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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
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26
32
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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
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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
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83
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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
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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
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(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
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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
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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.
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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.
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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
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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.
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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
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Your decision
What are the aims and objectives of
the TMC?
Aim:
Objective 1: ____________________
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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?
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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
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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)
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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)
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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
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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”.
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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.
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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?
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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.
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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)
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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.
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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.
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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.
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● 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.
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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.
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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.
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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.
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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.
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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
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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.
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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
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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”.
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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.
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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.
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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
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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.
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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.
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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.
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SELECTED MATHEMATICS CURRICULA ACROSS THE GLOBE
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● 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
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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.
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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).
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● 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.
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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
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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.
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● 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.
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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).
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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
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iii. –
i. –
3.2
ii. –
i. –
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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
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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.
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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.
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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.
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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
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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).
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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.
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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
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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.
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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
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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.
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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.
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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
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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?
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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
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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
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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.
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(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.
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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
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● 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
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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?
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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,
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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.
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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.
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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.”
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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.
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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
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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.
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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.
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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.
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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.
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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.
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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
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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);
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● 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
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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);
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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
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Can I describe some similarities and differences between
mathematics curricula in four countries, notably in terms of
their curriculum management structures?
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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
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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.
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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.
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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).
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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)
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● 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)
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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.
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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.
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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.
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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
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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.
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