# 2007 University of South Africa

All rights reserved

Printed and published by theUniversity of South AfricaMuckleneuk, Pretoria

PST201F/1/2008-2010PFC102R/2/2008-2010

98156284

3b2

Karin Style

CONTENTS

DEAR STUDENT (v)

INFORMATION ON THIS MODULE (v)

GENERAL OVERVIEW (vii)

SECTION 1Teaching mathematics: foundations and perspectives 1

Chapter 1: Teaching mathematics in the era of the NationalCouncil of Teachers of Mathematics (NCTM)Standards 2

Chapter 2: Exploring what it means to domathematics 6

Chapter 3: Developing concepts inmathematics 8

Chapter 4: Teaching through problem solving 12Chapter 5: Planning in the problem-based

classroom 15Chapter 6: Assessment in mathematics instruction 18Chapter 7: Teaching mathematics equitably to all learners 21Chapter 8: Technology and school mathematics 22

SECTION 2Development of mathematical concepts and procedures 23

Chapter 9: Developing early number concepts and numbersense 24

Chapter 10: Developing meanings for theoperations 30

Chapter 11: Helping children master the basic facts 33Chapter 12: Whole-number and place-value

development 36Chapter 13: Strategies for whole number

computation 40Chapter 14: Computational estimation with whole numbers 42Chapter 15: Algebraic thinking: generalisations, patterns and

functions 44Chapter 16: Developing fraction concepts 46Chapter 17: Computation with fractions 49Chapter 18: Decimal and percent concepts, and

decimal computation 52

PST201F/1/2008±2010PFC102R/2/2008±2010 (iii)

Chapter 19: Proportional reasoning 56Chapter 20: Developing measurement concepts 59Chapter 21: Geometric thinking and geometric

concepts 67Chapter 22: Exploring concepts of probability and data

analysis 69Chapter 23: Exploring concepts of probability 71Chapter 24: Developing concepts of exponents,

integers and real numbers 72

APPENDIX A: Learning outcomes and assessmentstandards for grades 4 to 9 73

BIBLIOGRAPHY 95

(iv)

DEAR STUDENT

This study guide is intended for mathematics teachers in the Intermediate andSenior Phases (grades 4±9). It is a wrap-around guide for the prescribed book Van deWalle, JA. 2007. Elementary and middle school mathematics: teaching developmentally. 6thedition. Boston: Pearson/Allyn & Bacon. It covers the whole syllabus, which issubdivided into number learning and spatial learning.

The aim of the study guide and the textbook is to prove to you and your learners thatmathematics makes sense and that you are capable of making sense of it yourself.

Please note the following:

. I use the same chapter titles and sectional divisions as the ones in the textbook. Ibelieve this will be easier for you.

. The page numbers referred to throughout this study guide are those in thetextbook, unless stated otherwise.

. The assignments are based on the textbook.

. The assignments are formulated in Tutorial Letters 101 and 102.

. The study guide has been adapted for use of the new 2007 edition of theprescribed book.

INFORMATION ON THIS MODULEThis module has been prepared by the School of Languages, Education, Arts andCommunication

Department: Teacher Education

Module: Mathematics and Mathematics Teaching

Module code: PST201F

Unit level: NQF level 5

Credits: 12 SAQA credits

Field and subfield of study: ETD Field Teacher Education

Date of issue: October 2007

Review date: April 2011

Purpose of this module: This module will prepare students to teach theconcepts, skills and values defined in thelearning area Mathematics in the Intermediateand Senior Phases.

Learning assumed to be in place: It is assumed that the student has a basicunderstanding of the stages of a child'sdevelopment and has leadership experience.

It is also assumed that the student has aknowledge of and has acquired skills in basicmathematics.

PST201F/1 (v)

SPECIFIC OUTCOME FOR THIS MODULEOn completion of this module, you, the student teacher, should be able to

. demonstrate that you understand

Ð modern trends and ideas in the teaching and learning of mathematicsÐ how learners learn mathematicsÐ how to teach mathematics

. apply modern teaching approaches and the knowledge of how learners learn invarious teaching and learning situations

. do continuous assessment in the learning area Mathematics in the Intermediateand Senior Phases

. plan the effective teaching of concepts, skills and values in the learning areaMathematics for various teaching and learning situations

The following quotation from the author of your textbook (Van de Walle 2007:xvii)applies to you, the teacher:

Learning how best to help children believe that mathematics makes sense andthat they themselves can make sense of mathematics is an exciting endeavorand a lifelong process. It requires the knowledge gained from research, thewisdom shared by professional colleagues, and the insightful ideas that comefrom your own daily experiences with students.

I trust that you will enjoy every step of this fantastic journey.

(vi)

GENERAL OVERVIEWBEFORE YOU BEGIN ...Welcome to the Department of Teacher Education at the University of South Africa(Unisa).

This module concerns the teaching of mathematics in the Intermediate and SeniorPhases.

The duration of the module is one year.

Please read this section carefully because it contains valuable general information onthe successful completion of this module.

STUDY MATERIALAmongst other material, your prescribed textbook and study guide for this moduleare the following:

. Prescribed textbook Ð Van de Walle, JA. 2007. Elementary and middle schoolmathematics: teaching developmentally. 6th edition. Boston: Pearson.

. This study guide Ð University of South Africa. Department of Teacher Education.2007. Professional Studies: Mathematics and Mathematics Teaching. Revised edition.Pretoria.

THE TEXTBOOKThe textbook forms the basis of this module and contains fundamental ideas on theteaching of mathematics, the development of mathematical concepts andprocedures, and other issues and perspectives relating to the teaching ofmathematics in the Intermediate and Senior Phases.

Read the preface in your textbook thoroughly. It contains a description of the contentof the textbook. It also gives you an opportunity to develop a strong theoreticalperspective on a learner's learning in mathematics.

THE STUDY GUIDEThe study guide was written to accompany the textbook.

In the study guide I used the same order of sections and chapters that appears in thetextbook.

The approach in the different sections is in accordance with the teaching approach ofthis module.

The textbook is your manual or workbook, and you will be working through itsystematically. It is an excellent book. No South African text such as this is currentlyavailable Ð therefore you should buy it immediately.

PST201F/1 (vii)

ADVANTAGES OF THE TEXTBOOKAs far as standards are concerned, the textbook meets current needs (of the country,people and curriculum) and requirements at all levels. It is written in a user-friendlystyle, uses an interactive approach and can, therefore, be used as a workbook.

The textbook is based on the problem-solving approach to teaching, which presentsthe teaching of mathematics in context. Mathematics teaching follows an integratedapproach which stresses mathematical thinking and making sense.

You, the reader are actively involved, since the textbook contains a large number ofself-assessment exercises which thoroughly and continually expose you to classroomrealities.

The textbook does not follow American standards slavishly, but offers enoughexamples of meaningful learning opportunities to all readers and students in SouthAfrica. Therefore, in using this book, I try to help and support you as a student in asystematic progression from whatever other approach you are accustomed to(perhaps the traditional one, or some personal and unrefined approach) towardsstudying and using a new approach: the problem-solving approach.

In 1998 the Republic of South Africa (RSA) introduced a new curriculum with a newapproach to teaching and learning, namely the outcomes-based education (OBE)approach. The teaching approaches that best suit this curriculum approach, andwhich are relevant to the mathematical and educational sciences, are the problem-solving and the problem-centred teaching approaches in mathematics education.

The textbook is based on the American education situation, but it is used becauseOBE in the RSA is close to the education situation that presently prevails in theUnited States of America (USA). (Refer to the American ``Principles and standardsfor school mathematics'' (in the textbook, p 510) and the South African NationalCurriculum Statement (NCS) Ð found at the back of this study guide.)

THE RELATIONSHIP BETWEEN DIDACTICS ANDMATHEMATICS EDUCATIONThe content of the course on Mathematics Education is founded partly on what isknown as Didactics.

Didactics is a subdiscipline of Education, which studies the components and facetsof teaching (and learning). Teaching and learning as didactic events in themathematics classroom concern the teacher, the learners, the outcomes of thelearning area, integration with other learning areas, the concepts, skills and valuesdefined for the learning area, the teaching methods used in the mathematicsclassroom, the media used in the teaching situation, and assessment and remedialwork in the mathematics classroom.

The general principles and pronouncements of didactics should be applicable to thedidactics of any subject. It may well be that the importance and emphasis may varyfrom one learning area to the next. I advise you to study general didactics bykeeping your specific learning area in mind and asking questions such as: ``How canthese general principles or pronouncements be applied in the teaching of concepts,skills and values in the learning area Mathematics?''

(viii)

ASPECTS OF THE COURSE THAT WILL BE EMPHASISEDIn this module on Mathematics Education, based on the learning area Mathematics, Iwill concentrate on elements that are closely related to the practical teaching of thecontents, skills and values defined for the Intermediate and Senior Phases.

In studying the teacher as a component of didactics, I will focus on the didacticknowledge necessary for successful teaching in the learning area Mathematics in theIntermediate and Senior Phases.

Since you are all graduate students and have successfully completed schoolmathematics, I assume that you have sufficient academic knowledge of the content ofmathematics. However, the question is whether your knowledge provides thenecessary background to school mathematics. The fact that so little graduatemathematical content appears in the learning area for mathematics, poses the questionwhether it would not be sufficient for teacher to have knowledge of only grade 12mathematics or perhaps a higher level. I agree with the statement that individualteachers should know considerably more than their learners and that they should beable to view the content of the learning area Mathematics from a wider perspective. Itis thus for a teacher to remember to steer the teaching-learning situation in such a waythat the learners' numerical, spatial and linguistic aspects of life will be developed.With this view in mind, it is essential that you read widely on mathematics.

You are expected to be familiar with the content of the learning area Mathematics forall three phases, namely the Foundation Phase, the Intermediate Phase and theSenior Phase. Although I will concentrate on the Intermediate and Senior Phases, allthe mathematical content defined in this learning area in all three phases is regardedas prescribed content for this module.

The emphasis in teaching approaches has recently shifted to the problem-solving(PSA) and problem-centred (PCA) approaches to teaching. We can refer to these asthe problem-based teaching approaches.

There is a close relationship between the teaching of mathematics and the way inwhich a learner learns mathematics. It is therefore not feasible to ignore thecomponent based on how a learner learns in the teaching of mathematics. This iswhy we refer to teaching-learning situations, problem-centred learning, or problem-based teaching and learning approaches.

In this module you will be confronted with the problem-solving teaching approach.You will also be introduced to a problem-centred approach. You will be exposed tomany situations in which you will have to make a choice between the traditionalapproach and the problem-based approach to teaching. Hopefully, you will beequipped with enough knowledge of the benefits of the problem-based approach tochoose this approach without hesitation.

The textbook is based on the problem-solving approach (PSA) to teaching andlearning.

Note that the role of the teacher in the two approaches (PSA and PCA) differsconsiderably, although his/her task and responsibility have not changed at all.

The main difference between the two approaches is that in the PCA theresponsibility of self-learning and self-development is shifted more towards thelearner than in the PSA. In the PCA a non-routine problem(s) is used as a vehicle oflearning and the teacher is seen as the facilitator of learning; whereas, in the PSA theteacher can still be a source of knowledge (although not the only source Ð as is the

PST201F/1 (ix)

case in the traditional classroom). In this approach thought-provoking activitiesusually act as vehicles of learning.

Please do not confuse the PCA or the PSA with the discovery teaching model Ðthere are major differences, as can be seen throughout the textbook.

I also expect you to reflect on the presentation of certain concepts in different gradesin the school. Because of the close ties between method and content, I will handlethem together.

Needless to say, the role of the learner in the teaching-learning process has becomecrucial when analysing and discussing the teaching-learning situation inmathematics education. It is important to understand that teaching and learning inmathematics are both based on certain strategies. These two sets of strategies shouldobviously be interlinked and interrelated. The questions, ``How do children learnmathematics?'' and ``How and what do children think when doing mathematics?''have become two of the most important questions in mathematics education today.The matter of strategic learning is crucial in the shift towards learner-centrednessand task-centredness.

In this module, some consideration will be given to certain aspects of practicalclassroom procedure. In this section, aspects such as the content of the learning areaMathematics, work programmes, record keeping of assessment, assessment, etcetera will be considered.

OUTCOMES-BASED TEACHINGIf you study the standards set in the USA well, you will see that, in principle, wefollow those standards. However, standards have been worked out with a firmSouth African orientation.

Van der Walle (2007) also uses the OBE teaching approach. The way in whichobjectives have been formulated and patterns of evaluation have been developed,illustrate this clearly. This reflects the standards in the USA.

You will note that the South African educational officials introduced newterminology for old concepts. They replaced ``subject'' with ``learning area'', and``aims and objectives'' with ``outcomes'' and ``learning outcomes''. This willdefinitely work in favour of the transformational process in order to establish a newframe of mind Ð it is not only because of political motives. Renewal should beaccompanied by a complete metamorphosis.

STUDY GUIDANCEStudy guidance is provided in the chapters in this study guide. It is here that I willtake you through the textbook step by step.

Use the study guidance in conjunction with your textbook. I will also use this part ofthe study guide to communicate to you what is expected of you.

Here are a few pointers on how to use the textbook. Before proceeding further, I willtake a look at three important techniques that you will use often at the beginning of anew chapter.

(x)

INDEPENDENT STUDYA crucial phase in the process of understanding and learning mathematics byproblem solving is to articulate your ideas about mathematics, and the teaching ofmathematics, both orally and in writing. Only when you have tried this process foryourself will you understand the full value of this exercise.

STUDY PROCESSI want to make you aware of your own studying process. I will explain a uniquelearning strategy to you Ð an approach you may find useful. Try to refine your ownapproach. The three techniques contained in this strategy are characterised asskimming, scanning and study-reading. You may find other subdivisions andemphases of these terms elsewhere, but I will use them here as follows:

Please note: This process requires you to work with a pen and a marker at everystage. Mark and write as you work through the material.

The exercise comprises three steps in total. In order to understand what these stepsinvolve, you should study the example given below. An instruction to carry out thisexercise will usually appear at the beginning of each chapter. This means that youshould follow the steps carefully as they are set out below. Wherever the activity isgiven, it will appear under the heading ``Overview and exploration''.

Overview and exploration (example)

(1) SKIMMING

. Page through, explore. Read the section or paragraph quickly, forming arough idea of the contents. Concentrate on headings and subheadings, words/text in bold and italics, boxes, tables and illustrations, and Ð in the case of achapter Ð introduction and summary.

. Make a cursory survey. While reading, ask yourself, ``What key terms occurin this division or chapter?'' Stop when you identify a key term and readcarefully what is said about it. Mark it in the textbook. What you are trying toascertain is: Where is it?

(2) SCANNING AND REFLECTING

. Scan the section or chapter.

. Start drawing a mind map (for the whole or parts of it, as in starting asummary). You are looking for items and concepts while reading theinformation in the section or chapter in a more evaluative way. Reflect oninterrelationships between concepts. The question now is: ``What is it?''``What is the meaning and the purpose?'' Visualisation is important and youare certainly going to start writing down key concepts. You can omit parts ofthe text.

. Deeper reflection. Start building a structure in your mind map; worktowards an entirety. As you work through the prescribed activities of thesection or chapter, keep on returning to the mind map to fill in the detail.Reflect on the value and meaning of the categories, concepts, motivations,variables and key terms.

PST201F/1 (xi)

(3) STUDY-READ

Study-read. This follows directly on steps 1 and 2 and should be done carefully,thoroughly and thoughtfully. The key terms and concepts you pinpointed have tobe linked up, and here the mind map and summaries are important. Pause whilereading, consolidate what you remember and consider how new information fitsin with what you already have.

Apart from the activity explained above, you will find an introductory paragraph (oroverview section) at the beginning of a chapter. The introductory paragraphexplains what the chapter is all about and how the content of the chapter is linkedwith other areas that are relevant to the specific chapter Ð read this carefully. Theaim here is to broaden your perspective and outlook, and to help you identifyproblems and resolve them in a new way in the context of teaching and learning.

MIND MAPSSome people have difficulty working out mind maps. You are urged to try it. Theidea that you should do some sort of summation of the content of the chapter on onepage is an important one. Please do your best.

ACTIVITIESWhenever you come to an activity, complete it in full on loose pages which you canthen insert in the plastic folders of your file and group together chapter by chapter. (Weshall henceforth refer to this file as your portfolio.) Supplement this with your ownnotes from your notebook. Proceed, using your textbook and this guide in tandem.

WORKING METHODYour best working method would be to work frequently and regularly on thissubject.

Use the following method to refer to Van Walle (2007):

Because the book is printed in a double-column format, I will refer to the twocolumns as (a) and (b) respectively. Thus, page 31(a) refers to the first columnon page 31 and page 31(b) to the second column. If there is no (a) or (b), thewhole page is intended.

SELF-ASSESSMENT EXERCISESAt the end of each chapter (in the textbook) there are self-assessment exercises called``Reflections on the chapter'', which are designed to

. assess the progress you have made towards achieving the chapter objectives

. allow you to determine your own level of competence and what you still have todo to reach the required standard

. reinforce and expand the knowledge and insights derived from the chapter

(xii)

This is the reason why I use summaries and mind maps. In this way I hope to fix theknowledge more firmly in your mind.

USE THE TEXTBOOKVan Walle (2007) is an excellent book. It has a comprehensive bibliography at theend of each chapter. Take a good long look at these bibliographies and order some ofthe articles listed from the library.

Van Walle (2007) is an excellent source to use when you do your teaching practice.Use it for lesson preparation and for new ideas regarding the concepts that should betaught!

PST201F/1 (xiii)

SECTION 1

TEACHING MATHEMATICS: FOUNDATIONS AND PERSPECTIVES

CHAPTER 1TEACHING MATHEMATICS IN THE ERA OFNATIONAL COUNCIL OF TEACHERS OFMATHEMATICS (NCTM) STANDARDS

gAfter working through this chapter, you should be able to

. decide for yourself whether the curriculum and evaluation standards for schoolmathematics formulated in the USA are meaningful and how they relate to theLearning Outcomes and Assessment Standards defined in the nationalcurriculum in our country

. give your own opinion on what the textbook discusses as ``the revolution inschool mathematics'' (Van de Walle 2007:2)

. demonstrate your understanding of thinking about mathematics teaching as:

Ð problem solvingÐ communicationÐ reasoning and proofÐ mathematical connectionsÐ representation

. give an account of the five shifts in the classroom environment

1.1 OVERVIEWIn this important first chapter, Van de Walle (2007) tries to make us aware of therevolution in school mathematics and the forces behind the revolution. He alsoaddresses the curriculum as the basis for teaching and learning school mathematics.

bActivity 1.1

Use step 1, Skimming, in your learning strategy to help you work through thechapter.

SKIMMING:

. Page through the chapter.

. Read the chapter quickly to form a rough idea of the content.

. Concentrate on the headings, the subheadings and the text in bold.

Note the following key terms:

. the revolution in mathematics education

2

. the forces behind the revolution

. the overview of the curriculum and evaluation standards for school mathematicsin the USA and how these correspond to the Learning Outcomes andAssessment Standards in our curriculum

. five shits in the mathematics classroom

The curriculum and assessment standards for school mathematics in the USA arereally targets (objectives) set for the teaching of mathematics at school. Can you seethe correspondence to the OBE approach implemented in the RSA?

bActivity 1.2

Use the step 2 (SCANNING) of your learning strategy to do the following:

. Scan the chapter.

. Start drawing a mind map.

. Reflect on the value, meaning and justifications in the different statements.

The following is part of your mind map for chapter 1. Complete the map on yourown.

Think what teaching mathematics is all about

What the reform movement in school mathematics is about

Think about the forces driving the reform movement, such as:

. the demands of society

. the influence of technology

. the direction of the National Council of Teachers of Mathematics (NCTM)

. the Third International Mathematics and Science Study (TIMSS) report

. the five content standards and their relationship to the Learning Outcomes inour own curriculum

. the five process standards and what our own curriculum has to say about them

. the five shifts in the classroom environment

PST201F/1 3

bActivity 1.3

Use step 3 of your learning strategy to do the following:

. Pause while reading, consolidate what you remember and consider how the newinformation ties in with what you already know.

. Think again about what is said about teaching and learning in the classroom oftoday and how it differs from the way in which you yourself were taught.

Please note the following:

. A report on the performance of South African learners at the Third InternationalMathematics and Science Study (TIMSS 1998) revealed that the overall scores ofSouth African learners (grades 7, 8 and 12) were significantly lower than those oflearners in more than 40 other countries.

. The results of this report and the research that follows it are of great significancefor the mathematics teacher in the RSA.

. You are advised to read this report because the demand for mathematical,scientific and technological understanding and expertise is greater than everbefore.

. The following are the key research questions in the TIMSS:

Ð What concepts, processes and attitudes relating to mathematics and sciencehave learners learnt, and what factors are related to their opportunity to learnthese concepts, processes and attitudes?

Ð How do educational systems differ in their intended learning goals formathematics and science, and what characteristics of educational systems,schools and learners relate to the development of these learning goals?

Ð What opportunities are provided for learners to learn mathematics andscience, how do instructional practices in mathematics and science varyamong educational systems, and what factors are related to this variation?

Ð How do the intended, implemented and attained curricula relate to thecontext of education, the arrangements for teaching and learning, and theoutcomes of the education process?

. The following are some of the results obtained in the TIMSS data:

Ð In South Africa learners' perceptions about their mathematical knowledge hadlittle effect on their achievements and, in general, they had unrealisticperceptions of their competencies.

The above indicates that the way in which we assess learners' knowledge,skills and attitudes should be adapted.

Ð Little value is added to learners' general scientific literacy by enrolling for anadditional mathematics or science subject. This is testimony to the poverty ofour current system.

The above asks for integration of the different learning areas as stated by OBE.

Ð On all the items presented, the performance of South African learners was thelowest.

Ð South African learners appear to have difficulty with graphic interpretation.Ð In general, South African learners experienced great difficulty in articulating

explanations for the free-response items (problem solving).

4

The above indicates that South African learners have difficulty relating theirmathematical knowledge to the real world (graphic interpretation) and lackproblem-solving skills and strategies. This indicates the importance of a problem-based approach to the teaching of mathematics, as advised in this module.

PST201F/1 5

CHAPTER 2EXPLORING WHAT IT MEANS TO DOMATHEMATICS

g After working through this chapter, you should be able to

. describe how your own perception of mathematics is influenced by the

description of mathematics which you studied in this chapter

. investigate the new educational goals for learners in mathematics and the verbs

related to the doing of mathematics

. discover, by doing the stated problems, what is meant by doing mathematics

and, at the same time, explore your own feelings about the strange attitude of the

author who does not supply you with the correct solutions to the problems

. describe an environment for doing mathematics and the role of the teacher who

has to create a classroom culture and environment in which learners do

mathematics

2.1 OVERVIEWThe content of this chapter will influence your attitude towards the teaching of

mathematics drastically.

The author warns us (mathematics teachers) not to accept outdated ideas about

mathematics and still expect to be outstanding teachers of mathematics.

The aim of this chapter is to open your eyes and mind to a new way of thinking

about the nature of mathematics and the way in which mathematics should be

taught.

b Activity 2.1

SKIMMING:

Page through the chapter.

Note the following:

. the difference in the traditional views on school mathematics and the description

of mathematics in the text

. the new goals for learners in mathematics and what it means to do mathematics

. the role of the educator in an environment for doing mathematics

6

STOP! Are you convinced that the ideas and statements in this chapter should havean influence on the attitudes of any mathematics teacher who intends to better his orher teaching?

If not read the chapter content again.

bActivity 2.2

Combine the SCANNING and STUDY-READ steps of learning to do thefollowing:

. Draw a mind map of the chapter.

. Write down how your attitude to the teaching of mathematics is influenced bywhat you read.

. Note the nature of the verbs on page 13. The verbs reveal a specific approach tolearning, understanding, doing and mastering mathematics. They help todevelop an environment for doing mathematics.

. Do some of the problems in the section ``Let's do some mathematics!'' (pp 14±20).

. Note the problem-solving strategies presented in the section ``An invitation todo maths'' (pp 14±20).

If you prefer to do so, you may do all the problems. I selected two, did them well(as was expected by the author) and then critically reflected on the commentsmade by the author on those problems and their solutions.

Ask yourself the question: How do I feel when I do not know whether myanswers are correct or incorrect? Do you agree with the ideas and statementsmade by the author regarding the provision of correct answers in the classroom?

. Write down the last paragraph in the section based on the verbs for doingmathematics on page 13: it contains powerful ideas.

. Have you made the required mind shift discussed in the paragraph?

Reflect on how the ideas and statements in this chapter correspond to the criticaloutcomes and learning outcomes defined for the learning area Mathematics (seeNCS Ð National Curriculum Statement) in the OBE approach in the RSA.

PST201F/1 7

CHAPTER 3DEVELOPING CONCEPTS IN MATHEMATICS

gAfter working through this chapter, you should be able to

. describe in your own words what it means to say ``we construct our ownknowledge'' as opposed to ``we absorb knowledge''

. discuss in detail the following statement made by the author: ``It is generallyaccepted that procedural rules should never be learned in the absence of aconcept, although, unfortunately, that happens far too often.'' (Van de Walle2007:28)

. develop your own idea on how mathematical concepts are learnt and understood

. reflect on and give an account of relational understanding and instrumentalunderstanding of mathematical concepts

. discuss conceptual and procedural knowledge and their interaction

. study and assess the benefits of relational understanding

. draw a diagram to illustrate five different representations of mathematical ideas

. investigate the use of models in the mathematics classroom

. examine six ideas on which a development approach to teaching and learning isbased

3.1 OVERVIEWIn the previous chapter the author discussed what it means to do mathematics. Youcan now ask yourself the question, ``How does a learner learn mathematicalconcepts?'' You can also ask the question, ``If I now know how my learners learn,what strategies can I apply to effectively teach these concepts?'' These questions areanswered in this chapter.

Changes in any educational system throughout the world normally occur extremelyslowly. The following is a statement made by Prof. Richard Kemp, a professor ofEducation at Warwick University, in 1971:

Readers for whom mathematics at school was a collection of unintelligible ruleswhich, if memorized and applied correctly, led to ``the right answer'' (thecriterion for which was a tick by the teacher) will probably agree that there hasbeen need for change.

This statement was made in 1971. Think about what happens in most classrooms inthis country. Do you really think that after more than 30 years major and radicalchanges have occurred?

Fortunately, radical changes in the teaching of mathematics have become inevitablein South Africa.

Enjoy the theories on how your learners learn and understand mathematical

8

concepts, and the discussion of the strategies on how effective teaching of theseconcepts and skills can take place, as presented in this chapter.

bActivity 3.1

SKIMMING

Page through the chapter and familiarise yourself with what is presented in it. Askyourself the following questions while you read:

. What is a constructivist view of learning?

. What is mathematical knowledge?

. How does a learner understand mathematics?

. What is the role of models in developing understanding?

. How can I teach developmentally?

. How can I select problems, tasks, activities, et cetera, that will effectively helpmy learners to learn mathematics?

Is this the first time that you have come across the theory of constructivism? You willhave to study the sections on constructivism in such a way that you can relate theconcepts in this chapter to the rest of the work in this course.

bActivity 3.2

SCANNING

Study the different sections with deeper reflection. Answer the following questionsin writing:

. Can I still think about mathematical knowledge as a wall built with bricks fromthe bottom of the wall upwards?

. What do the webs of associations for different mathematical concepts look like inmy own mind?

Ð Try to draw a web of associations that will contribute to your ownunderstanding of the number p.

Ð Now draw a web of associations that will contribute to a grade 7 learner'sunderstanding of the number p.

. How does a learner construct understanding?

. What is the difference between conceptual knowledge and procedural knowledgein mathematics?

. What are relational and instrumental understanding?

PST201F/1 9

bActivity 3.3

STUDY-READ

. Link the main ideas in the chapter and make a summary of the chapter.

. Use the following structures to help you:

Familiarise yourself with the ideas and statements in this chapter. This will helpyou to understand the work done in the rest of the textbook better because all thework in the next chapters can be linked to chapter 2.

10

A constructivist view of learning:What is it?

Constructing understanding:How is this done?

Examples of construction ofmathematical knowledge:

. relationships

. constructing computation methods

. developing a formula

What is knowledge of mathematics?

Conceptual knowledge Procedural knowledge

What is the role of proceduralknowledge?

What is meant by understanding mathematics?

Relational understanding

Understanding conceptual The individual nature ofknowledge understanding

Benefits of relational understanding:

. It is intrinsically rewarding.

. It enhances memory.

. There is less to remember, etc.

"

" ""

""" "

"

A:

B:

C:

Understanding procedural knowledge

bActivity 3.4

Models and the teaching of mathematics

STUDY-READ

. Draw figure 3.11 (p 33) in your workbook. This will indicate to you where andhow models are linked to mathematical ideas.

. Now make a summary to help you use models correctly in your classroom.

bActivity 3.5

Teaching developmentally

STUDY-READ the section on how to teach developmentally (p 34).

Use the following headings to summarise this section:

Teaching developmentallyWhat does this mean?On what ideas is it based?

bActivity 3.6

STUDY-READ

Study the work in the chapter again and then explain what is meant by drill androte learning.

What is the difference between these two actions?

Relate these two actions to actions in the classroom situation.

Let me again stress the author's warning:

``Do not accept outdated ideas about mathematics and still expect to be aquality educator.'' (Van Walle 2007:22)

All mathematics teachers should note the content of this chapter if they wish to takethe warning to heart.

PST201F/1 11

CHAPTER 4TEACHING THROUGH PROBLEM SOLVING

gAfter working through this chapter, you should be able to comment on a problem-based learning situation with regard to the following:

. a problem-solving and a problem-centred learning situation

. the non-routine problem used during the problem-centred learning situation

. the teaching actions before, during and after a problem-based learning situation

. how the teaching of problem solving itself took place during the learningsituation

. whether problem-solving goals were achieved while the learners were learningduring a problem-based learning situation

4.1 OVERVIEWKnowledge of constructivism implies a significant paradigm shift in howmathematics should best be taught.

In the traditional learning situations in our schools, teachers ``taught'' by conveyinginformation to learners who listened. Exercises followed to determine whether thelearners could do what had been taught. The learners' responsibility was to do asthey had been shown in the precise manner they had been shown.

The major pedagogical implication of the constructivist theory is that learners learnmathematics best by solving a certain kind of problem (called a ``non-routineproblem'') or by doing a thought-provoking activity which is used as a vehicle oflearning. How does a teacher teach through problem-based education? This is one ofthe questions answered in this chapter.

Please note the following again:

The textbook is based on a problem-solving approach to teaching and learning. Inthis approach a thought-provoking activity is used as the vehicle of learning.

A problem-centred approach uses a non-routine problem as a vehicle of learning.Each of these approaches is known as a problem-based approach to teaching andlearning.

The way in which you will study the content of this chapter differs from the way inwhich you studied the previous chapters.

You take the following steps in the study of this chapter:

Step 1: Study a description of a problem-centred learning situation.

Step 2: Apply the knowledge gained from the chapter to answer questions askedand to discuss statements made on the basis of the given learning situation.

12

Step 3: Reflect on what you have learned and form your own opinion on problem-centred learning versus the traditional learning situation.

The following learning situation took place in a grade 3 classroom. It is by no meansintended to be an ideal problem-solving learning situation but is based on aninteresting non-routine problem and illustrates most of the main aspects of problem-centred learning. It is a learning situation on which you, as a student, can reflect andcomment.

I enjoyed the description of the lesson and was amazed at the way in which conceptsand skills were learnt. I hope that you, too, will enjoy the work based on thislearning situation.

Although the learning situation took place in a grade 3 classroom in the FoundationPhase, all the aspects of the problem-solving approach to teaching are of the sameimportance to the teacher in the Intermediate Phase.

This is what the teacher in this classroom planned and did:

. The teacher wanted to help her learners realise the following learning outcomeand she used the given Assessment Standard to assist her.

LO 4: Measurement

Assessment standard: estimates, measures, compares and orders threedimensional object using non-standard measures.

. Length

. She used the following non-routine problem as a vehicle of learning:

She supplied each group of learners with a handprint (larger than the handprint ofan average man). She informed the learners that the handprint belonged to a giant.She wanted the learners to find out whether the giant with this handprint would beable to pass upright through the classroom door.

. The activities before, during and after the learning situation were as follows:

Ð She explained, by asking questions, the statement of the problem until she feltconfident that all the learners understood what was expected of them.

Ð She emphasised the point that that the use of standard units of measurementwas not allowed.

Ð She invited the learners to name a few nonstandard units of measurement,such as a foot or a handspan.

Ð She then let them discuss their problem-solving strategy in the differentgroups.

Ð Next, she let them solve the problem in groups.During this stage she paid a visit to each group and asked questions such as:``Explain to me how you plan to solve the problem.''

PST201F/1 13

Ð After the group solutions had been completed, the groups presented theirsolutions to the whole class with the help of the teacher.

Ð Using their preferred method of solution, individual learners then completedtheir solution to the problem in their exercise books or on the givenworksheets.

Ð Application of the lesson proceeded by doing measurements involvingnonstandard units of measurement. This forced the learners to reflect on theuse of a standard unit of measurement.

. The following is one solution presented by a group of learners:

Ð They made a handprint on paper of one of the girls in the group.Ð They let her lie on the floor and measured how many handprints can fit into

the length of her body.Ð They then measured out on the floor the same number of handprints, using

the giant's handprint.Ð This gave them an idea of the height of the giant.Ð They came to the conclusion that the giant cannot walk upright through the

classroom door.

Now go back to the contents of the chapter and do the following:

. Read the section on problems and performance tasks on page 37. Do you agreethat the educator in the above learning situation used a non-routine problem as aperformance task and as a principal means of promoting mathematical learning?

. Study the problem-solving strategies on page 57. Do you think that the learners inthis learning situation used these strategies in their solution?

. Study the teaching actions of a teacher in all three stages of the lesson on pages 41to 48. Read the description of the learning situation again and reflect on whetherthe teacher, in her presentation, did what is suggested in your textbook.

. You may ask yourself, ``Where did the teacher get hold of the problem?'' Readthrough the section on ``Designing and selecting effective tasks'' on pages 48 to 52to answer your question.

. The Assessment standard mentioned at the beginning of the learning situation isnot the only one realised during the learning situation. Reflect on the followingAssessment Standards for learning outcome 1.

. Explains own solutions to problems.

. Check the solutions given to problems by peers.

. Summarise the statements made on pages 54 to 56 on teaching tips and questions.

After completing the feedback you will be convinced that teaching through problemsolving completely blends problem-based with mathematics education. To helplearners with problem-solving skills, teachers should be aware of what goodproblem solving involves and should integrate the development of these skills intonearly every learning situation. Now read pages 57 on teaching about problemsolving itself and the important aspects related to it.

14

CHAPTER 5PLANNING IN THE PROBLEM-BASEDCLASSROOM

gAfter working through this chapter, you should be able to

. plan a problem-based lesson

. plan a teacher-directed lesson

. teach a heterogeneous group of learners

. use workstations and games when doing instruction

. use homework as a useful instructional tool

. use the textbook correctly in your classroom

. understand and apply drill and practice in the mathematics classroom

5.1 OVERVIEWThis chapter describes some basic structures for lessons and other instructionalstrategies, such as workstations and games.

The emphasis is on problem-based learning and teaching situations whichcharacterise OBE.

All interactive strategies and ideas given and expressed in this chapter can be linked tothe outcomes-based approach which is adopted in a learning and teaching situation.

5.2 PROBLEM-BASED LESSONSIn a problem-based lesson, the learner becomes responsible for his or her ownlearning (self-regulated learning). A problem-based lesson ensures the practicalinvolvement of the learner and is based on the constructivist view of learning.

bActivity 5.1

Problem-based lessons

SKIMMING: Read the introduction to the chapter carefully (p 61).SCANNING: Read the section on pages 61 to 65 on the planning of a problem-

based lesson.

Do the following STUDY-READ step:

. Rethink what you have learned so far and link it to the ideas and statementsmade in this section.

. Summarise the nine steps to follow when one plans a problem-based lesson.

PST201F/1 15

Components of a learner-centred lesson

Present the task

Discuss and reflect

Written work

Create opportunities for the learner to work.Make sure that you note down the roles of the educator and the learners

during this stage of the lesson.

!

!

!

!

. Use the following framework to summarise the components of a learner-centered

lesson.

. Variations on the three-part lesson. Note that the national curriculum

statements in our country suggests a wide range of different teaching and

learning strategies. Thus not every lesson that you present will be a problem-

based lesson.

5.3 PLANNING FOR DIVERSITYThis is one of the most difficult aspects of instruction. How does a teacher deal with

a heterogeneous group of learners?

This section of the chapter provides guidelines on how to answer this question.

b Activity 5.2

Dealing with diversity in the classroom

SKIMMING: Read the section quickly to form a basic idea (pp 64±67).

SCANNING and STUDY-READ: Summarise the advice given.

16

5.4 DRILL AND PRACTICEWhere and when should I review concepts? Am I allowed to run learners through adrill of any of the concepts learned?

These questions are answered in this section of the chapter.

bActivity 5.3

Drill or practice

STUDY-READ this section Ð make a few notes and write them down (pp 67±69).

Make sure that you understand the differences between these two actions in full.

5.5 HOMEWORKFirst think about what homework is and why we give learners homework. Then doactivity 5.4.

bActivity 5.4

Homework

STUDY-READ this section (p 70).

Write down your own notes on the nature and use of homework.

5.6 THE ROLE OF THE TEXTBOOKWhat is the role of the textbook in the modern classroom where problem-basedlessons are encouraged?

bActivity 5.5

Textbooks

STUDY-READ this section (pp 70±71).

Make your own summary of the main ideas and statements.

Please keep the following comments made by the author in mind when you use atextbook in future (Van Walle 2007:71):

If one considers the limitations of the print medium and understands that theauthors and publishers had to make compromises, the textbook can be a sourceof ideas for designing lessons rather than prescriptions for what each lessonwill be.

PST201F/1 17

CHAPTER 6ASSESSMENT IN MATHEMATICSINSTRUCTION

gAfter working through this chapter, you should be able to

. answer the question, ``What is assessment?''

. discuss the six statements (assessment standards) against which assessmentpractices can be judged

. reflect on the purposes of assessment

. reflect on what must be assessed

. explain how instruction and assessment can be combined

. understand how to collect data from performance tasks

. explore other assessment options

6.1 OVERVIEWYou should now be convinced that the way in which we teach should change. Butcan the way in which we assess the knowledge, skills and values of our learners staythe same? This is what is discussed in this chapter. The author again warns us tobreak the habit of testing only the lowest-level mathematics skills.

6.2 BLURRING THE LINE BETWEEN INSTRUCTIONAND ASSESSMENT

bActivity 6.2.1

What is assessment?

Start your study of this chapter by carefully reading the introduction to the chapterand the paragraph on the assessment standards on page 78.

Write down the definition of assessment given in this section and reflect on thecomments made.

18

b Activity 6.2.2

Assessment in instruction

SKIMMING

Page through the chapter and explore what it is about.

Note the following:

. the purposes of assessment

. what should be assessed

. how to combine instruction and assessment, and how to collect data fromperformance tasks

. the other assessment tasks and tools discussed.

Before drawing your mind map of the chapter, you should answer the followingquestions which will help you to compile a mind map.

Write down the answers to the following:

. What is assessment? Give a definition.

. How do I know that the assessment methods or tasks used are in line with what isexpected?

. What should the aims of my assessment be?

. What should be assessed?

. What is meant by blurring the line between instruction and assessment?

. What is an assessment task?

. What other assessment options are mentioned in the chapter?

. What is the meaning of the word ``rubric''?

. How do a portfolio, a journal and tests fit into an assessment plan?

Assess your understanding of the work done in this chapter by marking the relevantface:

Note:

By doing this simple exercise, you express your own feelings about the work in thischapter.

6.3 ASSESSMENT TASKS AS LEARNING TASKSRead about assessment tasks on page 80. Study the examples given on page 81.Reflect on what the textbook say and on how you yourself were assessed at school.

Compare the assessment tasks in your textbook with the assessment that you had inschool.

PST201F/1 19

6.4 OTHER ASPECTS OF ASSESSMENTThe part of the chapter based on other aspects of assessment gives us an in-depthlook at different methods, tools and the recording of assessment.

The most important section of the rest of this chapter is the work done on rubrics.

A rubric is a framework that can be designed or adapted by the teacher for aparticular group of learners or a particular mathematical task. A rubric consists of ascale of three to six points that is used as a rating of a learner's performance.

You as a student must be able to construct, adapt and use rubrics successfullybecause the use of rubrics as assessment tools is a necessity in today's instruction.

bActivity 6.3

Other aspects of assessment

. Study-read this section. Make sure that you will be able to construct a rubric foruse as an assessment tool on your own.

. Study-read the section on journals and tests and make your own notes onaspects that you think you might need in your classroom.

Do you, after completing your study of this chapter, agree with what the author sayson page 92 (Van de Walle 2007:92)?

The myth of grading by statistical number crunching is so firmly ingrained inschooling at all levels that you may find it hard to abandon. If one thing is clearfrom the discussions in this chapter, it should be that it is quite possible togather a wide variety of rich information about learners' understanding,problem solving processes, and attitudes and beliefs. To ignore all of thisinformation in favour of a handful of numbers based on tests, tests that usuallyfocus on low-level skills, is unfair to learners, to parents and to you as aneducator.

20

CHAPTER 7TEACHING MATHEMATICS EQUITABLY TOALL LEARNERS

It is impossible for a student to study all the work presented in the textbook.

For this reason, you should concentrate on reading the content of only someindicated chapters. Since this offers an overview of the contents of a particularchapter, it will also enable you to consult that section of the textbook againwhenever you need to.

It is said that one will find the most diverse groups of learners in the world inclassrooms in South Africa.

Read carefully through this chapter which deals with aspects of diversity.

PST201F/1 21

CHAPTER 8TECHNOLOGY AND SCHOOLMATHEMATICS

The use of technology in the mathematics classroom is encouraged. According toyour textbook, the problems of access and availability will undoubtedly fade in thenear future and useful technology will become as common in schools as chalkboardsand textbooks.

In the RSA the use of technology in the classroom is expanding. Access to theInternet, for example, is already a reality in many classrooms.

You should use the work addressed in this chapter as a source of reference.

For example: your school has facilities for learners to use computers but you feelunsure of how to set about utilising the technology to the full. Please consult thetextbook as a reference source and read what the author has to say about the use ofcomputers in the classroom.

22

SECTION 2

DEVELOPMENT OF MATHEMATICAL CONCEPTS AND PROCEDURES

CHAPTER 9DEVELOPING EARLY NUMBER CONCEPTSAND NUMBER SENSE

9.1 INTRODUCTIONRead the introduction to this section.

A teacher cannot successfully teach the concepts relating to number if he or she doesnot know and understand exactly how the different number systems are linked andif he or she does not have a sound knowledge of the properties of the numbers inthese number systems.

A teacher should have an in-depth knowledge of how learners learn the conceptsrelating to number, and should know and understand the sequence in whichlearners learn these concepts.

Most Intermediate Phase teachers have only a general idea of numbers.

For example, when they have to teach the concepts necessary to do operations withfractions, they do not understand that fractions are rational numbers and that thesystem of rational numbers is an extension of the system of whole numbers. Theyalso do not understand why the need for a system containing rational numbersforced mathematicians to extend the whole number system.

Here is a summary of the real number system. Test your knowledge of how thedifferent systems are linked and of the properties of the numbers in the differentnumber systems by answering the stated questions.

9.1.1 THE REAL NUMBER SYSTEMInterrelationship between number sets

Before discussing the interrelationship between different number systems, let us takea quick look at the evolution of our real number system.

Our real number system developed in the following stages:

. Stage 1: The very first numbers that humans ever used were natural numbers(N).

. Stage 2: The need for zero arose and zero added to the natural numbersformed Ð what we know today as the set of whole numbers (N0).

. Stage 3: In order to meet the need to subtract whole numbers at all times,negative numbers were invented. These, together with the wholenumbers, form the integers (Z).

. Stage 4: In order to meet the need to divide whole numbers by whole numbers(except 0), the numbers system had to be extended to the set of rationalnumbers (Q).

24

. Stage 5: In order to find exact results for numbers which cannot be expressed ascommon fractions, the irrational numbers (Q') were invented.

The expansion of rational numbers to include irrational numbers gave rise to the setof real numbers denoted by R. The following diagram captures this expansion:

THE REAL NUMBERS

Rational numbers Q Irrational numbers Q'

Integers Z[... -2; -1; 0; 1; 2 ...]

Whole numbers NO

[0; 1: 2: 3 ...]

Natural numbers N[1, 2, 3 ...]

From the diagram we can see that

. N is a subset of No (No includes N)

. No is a subset of Z (Z includes No)

. Z is a subset of Q (Q includes Z)

. Q and Q' are subsets of R (R consists of Q and Q')

NOTE that Q and Q' do not have any common elements.

Now answer the following questions:

. Are there so-called ``nonreal'' numbers? Answer: Yes.

. Is the answer to�������ÿ9p

a real number? Answer: No.. What kind of number is

�������ÿ9p

? Answer: non-real.. Why is

���3p

called an irrational number? Answer: It cannot be written as a fraction. Can you describe an irrational number in words? Answer: It is a non-recurring

decimal number.. Explain the difference between the following calculations and their answers:�������ÿ9p � �����

27p

; 3 � 27 and�������ÿ3p � ���������ÿ27

pAnswer: non-real, 30, non-real.

. Page through section 2 in the textbook. Note that the forming of concepts basedon number starts with natural numbers, extends to whole numbers and then torational numbers (fractions), integers and irrational numbers. Later, in a morealgebraic sense, we will meet the properties of number systems.

. Why do you think the teaching of these concepts follows this particular path?Answer: It follows the same pattern in which the real number system developed.

Let us now continue with our study of the development of number concepts andnumber sense.

PST201F/1 25

g After working through this chapter, you should

. demonstrate an understanding of how a learner develops his or her numberconcepts and number sense in the Foundation and Intermediate Phases

. be able to apply the knowledge obtained from your study of the chapter inlearning and teaching situations

. be able to apply the many interesting ideas on how to present the differentconcepts

. be able to form your own opinion on the teaching of number concepts andnumber sense after reading through the ideas and statements in the textbook

9.2 OVERVIEW. How do learners learn about numbers?. What is meant by number sense?. How can the teacher provide learners with a rich assortment of activities to help

them construct the many ideas about numbers?

All these questions are answered in this chapter for numbers up to 20.

b Activity 9.1

Number concepts and number sense

SKIMMING: Read through chapter 9 to get an idea of the different conceptsrelated to number and number sense.

Form an idea of how these concepts build on one another.

SCANNING: Start to reflect on the different sections of the chapter, in such away that you can draw a mind map which will show how theconcepts build on one another.

The author makes a few important statements on page 120 under the heading ``Bigideas''.

(1) Counting tells us how many things are in a group (set).

The basic mathematical knowledge used by all human beings consists of theskills used in counting.

Thus, teaching your learner to count various things in groups, withunderstanding, is to teach one of the basic concepts of mathematics.

(2) There is a wide variety of relationships between numbers.

Think about the following statements to see what is meant by this idea:

. Any number can be expressed in many different ways.

For example: 4 can be expressed as:

41 ; 8

2 ;�����16p

; 3 � 1

26

. Any operation with whole numbers can be done in many different ways.

For example: 2 + 11 can be calculated as follows:

. 2 + 11 = 13 (mentally)

. 2 + 11 = 2 + 10 + 1

= 13 (analyse one of the numbers)

. T 0

1 1 (group the numbers according to the place2 values of their digits)

1 3

. 2 + 11 L L L L L L L L L L L L L

= 13 (draw a picture to calculate the answer)

. 2 + 11 = 13 (use a calculator and the key sequence

2 + 1 1 =

Display 13.)

A problem-based approach to teaching does not suggest a prescribed solution to aproblem or method for doing a calculation. The learner is encouraged to use anymathematically correct method or solution. It is thus essential that teachersthemselves should be aware of the variety of different calculation methods and waysof solving problems.

bActivity 9.2

Development of number concepts

READ-STUDY:

. A suggested framework of the development of number concepts and numbersense is given below.

. Study-read the chapter to see whether you agree with the suggested frameworkand complete your own mind map with the assistance of the framework.

. Include activities which you can later give to learners to help them understandthe different concepts. This is suggested because the textbook contains a richselection of activities meant for use in your classroom. These activities have beenselected with great care.

Suggested framework

This framework illustrates how concepts relating to number and number sense areformed.

PST201F/1 27

Spatial relationships:

. What is meant by these?

. Examples of learners' activities

One and two more, one and two less relationships:

. What is meant by these?

. Examples of activities to use in a classroom

!

!

Starts withmeaningfulcounting

Concepts relatingto more, less andsame

Proceduralknowledge ofnumbers is nowobtained

""

Step 1:

The learner can now count meaningful, known concepts (such as more, less andsame), and can read and write the symbols representing numbers.

Step 2:

In this step learners create more relationships between numbers to develop what isknown as number sense.

A learner should develop the following types of number relationships:

. Spatial relationships. Learners should recognise the ``how many'' of a group ofobjects by recognising a common pattern without actually counting the objects.

Do you know the number represented by the following dots?

. ..

. .

. One and two more, one and two less. A learner should know that 7 is 2 more than5 but 2 less than 9.

. Anchors or ``benchmarks'' of 5 and 10. These are interesting concepts which canbe misunderstood by the teacher. Study the use of the 10-frame on pages 128±129.

. Part-part-whole relationship. This is the most important number relationship thata learner should develop.

Framework for different types of relationships

28

Anchoring numbers to 5 and 10:

. What is meant by these?

. The use of the 10-frame

Part-part-whole relationships:

. What is meant by these?

. Activities and material to use in teaching these relationships

!

. The concept of 10 and more:

What is said about it?

. Double and near-double relationships:

What are they and why are they important?

What is meant by number sense and the real world?

Activities discussed

Extensions to early mental mathematics

. What is meant by these?

. How is a little base-10 frame used to extend a learner's concepts to early mentalmathematics?

Step 3:

In this step the number concepts of the learner are extended to numbers larger than10.

Step 4:

Number sense and the real world

Relationships of numbers to real-world quantities and measures, and the use ofnumbers in simple estimations, can help learners to develop intuitive ideas about thenumbers that are most desired.

Step 5:

PST201F/1 29

CHAPTER 10DEVELOPING MEANINGS FOR THEOPERATIONS

gAfter working through this chapter, you should be able to

. explain the term ``operation sense'' and to show that you know how to teach therelevant concepts regarding operations to learners

10.1 OVERVIEWThis chapter is about learners' development of ``operation sense''.

For the successful mastery of the four basic operations, more than just anunderstanding of the operations is necessary.

bActivity 10.1

Developing meanings for the operations

. Read the section, ``Big ideas'' on page 135.

. Were you confronted with these ideas and concepts before you were expected todo the basic operations with whole numbers? Explain.

bActivity 10.2

SKIMMING: Read through the chapter quickly to form a rough idea of thecontent.

Note the following:

. the two tools used to construct understanding of operations

. word problems on addition and subtraction, with the different kinds ofrelationships involved

. the interesting models used for addition and subtraction

. the extension of the same ideas to multiplication and division

30

10.2 INTRODUCTION TO THE TEACHING OFMEANINGS FOR THE OPERATIONS

Do not assume that your young learners cannot add, subtract, multiply and dividebecause this is simply not true. The teacher is not expected to teach these concepts asif the learners have not constructed any concepts relating to these operations.Instead, the teacher is expected to build on what the learner already knows and tolink the concepts to word problems or word stories. Put the words ``problems'' and``stories'' into contexts that relate to the learners' own lives.

Note that the author thinks about models, word problems and symbolic equations asthree separate languages and that he encourages the use of equations at an earlystage.

10.3 TEACHING ADDITION AND SUBTRACTIONIn this section of the chapter, the author gives you insight into the teaching andlearning of addition and subtraction.

He also answers the important question: ``How should I use addition andsubtraction word problems in the classroom?'' (Van de Walle 2007:146)

Try the following approach to studying this section in the chapter.

Do not read what is said in the textbook. Make a mind map to illustrate how youwill teach addition and subtraction to young learners. This will give you an idea ofthe structures for teaching addition and subtraction in your own mind

Now, go to the relevant section in the chapter and do the next activity.

bActivity 10.3

Teaching addition and subtraction

. Study-read the section and make a mind map to assist you.

Compare this mind map to the one that you based on your own ideas.

10.4 REFLECTION ON ADDITION AND SUBTRACTIONA study of the work done in this section of the chapter will indicate that Van deWalle made an in-depth study of various word problems and their uses.

You may feel a little confused when you reflect on the different problem typesdiscussed in this section, but on pages 148 and 149 the important goal of the teachingof the concepts relating to addition and subtraction is stated and linked to classroomsituations. The main ideas are:

. The important point when teaching the main operations is not to obtain thecorrect answers, but to analyse relationships and to discuss approaches.

. Let your learners have access to a variety of materials which they can use to helpthem solve word problems.

. Your learners should be afforded the opportunity to present a problem in words

PST201F/1 31

and in written form. Give the learners the opportunity to discuss the differentideas and the solutions to the problems in pairs or in small groups. Request awritten response or explanation of the solution to the problem from each learner.If possible, let the learner include an equation that ``goes with the solution of theproblem''.

10.5 TEACHING MULTIPLICATION AND DIVISIONStudy-read the section on problem structures for multiplication and division as wellas the section on the teaching of multiplication and division.

You should appreciate this in-depth study of the teaching and learning ofmultiplication and division done by the author.

Your task is to study-read these sections in such a manner that you will know andunderstand them sufficiently to refer back to the textbook whenever you need toteach multiplication and division to young learners for the first time.

32

CHAPTER 11HELPING CHILDREN MASTER THE BASICFACTS

gAfter working through this chapter, you should be able to

. explain what is meant by the basic facts for the main operations

. demonstrate understanding of approaches to fact mastery

. discuss and apply strategies for addition, subtraction, multiplication anddivision facts

11.1 OVERVIEWYou have studied how learners develop number concepts and number sense, howthese concepts are related to the main operations and how the main operations arelearnt as an extension of number sense and number concepts.

A basic fact for addition and multiplication refers to combinations where both thefactors are less than ten. Subtraction and division facts correspond to addition andmultiplication facts, for example 15 ± 8 + 7 and 8 + 7 = 15.

After learners have mastered the main operations, they should construct efficientmental tools to help them master basic facts.

This chapter is about the mastering of basic facts and how they are learned.

bActivity 11.1

Orientation

Rethink the following statements and ideas before studying this chapter:

. Many learners in schools in South Africa still use counting to do calculations.This implies that they are still at the ``count all'' or ``count on'' stage of numberconcepts and lack the basic tools necessary for successful quick responses tocalculations. For example, to do 8 + 7, the learner counts the 8 ones and thencounts seven ones to obtain 15 as an answer.

. If efficient strategies for doing a certain calculation are not in place, practice anddrilling in the learning process amount to a waste of time and a frustration tothe learner.

. Note that the use of materials is to develop strategies Ð they are not answer-giving devices.

. The idea that drilling is not allowed in today's classroom is simply not true.

PST201F/1 33

11.2 HELPING LEARNERS MASTER THE BASIC FACTSMany teachers know that for successful teaching today, one needs a mind shift away

from what was done in the past.

The work in this chapter will clearly show what this statement means.

b Activity 11.2

Helping learners master the basic facts

SKIMMING: Read through the chapter.

Ask the following questions while you read:

. What is a basic fact?

. How do I teach these facts to learners? What approach must I use?

. What are the strategies that learners invent to master these facts?

. What learning and teaching materials are involved?

. What is effective drilling?

. How can I remediate basic facts in the higher grades?

11.3 APPROACHES TO FACT MASTERYIn this section (pp 165±168) you will become aware of the different approaches to

teaching mathematical facts, concepts and procedures because what is said in this

section does not relate only to basic facts teaching and learning.

Read this section carefully and compare your own ideas with those of the textbook.

11.4 STRATEGIES FOR ADDITION, SUBSTRUCTION,MULTIPLICATION AND DIVISION FACTS

Although this course is based mainly on the teaching and learning of mathematics in

the Intermediate Phase, the mathematics addressed in this section forms the basis of

mathematics for the Intermediate Phase.

Read through these sections. Note the following:

. The main aim of the teacher must be to assist the learner to make sense of basic

facts and the related concepts.

. Many interesting learning and teaching materials are used.

. Drilling of concepts and facts takes place only after conceptual knowledge is

formed.

. The author has interesting ideas about time tests.

34

11.5 FACT REMEDIATION WITH LEARNERS IN THEHIGHER GRADES

This is the section of the chapter which is important to you as an Intermediate Phaseteacher.

bActivity 11.3

Do the following:

. Think what you yourself will do when you encounter learners in the fifth andsixth grades, who have not mastered their basic facts.

. Summarise what the textbook says about the teaching and learning of theselearners.

PST201F/1 35

CHAPTER 12WHOLE-NUMBER AND PLACE-VALUEDEVELOPMENT

gAfter working through this chapter, you should be able to

. explain the number concepts and ideas in the early development of a learnertrying to master the place and number value of a digit in a number

. apply the use of models and number expansion cards to teach the concepts ofplace and number value of a digit in a number

. discuss a range of activities in which a learner can be engaged to learn the aboveconcepts

. explain how real-life situations can be used to teach concepts relating to placeand number values

. teach the concepts relating to numbers larger than 1 000

. assess the concepts learnt by learners, based on the place value and numbervalue of digits in numbers

12.1 OVERVIEWWithout a firm understanding of the place value and number value of a digit in anumber, the learner cannot master the basic calculations Teachers in South Africaalso have to teach concepts of the number value of a digit in a number and not onlyconcepts based on the place value of the digit. These are defined in the workprogrammes in the National Curriculum Statement for the learning areaMathematics, namely that the number value of the digit should form the basis of theteaching of the concepts relating to larger numbers.

12.2 BASIC IDEAS OF PLACE AND NUMBER VALUESWhat is the difference between the two concepts ``place value of a digit'' and`'number value of a digit''? In the number 2 714, the place value of the digit 7 isdefined as 7 hundreds determined by the place of the digit in the number. Thenumber value of the digit 7 is defined as 700.

The textbook contains a large variety of activities involving the place value of digits.

36

b Activity 12.1

The following activity explains the use of number expansion cards. These cards areused to analyse and compose a number. When a number is analysed, the numbervalue of each of its digits can clearly be understood.

(Please note: Number expansion cards are named differently in the literature.)

Description of an activity to teach the number values of the digits in the number2 714:

. Let the learner select the following cards from his or her set of numberexpansion cards:

2 0 0 0

7 0 0

1 0

4

. Let the learner overlap the cards to compose the number as shown:

2 7 1 4

. Let the learner write down:

2 000 + 700 + 10 + 4 = 2 714 (composition of the number)

. Let the learner separate the cards again and then let him or her write down theequation:

2 714 = 2 000 + 700 + 10 + 4 (to analyse the number)

. Then, discuss the number and place values of each of the digits in the givennumber.

b Activity 12.2

Number expansion cards

STUDY-READ

. Familiarise yourself with number expansion cards by solving the followingproblem.

. Use the following cards and then write down all the possible numbers that youcan find by overlapping the cards.

5 0 0 0 2 0 0 3 0 5

PST201F/1 37

12.3 MORE ON BASIC IDEAS AND THE USE OFMODELS

In this section the textbook gives us an idea of the thought processes of the younglearner before place value or number value concepts are in place, how he or shelearns the relevant concepts and how the concepts are linked.

b Activity 12.3

SKIMMING: Page through the chapter and do the following:

. Note the discussion on the early development of place value ideas.

. Note the relationship between groupings of 10 and other related concepts.

. Study figure 12.1 on page 189. (You can add the concept of analyses of thenumber given by the equation 53 = 50 + 3 to the figure.)

. Note the variety of models that can be used to teach the concepts of place value.

. Note the discussion of the range of activities that can be used in the classroom.

. Note the link with real-life situations.

In the next activity your understanding of the early development of the place valueand the number value of a digit will be tested.

b Activity 12.4

Early place value concepts

SCANNING: Reflect on the section on the early development of the place valueand the number value of digits on page 188.

Study the description of the following activity and then answer the questions basedon the activity.

A teacher filled a bottle with +100 beans. He instructed each of the learners in agrade 2 class to guess the number of beans in the bottle. He agreed to count thebeans with the help of the whole class. He instructed them to individually plan acounting strategy after guessing the number of beans in the bottle.

Answer the following questions based on the activity:

. How will a learner without any knowledge of the concept of place value countthe number of beans?

. Can you describe one other activity in which this learner can be engaged thatwill also indicate that his or her concept of place value is not yet in place?

. Can you name and describe three different ways in which learners can count thenumber of beans?

. Can you link the grouping of the beans in groups of 10 to numerals, the way inwhich the digits can be placed in indicated places, saying the number in wordsand eventually writing the number in standard form? Explain.

. How can you link the grouping of the beans in groups of 10 to the composition ofthe number of beans with the help of number expansion cards?

38

bActivity 12.5

Base 10 models

STUDY-READ the section of the chapter on the use of models to teach place valueconcepts. Summarise the section.

Note the following:

The textbook refers to counting in ones when a learner needs to count a largernumber of objects such as 53.

In our country we refer to ``counting all'' when a learner still counts the objects oneby one Ð we say the learner is in the ``count all'' stage.

When a learner counts a number of objects indicating, say, 20 objects and thencounts on from 20 until all the objects are counted, we say that the learner is in the``count on'' stage.

Most grade 1 learners are in the ``count all'' stage, whereas grade 2 learners mustmove on to the ``count on'' stage and eventually to the concepts related to numberand place values of digits in numbers.

As teacher in the Intermediate Phase, you will encounter many learners (even ingrades 6 and 7) who are still in the ``count all'' stage. This chapter gives you all theskills and knowledge needed to remediate these learners' work.

12.4 NUMBERS BEYOND 1 000The following sections in this chapter are of great importance to Intermediate Phaseeducation because they involve the teaching of concepts where larger numbers areinvolved.

bActivity 12.5

Large numbers beyond 1 000

STUDY-READ the section on pages 211 to 213.

. Make sure that you understand how to extend the different concepts.

. Reflect on the use of number expansion cards to analyse and compose numbersbeyond 1 000 to form the concepts discussed in this section.

PST201F/1 39

CHAPTER 13STRATEGIES FOR WHOLE NUMBERCOMPUTATION

gAfter working through this chapter, you should be able to

. explain words such as ``direct modelling'', ``algorithm'', and ``inventedstrategies''

. indicate what invented strategies are and what their benefits are

. explain how to develop learner-invented strategies

. teach traditional algorithms with understanding

13.1 OVERVIEWAs an Intermediate Phase teacher, one of your tasks will be to teach the four basicoperations (+, ±, 7, 6) with larger numbers. This will eventually end in teaching theprocedural knowledge needed to use the algorithms (recipes to multiply, divide, addand subtract).

After studying the previous chapters in the textbook you should now be convincedthat you cannot teach these operations by introducing the traditional algorithms toyour learners as the only methods of calculation. How then should all these conceptsand skills be taught? This is what this chapter is about.

bActivity 13.1

Do the following:

SKIMMING: Read through the whole chapter to form an idea of what is saidabout whole-number computation.

. Make sure that you understand how learners think about numbers and theoperations.

. Convince yourself that you cannot teach the traditional algorithms withoutletting your learners first invent their own strategies for doing calculations.

40

bActivity 13.2

Go back to the section on invented strategies again.

Now make a summary of what is said.

. Write down what an invented strategy means.

. Write down the benefits of invented strategies.

. Write down how you can assist learners to develop these strategies.

13.2 TRADITIONAL ALGORITHMSIn this section of the chapter the textbook gives guidance on how to teach theoperations up to the use of the traditional algorithms in such a manner as to enablethe learner to gain a meaningful understanding of the steps in each algorithm. Notethe explanations of the use of the algorithms for multiplication and division on pages235 (figure 13.22) and page 240 (figure 13.26). I am sure that you will find theseexplanations interesting.

bActivity 13.3

STUDY-READ this section of the chapter, then use a model, an invented strategy,an algorithm and an explanation of the algorithm to do each of the followingcalculations:

. 53 + 27

. 53 ± 27

. 53 6 15

. 53 7 15

Study the given statement (Van de Walle 2007):

What is the educational cost of teaching learners to master pencil-and-papercomputational algorithms? Here cost means time and effort required over theentire elementary grade span, in comparison with all other topics that are orcould be taught. How many algorithm skills or how much knowledge do youthink is really essential in an age in which calculators are readily available.

For a moment think back to how important these algorithms were when youyourself went to school. Can you still remember how these algorithms were taught?

After studying this chapter in the textbook, you are probably aware that the teachingof algorithms should only be done after number concept, the meaning of operations,mastery of basic facts, mental calculations and estimation have been done asprerequisites.

If you agree that the emphasis on these algorithms should diminish, how thenshould these algorithms be taught in the modern classroom? You should be able toanswer this question after studying this chapter.

PST201F/1 41

CHAPTER 14COMPUTATIONAL ESTIMATION WITHWHOLE NUMBERS

gAfter working through this chapter, you should be able to

. discuss the concept of estimation

. describe different ways of doing computational estimation

. discuss different estimation strategies

14.1 OVERVIEWThe number sense development of a learner (or any other person) is largelyinfluenced by his or her mental computation and estimation skills.

Any numerical calculation or measurement activity involves mental computationand estimation.

Study the description of the following activity and ask yourself whether you agreewith the statements made.

If you have to find the distance from your chair to the door of the room, reflect onwhat happens inside your own mind.

The following usually happens:

. You ``see'' the problem. You make a mental picture of the problem.

. You ``measure'' the distance with your eyes by looking at the distance.

. Next you decide on the unit of measurement that you should use to do themeasurement.

. You then use a tool (a measuring tape) to find the exact distance.

. When the exact distance has been measured, you verify the distance by comparingthe actual distance with the estimated distance.

Now ask yourself:

. Why do I first have to ``measure'' the distance with my eyes?

. Why do I compare the actual distance with the estimated distance?

The above discussion should lead you to the following conclusion:

You unconsciously estimated the distance which enabled you to select the correctunit of measurement and to verify the distance measured with the measuring tape.

How do we teach these estimation skills to our learners? The answer is in thischapter.

42

bActivity 14.1

Study the different types of estimation and the description of what estimation is onpage 246.

Write these down in your workbook.

14.2 UNDERSTANDING COMPUTATIONALESTIMATION

Think back to the way estimation was taught and learnt when you went to school.

Round the numbers to the nearest ten, hundred or thousand and then do thecalculation. The result will be the estimated answer.

If your estimation was not done exactly in this manner it was considered to bewrong.

bActivity 14.2

Carefully read through the chapter. Note the modern ideas presented to you by theauthor. You must study the content of this chapter in such a manner that you canrefer back to it whenever you need to teach learners computational estimation.

bActivity 14.3

Big ideas

Read the big ideas (important statements) on page 228. Note that the words``carries'' and ``borrows'' are used and that the statements/ideas are made with theplace values of numbers in mind. Read the big ideas/important statements givenbelow, in which the number values of the digits in the numbers are emphasised.

. Each of the traditional algorithms is simply a clever way of recording acalculation for digits with the same number/place value by regrouping groups oftens, hundreds and so on.

. Ten ones are regrouped to form one ten and ten tens are regrouped to form onehundred and so on.

The other way round, one ten could be regrouped to form one ten and onehundred could be regrouped to form ten tens or even a hundred ones.

The use of the words ``carry'' and ``borrow'' is no longer emphasised in SouthAfrican classrooms.

. Do the activities in this chapter to develop your own estimation skills. Are youfamiliar with these or do you find the activities difficult?

PST201F/1 43

CHAPTER 15ALGEBRAIC THINKING:GENERALISATIONS, PATTERNS, ANDFUNCTIONS

gAfter working through this chapter, you should be able to

. show that you understand what challenges concerning number lie ahead forlearners in the Intermediate and Senior Phases

. indicate why the statements are made about your own knowledge of the differentnumber systems at the beginning of this section

. upgrade your own knowledge about number where necessary

. give an account of the following concepts:

Ð functionÐ variableÐ graphic representations of functionsÐ equations that define functionsÐ solving equations

. upgrade your own knowledge of functions and relevant concepts

15.1 OVERVIEWYou are a teacher in the Intermediate Phase in the learning area Mathematics. Yourlearners have mastered the outcomes defined for number learning and haveacquired the skills and values necessary for number. How can you, as a teacher, helpto prepare your learners for the challenges that lie ahead? This issue is addressed inthis chapter.

You will again have to use this chapter as reference material. Whenever you need toteach concepts and skills related to algebra and functions to your learners, you areadvised to refer to the content of this chapter.

However you will have to do the following activities to give you a clear idea of thecontent of the chapter.

44

" "

bActivity 15.1

(1) Read, on page 260, what Kaput's description of algebraic reasoning is.(2) Read about variables and the solution of equations and inequalities on pp 262±

266.(3) Study the section on repeated patterns (p 268) and how to ``read'' these

patterns.(4) Study the pictures presenting growing patterns (p 271). Note the link from the

given pattern to the table and the general formula shown on page 273, figure15.11.

(5) Make sure that you understand that a function is a rule that uniquelyassociates elements of one set with elements of another set. Study the functionmachine below that illustrates the meaning of a function.

Rule

Input Output

x y = 2x + 1 y

(5)Use the machine to complete the given table.

Inputx 0 1

Output

y 1 3

(6) Write down the five representation forms of functions.(7) Read the section on the connection of the different representations (p 280).

PST201F/1 45

CHAPTER 16DEVELOPING FRACTION CONCEPTS

gAfter working through this chapter, you should be able to

. use three categories of fraction models to represent different concepts of fractions

. explain how a learner's number sense of fractions can be developed as aprerequisite for computation with fractions

16.1 OVERVIEWThe development of fraction concepts and the teaching of computations withfractions is a challenge for any Intermediate Phase teacher.

Note that the teacher himself or herself should know that a fraction is defined as arational number.

Learners are expected to understand that a fraction is an expression of a relationshipbetween a part and a whole.

The way to teach the relationship and to relate it meaningfully to symbols is thetopic of this chapter.

The textbook advises us to use three different categories of fraction models topresent different concepts.

You are advised to draw, make and handle these models yourself, which willenhance your own understanding of how to use them.

In this section on fractions (chapters 16, 17 and 18) we find the real meaning of theidea of teaching developmentally, namely:

. spending enough time on concept development (see chapter 16)

. approaching the new concept in more than one way (especially when it results inan algorithm)

. taking into account the way learners learn concepts

bActivity 16.1

SKIMMING: Quickly read through the chapter.

Note the following:

. the introduction of three different fraction models

. the use of these models to develop the concept of fractional parts

. the way the concepts of fractions are linked to symbols

. the introduction to a wide range of exercises to teach the various concepts

. what is meant by fraction number sense and how it is taught

. teaching the concepts of equivalent fractions

46

16.2 MODELS FOR FRACTIONSThis is the most important section of the chapter. You should understand the use ofthe three different models before moving on to the next section in the chapter.

bActivity 16.2

Models for fractions

STUDY-READ and reflect on the description of the models (pp 295±297).

Do the following practically:

. Fold a piece of paper to represent 1/6 (one sixth). What kind of model is this?

. Read the instructions on page 296 and make yourself a strip model as described.

. Use the model to indicate the fractional part which is one-sixth of the dark greenstrip.

. Read the instructions on page 297. Make yourself 20 counters that are colouredin two colours on opposite sides. Use the counters to represent one-sixth of a set.

. Draw a picture of a set of objects. Shade one-sixth of the objects in the set.

You have now used three different models to present fractional parts.

Do not continue with the work in this chapter if you do not fully understand the useof these models.

16.3 FROM FRACTIONAL PARTS TO FRACTIONSYMBOLS

bActivity 16.3

Fractional parts

STUDY-READ the section on how the concept of fractional parts should be taught(pp 297±303).

Write down the main ideas in a summary.

16.4 FRACTION NUMBER SENSEMany adults and most learners in the RSA will have difficulty with the activitiessuggested in this section. This is an indication that their number sense of fractions isnot well developed. One reason for this is that these concepts are not taught in ourschools.

PST201F/1 47

bActivity 16.4

Number sense and fractions

STUDY-READ this section on pages 303 to 308.

. Do the given activities yourself.

. Reflect on the comments made.

. Study the conceptual thought patterns for the comparison given on pages 304 to305.

. Write a summary, using your own words, to understand the conceptual thoughtpatterns better.

16.5 EQUIVALENT FRACTION CONCEPTSThe concepts of equivalent and similar (like) fractions are essential for successfulcomputation with fractions. You will encounter the teaching of similar fractions (likefractions) in the next chapter.

In this section of the chapter the teaching of the concepts of equivalent fractions isdiscussed. Make sure that you understand the difference between a learner'sconceptual knowledge of the concepts and his or her procedural knowledge.Remember that procedural knowledge is usually based on the use of a prescribedalgorithm.

bActivity 16.5

Equivalent fraction concepts

STUDY-READ this section on pages 308 to 313.

. Do all the given activities yourself.

. Note that the last method (p 312) to form fractions equivalent to a given fractionis the way in which an adult who has mastered all the concepts of fractions infull should form these fractions.

bActivity 16.6

When you teach Intermediate Phase learners, you will have to teach fractionalsense and computations with fractions.

After completing this chapter, start a summary on how to teach this to learners.

Write down the steps that you plan to follow from the basic concepts to the end ofcomputation with decimals.

Finish your summary after completing your study of chapter 18.

48

CHAPTER 17COMPUTATION WITH FRACTIONS

gAfter working through this chapter, you should be able to

. teach the concepts necessary for applying the well-known algorithms foraddition, subtraction, multiplication and division of fractions

. explain your understanding of the dangers of teaching rules (algorithms)without real understanding

. demonstrate (represent) all four of the operations with fractions by using thethree different kinds of models introduced in chapter 16

17.1 OVERVIEWMost adults and learners in our country cannot successfully add, subtract, multiplyand divide fractions. They lack the basic knowledge required for thesecomputations.

Test yourself. Can you fully explain all the concepts involved in the well-knownalgorithm:

If you need to divide one common fraction by another common fraction, changethe division sign to a multiplication sign. Find the reciprocal of the secondfraction and then multiply the two fractions.

If you could explain all the concepts involved in this calculation, you are one of theexceptions to the rule.

The lack of the necessary skills and knowledge to do computations with fractions isa direct result of the way in which these concepts and skills were taught in ourschools.

This chapter warns us about the dangers of teaching rules and algorithms whileteaching computations with fractions when learners do not really understand them.The author refers to rules which learners do not really understand.

This chapter gives us an alternative to the teaching of rules and algorithms withoutreal insight Ð on the part of both teacher and learners.

bActivity 17.1

Overview

. Read through the chapter. Note the emphasis on understanding and numbersense development before the introduction to the algorithms.

. Also note the detailed discussions that preceded the introduction of the actualalgorithm.

PST201F/1 49

bActivity 17.2

Big ideas

. Read the ``big ideas'' on page 316.

. Rethink how you yourself learnt the computation of fractions at school. Was anyattention paid to the ideas mentioned?

. Note especially that the estimation of computational answers was never done inschools in our days.

17.2 NUMBER SENSE AND FRACTION ALGORITHMSI wish to emphasise what the textbook states: focusing attention on fraction rulesand finding answers has two significant dangers. Firstly, none of the rules helplearners to do any thinking about the meanings of the operations or why they work.Secondly, learners who practise these rules may very well be doing rote symbolpushing in the purest sense.

bActivity 17.3

Number sense and fraction algorithms

STUDY-READ this section on pages 316 to 317.

Think about what the textbook says about the dangerous rush to rules. Do youagree with the statements made?

17.3 THE MAIN OPERATIONS AND FRACTIONS

bActivity 17.4

First think about how you yourself will teach the operations with fractions. Wherewill you start? What will you teach? What do you think is important to teach?

Now study a problem-based, number sense approach to the teaching of theoperations on page 317.

Some of the explanations of the use of the three different models used in this chapterare difficult to understand.

You are advised to do the activities practically with the use of your strip model andthe counters made in chapter 16. Use paper rectangles or circles to help you practise.

Enjoy the work done in this chapter to introduce the well-known algorithms Ð theideas are new to almost every reader of the chapter.

50

bActivity 17.5

Computation with fractions

STUDY-READ this section and do the different activities and examples yourself.

As far as possible, use the models you made during your study of chapter 16 andthe paper (or cardboard) circles and rectangles as discussed in the activities.

bActivity 17.6

Discussion and exploration

. If you have studied the statements made, the ideas and examples given, and havedone the various activities in this chapter, you will be equipped to discuss thequestions in the discussion and exploration section on page 332.

. Discuss and give your own opinion on the questions asked and statements madein questions 1 to 3.

PST201F/1 51

CHAPTER 18DECIMAL AND PERCENT CONCEPTS, ANDDECIMAL COMPUTATION

gAfter working through this chapter, you should be able to

. teach the concepts of decimal numbers and percent as an extension of theconcepts learned about fractions

. explain the use of different models to teach these concepts

. recognise and apply the importance of estimation when computations withdecimal numbers are introduced

18.1 OVERVIEWThe approach to the teaching of the concepts based on decimal numbers andpercentage is based on the assumption that what has been learnt about fractions isclosely related to the concepts based on decimal numbers. It is also assumed that theconcepts relating to decimal numbers and fractions can be extended to what learnersshould know about percentage and the relevant skills necessary to use percentcorrectly as a tool to better understand the relationships between numbers.

The author warns that a teacher should be aware that concepts based on decimalnumbers and percentages can be taught through the use of algorithms but thisserves no purpose and does not enhance any real understanding.

Please note: The work in this chapter is done mainly in the Intermediate Phase.

bActivity 18.1

Overview

. Read through the chapter quickly.

. While you read through the chapter, think about how the concepts are related.

18.2 CONNECTING TWO DIFFERENTREPRESENTATIONAL SYSTEMS

In order to have a clear idea of all the work done in this chapter, you (as a teacher)should have the following knowledge about rational numbers (fractions):

. Every fraction can be written as a decimal number.

52

Connecting fractions to decimal numbers

Base 10 fractions

. ...............

. ......................... etc

Extending the place value system

. ........................................

. ........................................

. ........................................etc

""

Familiar fractionsconnected todecimal numbers:How?

Approximationwith nicefractions: How?

Ordering decimalnumbers: How?

Other fraction-decimal equivalents:How?

Making the fraction-decimal connectionHow is this done? Give examples of activities.

Developing decimal Number senseWhat is meant by this?

"" How?

. Not all decimal numbers can be written as fractions Ð this gave rise to irrationalnumbers.

. Some decimal numbers are called recurring decimals, such as 0,333 ... or 0,3.

. Recurring decimal numbers can also be written as fractions and are thus alsorational numbers, for example 1/3 = 0,3Ç

. Decimal numbers, fractions and percentages are different ways of writing thesame rational number. For example: 3/4 is the same as 0,75, which is the same as75%.

. It is thus clear that the system of rational numbers enables one to write suchnumbers in more than one form.

bActivity 18.2

Connecting fractions and decimal numbers

SCANNING: Make a framework of a diagram similar to the one given below.

PST201F/1 53

The introduction of percent

As a third operator systemSummarise how.

Through realistic percentproblemsSummarise how.

"

""

How?

Estimation

STUDY-READ the section on linking the two different representational systemsand on the development of decimal number sense on pages 333 to 243.

Use your knowledge to complete the diagram.

Note the following:

. The use of the hundredths disk, the 10 6 10 square model for base 10 fractions anda decimal point face is important. (You are advised to use the material included inthe textbook to make these models and to use them when doing the activities.)

18.3 INTRODUCING PERCENTIn order to understand the work done in this section of the chapter you, the reader,should note the following:

. The concepts and skills relating to percentage are extensions of the concepts andskills learned when studying fractions and decimal numbers.

. Learners cannot progress and give real meaning to percentage if they do not havea sound grasp of fraction-decimal relationships.

. The idea of equivalent fractions is fundamental to the extension to decimalfractions and percentages.

bActivity 18.3

How to introduce percentage

STUDY-READ this section (pp 343±346).

. Write down the meaning of the term ``percent''.

. Do the given activities yourself.

. Note the use of the three types of models with fractions that are extended to theteaching of percent.

. Summarise the content of this section with the help of the following diagram:

54

18.4 COMPUTATION WITH DECIMALSRead the following far-reaching statement on computation with decimal numbers.``It is no longer necessary or useful to devote large portions of instructional time toperforming routine computations with decimal numbers by hand. Othermathematical experiences for the Intermediate Phase learner deserve emphasis.''

Think back to what happened when decimal numbers were taught to you at school.At that time, none of your teachers would have agreed with the above statement.

bActivity 18.4

Computation with decimal numbers

STUDY-READ this section (pp 346±349).

Summarise the main ideas and write down your own opinion on the work done inthis section.

18.5 THE ROLE OF ESTIMATIONIt is important for you to notice the importance of estimation of computationalanswers in all these sections. It is said that good estimation skills are necessary forgood number sense and good computation with numbers.

bActivity 18.5

Approximation with `'nice'' fractions: study-read this section on page 340.

Teach your learners these skills, because the ideas about nice numbers are new tomost learners and teachers.

PST201F/1 55

CHAPTER 19PROPORTIONAL REASONING

gAfter working through this chapter, you should be able to

. show your understanding of the concepts of ratio, rate, proportion and solving aproportion

. teach proportional reasoning to Intermediate Phase learners

. help learners solve proportions

19.1 OVERVIEWProportion, ratio and rate are taught in the Intermediate Phase. These topics areregarded as more advanced sections in the Intermediate Phase learning ofmathematics.

The words ``way of thinking'' imply not only that the learner should obtain andacquire knowledge, but also that the knowledge and skills should be such that theycan influence his or her thinking processes. This statement places a greatresponsibility on a teacher, and it is clear that the reasoning required for ratio andproportion cannot be acquired in three weeks of teaching these concepts. Thecontent of this chapter guides you, the teacher, as you teach concepts and skillsrelated to proportion and ratio.

I would like to advise you to teach the work in this chapter over time.

Start with informal activities to develop proportional reasoning. Give your learnersthese problems to do on a Friday or, for example, the last week of the term. Theproblems might encourage them to think and will hopefully assist your teaching.

bActivity 19.1

Overview

. Scan the chapter. Note that the first section deals with the different concepts andhow learners reason about proportion.

. The next section includes activities to develop proportional reasoning and waysof teaching learners to solve proportions.

. Finally, percent and proportion are linked.

56

b Activity 19.2

Proportion, ratio and rates

STUDY-READ the section on proportional reasoning.

. Write down the meaning of each of the following concepts:

Ð proportion

Ð ratio

Ð solving a proportion

Ð rate

. Link your knowledge to the statements made under ``Big Ideas'' (Van de

Walle 2007:353)

b Activity 19.3

Proportional reasoning (p 355)

STUDY-READ

. Study the ideas that are outlined Ð they will give you some direction on how to

help your learners develop proportional thought processes.

. Then study the rest of the section based on proportional reasoning and the

guidance given on how to teach these ideas.

19.2 INFORMAL ACTIVITIES TO DEVELOPPROPORTIONAL REASONING

The following three categories of activities are discussed in this section:

. selection of equivalent ratios

. scaling or table activities

. construction and measurement activities

The activities given in the textbook should be studied carefully.

b Activity 19.4

Activities to develop proportional reasoning

STUDY-READ

. Study the activities selected under each of the given categories.

. Select one example of each kind and make sure that you understand why this

activity was grouped into this category.

PST201F/1 57

19.3 SOLVING PROPORTIONSHere is an example of a proportion containing an unknown value and a method tosolve the proportion in order to find the value of the unknown.

The price of a box of two dozen sweets is R4,80. John wants to buy five sweets. Whatwill he have to pay? The ratio of sweets to money is given as 24 sweets to R4,80.

The ratio of the sweets bought by John to the price of the sweets is the same as thefirst ratio: 5 sweets to Rx.

If 24 sweets cost 480c

1 sweet will cost 20c

and 5 sweets bought by John will cost 20c 6 5 = R1,00.

Finding the unknown value in the proportion is called solving the proportion.

bActivity 19.5

Solving proportions

STUDY-READ the section based on the solution of proportions (p 366).

. Different methods are suggested. Select the method that you prefer.

. I prefer not to introduce cross-multiplication to learners in the IntermediatePhase because the algorithm can lead to mistakes made in later years whenequations in algebra should be solved.

. I prefer to go to a unit rate and use this knowledge to solve the proportion. But,as mentioned before, you are advised to select the method you prefer in the sameway as your learners should be allowed to select theirs.

19.4 PERCENT PROBLEMS AS PROPORTIONSThe concept of percent is linked to the concept of proportion. The integration ofrelated ideas is an excellent example of the network of ideas that increasesunderstanding.

bActivity 19.6

Percent and proportions

STUDY-READ this section on page 370.

Note the way in which a line segment is used to enhance and simplify the actualcalculations to solve the proportions. Also note the role of equivalent fractions.

58

CHAPTER 20DEVELOPING MEASUREMENT CONCEPTS

gAfter working through this chapter, you should be able to

. help the young learner develop measurement concepts and skills

. discuss the teaching of the concepts and skills relating to length, from thereception year to a stage where the learners can use a ruler with confidence

. use the knowledge gained in the first part of the chapter to teach the skills andknowledge needed for measurement of

Ð areaÐ volume and capacityÐ weight and massÐ angles

. show that you appreciate the amount of time spent on concept development inthis topic

. know when and how to introduce formal formulae to your learners

. be able to help learners with time and clock reading

20.1 OVERVIEWIn the RSA, many adults and school learners lack the basic knowledge and skillsnecessary for successful measurement.

The reasons for this is the following:

. They have never learnt to estimate quantities.

. They lack the basic knowledge of standard units of measurement and their use.For example, very few adults or learners can explain the mass of 1 gram.

. They do not link the correct unit of measurement to the ``quantity'' needed to bemeasured. For example, they do not link square centimetre to the measurement ofarea. When asked what a unit for measurement of area is, many learners willanswer that it is a centimetre.

Fortunately, the teaching of the concepts and skills relating to measurement involvesreal-life situations which can make the teaching and learning of these great fun.

bActivity 20.1

Overview

SKIMMING: Read through the first part of the chapter on pages 375 to 378quickly.

This section forms the basis of the rest of the work in the chapter.

PST201F/1 59

""

""

General plan of instruction

Step 1: Teach the understanding of the attribute.

Suggested activity

. Making comparisons Ð

Step 2: Develop the understanding that the comparing of anattribute to a measuring unit produces a numbercalled a measure.

20.2 THE MEANING AND PROCESS OF MEASURINGIn order to measure successfully, you should perform three basic steps:

. Decide on the attribute to be measured (decide what should be measured Ðlength, volume, area, etc).

. Select the unit that has that attribute (select the unit that can be linked to whatshould be measured by using a rough estimation).

. Do the measurement using the selected unit of measurement by using the skill ofcomparison.

20.3 DEVELOPING MEASUREMENT CONCEPTS ANDSKILLS

20.3.1 A GENERAL PLAN OF INSTRUCTIONA general plan to teach measurement concepts and skills is discussed in this section.The plan discussed in the textbook probably did not even feature when you werefirst introduced to skills and the concepts of measurement at school.

The plan for measurement instruction given on page 376 is very important.

Study the table 20.1 given on p 376. Base all your teaching of measurement on thesteps described in this table.

bActivity 20.2

General plan of instruction

STUDY-READ the general plan of instruction.

Use the following framework to summarise the main ideas:

60

Suggested activities:

. Use informal units of measurement.Reasons for their use:

. Estimate the attribute.Why is it important?

""

Step 3: Learners will now use common measuring tools withunderstanding and flexibility.

Suggested activities:

. Make measuring instruments.Why?

. Make a direct comparison between the learner-made toolsand the standard tools.

. Do estimation using the standard unit of measurement.

20.4 MEASURING LENGTHThe discussion of the teaching of these concepts is done in great detail. Note that theway in which the concepts were developed corresponds to the general plan ofinstruction given in the first part of the chapter. Also note how the teaching of theconcepts differs from what happens in classrooms all over the country where theruler or measuring tape is used as a starting point in the measurement of length.Proper concept development is of vital importance.

bActivity 20.3

Measurement of length

SKIMMING: First read through this section (pp 378±382) and compare thedifferent steps used with the general plan of instruction.

STUDY-READ the section. The main steps (headings) are given. Under each ofthese steps explain what is done in the step and write down one activity toillustrate the actions in each step.

20.5 MEASURING AREAThe steps discussed in the general plan of instruction are again followed whenteaching the concepts of measurement and area.

PST201F/1 61

The space occupied bythis 1 cubic centimetrecube is its volume:

The unit ofmeasurement for volumeis a cubic centimetre.

The capacity of the small box(the one cubic cm cube) is 1millilitre.

It can hold 1 ml of water.

The mass of the purewater inside the smallbox is about 1 gram.

"

Note the great range of activities linked to area in this section. The teaching of areaand its measurement can be enjoyed by both the teacher and the learner.

Also note the clever use of triangles, tangram pieces, tiles and grids which enhancesthe learners' enjoyment while doing the activities.

bActivity 20.4

Area

. Read the section first to form an overall idea.

. Read the section again and, where possible, do the given activities. (You shouldmake yourself a tangram [shown in fig 20.8 on p 384].)

. Note, again, that the general plan of instruction was followed.

. Write down the headings. Under each heading, explain what is meant by theactivities and write down one activity to illustrate the explanation.

20.6 MEASURING VOLUME AND CAPACITYA description of volume refers to the ``size'' of a three-dimensional object (an objectthat has length, breadth and height) which can be described as the space occupied bythe object.

The capacity of a container can be described as the amount of liquid the containercan hold. What then is the relationship between the volume and the capacity of anobject?

Study the picture below to help you understand the relationship between 1 cubiccentimetre, 1 millilitre and 1gram (in the case of pure water).

b Volume: 1 cubic cm

62

bActivity 20.5

Measuring volume and capacity

. SKIMMING: Read through the section first (p 327±328).

. STUDY-READ the section. Pay attention to the general plan of instruction andwrite down the main ideas.

20.7 MEASURING WEIGHT AND MASSBefore you can teach the relevant concepts, you should be familiar with thedifference between mass and weight.

Weight is a measure of the pull or force of gravity on the object. I prefer to confinemy attention to mass and to give a comprehensive account of the related conceptsand skills. Let the science teacher work with gravity and its forces.

bActivity 20.6

Weight and mass

SKIMMING: Read through the section.

If you keep your discussion exclusively to mass, you will avoid confusion.

STUDY-READ and compare the material with the steps mentioned in the generalplan and write your own notes.

20.8 MEASURING ANGLESNote that, once again, the author advises us to follow the three steps suggested forinstruction. I agree with the following statement about the use of a protractor tomeasure the size of an angle (Van de Walle 2007:393): ``The protractor is one of themost poorly understood measuring instruments found in schools, and yet it iscommonly and frequently used. Part of the difficulty arises because the units(degrees) are so small. It would be physically impossible to cut out a single degreeand use it.''

bActivity 20.7

Measuring angles

. First read the section quickly.

. Read the section again and, where possible, do the different activities.

Ð Make yourself a wax paper ``sixteen-wedge'' protractor and explain its usein an activity. (You will have to include this homemade protractor in yourportfolio.)

. Write down the main ideas in this section.

Ð Do you fully agree with the way in which the measurement of angles is

PST201F/1 63

. Ð taught? (I have a slightly different idea of how to teach these concepts andskills.)

. Ð Think about how you would teach these concepts and write down your ownideas.

Notes on activity 20.7

You have to write down a step-by-step plan on how to teach the measurement ofangles. The skills and concepts necessary for successful measuring of angles aretaught in the Intermediate Phase classroom.

Read the following examination question. Then think about its answer. Compareyour thinking to the limits given.

A learner in grade 8 cannot measure the sizes of angles accurately.

What do you think went wrong with the teaching and learning of these skillsand concepts?

How will you teach the measurement of an angle. Clearly state your step-by-step plan.

Here are the hints to assist you with your answer:

(1) Use real-life situations Ð let your learners rotate their bodies to form differentangles. Teach them concepts such as a rotation, a straight angle, a right angle, anacute angle, an obtuse and a reflex angle.

(2) Use the rotation of the learners' bodies or the rotation of your own to guess thesizes of angles. Explain to them that a revolution is 3608, a straight angle is 1808and a right angle is 908.

Now proceed with what is done in your textbook, for example:

. Use units of angular measure.

. Make a protractor yourself, and so on.

20.9 INTRODUCING STANDARD UNITS OFMEASUREMENT

I will rewrite what the textbook says about standard units of measurement, bringingthe information more into line with the standard units of measurement used in theRSA.

Standard units of measurement can only be taught when the concepts of and skills inmeasurement itself are firmly in place.

When teaching the concepts relating to standard units of measurement, thefollowing broad goals should be identified:

. Familiarity with the unit. When the learner encounters a standard unit ofmeasurement, he or she should have a basic idea of its size.

For example. If he or she encounters 1 litre as a standard unit of measurement, heor she should have a clear idea of this amount of liquid. (My own idea is linked tothe amount of liquid in a two-litre coke bottle.)

64

Why estimate?

Estimation techniques used by skilled people

Tips on how to teach estimation skills

An activity (or activities) to help teach estimation skills

!

!

!

Note: Without this association with a specific familiar unit of measurement, it isimpossible to make any sense of the concept. This is the main reason why learnersin our country cannot measure things meaningfully.

. Choosing appropriate units. This is related to a person's familiarity with the unit. Itis also linked to a kind of estimation and common sense.

A lack of the skills needed to select the appropriate unit of measurement isanother reason for the poor performance of learners in the RSA.

. Knowledge of important relationships between units. Lengthy conversions from oneunit to another are done in our schools, but the exercise serves no purpose becauseit is done in isolation and without real understanding on the part of the learners.

b Activity 20.8

Introducing standard units

. Read through this section (p 394).

. Then, in short sentences, write down the main ideas relating to the units ofmeasurement used in South Africa.

20.10 ESTIMATING MEASURESThe importance of estimation involving measurement cannot be overemphasised. Iwish to reiterate that a prerequisite for successful measurement is the skill ofestimating the ``amount'' that should be measured. This section (based onestimation), once again features the importance of estimation and provides hints(tips) on how to teach these skills.

b Activity 20.9

Estimation and measurement

. SKIMMING: Read the section quickly (p 397).

. STUDY-READ the different ideas and complete the following diagram:

PST201F/1 65

20.11 DEVELOPING FORMULASThe learning programmes for the learning area Mathematics in the RSA mainlydefine outcomes and assessment standards guiding the teaching of formulas in theSenior Phase.

I would advise Intermediate Phase teachers to teach their learners all the skills andconcepts necessary for understanding formulae, but to leave the teaching of theformulae themselves to Senior Phase teachers.

Read what the textbook says about this in the section on common difficulties (p 399).

bActivity 20.10

Formulae

. You are advised to read through this section on page 398.

. This will give you an insight into what you should work towards when you planactivities in the Intermediate Phase, based on concepts necessary for thedevelopment of formulae.

20.12 MEASURING TIMEA unit of time is selected and used to ``fill'' the time to be measured. Time is also theduration of an event from its beginning to its end.

Well-known units of time are minutes, hours, days and years. This section discussesthe teaching of concepts related to time.

bActivity 20.11

Measuring time

STUDY-READ

. Read through this section (p 390).

. Write down the suggestions for helping a learner to understand clocks and howto read them. Use your own words. I am sure that one day you will return tothis section for guidelines on how to teach these concepts in your own classroom.

66

CHAPTER 21GEOMETRIC THINKING AND GEOMETRICCONCEPTS

g After working through this chapter, you should be able to

. indicate why the method we use to teach spatial learning should changedrastically to adapt to the modern approach discussed in this study material

. understand and apply the ``Van Hiele levels of geometric thought'' whenteaching geometric concepts

. explain why a study of geometric concepts and geometric thinking should startin the three-dimensional world

. assess the thinking skills and geometric concepts of your learners

21.1 OVERVIEWThe work prescribed for this course, which is based on spatial learning (geometry),requires a great deal of reading.

This material is new to the average reader and the method suggested for introducinglearners to spatial concepts and thinking is also totally different from the way inwhich any of the adults in the RSA were taught.

In order to understand the reasoning behind the teaching approach, you (as astudent teacher) should understand the following:

. A learner is born into a three-dimensional world. Therefore, if we take theconstructivistic view of learning into consideration, we should start the teachingof spatial concepts and geometric thinking with three-dimensional objects(shapes).

Can you think of an example of two-dimensional shape in nature?

. In the past, geometric concepts such as a point, line, line segment et cetera weredefined or described, and concepts were built on these concepts, which werelearnt by the learner. There was no real understanding, and most of thedescriptions and definitions were learnt by heart through memorising recipeswhich do not lead to meaningful learning.

. In the past the findings of the Van Hiele husband and wife team (the Van Hielelevels of geometric thought) were not taken into consideration in the teaching ofgeometric thinking and concepts.

. A study of the content of this study material will convince you that a hands-onapproach to the teaching of these concepts is the only way to go about it. This wasnever explored before in South African schools.

PST201F/1 67

If you agree with the above statements, you will understand why most adults in thiscountry found geometry extremely difficult and why only a few adults can applybasic geometrical knowledge in their everyday-life situations.

21.2 BACKGROUND KNOWLEDGETo fully understand the work done in this chapter and to apply it in your classroom,you need to do the following activities.

bActivities 21.1

(1) The Van Hiele thought levels. All teachers in classrooms today must knowabout and base their teaching of geometry on these levels.

STUDY-READ these levels and make sure that you understand every aspectof the work l.

(2) Activities for each of the levels. You as a teacher must be able to identifythe level of thought of a certain activity. To be really able to do this you,yourself, have to do all the activities given in this chapter.

(3) Addressing different aspects of geometry. The work done in the sections onlocation, transformations and visualisation is crucial to mathematics teachersin the RSA. Refer to the Learning Outcomes and Assessment Standards givenat the end of this study guide Ð all these concepts and skills are required bythe Learning Outcomes.

(4) Assessment. It is vital for all teachers to be able to assess the thought level ofeach of their learners. This knowledge will guide future teaching and learning.You must study the work in such a manner that you will be able to assess yourlearners' thought levels.

68

CHAPTER 22EXPLORING CONCEPTS OF PROBABILITYAND DATA ANALYSIS

gAfter working through this chapter, you should be able to

. explain the basic concepts of data analysis

. present data in a variety of contexts

22.1 OVERVIEWThe public are bombarded with graphs and statistics in advertising, opinion polls,population trends, health risks, et cetera.

To be able to deal with this information, learners should have the opportunitythroughout their school careers to have informal yet meaningful experiences withthe basic concepts involved.

When next you study the outcomes in the learning area Mathematics defined for theIntermediate Phase and the Foundation Phase learner, you will notice that the basicconcepts of probability, chance and statistics should be addressed.

In your classroom you are advised to do different activities based on the basicconcepts. The reasons for this are that the activities can easily be linked to real-lifesituations known to the learner and that learners find doing these activities anenjoyable experience.

Many South African teachers have not yet heard about

. cluster graphs

. box-and-whisker plots

. Stem-and-leaf plots

. and so on

All these representation forms are addressed in this chapter.

bActivity 22.1

Overview

SKIMMING: Quickly read through the chapter.

Note the following while you skim:

. the introduction to the concepts

. the material used to develop these concepts

. the graphic representation of data

PST201F/1 69

22.2 GRAPHIC REPRESENTATIONSOnce data have been gathered, what do we do with them? We represent them indifferent ways to make them more accessible to us. Once a graph has beenconstructed, what do you do with it? We deduce information from it. The answers tothese questions can be found in this section of the chapter.

bActivity 22.2

Graphic representation

STUDY-READ this section (p 458).

Note the unique way in which the author involves the learner in the learningprocess.

Whenever necessary, use the rest of the chapter as reference material.

Teachers in South Africa are expected to teach all the concepts discussed in thischapter to learners in the Foundation Phase, Intermediate Phase and the SeniorPhase.

70

CHAPTER 23EXPLAINING CONCEPTS OF PROBABILITY

gAfter working through this chapter you should be able to

. define and teach concepts related to chance and probability

. explain the difference between experimental and theoretical probability

. compute theoretical probabilities for independent and dependent events

23.1 OVERVIEWWhen you study the learning outcomes and assessment standards defined for theIntermediate Phase you will notice that you have to teach concepts and skills relatedto probability.

Learners in our country find the concepts and skills related to probability difficult.This is because these concepts are not addressed from grade R: the learnerencounters these concepts for the first time in grade 6.

bActivity 23.1

SKIMMING: Read through the chapter. Note the activities suitable for younglearners.

23.2 PROBABILITYThe content of this course covers a wide range of mathematical concepts and skills.You will have to use this chapter as a resource, and to refer to it in the futurewhenever you need to teach concepts relating to chance and probability.

bActivity 23.2

For handy use of this chapter for reference purposes, you need to study-read it insuch a manner that you can define the following concepts. Write down theexplanations or definitions of the following:

. chance

. probability

. probability on a continuum

. theoretical and experimental probability

. sample space

. independent events

. dependent events

. simulations

PST201F/1 71

CHAPTER 24DEVELOPING CONCEPTS OF EXPONENTS,INTEGERS AND REAL NUMBERS

gThis chapter pertains to the Senior Phase, especially grades 8 and 9.

After working through this chapter, you should be able to describe the followingconcepts:

. exponents

. representation of large numbers: scientific notation

. integers

. operations with integers

. rational numbers

. real numbers for example roots

bActivity 24.1

STUDY-READ the work in this chapter to familiarise yourself with theseconcepts.

Pay special attention to the new approaches to the teaching and learning of theseconcepts.

To what extent do these methods differ from or correspond to the way in which youwere taught? Do you feel that these approaches would prove more successful?Substantiate your answer.

72

PST201F/173

APPENDIX ALearning outcomes and assessment standards forgrades 4 to 9Learning Outcome 1: Numbers, Operations andRelationships

74

PST201F/1 75

76 Learning Outcome 2: Patterns, functions andAlgebra

PST201F/177

Learning Outcome 3: Space and shape

78

PST201F/179

Learning Outcome 4: Measurement

80

PST201F/181

Learning Outcome 5: Data Handling

82

PST201F/183

Learning Outcome 1: Numbers, Operationsand Relationships

84

PST201F/1 85

86 Learning Outcome 2: Patterns, functions andAlgebra

PST201F/1 87

88

Learning Outcome 3: Space and shape

PST201F/189

90

PST201F/191

Learning Outcome 4: Measurement

92

PST201F/193

Learning Outcome 5: Data handling

94

BIBLIOGRAPHYDepartment of Education. 2003 Revised National Curriculum Statement Grades R±9

(Schools). Mathematics. Pretoria: Department of Education.Van der Walle, JA. 2007. Elementary and middle school mathematics: teaching

developmentally. Boston: Pearson Education.

PST201F/1 95