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FUNDAMENTALS OF
MATHEMATICS
TEACHING AND

Faculty of Education

ÌNWU-RMAT171SG8.1¶¹´¸μ¹Î

RMAT 171 PED

FUNDAMENTALS OF MATHEMATICS
TEACHING AND LEARNING IN GRADE R

Faculty of Education

Study guide compiled by: C. Kruger

North-West University, Potchefstroom.

No part of this document may be reproduced in any form or in any way without the written permission of the publishers.

MODULE CONTENTS

Module information .........................................................................................................................iii
Word of welcome ............................................................................................................................iii
Who is this programme for? ............................................................................................................iii
What is the purpose of this programme?.........................................................................................iv
How is the programme structured? .................................................................................................iv
Work-integrated learning: (RWIL)...................................................................................................vii
Rationale for this module ...............................................................................................................vii
The structure of this module...........................................................................................................vii
Study material...............................................................................................................................viii
How to study the contents...............................................................................................................ix
Getting started ................................................................................................................................ x
Module outcomes............................................................................................................................xi
Assessment ....................................................................................................................................xi
Module mark ..................................................................................................................................xii
Action verbs ..................................................................................................................................xiii
Icons ..............................................................................................................................................xv
Icons for Grade R learning .............................................................................................................xv
Warning against plagiarism...........................................................................................................xvi

Study unit 1 Why mathematics in Grade R? .................................................................... 1
1.1 Introduction ..................................................................................................... 1
1.2 What is mathematics?..................................................................................... 2
1.3 The role of mathematics in Grade R................................................................ 3

Study unit 2 Early concept development in mathematics .............................................. 7
2.1 Introduction ..................................................................................................... 7
2.2 How concepts develop .................................................................................... 8
2.3 How concepts are acquired........................................................................... 10
How concepts are remembered .................................................................... 12
2.3.1 The role of different types of knowledge in mathematical concept forming .... 12
2.4 Diversity in mathematical concept development............................................ 13
2.5 Teaching for mathematical concept............................................................... 15
2.6 The role of problem solving in mathematical concept development............... 16
2.7 Assessing for mathematical concept development ........................................ 19
2.8 Providing for LSEN to develop mathematical concept ................................... 21
2.9

i

2.10 Implementation of a mathematics learning centre ......................................... 26

Study unit 3 Fundamental mathematical concepts and skills ...................................... 27
3.1 Introduction ................................................................................................... 27
3.2 Overview of the main areas of mathematics.................................................. 28
3.3 Mathematical concepts and skills implemented in Grade R........................... 30
Matching / one-to-one correspondence ......................................................... 31
3.3.1 Classification................................................................................................. 32
3.3.2 Comparing .................................................................................................... 34
3.3.3 Ordering or seriation ..................................................................................... 35
3.3.4 Estimation ..................................................................................................... 35
3.3.5 Relationship between mathematics and other areas of learning.................... 37
3.4

Reference list ...................................................................................................................... 43

ii

Module information RMAT 171

Module code 16
Module credits
Module name Fundamentals of mathematics teaching and learning in

Contact details e-mail Contact number
Call centre [email protected] 018 285 5900
C. Kruger [email protected]

Word of welcome

Welcome to the Diploma in Grade R teaching. This is a professional training course
and you should make full use of the opportunities presented to you.

We trust that you will find the course rewarding and hope that you will integrate the
knowledge and skills you attain in this module to become more successful in your
teaching career.

The Department of Basic Education stated that all learners should attend a Grade R
class (Reception Year) the year before they start Grade 1. Presently most of the
learners in our country do not have the privilege to attend quality Grade R
programmes. Grade 1- and other teachers experience numerous problems because of
the fact that most of the learners lack essential skills when they commence reading,
writing and doing Mathematics. While the early childhood initiative of the National
Department of Education values quality learning readiness programmes for all young
learners, a high standard for the professional development of all Grade R teachers is a
national priority. Professional Grade R teachers are expected to act as agents of
change in a country where poor standards of education can no longer be tolerated. A
Grade R teacher should have the necessary skills and knowledge to lay a solid
foundation for all later learning, be able to identify problems Grade R learners may
experience, plan learning and teaching activities which will enable the learners to
overcome these problems as well as assisting Grade R learners in acquiring the skills
they need to reach their maximum potential.

It is important that you familiarise yourself with the content of this module to enable you
to complete the exercises and activities successfully. The information and tasks
contained herein need to be mastered and completed in order for you to achieve
success.

Who is this programme for?

The Diploma in Grade R teaching is intended for all students who wish to acquire a
professional teaching qualification recognised by the Department of Basic Education
and Training (DoBET). Currently a National Senior Certificate with diploma
endorsement or a Level 4 or Level 5 Certificate or Diploma in Early Childhood
Development (REQV 10) is the minimum requirement for admission.

iii

What is the purpose of this programme?

This qualification is primarily vocational, occupational or industry specific. The
knowledge emphasizes general principles and application within the field of Grade R
practice. The purpose of the Diploma is to develop diplomats who can demonstrate
focused knowledge and skills within the field of Grade R education. Students will have
to gain experience in applying such knowledge and skills in the Grade R classroom
context. In-depth and specialized knowledge, together with practical skills and focused
experience in the Grade R classroom, will enable successful students to embark on a
number of career paths and to apply their learning to particular employment within the
Grade R context. This Diploma also prepares students for further studies within the
field of Early Childhood Development (Foundation Phase) at NQF-level 7.

How is the programme structured?

The Diploma in Grade R teaching has been registered at NQF level 6 (REQV 13). It
has been built up from existing unit standards.

Curriculum structure: The curriculum for the Grade R Diploma in Teaching consists
of the following modules offered over three years:

New curriculum map for relevant year level(s)

1ST YEAR

YEAR MODULES (Semester 1 & 2)

RSLD171 Disabilities and Learning Barriers 16
16
RTAL171 Teaching and Learning 16

RMAT171 Fundamentals of Mathematics Teaching and Learning Gr R 8

SEMESTER 1 SEMESTER 2 8

RWEL111 Life Skills: Personal Well- 8 RWEL121 Life Skills: Social Well- 8
being being 8

RWIL111 Work-integrated Learning 8 RWIL121 Work-integrated Learning 8
8
RFLS111 Fundamental academic 12 CHOOSE ONE OF THE FOLLOWING 8
Literacy and Support LANGUAGE OF TEACHING AND 8
LEARNING (LOLT) 8
8
RHWP111 Handwriting proficiency 8 RELS121 English 124

RTCL111 Technology & Computer 8 RLSA121 Afrikaans
literacy for Educators

RLST121 Setswana

RLSX121 isiXhosa

RLSZ121 isiZulu

RLSO121 Sesotho

RLSP121 Sepedi

RLSW121 Siswati

TOTAL CREDITS YEAR 1

iv

2nd YEAR

YEAR MODULES (Semester 1 & 2)

RLCA271 Creative Arts 16
16
RRTL271 GR R Teaching and Learning 16
16
RLBK271 Life Skills: Beginning Knowledge
16
REDM271 GR R Education Management
8
SEMESTER 1 SEMESTER 2
8
RLSS211 Social and Health 16 RMAT121 Fundamentals of 8
Barriers Mathematics Teaching and 8
Learning Gr R 8
8
RWIL211 Work-Integrated Learning 8 RWIL221 Work-Integrated Learning 8
8
in GR R in GR R 8
144
RCDP211 Child development and 16
perceptual skills

CHOOSE ONE OF THE FOLLOWING CHOOSE ONE OF THE FOLLOWING
LANGUAGE OF TEACHING AND CONVERSATIONAL LANGUAGE
LEARNING (LOLT) PROFICIENCY:

**ROLT211 English 8 RCLP221 English

RALT211 Afrikaans 8 RCLS221 Setswana

RSLT211 Setswana 8 RCLX221 isiXhosa

RXLT211 isiXhosa 8 RCLZ221 isiZulu

RZLT211 isiZulu 8 RCLO221 Sesotho

RELT211 Sesotho 8 RCLE221 Sepedi

RPLT211 Sepedi 8 RCLW221 SiSwati

RWLT211 SiSwati 8 RCLA221 Afrikaans

TOTAL CREDITS YEAR 2

**Compulsory If English was selected in year one

v

3rd YEAR

YEAR MODULES (Semester 1 & 2)

RLSI371 Policy Perspective on inclusive Education 16
16
RLSP371 Life Skills: Physical Education
8
SEMESTER 1 SEMESTER 2 8
16
REMS311 Education Management 8 REDL321 Educational Law 8
and Systems

RWIL311 Work-integrated Learning 8 RWIL321 Work-integrated Learning

in GR R in GR R

RIRS311 Introduction to Research 8 RLSE221 Emotional and Social
Skills Barriers

RMAT211 Teaching, learning and 16 RFAL221 First Additional English
assessment of Language
Mathematics in GR R

RLSM221 Life Skills: Music 8

TOTAL CREDITS YEAR 3 112

CHOOSE ONE COMBINATION IF ENGLISH (RELS121 & ROLT211) was selected in
previous years:

RLST121 Setswana First Language 8 RSLT211 Setswana 1st Language 8

RLSA121 Afrikaans First Language 8 RALT211 Afrikaans First Language 8

RLSX121 isiXhosa First Language 8 RXLT211 isiXhosa First Language 8

RLSZ121 isiZulu First Language 8 RZLT211 isiZulu First Language 8

RLSO121 Sesotho First Language 8 RELT211 Sesotho First Language 8

RLSP121 Sepedi First Language 8 RPLT211 Sepedi First Language 8

RLSW121 SiSwati First Language 8 RWLT211 SiSwati First Language 8

Or one of the following in the 2nd semester

Language

Language

RFAS 221 Setswana First Additional 8
Language

Language

Language

Language

Language

TOTAL CREDITS YEAR 3 380/

388

vi

Work-integrated learning: (RWIL)

Students also receive Tutorial Notes, which should be used in conjunction with the
Study Guide, Textbook and Manual

PER SEMESTER PER YEAR OVER 3 YEARS

8 credits 16 credits 48 credits
80 hours 160 hours 480 hours
15 days of teaching in 30 days of teaching in 90 days of teaching in
school school school
3 weeks 6 weeks 18 weeks

Rationale for this module

The poor Mathematics performance of learners in South African schools is blamed on
poor foundations for mathematical concept forming in the early years. Mathematics is
viewed as a critical skill for all learners in the 21st century and the Department of
Education envisions learners who are mathematically literate when they exit grade 9.
Educationists worldwide acknowledge the importance of quality early mathematical
experiences on later Mathematics performances. The relationship between teacher
mathematical context specific knowledge and effective teaching and learning of
Mathematics necessitates that a Grade R teacher education programme provides the
teacher with a sound knowledge of all aspects of effective Mathematics education.

This module focuses on the critical context specific knowledge of Grade R Mathematics
teaching and will equip teachers regarding the way young learners master
mathematical concepts, the mathematical concepts which should be mastered by the
Grade R learners and the way a Grade R teacher should facilitate the concept forming
of fundamental mathematical concepts in Grade R.

This module is based on theories of effective mathematical concept forming by young
learners, the implications of learning theories on the facilitation of integrated
mathematical learning experiences in Grade R as well as knowledge of the main areas
of mathematics, including key terms, concepts, facts, rules within the field of basic
mathematics.

The structure of this module

This is a 16-credit module, implying that you are to spend 160 hours on it. This
includes contact time, reading time, times set aside for viewing video clips and for
reflection and discussions of visual material, research time and the time required to

In this module the study material is unlocked by dividing it into three study units, each
dealing with a specific theme. The Study Guide will refer you to the relevant sections in
the study material.

Target dates for completion: Use this for structuring and monitoring your personal
progress.

vii

CONTENT AREA Time allocated Target Dates for
(in hours) completion

Study Unit 1 Why Mathematics in Grade 30
R? 45
45
Study Unit 2 Early concept development 40
in Mathematics 160

Study Unit 3 Fundamental Mathematical
concepts and skills

Assignment

Total hours allocated to this module

Study material

In order to attain the outcomes of this module you need relevant information. The
following are the main sources at your disposal:

Prescribed manual (compulsory):

x Prescribed manual: RMAT 111, 121, 211 C. Kruger. 2013. Mathematics teaching

x CHARLESWORTH, R. & LIND, K. K. 2013. Math & Science for young children.
5th Ed. New York: Thomson

Or:

x CHARLESWORTH, R. 2016. Math and Science for young children. 8th Ed.
Cengage learning

x BREWER, J.A. 2007 or 2012. Early Childhood education, preschool through

x Curriculum and Assessment Policy Statement (CAPS) – Foundation Phase
African Department of Basic Education, 2011) . The Curriculum and Assessment
Statement (CAPS) (obtainable from your current school, Education Specialists
and APO). NB. A copy of this document can be downloaded from the internet
www.education.gov.za

x DVD – RMAT 111

x Seefeldt, C. & Galper, A. 2008. Active experience for active children –
Mathematics. 2nd Ed. New Jersey: Pearson

x Sperry Smith, S. 2009. Early childhood Mathematics. 4th Ed. Boston: Pearson

viii

How to study the contents

In the case of distance learning this study guide fulfils the role of the lecturer or
facilitator. The study guide aims at eliminating all unnecessary telephone calls
students often have to make because they do not know what to do. It indicates when
to do what and how. Always keep it next to you when you are busy studying. The
study guide should enable you to master the study material quite independently by
means of self-study. You are therefore expected to master the factual knowledge, to
apply it in practice whenever required and present problem-solving activities. It is your
responsibility to study the module according to a well-planned study timetable. The
study guidelines are guidelines only, not hard and fast rules - and you have to adapt
them to your personal circumstances so as to derive maximum value from the module.

To make a success of this module, you should first study the different study units. The
focus of each study unit is not necessarily the content prescribed for the Foundation
Phase, but will contribute to enhancing your ability to instruct the learners in how to
deal with the contents in the school curriculum. You should thus constantly focus on
application! In addition to the gathering of information, you will also have to compile
constructive activities with the compiled and existing information and contents.

For this you are expected to integrate theoretical and practical knowledge to complete

Theoretical activities include, amongst others, reading, reflection, summarising of
ideas, preparation, etc. Practical activities comprise the planning of various activities
for Grade R. Learners should also be able to apply their knowledge concerning
assessment in the classroom.

You must, above all, accept full responsibility for your own study, thinking, planning,
doing and monitoring yourself as you progress with this Diploma course.

ALWAYS USE THE STUDY GUIDE FOR EACH STUDY UNIT WHEN YOU START
THE UNIT BUT ALSO WHEN YOU EVALUATE YOUR PROGRESS AND
COMPREHENSION OF THE UNIT.

Much of what you learn in this module will be dictated by your own effort and
commitment. The most successful student is the one who is most disciplined and
organised and able to apply theory to practice. A list of action words is given which
should assist you in your interpretation of the work. It is important to remember that
self-study is the key to success. Contact classes are presented at tuition centres and
you are encouraged to attend as many of these as you possibly can. When attending,
you should at least have read the study guide and familiarise yourself with the
instructions regarding the assignment. Come to the contact classes well prepared so
that you can participate in class discussions. Identify all problems before the group
meetings and specify them clearly. Complete and submit assignments on time.

NB. CAPS: Although a professional teacher should be able to apply knowledge
and skills in classroom context according to any national curriculum, this
programme specifically refers to the South African school curriculum. The
current curriculum is called the Curriculum and Assessment Policy Statements
and is generally referred to by the use of the acronym ‘CAPS’.

ix

Getting started

Page through the study guide first to gain an overview of what is expected of you. Go
to the first study unit and read its outcomes. The most important outcomes are stated
at the beginning of each study unit. Take careful note of these outcomes to focus your
thoughts on the end result by acquiring a holistic view of the relevant study unit. Follow
the leads given by the study guide. Carefully study the relevant content in your
textbook/reader/manual or DVD by applying study strategies that are best suited to
your study methods and personal circumstances.
Search for and utilize additional information whenever necessary. Make use of the
internet, resource centre or NWU library. Completion of the given exercises and self-
evaluations will contribute significantly in assisting you in passing this module.
When studying Fundamentals of Mathematics Teaching and Learning in Grade R, you
need to:
x page through the study guide once in order to get an overview of the content;
x buy a workbook/file in which you can answer questions and/or identify specific

problems;
x thoroughly study the outcomes in the study guide to ensure that you know what is

expected of you (acquire the necessary knowledge of the module), and upon
completion of your study thereof, you will know exactly which knowledge/skills
you have mastered. All questions in the examinations will be formulated around
the study outcomes;
x page through the prescribed study material to get an overview thereof;
x study the learning/study content according to the directions given in the study
guide, keeping the outcomes in mind;
x plan your time for study per study unit according to the set outcomes and a
worked-out personal study timetable;
x complete the questions of every study unit, as you can expect similar questions in
the examination papers;
x complete and check the self-evaluation exercises according to the answers
supplied in the study guide/textbook and/or during the contact sessions;
x reflect on your own professional development after each module by completing
the professional development journal (Study Guide addendum and WIL study
guide). Make copies of these pages or use the electronic version on the DVD.
Include these pages as the last section of your WIL portfolio. Start this section of
your WILL portfolio with a divider that clearly shows the beginning of your
Reflective Journal. (See your WIL study guide)
x ceep to the scheduled dates for handing in assignments.

x

Module outcomes

On successful completion of this module the student should be able to
demonstrate
x knowledge and concept of different forms of mathematical knowledge, various

views on effective Mathematics teaching and learning and an understanding of
mathematical knowledge production processes;
x knowledge in the main areas of Mathematics, including key terms, concepts,
facts, rules and theories within the field of basic Mathematics; and
x detailed knowledge of Mathematics as implemented in Grade R and of the way
Mathematics relates to other areas of learning.

Assessment

Pay special attention to the following assessment arrangements for this module:
There will be a prescribed assignment and an examination.
One assignment must be submitted after completion of this module. The exact date
for submission will be found in your current Information Booklet. The assignment is a
major component of the assessment process of the module. Consequently, you must
make a worthwhile effort to produce a quality work. Plan your assignment by
meticulously following the prescriptions for the assignment in your Tutorial Notes. The
following assessment methods apply:
1. Continuous self-assessment:

Certain study units have self-assessed activities or questions to test your
knowledge. These self-study tasks and questions also prepare you for
examinations.
2. Informal assessment:
During the contact sessions the facilitator will guide the student on the preliminary
work done on the assignment.
3. Formal assessment:
An assignment of 100 marks which should be handed timeously, consult the
information booklet to confirm the exact date of submission.
4. Formal examination: An examination of two hours with a total of 100 marks will
be written after completion of this module.

xi

Module mark

Students need a participation mark (the assignment mark) to be able to participate in
the examination.

The participation mark + the examination mark (sub minimum of 40%) contribute to a
module mark. Please note that the module mark must be at least 50%:

40% of assignment mark = Module mark (final Sub minimum for module:
(no sub minimum) mark) out of a 100% 50%

60 % of examination mark
(sub minimum = 40%)

Thus, the sub-minimum required for passing the examination is 40% and the final

pass mark is calculated by taking into account your participation mark and the
examination mark in a ratio of 40 (assignment) to 60 (examination). A pass
requirement of a minimum of 50% for the module (final) mark applies.
Comprehensive details concerning all the pass requirements appear in the official
calendar of the Faculty and University, as well as in your Tutorial Notes.

x IMPORTANT: Attendance of contact sessions and/or vacation schools and
make use of these if possible.

x You will receive Examination Information with your marked assignment to guide

Module RWIL 111 will require of you to apply knowledge gained through the semester
modules in an integrated way. Refer to the RWIL 111 study guide regarding
requirements of the work integrated learning task.

Although all tasks indicated in the RWIL 111 study guide should be completed and
presented in a professional way in your portfolio, you should also feel free to use the
school policies, lesson plan formats, assessment tools, etc.

In this module you will also be asked to observe a lesson presented by your mentor.

As stated above you also need to include your reflections on your own professional
development in your portfolio in a section clearly marked ‘Reflection on my
professional development’. (See Addendum of this Study Guide and RWIL study
guide).

xii

Take note:

This icon refers to some modules you will also need to complete in your second and
third year of study. The second and third year study material that is referred to will
not be taken into consideration for this module’s assignments and examinations.

Action verbs

The following action verbs are defined to ensure that you know exactly what is
expected from you each time they are used. Make sure you understand the definition of
each one and that you will be able to complete an instruction or answer a question
correctly.

x Analyse
Identify parts or elements of a concept. Examine a requested aspect or concept in
order to learn/explain what it is composed of.

x Argue
To put forth reasons for or against something

x Clarify
Make something clearer or easier to understand.

x Classify
Arrange certain aspects systematically in groups, classes, or categories according to a
given instruction.

x Comment
Briefly, state your own opinion on a subject.

x Compare
Point out the similarities (things that are the same) and the differences between
objectives, ideas, or points of view. When you compare two or more objectives, you
should do so systematically by completing one aspect at a time. It is always better to
do this in your own words.

x Criticise
This means that you should indicate whether you agree or disagree with a certain
statement or view. You should then describe the aspects you agree/disagree with and

x Define
This means you have to provide the accurate meaning of a concept.

x Demonstrate
Include and discuss examples. You have to prove that you understand how a process
works or how a concept is applied in real-life situations.

x Describe
Say exactly what something is like, give an account of the characteristics or nature of
something, and explain how something works. No opinion or argument is needed.

x Discuss
Comment on something in your own words. This often requires that two viewpoints or
two different possibilities be debated.

xiii

x Distinguish
Point out and emphasize the differences between objects, different ideas, or points of
view.

x Essay
An extensive description of a topic is required.

x Evaluate
Make a thorough study of the required content (an argument or point of view). Analyse
the aspect(s) under discussion and decide on the value of specific aspects. Motivate

x Example
A practical illustration of a concept is required.

x Explain
Clarify or give reasons for something, usually in your own words. You must prove that
you understand the contents. It may be useful to use examples or illustrations.

x Identify
Give the essential characteristics or aspects of a phenomenon.

x Illustrate
Draw a diagram or sketch that is a representation of a phenomenon or an idea.

x Indicate
Point out, make known, state briefly.

x List
Simply provide a list of names, facts, or items required. A particular category or order
may be specified. Neither a discussion nor an explanation is necessary.

x Motivate
You should explain the reasons for your statements or views. Try to convince the

x Name or mention
Briefly, name/mention something without giving details.

x Outline
Emphasise the major features, structures, or general principles of a topic, omitting
minor details. Slightly more detail than in the case of naming, listing or stating of
information is required.

The subject content must be read with concentration. Focus your attention on the
relevant content to get a clear overview of the main aspects and facts.

x Show
To indicate by explanation as well as example

x State
Supply the required information without discussing it.

x Study
This implies the devotion of time and thought to gain knowledge of a particular subject.
The relevant study content must be study-read to gain a high level of knowledge,

xiv

comprehension, and insight, and to be able to effectively reflect on the content in a
systematic manner.
x Tell
To explain, to indicate, communicate information?

x Summarise
Give a structured overview of the key (most important) aspects of a topic. This must
always be done in your own words.

Icons

Time allocation Learning outcomes

Study material Assessment /
Assignments

Individual exercise Group Activity

Example Reflection

Integration between modules

xv

Warning against plagiarism

ASSIGNMENTS ARE INDIVIDUAL TASKS AND NOT GROUP ACTIVITIES.
(UNLESS EXPLICITLY INDICATED AS GROUP ACTIVITIES)
Copying of text from other learners or from other sources (for instance the study guide,
prescribed material or directly from the internet) is not allowed – only brief quotations
are allowed and then only if indicated as such.
You should reformulate existing text and use your own words to explain what you
have read. It is not acceptable to retype existing text and just acknowledge the source
in a footnote – you should be able to relate the idea or concept, without repeating the
original author to the letter.
The aim of the assignments is not the reproduction of existing material, but to ascertain
whether you have the ability to integrate existing texts, add your own interpretation
and/or critique of the texts and offer a creative solution to existing problems.
Be warned: students who submit copied text will obtain a mark of zero for the
assignment and disciplinary steps may be taken by the Faculty and/or
University. It is also unacceptable to do somebody else’s work, to lend your work
to them or to make your work available to them to copy – be careful and do not
make your work available to anyone!
For the NWU link for plagiarism, go to http://www.nwu.ac.za/webfm_send/25355

xvi

Study unit 1

Study unit 1

Study time

It is recommended that you allow approximately 30 hours for completing this Study
Unit successfully.

Study outcomes

After completion of the Study Unit the student should be able to

x describe what Mathematics entails – inside and outside of the school context;
x critically evaluate and discuss the way mathematics is presented in Grade R,

and
x explain and argue the role of Mathematics as implemented in Grade R .

1.1 Introduction

This module aims to ensure that Grade R teachers have a sound knowledge regarding
all aspects of Mathematics as illustrated in the diagram below. The following diagram
illustrates the important aspects of a teacher’s context specific knowledge of
Mathematics, which will be covered in the respective Study Units of this module. Study
Unit 1 focuses on the teacher’s beliefs of what Mathematics is and the role of
Mathematics in the development of the young child.

The teacher’s belief of what Mathematics is will determine how he/she teaches
Mathematics, which will in turn have a direct influence on what Mathematics the
learners learn and how they learn Mathematics. Teacher beliefs will also serve as
motivation for the teacher to get to grips with the aspects of Mathematics relevant to
Grade R and beyond to ensure that all teaching is based on sound context specific
knowledge. Many early childhood teachers are reluctant to facilitate the learning of
mathematical concepts without realising that Mathematics plays a major role in the
young learner’s everyday life. A sound mathematic foundation provides multiple
opportunities for informal concept forming of Mathematics. Teachers, who understand
the critical role they play in laying the foundation for mathematical learning by our
young learners, will want to come to grips with the foundational aspects of Mathematics
to ensure that all teaching is based on sound context specific knowledge. In this Study
Unit we will focus on what Mathematics is and the subsequent role of Mathematics in

1

Study unit 1

1.2 What is mathematics?

Mathematics is more than completing sets of exercises or mimicking processes the
teacher explains. Doing Mathematics means generating strategies for solving
problems, applying those approaches, seeing if they lead to solutions, and checking to
see if your answers make sense. Doing Mathematics in classrooms should closely
model the act of doing Mathematics in the real world. Mathematics is a science of
concepts and processes that have a pattern of regularity and logical order. Finding and
exploring this regularity or order, and then making sense of it, is what Mathematics is
all about. Even the youngest learners can and should be involved in the science of
pattern and order (Van de Walle et al. 2010:13).

Study material

Study the relevant sections:

Manual: Unit 1: Early concept development in Mathematics

Brewer: 2007/2012. Introduction: Why Mathematics?
CAPS
x Pre-Kindergarten Math Concepts - Bixler, M. 2006.
x Report on Mathematics Education - Horowitz, S. H.

2008.

Relevant chapter on mathematics: Manipulation and
discovery through Mathematics - Defining Mathematics

Chapter 2: Mathematics – Aims, skills and content

2

Study unit 1

x Early childhood teachers' misconceptions about mathematics education for

young children in the US - Sun Lee and Ginsburg (2012)
x www.earlychildhoodaustralia.org.au

Individual activity

1. How do Van de Walle et al. (2010:13) describe Mathematics?
2. What are the implications of this view expressed by Van de Walle et al. for the

teaching and learning of Mathematics?
3. Make a summary of the reasons given by Bixler (2006) and Horowitz, (2008)

why children should engage with Mathematics at an early age. (How does this
compare with the findings of Sun Lee and Ginsburg regarding teachers’
misconceptions?)
4. Define Mathematics within the context of the South African school curriculum.
5. Draw a flow chart to illustrate the mathematical aims and skills set by the
South African National School Curriculum for learners from Grades R to 9.

1.3 The role of mathematics in Grade R

Various authors and research refer to the value of Mathematics in the early years.

Study material

Study the relevant sections.

x Math matters - Stipek, D., Alan Schoenfeld, A. & Gomby, D.
2012.

x Report on Mathematics Education - Horowitz, S. H. 2008.

x To Teach Mathematics to home-schooled children - Bowers, A.
2012.

Seefeldt, C. & Galper, A. (2008) Active
experiences for active children –
Mathematics (2nd Ed)

(Available at resource centres)

3

Study unit 1

Individual activity

1. Based on the literature above, explain the role Mathematics plays in the world
outside the school context.

2. Discuss in your groups or with a colleague how the Grade R teacher can
contribute towards the realisation of the aims for Mathematics learning as set
out in the CAPS.

3. Find one more source from the Internet, library/resource centre, which confirms
the critical role of Mathematics learning in the early years. Write down the
author, title and short summary of this source. You may refer to a book,
academic article, newspaper article, webpage or any other relevant literature.

Individual activity

First study the mathematical components listed in the table below, which form the
focus of this exercise. View the video: Exploring Mathematics in Grade R.(videos 2.1
– 2.4) How would you describe the Grade R learning experiences regarding each of
the following aspects?

1. What Mathematics are learners learning? Keep a pencil ready and use the
table to jot down the Mathematics, which you can identify in the video.

Mathematics Activity Resources

Counting

Number recognition

Number operations:
x subtraction
x division
x multiplication

Patterns

Sorting

Measuring

Other

4

Study unit 1

1. Is the teacher instructing, demonstrating or assisting? Explain the role of the
teacher.

2. Learner’s role – Are learners active? Passive? Cooperating? Exploring? Having
fun? What is your observation of the learners’ role and attitude towards the

3. Learning tasks – What kind of learning tasks are implemented? Are learners
working individually, in groups or both?

4. Which activities provided for exploration of more than one mathematical concept?
5. How would you describe the classroom atmosphere? Relaxed? Rigid and

structured? Open to own exploration?
6. Integration: Can you identify integration of other subjects such as Life Skills,

Science, and Language? While you watch the DVD, make notes of which
subjects are integrated and how they are integrated.
7. Resources used: Are these resources easily available? Durable? Applicable? Are
there alternatives for the resources used? Can you think of ways to design
resources, which would serve the same purpose as more expensive resources
used?

Individual activity

following questions:
1. How would you describe the socio-economic circumstances of these learners?
2. What mathematical concepts were covered in these Grade R classes?
3. What would the lesson planning look like?
4. What did you like about the teaching and learning approach?
5. What would you have done differently in your teaching context?
6. Will you describe the Mathematics learning implemented in the video as

developmentally appropriate for the Grade R learners?
7. Do these activities provide for diversity?
8. Will these teaching strategies be applicable to all Grade R classes in South

Africa?

Reflection

TIME FOR REFLECTION

Use the Professional Development Journal (see Study Guide Addendum) to reflect
on what you have gained in this unit regarding knowledge, skills and attitude. You
can use the electronic version on the DVD or make copies of the journal format
provided in the Study Guide Addendum.
Also feel free to express your feelings about barriers you experienced in mastering
the outcomes of the unit.

5

Study unit 1

Important information

Please make sure that all exercises are completed before you continue with the next
unit and that you have achieved the outcomes.
Spot checks may be done by NWU assessors at students’ schools to monitor the
application thereof.

6

Study unit 2

Study unit 2
EARLY CONCEPT DEVELOPMENT IN MATHEMATICS

Study time

It is recommended that you allow approximately 45 hours for completing this Study
Unit successfully.

Study outcomes

After completion of the Study Unit the student should be able to

x identify, compare and evaluate the various views on effective Mathematics
teaching and learning;

x outline, compare and explain theories on the different ways young learners
form concept of Mathematics;

x describe the different forms of mathematical knowledge; and
x explain and demonstrate how knowledge is generated, adapted and

incorporated in effective teaching and learning of Mathematics in Grade R.

2.1 Introduction

Early childhood experiences and settings are main determinants of later achievement
in Mathematics. Children enter school with a great deal of innate knowledge about
Mathematics. When early childhood teachers embrace the learners’ existing
knowledge, learners develop new mathematical concepts and skills in a spontaneous
and unforced way. Early mathematical knowledge lays the foundation for later
achievement as it gives young children the confidence to engage in mathematically
challenging concepts. As we saw in Study Unit 1, the teacher’s belief of what
Mathematics is will determine how he/she teaches Mathematics, which will in turn have
a direct influence on what Mathematics the learners learn and how they learn
Mathematics. All aspects in the diagram below are interrelated. This unit will focus on
the young learners’ cognition (how learners learn Mathematics), and the implications of
learner cognition for Mathematics teaching in Grade R. This pedagogical knowledge is
critical to ensure that you are able to design and implement meaningful mathematical
learning experiences in the Grade R classroom.

7

Study unit 2

2.2 How concepts develop

In order to plan and implement effective and meaningful Mathematics learning
experiences in Grade R, the teacher has to know and understand how young learners
develop mathematical concepts.

Study material

Study the relevant sections. 7th Ed Unit 1 or 8th Ed Chapter 1 (1-1e;
Textbook: 1-1f):
Charlesworth and Lind. 2013. How concepts develop –
Math and science for young children (7th x Piagetian periods of concept
Ed)
Or development and thought
Charlesworth, 2016 (8th Ed) x Piaget’s view of how children acquire

Manual: knowledge
x Vygotsky’s view of how children learn

and develop
x The learning cycle and Traditional vs.

Reform instruction

Unit 1: Early concept development in
Mathematics

x Teaching and learning of
Mathematics

Mathematics - Papandayan, J.I. 2009.

8

Study unit 2

Brewer Brewer. 2007/2012. Relevant chapter: Young children
Introduction to Early Childhood growing, thinking and learning
Education: Pre-school through
practice
CAPS
x Theories of development; Section
on Children’s development

Relevant chapter : Manipulation and
discovery through Mathematics

x Learning Mathematics and Process
Strands

Chapter 2: Mathematics – Aims, skills
and content - Grade R focus

Integration between modules

Module RTAL 171 Study Units 1 & 2: Learning theories: relevant sections

Smith, S.S. 2009. Early childhood Chapter 1: Foundations, Myths and Standards
mathematics. x Developmentally appropriate Education
x Piaget, Vygotsky, Bruner, and Dienes.
(Available at resource centres)
Part one – theory of Experiences:
Seefeldt, C. & Galper, A. 2008. Age appropriateness

Active experiences for active
children – Mathematics (2nd Ed)

(Available at resource centres)
From the internet:

http://www.naeyc.org/files/naeyc/fil Developmentally Appropriate Practice in Early

e/positions/position%20statement% Childhood Programs Serving Children from

20Web.pdf Birth through Age 8

Individual activity

1. Define the concept development in your own words.

2. Identify mathematical concepts young learners should develop before
commencing with formal schooling.

3. According the CAPS document the acquisition of emergent Mathematics
and related mathematical concepts should move through three stages of
learning, namely the kinaesthetic stage, the concrete stage (3D), and the
paper and pencil representation stage. Discuss the relationship between these
stages and Piaget’s view on how children acquire knowledge in the early years.

9

Study unit 2

2.3 4. Discuss the implications of Vygotsky’s ZPD model for the teaching and learning

5. Explain the learning of Mathematics by young children by referring to the
learning cycle. Demonstrate this cycle by means of an example within the

6. “The reform of classroom instruction has changed from traditional drill and
practice memorization approach to adoption of the constructivist approach”.
Discuss this statement by referring to the implications of the reform approach

7. Discuss the implications of Dewey’s philosophy of learning and teaching for
Mathematics learning in the early years.

How concepts are acquired

A teacher, who has knowledge of and understands young learners’ development and
how they learn, will be able to provide opportunities for learners to acquire concepts.
The term ‘acquire’ is defined by TheFreeDictionary as follows: “to get or gain
(something, such as an object, trait, or ability), more or less permanently”.

Acquire is also described as the process whereby a person

x gains possession of [something concrete or abstract];

x gets by one's own efforts: acquire proficiency in math; and

x gains through experience; come by.
The above definitions show that people acquire knowledge through experience and
that, when acquiring mathematical knowledge, the knowledge/concept is ‘possessed’
by the person. This implies that when a learner acquires mathematical knowledge it
becomes part of the learner’s internal knowledge network and that the learner will be
able to use the knowledge in his/her life to solve problems in a meaningful way (Recap:
Module RTAL 171: Study Unit 1 and 2). This is thus a completely different process than
when a learner learns by rote.

Study material

Study the relevant sections.

Textbook: 7th Ed Unit 2 or 8th Ed Chapter 1 (1-2):
Charlesworth and Lind. 2013.
Math and science for young children (7th x How concepts are acquired –
Ed)
Or naturalistic, informal and

Charlesworth, 2016 (8th Ed) structured/adult guided learning

experiences

Manual: Unit 1: Early concept development in
Mathematics

Children are born mathematicians -
Geist, E. 2006.

10

Study unit 2

Brewer.2007/2012.. Introduction to Relevant chapter: Manipulation and
Early Childhood Education: Pre- discovery through Mathematics
school through Primary Grades. x Learning Mathematics
x Teaching Mathematics
Integration between modules x Developmentally appropriate

RTAL 171 practice
Teaching and learning in Grade R
Study Units 1 & 2: Relevant section on
cognitive views on learning; Information
processing

Smith, S.S. 2009. Chapter 1: Foundations, Myths and
Early childhood mathematics. Standards
(Available at resource centres)
Part one – Theory of Experiences:
Seefeldt, C. & Galper, A. 2008. Section 1: Deep personal meaning
Active experiences for active children – Part two – Active children, active
Mathematics (2nd Ed) environments
(Available at resource centres)

Individual activity

1. Define and discuss the three types of learning experiences described by
Charlesworth and Lind (2013) / Charlesworth (2016).

2. Provide examples of learner tasks within each type of learning experience,

3. What are adult-guided learning experiences? Do you think there is a
relationship between adult-guided learning experiences and Vygotsky’s ZPD

4. Distinguish between divergent questions and directions and convergent
questions and directions.

5. Snack time or lunchtime can be a perfect context for learning about
mathematical concepts such as shapes, number, fractions and division.
Provide examples of divergent questions you can ask learners to mediate
mathematical learning during snack time.

6. Discuss the role of group work and projects in acquiring mathematical
knowledge in the early years.

7. Geist discusses six aspects teachers should keep in mind when treating young
learners as mathematicians. Shortly summarise these aspects.

11

Study unit 2

2.3.1 How concepts are remembered

Integration between modules

It is important that you study this section together relevant sections in RTAL 171,
Study Unit 2 - The information processing model of memory. Use the diagram in
RTAL 171 for deeper understanding.

According to the Information Processing Model in order for learners to remember new
information, it has to be sent to the long term memory (LTM). In order for information
to be stored in the LTM, new information has to get past the sensory buffer of a child.
As you have realised by now, the young child will pay attention to new knowledge
when learning includes the senses. The more senses involved in the learning
process, the greater the chance for knowledge to get past the sensory buffer, move
on to the working memory for interpretation and accommodation and to be stored in
the LTM. Only when knowledge is stored in the LTM can the learner use this
knowledge in future as needed e.g. to solve similar problems in real life situations.
Knowledge that does not pass the sensory buffer is lost for always unless the learner
gets another opportunity to learn the concept.

This model implies that the teaching and learning of mathematics in the Foundation
Phase needs to include the senses in order to pass the sensory buffer of the child.
Learners need not only ‘see’ but also ‘touch’ the objects they are counting,
subtracting, adding, sharing, etc. This is one of the main reasons worksheets are not
developmentally appropriate practice in Grade R. Learners will not be able to
understand the mathematical concepts by only completing worksheets on two
dimensional level and as a result the knowledge will not be sent to the long-term
memory for later recall by the child. As new mathematic knowledge needs to build
onto prior knowledge stored in the LTM, this will eventually result in poor
mathematics performance of the child.

2.4 The role of different types of knowledge in
mathematical concept forming

We already know that social knowledge, physical knowledge and logical-mathematical
knowledge play an important role in the way young learners “come to know”
Mathematics. In order to facilitate meaningful Mathematics learning a Grade R teacher
also needs to be aware of the difference between conceptual and procedural
knowledge.

Study material/Integration between modules

Study the relevant sections.

Manual: Unit 1: Knowledge Types – Procedural and Conceptual knowledge
RTAL 111
Study Unit 2: Variables that affect learning

Reader:: Types of knowledge - The Attic Learning Community
(2012)

12

Study unit 2

Individual activity

1. Explain the difference between procedural knowledge and conceptual
knowledge by referring to classroom-relevant examples.

2. Critically reflect on your own school career – did your Mathematics teachers
accommodate both kinds of knowledge in their teaching strategies? Can you
provide an aspect of Mathematics, which you could have understood better if
conceptual knowledge was applied? Why do you think research showed
improved mathematical performance where conceptual knowledge formed the
basis of learning?

3. What is the relationship between conceptual and procedural knowledge on the
one hand and social, physical and logical-mathematical knowledge on the
other?

4. What role does procedural knowledge play in Mathematics learning in Grade
R?

2.5 Diversity in mathematical concept development

Teaching does not necessarily guarantee that knowledge will be acquired. External as
well as internal factors also play a role in acquiring mathematical concepts. These
factors may originate from the child’s environment, such as the background and
learning environment of the child (external factors) or may originate from an innate
characteristic of the child (internal factors). Internal factors include learning style, type
of intelligence, as well as the temperament of a child. A Grade R teacher should
recognise diversity resulting from these external and internal determinants of learning.

Parents and teachers play a major role in creating an environment where young
learners can acquire knowledge and concept in a developmentally appropriate way. It
is important that teachers and parents know that learners do not all acquire knowledge
in the same way. When creating a learning environment that will allow all learners
equal opportunity to acquire mathematical knowledge and concept, the teacher not
only needs knowledge of how learners develop concepts, but he/she should also have
knowledge of the learners’ background and know how to provide learning opportunities
for the diverse learning needs of all learners.

Learners learn many fundamental concepts in Mathematics before they enter school.
Bronfenbrenner’s Ecological Systems Theory (Brewer, 2007) stresses that the context
in which learners learn and develop out of school, plays a major role in the way they
learn and develop in school. Research showed that if mathematical learning
experiences are embedded in the learners’ background and cultural world, learners are
able to relate to new knowledge and Mathematics is experienced as a meaningful
activity. While it is generally accepted that the environment in which the child develops
has an external influence on how and what a child learns, Howard Gardner further
focuses our attention on the influence of inherent determinants to the way concept and
knowledge develop, namely the type of intelligence of each child.

13

Study unit 2

Individual activity

View the video clip: Multiple Intelligences at Smartville (available at
http://www.edutopia.org/multiple-intelligences) and discuss the implementation of
Howard Gardner’s theory in a Grade R classroom context during a contact session or
discussions:

1. How does Smartville ensure the accommodation of all intelligences?

2. Which intelligences did the learners in the video highlight as their own special
intelligences?

3. Reflect on the following scenario: A boy who is a gifted mathematician has a
language barrier, which influences his ability to formulate his thoughts on
paper. How do you think this boy would be accommodated in Smartville? How
would you describe this boy’s emotions when walking underneath the “Logic
Smart” (Mathematical Intelligence) banner in the hallway?

4. Which banner would have made you feel smart?

5. In view of the comments and expressions of the learners in the video, what do
you think is the value of implementing Gardner’s multiple intelligence theory for
Mathematics learning in schools in general? And for Mathematics learning in

6. Smartville is a school from a developed country. Which barriers may hinder the
implementation of the Multiple Intelligence theory in the South African Grade R
classroom? How can these barriers be overcome?

Study material

Study the relevant sections. 7th Ed Unit 2 or 8th Ed Chapter 1 (1-2d):
Textbook: How concepts are acquired –
Charlesworth and Lind. 2013. Learning styles / Ethnomathematics
Math and science for young children
Or x Howard Gardner and multiple
Charlesworth, 2016 (8th Ed)
Manual: intelligences – Smith, M. K. (2002, 2008)
x How to Address Multiple Intelligences in
Brewer. 2007/2012. Introduction to
Early Childhood Education: Pre- the Classroom - Tips and resources for
school through Primary Grades. putting MI theory into practice - Bernard,
S. (2009)

Relevant chapter:

Multiple Intelligences Theory (Gardner)
Ecological Systems Theory (Bronfenbrenner)

14

Study unit 2

Individual activity

1. What are the implications of Bronfenbrenner’s Ecological Systems Theory for
the effective teaching of Mathematics in Grade R?

2. What are the implications of Bronfenbrenner’s theory for planning of
mathematical learning tasks for Grade R learners from rural areas? And for
learners attending a school in urban areas?

3. Discuss the implications of each intelligence type as identified by Gardner for
the facilitation of mathematical concept forming in Grade R. Which intelligence
do you think is your most dominant intelligence type? Why do you think so?

4. What is Ethnomathematics and what are the implications of Ethnomathematics
for the Grade R teacher in the South African context?

2.6 Teaching for mathematical concept

The teacher as manager of all classroom events will determine greatly if learners will
learn Mathematics in a way that promotes understanding. This will in turn determine
how learners view their own ability to learn Mathematics and solve mathematical
problems. If learners’ first experiences of Mathematics help them to perceive
Mathematics as a meaningful tool, which lies within their capability, they will be
motivated to explore Mathematics with an open mind. Such a feeling of self-efficacy (“I
can do Mathematics”) will contribute to a positive attitude towards Mathematics, which
will make a huge difference to the learners’ academic career.

Study material 7th Ed Unit 3 or 8th Ed Chapter 1 (1-3g):
Promoting young children’s concept through
Study the relevant sections. problem solving (all sections)
x Four steps in problem solving
Textbook:
Charlesworth and Lind. 2013. x criteria for problem based approach
Math and science for young
children (7th Ed) x Eleven problem solving strategies suggested
Or by Reys et al. 2001/2004
Charlesworth, 2016 (8th Ed)
Unit 1: Teaching for effective learning of
Manual: Mathematics

Individual activity

1. Identify examples of the six steps in instruction, which you would implement
when mediating Mathematics concept forming in Grade R.

2. Describe the advantages of using the six steps in instruction.
3. Which four aspects should adults keep in mind when planning mathematical

learning experiences for Grade R learners?

15

Study unit 2

2.7 4. When should abstract experiences be introduced?

5. What are the three main things, which should be considered when selecting
mathematical learning materials?

6. According to Charlesworth and Lind (2013:37) / (Charlesworth (2016:23,24) it is
critical that mathematical concept starts with concrete experiences. Review the
knowledge types and Piaget’s theory on child development and explain why
this step should not be skipped.

7. Show how you would follow the five steps from concrete materials to paper and
pencil when mediating mathematical concept of the young child.

8. What is intentional teaching?

9. What is the purpose of the evaluating stage in the learning experience?

10. Explain how this step is implemented as part of a mathematical learning

The role of problem solving in mathematical
concept development

People often view problem solving as a characteristic of Mathematics only. This is
probably because many people see Mathematics as a problem-solving tool. Problems
form part of our everyday lives and learners who learn how to solve mathematical
problems in a logical and meaningful way, are able to apply this very important life skill
to all spheres of life. Being an effective problem solver further enhances a child’s self-
image and boosts confidence, which is transferred to all areas of learning.

Individual activity

Review the video: Exploring Mathematics in Grade R (videos 2.1 – 2.4). Pay
attention to the following: Problem solving through play.

1. Can you identify problem-solving activities in the visuals?
2. Do you think these teachers were successful in creating problem-solving

learning tasks? What do you think the teacher’s role is in creating problem-
solving activities?
3. What is the learner’s role in these learning tasks?
4. Critically evaluate if these activities are on the level of the learners (DAP).

Study the following attentively

Various sources stress the importance of a problem-solving focus in Mathematics
learning, but according to Lesh et al. (as quoted by Smith, 2009), not all knowledge
needs to be constructed or discovered by the learner. Although constructing of own
knowledge leads to improved understanding, constructing is but one way to arrive at

16

Study unit 2

mathematical understanding. For example, a master chef or a master carpenter passes
down fundamental skills and known relationships to her or his apprentices. A teacher
should, however, refrain from demonstrating solutions or providing the answers to
problems where learners could have gained more by rather grappling with the problem
to find their own solutions. A teacher should only resort to direct teaching where the
learning content does not lend itself to self-exploration, for example learners will not be
able to “discover” that number “1” is the visual symbol that represents one object. The
teacher will have to show the number and link the number with one object. After the
learner knows that 1 represents one apple, he/she can now further explore how many
1’s he/she will need to represent 3 apples. The models and modelling perspective also
recognise that there are at least four independent objectives in the domain of
Mathematics education:

1. Behavioural objectives, such as learning the facts;

2. Process objectives, such as analytical skills like sorting, analysing, making
conjectures and organising;

3. Affective objectives, such as how one feels about maths; and

4. Cognitive objectives, such as being able to model an answer, orally explain one’s
thinking, and extend the model to a new situation (Lesh & Doerr, as quoted by
Smith, 2009:12).

Individual activity

Review your observations regarding the visuals in the DVD: Exploring Mathematics
(videos 2.1 – 2.4) in Grade R.

1. Which tasks could be mastered through problem solving, own knowledge
construction and discovery?

2. Which tasks also required modelling and instruction by the teacher?

3. Which of the objectives stated by Lesh and Doerr (as quoted by Smith, 2009)
do you think were reached through the activities?

4. Identify activities in the DVD which would have contributed to reaching these
objectives.

Objectives Activities contributing to each objective

Behavioural objectives

Process objectives

Affective objectives

Cognitive objectives

17

Study unit 2

Study material 7th Ed Unit 3: Promoting young children’s
concept through problem solving – Problem
Study the relevant sections: solving (all sections)
8th Ed Chapter 1: (All sections on problem
Textbook: solving (1-3g)
Charlesworth and Lind. 2013.
Math and science for young children Relevant chapter: Manipulation and
(7th Ed) discovery through Mathematics
Or
Teaching Mathematics: The process strands
Charlesworth, 2016 (8th Ed) – Problem solving

Brewer. 2007/2012. Introduction to
Early Childhood Education: Pre-

Integration between modules Study Units 1 and 2: Relevant sections
Study Unit 5: Teaching methods
RTAL 171 : Teaching and learning

and subtraction
Smith, S.S. 2009.
Early childhood mathematics. Part two – Section ten: Learning to
(Available at resource centres) problem-solve
Seefeldt, C. & Galper, A. 2008.
Active experiences for active children –
Mathematics (2nd Ed)
(Available at resource centres)

Individual activity

1. Define problem solving in your own words.

2. Young children use several kinds of representations to explain their
mathematical ideas. Name these and give an example of how you would
accommodate each in the Grade R class.

3. What are the four essential steps it takes for a person to solve a problem?

4. What is the teacher’s role in problem solving mathematical activities?

5. Discriminate between routine and non-routine problems and discuss the role of
each in the Grade R Mathematics curriculum.

6. Explain by means of an example how you would lead young learners to follow
Polya’s four-step procedure to solve a problem. Choose a non-routine problem
to which the Grade R learner can relate.

18

2.8 Study unit 2

7. Computational estimation should only be introduced at the end of the
Foundation Phase. What is the role of estimation in problem solving in Grade
R?

8. Discuss how you could implement multicultural problem solving in the Grade R
classroom.

9. List the strategies for problem solving as suggested by Reys et al (2001) and
provide an example of each for the Grade R context.

Assessing for mathematical concept
development

Assessment is dealt with in more detail in RTAL 171. You need to study relevant
sections in RTAL 171, Charlesworth and Lind (2013) / Charlesworth (2016) and
Brewer (2007, 2012) in an integrated way to ensure you have a good knowledge of this
important aspect of teaching and learning. This will enable you to implement the most
relevant assessment principles when assessing the Mathematics competencies of
Grade R learners. Where this Study Unit supports your exploration regarding the why,
what and how of assessment in the early years, modules RMAT 121 (Planning for
emergent Mathematics in Grade R) will guide your actual planning for assessment,
while RMAT 211 (Teaching, learning and assessment of Mathematics in Grade R) will
assist you in implementing your planned assessment strategies during a Mathematics
learning activity.

Assessment forms an integral part of each component of the learning cycle and cannot
be separated from the teaching-learning process. The following diagram shows the
implementation of formative assessment while learners are learning Mathematics.
Evidence of the learners’ understanding will have a direct influence on the learning
cycle as it will guide the teacher on the implementation and revising of teaching
strategies. Based on the assessment outcome the teacher will find answers to
questions such as: “How can I best assist the learners to master the knowledge?” “Are
they ready to proceed to a new level?” The teacher thus uses the knowledge gained
through assessment to adjust and restructure the learning experience. Halfway through
a learning experience formative assessment will guide the teacher to revise his/her
teaching strategies if the planned strategies are not assisting the learners in the most
relevant way to form concept of the Mathematics as he/she hoped it would. It is
senseless to carry on with a lesson as planned if your continuous assessment shows
that learning is ineffective and may result in poor concept and confusion.

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Study unit 2

Study material/Integration between modules

Study the relevant sections: 7th Ed:
Textbook: Unit 3: Promoting young children’s concept through
Charlesworth and Lind. 2013. problem solving - Relevant sections on
Math and science for young assessing/assessment;
children Unit 4: Assessing the child’s developmental level
Or 8th Ed Chapter 1 (1-4) National assessment
Charlesworth, 2016, 8th standards (all sections a to e)
Edition NB. Pay attention to the integration of assessment in
all units; Most units in the text book continuously
Brewer. 2007/2012. incorporate assessment strategies, which guide
Introduction to Early teachers in applying relevant assessment techniques
Childhood Education: Pre- to support the learning of the different Mathematics
school through Primary content areas.
Relevant chapter: Manipulation and discovery
through Mathematics Assessment of Mathematics

Relevant chapter: Assessment and reporting
(relevant sections)

RTAL 171: Study Unit 6: Relevant sections on assessment with
RLSD 171: focus on integrated assessment.

Relevant sections on assessment of LSEN

20

Study unit 2

Individual activity

1. Define assessment in your own words.
2. What is the difference between assessment of Mathematics concept and

assessment for Mathematics concept?
3. What are the purposes of assessing mathematical learning in Grade R?
4. Name the elements, which should be included in all assessment.
5. Explain how to find a child’s level of Mathematics concept development.
6. Summarise the different assessment methods a Grade R teacher can

implement to assess the Mathematics developmental level of the learners.
7. Explain the role of equity in the assessment of Grade R learners and how you

would ensure this through your assessment strategies.
8. Define the following concepts:

a. Record folder/file
b. Rubric
c. Holistic evaluation

2.9 Providing for LSEN to develop mathematical
concept

Teaching in an inclusive Grade R classroom is an enormous challenge due to a variety
of approaches a teacher needs to apply in order to meet all the learners’ needs.
Assisting LSEN to develop to their optimal ability is dealt with in depth in module RLSD
171 and you have to study these modules in an integrated manner. The scope of this
module does not allow for an in-depth study of how to assist LSEN in mastering the
fundamental concepts of Mathematics. The textbooks by Charlesworth & Lind (2013) /
Charlesworth (2016)(8th Ed) and Brewer (2007, 2012) guide teachers in this difficult
task by continuously providing methods whereby the teacher can assist LSEN. These
methods will support the teacher in developing and employing a variety of strategies to
assist special learning needs. Accommodating LSEN requires a lot of extra effort, self-
regulated study and research by the teacher in order to gain sufficient knowledge to
assist LSEN in the most effective manner. However, the personal reward and
satisfaction when these learners eventually master the knowledge and skills cannot be
substituted by any financial compensation and provides a feeling of fulfilment that
cannot be described in words.

21

Study unit 2

Study material/Integration between modules

Study the relevant sections. 7th Ed Unit 2 or 8th Ed Chapter 1 (1-2e):
Textbook: Children with special needs
Charlesworth and Lind. 2013.
Math and science for young 7th Ed Unit 3 or 8th Ed Chapter 1 (1-3g):
children (7th Ed) x Meeting special needs
x Multicultural problem solving
Or NB. Pay attention to the integration of LSEN in all
Charlesworth, 2016 (8th Ed) units; Most units in the text book continuously
incorporate strategies to accommodate LSEN,
Brewer. 2007/2012. which guide teachers in applying relevant
Introduction to Early techniques to support the learning of the different
Childhood Education: Pre- Mathematics content areas by LSEN.
school through Primary x Relevant chapter: Manipulation and discovery
through Mathematics Children with special
needs

x Celebrating diversity

RTAL 171 Study Unit 2: Relevant sections - Variables that
RLSD 171 affect learning.

Relevant sections on teaching LSEN.

Smith, S.S. 2009. x Equity for every child
Early childhood mathematics. x Serving children in inclusive settings
(Available at resource centres) x The child who is gifted
x The child with learning disabilities
Seefeldt, C. & Galper, A. 2008. x The child who is cognitively disabled
Active experiences for active
children – Mathematics (2nd Ed) Part one
(Available at resource centres) x Active children, Active environments
x Planning for inclusion

Individual activity

1. Accommodating special learning needs in Mathematics learning is challenging.
How would you assist a child who is not fluent in the language of teaching and
learning to join in the Mathematics learning experience? Can you think of more
ways to accommodate other special needs in the learning of Mathematics?

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Study unit 2

2. What is MLD?
3. What methods are suggested by Geary (Charlesworth & Lind, 2013, p. 28)/

Charlesworth 2016:19, 20) to help learners who have MLD?
4. Karp and Howell (Charlesworth and Lind, 2013:29) / Charlesworth 2016:20)

emphasise the importance of individualised approaches for children with
learning disabilities. Discuss the four components of individualisation as
suggested by Karp and Howell.
5. Charlesworth and Lind (2013:47) / Charlesworth (2016:32) refer to behaviourist
approaches to Mathematics instruction in ECSE. Why do you think this
approach is often used for LSEN? What is the view of Charlesworth and Lind

Remember to file mentor assisted tasks and any practice based self-study tasks in
Section 4 of your Portfolio. Create relevant subsections e.g. Mathematics,
Assessment, etc. to organise this section in a meaningful way.

Ask your mentor for examples of the various assessment instruments/tools he/she is
using. Include a copy of each tool/instrument in your portfolio and write on a separate
page what the purpose of each of these tools are and your mentor’s advice on the
best way to implement each. If possible, ask the mentor if you may have an
electronic copy which you will be able to adapt to suit your own specific teaching
needs and context. Create a division in your portfolio for ‘Assessment
tools/instruments’ and file these examples in Section 4 of your Portfolio. Add more
examples to your file whenever you come across usable tools such as rubrics,
assessment sheets, peer and self-assessment worksheets, etc.

You need to observe a Mathematics lesson presented by your mentor. Discuss
entails.

1. After the lesson observation, complete the observation sheet below (Table 1).
Keep the module outcomes in mind and provide evidence of your knowledge or
the module content through your observations.

2. Use Table 2 to identify if and how the lesson accommodates the learning
theories as set out in column one. Create a section in Section 4 of your portfolio
for “Lesson observations’ and file a copy of the lesson plan, observation
sheet as well as your observation of the way this lesson adhered to the
principles of the various learning theories in your portfolio.

3. File above tables in Section 4 of your Portfolio as evidence of implementation.

23

Study unit 2 TABLE 1: RMAT 111 – WIL FORMATIVE ASSESSMENT TASK

Aspects observed MATHEMATICS LESSON - OBSERVATION SHEET
Introduction phase
Observations What did I learn?
Teacher’s voice – what role does the
teacher’s voice play to promote
learning/get the learners’ attention?
Resources used
Getting the attention of learners
How does the introduction prepare
learners for learning experience?

Teaching and learning phase
What role does the teacher play? Refer to
teaching as well assessment roles.
Learners’ role
DAP teaching and learning
Integration of other subjects
Choice and implementation of resources
Classroom atmosphere
Closure phase
Teacher’s role
Learners’ role
Resources

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Study unit 2

TABLE 2: RMAT 111 – WIL FORMATIVE ASSESSMENT TASK

LESSON OBSERVATION WITH FOCUS ON LEARNING THEORIES

Main characteristics/principles Principles of Reflection on lesson:

of each theory theories identified In which way did/could
principles contribute to
THEORIES effective Mathematics
learning?

Mostly
Fairly
Seldom

Piaget Constructivist theory

colour, size, shape and texture
can be used to construct
logico-mathematical
knowledge.

Vygotsky Socio-cultural theory:

constructing knowledge within
the context of interaction

ZPD theory

Bronfenbrenner Ecological Systems Theory

A child develops within a
complex system of
relationships and all
relationships are influenced by
surrounding environment.

Gardner Multiple Intelligences Theory

A child should get opportunity
to learn through his/her
strongest skills

Ethno- Mathematics learned outside
mathematics of school (cultural, historical

background) serves as
knowledge base for new
knowledge

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Study unit 2

2.10 Implementation of a mathematics learning centre

Literature clearly shows that the young child learns best when allowed to explore
concepts in an informal classroom environment through play. Creating an area where
learners can investigate Mathematics is an effective way to ensure that mathematics
learning is mediated in the most developmentally appropriate way. In this area the
teacher can place various objects through which the learner can discover concepts on
his/her own or in groups. These objects can relate to the topic under discussion such
as “My body” but can also include objects such as puzzles or educational resources
that support mathematical concept development.

RWIL 111 - Formative assessment task for RMAT 111

Refer to RWIL 111 for information on the mentor assisted task based on
Mathematics. You need to apply the knowledge gained through this Study Unit of
RMAT 111 in the implementation of the formative assessment task as proof of work
integrated learning.

NB. This task needs to be implemented during the three weeks practical teaching.
Students need to file any formative assessment and feedback by the mentor in
Section 2 of the portfolio and also provide evidence of implementation of the WIL
task in Section 3 of the portfolio.

Reflection

TIME FOR REFLECTION

Use the Professional Development Journal (see Study Guide Addendum) to reflect
on what you have gained in this unit regarding knowledge, skills and attitude. You
can use the electronic version on the DVD or make copies of the journal format
provided in the Study Guide Addendum.

Also feel free to express your feelings regarding barriers you experienced in
mastering the outcomes of this unit.

Important information

Please make sure that all exercises are completed before you continue with the next
unit and that you have achieved the outcomes.

Spot checks may be done by NWU assessors at students’ schools to monitor the
application thereof.

26

Study unit 3

Study unit 3
FUNDAMENTAL MATHEMATICAL CONCEPTS AND SKILLS

Study time

It is recommended that you allow approximately 45 hours for completing this Study
Unit successfully.

Study outcomes

After completion of the Study Unit the student should be able to

x define and describe the main areas of Mathematics;
x recognise and describe mathematical concepts implemented in Grade R; and
x describe, demonstrate and argue the relationship between Mathematics and

other areas of learning in Grade R.

3.1 Introduction

The Department of Education sets a specific framework for content to be covered in the
form of a Mathematics curriculum. This ensures that all learners have equal opportunity
to master the relevant mathematic content in more or less the same time frame. The
teacher needs to have a good knowledge of the main areas of Mathematics and the
key terms, concepts, facts and rules in relation to each main area.

In Study Unit 1 we looked at what Mathematics is and in Study Unit 2 we explored
how learners learn Mathematics and the implications of learner cognition for
teaching Mathematics (pedagogy) in the early years. We also learnt that the
teacher’s view of Mathematics and how learners learn Mathematics will determine
greatly what happens in Mathematics learning experiences. In this Study Unit we will
be exploring mathematical content relevant for Grade R learners. Modules RMAT121
and RMAT 211 will deal in more depth with the mathematical content areas and the
implementation of Mathematics teaching and learning in the planning of learning
experiences for Grade R learners. It is thus necessary that you make a thorough study
of all content areas in a self-regulated way. Although Grade R learners are not
expected to learn all formal mathematical terminologies, a Grade R teacher should
know what concepts the child should master and the relationship between these
concepts and more formal Mathematics for which the Grade R curriculum is preparing
the learners.

27

Study unit 3

3.2 Mathematics is based on a universal mathematical language. Although approaches to
the teaching and learning of Mathematics may differ from one country to another, one
school phase to another or one classroom to another, Mathematics in schools all over
the world focuses more or less on the same main content areas.

Overview of the main areas of mathematics

Important information

Students practicing in countries other than South Africa: Apply this section to the

Study material

Study the relevant sections: 7th Ed Unit 1 or 8th Ed Chapter 1 (1-b)
Principles of school Mathematics
Textbook: Standards of school Mathematics
Charlesworth and Lind. 2013.
Math and science for young children (7th Unit 2: Content areas for Mathematics
Ed) Education
Or
Charlesworth, 2016 (8th Ed) Relevant chapter: Manipulation and
discovery through Mathematics –
Manual: Teaching Mathematics – Content
strands
Brewer. 2007/2012. Introduction to Early
Childhood Education: Pre-school through

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Study unit 3

CAPS Chapter 2: Mathematics – Aims, skills
and content

Individual activity

1. Compare the Strand Model as implemented in the USA with the Content Areas
as set out in the South African Mathematics school curriculum (CAPS). Discuss
the differences and similarities between the two conceptualisations of the
important elements of Mathematics education with your colleagues or during
contact sessions.

2. Why does the Strand Model place Numbers and Operations at the centre of the
diagram? Do you agree with this outlay? What is the position of this Content
Area in the South African Mathematics curriculum?

3. Why are the Process Strands placed in the outer ring of the Strand model? Can
you find concepts in the South African Mathematics school curriculum that
correlate with the Process Strands? Which processes in Mathematics teaching
and learning can you identify in the SA curriculum?

4. Diagrammatically illustrate the main Content Areas as set out in the South
African Mathematics school curriculum.

Individual activity

Review the DVD: Exploring Mathematics in Grade R (videos 2.1 – 2.4): Integrating
within Mathematics. Pay attention to the following: Holistic teaching and learning of
Mathematics.

1. While watching the DVD, use the graph below and indicate with an X how
many times you can identify the various content areas to be implemented
during the learning experience.

10 Number and
9 Operations
8 Patterns
7 and Algebra
6 Shape and
5 Space
4 Measuring
3 Data
2 handling
1

Times
implemented

CONTENT AREAS

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Study unit 3

2. How many times did the learners use Numbers, Operations and Relationships?
And the other Content Areas? What does this data tell you about the Content
Areas as applied in early Mathematics learning?

3. Compare your findings with the weighting of the various Mathematics Content
Areas for Grade R as suggested by the school curriculum (CAPS – inserted
below).

3.3 Mathematical concepts and skills implemented in