing a model to the curriculum development process to the decision about what to include in school mathematics and how to organise it. We presented it as follows: (Nature of) Mathematics Curriculum development process School Mathematics We can consider three philosophical perspectives on mathematics, which then translate into the curriculum development process and then into what might be expected of school mathematics. These are the Platonist philosophy, the formalist philosophy and the constructivist (or intuitionist) philosophy, as has been discussed in previous learning units of this module. The spiderweb (Van den Akker 2006) guides us on the components of the curriculum. We start with the formulation of aims and objectives (to be explored in 5.1.1), the decision about content and learning activities (in 5.1.2), the conceptualisation of the teacher’s role, pedagogical principles and educational resources (in 5.1.3), time and location arrangements (discussed in 5.1.4) and finally assessment (in 5.1.5). PDM4801 89 LEARNING UNIT 5 DESIGNING MATHEMATICS CURRICULA IN CONTEXT Duplicate of figure 1.2: The curricular spiderweb (Van den Akker 2003 in Thijs & Van den Akker 2009:11) You may draw on learnings from the previous learning units and outside readings to design you own aims and objectives for the curriculum that you envisage for your particular context, which we shall call your Community Mathematics Education Curriculum (CMEC). 5.2.1 Aims and objectives What are the aims and objectives of the curriculum for your CMEC? The aims and objectives guide everything that follows. They require looking beyond the present. What broad mathematical ideas would you want the learners in your educational project to have? The learners who pass through your CMEC will go out into the world of work. What kind of learners do you envisage stepping into the workplace? ACTIVITY 5.2.1 MY CMEC – AIMS AND OBJECTIVES Write a broad overarching aim for your CMEC and focus on the mathematics component of the curriculum. Write five objectives covering the following: ● The mathematical content that will be included in your CMEC ● The learner disposition (look up this term, we are going to use it again) 90 DESIGNING MATHEMATICS CURRICULA IN CONTEXT LEARNING UNIT 5 ● The teacher’s role ● The classroom environment ● The assessment programme FEEDBACK ON ACTIVITY 5.2.1 With this short exposure to curriculum design, you can see that a context can make a huge difference in terms of the curriculum components, mathematical ideas, learner disposition, teacher’s role, classroom environment and assessment approach. There is also the ever-present element of resources to be taken into account. 5.2.2 Content and learning activities This section focuses on the learning content to be included and the learning activities that will best enable the learning of concepts. Here we focus on the intermediate phase and we chose one of the five content domains, namely number, operations and relations, patterns, function and algebra, space and shape, measurement or data handling. You will write down what content you would cover and what learning activities you think would achieve the learning of these concepts, for two weeks (ten hours) in measurement at Grade 6. ACTIVITY 5.2.2 MY CMEC – CONTENT AND LEARNING ACTIVITIES Grade 6 Mathematics: Measurement (1) Week 23 and 24 (2) Topics: Complete the table with two topics and two corresponding activities for each topic Topic 1. Activity (a) Activity (b) Topic 2. Activity (a) Activity (b) PDM4801 91 LEARNING UNIT 5 DESIGNING MATHEMATICS CURRICULA IN CONTEXT FEEDBACK ON ACTIVITY 5.2.2 In this activity you kept in mind the context of your CMEC when you choose two topics, those that would be relevant for them in their own situation (e. g. time, volume, mass, length, temperature, area). 5.2.3 Teacher’s role, pedagogical principles, materials and resources Teachers are human beings with intelligence, independent thought and autonomy, they are not robots. Their role is what is expected of them and must be flexible to cater for differences. General guidelines provided by the curriculum within an educational system are helpful though, both for the system and for the teacher. Pedagogical principles, such as those provided in the constructivist philosophy or the realistic mathematics education approach to teaching mathematics are helpful if one understands the rationale behind the principle. In Activity 5.2.3, we would like you to broadly describe the teacher’s role and then formulate some pedagogical principles. You can draw on the principles from previous learning units, but then adapt them to suit your CMEC. ACTIVITY 5.2.3 MY CMEC – TEACHER’S ROLE AND PEDAGOGICAL PRINCIPLES Describe the role of the teacher (in your community project CMEC) in five lines. Draw on progressive educational principles to formulate the five main principles that could guide your teachers in this community project. Think of these principles as what you would communicate with them when you employ them and then use in your appraisal meeting with them, to evaluate their performance. 1. 2. 3. 4. 5. FEEDBACK ON ACTIVITY 5.2.3 In formulating the role of the teacher and selecting the guiding principles for their performance in your community project, you are setting standards according to international best practice; however, you shape and adapt all of this according to the context of this particular CMEC. 92 DESIGNING MATHEMATICS CURRICULA IN CONTEXT LEARNING UNIT 5 5.2.4 Location and time Most educational activities happen in a classroom. For some activities, however, a classroom may not be the best location. For example, in one school the learners were being taught about making bread. They first sowed the wheat, watered it regularly, and when it was ready, they harvested the wheat, ground it into flour and made the bread. Some activities may best be conducted in places other than a classroom. In most schools, the day is carved up into 30-, or 40- or 60-minute periods, although some schools have a different philosophy. For example, the Waldorf schools have a main lesson that lasts for about two hours every morning. The lesson focuses on a single theme for two to three weeks. This arrangement enables them to cover concepts in depth. Activity 5.2.4 is about how you would like to organise the location of teaching activities and the time allocation. ACTIVITY 5.2.4 MY CMEC – LOCATION AND TIME Identify two alternative locations outside a classroom where learning can take place. 1. 2. Plan an alternative timetable from 08:00 to 14:00 for one day of the week. FEEDBACK ON ACTIVITY 5.2.4 This activity took us out of the box! Maybe we think the way we habitually do things in school, is the only way – no, we have seen that within various contexts, even location and time management may differ – and fruitfully so. 5.2.5 Assessment There is a view that assessment drives what is taught in the classroom, rather than the intended curriculum. Jennings and Bearak (2014) elaborate on this idea and on what has been discussed previously in this module. Another view is that while it is inevitable that assessment will drive what it is taught, the assessment must test what is worthwhile. This situation is a win-win situation. The point is that assessment should be meaningful and should guide the teacher in what is taught. If the assessment requires short, superficial answers, the teaching will inevitably focus on lower order skills, such as memorisation and routine procedures. If the assessment requires extended in-depth responses, the teacher will also teach in such a manner. ACTIVITY 5.2.5 MY CMEC – ASSESSMENT Identify two pitfalls with assessment that should be avoided. a) – b) – Describe in ten lines an assessment approach that will promote in-depth learning. PDM4801 93 LEARNING UNIT 5 DESIGNING MATHEMATICS CURRICULA IN CONTEXT FEEDBACK ON ACTIVITY 5.2.5 Through our discussion and by completing this activity, it became clear that the way you teach, predicts the way you assess; and the way you assess, reflects the way you teach. While these components of teaching are intertwined and inseparable, we become aware that we have a significant role to play to align the way learners learn, with the best practices in teaching and assessment – always keeping the context in mind within which the learning takes place. 5.3 AN INTEGRATED VIEW OF A MATHEMATICS CURRICULUM An integrated view of a mathematics curriculum will have coherence from aims and objectives through to the assessment. Vision drives integration and coherence. If an education ministry, or even a school principal or a teacher, has a vision of what mathematics education could be, they will more likely align the content, the activities and the assessment towards that goal. 5.3.1 Vision for a mathematics classroom Think back to your own schooling. What were the highlights of your mathematics education experience? Can you describe these important moments? Talk to a friend. What were his or her positive experiences? You could also watch some videos and describe what you think is a productive mathematics classroom. ACTIVITY 5.3.1 VISION FOR A MATHEMATICS CLASSROOM “The classroom below is my ideal mathematics classroom.” Draw and describe some of the happenings in this classroom. 94 DESIGNING MATHEMATICS CURRICULA IN CONTEXT LEARNING UNIT 5 FEEDBACK ON ACTIVITY 5.2 If we could compile the responses to this activity in one volume, imagine how teaching would be enriched as we source ideas from one another. 5.3.2 The learner and a productive disposition Kilpatrick Swafford and Findell (2000) regard a productive disposition as a critical strand of mathematical proficiency. Without this disposition, it is difficult to tackle problems. Productive disposition “refers to the tendency to see sense in mathematics, to perceive it as both useful and worthwhile to believe that steady effort in learning mathematics pays off and to see oneself as an effective learner and doer of mathematics” (Kilpatrick et al 2001:131). Kilpatrick, J, Swafford, J & Findell, B. 2001. Adding It Up: Helping Children Learn Mathematics, National Research Council, ISBN: 0-309-50524-0, 480. Available at http://www.nap.edu/catalog/9822.html. What do you understand by a productive disposition? ACTIVITY 5.3.2 THE LEARNER AND A PRODUCTIVE DISPOSITION Contrast two learners. Learner P sees him- or herself as an effective learner and a doer of mathematics and Learner N sees no sense in doing mathematics. Learner P Learner N How would you encourage Learner N to work on their mathematics? Write down the encouraging words you would have for Learner N. FEEDBACK ON ACTIVITY 5.2.2 Some learners never hear their names other than in an annoyed tone and going with a big NO! NO THAMI, DON’T … It is in our power as teachers to let them associate mathematics with approval, a pleasant atmosphere and success. PDM4801 95 LEARNING UNIT 5 DESIGNING MATHEMATICS CURRICULA IN CONTEXT Check yourself: Have you reached the goal and the specific Tick the box outcomes of learning unit 5? Have you been exposed to designing a mathematics curriculum for a particular context in this learning unit? 1 Can I analyse the structure of mathematics curricula in selected countries? 2 Can I formulate an integrated view of a mathematics curriculum? 5.4 CONCLUSION The purpose of this module was to provide you with a variety of the components of mathematics curricula from different countries to analyse and compare these curricula to reflect on the impact they have in the development of these countries. The different mathematics curricula in selected countries were meant to enhance awareness of a contextual curriculum. In comparing curricula, we selected New Zealand, Kenya, Singapore and the Netherlands. We reflected on the analysis of large-scale international assessment studies, TIMSS and SAQMEQ, which render comparisons of mathematical achievement across the globe and Africa. We explored our own mathematics curriculum, CAPS. In so doing, we developed a sound understanding of the role players of curriculum implementation in South Africa. We also interrogated topics in the mathematics curriculum to evaluate the trajectory of learning mathematics and making crucial conceptual transitions. We tried our hands at designing a mini curriculum for a particular context. We hope that you found this module useful, enjoyable and stimulating. All the best for your future endeavours in studying mathematics education. 96