FUNDAMENTALS OF

MATHEMATICS

TEACHING AND

LEARNING IN GRADE R

Faculty of Education

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RMAT 171 PED

FUNDAMENTALS OF MATHEMATICS

TEACHING AND LEARNING IN GRADE R

Faculty of Education

Study guide compiled by: C. Kruger

Copyright © 2018 edition. Review date 2018.

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

Portfolio tasks ................................................................................................................................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

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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

Study guide addendum .............................................................................................................. 40

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

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Module information RMAT 171

Module code 16

Module credits

Module name Fundamentals of mathematics teaching and learning in

Grade R

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

RFAA221 Afrikaans: First Additional 8

Language

RFAX221 isiXhosa First Additional 8

Language

RFAS 221 Setswana First Additional 8

Language

RFAZ221 isiZulu First Additional 8

Language

RFAO221 Sesotho First Additional 8

Language

RFAP221 Sepedi First Additional 8

Language

RFAW221 SiSwati First Additional 8

Language

TOTAL CREDITS YEAR 3 380/

388

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Work-integrated learning: (RWIL)

See Work-integrated Learning Study Guide for more information on Work-integrated

learning and learning tasks.

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

write your assignment.

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.

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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

and learning in Grade R

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

primary grades. Boston: Pearson

x Curriculum and Assessment Policy Statement (CAPS) – Foundation Phase

Mathematics Grade R-3 and Foundation Phase Mathematics Grade R (South

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

Additional literature

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

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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

your written assignment for assessment.

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.

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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

interactive white board sessions is to your advantage and you are advised to

make use of these if possible.

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

you through your examination preparation.

Portfolio tasks

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

portfolio to showcase your professional development. Use your own discretion by

creating additional sections in your file for aspects such as forms for administration,

school policies, lesson plan formats, assessment tools, etc.

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

Your observation form should also be included in your portfolio.

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

give reasons for your view.

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

your stance in this regard.

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

reader of your view.

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.

x Read

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

Icons for Grade R learning Mentor task

Portfolio task

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

WHY MATHEMATICS IN GRADE R?

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

Grade R education.

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

Addendum 1A:

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

Additional Reading:

Follow the link to Google and study the following article:

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

Complete self-study task 1.1:

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.

Manual: Addendum 1A:

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.

Additional reading: Part one – theory of Experiences

Seefeldt, C. & Galper, A. (2008) Active

experiences for active children –

Mathematics (2nd Ed)

(Available at resource centres)

3

Study unit 1

Individual activity

Complete self-study task 1.2:

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

Self study task 1.3:

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 addition

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

tasks?

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

Self study task 1.4:

Discuss the visuals with your co-students/colleagues/mentor and find answers to the

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

Addendum 1A: How children learn

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

Primary Grades. x Developmentally appropriate

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

Additional reading

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

Complete self-study task 2.1:

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

of Mathematics in Grade R.

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

Grade R context.

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

for the Grade R teacher.

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

Addendum 1A:

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

Additional reading

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

Complete the self-study task 2.2:

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,

relevant to Grade R.

3. What are adult-guided learning experiences? Do you think there is a

relationship between adult-guided learning experiences and Vygotsky’s ZPD

theory? Motivate your view.

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

Complete self-study task 2.3:

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

Self study task 2.4

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

with your colleagues. Find answers to the following questions through group

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

Grade R?

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

(7th Ed) Addendum 1A:

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

Complete self-study task 2.5:

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

Complete self-study task 2.6:

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

experience in Grade R.

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

Self study task 2.7

Review the video: Exploring Mathematics in Grade R (videos 2.1 – 2.4). Pay

attention to the following: Problem solving through play.

Discuss in your groups or with your mentor/colleagues:

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).

Motivate your answer.

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

Self study task 2.8:

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-

school through Primary Grades.

Integration between modules Study Units 1 and 2: Relevant sections

Study Unit 5: Teaching methods

RTAL 171 : Teaching and learning

Additional reading Chapter 10: Problem solving – addition

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

Complete self-study task 2.9:

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.

19

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.

Grades.

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.

Addendum article: Katz, LG. (1997)

Relevant sections on assessment of LSEN

20

Study unit 2

Individual activity

Complete self-study task 2.10:

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

Grades.

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.

Additional reading Chapter 1: Foundations, Myths, and Standards

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

Complete self-study task 2.11:

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?

22

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

regarding this approach? And your view? Motivate your answer.

Portfolio task

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.

Mentor assisted task

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.

Mentor assisted task/ Portfolio task

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

the task with your mentor and make sure the mentor knows what the task

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

Linked to introduction?

What role does the teacher play? Refer to

teaching as well assessment roles.

Learners’ role

Learning tasks

DAP teaching and learning

Content addressed

Integration of other subjects

Choice and implementation of resources

Classroom atmosphere

Closure phase

Teacher’s role

Learners’ role

Resources

24

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

Physical knowledge about

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

25

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.

Mentor assisted task/Portfolio task

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

prescribed curriculum/syllabus of your country; for portfolio tasks – please indicate

which curriculum you base your answers on.

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

Primary Grades.

28

Study unit 3

CAPS Chapter 2: Mathematics – Aims, skills

and content

Individual activity

Complete self-study task 3.1:

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

29

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

Grade R

It is import to implement Mathematics in an informal way through play in Grade R.

Learners need to experience Mathematics as exciting and meaningful. As play and

games are a serious business in the early years, young learners will use Mathematics

to solve problems which they encounter through play and also apply these skills in real-

life problem solving. Grade R learners explore various Mathematics concepts through

engaging with activities presented as part of the daily programme. Concepts such as

matching, classification, ordering, comparing and estimation are often used in an

integrated way in Mathematics and Science.

30