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MO

DU

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Con�nuous Professional Development Cer�ficate in

Educa�onal Mentorship and Coaching for Mathema�cs Teachers (CPD-CEMCMT)

STUDENT MANUAL

Pedagogical Content Knowledge and Gender in Mathematics Education

2nd EDITION

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UR-CE (2019) Continuous Professional Development Certificate in Educational Mentoring and Coaching for Mathematics Teachers, Student Manual, Module 2, Kigali.

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Continuous Professional Development Certificate

in Educational Mentoring and Coaching

for Mathematics Teachers(CPD-CEMCMT)

Module 2Module code: PDM1142 Module Title: Pedagogical Content Knowledge and Gender in Mathematics

Education

Student Manual2nd Edition,

Kigali, April 2019

MO

DU

LE 2

Con�nuous Professional Development Cer�ficate in

Educa�onal Mentorship and Coaching for Mathema�cs Teachers (CPD-CEMCMT)

STUDENT MANUAL

Pedagogical Content Knowledge and Gender in Mathematics Education

2nd EDITION

UR-CE 2019Continuous Professional Development Certificate in Educational Mentoring and Coaching for Mathematics TeachersCPD-CEMCMT

iii

TABLE OF CONTENTS

LIST OF FIGURES VI

LIST OF TABLES VIII

LIST OF ACRONYMS IX

ABOUT THE AUTHORS X

ACKNOWLEDGEMENTS XII

INTRODUCTION 1

MODULE LEARNING OUTCOMES 2

UNIT 1: ANALYSIS OF THE MATHEMATICS CBC FOR PRIMARY SCHOOLS 3

Introduction 3

Learning Outcomes 4

Section 1: Competences in the Curriculum 5

Section 2: Mathematics Syllabus 8

Section 3: Lesson Planning 9

UNIT 2: KEY CONCEPTS IN MATHEMATICS EDUCATION 14

Introduction 14

Learning Outcomes 15

Section 1: Pedagogical Content Knowledge for Mathematics 16

Section 2: Mathematical Proficiency 24

Section 3: Mathematical Literacy 29

Section 4: Learner-Centred Pedagogy (LCP) 33

UR-CE 2019 Continuous Professional Development Certificate in Educational Mentoring and Coaching for Mathematics Teachers CPD-CEMCMT

iv

UNIT 3: KEY ASPECTS OF MATHEMATICS INSTRUCTION 35

Introduction 35


Section 1: Questioning 36

Section 2: Mathematics Conversations 49

Section 3: Developing Problem Solving Skills 55

Section 4: Addressing Learner Errors and Misconceptions 67

Section 5: Connecting Concrete, Pictorial and Abstract Representations of Mathematical

Concepts 74

Section 6: Games 86

Section 7: Inclusive Education 96

Section 8: Group Work 109

UNIT 4: GENDER AND MATHEMATICS EDUCATION 116

Introduction 116


Section 1: What Is Gender? 118

Section 2: Key Terms 119

Section 3: Gender Responsive Pedagogy for Mathematics 120

UNIT 5: ASSESSMENT 126

Introduction 126


Section 1: Formative and Summative Assessment 127

Section 2: Conducting Formative Assessment 130


v

UNIT 6: ACTIVITIES PER CONTENT AREA 138

Section 1: Numbers and Operations 138

Section 2: Fractions, Decimal Numbers and Percentages 152

Section 3: Elements of Algebra 164

Section 4: Probability and Statistics 178

MODULE REFERENCES 182

APPENDIX 189

Appendix 1: Self-Evaluation for Primary Mathematics Teachers 189

Appendix 2: Sample Lesson Plan of primary mathematics lesson 192

LIST OF FIGURES

Figure 1: Links between competences elaborated throughout the CBC (REB, 2015) 7

Figure 2: Basic information part of the CBC lesson plan (REB, 2015) 12

Figure 3: Specific Part of the CBC Lesson Plan (REB, 2015) 13

Figure 4: PCK for maths at the intersection of teaching maths and teaching people (VVOB) 16

Figure 5: Components of Mathematical Proficiency (National Research Council, 2002) 24

Figure 6: Examples of mathematics in daily life 30

Figure 7: Applications of mathematics in daily life 31

Figure 8: What makes a good question? Example of a Concept map (VVOB, 2017) 36

Figure 9: The importance of questioning (VVOB, 2017) 37

Figure 10: Verbs associated with higher levels of Bloom’s Taxonomy (Belshaw, 2009) 45

Figure 11: Importance of correct mathematical language in division operations 52

Figure 12: Multiple representations of seven (bstockus) 75

Figure 13: Three main types of graphs (Burns, 2015) 79

Figure 14: Representation of tiles in the bag activity (Burns, 2015) 80

Figure 15: Example of a fraction kit (Burns, 2015) 81

Figure 16: Game Board example for The Greatest Wins game (Burns, 2015) 89

Figure 17: Game board variations for The Greatest Wins game (Burns, 2015) 90

Figure 18: Overview of the Uncover game (Burns, 2015) 93

Figure 19: Example of learners’ recordings from the Uncover game (Burns, 2015) 93

Figure 20: Circles and Stars Game 94

Figure 21: Components of Inclusive Education (Ainscow, 2005) 97


vii

Figure 22: Differentiation is & Differentiation is not (ASCD, 2015) 98

Figure 23: Equality versus Equity (Save the Children, Mureke Dusome project, 2017) 100

Figure 24: Approaches to differentiation 102

Figure 25: Example of Open Task (Beckmann, 2013) 104

Figure 26: Example Solution for Open Task (Beckmann, 2013) 105

Figure 27: Example of open learning task 105

Figure 28: Talking Points on fractions 114

Figure 29: Formative and Summative Assessment 127

Figure 30: Formative versus Summative Assessment 129

Figure 31: Traffic Light Cards and Voting Cards (TES, 2013) 135

Figure 32: Voting cards with letters 135

Figure 33: Using a number line for subtractions 143

Figure 34: Area model for multiplication 146

Figure 35: Using double number lines to represent multiplications 147

Figure 36: Scaling on a number line 147

Figure 37: Various units for fractions 153

Figure 38: Various meanings of one quarter 154

Figure 39: Example of a spinner 178


viii

LIST OF TABLES

Table 1: Example of a Unit structure from the mathematics syllabus 8

Table 2: Teacher-centred versus Learner-centred education 34

Table 3: Verbs that elicit higher levels of Bloom’s Taxonomy (Bloom, 1968; Krathwohl, 2002) 44

Table 4: Examples of physical models to illustrate mathematical concepts 82

Table 5: Learning Challenges and Possible Classroom Strategies 107

Table 6: Actions to make classroom interactions more gender responsive (Mlama, 2005) 125

Table 7: Templates for Exit Tickets 134

Table 8: Meanings of Multiplication 146

Table 9: relation between multiplication and division 148

Table 10: Colour in the spinners to show the different probabilities. 179

Table 11: Colour in the spinners to show the probabilities that you define 180


ix

LIST OF ACRONYMS

CBC Competence Based Curriculum

CPD Continuous Professional Development

CoP Community of Practice

DDE District Director of Education

DHT Deputy Head Teacher

HoD Head of Department

HT Head Teacher

ICT Information and communications technology

LCP Learner-centred Pedagogy

NT New Teacher

OECD Organisation for Economic Cooperation and Development

PCK Pedagogical Content Knowledge

PP Policy Priority

RAWISE Rwandan Association for Women in Science and Engineering

REB Rwanda Education Board

SBI School Based In-service

SBM School Based Mentor

SEO Sector Education Officer

SSL School Subject Leader

TDMP Teacher Development and Management Policy

TTC Teacher Training College

UR-CE University of Rwanda – College of Education


x

ABOUT THE AUTHORS

Dr Alphonse UWORWABAYEHO obtained his PhD in Mathematics Education, specializing on

integration of ICT in the teaching and learning of mathematics from the University of Bristol, United

Kingdom. His research interest lies in teacher professional development on enhancing active teaching

learning. Currently, he is leading the Department of Early Childhood and Primary Education at the

University of Rwanda-College of Education (UR-CE). He is a member of African Centre of Excellence

for Innovative Teaching and Learning Mathematics and Science (ACEITLMS) based at the UR-CE.

Théophile NSENGIMANA is specialized in Mathematics and Science Education and holds a Master’s

degree in Education from Naruto University of Education. He is currently an Assistant Lecturer in the

Department of Mathematics, Science and Physical Education and a PhD student in Science Education

in the University of Rwanda.

Sylvain HABIMANA is an Assistant Lecturer in the Department of Mathematics, Science and Physical

Education at the University of Rwanda -College of Education (UR-CE). From 2016, he is pursuing

PhD studies at UR-CE with mathematics education as specialization. He holds a Master’s degree of

Education from Kampala International University (KIU), Uganda. He also holds a Bachelor’s degree

in Mathematics-Physics-Education (MPE) and a Postgraduate Certificate in Learning and Teaching in

Higher Education (PgCLTHE) from UR-CE/ former KIE, Rwanda.

Théoneste HAKIZIMANA holds a Master’s degree in Mathematics and a certificate of teaching

mathematics in higher learning institutions and secondary schools delivered by People’s Friendship

University of Russia in 1992. He has 27 years of experience in teaching mathematics in various higher

learning institutions. He is currently a lecturer and subject leader of mathematics at the department

of Mathematics, Science and Physical Education, College of Education, University of Rwanda.

Dr Védaste MUTARUTINYA (PhD) pursued his studies in the Faculty of Sciences of Friendship People’s

University of Russia (FPUR) at Moscow, where he successively obtained a BSc’s degree (1994), a

Master’s degree (1996) in Mathematics and Physics and a PhD (2000) in Mathematics. He is the

author of several scientific articles in national and international journals. His fields of research are:

Functional Analysis with applications and Mathematics Education.

Nehemiah BACUMUWENDA holds a Master’s degree in Public Health from the University of Rwanda,

a Bachelor’s Degree in Management from Independent University of Kigali and a Bachelor’s Degree in

Educational Psychology from Adventist University of Central Africa. He is specialised in the Rwandan

education system and has more than 30 years of experience in the education sector. He is currently

a curriculum developer at Rwanda Education Board.


xi

Dr Leon MUGABO Rugema (PhD) is an experienced science teacher and science teacher educator.

He holds a PhD in Science Education. Currently, he is a lecturer in the Department of Early Childhood

and Primary Education, School of Education at UR-CE. He is also Head of Teaching and Learning

in the Africa Centre of Excellence for Innovative Teaching and Learning Mathematics and Science

(ACEITLMS).

Dr Jean Francois MANIRAHO (PhD) is a lecturer and researcher at UR-CE. Over the last 10 years,

Dr Maniraho has been lecturing in different higher learning institutions both inside and outside

Rwanda. He is specialized in mathematics education with knowledge of teacher education. Based

on his background with a Master of Applied Mathematics, he is also a data analyst, with experience

in MATLAB.

Clementine Gafiligi UWAMAHORO holds an MBA in International Business at Amity University of

India and a BA in Educational Psychology from Adventist University of Central Africa. She is specialized

in the Rwandan education system. She has more than 17 years of experience in the education sector,

especially in curriculum development, education planning, school leadership, teacher training,

quality insurance and assessment of Education system. She is currently an education adviser for

VVOB.

Stefaan VANDE WALLE is education advisor school leadership and STEM education with VVOB. He

holds Master’s degrees from the University of Leuven, Belgium (geography), Radboud University

Nijmegen, The Netherlands (project planning) and the Open University, UK (online and distance

education). He has been working for VVOB since 2008 in Cambodia, South Africa and Rwanda. His

areas of specialization include school leadership, teacher education, STEM education and online

learning.


xii

ACKNOWLEDGEMENTS

We owe a large debt of thanks to the authors of this guide: Dr Alphonse Uworwabayeho, Théophile

Nsengimana, Sylvain Habimana, Théoneste Hakizimana, Dr Védaste Mutarutinya, Dr Mugabo

Rugema Leon, Dr Jean Francois Maniraho, Clementine Gafiligi Uwamahoro and Stefaan Vande Walle.

We also like to thank the following TTC Tutors and mathematics subject leaders who have played

crucial additional roles in developing and reviewing the course text: Jean Damascene Habimana

(TTC Kabarore), Theophile Ngizwenayo (TTC Matimba), Jean Pierre Kuradusenge Rukeribuga (TTC

Gacuba), Jean de Dieu Rutagengwa (TTC Zaza), James Bayingana (Nyagatare District), Tharcisse

Uwimanimpaye (Nyagatare District), Augustin Gabiro (Kirehe District), Barthelemie Niyitegeka

(Kirehe District), Joseline Nyiramatabaro (Rusizi District), Dieudonne Nayituriki (Rusizi District),

Jean de Dieu Mugabo (Gastibo District), Francois Mbarushimana (Gastibo District), Simeon Nikuze

(Nyabihu District), Vedaste Mufaransa Rushema (Nyabihu District), Solonga Nzahemba (Kayonza

District), Theophile Ndagijimana (Kayonza District).

This course would not have been possible without the financial support from the Belgian Government

and ELMA Foundation.

Finally, we like to thank the Ministry of Education in Rwanda (MINEDUC), Rwanda Education Board

(REB) and the University of Rwanda - College of Education (UR-CE) for their continued support to

education in Rwanda in general and to this CPD Certificate Programme on Educational Mentoring

and Coaching for mathematics school subject leaders (MSSLs).


1

INTRODUCTION

The first module of this course focused on coaching, mentoring and communities of practice (CoPs).

This second module on Pedagogical Content Knowledge (PCK) and gender in mathematics education

aims at improving the quality of your coaching and mentoring to teachers and enrich the discussions

in CoPs by focusing on the various dimensions of mathematics teaching.

We start this module with a discussion of the recently implemented competence-based curriculum

for primary mathematics. Secondly, we introduce the key concepts of the module: Pedagogical

Content Knowledge (PCK), Mathematical Proficiency, Mathematical Literacy and Learner-Centred

Pedagogy. In the third unit, we will introduce key aspects of successful mathematics instruction. For

each aspect, we briefly introduce the concept and relevant research before we move to concrete

classroom-based techniques. Unit 4 and Unit 5 are dedicated to the cross-cutting themes of gender

and assessment. Both units start from general ideas before moving to concrete techniques and

approaches to teach in a gender responsive way and use assessment as a tool for learning by students

and teachers. The final unit contains a variety of classroom activities, questions and problems that

you can use in your lessons, arranged per content area.


2

MODULE LEARNING OUTCOMES

By the end of the module participants should be able to:

Explain the principles of the Competence Based Curriculum for primary maths.

Understand the concepts of Pedagogical Content Knowledge, mathematical proficiency,

mathematical literacy and learner-centred pedagogy;

Demonstrate understanding of key principles of successful mathematics instruction;

Successfully mentor fellow teachers in teaching mathematics;

Apply a variety of techniques and approaches to develop knowledge, understanding, problem

solving and reasoning skills and appreciation for mathematics with learners;

Organize professional development activities for mathematics teachers, including providing

effective feedback to peers.

Address gender stereotypes associated to the teaching of mathematics at primary school

level.

Select and develop appropriate and inclusive teaching and learning materials and methods

for teaching and learning mathematics;

Make learning mathematics enjoyable for all learners by Integrating daily life in mathematics

lessons;

Adapt interventions to meet personal and professional development needs in teaching and

learning mathematics;

Create a culture of on-going reflection and learning for improvement;

Developing an action plan for improving teaching and learning mathematics in their school;

Believe that all learners can achieve reasonable levels of mathematics proficiency;

Appreciate collaboration, team work and joined leadership within the school;


3

UNIT 1: ANALYSIS OF THE MATHEMATICS CBC FOR PRIMARY SCHOOLS

Introduction

Activity 1

Think about the differences between the Knowledge-based curriculum (KBC) as applied up to

2015 and the competence-based curriculum (CBC) currently applied in the Rwandan education

system.

Write down your ideas and discuss them with your neighbour.

In 2016, REB started with the implementation of a competence based curriculum (CBC) in pre/

primary and secondary education (REB, 2015). A competence is the ability to use an appropriate

combination of knowledge, skills, attitudes and values to accomplish a task successfully. In other

words, it is the ability to apply learning with confidence in a wide range of situations (REB, 2015).

Within the CBC framework, teaching and learning are based on competences rather than focusing

only on knowledge.

Learners work on acquiring one competence at a time in the form of concrete units with specific

learning outcomes. The student is evaluated against these standards. Learning activities should be

learner-centred, balancing individual and social learning. Therefore, mathematics teachers need

to have the resources and skills that enable them to respond to curriculum requirements in the

classroom. REB (2015) states that mathematics equips learners with the competences to enable

them to succeed in an era of rapid technological change and socio-economic development. Mastery

of basic mathematical ideas and operations (mathematical literacy) should make learners confident

in problem-solving in life situations. A high-quality mathematics education therefore provides a

foundation for understanding the world, the ability to reason mathematically, an appreciation of the

beauty and power of mathematics, and a sense of enjoyment and curiosity about the subject.


4

Learning Outcomes

By the end of this unit, participants should be able to:

Explain the structure of the competence-based curriculum of primary mathematics

education;

Explain the use of different components of the competence-based curriculum of primary

mathematics education;

Continuously reflect on teaching approaches in line with mathematics competence-based

syllabus;

Plan mathematical learning activities that enhance learners’ competences and move

beyond transferring knowledge;


5

Section 1: Competences in the Curriculum

A competence means the ability to do something successfully or efficiently. A competence-based

curriculum implies that learning activities are chosen so that learners can acquire and apply the

knowledge, skills and attitudes to situations they encounter in everyday life. Competency-based

curricula are usually designed around a set of key competences/competencies that can be cross-

curricular and/or subject-bound. A competence-based curriculum is less academic and calls for a

more practical and skills-based approach and more orientation to a working environment and daily

life.

The CBC distinguishes between two categories of competences: basic competences and generic

competences. Basic competences are key competences that were identified basing on expectations

reflected in national policy documents. These competences are built into the learner’s profile in each

level of education and for all subjects and learning areas. Basic competences have been identified

with specific relevance to Rwanda. These are literacy, numeracy, ICT, citizenship and national

identity, entrepreneurship and business development, science and technology, and communication

in the official languages (REB, 2015).

Generic competences are competences which are transferable and applicable to a range of subjects

and situations (REB, 2015). They promote the development of higher order thinking skills. In doing

so they strengthen subject learning, but they are also valuable in themselves. They are generic

competences because they apply across subjects.

To guide teachers in sequencing teaching and learning activities, competences have been elaborated

at every level of the curriculum from the learner profile down to the Key Unit Competences. The

learner profiles describe the general learning outcomes expected at the end of each phase of

education. Teachers are responsible to design lesson plans with instructional objectives linked to

the Key Unit Competences and leading to all competences above.


6

The key unit competence is the most important element to pay attention to while designing a lesson plan as it determines the instructional objective(s) of each lesson within the unit.

Figure 1 shows the links between the various competences in the CBC.

Broad Competences are formulated for the end of each learning cycle (at the end of Pre-Primary, Lower Primary, Upper Primary, Secondary 3, and Secondary 6). National Exams assess the achievement of these broad competences according to National Assessment Standards.

Key competences are formulated for the end of each grade. Districts and schools design assessment strategies to ensure learners have achieved the necessary competences and qualify for advancement or need further remediation to meet National Assessment Standards.

Key unit competences are formulated throughout the subject syllabus. The syllabus is divided into units of study to organize learning and encourage teachers to focus on specific content related to learners’ daily life and cross cutting issues. Each unit aims to develop basic and generic competences which are evaluated through end unit assessment according to National Assessment Standards.

Learning objectives are specific knowledge, skills, attitudes and values learners should gain within lessons to build progressively the key unit competences. Teachers are responsible to prepare lesson plans based on the subject syllabus.


7

Figure 1: Links between competences elaborated throughout the CBC (REB, 2015)

The key unit competence is the most important element to pay attention to while designing a lesson

plan as it determines the instructional objective(s) of each lesson within the unit.

While setting lesson instructional objectives, teachers are advised to balance Lower Order Thinking

Skills (LOTS) and Higher Order Thinking Skills (HOTS). Higher Order Thinking Skills (HOTS) are a

central element in a competence-based curriculum because they develop the understanding that

enables the effective application of knowledge.


8

Section 2: Mathematics Syllabus

Mathematical concepts are applied in other subjects such as science, technology and in business. Mathematics subject content enhances critical thinking skills and problem solving. Mathematics teaches learners to be systematic, creative and self-confident in using mathematical language and techniques to reason deductively and inductively.

The primary mathematics curriculum is structured into topic areas, sub- topic areas (where applicable) and in units. Table 1 shows the structure of each unit.

Table 1: Example of a Unit structure from the mathematics syllabus

source: MINEDUC, 2015


9

Section 3: Lesson Planning

Planning a lesson is an important responsibility for a teacher. A lesson plan is a teacher’s description

of the ‘learning trajectory’ for a lesson. A lesson plan is the teacher’s guide for running a lesson:

It includes the goal (what the students need to learn), how the goal will be reached (methods,

procedures) and a way of measuring if the goal was reached (test, activity, homework etc. (REB,

2015). Key elements in developing a lesson plan are summarized in Figure 3.

1. Check your scheme of work

At the start of every academic year, teachers develop a Scheme of Work based on the subject syllabus,

the school calendar and the time allocated to the subject per week. For lesson plan preparation,

consider the following questions:

What lesson have you planned to teach in a period, such as a term, a month and a week?

What key competences do you hope to develop by the end of unit?

2. Identify relevant generic competences and crosscutting issues

Each lesson must address generic competences and crosscutting issues. In the lesson plan template,

there is a section titled ‘Competence and crosscutting issues to be addressed’. In this section, you

can describe what learners should be able to demonstrate and how the teaching and learning

approaches will address these crosscutting issues.

3. Set instructional objectives

An instructional objective should have five components. The following steps can guide you to

formulate an instructional objective:

a. Reflect on the conditions under which learners will accomplish the assessment task

(teaching aids, techniques, outdoors or indoors);

b. Determine who you are talking about (learners);

c. Identify at least one measurable behaviour (knowledge, skills, attitude or values) that you

are looking for – evidence of learners’ activity. Use a verb which describes the result of


10

learning activities. (e.g. read, write, explain, and discuss). Aim for Higher Order Thinking

Skills;

d. Include the content of the activity. You can take this from the subject syllabus.

e. Set standards of performance. Write down the criteria for minimum acceptable

performance (for example time, number of correct answers, presence of expected/

shared values);

f. Identify the types and number of learners with learning disabilities in the section ‘Type of

Special Educational Needs and the number of learners in each category, insert the type

of disability that you have identified in your class and the number of learners with that

disability. In addition, note how these learners will be accommodated in the learning

activities.

Education policy targets learners with disabilities (Special Educational Needs or SEN), who qualify

(through standardized SEN assessment) for adjusted educational provisions, or/and who meet

barriers within the ordinary education system (REB, 2015). The group includes:

a. Learners with functional difficulties, including physical and motoric challenges, intellectual

challenges, visual impairments, hearing impairments, speech impairments;

b. Learners with learning disabilities, including specific and general learning difficulties

(dyslexia, dyscalculia…);

c. Learners with social, emotional and behavioural difficulties (Attention Deficit Hyperactivity

Disorder, Asperger’s Syndrome…);

d. Learners with curricula-related challenges and difficulties to comprehend or use the

teaching languages (including linguistic minorities);

e. Learners with health challenges.

4. Identify organizational issues

This part of the lesson plan as about creating positive learning environments, specifically related to

physical safety and inclusion. In the section titled “Plan for this class (location: in / outside)”, you can

write down where you will hold the lesson, seating arrangements etc.


11

5. Decide on teaching and learning activities

In this part, the teacher summarizes the learning and teaching process including main techniques

and resources required. In the column “teacher’s activities”, you describe the activities using action

verbs. Questions and instructions from the teacher are also written in this column. In the column

“learner activities”, the teacher describes the learner activities, findings and answers. Activities or

answers which don’t fit in the column, can be added in an appendix. The teacher specifies whether

activities are carried out individually, in small groups or with the whole class.

In the column of steps and timing in the lesson plan format, there are three main steps: introduction,

development of the lesson and conclusion.

Introduction is where the teacher connects the lesson with the previous lesson. For example,

the teacher organizes a short discussion to encourage learners to think about the previous

learning and connect it with the current instructional objective.

Development of the lesson. Depending on the lesson, the development of the content will

go through the following steps: discovering activities, presentation of learners’ findings,

exploitation and synthesis/summary. In discovery activities, teachers give a task to learners to

identify the prior knowledge in relation with the new topic. The teacher and learners analyse

their findings towards understanding and construction of the new concept. Thereafter the

teacher deducts the learning facts which are the summary of the lesson.

In conclusion, the teacher assesses the achievement of instructional objectives and guides

learners to make the connection to real life situations. You may end with homework.

6. Decide on the timing for each step

You need to allocate time for each step of the lesson. It is advised to reserve time for learners to

write down key words or a summary of the content in their notebooks.

The lesson plan has two main parts: a basic information part (Figure 2) and a specific part (Figure 3).


12

Figure 2: Basic information part of the CBC lesson plan (REB, 2015)


13

The teacher should take into account all learners While formulating the instructional objectives, the type of activities will be mentioned

The teacher mentions generic competences and cross-cutting issues to be developed in relation to learners’ activities and lesson content. The teacher provides short explanations justifying how these competences and cross cutting issues are addressed.

E.g.: the teacher asks effective questions on how learners perceive the lesson, how it’s connected to their life experience and how they will use the acquired competences.

Summary of the teaching and learning process.

Teacher indicates the learning material needed and specifies how all learners will be involved

The teacher describes the activity using action verbs. Questions and instructions are also indicated

The teacher describes the learners expected activities, findings and answers

The teacher indicates the steps to follow: - Discovery activities, - Presentation of findings, - Exploitation and - Synthesis/summary

Figure 3: Specific Part of the CBC Lesson Plan (REB, 2015)


14

UNIT 2: KEY CONCEPTS IN MATHEMATICS EDUCATION

Introduction

This training material is conceived for mathematics subject leaders in basic education who support

and coach their fellow teachers, especially newly qualified ones. Several key concepts on teaching

mathematics for the basis for this course: Pedagogical Content Knowledge (PCK) for mathematics,

Mathematical Proficiency, Mathematical Literacy and learner-centred pedagogy. Familiarity with

these concepts will enable you to improve your teaching and your support to your fellow teachers.


15

Learning Outcomes


Understand key concepts of mathematics education;

Understand that procedural fluency, conceptual understanding, strategic competences,

adaptive reasoning and productive disposition are interrelated and all equally important to

achieve mathematical proficiency;

Understand the importance of learning mathematics for students’ daily life;

Recognize key concepts in mathematics education in classroom situations;

Support fellow teachers to use mathematics to solve problems related to learners’ daily life;

Respect the diversity in feelings, opinions and prior knowledge in learners;


16

Section 1: Pedagogical Content Knowledge for Mathematics

Activity 2

Think individually about the following question:

What does a teacher need to know, be able to do and care about to be a good maths teacher?

Write down your ideas and discuss them with your neighbour.

Research on the relation between teacher knowledge and student learning found no relation

between more teacher mathematical knowledge and more student learning (Ball, Thames, &

Phelps, 2008). In other words, having more knowledge of mathematics does not automatically lead

to better teaching of mathematics. A mathematics teacher does not just teach mathematics, but

teaches mathematics to learners (Figure 4).

PCK

Figure 4: PCK for maths at the intersection of teaching maths and teaching people (VVOB)

Shulman (1986) identified different types of knowledge that teachers need to teach well: content

knowledge, curriculum knowledge and pedagogical content knowledge.

For example, a teacher who plans to teach a lesson on multiplying decimals needs to know a lot

more than how to do the multiplication (Ball, 1990, p. 448):

“The teacher had to know more than how to multiply decimals correctly herself.

She had to understand why the algorithm for multiplying decimals works and what


17

might be confusing about it for students. She had to understand multiplication

as repeated addition and as area, and she had to know representations for

multiplication. She had to be familiar with base-ten blocks and know how to

use them to make such ideas more visible to her students. Place value and the

meaning of the places in a number were at play here as well. She needed to see

the connections between multiplication of whole numbers and multiplication of

decimals in ways that enabled her to help her students make this extension. She

also needed to recognize where the children’s knowledge of multiplication of whole

numbers might interfere with or confuse important aspects of multiplication of

decimals. And she needed to clearly understand and articulate why the rule for

placing the decimal point in the answer – that one counts the number of decimals

places in the numbers being multiplied and counts over that number of places

from the right – works. In addition, she needed an understanding of linear and

area measurement and how they could be used to model multiplication. She even

needed to anticipate that a fourth-grade student might ask why one does not do

this magic when adding or subtracting decimals and to have in mind what she

might say.”

Some research findings on PCK are (Ball, 1997; Ball et al., 2008; Hill, Ball, & Schilling, 2008):

Expert knowledge of subject matter alone is inadequate for good teaching.

Teaching mathematics requires the capacity to “deconstruct” one’s own knowledge and

identify the critical components or steps for learners to acquire this knowledge.

Teachers need to be able to reason through and justify why certain procedures and properties

hold true, to talk about how mathematical language is used, to see the connections between

mathematical ideas and to understand how they build upon one another.

“What you do when you’re teaching is you think about other people’s thinking. You don’t think about your own thinking; you think what other people think. That’s really hard.” -Deborah Ball


18

For example, many people will be able to solve the multiplication below. The teacher’s task is not

only to provide the correct answer, but to recognize why learners make mistakes and adapt their

teaching accordingly.

However, consider the following incorrect answers from learners. How was each answer produced?

What misunderstandings might lead a student to make these errors? This specialized knowledge

is less likely to be present with people who are good at doing mathematics, but don’t have any

teaching experience. Recognizing the underlying thoughts from learners that cause these errors is

a crucial skill for teachers. A teacher who can only say: “Your answer is wrong”, is not more helpful

than a doctor who says that you’re sick but can’t make a good diagnosis.

What mathematical misunderstandings could lead to each of these three answers?

(a) 1485

What mathematical steps are involved? Multiply 9 x 5, which produces 45. Write down the 5

and carry the 4. Add the 4 to the other 4 in the tens column, which yields 8, and multiply 8 x

5, which is 40. Write down 40. Next, multiply 9 x 2, which equals 18. Write down 8 and carry

the 1; as before, add the 1 to the 4 before multiplying, i.e., 5 x 2, which equals 10.


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What is the main issue to understand? This process adds the carried ten in before multiplying, instead of afterwards.

(b) 325

What mathematical steps are involved? Multiply 25 x 9 first (bottom up). This yields 225. Then multiply 25 x 4, which equals 100.

What is the main issue to understand? This process starts with the bottom number instead of with the top as is conventional. This is mathematically valid because multiplication is commutative and so the order in which one multiplies does not matter. However, 25 x 4 is really 25 x 40, which would produce 1000.

(c) 1275

What mathematical steps are involved? Round 49 up to 50, then multiply 50 x 25, which is 1250. Then add 25 to 1250 because 49 is less than 50.

What is the main issue to understand? This process compensates in the wrong direction –– i.e., adds 25 to the 1250 instead of subtracting. Someone might do this because with the conventional procedure one adds together the two separate answers.

The knowledge that teachers need for teaching maths goes beyond the mathematics content and includes:

Guiding instruction starting from learners’ prior knowledge and development level

Selecting appropriate examples in the right order

Using correct mathematical language and notation

Using learners’ errors as rich sources of information

Anticipating and reacting immediately to learners’ responses (“learning to see more in the moment”)

Selecting appropriate routine and non-routine word problems

Asking learners questions that guide them in their learning process, mainly using open questions

Choosing, using and connecting different representations of a concept

Linking mathematics with daily life applications


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Example 1: Claire’s Lesson on introducing ratios (Chick & Harris, 2007)

Claire began by showing the students a 2cm×3cm rectangle and reviewing the definition of

area and perimeter, highlighting meaning and units. She then demonstrated how to colour the

rectangle in such a way that for every square that was coloured in red, two squares had to be

coloured in blue, before having a student repeat this process for another 2cm×3cm rectangle. She

asked students how many squares were coloured red and how many were blue, but before this

had been answered one student pointed out that 1/3 of the rectangle was red and 2/3 was blue.

This unexpected response allowed Claire to explore the connection between fractions and the

situation that they had, highlighting that the 1/3 came from the fact that 2/6 of the squares in

the rectangle had been coloured red. After showing students that they could also colour half

squares while still achieving one red colouring for every two blue, and demonstrating such an

example, she asked students to find the different colourings of the 2cm×3cm rectangle using the

“one red for every two blue” scheme. As they started work she drew from them the need to work

systematically, suggesting that they start with whole square colourings first, and attend to the

different possible positions of the red coloured squares. After allowing students to explore the

problem for about 15 minutes she had students talk about how they had worked through all the

possible arrangements of red and

blue colourings, incorporating some discussion about how equivalent arrangements can arise

by “flipping” (reflecting) arrangements already found. She concluded her use of this example by

emphasising to students that although they had produced many different arrangements, the area

of red in all cases was 2cm2 and the area of blue was 4cm 2.

Claire did not mention ratio at all during the first 25 minutes of the lesson; the emphasis seemed

to be on area, working systematically, and then, briefly, ideas of symmetry.

However, her choice of the 2cm × 3cm rectangle, and the simple proportion “one red to two blue”

allowed students to consider the area, problem solving, and symmetry ideas—with fractions

receiving some consideration as well—while building a foundation for talking about ratio. It was

only after this exploration of a single example that she defined ratio, using the 2 red to 4 blue

idea, helping students to see the connection to fractions, identifying the connection between the

parts and the whole, getting students to simplify the ratio 2:4 to the “basic” ratio 1:2, and linking

this back to her original colouring instruction to colour 1 red and 2 blue. The example used—the

2cm×3cm rectangle and the ratio 1:2—was used for teacher demonstration with a conceptual

focus, but was also used as a student task, and the focus was on conceptual ideas rather than

procedural ones.


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Example 2: Jean’s Lesson on introducing ratios (Chick & Harris, 2007)

Jean began his lesson by asking the class if anyone knew what a ratio was, with the students’

responses suggesting that some had heard the term but had little idea about its meaning. One

student used the expression “a ratio of one to two” but could not illustrate its meaning. Jean then

explained that ratio is associated with fractions or proportion and is used to show the amounts that

comprise a whole. His explanation was, at this stage, imprecise and given without an illustrative

example. He then invited ten students to stand at the front of the class, highlighted that the ten

was the whole, and asked students to determine what proportion of boys and what share were

girls. He showed students how to write this as 3:7 and emphasised that 3+7 gives ten, the total in

the group. He had students rearrange themselves to show the ratio of their favourite colour (blue

or red), which turned out to be 5:5.

Based on these examples, Jean then gave the students some notes about ratio. He used three

different examples based in the same context: a discrete collection that he divided into two

groups in two different ways. He used the examples to demonstrate the notation of ratio and

how to say it, and to remind students about the whole and that ratio compares two numbers. His

emphasis during the introductory exposition was partly conceptual, but with a strongly procedural

emphasis in the discussion of the way to write and say ratios. His notes on the board for students

to copy included an extra example, 1:5, which initially had no physical context, and which his

notes suggested could also be written as 1/5. This was done without comment or additional

explanation.

He later illustrated the 1:5 example in the context of making lemonade, where he highlighted that

one part of fruit juice and five parts of water should be used, to give a total of six equal parts. He

also clarified that the actual size or amount of these equal parts did not matter, provided all parts

are equal, and emphasised that the order of the numbers in the ratio matters.

Activity 3

Read the case studies in the boxes above. Do both teachers demonstrate strong PCK in their lesson? List good points and points for improvement.


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Discussion of case stories

One of the best things about Claire’s lessons was her capacity to make connections among a range

of mathematical topics. She made strong use of area understanding (and referred to perimeter

in passing), was careful in establishing the links to fractions, and used correct terminology such as

“factor”. Claire’s work with the rectangles and letting students build up an equivalent ratio from the

simpler one and allowing students to see multiple configurations of square colourings all of which

show 2:4, have made it much easier for them to understand the idea of simplifying the ratio.

Jean certainly appeared to understand the content but was not explicit about the connection

between a ratio and its simplified form. Although he recognised that fractions and ratios are linked,

he did not address the connection between the ratio 1:5 and the fraction 1/5.

The situations highlight the difficulty of selecting appropriate examples and using them effectively to

illustrate general principles. The need to choose suitable representations is particularly important.

The discrete representations of ratios apparent in the groups of people used in Jean´s class restrict

full understanding of ratio, when compared to the continuous area model used in Claire’s class,

especially as she allowed students to colour half squares and to consider quarters as well. The

sequencing of examples was also important, with Claire building up non-simplified ratios before

considering simplification and equivalence, and she also tried to ensure that students were prepared

for the problems on the worksheet.

Strengthening PCK is a key instrument to improve the quality of teaching and learning. PCK develops

with teaching experience. However, it doesn’t come automatically, but requires continuous

professional development and reflection. You can strengthen your PCK as a teacher by doing the

following:

Figure out why procedures work, not just how to do them;

Try to solve problems in more than one way;

Listen to and ask questions to learners about their work, especially when they are struggling;

Study learners’ thinking and work;

After a lesson, reflect critically on what went well and what could be improved, preferably

with a colleague;

Prepare lessons together with peers, observe lessons from your colleagues and discuss them

afterwards.

Invite colleagues to observe your lessons and give you feedback.


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Activity 4

Review the key aspects of good mathematical instruction that we discussed in unit 3. Can you

find examples of PCK? Explain your choices.


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Section 2: Mathematical Proficiency

Mathematics teachers should not only focus on making sure that students can perform the necessary

procedures. Equally important aspects of teaching maths are to help them see the relations

between concepts and to motivate them to learn mathematics. The National Council of Teachers of

Mathematics (NCTM) in the US has developed the concept of mathematical proficiency (National

Research Council, 2002).

Mathematical proficiency has five components (Figure 5):

1. conceptual understanding

2. procedural fluency

3. strategic competence

4. adaptive reasoning

5. productive disposition (motivation)

Figure 5: Components of Mathematical Proficiency (National Research Council, 2002)


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These five components are related to each other and are relevant for competence-based mathematics curriculum implementation. Each component of mathematical proficiency strengthens the other ones to make learners proficient in mathematics.

For example, procedural fluency and conceptual understanding strengthen each other. As a learner achieves conceptual understanding, he/she will remember procedures better as well. In turn, as a procedure becomes more automatic, the learner can start to think about other aspects of a problem and tackle new kinds of problems, which leads to new understanding.

Many teachers think that they must choose between focusing on procedural fluency or on conceptual understanding. However, good maths teachers combine the two components in their lessons. Understanding makes learning skills easier and learners will forget a procedure less quickly if they understand why it works.

Let’s have a closer look at what each component means:

1. Conceptual Understanding

Students with conceptual understanding know more than isolated facts and methods. They can learn new ideas by connecting them to what they already know.

2. Procedural Fluency

Procedural fluency is very important. In daily life, you need to be able to solve certain problems such as additions and multiplications quickly without thinking through the underlying concepts or using a calculator. Also, learners need basic fluency with procedures when solving more complicated problems (Burns, 2015).

Such belief is strong with children in the early grades when, for example, they learn one procedure for subtraction problems without regrouping and another for subtraction problems with regrouping. Another consequence when children learn without understanding is that they separate what happens in school from what happens outside. They think that mathematics is something from school, not something from their daily life.

Example of integrating procedural and conceptual understanding

Consider the multiplication of multi-digit whole numbers. Many algorithms for computing 47×268 use one basic meaning of multiplication as 47 groups of 268, together with place value knowledge of 47 as 40+7, to break the problem into two simpler ones: 40×268 and 7×268. For example, a common algorithm for computing 47×268 is written the following way, with the two partial products, 10720 and 1876, coming from the two simpler problems:


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Familiarity with this algorithm may make it hard for adults to see all the underlying knowledge that

is needed. It requires knowing that 40×268 is 4×10×268; knowing that in the product of 268 and 10,

each digit of 268 is one place to the left; having enough fluency with basic multiplication combinations

to find 7×8, 7×60, 7×200, and 4×8, 4×60, 4×200; and having enough fluency with multi-digit addition

to add the partial products. As students learn to execute a multi-digit multiplication procedure such

as this one, they should develop a deeper understanding of multiplication and its properties. On the

other hand, as they deepen their conceptual understanding, they should become more fluent in

computation. A learner who forgets the algorithm but who understands the role of the distributive

law can reconstruct the process by writing 268×47=268× (40+7) = (268×40) + (268×7) and working

from there. A learner who has only memorized the algorithm without understanding can be lost

when memory of the procedure fails.

3. Strategic Competence

This component includes problem solving and problem formulation.

Problem solving is not just giving learners problems to solve. Outside of school a big part of

the difficulty is to figure out what the problem is and formulate the problem in such a way

that a learner can use mathematics to solve it.

4. Adaptive Reasoning

Adaptive reasoning refers to the capacity to think logically about the relationships between

concepts and situations.

It includes estimating the result of a mathematical problem and identifying unrealistic

answers.

It includes being able to justify one’s work with correct mathematical language.

You can develop adaptive reasoning by giving learners regular opportunities to talk about the

concepts and procedures they are using and let them explain what they are doing and why.


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Questions that develops learners’ reasoning skills

Example 1

If 49 + 83 = 132 is true, which of the following is true (without calculating) and explain why.

a. 49 = 83 + 132

b. 49 + 132 = 83

c. 132 – 49 = 83

d. 83 – 132 = 49

Research found out that only 61% of American 13-year-olds chose the right answer on this question,

which is lower than the percentage of students who could correctly compute the result (National

Research Council, 2002).

Example 2

Without calculating, estimate which number is closest to this sum:

12/13 + 7/8

a. 1

b. 2

c. 19

d. 21

Fifty-five percent of American 13-year-olds chose either 19 or 21 as the correct response. Even small

levels of reasoning skills should have prevented this error. Simply observing that 12/13 and 7/8 are

numbers less than one and that the sum of two numbers less than one must always be less than two

would have made it clear that 19 and 21 were unrealistic answers (National Research Council, 2002).


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5. Productive Disposition

Productive disposition means seeing mathematics as both useful and worthwhile to study,

believing that effort in learning mathematics will be rewarded and seeing oneself as an

effective learner and doer of mathematics.

Productive disposition develops together with the other components and helps each of

them develop. For example, as learners build strategic competence in solving problems, their

attitudes and beliefs about themselves as mathematics learners become more positive.

It is important that students regularly have success experiences that strengthen their

confidence as mathematics students. Integrating mathematical games and situations from

real life show learners that mathematics can be fun and is relevant for their lives.

Box: Further reading

https://buildingmathematicians.wordpress.com/2016/07/31/focus-on-relational-under-

standing/

http://www.nixthetricks.com/NixTheTricks2.pdf (pdf in maths resources)

Activity 5

Review the activities per content area in unit 6. Can you find examples of activities that you can use to strengthen each component of mathematical proficiency? Explain your choices.


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Section 3: Mathematical Literacy

Activity 6

What do you understand by mathematical literacy?

What aspects that you have learnt in the course, can you use to develop mathematical literacy

with your learners?

The Organisation for Economic Cooperation and Development (OECD) defines mathematical literacy

as a learner’s capacity to identify and understand the role that mathematics plays in the world, to

make well-founded judgements and to use and engage with mathematics in ways that meet the needs

of that person’s life as a constructive, concerned and reflective citizen (OECD, 2006). Mathematical

literacy is therefore the ability to use mathematics to solve real-world problems or use mathematics

in daily life situations, such as calculating how much you need to pay in the market.

Activity 7

Can you give examples of how you develop mathematical literacy with your learners?

Mathematical literacy is a key learning outcome for all students, alongside literacy. The term “numeracy’ is used as well, which refers to having basic competences in numbers and operations.

Some students struggle to apply knowledge and skills in real life situations, as mathematics requires abstract thinking which can be a difficult transition. Many students also find it challenging to interpret word problems—figuring out exactly what the problem is and identify the steps to find the answer.

Students mustn’t think of mathematics as something that they will only use in the classroom. If students are shown real-world examples of how math is used in our daily lives, this can help to motivate them to make the effort needed to become mathematically literate. Mathematics is everywhere, and it is used in everyday life from cooking, sports, home construction, agriculture, nursing and driving (Figure 6 and Figure 7).


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Figure 6: Examples of mathematics in daily life


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Figure 7: Applications of mathematics in daily life

Source: jovsan fernandes

Activity 8

List situations outside of school during the past month for which you have used mathematics.

--------------------------------------------- ---------------------------------------------

--------------------------------------------- ---------------------------------------------

--------------------------------------------- ---------------------------------------------

--------------------------------------------- ---------------------------------------------


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As a mathematics teacher, you can play an important role by making meaningful connections between mathematics, the real world and other subjects. This will help learners to realize that mathematics is not something that is separated from the real world, but that it is a way to describe the real world.

A lack of confidence or motivation may get in the way of students achieving mathematical literacy. Therefore, it is very important for students to have positive experiences with mathematics from an early age with many opportunities to achieve success.

Source: https://www.oxfordlearning.com/what-does-math-literacy-mean/


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Section 4: Learner-Centred Pedagogy (LCP)

What is Learner-Centred Pedagogy?

Activity 9

Can you give examples from this course of learner-centred pedagogy? Why are they examples

of LCP?

Learner-centred pedagogy is an approach with its origins in constructivist theories of learning.

These theories start from learners’ individual needs, interests, abilities and backgrounds, and aim at

creating an environment where learning activities encourage learners to construct the knowledge,

skills and attitudes either individually or in groups in an active way (Nsengimana, Habimana, &

Mutarutinya, 2017). In learner-centred classrooms, learners co-influence the teaching and learning

process, in contrast to a teacher-led classroom whereby the teacher is fully in charge of the

content, the teaching and learning process. In a teacher-centred (or teacher-led) classroom, it is

only the teacher who has the authority to deliver knowledge, skills and attitudes as if the learners

are empty vessels to be filled. In Rwanda, learner-centred pedagogy is characterized by features

such as discovery approach, active participation of students, and engagement in experimentation

(Nsengimana, Ozawa, & Chikamori, 2014)

There is a misconception that learner-centred pedagogy always means working in groups (Nsengimana

et al., 2017). Learner-centred pedagogy includes a variety of techniques and approaches, including,

but not limited to, group work. In this guide, we will discuss a variety of learner-centred techniques

such as open-ended questioning, games, mathematics conversations and problem-solving activities.


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Table 2: Teacher-centred versus Learner-centred education (Altinyelken, 2010; Schweisfurth, 2011;

Vavrus, Thomas, & Bartlett, 2011; Weimer, 2012)

Teacher-centred education Learner-centred education

Teacher organizes and controls the content

and the teaching and learning process.

Teachers ask the questions, call on learners,

add detail to their answers, offer examples

and do the review

Learners co-control the content and the learning

process. Teachers teach students to reflect,

analyse and critique what they are learning and

how they are learning it.

Techniques like lecturing and whole class

drilling dominate.

Variety of techniques are used (including

lecturing) and include collaborative interactions

between teacher and learners.

Fixed curriculum Room for individual interests, learning preferences

and needs. Students may get some choice about

assignments, classroom policies, deadlines or

assessment criteria.

Teacher is the only authority and source of

knowledge

Teacher is a facilitator of learning. Learner-

centred teachers recognize that students can

learn from and with each other and that teachers

can learn from students as well.

Focus on teaching and covering the

curriculum

Focus on learning by all learners. Teachers

teach students skills such as how to think, solve

problems, evaluate evidence, analyse arguments,

generate hypotheses etc.


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UNIT 3: KEY ASPECTS OF MATHEMATICS INSTRUCTION

Introduction

Based on the main ideas that underlie quality teaching of mathematics (Unit 2), we go in this unit

deeper into key aspects of mathematics teaching. We have divided this unit into 8 sections. In each

section, we introduce one aspect of mathematics instruction. In each section, we start with basic

principles, followed by concrete methods that you can use in your teaching.

Learning Outcomes

By the end of this unit, you will be able to:

Have insight in approaches to build mathematical proficiency with learners;

Apply appropriate and inclusive methods for teaching and learning mathematics;

Support fellow teachers with a focus on learner-centred techniques for mathematics teaching;

Respect of feelings, opinions, people diversity and initiatives of other

Value social justice and sustainability;

Appreciate the need for lifelong learning;

Value collaboration, team work and joined leadership within the school.

Activity 10

Individually, complete the self-evaluation that you can find in Appendix 1. After completing the

self-evaluation, identify for yourself 3 elements of your mathematics teaching that you want to

improve upon.


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Section 1: Questioning

Introduction

Questioning is a key skill for teachers. During an average lesson, teachers ask tens of questions

(Lemov, 2015). But what makes a question effective? And how can you use questioning to stimulate

thinking, collaboration and motivation in your mathematics lessons?

Activity 11

Think in small groups about what a good question means. Make a concept map with criteria of

a good question.

Put the concept maps on the wall, look at the maps from the other groups and discuss areas of

agreement and disagreement.

Figure 8: What makes a good question? Example of a Concept map (VVOB, 2017)


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Asking and answering questions is the essence of learning. Questions are important in a lesson for different reasons. They enable you to involve learners (foster commitment), focus attention on what is important, encourage thinking and exploration and let them develop new ideas, connecting old with new knowledge (Figure 9) (Lemov, 2015; Martino & Maher, 1999).

Figure 9: The importance of questioning (VVOB, 2017)

Unfortunately, many teachers don’t use the power of questioning to stimulate thinking and learning

fully. Upon hearing a correct answer, many teachers are happy to move on. Upon hearing a wrong

answer, they correct it or ask another learner to give the correct answer. Some teachers consider

a wrong answer as something that needs to be avoided as much as possible. Often, teachers move

on without knowing why a learner gave an answer or if anybody else had other thoughts. However,

answers of confident students are a bad guide to what the rest of the class is thinking (Wiliam, 2016).

In this section, we will underline the importance of slowing down and asking further questions no

matter if the response is correct or not. Questions are not only about getting the right answer

from learners but are about developing reasoning skills and the capacity to formulate one’s thinking

accurately.


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The answers of confident students are a bad guide to what the rest of the class is thinking - Dylan Wiliam

Skilful questioning enables learners to “construct” explanations to solve the task at hand, prompting

them to build on and improve their current knowledge. Questioning helps learners to identify

thinking processes, to see the connections between ideas and to build new understanding as they

work their way to a solution that makes sense to them. Research has shown that teachers ask many

questions to check understanding (knowledge questions), whereas they ask few questions to make

learners think (Wiliam, 2016). Good questioning stimulates the ‘student voice’ and reduces the

‘teacher voice’(Burns, 2015).

Example: What would you say in this situation?

3, 12, 21, 30, …

Teacher: What do you think is the tenth number in this pattern?

Student: I think it’s 12

The question above looks like a straightforward growing pattern where 9 is added every time. Many

teachers would react on the student’s answer by correcting the answer. A better reaction is to ask:

“why did you come up with that answer?”. This might reveal unexpected but valuable reasoning. In

this example, the learner may have assumed a repeating pattern with 4 units. The purpose of such

questions is to create a classroom culture where it is safe to share alternative answers or a different

reasoning.

In this section, we will introduce and discuss a variety of techniques to help you with effective

questioning in your lessons.


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Techniques for Effective Questioning

1. Don’t let only learners answer who have raised their hands

When you only let learners who have raised their hands, answer questions, it is easy for other learners

not to be involved (Lemov, 2015). Also, as boys are often more vocal and eager to raise their hands,

you risk giving girls fewer opportunities to answer (Consuegra, 2015). It is better to choose yourself

who answers a question or you can let all learners answer at once raising a card or their hands.

Not focusing on learners who raise their hands (a “no hands” approach) has four advantages:

It allows you to effectively and systematically check for understanding with all learners. You

don’t just check the students who volunteer. You also want to know how the other students

are doing.

All learners need to think and have an answer ready in case the teacher calls on them to

respond. It increases engagement because students don’t know when they will be called on.

It increases the pace of questions and answers. You don’t ask, “Who can tell me how much

is 198 + 65?” and then look around the classroom for hands. You no longer provide hints to

get learners’ participation.

It distributes work more equally among learners. It encourages those students who would

not volunteer, but know the answer, to participate. You also let them know that you value

their contribution. It allows you to make sure that boys and girls get equal opportunities to

answer.

Example: https://www.youtube.com/watch?v=g-SUzv1t78k

2. Have learners use exercise notebooks or voting cards

Exercise notebooks or voting cards on which learners can write answers to questions can be a very

powerful pedagogical tool. After posing the question, the teacher counts down and on ‘zero’ all

learners raise their book or card simultaneously. Such exercise notebooks can be useful resources

because:

When learners hold their ideas up to the teacher, he/she can see immediately what every

learner thinks.


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During class discussions, they allow the teacher to ask different kinds of questions (typically

beginning with ‘Show me . . .’).

They allow learners to simultaneously present a range of written and/or drawn responses to

the teacher and each other, thereby stimulating all learners to think.

Examples of questions that you can use for the exercise notebooks are:

Give me two fractions that add to 1. Now show me another pair of fractions.

Give me a number between 1/3 and 1/4. Now a number between 1/3 and 3/7.

Draw a quadrilateral with two lines of symmetry.

You can use multiple-choice questions, where learners write their response (letter) in the

exercise notebook.

As a follow-up, it can be helpful to write a few of the learners’ answers (anonymously), both correct

and incorrect, on the board for discussion with the whole class. When answers are written on the

board, learners feel less threatened when the answers are criticised by others. This encourages risk

taking. You can let learners vote about what they think the correct answer is and discuss in pairs.

Some teachers also introduce answers that are not given by learners but which bring out some

important learning points (frequent mistakes, misconceptions) that they wish to emphasise (Swan,

2005).

3. Let Learners Vote

This technique gets students to actively think and make judgements about their peers’ answers.

“Stand up if you agree with Alexis” or “Thumbs up if you think Jean-Claude is right.” The answers

will help to inform your teaching, especially if you ask students to defend their answers, “Why is your

thumb down, Gilbert?”

Voting helps students process content and helps a teacher check for understanding (formative

assessment). The technique brings students’ answers to the forefront and keeps them involved.

You can involve all learners by using multiple choice questions and let all learners raise their hands,

for example, one finger for the first answer, two fingers for the second etc. You can ask a learner in

each answer category to justify her answer. You can also use this technique by asking a question and

collecting the various answers from learners.


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A good way to use voting is to let learners evaluate mathematical statements or generalizations.

Learners are asked to decide whether the statements are ‘always’, ‘sometimes’ or ‘never’ true, and

(important!) give explanations for their decisions. Explanations involve generating examples and

counterexamples to support or refute the statements.

Statements can be formulated at any level of difficulty. Some examples of statements:

If you divide a number by 2, the answer will be less than the original number.

If you divide 10 by a number, your answer will be less than or equal to 10.

Numbers with more digits are greater in value

Multiplying makes numbers bigger

When you multiply by 10, you add a zero

You can’t have a fraction that is bigger than one

Five is less than six so one fifth must be smaller than one sixth

Every fraction can be written as a decimal

Every decimal can be written as a fraction

If you double the radius of a circle, you double the area.

Shapes with larger areas have larger perimeters

A rectangle is also a trapezium

if you double the lengths of the sides of shapes, you double the area;

In January, bus fares went up by 20%. In August, they went down by 20%. Michel claims that:

“The fares are now back to what they were before the January increase.” Do you agree?

Throughout this process, the teacher’s role is to:

encourage learners to think more deeply, by suggesting that they try further examples (“Is

this one still true for decimals or negative numbers?”; “How does that change the perimeter

and area?”);


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challenge learners to provide more convincing reasons (“I can see a flaw in that argument”;

“What happens when …?”);

help learners formulate their thoughts in a mathematically correct way;

play ‘devil’s advocate’ (“I think this is true because . . .”; “Can you convince me I am wrong?”).

4. Let students formulate questions

By asking students to look at some information and think of questions to ask each other, they have to

make all sorts of connections to their prior knowledge. Of course, they will likely start with obvious

questions, but with practice they will get more creative. A good practice is to give problems that

have gaps in them and ask learners to help you fill in those gaps. The most interesting problems are

co-developed by teachers and students, not merely assigned by the teacher.

“I like providing students situations with lots of information and asking

students to pose the questions we might solve based on this information

(Bushart, 2016)

For example:

Anita has five oranges, Angelique has 20 oranges and Andrew has 15 apples. Elise has no fruit, but

has 2000 Frw. The price of an apple is 700 Frw, the price of an orange is 300 Frw and the price of an

avocado is 400 Frw. What questions can you make from this information?

5. Not Tennis but Volleyball

This is a variation on the previous technique. When you ask a question and a student answers,

you can stop all momentum by saying “correct,” and move on. But imagine if you say instead, “Oh,

Eduard thinks the answer is 24. John, do you agree or disagree with this answer?” followed by, “Oh,

John says he agrees with the answer of 24. Lydia, why do you think both students are saying the

answer is 24?” The students’ answers pass through you but you immediately pass them on in the

form of a new question to another student in the class. Of course, you don’t have to do this if the

question is simple.

“If I’m teaching P5 graders and for some reason I ask the sum of 12 + 12,

then I’m not going to engage in a lengthy discussion, but if the students are


43

evaluating a situation using concepts we’re currently working on, then you

better believe we’re going to talk it out, and they’re not going to think the

answer is correct because I told them so, but because we built consensus as

a class.” (Bushart, 2016)

This will stimulate learners to listen to each other, think actively about each other’s responses and

develop learners’ reasoning skills.

Here is a video from Dylan Wiliam illustrating the technique: https://www.youtube.com/

watch?v=029fSeOaGio

6. Pose Open Questions

Open questions are questions where the answers are not limited to a few possible answers. They are

often a good way to initiate thinking and start a deeper conversation. An open question encourages

a variety of approaches and responses. Consider “What is 4 + 6?” (closed question) versus “Is there

another way to make 10?” (open question) or “How many sides does a quadrilateral figure have?”

(closed question) versus “What do you notice about these figures?” (open question).

Open questions are questions where the answers are not limited to a few possible answers.

Open questions help teachers build self-confidence with learners as they allow them to respond at

their own level of development. Open questions allow for differentiation, as responses will reveal

individual differences. These may be due to different levels of understanding or readiness, the

strategies to which the students have been exposed and how each student approaches problems

in general. Open questions signal to students that a range of responses are expected and, more

importantly, valued. By contrast, yes/no questions tend to limit communication and do not

provide teachers with as much useful information. Learners may respond correctly but without

understanding.


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Some examples of prompts for open-ended questions are:

How else could you have …?

How are these the same/ different?

What would happen if …?

What else could you have done?

Is there any other way you could …?

Why did you …?

How do you know?

Could you use some other materials to …?

How did you estimate what the answer could be?

Show me an example of…

What is wrong with the statement? How can you correct it?

Is this always, sometimes or never true?

• How can we be sure that…?

Open questions often address higher levels of Bloom’s Taxonomy (Table 3 and Figure 10). Verbs

such as connect, elaborate, evaluate and justify stimulate learners to communicate their thinking

and deepen their understanding.

Table 3: Verbs that elicit higher levels of Bloom’s Taxonomy (Bloom, 1968; Krathwohl, 2002)

Observe Evaluate Decide ConcludeNotice Summarize Compare RelateContrast Predict Connect InterpretDistinguish Justify Explain Elaborate


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Figure 10: Verbs associated with higher levels of Bloom’s Taxonomy (Belshaw, 2009)

Using these kinds of prompts will lead to class discussions about how the solution relates to prior

and new learning. Stimulate mathematical conversations among learners and not only between the

teacher and the learner.

After asking an open question, it is important to welcome and encourage answers, but not

immediately judge them. “Thanks, that is a really interesting answer. Does anyone have something

different?” will generate discussion, whereas “That is a really good answer.” will inhibit discussion,

because learners with alternative ideas tend to remain silent. Therefore, judgements should be

reserved for the end of a discussion.


Resources on Bloom’s Taxonomy: http://larryferlazzo.edublogs.org/2009/05/25/the-best-

resources-for-helping-teachers-use-blooms-taxonomy-in-the-classroom/


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7. Provide Wait Time after a Question

Many teachers are uncomfortable with silence. So, they let the first student who raises the hand

answer. By waiting a few seconds, several things happen (Lemov, 2015):

The length and correctness of students’ responses increase.

The number of failures to respond (“I don’t know”) decreases.

The number of students who volunteer to answer increases. Many students simply need

more time to formulate their thoughts into words.

The use of evidence in answers increases.

Wait time is not as simple as just counting to three in your mind though. The teacher also needs

to tell students why they are waiting, so it becomes waiting with a purpose. Some things a teacher

might say to coach students are:

“I’m waiting for more hands before I take an answer.”

“I’m waiting for someone who can connect this question to what we have seen yesterday.”

“I like all the people I see checking their notes for help with a good answer.”

This can be combined with strategies like turn and talk (see: p. 89) and think-pair-share (see p.

136), which give learners time to clarify and articulate their thinking in pairs or small groups before

answering.

Example: https://www.youtube.com/watch?v=dBnuSUL0ymM

8. Right is right

“Right is right” means that when teachers ask a question, they hold out for a complete answer, or one

that would be acceptable on a test, with that student. Students often stop thinking when they hear

that their answer is “right.” However, many teachers accept answers that are partially correct or not

complete. They affirm these answers by repeating them and then adding information to make the

answer completely correct. The key idea behind this technique is that the teacher should set a high

standard of correctness by only naming “right” those answers which are completely right (Lemov,

2015). If the answer is not completely correct, the teacher should continue asking questions.


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For example,

Teacher: Can you someone give the definition of volume?

Student: The volume is equal to L x W x H

Teacher: That’s the formula. I’m asking for the definition.

In many cases, teachers would praise the answer of the student even though they did not answer

the question directly.

There are four ways to use the “right is right” technique:

1. Hold out for all the way. When students are close to the answer, tell them they are almost there.

While great teachers don’t confuse effort and mastery, they do use simple, positive language

to appreciate what students have done and to hold them to the expectation that they still have

more to do. For example, “I like what you’ve done. Can you get us the rest of the way?”

2. Focus on answering the question. Students learn

quickly that if they don’t know an answer they can

answer a different question, particularly if they relate

it to their own lives.

3. Right answer, right time. Sometimes students get

ahead of you and provide the answer when you are

asking for the steps to the problem. While it may be

tempting to accept this answer, if you were teaching

the steps, then it is important to make sure students have mastered those steps, “My question

wasn’t about the solution. It was, what do we do next?”

4. Use technical vocabulary. Good teachers accept words students are already familiar with as right

answers, “Volume is the amount of space something takes up.” Great teachers push for precise

technical vocabulary, “Volume is the cubic units of space an object occupies.” This approach

strengthens a student’s mathematical vocabulary.

Example: https://www.youtube.com/watch?v=DYZjfEOg-lI


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9. Stretch learners to extend or deepen their answers

Rather than stopping after a student gives you the correct answer, follow up with questions that

extend knowledge and check for full understanding. You can do this by asking students how they

got the answer, what is another way to get the answer, why they gave the answer they gave, how to

apply the same skill in a new situation, or what more specific vocabulary they could use. This both

challenges students to extend their thinking and checks that students don’t get the correct answer

by luck, memorisation or partial mastery. This technique sends the message that learning does

not end with a right answer. This technique is especially important for differentiating instruction

(Lemov, 2015).

Prompts or questions that you can use to stretch your students are:

Asking how or why

Ask for another method to get to the answer

Ask for a better word or a more precise expression

Ask for evidence

Saying “tell me more” or “develop that”

This technique works best when you use it frequently. Avoid using it only when a learner has made

a mistake. Learners will quickly realize that you ask these questions to indicate that the learner has

made a mistake. You should be asking this question regardless of whether the answer is correct or

not.

Example: https://www.youtube.com/watch?v=8P1o8y9ZXWY

Activity 12

Select a topic from the CBC and develop a short (5 minutes) questioning sequence for that

topic, applying techniques for effective questioning.


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Section 2: Mathematics Conversations

Introduction

Mathematics teachers are often concerned with concepts and numbers but lack the language and

argumentation skills to support or challenge learners’ answers. Similarly, learners often get right

answers to problems but cannot explain how they came up with those answers. Although young

children may have a beginning understanding of mathematical concepts, they often lack the language

to communicate their ideas. By modelling and stimulating discussions and paying attention to using

correct mathematical terms, teachers can help students to express their ideas. It is also important

to encourage talk among students as they explain, question and discuss each other’s ideas and

strategies.

Group work and other student-centred methods are less effective when the quality of mathematical conversations in groups is low.

Activity 13

Think individually about following questions:

What elements make a good mathematics conversation?

How can you stimulate your learners to engage in mathematics conversations?

After a few minutes of thinking, discuss your ideas with your neighbour.

Productive conversations are a crucial aspect of mathematics lessons. Group work and other student-

centred methods are less effective when the quality of mathematical conversations in groups is low.

Klibanoff and colleagues discovered that teacher-facilitated “mathematical talk” in the early years


50

significantly increased children’s growth in understanding of mathematical concepts (Klibanoff,

Levine, Huttenlocher, Vasilyeva, & Hedges, 2006).

Students who engage in meaningful mathematical discussions increase their conceptual

understanding and deepen their content knowledge. Students also learn to accept one another’s

ideas. When all students contribute in mathematical discussions, everyone feels that his or her

ideas are welcome.

In this section, we will discuss how to stimulate productive mathematics conversations with your

learners.

Tips for mathematics conversations

1. Revoicing or Paraphrasing

Revoicing or paraphrasing is a technique that is very useful when a student’s explanation is confusing

or hard for others to understand. Revoicing means that the teacher repeats all or some of what the

learner said and then asks for clarification, which in turn provokes the learner to clarify and offer

further explanation. This also gives the teacher an opportunity to embed mathematics vocabulary

in the conversation so the child can further explain his/her thinking (Chapin, O’Connor, O’Connor, &

Anderson, 2009, p.14).

An example of a revoicing response is: “So you’re saying that [it’s an odd number?]”. When revoicing,

the listener repeats part or all the speaker’s words and asks the speaker to say whether the repeated

words are correct (Chapin et al., 2009).

Example: https://www.youtube.com/watch?v=X2Oyhrt0hoU

Revoicing can also be applied by students when engaging in group work or discussing in pairs.

Revoicing provides students with another means of responding to each other appropriately.

2. Repeating and reasoning

You can stimulate mathematical conversations by letting students repeat or reason, based on another

student’s answer. Possible prompts are:

Can you repeat what he just said in your own words?

Do you agree or disagree and why?


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Would someone like to add something more to that?”

What does this make you think of?

What other mathematics can you connect with this?

When do you use this mathematics at home? At school? In other places?

How is this like something you have done before?

3. Asking why

Tell your learners that “because” is the magical word you want to hear in every answer! When they give an answer, they develop the habit of adding “because” and explaining their answer.

Example: Where does 1.6 go on a number line?

‘I draw a number line that goes from 0 to 2, and I say 1.6 goes here.’ (-)

‘I draw a number line that goes from 0 to 2 BECAUSE ……. , and I say 1.6 goes here BECAUSE ….’ (+)

4. Stimulating Precise Use of Mathematical Language

Lack of precise language can impede students’ understanding of a concept and may even lead to the development of misconceptions in mathematics. Using precise mathematical language expands students’ mathematics vocabulary and builds capacity to learn new terms. It will also support them in thinking more carefully about their ideas and their peers’ ideas.

Examples:

1. Multiplying by 10 = adding a 0 to the right.

Why is putting a zero to the right of the unit not good instruction for multiplication? What can be consequences for students’ understanding?

Adding a zero to the right doesn’t work for multiplications with decimal numbers. The same goes for “multiplication makes bigger”. This is also not always true. Better is to say: “multiplication makes bigger when/ if …” “Sharing means less” is also not always true.

These statements are introduced in early grades and cause confusion in later grades.


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2. Meaning of the “=” sign

Learners need to understand that the equal sign shows that quantities on either side of the sign

have the same value. However, students often think it means that they should do something to the

numbers before it and write the answer after it. They often read an equation like 6 + 1 = 7 as “six

plus one makes seven.”

3. “Reducing” Fractions

Many students are confused by statements such as:

Visually, students can see that the value of the digits in both the numerator and the denominator

have become smaller, so it is initially difficult to understand that these fractions are equivalent.

Combine that with the fact that a teacher may be saying, “the fraction 3 sixths can be reduced to 1

half,” and you can see why a student could be confused by the equation shown. Therefore, phrases

like, “rename it in its simplest form” or “rename it using the largest units possible” are more likely to

help students deepen their understanding of fractions.

Figure 11 illustrates the importance of being precise with mathematical language

Figure 11: Importance of correct mathematical language in division operations


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Why is putting a zero to the right of the unit not good instruction for doing multiplications? The rule

doesn’t work with decimal numbers. The same goes for “multiplication makes bigger”. This is also

not always true. It is better to use statements like: “multiplication makes bigger when/ if …”

Activity 14

Think individually about the questions below:

1. Can you find other examples of how unprecise use of language may cause confusion in

mathematics?

2. How do you stimulate precise use of mathematical language with your students?

Techniques to promote the use of precise mathematical language with your students are:

Use mathematical vocabulary yourself correctly and regularly.

Ask students to paraphrase (say in their own words) what you or other students have said,

using the correct vocabulary.

Introduce and model new vocabulary through explanations, examples, and illustrations.

Point to symbols when saying the words that the symbol represents.

Engage your class in discussing and defining terms.

When a student uses a new mathematical word correctly, point it out (for the benefit of the

whole class, not just that student).

Point out when the common definition of a word is different from the mathematical meaning.

Write new vocabulary on a Mathematics Words flipchart and have students keep their own

lists of mathematics words in their exercise books.


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http://www.edu.gov.on.ca/eng/literacynumeracy/inspire/research/CBS_Maximize_

Math_Learning.pdf

http://www.edu.gov.on.ca/eng/literacynumeracy/inspire/research/CBS_AskingEffec-

tiveQuestions.pdf

http://www.nctm.org/Publications/Teaching-Children-Mathematics/2015/Vol22/Is-

sue4/Creating-Math-Talk-Communities/

Activity 15

Select a lesson topic from the CBC and develop a mathematical conversation for that topic,

integrating the tips for mathematical conversations and questioning (section 1).


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Section 3: Developing Problem Solving Skills

Introduction

Being able to solve problems is a key objective of studying mathematics (REB, 2015). Solving

problems is at the core of what doing mathematics means (Burns, 2015). Learning mathematical

rules and facts is important, but they are the tools with which we learn to do mathematics fluently,

not the final objective of teaching mathematics.

Problem solving is about engaging with real problems; guessing, discovering, and making sense of

mathematics (Polya, 1945). For Polya, problem solving is:

seeking solutions not just memorising procedures.

exploring patterns not just memorising formulas.

A three-phase structure for problem-solving lessons (Burns, 2015, p. 135)

A structure in three phases is useful for planning lessons that include problem solving. The three

phases are introducing, exploring and summarizing. Introducing for launching the investigation,

exploring for students to work independently or in groups and summarizing for a classroom

discussion to share results and talk about the mathematics involved.

Phase 1: Introduction

The goal of the introduction is to help learners understand what they are going to investigate and how

they will work. This is best done with the whole class so that everyone gets the same information.

You can follow these four steps when introducing an investigation:

present or review concepts

pose a part of the problem or a similar but smaller problem

present the investigation

discuss the task to make sure that learners understand what they need to do.


56

Phase 2: Exploring

Once learners understand what to do, they engage with the investigation, usually in pairs or small

groups. During the exploring phase, the role of the teacher is:

observe interactions within groups and help learners to get under way with the investigation;

assist groups as needed, either when all members raise their hands or when a group is not

working productively;

provide an extension to groups that finish more quickly than others.

Phase 3: Summarising

The summarising part of a problem-solving sequence is very important and should not be skipped

or shortened. It is crucial for students to reflect on their learning, hear from others and connect

others’ experiences to their ideas. To prepare for the summarising, you can let learners write down

summary statements about their experiences: what they noticed, conclusions they made etc. Use

the summarising discussion to talk about how the solutions can be generalized. Generalising involves

extending a solution to other situations.

Example: introducing division grouping problems (Burns, 2015, p. 392)

Victoria and Sam are about to have a snack. Their mother has baked a dozen cookies. Just as they

are about to divide the cookies, two friends arrive. The, just before the four children begin to eat

cookies, two more friends arrive. And once these six children are about to have their snack, six

more friends arrive. Now, there are twelve children and twelve cookies. The children freeze when

another person arrives, but this time, it is grandmother with a new plate of freshly baked cookies.

Distribute twelve colour tiles or cubes to each pair of students and let them use the tiles or cubes to

represent each stage of the story.

After the work in pairs, summarize the problem, focusing on:

mathematical vocabulary: dividend, divisor, quotient

relation between multiplication and division. Point out the related multiplication equation

for each problem. For example, if only 2 children share all twelve cookies, they would each

have 6 cookies, which can be represented with multiplication by 2 x 6 = 12. Make sure to

read this as “two groups of twelve” and as “two times twelve”.


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Example: the consecutive sums problem (Benson et al., 2004; Burns, 2015)

The consecutive sums problem is a good example of a problem-solving activity because it challenges

learners to investigate patterns between numbers, make hypotheses, test theories and communicate

ideas.

During the introduction phase:

1. Present or review concepts: review consecutive numbers.

2. Pose part of problem or similar but smaller problem: use a question such as, who can

think of a way to write the number 9 as the sum of consecutive numbers? Record on the

board, and underneath it, write another equation:

9 = 4 + 5

9 = 2 + 3 + 4

This shows that it is possible to write 9 as the sum of consecutive numbers in at least two different

ways. You may introduce another example such as 15, which can be represented as the sum of

consecutive numbers in three ways:

15 = 7 + 8

15 = 4 + 5 + 6

15 = 1 + 2 + 3 + 4 +5

3. Present the investigation: ask learners in groups to find all the ways to write each

number from 1 to 25 as the sum of consecutive numbers (addends). Tell them that

some of the sums are impossible and challenge them to see if they can find a pattern

of those numbers. Challenge them to find other patterns as well, such as how many

different sums there are for different numbers. Ask groups to write down their findings,

equations and patterns.

4. Discuss the task to make sure that learners understand what they need to do.


58

During the exploration phase:

1. Observe the interactions in the groups, how members divide tasks and what strategies

they are using.

2. Assist where needed or when they are headed the wrong way. For example, groups

sometimes make erroneous generalisations. They discover that it is impossible to write 2

and 4 as the sum of consecutive numbers and they conclude that 6 would fit the patterns

and also be impossible. In such a situation, confront them with a contradiction. Ask the

group to consider 1 + 2 + 3. When they realise that the sum of those numbers is 6, you

can leave them to rethink their work.

Questions you can ask to stimulate thinking in groups are:

How could you describe the pattern of numbers that are impossible to write as

the sum of consecutive addends?

What do you notice about the numbers that had three possible ways?

Which numbers had only one possible way?

Which numbers cannot be written as the sum of consecutive numbers?

During the summarising phase:

1. Make sure that groups are prepared to report in the classroom discussion.

2. Initiate a classroom discussion about the findings:

Start with asking how groups organized the work.

Ask what strategies they used. Some groups may have used the guess and check

strategy to find ways that worked. Other groups may have started from writing

consecutive addends such as 2 + 3 + 4 and then write each expression under the

appropriate sum. It is important to discuss different strategies so that learners

become aware that there is often a variety of ways to approach a problem.

3. Have groups report their results, explaining their reasoning or strategies. Discuss any

differences and similarities in the solutions.


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4. Generalise from the solutions. Ask one group to share one of their statements. Other

groups share whether they have found the same statement. Then, a second group

shares a statement and so on.

Questions you can ask to stimulate learners to generalise:

which sums were impossible to write with consecutive addends?

what patterns did you notice for sums that you could write in two ways? And

three ways? Four ways?

what patterns did you notice for sums that could be written as the sum of two

addends? And three addends?

what do you notice about sums that are prime numbers?

how you can you link the problem with what you know about multiplication?

More info about the consecutive sums problem:

https://nrich.maths.org/summingconsecutive/solution

http://mathpractices.edc.org/pdf/Consecutive_Sums.pdf

Example: the three squares problem (https://nrich.maths.org/143/note)

This problem helps learners to reinforce their understanding of the properties of a square. The

interactivity enables learners to access the task immediately, so they can easily begin to explore.

This in turn means that they are much more likely to become curious about the challenge of finding

as many squares as possible, so are motivated to work mathematically. The interaction not only

supports the exploratory nature of the problem, but also helps to deepen children’s understanding

of what makes a square a square.

What is the greatest number of squares you can make by overlapping three squares of the same size?


60

During the introduction phase:

You could begin by using the interactivity to arrange just two squares in different ways and asking

children to count the number of squares made in each case. You may ask learners to review a square’s

properties.

Once they are familiar with the idea, introduce the main problem and suggest they work in pairs.

You can provide square frames cut from paper/card or made using construction equipment/straws.

It would also be useful to have squared paper available for recording.

During the exploration phase:

Similar to the previous example, you observe interactions in the groups and assist where needed.

As an extension, some children could try using four squares in the same way, or they could use

equilateral triangles instead.

During the summarising phase:

Talk with learners how they went about solving the problem. Did they record as they went along? If

so, what and why? You may find that some learners drew an arrangement so that they could count

the squares more easily by marking in colour. Others might have recorded an arrangement as a

reminder of the largest number of squares they had found so far.

How many squares can you make by overlapping two large squares?

How do you know that is a square?

Can you move the large squares so that you create more squares?

How do you know that it isn’t possible to make more squares?

Example: Measuring round things (Burns, 2015, p. 152)

The goal of this activity is to make learners familiar with the relation between the circumference and

area of circular objects.

For this investigation, learners need a range of circular objects (plates, glasses, jar lids, circles cut out

of paper…), a ruler or measuring tape and a string for measuring the circumference.


61

Choose a circular object and demonstrate measuring its diameter and circumference in several ways:

measuring with a string, using a measuring tape and rolling he objects over a piece of paper to mark

the length of one rotation.

Draw a table on the board with columns: object, diameter, circumference.

Organize learners in small groups and give following instructions:

draw a table as shown by the teacher.

choose a circular object and measure its circumference and diameter.

record the results in the table.

measure at least 5 objects.

review measurements to look for patterns that describe the relationship between the

diameter and circumference of each object.

After the group work, bring learners together for a class discussion. At some point, focus learners

on looking at the result of dividing the circumference by the diameter for each circle they measured.

Next, connect the investigation to pi. Let learners calculate the circumference of any cIrcle by its

diameter and add the value in the table. Explain that when you divide the circumference of any

circle by its diameter, the result is always a little more than 3. Another way to say this is that pi and

the ratio of the circumference to the diameter of the circle. This holds true for all circles, no matter

how small or large.

Box: More ideas for problem solving activities:

https://nrich.maths.org/primary

http://mathpractices.edc.org/browse-by-mps.html

Activity 16

Develop a short teaching sequence with questioning and mathematical conversations for one of

the three examples of problem solving that are discussed in this section.


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Word Problems

The main method to introduce problem solving in primary mathematics is through word problems.

They are often perceived as difficult by learners because of the combination of language and

mathematics skills they require. The best way to prepare learners for word problems and to develop

problem solving skills is to expose them regularly to various types of word problems. In this section,

we discuss strategies for getting the most out of word problems.

Word problems help students to connect situations to arithmetic operations. As such, they help

students understand the meanings of addition, subtraction, multiplication and division. However,

in real-life problems, students will rarely have all the information they need in a nice package in the

way most word problems provide. Instead, students need to collect the data themselves and there

is often more than one possible method.

An effective way to develop mathematical understanding is to present word problems as authentically

as possible. Authenticity does not just mean that the context for the word problem comes from

children’s daily lives (e.g. dividing candy, buying milk). Authentic word problems have a different

structure. All key information is not necessarily included into the problem. In other cases, too much

information is available, and students need to select what is relevant to solve the problem. Because

of this, students engage with the problem by asking questions, testing ideas, and organizing what

they think they need to know.

Almost any question can be used for problem solving (Lemov, 2015). However, a routine problem for

one class, may be new for another class. Something challenging for one student can be familiar for

another. Therefore, it is important to start from the prior knowledge of students and differentiate.

The main criterion of a good word problem is that it should be non-routine to the student.

There are two wrong ways of using word problems. In the first case, a teacher presents learners with

a word problem that has all the necessary information already included in it. The learner must read

the problem, extract the key numbers and solve (and repeat the same process with the next word

problem). Sometimes, this is called the “cookbook” way of solving word problems.

In the second case, teachers provide minimal guidance and let learners struggle with the problem.

Some teachers think that fostering struggle with learners helps them with learning. In fact, there

is no evidence to support this (Hattie, 2009; Sweller & Cooper, 1985). On the contrary, there is

research that shows that this is demotivating for learners and increases inequality by favouring

stronger learners.


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Strategies to develop problem solving skills of learners

Let’s now look at some strategies to promote learners’ problem-solving skills.

1. Stimulating learners to develop and use a variety of problem-solving strategies

When learners use interesting or useful strategies to solve a problem, teachers should acknowledge and discuss them with the whole class. Explicitly describing and labelling a strategy is a useful way for learners to talk about their methods, learn methods from each other and for you, as a teacher, to provide suggestions. This strengthens student’ belief that their contributions are valuable and that there may be several strategies to solve a problem.

Box: Example problem solving strategies

Show all the way that fifteen objects can be put into four piles so that each pile has a different

number of objects in it.

possible or reasonable strategies?

which strategy or combination of strategies will you use first?

did you change strategies or use others as well? Describe.

Box: Example problem solving strategies

Marie and David are playing a game. At the end of each round, the loser gives the winner a

coin. After a while, David has won three games and Lisa has three more coins than she did

when she began playing the game. How many rounds did they play?

possible or reasonable strategies:

which strategy or combination of strategies will you use first?

solution:

did you change strategies or use others as well? Describe.


64

These are strategies that learners may use during problem solving (Burns, 2015):

Drawing a picture, using a model.

Looking for a pattern.

Making a table or chart.

Using examples to find a general rule.

Trying a simpler form of the problem (e.g., with smaller numbers). By solving the easier

problem, learners may gain insights that can then be used to solve the original problem.

Try out a possible solution and check if it is correct. Next, narrow down possible solutions

and check again.

2. Let learners ask questions and develop the problem

You can start a lesson by posting a sentence on the board and ask learners to record the missing

information to solve the problem. Only after all students have participated and understand the

scenario thoroughly do you reveal the question.

Leaving out the question increases participation from struggling students because there is no

right answer and no wrong observations. It keeps fast students engaged in creative brainstorming

rather than closed-ended problem solving. And having a question to solve that students generated

increases all students’ understanding of the task and their engagement.

Source: http://mathforum.org/workshops/universal/documents/notice_wonder_intro.pdf

For example, consider the word problem below.

“Francis has 5 boxes of chocolate bard for his class. Each box has 6 chocolate bars. How many

chocolate bars are there altogether?

You can transform the problem into the following:

“Francis has boxes of chocolate bars to share with his class.”

How can you use this statement to have a productive mathematics conversation with your students?


65

Some possible reactions from learners:

“We don’t know how many boxes of chocolate bars there are.”

“There isn’t enough information to know what’s going on.”

“We don’t know if it is adding, subtracting, multiplying, or dividing.”

“There are multiple people in the class, because it says boxes and share.”

“How many chocolate bars are in each box?”

“How many boxes did he bring to class?”

“How many kids are in his class?”

“What kind of chocolate bars are they?

More information: http://www.teachingchannel.org/blog/2016/04/07/math-word-problems/

3) Moving word problems to the start of the lesson

A strategy which is easy to apply is to shift a problem to the beginning of the lesson. Choose a

challenging question that includes all theory that you want to teach and start the lesson with that

question. This question serves as a kind of key question for the lesson. Let students try out and find

the missing information or at least have a clear(er) idea of what knowledge they are still missing. In

this way, students will realize why certain knowledge is useful and it will help them to connect new

knowledge to prior knowledge. As a teacher, you will get useful information with this technique

about what students find difficult and you could give these aspects more information during the

lesson.

For example, in a lesson on algebraic reasoning, you could start with this problem:

Imagine that you are at a huge party. Everyone starts to shake hands with other

people who are there. If 2 people shake hands, there is 1 handshake. If 3 people

are in a group and they each shake hands with the other people in the group,

there are 3 handshakes.

How many handshakes are there if there are 4 people? 10 people? Can you develop a rule to help

you figure this out for any number of people?


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Activity 17

Using one of the techniques described above, develop a word problem that you can use in your

class to develop learners’ problem-solving skills.

After 15 minutes, you will receive someone else’s problem to review.

Box: Further Information

Practical ideas for problem solving activities in primary mathematics:

https://nrich.maths.org/10367

http://www.teachingideas.co.uk/subjects/problem-solving


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Section 4: Addressing Learner Errors and Misconceptions

Introduction

In this section, we will show how you can use errors that students make to improve your teaching

and learning. The reaction of many teachers when students make an error is to correct it as quickly

as possible. However, errors provide teachers with valuable information about students’ thinking

and can be used as starting points for powerful teaching.

Learners as well need to see errors as opportunities for learning, not as things to avoid. They must

feel that it is okay to offer an idea that might be incorrect and know that they will have the support

of their teacher and fellow learners to resolve errors in their thinking.

Activity 18

Edouard wrote 7.10 in the empty box on the number line below. Why would he write this? Describe how you could help Edouard to find the correct answer.

Errors and misconceptions

Not all errors are the result of misconceptions of learners, for example calculation errors. Misconceptions are conceptual structures constructed by learners that make sense to them in relation to their current knowledge (Brodie, 2014). Many studies have identified misconceptions for a wide range of mathematical topics and showed how these make sense to learners (Smith III, Disessa, & Roschelle, 1994). This is a key characteristic of misconceptions: from the student’s point of view, they make perfectly sense. Many misconceptions arise from learners’ overgeneralisation of a concept from one domain to another (Smith III et al., 1994). For example, from their knowledge of natural numbers, many learners think that adding always means more and that multiplication by 10 is equal to adding a zero to the right. Mathematical knowledge that works in one domain (e.g. natural numbers) does not necessarily work in new domains as well (e.g. decimals and fractions).


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For example, this example of student writing is extracted from the work of Ball and colleagues

(2008).

For example, this example of student writing is extracted from the work of Ball and colleagues

(Ball et al., 2008).

Can you identify the most likely cause of the error in the above learners ’work? How could you

address this error?

Activity 19

In this exercise, you will practise recognizing some common misconceptions in numbers and operations. For each exercise in the table below, think about an incorrect answer that you have seen your learners make and that reflects a misconception.

1. 3 + __ = 7 2. 35 + 67 -------

3. 42 - 17 -----------

4. 300 - 136 -----------

5. 3840 : 12 (long division) 6. 1/2 + 2/3 =

7. 2.06 + 1.3 + 0.38 = 8. 5.40 X 0.15 ------------------


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Following errors and explanations are examples of students’ incorrect thinking (Burns, 2015).

Error as a result of a

misconception

Underlying misconception

1. 3 + 10 = 7 A plus sign means to add.

2. 35

+ 67

-----------

912

Add the numbers in each column and write the sums under the

line

3. 42

- 17

------

35

When you subtract, you take the smaller number from the larger

4. 300

- 136

---------

163 OR 174

You can’t subtract a number from zero, so you change the

zeros into nines OR You can ‘t subtract from zero, so you

borrow from the three and the zeros become tens.

5. 3840 : 12 = 32 You can drop the zero at the end of the problem.

6. 1/2 + 2/3 = 3/5 When you add fractions, you add across the top and across

the bottom.

7. 2.06 + 1.3 + 0.38 = 2.57 Line the numbers up below each other and add.

8. 5.40 x 0.15 = 81 After you solve the product, bring down the decimal point.

Thinking about possible causes for these mistakes is important for teachers. When learning about

numbers and operations, students focus on the numbers and symbols in the problems and see

mathematics as doing something with them to get the right answer. Learners who make these

errors are not focusing on the meaning of the problem but on the symbols in the problem. For

example, they learned that a plus sign means to add, so they combined 3 and 7 to get the sum

instead of figuring out how much more was needed to add to 3 in order to get 7. However, the

same error does usually not occur when the same problem is presented as a word problem. For

example, you have 3 candles, but you need 7 altogether. How many more candles do you need?

Students generally interpret this problem correctly and find that they need three more candles. As

a teacher, it is important that learners can explain the meaning of a mathematical problem and not

just perform the procedure.


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Activity 20

In pairs or small groups, you will receive a curriculum topic from the list below. Come up with

two frequent errors that your learners make about that topic that reflect a misunderstanding

rather than a calculation error. For each misconception, consider why is the chosen example is

a misconception.

Topics:

1. Place value and number sense

2. Addition and subtraction

3. Multiplication and division

4. Fractions

5. Decimals and percentages

6. Geometry

7. Probability and statistics

There are many misconceptions in the understanding of mathematics. The following misconceptions are common in numbers and operations.

1. Rounding numbers: When asked to round a value to the nearest 1000, some students mistakenly round to the nearest 10, then the nearest 100 and finally to the nearest 1000.

2. Multiplication: Many students hold an idea that multiplication always increases the size of a number.

3. Multiplying decimals: Mathematical language can be a source of misconceptions. For example, the term “times” is mixed up with “of”, thus one-tenth of one-tenth is equal to one-hundredth.

4. Decimals and their equivalent fractions: There a misconception that decimals and fractions are different types of numbers while most fractions can be expressed with denominators of 10, 100 or 1000 to find their decimal equivalent.


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5. Dividing whole numbers by fractions: Many students think that dividing a whole number decreases its size which is not always the case depending on the type of fraction. For example, dividing three by one quarter, as the number of quarters that fit into three.

6. Adding with negative numbers: To help correct misconceptions regarding how to solve a calculation involving negative numbers, the term taking away is used to represent minus and giving back for plus. Thus -8 +6 is read as taking away 8 and giving back 6, which is equivalent to taking away only 2. Taking away conveys only one interpretation of subtraction. Subtraction as the difference between two numbers is another one.

Source: https://www.stem.org.uk/elibrary/resource/32755

Box: Further Reading: more information on mathematics misconceptions:

http://www.westada.org/cms/lib8/ID01904074/Centricity/Domain/207/Misconceptions_

Error%202.pdf

This document (40 pages) gives an overview of common misconceptions per topic (numbers

and operations, fractions, geometry, measurement, probability…).

https://morelandnumeracyaiznetwork.wikispaces.com/file/detail/

COMMON+MISCONCEPTIONS240810.ppt

This document (presentation) contains an overview of common misconceptions in geometry

and measurement.

Techniques to address learner misconceptions

Misconceptions and the errors they produce cannot be easily “removed” or “replaced” through instruction since they make sense in the light of the learners’ current knowledge. As misconceptions arise in the connections between different ideas, teachers’ best strategy of dealing with them is to understand and deal with these connections, rather than to re-teach concepts (Brodie, 2014).

1. Waiting to give the correct answer, followed by Turn and Talk

A simple technique to address misconceptions is “Withholding the answer”. When learners make an error, many teachers immediately correct the learner by asking another learner or by giving the correct answer themselves. By doing this, they miss the opportunity to use such errors as “teachable moments”. Teachable


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moments are moments in a lesson that provide great opportunities for learning. Errors can expose valuable information for the teacher, such as misconceptions or incomplete understanding.

Write all answers that learners give on the blackboard. Then, let learners discuss the question in pairs for a few minutes. This is called “Turn and Talk”. Next, you may let learners vote on what they think is the correct answer. Let learners with different answers explain their reasoning and through questioning, guide them towards the correct answer. Such a discussion will give you information about how many learners have the wrong understanding and what misconceptions are present. Of course, you don’t need to do this for every error a learner makes, only for those that reflect deeper misconceptions.

Source: https://onderzoekonderwijs.net/2016/12/10/teach-like-a-champion-8-culture-of-error/

2. Normalize learners’ errors

Too often children won’t answer a question because they are afraid of being wrong. Where did they

learn that? Most of us learn that making mistakes is a part of learning, and the more risk you take,

the more likely you will succeed. So, this technique has two parts: a part for the wrong answer and

a part for the right answer.

Dealing with wrong answers

Avoid blaming students for wrong answers: i.e. “We went over that last week. I can’t believe you

don’t know that.” Don’t excuse the mistake either, i.e. “That’s okay Agnes. You’ll probably get it right

next time.”

You want students to know that it is normal to get an answer wrong, just as it is to find the right

answer. In fact, if errors are a normal, healthy part of learning, then they don’t need much attention.

Spend less time naming “the wrong” and more time on moving on to getting the right answer. You

might just respond “not quite,” and go back. “What is the first thing we need to do to find out how

many each person took home? That is right, find the total.” If you leave the response ambiguous,

students will be eager to find out the correct answer.

3. Analyse the root causes of students’ mistakes

An important skill as a mathematics teacher is to recognize which learner errors reflect deeper

misconceptions. For such errors, it is not enough that students know that they have made a mistake.

They also need to receive feedback on where the mistake lies. Discussing explicitly the root causes

of their mistakes with targeted support is the best way to change students’ mental frameworks and

prevent students from making the same mistake again.


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Activity 21

Which of the strategies above have you used already? What are your experiences? Which will

you try out in your lessons? Do you use other strategies to expose and correct misconceptions

with your learners?


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Section 5: Connecting Concrete, Pictorial and Abstract Representations of Mathematical Concepts

Introduction

Concrete materials are important tools for helping children make sense of mathematics. They can

support learning and be effective for engaging students’ interest and motivating them to explore

ideas (Burns, 2015; Carbonneau, Marley, & Selig, 2013). However, they are not a miracle solution.

Just handing out manipulatives to learners will not make any difference. It is important to have a

good insight in how concrete materials can help children learn (Van de Walle et al., 2015, p. 30).

Mathematics is a language that uses many representations of ideas. Because of the abstract nature

of mathematics, people access mathematical ideas through the representations of those ideas in

symbols (National Research Council, 2002). There is no inherent meaning in symbols. Symbols

always stand for something else. The meaning a symbol has for a child depends on what the child

knows and understands about the concepts the symbol represents (Richardson, 2012).

The following symbols may have absolutely no meaning to you. They are inaccessible.

你好嗎?

我很高興跟你見面

You can compare mathematical symbols with Chinese characters. Without knowledge of the Chinese

language, the symbol does not mean anything to you.


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Concrete – Pictorial – Abstract Sequence

Activity 22

Observe the picture showing different representations of seven. Then discuss with your colleagues the given questions.

Figure 12: Multiple representations of seven (bstockus)

What do these representations say to you about the meaning of the number 7?

Do they all represent the same thing about the number seven?

Do some representations give you different understandings than others?

How many different things can you learn about the number seven from these represen-tations?

How could you make this activity suitable for learners with visual impairments?

For example, the idea of 7 is represented by the symbol “7”. How does the symbol “7” convey the

meaning of seven?

The symbolic form of this number does not say anything about the number seven. Even if someone

told you this is the number seven, what that means to you will vary depending on what you already


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understand about that number. Just being able to see this symbol and say the word, “Seven,” does

not necessarily mean someone understands anything about the number seven or the quantity it

represents.

Here are some things these representations may say:

7 can be made with combinations of smaller numbers: 1 and 6, 2 and 5, 3 and 4.

You may see a specific combination within a representation, like 4 and 3 in the domino or 5

and 2 in the math rack. After spending time looking at them, do you start to notice multiple

combinations within some representations? The teddy bears show 4 and 3 if you look at the

rows. However, you might also see 6 and 1 if you look at the group of 6 with 1 teddy bear

hanging off the end.

You may also see that 7 can be made with combinations of more than two numbers: 3, 3, and

1 for example as shown in the matches and the teddy bears.

The number track shows you where 7 is in relation to other numbers. You can see that 6 is

just before 7 and 8 is just after 7.

You can see how 7 is related to 10. The math rack, number path, and fingers all show that 7

is 3 less than 10.

This is a short list of ways how the meaning of 7 is conveyed to demonstrate that the more

representations of 7 you give students access to, the more robust their understanding of the number

7 will become. The same applies for any mathematical concept.

Teachers need to ensure that they are providing students access to these concepts via multiple and

varied representations and don’t rush to the use of a symbol. Without a range of representations,

a symbol does not make sense to learners. There is nothing inherently more mathematical about

a symbol like 7 than a collection of dots on a domino or seven fingers on my hands. What numeric

symbols allow for is efficiency of representing a quantity, especially once the place value system

comes into play. But that efficiency is lost on students, especially those who struggle, if they do not

have a solid foundation in the concepts the symbols represent.

Students with learning disabilities may be weaker in their use of some representations. For these

students, it is especially important to use multiple representations. For learners with visual

impairments, representations that include touch (using concrete objects) or sound (tapping with

the hand) can be used.


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The purpose of teaching through a concrete-to-semi-concrete-to-abstract sequence of instruction

is to ensure students develop a deep understanding of the mathematical concepts. When students

are supported to first develop a concrete level of understanding for a mathematics concept, they

can use this foundation to link their conceptual understanding to abstract mathematics learning

activities. Having students represent their concrete understandings by drawing simple pictures

that reflect their use of concrete materials provides them with a supported process for gradually

transferring their concrete understandings to the abstract level.

Concrete. A mathematical concept is first modeled with concrete materials (e.g. chips, cubes,

base ten blocks, beans, pattern blocks). Students are provided with many opportunities to

practice and demonstrate mastery using concrete materials.

Semi-concrete. The mathematical concept is next modeled at the semi-concrete level,

which involves drawing pictures that represent the concrete objects previously used (e.g.

tallies, dots, circles…). Again, students are provided with many opportunities to practice and

demonstrate mastery by drawing solutions.

Abstract. The mathematical concept is finally modeled at the abstract level, using only

numbers and mathematical symbols. These numbers and symbols are explicitly linked to

the semi-concrete representations, so that learners can clearly see what the abstract

representations means. Students are provided with many opportunities to practice and

demonstrate mastery at the abstract level. If necessary, they can return to the semi-concrete

or concrete levels to develop further conceptual understanding.

When learners fail to understand a concept, the primary intervention is often to reteach procedures

and give additional practice in the hope that the learner will understand. When learning mathematics,

and especially complex concepts, it is crucial that students have an opportunity to explore multiple

representations of this concept, starting with concrete materials. The (extra) time invested in sense-

making experiences at the pre-formal level will substantially reduce the time needed to reteach and

practice at the formal level.

More formal (abstract) representations build on informal (concrete) ones and pictorial (semi-

concrete) ones. A student should be able to revisit representations, especially when new and

unfamiliar contexts are encountered. In fact, it is perfectly possible that some learners may make

the step to formal representations, but still can solve problems using concrete or semi-concrete

representations.


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Example:

Concrete Pictorial/ Semi-concrete Formal/abstract

7

Example: introducing fractions using real-world contexts

Learners need many opportunities to talk about fractional parts, work with concrete materials and relate their experiences to the mathematical notation (Burns, 2015, p. 418). It is important that fractions as parts of a whole and fractions as parts of sets are introduced.

Use a variety of objects: a bunch of 7 bananas, a set of 12 beans, 5 plastic bottles, a set of 8 bottle caps, some red and some white, and ask questions such as:

what fractional part is one banana? Two?

what fractional part are 2 plastic bottles?

what fraction of the bottle caps is red?

what fraction of the learners are boys? and girls?

what fractional part is 4 beans? and 6?


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Example: introducing graphs (Burns, 2015, p. 174)

In the early grades, graphs are best introduced with concrete objects. A pictorial representation

of that relationship can be introduced at a later stage and still later, a symbolic representation can

be made. The possibilities for things to make a graph of should be taken from the interests of the

learners and can draw on experiences that occur in the classroom.

Figure 13: Three main types of graphs (Burns, 2015)

Real graphs use actual objects to compare and build on learners’ understanding of more and less.

Topics that you can use to make a concrete graph are:

colours of counters or any other materials

shoes with and without laces

male and female learners

year of birth of learners

month of birth of learners

Picture graphs use pictures or models to represent real objects. Examples include circles representing

counters, drawings of shoes and symbols of people to represent learners.

Symbolic graphs are the most abstract because they use symbols, such as a coloured square or a

tally mark, to represent real things.


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A variation on introducing graphs is to use a small paper bag and ten tiles or counters in different

colours, for example red and blue. Tell learners how many objects there are in the bag, but not

how many of each colour. You can put seven or eight of one colour in the bag. Ask learners to take

an object from the bag without looking, note its colour, then replace it. Have a learner record the

colours on the board using different representations (Figure 14).

Figure 14: Representation of tiles in the bag activity (Burns, 2015)

After a few drawings, ask learners of which colour they think that there are more objects in the bag.

Ask them whether they are sure and when we can be sure about the answer.

An important aspect of a lesson on graphs is the discussion and interpretation of the information.

You can use following questions to discuss:

which column has the least/ most?

are there more/ fewer …?

how many more/ fewer are there …?

Example: Fraction Kits

A fraction kit introduces students to fractions as parts of a whole. Learners can develop their own

fraction kit.

To make a fraction kit, you need five strips of paper (approx. 7 cm x 40 cm). If possible, use thick

paper such as Manila paper and use for each strip a different colour or let learners colour each strip.

Each learner should have an envelope to keep the strips.

Give each learner a set of five strips and provide directions to cut and label them. They leave one

strip whole and cut the others into halves, fourths, eights and sixteenths. Decide on which colour to

use for each strip so that all the fraction kits are the same.

Choose a colour and model for the learners how to fold it in half, open and label each sections 1/2,


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cut on the folds so they have two pieces and write the learner’s initials on the back of each piece.

This will be helpful when pieces get misplaced. Review the rationale for the notation ½ by explaining

that they divided the whole into two sections of the same size, that each piece is one of the two

sections and that ½ means one of two equal pieces. Next, choose a colour for the second strip and

model for the students how to fold it in half and then half again, open and label each section ¼, cut

on the folds so they have four pieces and write their initials on the back of each piece. Talk about

each piece being one of four, or one-fourth. Next, repeat the process for 1/8 and 1/16.

Each student now has a fraction kit to use. Having learners cut and label the pieces helps them

relate the fractional notation to the concrete materials and compare the sizes of the fractional parts.

They can see that ¼, for example, is larger than 1/16 and they can measure to prove that 2 of the

1/8 pieces are equivalent to ¼.

You can extend the set with other fractions, such as 1/3, 1/6, 1/9 and 1/12. You can make a fraction

die with the faces labelled 1/2, 1/3, 1/4, 1/6, 1/6 and 1/12 to play Cover and Uncover and record

equations (see: Section 6 Games).

Figure 15: Example of a fraction kit (Burns, 2015)

Example: Growing Patterns

You can find a description of the activity in unit 6, section 3 (Elements of Algebra).


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Activity 23

Think individually about the questions below. After about 10 minutes, discuss your ideas in

small groups.

Identify existing teaching resources for mathematics in your school. Classify these re-

sources (physical/concrete, pictorial).

Which resources do you find particularly useful and why?

What resources can you make yourself or find in your environment?

Activity 24

Work in small groups and review the resources listed in

Table 4. For which lessons are these materials particularly useful? Do you use other materials?

Give examples of how you use these or other materials, as in the example below.

Examples of low-cost resources

There are many low-cost resources that are useful for the primary mathematics teacher.

Table 4: Examples of physical models to illustrate mathematical concepts (Van de Walle et al., 2007, p. 32; adapted by VVOB)

Countable objects can be used to model

“number” and related ideas such as “one more

than”. They are useful to explain place value

and decimals.

Base-ten concepts (ones, tens, hundreds) are

frequently modelled with strips and squares.

Sticks and bundles of sticks can also be used.


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“Length” involves a comparison of the length

attribute of different objects. Rods can be used

to measure length.

“Chance” can be modelled by comparing

outcomes of spinners with various colours.

Number track/path: no zero, shows counting

numbers, ideal for young children because

it shows ‘distinct steps’ that they can count.

Precursor to the number line.

Fraction kits are useful to help learners

understand the relative sizes of fractions.

Cards in different colours and shapes can be

used to introduce patterns

Toothpicks (or mud sticks) and rubber bands

can be used to teach place value


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ICT

There is a wide variety of ICTs available for the mathematics teacher. Discussing them in detail is

beyond the scope of this manual. Roughly, ICT resources can be classified in two categories:

Tools that help you prepare a lesson;

Tools that you can use while teaching.

In the Rwandan context, the former category is probably the most feasible in the short term. Apart

from general search engines and video sites, there are some dedicated repositories of interesting

mathematics content.

Technology offers us many exciting new resources that engage and motivate learners to work on

mathematics.

There are many reasons for using ICT in mathematics lessons:

They are interactive. They enable learners to explore situations by changing something on

the screen and observing the effect.

They provide instant feedback. Learners can immediately see

the consequences of decisions they make. This makes them very

useful for formative assessment.

They are dynamic. Learners can visualise concepts in new ways. For example, they allow

graphs or geometrical objects to be generated and transformed.

They link the learner with the real world. For example, real data may be downloaded and

used in sessions.

Khan Academy

Khan Academy contains lots of free mathematics instructional videos. These are particularly useful

for teachers to revise their content knowledge and to get inspiration for explanations and examples

to use in their lessons. The site also contains self-assessment tests.


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Animations

The Nrich website (https://nrich.maths.org/) from the University of Cambridge contains a wide

range of freely available animations for primary and secondary mathematics. You can search per

topic on: https://nrich.maths.org/public/leg.php .

Many online animations are designed to instil curiosity and interest for mathematics with children,

for example: https://nrich.maths.org/7044.

Activity 25

Develop a teaching sequence in which you move from a concrete to a pictorial stage and

introduce the abstract concept. Pay attention to:

Explicitly linking the different stages;

Differentiation: some learners may need more time or opportunities in the concrete or

pictorial stage.

Use good questions (open questions, thinking questions) and conduct mathematical

conversations.


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Section 6: Games

Introduction

Games can be very useful to capture students’ interest and provide alternative ways for engaging

them in learning mathematics. Games are also ideal for letting students work independently and

productively. Games can address various skills such as strengthening procedural knowledge, strategic

thinking and creativity.

Good mathematics games for the classroom are:

Easy to teach

Accessible to all students

Reinforce understanding and/or provide practice

Encourage strategic thinking

Rely only on a few materials in addition to paper and pencil

Can be played in different versions for differentiation

Examples of Games

1. Four Strikes and You Are Out (Burns, 2015, p. 89)

This game helps learners to practice numbers and operations. You can play the game at different

levels, choosing numbers and operations that are appropriate for the level of your class.

First, explain the game by playing it with the whole class.

Write on the blackboard:

___ ___ + ___ ___ = ___ ___ 0 1 2 3 4 5 6 7 8 9

Explain that each blank contains one number and the purpose of the game is to find the numbers

in the problem. A learner guesses a number and if it is in the problem, you write it in all the places

where it belongs. If the learner guesses a number that is not in the problem, he/she gets a strike.


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For example, take the sum, 35 + 10 = 45

The first learner guesses 3, so the teacher writes:

3 __ + ___ ___ = ___ ___ 0 1 2 3 4 5 6 7 8 9

The next learner guesses 2, so the teacher writes (X means 1 strike):

X

3 __ + ___ ___ = ___ ___ 0 1 2 3 4 5 6 7 8 9

The next learner guesses 9, so the teacher writes:

XX

3 __ + ___ ___ = ___ ___ 0 1 2 3 4 5 6 7 8 9

The next learner guesses 5, so the teacher writes:

XX

3 _5 + ___ ___ = ___ 5_ 0 1 2 3 4 5 6 7 8 9

Now, there are some clues in the problem, that can help learners to make the next guess. You can

ask learners to briefly discuss in pairs what number should be guessed next. Some learners may

realize that the two 5s means that there had to be a zero in the ones position of the second number.

After a minute, ask a learner to guess the next number.

If the learner guesses a zero, ask why. As learners play the game a few times, they start to reason

numerically about how clues can help. This requires mental maths skills and develops their number

sense.

XX

3 _5 + ___ _0_ = ___ 5_ 0 1 2 3 4 5 6 7 8 9

Again, you can let learners discuss what numbers could work and which ones are impossible. For

example, 8 is no longer possible. The next learner guesses 7.


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XXX

3 _5 + ___ _0_ = ___ 5_ 0 1 2 3 4 5 6 7 8 9

Discuss with learners the remaining possibilities: 1, 4, 6 and 8. Which ones are possible? Learners

may find out that the remaining numbers are 1 and 4. In that case, they solved the problem with

only 3 strikes, so they won the game.

You can repeat the game with other examples, such as 50 + 26 = 76 and 29 + 13 = 42. Later, you can

move to 3 digits and include subtraction, for example 37 + 87 = 124 and 70 – 12 = 58. You can also

introduce problems that involve multiplication and subtraction.

When learners understand how to play, the can learn to play the game independently. You can let

pairs of learners play against other pairs to allow for discussion among learners. Having learners play

in pairs allows for both cooperation and competition.

2. Numbers and operations game

This activity can be used as a game to practise learners’ skills in basic operations. You can make the

sequences as difficult as you like.

Given a set of 5 numbers, try to get as close as possible to the number on the top by using addition,

subtraction and multiplication with the numbers below:

30 45 619 9 91 6 83 11 77 3 34 2 11

3. Seven Up (Burns, 2015, p. 93)

This game helps learners to develop fluency with combinations of 10. For this game, you need 40

cards, each numbered 1 to 10. To play, learners deal 7 cards faceup in a row. They remove all 10s,

either individual cards with the number 10 on them, or pairs of cards that add to 10, and place those

cards in a pile separate from the deck. Each time they remove cards, they replace them with cards

from the remaining deck. When it is no longer possible to remove any cards, they deal a new row

of 7 cards on top of the ones that are there, covering each of them and any blank spaces with a new

card. When those cards are removed, it is possible to use the cards underneath. The game ends


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when it is no longer possible to make 10s or all the cards in the deck are used up.

First, play the game with a few learners in front of the class. When learners understand how to play

the game, they can play it in pairs or small groups. One learner has the job of removing cards and

the other puts out the 7 cards to start and adds new cards to fill the spaces or when they are stuck.

4. The Greatest Wins (Burns, 2015, p. 94)

This is another game to practice learners’ skills in basic operations. It can be adapted for various

grade levels. For this game, learners need a die with the numbers 1-6 on it.

You start the game by drawing a game board for each player on the blackboard, for example:

Figure 16: Game Board example for The Greatest Wins game (Burns, 2015)

Learners take turns rolling the die and writing the number in one of the boxes on the game board.

Once a learner writes a number in a box, that number can’t be changed. Students use the “reject”

box to write one number that they think is not helpful. After all players have filled the boxes, learners

do the calculation and compare to see who has the greatest answer.

Introduce the game by asking two volunteers to come up to the front of the class and play the game

with the teacher. When students understand the game, they can play it in small groups.

After playing a few rounds, organize a discussion about the strategies that learners use to play the

game, asking questions such as: How did you decide where to place a 1 or 2? What about a 5 or 6?

Who has a different idea about where to place those numbers?

Below are some variations on the game board that you can use. Notice that for the first game board,

no computation is needed. For this game setup, it is important that students read the resulting

number aloud. You can also change the game into the smallest wins. Instead of using a die, you can

also use a spinner with the numbers 1-9.


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Figure 17: Game board variations for The Greatest Wins game (Burns, 2015)

5. Target 300 (Burns, 2015, p. 96)

This game develops learners’ number sense, gives them practice in multiplying by 10 and multiples

of 10. The objective of the game is to get a total closest to 300 after six rolls of a 1-6 number die.

The total can be exactly 300, lower than 300 or higher than 300 but players must use all six turns.

The first player rolls the die and decides whether to multiply the number that comes up with 10, 20,

30, 40 or 50. Learners record their own and each other’s problems. For example, if player 1 rolls a

2 and multiplies it by 20, both players record 2 x 20 = 40. Then player 2 takes a turn. Players keep a

total of their scores. After each player has had 6 turns, they record the following:

______________ won

______________ was _____ points away from 300.

______________ was ______ points away from 300.

Again, you can make variations to the game depending on the grade level or as a way to differentiate

within your class. For some learners, you can change the game into target 200 and for others, you


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can make it target 600. Instead of a die, you can use 2 dice or a 1-9 spinner to increase the range of

possible numbers.

6. Leftovers (Burns, 2015, p. 410)

This game helps learners to practice division. You need two 1-6 dice per pair of students to play the

game. The goal of the game is to get the highest possible score. Play the game using the following

rules:

▪ Agree on a starting number between 200 and 500.

▪ One player rolls the two dice and uses the number to make a two-digit divisor. For example,

if the learner throws a 3 and a 5, he/ she can use 35 or 53 as the divisor. The learner divides

the starting number by the divisor and keeps the remainder as his/her score.

▪ The other player records the division sentence, marking the division sequence with the first

player’s initial.

▪ Both players subtract the remainder from the starting number to determine the next starting

number.

▪ Learners change roles and repeat steps 2-4.

▪ Continue switching roles and playing until the starting number becomes zero or it is no longer

possible for either player to score.

▪ Calculate the total remainders for each player. The player with the greater total is the winner.

7. Hit the Target (Burns, 2015, p. 412)

This game helps learners to practice multiplication. To play the game, learners need one 1-6 die

per pair. the goal of the game is to hit the target range in as few steps as possible. Play the game

according to following rules:

▪ To choose a target range, throw the die three times (or four times to play with greater numbers). Arrange the three or four numbers into the highest possible number. This is the lower end of the target range. For example, if you roll 3, 2 and 6, then the number you make is 632. Add 50 to the original number (or a smaller number to make it more difficult) to determine the upper end of the target range. In the example, the target range becomes 632-682.


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▪ Player 1 chooses a number between one and hundred (14 for example).

▪ Player 2 chooses another number between one and hundred to multiply the first number with, for example 50.

▪ If the product doesn’t hit the target range, player 2 goes back to the original number (14 in this example) and multiplies it by another number. Player 1 verifies and records the result.

▪ Players repeat step 4 until the product falls within the target range.

▪ Learners repeat the game, switching roles.

8. Cover up and Uncover (Burns, 2015, p. 424)

This game enables learners to practice simple fractions. For this game, learners need their fraction

kit (see: section 5), a coloured whole blue strip per player and a die with the faced labelled: 1/2, 1/4,

1/8, 1/8, 1/16 and 1/16. Learners can play the game following these rules:

▪ Both players start with a blue whole strip to cover up.

▪ One player rolls the fraction die. The fraction on the die tells what size piece to place on the

whole strip.

▪ The player gives the die to the other player, who now rolls the die and repeats the process.

▪ continue until a player has completely covered the strip with no overlaps. If you roll a fraction

that is too big, you give the die to the other player to throw.

A variation on this game is called Uncover. For this game, learners need a fraction kit and the same

fraction die as in the Cover Up game. Play the game according to the following rules:

each player starts with the whole strip covered with the two 1/2 pieces. The goal of the

game is to uncover the whole strip completely.

one player rolls the fraction die. The fraction on the die shows what size piece to remove

from the whole strip. There are three options: remove a piece (only if there is a piece that

is the size indicated by the fraction die), exchange any of the pieces left on the strip for

equivalent pieces or do nothing and pass the die to the next player.

the next player throws the die.

continue until a player has uncovered the whole strip, without any overlaps.


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Figure 18: Overview of the Uncover game (Burns, 2015)

Stimulate learners to use correct mathematical language during this game. On the board, record

some examples. For example, if a learner used three 1/4 pieces and two 1/8 pieces, write: 1/4 + 1/4

+ 1/4 + 1/8 + 1/8 = 1. Next, explain how to shorten the equation by counting the fourths and the

eighths and writing: 3/4 + 2/8 = 1

Figure 19: Example of learners’ recordings from the Uncover game (Burns, 2015)

9. Circles and Stars (Burns, 2015, p. 378)

This game introduces learners to multiplication as combining equal groups (repeated addition

interpretation of multiplication). Students move from a pictorial representation (drawing circles

and stars) to a symbolic representation (writing and reading the equations).

You need one 1-6 die per pair of students, a piece of paper and a pencil. Students play the game in

pairs.


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Rules of the game:

Fold a piece of paper in 8 sections. Let students write their name in the first section and use

the other sections to draw circles and stars.

Roll the die. Draw that many circles.

Roll the die again. Draw that many stars in each circle.

Record the total number of stars that you drew.

Give the die to the other player.

Continue playing until you have drawn circles and stars in each section of the paper.

The player with the most stars and circles drawn on the whole sheet is the winner.

During the class discussion, draw a sample page of three circles with two stars in each and

underneath write: 3 x 2. Explain to students that this is a way to write three groups of two with

maths mathematical symbols. Tell them that you can also read it as “three times two” and it means

the same thing. Write = 6 and explain this complete the equation to tell how many stars there are

in all for that round. Write on the board the different ways to read 3 x 2 = 6.

Figure 20: Circles and Stars Game

Let students work in pairs to write a mathematical equation for each section of their paper.


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Use following questions to help students understand multiplication:

Can you find two rounds with the same total but with different arrangements of circles and

stars?

What is the smallest number of stars possible in one round? What is the greatest number?

Who drew twelve stars in one round? Describe how many circles you drew and how many

stars you drew in each?

The purpose of this game is to teach the concept of multiplication, not to let them practice

multiplication facts.

Activity 26

In groups of 4, try out one game and play it with your group.

In the second phase, we will mix the groups. Explain the game that you studied to the other

group members and play the game.

Tips for using mathematical games:

It is good to have a mix of both competitive and cooperative games throughout the year.

Competitive games help students test their skills take risks and learn to be graceful winners

and losers. However, it is also important to develop communication and cooperation among

students. Having students play in pairs or small groups allows for both cooperation and

competition.

Often, you can play the game first with the whole class, so all students are familiar with the

rules. Play the game step by step and say aloud what you are doing and why. Next, they can

play the game in small groups.

Encourage students to play the games at home.

Games are ideal for engaging students in your class, freeing up time for you to engage learners

who require additional support.


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Section 7: Inclusive Education

Introduction

Inclusive education is based on the idea that all learners are different but have the capacity to

achieve the learning outcomes. Inclusive education recognizes that diversity is essential and valued.

Inclusive education means adapting teaching to meet the needs of each individual learner. For many

mathematics teachers, the most difficult issue they face daily is how to meet the needs of so many

students that vary greatly in terms of what they currently know, what they can do, their motivation,

their personalities…

The CBC identifies special needs as a cross-cutting issue in all subjects. Therefore, teachers are

called to identify students who are struggling mathematically and adjust the learning environment

to enable them to learn. Inclusive education is about treating all learners as individuals. It is about

making sure that all learners can learn. Therefore, inclusive education is much broader than special

needs education, which focuses on learners with disabilities.

Activity 27

Describe in one sentence what inclusive education means to you practically in your daily teaching.

Compare and discuss with your neighbour. Try to come to an agreement.

Components of Inclusive Education

When we think about inclusive education, often we think about getting children into school,

i.e. making sure they are present in school. However, we also need to ensure that children are

participating in lessons and school life, and that they are achieving academically and socially as a

result of coming to school.

When thinking about inclusive education, always consider Presence, Participation and Achievement

(Figure 21) (Ainscow, 2005).


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Presence

AchievementParticipation

Inclusiveness

Figure 21: Components of Inclusive Education (Ainscow, 2005)

It is not enough that they simply attend the lessons; all children should be given the same opportunities

to fully participate and achieve.

Equal Presence: Teachers should be instructed to do daily attendance of all children. If there

is an attendance issue related to sex, disability or other reason, talk with parents through

School General Assembly meetings. Invite the concerned parents at school to speak about

why all learners should be provided with equal learning opportunities and how to support

their learning needs.

Equal participation: Teachers should ensure that all learners are participating actively and

given chances to lead in classroom activities, classroom discussions, and different clubs.

Equal achievement: Parents, teachers and school leadership should ensure that all learners

have equal opportunities to access learning materials and that there are not any systematic

achievement gaps. You may think it is too difficult to address the needs of a diverse range

of children, as there are so many challenges. However, by working as a team within your

school, with support from families and local communities, and by making small changes to

your teaching methods, you will be able to meet the needs of all children.

Differentiation

Differentiation is a key classroom strategy to make teaching and learning more inclusive. But what

does differentiation really mean? Is it feasible in classrooms with many leaners and how do you go

about it?


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Activity 28

Discuss the following statements (agree/not agree/ depends)

1. Differentiation is an idea as old as effective teaching

2. Differentiation means grouping students by ability

3. Differentiation is mostly aimed at students with identified learning challenges

4. Differentiation is about valuing and planning for diversity

5. Differentiation means that all students do different things

6. Differentiation is not possible in classes with more than 50 learners.

You can find the solutions in the infographic below.

Figure 22: Differentiation is & Differentiation is not (ASCD, 2015)

Source: http://www.ascd.org/ASCD/pdf/siteASCD/publications/Differentiation_Is-IsNot_infographic.pdf


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Every classroom at every grade level contains a range of students with varying abilities and backgrounds. In Rwanda, many students’ mastery of learning is several grade levels below the grade they are in. Perhaps the most important work of teachers is to identify students’ level of prior knowledge in mathematics and then plan lessons that support and challenge all students to learn. This will enable teachers to differentiate instruction effectively through, considering the large class sizes, providing remediation for struggling learners so they can catch up with the rest of their peers.

Activity 29

A mathematics teacher teaches in grade 5, but notices that some students do not have the

required prior knowledge on fractions that they should have learned in grades 3 and 4.

What would you do?

Activity 30

Read the case story below. Which teacher practices differentiation? List reasons why one teacher

teaches inclusively and the other not.

Case Story

I encountered two teachers whose impact on me extended beyond the year they taught me—even

until today. It would be correct to say that one of them taught math, the other taught me English.

There is a subtle but important difference in the way those clauses are written. Ironically—or

perhaps not—I cannot recall the math teacher’s name, although I have a clear image of her

standing at the blackboard, mobbing quickly through the math text. She was a serious math

teacher. She covered math with a focus that was evident even to P6 learners. She explained the

math in one way and one way only. She taught each topic one time and one time only. She used

one form of assessment and one form only. She knew math, but she didn’t know about me at

all. That I understood almost nothing she was talking about was either off her radar or not of

her interest. She kept going on and on, covering the curriculum. I got more profoundly lost—

more profoundly desperate. My sense of hopelessness was made worse by the fact that a good

number of my friends seemed to be hanging on to various degrees while I sank by the day. One

way of looking back at the math episode is simply to say I didn’t do well that year. In truth, my


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grade was the least of my problems. My uncertainty about myself grew in direct proportion to

the math confusion that grew inside me day by day. Not only did I become a seventh grader who

“couldn’t do math”, but I remain to this day a person who regards all things mathematical with a

feeling in my stomach that takes me directly back to the worst aspects of early adolescence. I do

remember my English teacher’s name. He was Mr. Alfred. He was a new teacher and wasn’t very

good yet. He was not strong in either the charisma or the classroom management categories. But

he worked hard to know us as individual students and to make the class work for us as individual

students. He met during class with small groups of students who needed help with an assignment.

He connected our various interests and personalities to literature we read. He picked out books

for individuals’ book reports, dignifying us with that bit of personal attention. n He gave careful

thought to student groupings and told us what he thought would make the class work for us. He

saw that I needed to learn at a different pace and even in different directions than did some of my

peers in his class—and he saw to it that my needs were a part of his plans, as were the needs of

my various classmates. I found young adolescent hope in literature and writing in the same way

that mathematics came to strengthen my young adolescent despair. It took years to undo what

that mathematics class did and some of it has not yet gone away.

I don’t think anyone used the word “differentiation” in those days, but they could have. At the time

in my life when I was seeking identity, a one-size-fits-all approach to mathematics proved to me

daily that I was a loser. A much more student-focused and personalized English class planted the

seed for my future, even though I could not see it at the time.

Carol Ann Tomlinson, 2005

Differentiation is not about treating every learner equally. It is about giving each learner the support

he or she needs to achieve the learning outcomes. This implies that some learners will need more

or different support than others (see Figure 23).


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Figure 23: Equality versus Equity (Save the Children, Mureke Dusome project, 2017)

Differentiation is not about having students do different things all the time, nor is it about teachers

choosing the learning for them, it is about students doing the same thing in different ways. By

sharing our differences, we learn from and with each other.

Source: https://buildingmathematicians.wordpress.com/2017/03/12/the-same-or-different/

Differentiation is not about treating every learner equally. It is about giving each learner the support he or she needs to achieve the learning outcomes.

How to differentiate

A first step in differentiating teaching is taking the knowledge that learners bring to class into account.

There is evidence that learning is improved when teachers pay attention to the prior knowledge

and beliefs of learners, use this knowledge as a starting point for instruction and monitor learners’

changing conceptions as the lesson proceeds. If their initial understanding is not engaged, they may

fail to understand the new concepts.


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Secondly, differentiating instruction means engaging all students. All students need sufficient time

and a variety of problem-solving contexts to use concepts, procedures and strategies and to develop

and consolidate their understanding. Teachers should consider the different ways that students

learn by introducing a variety of teaching strategies without pre-defining their capacity for learning.

Thirdly, differentiating instruction involves continuously assessing your learners and designing tasks

and activities that cater both for learners that are at risk of falling behind (remediating instruction)

and those that are ready for more challenging problems.

There are four approaches to differentiation (Figure 24):

Figure 24: Approaches to differentiation

Differentiate by quantity

This approach assumes that higher performing learners will work faster and extra work should be

prepared to cater for this. However, ‘more work’ is unhelpful when this only means ‘more of the

same’. These learners need to explore ideas in more depth, not merely cover more content.


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Differentiate by task

In this approach, learners are given different problems or activities, according to their learning

needs. This approach is difficult to implement well, because it assumes that the teacher can prejudge

the performance of each learner accurately and that there is also a supply of suitable problems

or activities. If you decide in advance that some learners will not be able to cope with particular

concepts and ideas, you deny them the opportunity to engage with these ideas. It is therefore not

a good strategy to simplify activities for some learners in advance.

A better approach is to give learners some choice in the activities they undertake. For example,

learners can be asked to choose between an easy, a challenging and a very challenging task. Research

showed that few learners choose the easy task and that most prefer a challenge (Swan, 2005). This

approach assumes that learners can make a realistic assessment of their own ability to solve the

problem. It works less well with less confident learners.

Differentiate by level of support

In this approach, all learners are given the same task, but are offered different levels of support,

depending on the needs that arise during the activity. This avoids the danger of prejudging learners.

For example, you may give carefully chosen hints during a group work activity.

Differentiate by outcome

Open activities that encourage a variety of possible outcomes offer learners the opportunity to set

themselves appropriate challenges. This approach is used in many of the activities in this guide. For

example, some activities invite learners to create their own classifications or their own problems and

examples. Teachers may encourage learners to ‘make up questions that are difficult, but that you

know you can get right’.

Activity 31

Think individually about the following questions:

Review the four ways to differentiate teaching. Give an example from your mathematics teaching

for each of them. Which one do you use the most? Which one the least? Why?


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Strategies to implement differentiation in mathematics

1. Identify and focus on key concepts

Determine for each lesson what the key concepts are that each learner should master at the end of

the lesson. The CBC provides you with a starting point for what the key concepts are.

For example, a Grade 6 teacher is planning a lesson on multiplying whole numbers by decimals.

Although the goal of the instruction is performing a computation like 1.5 x 3, the key concept that

students need to understand is that multiplication has many meanings (e.g., repeated addition,

counting of equal groups, objects in an array, area of a rectangle).

2. Designing Open Tasks

Suppose a P4 teacher wants to teach the key concept that any subtraction can be thought of in

terms of a related addition. P4 students should be able to solve addition and subtraction problems

involving multi-digit numbers, using concrete materials and standard algorithms, as well as use

estimation to help judge the reasonableness of a solution. Some students may not be ready to deal

with three-digit numbers, even with the use of concrete materials. A teacher might change the

planned task to turn it into an open task (Figure 25). Open tasks are also called “Low threshold,

high ceiling tasks”. The low threshold means that the task can be done by learners who still have a

low understanding of the concept. The high ceiling means that the task can still be challenging for

learners who have already a good understanding of the concept.

Figure 25: Example of Open Task (Beckmann, 2013)

With the open number task, students have a choice in the numbers they use, choice in the strategies

they use and a choice in how they interpret the meaning of the problem. Students who can only

handle numbers below 20 can do so. Students who can handle numbers below 100 in a concrete

way can do so too. Students who are ready to work with very large numbers can do so as well. Also,


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in the revised task, some students will interpret the phrase “most of the books” to mean more than

half. Others can simply interpret it as meaning that more books are about dogs than other animals;

they might make a list of different animals with a total number of books about each animal, ensuring

that the number for dogs is the greatest number on the list. These variations really don’t matter.

All students will be considering a subtraction situation; all of them are relating it to an addition

situation; all of them have an opportunity to understand and solve the problem using their own

student-generated strategies and appropriate materials. Whether students are working with large

or small numbers, the sharing of their mathematical thinking is valuable for the collective learning

of the class.

Figure 26: Example Solution for Open Task (Beckmann, 2013)

In fact, there might be more mathematically sophisticated thinking from a student who uses a

smaller value than one who simply uses a standard algorithm to subtract 118 from 316. With several

differentiated student responses to the problem, it is valuable for students to share their thinking

and compare strategies. In this example, the teacher can co-ordinate a class discussion about the

use of different models of representations to show different mathematical thinking:

Some students might use an empty number line. This has the benefit of flexibility; students

can use numbers in whatever increments make sense to them.

Other students might use base ten blocks and focus on place value concepts. These students

practise the important skill of decomposing numbers into their hundreds, tens and ones

(units) components.


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Some students might draw diagrams. For example, the student might draw a model for 316

– 118. The model reinforces the mental concept that to subtract 118 from 316, you can think

of subtracting 116 and then another 2, to get 316 – 116 = 200 and 200 – 2 = 198.

In the example below, children can choose various combinations of numbers to solve the problem.

Figure 27: Example of open learning task

Another example: Sarah and Mike ran each day this week. Each day Sarah ran 3 kilometres in 30

minutes. Mike ran 6 kilometres in 72 minutes. Here are the answers: 42, 2, 294, 3 ½. What can be

the questions for each answer?

Possible responses:

42: How many more minutes did Mike run than Sarah each day?

2: How many more minutes does it take Mike to run a kilometre?

294: How many more minutes did Mike run this week than Sarah?

3,5: How many hours did Sarah run this week?

3. Regularly checking students’ understanding

Structure your lesson in such a way that there are frequent moments for checking learner

understanding. Avoid long series of exercises where students may get stuck for a long time. Some

struggle is fine for students, and even helps learning and retention, but avoid that they get completely

stuck and become demotivated.

Build in moments during the lesson when learners show their learning, before they can move to the

next step. This is part of the process of collecting learner data. The objective is to closely monitor

learning progress so quick remediation is possible. For example, learners make a few exercises.

When they are finished, they raise their hand for a quick check. If ok, they can proceed with the next

exercises. If the same errors keep coming back, you can build in a moment of whole-class instruction.


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These control moments should be short to limit wait time. The risk of the technique is that learners

must wait too long before they get feedback. More exercises between control moments increase

the time difference that learners finish the exercises. This technique allows for accommodating both

faster and slower learners.

4. Involving learners with disabilities

Differentiation does not require the specialized knowledge to deal with specific learning disabilities.

However, as a teacher you can take some simple steps to help learners with learning disabilities.

Table 5 lists some classroom strategies to help learners with various learning challenges.

Table 5: Learning Challenges and Possible Classroom Strategies (Save the Children, Mureke

Dusome project, 2017)

CHALLENGE CLASSROOM STRATEGY TO ADDRESS

Hearing Try to convey information to the child using sign language or informal signs and hand gestures.

Seat the child in the front row. Speak loudly and clearly.

Ensure the child can see your mouth when you speak.

Provide the child with a detailed outline of the lesson/objectives.

Use charts, pictures and icons to convey information.

Assign the child a learning buddy.

Speak with the child’s parents to identify and build on communication techniques used at home.

TALKING Encourage the child to continue when he/she is trying to communicate.

Be attentive while he/she is talking.

Provide opportunities to use different ways of communication such as role play, gestures, drawing, writing, etc.

Speak with the child’s parents to identify and build on communication techniques used at home.


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PHYSICAL ACCESS Ensure the child is physically able to access his/her classroom and seat.

Ensure the child can access learning materials.

Assign a student helper or circle of friends to help the child navigate the classroom.

Shift classroom furniture so that there are clear passage ways.

READING Ask the child to follow along with a finger.

Provide a piece of paper or other material and instruct the child to uncover one sentence at a time while reading.

Provide extra reading practice time in school and at home.

Pair the child with a reading buddy who reads with him/her daily.

SEEING Ensure that the classroom has good lighting.

Write in large clear letters on the blackboard.

Assign the child a learning buddy.

Seat the child in the front row.

Refer the child for glasses, if possible.

Activity 32

Review Table 5. Can you give examples of the strategies that are listed? Do you use other

strategies to help learners with specific learning disabilities learn? Discuss your experiences and

ideas in small groups. If time permits, groups can briefly present and discuss their findings.


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Section 8: Group Work

Introduction

Research has shown that cooperative small group work has positive effects on both social skills

and mathematics learning, but this effect is dependent on (1) shared goals for the group and (2)

individual accountability for the achievement of the group (Askew & Wiliam, 1995). This section

discusses what group work is, when to use it and how to use it in your lessons.

Activity 33

Why do you use group work in your lessons? Are there situations that group work is not

useful? Explain your ideas.

There is a clear difference between working in a group and working as a group (Swan, 2005)

Conditions for Successful Group Work

Learners working in groups is a key component of learner-centred pedagogy. However, group work

is not always appropriate. When the purpose of the lesson is to develop fluency in a skill and there is

little to discuss, then individual practice is more suitable. Group work is useful when the purpose of

the lesson is to develop conceptual understanding or problem-solving skills. In these cases, learners

need to share their interpretations and approaches.

There is a clear difference between working in a group and working as a group (Swan, 2005). It is

common to see learners working independently, even when they are sitting together. Sometimes,

one group member does the work and others copy the solution. In this case, learners work in a

group, but not as a group.

Students need practice, discussion and encouragement to learn to work productively in a group.


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Critical sharing, participation, listening and communication skills include:

allowing all members in the group adequate opportunity to express their ideas, and being

patient when they have difficulty doing so;

overcoming shyness and being willing to cooperate with the group;

listening rather than simply waiting to offer one’s own point of view;

taking time to explain and re-explain until others understand.

Learners need to take time to learn to work in these ways but, when they do so, the benefits are

high, as we will discuss in the next section.

Advantages of group work

Research into group work (DfES, 2004; Stewart, 2014) highlight many benefits for both teachers and

learners.

For teachers, it can help them to:

empower learners in group situations to engage in peer teaching, learning and assessment

to show what they know, understand and can do and identify what they still have to learn;

get information about how learners are understanding and applying the learning content.

For learners, collaborative learning can help them to develop their thinking and problem-solving

skills by encouraging them to:

explain and negotiate their contributions with others in a group;

take turns in discussion while exploring a topic;

apply their knowledge to practical situations;

develop mathematical language skills:

support and build on each other’s ideas.


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Additional benefits of group work include:

Academic achievement: Research has shown that students who work in cooperative groups

do better on tests, especially regarding reasoning and critical thinking skills.

Motivation: One reason for improved academic achievement is that students who are

learning cooperatively are more active participants in the learning process. They care more

about the class and the material and they are more personally engaged.

Life Skills: team work is essential in modern workplaces. Group work helps them to develop

argumentation and listening skills.

In summary, explaining something to your peers requires putting your ideas into a coherent story,

which requires formulating, reflecting and clarifying, all processes that stimulate learning (Burns,

2015).

Role of the teacher during group work

What should teachers do during group discussions? Here is some guidance (Mercer & Sams, 2006):

1. Make the purpose of the task clear

Explain what the task is and how learners should work on it. Also, explain why they should work

in this way. For example, “Don’t rush, take your time. The answers are not the focus here, but the

reasons for those answers. You don’t have to finish, but you do have to be able to explain something

to the whole group.”

2. Set clear rules for group work

It helps to prepare students to work together by establishing rules. These rules can be very helpful:

Learners must be willing to help any group member who asks. When someone asks a

question, don’t just give the answer, but help by asking questions that helps the learner

focus on the problem at hand.

You may only ask the teacher for help when everyone in your group has the same question.

This rule forces learners to discuss questions first among themselves. It motivates learners

to rely more on each other and less on the teacher.


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3. Divide the groups

There is no optimal group size. It depends on the students and the task at hand. The most important

thing is to make sure and monitor that all learners are involved. It is good for learners to have the

opportunity to work with all their classmates over the course of time and therefore it is best to

change group composition regularly.

Grouping children by ability is usually not recommended as it reinforces the idea that there are

strong and weak learners which are best kept separate. Grouping by ability may lead to labelling

students and place them always in the same low, middle, or high group.

However, in some cases ability grouping can be useful:

Students are not forced to wait or rush: When you place students of the same ability

together, they usually are able to work at about the same pace. This means the students

that understand the concept you are teaching can move on to a more advanced stage and

the ones that need extra guidance can slow down and get extra help.

Teachers can work more intensely with those that need help: since they are seated and working

together, you can take this opportunity to sit with the ones that need extra instruction.

4. Listen before intervening

When approaching a group, stand back and listen to the discussion before intervening. It is all too

easy to interrupt a group and give the right answer.

The purpose of an intervention is to increase the depth of reflective thought. Challenge learners to

describe what they are doing (quite easy), to interpret something (”Can you say what that means?”)

or to explain something (”Can you show us why you say that?”).

When a learner asks the teacher a question, don’t answer it (at least not immediately). Rather ask

someone else in the group to answer.

5. Don’t be afraid of leaving discussions unresolved

Some teachers like to resolve discussions before they leave the group. When the teacher leads

the group to the answer, then leaves, the discussion has ended. Learners are left with nothing

to think about, or they go on to a different problem. It is often better to reawaken interest with


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a further interesting question that builds on the discussion and then leave the group to discuss it

alone. Return some minutes later to find out what has been decided.

Techniques for effective group Work

1. Think- Pair-Share

In a Think-Pair-Share approach means that learners first work alone, writing down their ideas or

solutions, then pair and exchange ideas with a partner. Finally, the sharing is done during the class

discussion.

Video: https://www.teachingchannel.org/videos/think-pair-share-lesson-idea

Many teachers find that asking learners to work in pairs or threes is most effective. In larger groups,

there is the risk of ‘passengers’, members who rely on the others to do the work. In a think-pair-

share, learners begin by responding to a task or question individually. Usually, this does take only a

few minutes. By letting students first think and prepare the question individually, you ensure that

everyone can contribute to group discussions.

In pairs, learners can provide each other with a different explanation or perspective. In some

cases, you can join pairs together into fours so that a broader consensus can be reached. Each pair

chooses one item to share with the whole group. Quickly go around the room hearing each pair’s

items. Finally ask, “Did anyone have any other findings they wanted to share?” and collect those. In

this fashion, each student is stimulated to think before hearing from others, and students who are

thoughtful and move slowly get a chance to organize their thoughts before sharing. Finally, collect

some examples of different responses and write these on the board anonymously.

Further reading:

http://mathforum.org/workshops/universal/documents/notice_wonder_intro.pdf

https://www.cultofpedagogy.com/think-pair-share/

2. Talking Points

Talking points are an effective method to stimulate mathematical conversations in groups (Lemov,

2015). You can use the technique at the beginning or end of a lesson to collect prior knowledge or


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review the topic. You prepare a set of statements that reflect the lesson objectives, standards or

misconceptions about the topic.

During the group discussions, learners follow a fixed routine:

1. Go around the group, with each person saying in turn whether they agree, disagree or are

unsure about the statement and why. Even if you are unsure, you must state a reason why

you are unsure. No comments on each other’s answers are given. You can change your mind

during your turn in the next round.

2. Go around the group again, with each person whether they agree, disagree or are unsure

about their own original statement or about someone else’s statement they just head and

say why. No comments on each other’s answers are given. You are free to change your mind

during your turn in the next round.

3. Take a tally of agree, disagree and unsure and make notes on your sheet. No comments are

given.

4. Move to the next talking point.

Consider this example of a talking points sheet on fractions:

Figure 28: Talking Points on fractions


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Each statement refers to a specific lesson objective. You can follow up on the group activity with a

class discussion. Questions you can use are:

Which talking point did your whole group agree with and why?

Which talking point did your whole group disagree with and why?

About which talking point were you most unsure and why?

Which talking point do you know you are right about and why?

Could any of the talking points be true and false?

Source: https://kgmathminds.com/2017/02/05/fraction-talking-points-3rd-grade/

“As a class, we reviewed the process and practiced Talking Point #1 together as a

class. From there I let them go and circulated the class to hear the conversations!

It was the absolute highlight of my first week!” (Kristin Gray, source below)

Source: https://kgmathminds.com/2014/09/06/week-one-talking-points-math-mindset/


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UNIT 4: GENDER AND MATHEMATICS EDUCATION

Introduction

Areas where consistent gender differences have emerged are children’s beliefs about their abilities in

mathematics, their interest in mathematics and their perceptions of the importance of mathematics

for their future. In all three domains, girls have found to be scoring lower than boys. Researchers

have found that girls often have less confidence in their mathematics abilities (Zuze & Lee, 2007).

This is a problem because research shows that children’s beliefs about their abilities are central to

determining their interest and performance in different subjects and the career choices they make

(Beilock, Gunderson, Ramirez, & Levine, 2010).

These gender differences contrast with research that males and females generally show similar

abilities in mathematics (Hyde, Fennema, & Lamon, 1990). National data show that both girls

and boys face gender-related barriers to learning. Based on national examination results, boys

outperformed girls in almost all districts at P6 and S3 levels during the period 2008-2014 (MINEDUC,

2015). This indicates that girls face more challenges than boys.

An analysis of data shows the percentage of children making it from P1 to P6 in the previous six years

was only 10% on average; for boys, the percentage was slightly lower than for girls (NISR, 2015). This

shows that, while girls face many challenges related to learning, progression and completion, also

boys face challenges that include repeating and dropping out of primary school (NISR, 2015).

To eliminate all the causes and obstacles which can lead to inequity in education, the Ministry

of Education included gender as one of the crosscutting issues in the pre/primary and secondary

Competence Based Curriculum framework (Rwanda Education Board, 2015).

This section aims therefore at equipping mathematics teachers with the competences to apply a

gender responsive pedagogy in their teaching.


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Learning Outcomes


Understand the meaning of gender and related concepts;

Apply gender responsive pedagogy in the classroom;

Reflect on how to apply gender concepts to teaching and learning mathematics;

Design learning activities that will equally interest and engage girls and boys in mathematics;

Support fellow teachers in applying gender responsive pedagogy in the classroom;

Make learning of mathematics enjoyable for both girls and boys;

Acknowledge the presence of gender stereotypes in mathematics instruction;

Appreciate that boys and girls have equal abilities to achieve proficiency in mathematics;

Commit to working towards gender equity in their school.


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Section 1: What Is Gender?

Gender is a concept that is widely used and perceived by many to mean “women’s issues”. In

reality, gender refers to the socially determined roles and relations between males and females

(Subrahmanian, 2005) in this regard. These two goals are distinguished as gender parity goals

[achieving equal participation of girls and boys in all forms of education based on their proportion

in the relevant age-groups in the population] and gender equality goals [ensuring educational

equality between boys and girls]. In turn these have been characterised as quantitative/numerical

and qualitative goals respectively. In order to consider progress towards both types of goal, both

quantitative and qualitative assessments need to be made of the nature of progress towards gender

equality. Achieving gender parity is just one step towards gender equality in and through education.

An education system with equal numbers of boys and girls participating, who may progress evenly

through the system, may not in fact be based on gender equality. Following Wilson (Human Rights:

Promoting gender equality in and through education. Background paper for EFA GMR 2003/4, 2003.

Gender is different from sex. Sex refers to purely biological differences between men and women.

Gender roles, on the other hand, are created and sustained by the society, which assigns different

responsibilities to men and women, e.g., cooking for women and decision-making for men.

Gender roles can therefore be changed and vary over time and from community to community.

These gender roles are consciously or unconsciously carried into the classroom by teachers, students,

school leaders, parents and other stakeholders. In children’s textbooks, for example, women are

often represented as cleaners, caregivers and nurses, and men are drivers, doctors and leaders. The

images reinforce gender roles, which are socially constructed.

Activity 34

What is the influence of cultural norms and practices on girls’ participation in mathematics

classes in your school?


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Section 2: Key Terms

Several related concepts underlie the development of a clear understanding of gender:

Gender discrimination: Denying opportunities and rights or giving preferential treatment to

individuals on the basis of their sex. For example, only giving boys the opportunity to be a

team leader.

Gender equality: The elimination of all forms of discrimination based on gender so that

girls and women, boys and men have equal opportunities and benefits (OECD, 2015). For

example, giving an equal chance to boys and girls to be a team leader.

Gender equity: Fairness in the way boys and girls, women and men are treated. In the

provision of education, it refers to ensuring that girls and boys have equal access to

enrolment and other educational opportunities (Subrahmanian, 2005)in this regard. These

two goals are distinguished as gender parity goals [achieving equal participation of girls and

boys in all forms of education based on their proportion in the relevant age-groups in the

population] and gender equality goals [ensuring educational equality between boys and

girls]. In turn these have been characterised as quantitative/numerical and qualitative goals

respectively. In order to consider progress towards both types of goal, both quantitative and

qualitative assessments need to be made of the nature of progress towards gender equality.

Achieving gender parity is just one step towards gender equality in and through education.

An education system with equal numbers of boys and girls participating, who may progress

evenly through the system, may not in fact be based on gender equality. Following Wilson

(Human Rights: Promoting gender equality in and through education. Background paper for

EFA GMR 2003/4, 2003. For example, giving additional support to girls so they can become

confident to volunteer for team leader.

Gender stereotype: The constant presentation, such as in the media, conversation, jokes or

books, of women and men occupying social roles according to a traditional gender role or

division of labour (OECD, 2015). For example, a textbook where always boys names are used

to describe team leaders.

Gender sensitive: The ability to recognize gender issues. It is the beginning of gender

awareness (UNICEF, 2017). For example, a teacher who is aware that boys are always team

leaders and that something should be done about this.

Gender parity: This refers to the equal representation of boys and girls (UNICEF, 2017). For

example, in a class, there is an equal number of male and female team leaders.


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Section 3: Gender Responsive Pedagogy for Mathematics

Introduction

Observations of classroom practices show that teaching and learning is often gender biased (Aikman

& Underhalter, 2007; Consuegra, 2015). Many teachers apply teaching methodologies that do not

give girls and boys equal opportunities to participate and learn. They also use teaching and learning

materials that perpetuate gender stereotypes. Therefore, it is important for teachers to apply a

gender responsive pedagogy.

Gender responsive pedagogy means that teaching and learning processes pay attention to the

specific learning needs of girls and boys (Mlama, 2005). It does not mean treating boys and girls

equally. It includes lesson planning, teaching, classroom management and evaluation.

“Gender responsive pedagogy refers to teaching and learning processes that pay attention to the specific learning needs of girls and boys.” (Mlama, 2005)

In this section, we discuss some strategies that teachers can use to promote the involvement and

learning of girls in mathematics lessons.

Keep in mind thought that many techniques that we discuss in this course aim at involving all

learners. None of these strategies, however, is automatically gender-responsive. Often, boys

dominate learning processes in the class. Therefore, teachers need to consider the specific gender

needs of girls and boys in planning their lessons. Being gender responsive does not means treating

all learners equally but making sure that all learners have equal opportunities to learn.


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Activity 35

Think individually about what teaching approaches you have used to encourage equal

participation and achievement of boys and girls in your lessons. Afterwards, discuss your ideas

with your neighbour. Next, the facilitator will organize a plenary discussion.

In lesson plans, teachers should consider how all students can participate in learning activities.

They should ensure that there is equal participation in activities such as making presentations,

conversations and practical activities. In group activities, ensure that girls and boys are given

leadership positions and roles. Consider how learning materials will be distributed equally to both

girls and boys, especially in case of shortages.

Things that you can do to make classes gender equitable

1. Using gender neutral language

Gender responsive pedagogy includes gender neutral language use by the teacher. Inappropriate

language use can transmit negative messages and inhibit learning. A boy or girl whose teacher

constantly tells them “you are stupid”, will come to believe this to be true. Language can also reinforce

gender differences and inequalities and in the classroom often reflects male dominance and reduces

females to an inferior position. By contrast, a teacher can enhance students’ performance by using

encouraging, inclusive language in the classroom.

Teachers often discourage girls from doing mathematics by telling them that such subjects are for

boys or too difficult for girls. When a girl is assertive, she is told to stop behaving like a boy, and

when a boy cries, he is cautioned to stop behaving like a woman.

Much gender insensitive communication is non-verbal. An indifferent shrug of the shoulders or

rolling of the eyes suggests that the student is too foolish or annoying to deserve attention. Other

gestures and body language, such as winking, touching, brushing, grabbing and other moves may be

overtly sexual. This type of communication may go unnoticed by others for a long time, but it can

be very damaging to classroom participation to the one at whom the communication is targeted.


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2. Classroom arrangement

The typical classroom arrangement – desks lined up in an array of neat rows facing the teacher -

often reinforces the traditional socialization processes (Mlama, 2005). Since girls are not brought up

to speak out – or rather, are brought up not to speak out – when they sit at the back of the class, they

are less likely to participate unless the teacher makes a special effort to involve them. Remember

the distinction between equality and equity.

A different arrangement such as breaking the class into smaller groups may encourage the girls to

participate.

3. Teach learners that learning abilities are improvable

To enhance girls’ beliefs about their abilities, teachers should understand and communicate this

understanding to students:

Mathematics abilities can be improved through consistent effort and learning. Research shows that

even students with high ability who view their cognitive skills as fixed are more likely to experience

discouragement, lower performance and reduce their effort when they encounter difficulties. Such

responses are more likely in the context of mathematics, given stereotypes about girls’ mathematics

abilities (Dweck, 2006). Negative stereotypes can lead girls to choose unchallenging problems to

solve, lower their performance expectations and not consider mathematics as a career choice.

In contrast, students who view their abilities as improvable tend to keep trying in the face of difficulty

and frustration to increase their performance. To help girls resist negative reactions to the difficulty

of mathematics work, it is very important to stress for them to learn that their mathematical abilities

can improve over time with continuous effort and engagement.

More information: https://www.youtube.com/watch?v=fC9da6eqaqg

4. Expose girls to female role models

Girls who only encounter men as maths and science teachers may be confirmed in their beliefs that

mathematics and science are for men. If there are no female mathematics and science teachers in

your school, you can still introduce them to examples of women who achieved a lot in mathematics

and science.


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Teachers can invite women or older students as guest speakers or tutors. These role models should

communicate that becoming good at mathematics takes hard work and that self-doubts is a normal

part of the process of becoming an expert.

Activity 36

Joining the ranks of Neurosurgery: My Impossible Dream | Claire Karekezi.

After 15 years of intense training and studying that has taken her across three continents, Dr

Claire Karekezi returns home to Rwanda as the only female neurosurgeon in the country.

Video: https://www.youtube.com/watch?v=96wNdg-8t2o

Discussion questions:

1. Why is it important to expose girls to women who achieved a lot in mathematics?

2. How can female role models help with achieving gender equity in mathematics in your

school?

Examples of role models for mathematics and science in Africa:

1. Apps and Girls

Apps & Girls is a Tanzanian registered social enterprise that was founded in July 2013 by Carolyne

Ekyarisiima. It seeks to bridge the tech gender gap by providing quality coding training (web

programming, mobile app development game development and robotics) and entrepreneurship

skills to girls in secondary schools via coding clubs and other initiatives such as mentorships and

scholarships. So far, they have created 25 coding clubs in Tanzania and they have trained 269 teachers

and 2656 girls. They want to train 1 million girls before 2025.

Link website: http://www.appsandgirls.com/

Link YouTube: https://www.youtube.com/watch?v=yNNrVqUvkjg


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2. RAWISE

The Rwandan Association for Women in Science and Engineering (RAWISE) is a non-profit

organization founded by a group of Engineers and Scientists women from Rwanda. The association

aims at increasing the number of girls in science, technology, engineering and mathematics (STEM);

provide a platform for engineers and scientist women in Rwanda to meet, discuss and collaborate;

and increase female participation in scientific and technology-related professions in Rwanda.

The association aims to increasing women scientists’ participation in decision making and

development our country Rwanda and provide a hub for Rwandan women scientists where they can

meet, network, collaborate and further their research.

5. Gender-responsiveness in classroom interactions

Many of the techniques that we have discussed in this guide aim at improving the quality of

interactions in the mathematics classroom, both between teacher and learners and between

learners. In managing these interactions, it is important as a mathematics teacher that you are

aware of potential gender bias and that you can act to address this. In Table 6 we list some guidelines

to ensure that conversations and group activities are gender responsive.


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Table 6: Actions to make classroom interactions more gender responsive (Mlama, 2005)

Methodology Action

Conversations

(questions and

answers)

Give equal chances to both girls and boys to answer

questions, including more difficult questions.

Give positive reinforcement to both girls and boys.

Allow sufficient time for students to answer questions,

especially girls who may be shy or afraid to speak out.

Assign exercises that encourage students, especially girls, to

speak out.

Distribute questions to all the class and ensure that each

student participates.

Phrase questions to reflect gender representation – use

names of both men and women, use both male and female

characters.

Group activities Ensure that groups are mixed (both boys and girls).

Ensure that everyone has an opportunity to talk and lead

the discussion.

Ensure that group leaders are both boys and girls.

Encourage both girls and boys to present results.

Ensure that both girls and boys record outcomes.

Activity 37

Have you tried one of the approaches discussed above in Table 6? If so, what have been your

experiences? If no, is there anything that stops you from trying them out?


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UNIT 5: ASSESSMENT

Introduction

Assessment is a crucial element in teaching and learning (Hattie, 2009). Quality assessment provides information to students, teachers, parents and the education system in effective and useful ways. To be helpful, however, it must be broad ranging, collecting a variety of information using a range of tasks before, during and after a teaching sequence. Assessment is more than the task of collecting data about students’ learning. It includes the process of drawing conclusions from the collected data and acting upon those judgements during teaching. Such actions may occur at many levels, but the key focus considered here is the classroom.

Learning Outcomes

By the end of this unit, participants will be able to:

Explain principles of formative and summative assessment in the competence-based

approach;

Understand the role of formative assessment in improving learners’ performances;

Conduct formative and summative assessment with the objective to improve learner’s

performance;

Support fellow teachers to organise formative and summative assessment activities and

use data from the assessment to improve learners’ performance;

Appreciate the role of assessment within quality mathematics teaching and learning.


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Section 1: Formative and Summative Assessment

Formative Assessment

Activity 38

Can you explain the difference between formative and summative assessment using the cartoon

below (Figure 29)?

Figure 29: Formative and Summative Assessment

The goal of formative assessment is to monitor student learning frequently to provide feedback

for teachers to improve their teaching and for students to improve their learning (Black & Wiliam,

2001). More specifically, formative assessment:

helps students identify their strengths and weaknesses and target areas that need attention;

helps teachers recognize where students are struggling and address problems immediately;

enables teachers to build on learners’ prior knowledge, and match their teaching to the

needs of each learner

Formative assessments are low stakes, which means that they have no or a low impact on students’


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final grades. Examples of formative assessments include asking learners to:

draw a concept map in class to represent their understanding of a topic;

complete a short quiz at the start or end of the lesson;

write short notes summarizing the main ideas of the lesson;

work in groups to make a poster or presentation on a topic.

use voting cards to answer the teacher’s questions

Black & Wiliam (1998) argue that teachers need to focus more on ‘formative’, rather than ‘summative’

assessment. They recommend small, frequent tests that require good feedback. It is the feedback on

what they don’t know, not that which the student got right, that leads to learning (Black & William,

1998).

Summative Assessment

Bloom, Hastings, & Madaus (1971) define summative evaluation as assessment given at the end of

units, mid-term or at the end of a course, and which is designed to judge the extent of students’

learning of the material in a course, with the purpose of grading, certification, evaluation of progress

or even for researching the effectiveness of a curriculum. The goal of summative assessment is to

evaluate student learning at the end of a unit or term by comparing it against standards or outcomes

(Black and Wiliam, 2001).

Examples of summative assessment include:

a midterm exam

P6 national examination

a final project

Unlike formative assessment, summative assessment is not part of the instructional process.

Summative assessments happen too far down the learning path to provide information at the

classroom level and to adjust and intervene during the learning process. Another distinction

between formative and summative assessment is student involvement. If students are not involved

in the assessment process, formative assessment is not practiced or implemented effectively.

However, formative and summative assessment are connected (Figure 30). Information from


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summative assessment can be used formatively when students or teachers use it to guide their

efforts and activities in their teaching.

Figure 30: Formative versus Summative Assessment

Source: https://improvingteaching.co.uk/2016/12/11/a-classroom-teachers-guide-to-formative-assessment/


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Section 2: Conducting Formative Assessment

Activity 39

Think individually about what techniques that you can use from unit 3 to conduct formative

assessment in your class. After a few minutes, exchange your ideas with your neighbour.

Techniques to practice formative assessment in your teaching

1. Share learning objectives with learners

Formative assessment involves both the teacher and the learners. Therefore, the first step is that

learners know what the learning objectives of the lesson are. Often, the teacher knows why the

students are engaged in an activity, but the students are not always able to differentiate between the

activity and the learning that it is meant to promote. Explicitly sharing the learning objectives will

direct students’ attention to the learning. The learning intention is expressed in terms of knowledge,

understanding and skills, and links directly with the curriculum.

The design of learning intentions starts with the answers to these questions.

What do I want students to know?

What do I want students to understand?

What do I want students to be able to do?

When students know the learning objectives of a lesson, they are helped to focus on the purpose of

the activity, rather than simply completing the activity.

The teacher shares these learning objectives with her students, either verbally or in writing.

Sometimes the learning objectives are written on the board and shared with students at the

beginning of a lesson or unit. At other times, it is not mentioned until after the activity.

2. Plan assessment opportunities during lessons

Researchers recommend small, frequent tests that result in good feedback. It is the feedback on

what they don’t know, not that which the student got right, that leads to learning (Black & William,


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1998). As well as informing teachers, planned assessment should also help learners become more

aware of what they still need to learn and how they might go about learning these things.

Research in Rwanda found that there is little or no time to gather, analyse and use assessment

information to improve learning and inform planning. This prevents teachers’ ability to get to know

their learners personally, differentiate appropriately, as well as improve the effectiveness of teaching

(REB, 2017). A common feature of bad lessons is the failure of teachers to make regular checks

on students’ learning and their determination to continue with the planned work even when the

students clearly do not understand it.

3. Encourage self-assessment and peer-assessment

Studies on formative assessment point to the value of learners assessing themselves. Through this

process learners become aware of what they need to know, what they do know, and what needs

to be done to narrow the gap. One way of achieving this is to give copies of learning objectives to

learners, ask them to produce evidence that they can achieve these objectives and, where they

cannot, discuss what they need to do next. Over time, it is also possible to foster a collaborative

culture in which learners take some responsibility for the learning of their peers. This involves making

time for learners to read through each other’s work and to comment on how it may be improved.

4. Give feedback that is useful to learners

Evidence suggests that the only type of feedback that promotes learning is a meaningful comment

(not a numerical score) on the quality of the work and constructive advice on how it should be

improved (Nicol, 2007). Indeed, grades usually detract learners from paying attention to qualitative

advice.

The research evidence (Black & Wiliam, 2001; Nicol, 2007; Hattie & Timperley, 2007) clearly shows

that helpful feedback:

focuses on the task, not on grades;

is detailed rather than general;

explains why something is right or wrong;

is related to objectives;


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makes clear what has been achieved and what has not;

suggests what the learner may do next;

describes strategies for improvement.

This doesn’t necessarily mean writing long comments at the bottom of each piece of work. It is

helpful to give comments orally and then perhaps ask learners to summarise what has been said in

writing.

5. “My Favourite No” Technique

Students will answer a question provided by their teacher and then analyse a wrong answer given by

a classmate (Lemov, 2015). The purpose of this activity is for the teacher to quickly assess how many

students are understanding the concept and for those who are not, what exactly is causing their

misunderstanding. It is essentially a formative assessment that works well as a warm-up activity.

It is important to foresee enough time for the analysis of the wrong answer. “My Favourite No” is

a teacher’s strategy that helps students to realize that wrong answers are an important part of the

learning process.

Key elements of this technique are:

Select an error that is commonly made by students or that reflects important misconceptions

for the topic.

Start with what is good in the answer

Move to what is incorrect in the answer and create a dialogue about the error.

Activity 40

Example: https://www.teachingchannel.org/videos/class-warm-up-routine#

Questions for discussion:

What criteria does the teacher use to pick her “favourite no”?

How does the teacher use assessment data to inform her teaching?


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6. “Exit ticket” Technique

An exit ticket is a brief evaluation that students write and turn in before the end of the class (Lemov,

2015). It should have only 2 or 3 short questions or problems and show what they have remembered

from the day’s class. This can provide valuable information on who learned what and who needs

more help. It can help you respond to individual students’ needs and decide on what to focus in

the next lesson. It is a kind of formative assessment that informs the teacher, but also the learners

about how well they have understood the key outcomes of the lesson.

Good exit tickets:

Contain just a few questions.

Contain questions of different types (e.g., one multiple choice, one open-ended question)

Answers can be analysed quickly by the teacher.

Questions relate to the key objective(s) of the lesson.

Questions that encourage student self-reflection can also be used, possibly in combination with

content-oriented questions:

What did you find the most important idea of the lesson?

What did you find difficult and would like more exercises or explanation on?

How does the lesson relate to what you have learned before?

Write one question you still have

Table 7 shows two possible templates for an exit ticket. The main idea is that it is short and allows

you to get a quick insight in students´ mastery of the key outcomes of the lesson.


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Table 7: Templates for Exit Tickets

An exit ticket should give you quick data. It is important that you follow up on the results of the

exit ticket. If most students have a problem with the first question, look at the kinds of problems

students encountered, and model the way to correct the problem (Lemov, 2015). You select some

common mistakes for discussion the next day or you can put some students in a separate group for

remedial instruction or exercises.

More information on exit tickets: https://buildingmathematicians.wordpress.com/2016/07/04/

exit-cards-what-do-yours-look-like/

7. Using “Traffic Light Cards” and “Voting Cards”

Traffic light cards and voting cards are cards that are used by learners to respond to questions from

the teacher (Figure 31).

Traffic light cards are used by learners to communicate their understanding about a topic:

1. Raising a red card means: “I’m stuck, I need some extra help”

2. Raising an orange cards means: “I’m not quite sure, I need a little help”

3. Raising a green card means: “I fully understand, I don’t need any help”

A teacher can use the technique at the end of parts within a lesson. A lot of red cards mean that

many learners are still struggling. It shows the need for additional instruction or more exercises. A

situation with few yellow or red cards shows the teacher that some learners do still have problems.


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They may be taken apart by the teacher for additional explanation. If there are some learners with

green cards, the teacher may ask them to explain the concept to those with red or yellow cards.

Learners may vote not according to what they think, but what others do. Therefore, it is good

to follow up the voting with a few questions like: “Emile, you voted red, what is it that you find

difficult?”, or, “Emmanuel, you voted green, can you explain the key idea to the others?”

Figure 31: Traffic Light Cards and Voting Cards (TES, 2013)

Voting cards are used by learners to vote for a specific answer on a question by a teacher. This can

be a true-false question (Figure 31) or a multiple-choice question (Figure 32).

Figure 32: Voting cards with letters


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You can print these cards for your learners. If possible, laminate them, so they will keep longer. You

can combine colours and letters on the front and back side.

Example of a formative assessment technique: “My Favourite No”

1. Introduce the activity

At the beginning, it is important to reinforce the purpose of the activity for the students. Over time,

this will become unnecessary as students get used to the activity. Share the purpose with students

and stress that analysing the wrong answer is a great opportunity for learning and that it is not about

punishing students for wrong answers.

Share the question with students and, if possible, distribute small papers.

It is important that your question has a right and wrong answer and be complex enough for it to

justify an analysis. For example, have students solve a multistep problem, create a diagram to

represent a word problem and solve it, develop a definition or describe a process.

2. Students answer the question

This should be a timed activity. Keeping it to less than five minutes as a warm-up/do now activity is

a general recommendation.

3. Collect the answers and tally the results

Sort the answers aloud into simple “Yes” and “No” piles. Share the data with students, e.g., 32

students were correct, 20 students were not.

4. Select the Favourite No

As you sort the index cards think about what the “Yes” pile did that the “No” pile did not do. What

is the mistake most students are making? Which student’s answer would help you get to the heart

of the misunderstanding the best? What answer can serve to address a common misconception or

enhance a fragile understanding of the topic? You will have to decide quickly. One way to help in

this process is to prepare for this when writing the question. Have specific things in mind that you

are looking for in the right answers and anticipate where students may show misunderstandings.


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If you find it difficult to do this immediately, you can also review the answers and come up with a

Favourite No during the next lesson.

5. Share the Favourite No

As you share the Favourite No, it is important to emphasize two things. First, that this is the wrong

answer. Second, that everyone makes mistakes and it is about learning from our mistakes. Don’t

mention the name of the student whose answer you share.

6. Analyse the positives of the answer

Ask the class to analyse what this student did right in the answer. Sample questions include: · What

in this problem am I happy to see? · What is right? · What do you think I like about this answer?

It could be that the student had part of the calculations correct or knew he had to multiply.

7. Analyse what made the answer wrong

What made this answer incorrect? ·Where did this student make a mistake? ·How do you know that

it is the wrong answer? You want students to explain their thinking as they analyse the answer.

8. End on a positive note

Do something that acknowledges the difficulty in having a student’s wrong answer analysed by

the class. An example might be a quick applause for the anonymous person whose answer was

analysed. It could be a simple statement of encouragement from the teacher, “We’re all working to

get better…”

Activity 41

In pairs, prepare a “My favourite no” activity on a mathematics topic of your choice.


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UNIT 6: ACTIVITIES PER CONTENT AREA

Section 1: Numbers and Operations

Number Lines

Number lines are a useful tool to help learners develop a sense of the meaning of numbers in the

early primary years. They are also useful to gradually let learners develop the concept of place value,

one of the most important concepts in primary maths. A number line is a graduated straight line

that serves as representation for real numbers. In this section, we introduce a variety of questions

and activities with number lines.

1. Comparing numbers on a number line

Number lines are useful to develop a sense of the relative size of numbers with learners.

Which number is bigger?

2 or 5 11 or 9 -2 or 5 -5 or 2

How do we decide? Place both numbers on a number line:

4.37 or 3.5737 1.8 or 1.08 -4.3 or 3.7

2. Using an empty number line

How do we label the number the arrow is pointing at? How do we use place value to help us with

the label?

Label 3,2 5,9 6,7 on this number line. Label a new number line with 6.7 6.17 and 6.71.


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3. Use an empty number line to show that:

Number lines can be used to develop learners’ understanding in place value, such as the meaning of

unit, tenths, hundredths and thousandths.

1.7 lies between which two successive units?

1.73 lies between which two successive tenths?

1.738 lies between which two successive hundredths?

4. Dealing with misunderstandings in number sense

This activity exposes a frequent misconception with learners: that 7.10 is bigger than 7.9. Edouard

wrote 7.10 in the empty box on the number line below. Why would he write this? Describe how you

could help Edouard to find the correct answer.

5. Subtraction and use of number lines

Number lines can be used to teach mental calculation strategies for addition and subtraction.

Jeanne and Thomas want to calculate 253 – 99 by first calculating

253 – 100 = 153

Jeanne says that they must now subtract 1 from 153, but Thomas says that they must add 1 to 153.

Draw a number line to help you explain who is right and why.

Place Value

Place value is a key concept in primary mathematics, as it forms the basis for numbers and operations.

The following activities can help learners acquire a sound understanding of place value.


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6. Counting Strategies

The first step in developing number sense is developing strategies for counting objects faster. In

this activity, you need toothpicks (or another counting object) and a die. The activity helps learners

explore how they can count faster by grouping. It is a good way to introduce the concept of tens.

Counting collections also introduces ideas about how the place value system helps counting.

In small groups:

1. Toss the die, then multiply the number by 6

2. Represent this total with bundled toothpicks

3. Toss the die again, then multiply the number by 5

4. Represent this total with bundled toothpicks

Combine the two bundles (or dried beans, bottle caps…) and calculate the overall total. After

students have done their counting, discuss strategies children used for counting. Was it easier to

count by 2s? By 10s? What other strategies did learners use? Did all the groups who counted the

same thing get the same answer? Which counting methods are most accurate? Which are easiest?

7. Rounding off to the nearest… activity

Many learners think that rounding off means always rounding to a higher number. Using a number

line for the exercise below can help learners understand that rounding off can result in a lower of in

a higher number.

Round 34.617 to:

The nearest five

The nearest ten

The nearest hundred

The nearest tenth

The nearest hundredth


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Round 12.56 to these values as well.

Round 999 253. 34 to all these values as well.

8. Draw a secret number

Draw a number line with a 0 and 200 at opposite ends of your line. Mark a point with a question

mark that corresponds with your secret number. Estimate the position the best you can. Students

guess your secret number. For each guess, place and label a mark on the line that corresponds with

the number guessed.

Continue marking each guess until your secret number is discovered. You can vary in the endpoints.

For example, try 0 and 1000, 200 and 300 or 500 and 1000. It is important that you mark the guesses

of the learners. Labelling those numbers at the correct locations will support students’ reasoning in

the process of identifying the secret number.

After you played the game with the whole class, learners can play it in small groups.

9. Close, far and in between

This activity is useful for developing learners’ sense of place value and their skills in basic operations

(Van de Walle et al., 2015).

Put any three numbers on the board. Use numbers that are appropriate to the learners’ level (for

example, 257, 344 and 405). Starting from these 3 numbers, ask questions such as the following and

encourage discussion, for example through voting.

Which two are closest? Why?

Which is closest to (200)? To (450)?

Name a number between (257) and (344).

Name a multiple of 25 between (257) and (344).

Name a number that is more than all these numbers.

About how far apart are 257 and 500? 257 and 5000?


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10. Which one doesn’t belong?

Activities like this one are useful to organize mathematical conversations. Let learners with different

answers reason about their answer. You can also let learners work in groups to develop their own

“which one doesn’t belong” questions and let them solve each other’s questions.

Provide students with lists of numbers and asking them to argue why one of the numbers doesn’t

belong to the list. There can be different valid solutions, as long as the arguments are sound. For

example,

1/2 5/3 2/10 1/5

Another example:

0.25 ¾ 0.8 0.5

Another example:

Addition and Subtraction

11. Drawing number lines for addition and subtraction problems

It is important to help learners notice the different situations in which to use subtraction and the

language that you use when talking about subtraction (Page, 1994). Many students in early grades

only know the take-away meaning for subtraction. For problems such as 100 – 3 = ___. Thinking

in terms of take-away serves many students well. A popular strategy is to start from 100 and count

down (99, 98, 97), often using fingers. For problems like 100 – 3, this way of reasoning is good,

because the subtrahend is small (the student must take away only 3). However, in 201 – 199, the

difference is small, but the subtrahend is large. In these situations, thinking about subtraction as


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take-away can be highly inefficient, whereas thinking of the difference as the distance between the

given numbers is more useful.

Drawing number lines help learners view addition and subtraction problems as distances between

numbers and make connections between the ideas of addition and subtraction, counting forward

and backward and even linear measurement. It reinforces their insights in the relationships

between numbers and their mental mathematics competences. Seeing differences as distances

between numbers also works better when working with negative integers, for example for 3 – (–5).

Finally, reasoning about differences in terms of distance is good preparation for the transition from

arithmetic to algebra.

Use a number line posted on the wall of your classroom when discussing subtraction problems and

strategies or make it a habit to draw a number line with addition and subtraction problems.

For example, consider 81 -29 (Figure 33).

Figure 33: Using a number line for subtractions

12. Use a variety of word problem types for addition and subtraction

It is important that learners can recognize various problem types (Carpenter & Lehrer, 1999) in word

problems, including

Joining Situations (variations: result unknown, change unknown, start unknown)

Separating Situations (variations: result unknown, change unknown, start unknown)

Part-Part-Whole Situations (Whole Unknown, Part Unknown)

Comparison Situations (Difference Unknown, Larger Quantity Unknown, Smaller Quantity

Unknown)


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For example, the following word problems contain a combination of addition and subtraction

situations. Discuss with your learners which are addition and subtraction problems and why. Use

number lines (and let them draw number lines) to visualize each problem.

▪ Eline has 23 apples. She got 18 more apples. How many does she have now?

▪ Pierre has 23 apples and 29 bananas. How many pieces of fruit does he have?

▪ Alex had 38 apples. He gave away 19 apples. How many does he have now?

▪ Chris has 17 tomatoes. Pierre has 15 tomatoes more than Chris. How many tomatoes does

Pierre have?

▪ Fabrice has 16 apples. Benny has 46 apples. How many fewer apples does Fabrice have than

Benny?

▪ Elsie has 12 mangoes. How many more mangoes does she need to have 30 mangoes

altogether?

▪ Marie has 12 red triangles and 3 blue triangles. How many more red triangles does Marie

have than blue triangles?

▪ Farida had some pencils. After she got 5 more pencils, Farida had 22 pencils altogether. How

many pencils did Farida get?

▪ Eugene is reading a book that has 462 pages. He has 148 pages left to read. How many pages

has he read?

▪ In a bag of 74 marbles, 45 belong to Pierre and the other belong to Marie. How many

marbles does Marie have?

13. Let students create their own word problems

A powerful, inclusive exercise is to let students create their own word problem based on a given

addition or subtraction. Discuss the variety of responses with the learners and try to include different

types of addition and subtraction problems in the discussion. Apart from developing learners’

problem-solving skills, this kind of exercises also strengthens their correct use of mathematical

language. You can extend this exercise to include multiplication and division, as well as decimal and

negative numbers.


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1. Create a problem for 5+ 3, 12 -4…

2. Create change problems and part-part-whole problems for 5+ 3, 12 -4

3. Create a compare problem (comparison of a larger quantity and a smaller quantity) for 5+ 3,

12 -4

Box: Further Reading

https://buildingmathematicians.wordpress.com/2016/11/25/subtracting-integers-do-you-see-

it-as-removal-or-difference/

http://mathforlove.com/lesson/pyramid-puzzles/

Multiplication and Division

Introduction

The key difference between additive and multiplicative reasoning is that additive reasoning is

based on thinking about how quantities are related in terms of how much more or less, whereas

multiplicative reasoning is based on thinking about how quantities are related in terms of how many

times more or less (Beckmann, 2013).

In the early grades, the emphasis should be on making sense of multiplication and division situations

and represent them. Make explicit connections between skip counting (addition) and multiplication

situations. Use various multiplicative situations like scaling up (e.g. doubling or ‘three times as many

children’) and scaling down (halving or ‘a quarter of the chocolate bar’) and linking them to students’

daily lives.

In upper primary, the emphasis should be on introducing various models that support children

with multiplication and division. In this, it is important to focus on sense-making (conceptual

understanding), rather than only on the procedures. Students need to get familiar with various

situations that can be modelled through multiplication (as repeated addition, rate, scaling) and

division (sharing and grouping) (See Table 8).


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Table 8: Meanings of Multiplication

SIMPLE RATIOS If a mango costs 500 Frw, how much will I pay for 5 mangoes?

This question has an implicit ‘per item’ built into it: 500 Frw

per mango, and so is a very simple proportion problem: 1

(mango) is to 500 (Frw) as 5 (mangoes) is to 2500 (Frw).

REPEATED ADDITION On Monday Michel saved 800 Frw. On Tuesday, he saved 800

Frw and on Wednesday he saved 800 Frw. How much did

Michel save altogether?

CARTESIAN PRODUCT Tom has 4 t-shirts and 3 pairs of jeans. How many days can he

go out and wear a different combination of t-shirt and jeans?

SCALING MEASUREMENTS On Monday, Alice’s beanstalk was 15 cm tall. On Friday, it was

5 times as tall. How tall was the beanstalk on Friday? How

many times bigger (or smaller)?

MULTIPLE PROPORTIONS A jug of milk provides enough milk to fill five saucers. A pail of

milk will fill four jugs. How many saucers of milk can be filled

from a pail of milk?

14. Using various visual models of multiplication situations

Introduce various visual interpretations of multiplication. Depending on the problem, one

representation might be more suitable than others. Multiplication situations can be represented by

an area model, a double number line or a simple number line.

1. Area Model

Figure 34: Area model for multiplication


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2. Double number line

For example:

1. Petrol costs 1000 Frw for 1 litre. What is the cost of 15 litres?

2. If James earned 12 000 Frw in 8 hours, how much would he earn in 3 hours?

Figure 35: Using double number lines to represent multiplications

3. Scaling on a number line

Last month, Fabien had 14 marbles. Now he has 3 times as many marbles. How many marbles does he have?

Figure 36: Scaling on a number line

Clarifying the relation between multiplication and division

For their conceptual understanding, it is important to let learners discover the relation between

multiplication and division. For example, 6 bags each hold 7 mangoes. How many mangoes are

there altogether? This is an example of a multiplication question. 42 mangoes are shared equally

into 6 bags. How many mangoes does each bag contain? Now, the problem has become a division

problem (see Table 9.


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Table 9: relation between multiplication and division

Multiplication Division as sharing

Let learners practise reformulating multiplication problems so they become a division problem. Let

them distinguish between division as sharing (number of groups is known and size of each group

is unknown) and grouping (number of groups is unknown, but size of each group is known). For

example, write a simple word problem and make a math drawing to help children understand what

10 ÷ 2 means (Beckmann, 2013).

Many students find it difficult to understand the “how many groups” interpretation of division.

However, this is the model that makes the most sense for the division of fractions (Beckmann, 2013).

This interpretation of division can also impact a student’s ability to be successful with long division.

For example, students need to be able to think, “How many groups of 30 are there in 1429?”

15. Identifying patterns in multiplications

Although learners should be able to solve multiplication problems with the standard algorithm,

it is useful to let them also look at multiplications (and divisions) without immediately using this

algorithm. Sometimes, there are easier and faster ways to solve a multiplication problem. It is good

when learners master different procedures to solve a problem. Not only can they select a procedure

according to the context, they can also verify a result obtained with one procedure by using another

procedure.

Present learners with these sequences of multiplications. How can learners solve each one based

on the result of the previous one? It is important that you give these problems in series, so learners

can discover the relations between them. Discuss the relations with your learners. Let learners use

correct mathematical language to describe patterns and relationships they notice.

48 x 26 448 x 25 448 x 2,5 2,5 x 8,4

10 x 8 2 x 8 2 x 8 6 x 16

6 x 10 6 x 40 6 x 39 6 x 41


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18 x 10 18 x 9 19 x 18 21 x 18

6 x 8 12 x 4 3 x 16 32 x 1,5

21 ÷ 7 70 ÷ 7 91 ÷ 7 42 ÷ 14

24 ÷ 2 24 ÷ 4 24 ÷ 8 24 ÷ 16

12 ÷ 2 24 ÷ 4 48 ÷ 8 48 ÷ 16

160 ÷ 16 320 ÷ 16 640 ÷ 16 1240 ÷16

6 x 20 6 x 100 6 x 120 6 x 119

Other examples to practise reasoning skills in mathematics are estimation questions:

What is the rough cost (no detailed calculation) of 21 cans of coke costing 360 Frw each?

Other examples:

2.6 x 4,8

1.26 x 0.5

Estimate how much is 102 x 102.

Estimate how much is 102 x 98.

Estimate how much is 21/23 x 8/9

16. Numbers and operations game

This activity can be used as a game to practise learners’ skills in basic operations. You can make the

sequences as difficult as you like.


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Given a set of 5 numbers, try to get as close as possible to the number on the top by using addition,

subtraction and multiplication with the numbers below:

30 45 619 9 91 6 83 11 77 3 34 2 11

17. Word problems with multiplication and division

Here are some examples of simple word problems for multiplication and division. It is good to mix

word problems (different meanings of multiplication and division), so that learners are stimulated to

think for each problem. Stimulate learners to make drawings of the word problem. Let them explain

the problems to each other and let them construct their own problems.

6 bags each hold 14 mangoes. How many mangoes are there altogether?

1/3 of the children in a class have a white shirt. ½ of those children also have black trousers.

How many children in the class have black trousers and a white shirt?

A recipe needs 2/3 kg of sugar. You only want to make ½ of the recipe. How much sugar

should you use?

A pharmacist has 7,5 l of a cough mixture. She wants to distribute it in bottles of 0,25 l each.

How many bottles can she fill with the mixture?

You have 2/3 of a pie left over from Christmas. You want to give 1/2 of it to your sister. How

much of the whole pie will this be?

A pharmacist has 2.5 l of a cough mixture that she wants to distribute equally over 6 bottles.

How much can she put in each bottle?

Elisa prepared 12.4 l of mango juice. She wants to distribute the juice equally among 30

children. How much juice will each child get?

A kilogram of potatoes costs 400 Francs. How much will you pay if you buy 6 kg of potatoes?

A water tank holds 235 l of water. Albert wants to divide the water into pots of 5 l. How

many pots can he fill?

The district has a fence of 740 m. It wants to plant a tree every 12 m. How many trees can

the district plant?


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The government donates 52 198 books to 5 schools with 9 classes each. It wants to give each

class the same number of books. How many books will each class receive?

The village has harvested 14 820 kg of beans. It wants to give families 250 kg of beans. How

many families can be given beans?

A field is 320 metres long and 78 metres wide. What is the field’s perimeter? What is its area?

6 lengths of fencing are each 8 metres long. How much fencing is there altogether?

A litre of petrol costs 940 Frw. How much would 8 litres cost?

A man’s shadow is 3.5 times as long as his height. If he is 1.73 metres tall, how long is his

shadow?

How many teams of 15 can be formed from 263 children?

How many coaches (allowed to carry a maximum of 42 passengers) will be needed to

transport all the children?

Using place value and number lines in calculations

This activity develops learners’ skills in using place value for mental mathematics. These basic operations can be solved with the standard algorithms but can also be solved more quickly using place value and number lines. When students are familiar with the strategy, they can use the number lines only in their head instead of drawing them. It is important that learners are familiar with different strategies to solve basic operations problems. In some cases (such as with the examples below), using place value and flexible grouping strategies involving the use of 5/10 (“friendly numbers”) is quicker than using the standard algorithm. Encourage students to notice when this strategy is helpful, depending on the numbers in the problem.

199 + 199

265 + 197

199 + 299

104+98

4265 + 147 + 949

4307 – 609

48 x 6

98 x 19

642 ÷ 3


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Section 2: Fractions, Decimal Numbers and Percentages

Introduction

When first learning to count, children often use their hands or physical objects as tools to help

the process of counting with whole numbers. However, when the time comes to expand one’s

number concept, the safety of the fingers or physical objects reaches an end. Linked to this, there is

the phenomenon of Natural Number Bias in which students continue applying the rules of natural

numbers (e.g., larger digits mean larger numbers) to rational numbers (e.g., larger digits can also

be an indicator of smaller numbers: 2/3 > 5/9; 2.20>2.025), even when these rules conflict with

each other (Vamvakoussi, Van Dooren, & Verschaffel, 2012). Research shows that Natural Number

Bias is the biggest difficulty to overcome in understanding rational number concepts (Kainulainen,

McMullen, & Lehtinen, 2017).

Fractions, decimal numbers and percentages require big changes in learners’ concept of numbers

in aspects as their symbolic representation (discrete numbers vs. fractions, decimals, percentages),

their size (larger quantity of digits makes number larger vs. larger quantity of digits can make number

smaller or larger), and operations such as multiplication (makes numbers larger vs. can make numbers

larger or smaller) and division (makes numbers smaller vs. can make numbers smaller or larger).

For many teachers, the topic of rational numbers is a difficult topic for teaching, because children’s

ways of understanding rational numbers may be very difficult for teachers to observe (Moss

& Case, 1999; Nunes & Bryant, 1996). Students may go through school without understanding

the qualities of fractions, without anyone noticing it. Nevertheless, fractions are a key concept in

primary mathematics. They form the basis for understanding decimals and percentages, algebra

and probability.

Ideas to introduce fractions in the early grades

Introduce a variety of situations where learners need to share something equally. Use various types

of units (chocolate bars, bananas, pencils) and contexts.

For example,

Elsie and David want to share 3 chocolate bars equally. Show them how to do it.

Elsie, David and Laurence want to share 4 chocolate bars equally. Show them how to do it.


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Elsie, David, Laurence and Fabien want to share 5 chocolate bars equally. Show them how to

do it.

Comments:

There are more objects (chocolate bars) than children

Allow children to make sense of the situation and to draw the solution – they do not need

the fraction names or notations yet.

Discuss the different plans that children in the group made.

Reason for the choice of chocolate bars: rectangular objects.

Young children have already been introduced to the idea of a ‘fraction’ before formally

learning about the concept in school using language such as: ‘a small piece’, ‘a little bit’.

Fractions should be introduced to young children using real problems that involve dividing or

breaking – which support them to come up with their own solutions. Fractional terms like

one half can be introduced as the need arises.

Introduce various possibilities for what a unit is:

Figure 37: Various units for fractions


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Figure 38: Various meanings of one quarter

Why do learners find fractions difficult?

1. They express a relative rather than fixed amount

2. The same fraction can refer to different quantities

3. The same quantity can be expressed by different equivalent fractions

4. Any fraction can refer to objects, quantities or shapes

5. The rules for whole numbers do not always apply

6. A fraction can be a part of a shape or shapes, a part of a set of discrete objects or a

position on a number line (a number in its own right).

¾ can mean many things:

1. Three parts of a pizza cut into four equal parts.

2. The result of four hungry children equally sharing three pizzas.

3. The fraction of counters that are red if there are four counters on the table, three red

and one white.

4. The likelihood of turning over an even number card if cards with 1, 2, 4 and 6 on are

face down on the table.

5. The fraction of a puppy’s length if it is 12 cm long and its mother is 16 cm long.


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¾ can also be expressed in (an infinite) number of different ways (equivalent fractions), including,

6/8

30/40

0.75

75%.

Use of Double Number Bars and Ratio Tables

Double number bars can be used when two quantities that are in relation to each other are measured

in different units.

For example, I buy 2 mangoes for 900 Frw. How many mangoes can I buy with 1800 Frw?

Double number bars help children make the move from additive to multiplicative reasoning.

They can also be used to link multiplication and division. For example:

I am putting apples into bags. There are six apples in each bag. I fill seven bags. How many apples is

that? (Multiplication).

I am putting apples into bags. There are six apples in each bag. I have 42 apples. How many bags can

I fill? (Division as repeated subtraction/grouping)

Rene was putting photos into an album. He put the same number on each page. He put 6 photos on

each page. He had 42 photos. How many pages did he fill?


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Sarah was putting stickers into an album. She put the same number on each page. She filled 7 pages.

She had 35 stickers. How many stickers did she put in each page?

Activities

1. Sketch these diagrams and shade in one tenth of the diagram in each case. Sketch the

diagrams again and shade one fifth of the diagram. How can we write the answer in each

case?

This exercise introduces various units of fractions. In discussing the cases, you can move between

part/whole relation, fraction, decimal notation and %.

2. What fraction is coloured blue?

Use representations such as the ones below to familiarize students with various fraction units.


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3. One tenth is always smaller than one fifth. Correct?

This kind of question lets students actively engage with and discuss frequent misconceptions, based

on differences between fractions and integers (Beckmann, 2013).

4. Shade a quarter

Use various shapes and units and let learners shade various fractions.


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5. Ordering fractions

Letting learners order fractions from smallest to largest is a good exercise to develop their

understanding of fractions. You can do this as a think-pair-share. During the class discussion,

stimulate reasoning by students. Examples are:

1/2, 1/5, 1/3, 1/7 and 1/10

1/4, 11/6, 3/8, 1/16 and 3/4

3/4, 5/3, 6/7 and 1/6

6. Fractions and Proportions

What fraction of the square do A, B and C represent? What fraction do we get when we put A and C

together? A and B together? B and C together?

How many times bigger is A than B?

A = __ of B

B = __ of A

What fraction is half of B? And half again? What fraction is one third of A? What proportion of the

whole do A and B make together?

This question directly aims to address common misconceptions about fractions – that equal shares

means identical appearance, or same shape, rather than same proportion of the overall unit. In this

question, the areas A and C can be represented by the same fraction (1/4).


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7. Using a Fraction Wall to relate fractions to decimal numbers and percentages

Fractions walls are a useful instrument to practise with students the operations with fractions, and

the relations between fractions, decimal numbers and percentages.

One possibility is to let students write each fraction as a decimal number and a percentage. Discuss

which ones are easy to write as decimals and %. Why are they easy? Secondly, you can pose

questions like: Can you find a fraction/decimal in between two other fractions and decimals? You

can let learners play a game. One learner chooses two fractions or decimals on your line. The other

learner must name a fraction between the two. For example, find a fraction between 2/5 and 3/5,

between 5/8 and 6/8, between 1/3 and ¼.

Using a fraction wall, let learners solve problems like:

Which fraction is bigger?

5/6 or 4/6

3/7 or 3/8

7/8 or 8/9


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8. Thinking in proportions: making juice example

There are four mixtures of juice and water (A, B, C, D). Which juice is the tastiest (which mixture has

proportionally the most juice in it)?

With this question teachers can link part-part language of ratios to part-whole language of fractions.

You can take the question further. What happens when we make more juice (see figure below)? Do

both juices still taste the same?

A and B make some juice. 1-part juice to 2 parts water. But B decides he wants some more juice, so

he adds two more parts of juice and 2 more parts of water to his juice. Will his juice still taste the

same as the first juice he made?


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9. Word problem: Which is better?

Afrodis got 17/20 on a mathematics test. He got 22/25 on a science test. Joe says he is as good at

mathematics as he is at science because he got 3 questions wrong on each test. Draw a diagram to

show whether Afrodis is correct.

10. Word Problems on proportions

A muffin recipe needs flour and milk in the ratio 9: 2. How many cups of milk would be needed to go

with 21 cups of flour?

Therese has 8 tins of cool drink. How many glasses can she fill from the 8 tins, if one glass takes

exactly three fifths of a tin? Use a diagram to work out the answer.

11. Proportions involving Fractions

Use double number bars to work out your answers for the missing values in the table


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12. Fill in the blanks

The questions below are examples of open questions that learners can solve at different levels.

Therefore, they enable differentiation at the task level.

13. Fractions and Number Lines

Number lines are useful to help learners understand the relative sizes of fractions. Use examples

such as the figures below.


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14. Word problem

The rectangle of *’s below is 4/5 of the original rectangle of *’s. Draw or mark the original rectangle.

The same rectangle of *’s below is now 5/4 of the original rectangle of *’s. Draw or mark the original

rectangle.

This question lets learners think about what the unit is in each fraction. It underlines the importance

of keeping the unit in mind when comparing fractions.


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Section 3: Elements of Algebra

Introduction

Algebra is a key tipping point in the study of mathematics for many children (Mason, 2008). Before,

mathematics makes sense to children, but algebra does not make sense to them anymore. They don’t

see a link between algebra and their daily life. More often, they experience a strong gap between

the concrete work with numbers and operations and the abstract nature of algebra (Kainulainen et

al., 2017). Therefore, preparing learners for algebra (algebraic thinking) should start in the early

grades, not through using x and y, but by introducing the ideas behind algebra, such as identifying

patterns. Algebraic thinking needs to be a logical and cohesive thread in the mathematics curriculum

from pre-school to high school (The National Council of Teachers of Mathematics (NCTM), 2007).

However, rushing students to represent patterns with letter symbols is counterproductive. Research

on patterns suggests that it is generally more profitable for young students to explore for long

periods of aspects of the generality in their patterns than to be exposed too quickly to the symbolic

representation of this generality (Moss, Beatty, McNab, & Eisenband, 2006).

“Algebra is a key tipping point in the study of mathematics for many children. Before, mathematics makes sense to children, whereas algebra does not make sense to them anymore.”

(Mason, 2008)

In the early grades, algebraic thinking comprises:

sorting and classifying

recognising and analysing patterns

observing and representing relationships

making generalisations

analysing how things change


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Young children are naturally curious about patterns and teachers can build on this curiosity. Children’s

work with patterns is an important developmental step on their journey towards algebraic thinking.

For example, let learners generalise about things that are the same and different in patterns. As

children explore and understand basic operations, they can look for patterns that help them learn

procedures and facts such as exploring patterns in the multiplication tables. These are interesting

to children and help them learn their multiplication facts and understand the relationship between

facts.

In this section, we will discuss some key idea of algebraic thinking and suggest some activities that

teachers can use to move from the concrete work with numbers and operations to the more abstract

nature of algebra.

Relational Reasoning

Relational reasoning is about finding an unknown quantity without calculating, but by using the

relationship between the numbers. For example, what should the missing number be on the second

line to keep the size of the gap the same?

We keep the gap the same by …. .[increasing both numbers in the initial relationship by the same

amount]. We have made a general statement. We can use the general statement to calculate missing

numbers in problems without calculating. For example: 68 - 39 = 69 - __ = 70 - __ = ...

… = 271 - __ = __ - 140.

For example, take 375 + 99. It is easier to adjust the sum to make it 374 + 100. This is algebraic

thinking. Research in UK with 11-year olds showed that only few learners used algebraic thinking,

most worked out the sum (Mason, 2008).


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Using quasi-variables helps learners to make bridges from existing arithmetic knowledge to algebraic

thinking without having to rely on knowledge of algebraic symbols (Fujii & Stephens, 2008). Examples

are open number sentences like 647 – 285 = [ ] – 300 or using a drawn cloud to represent the

unknown (rather than x and y, which should only be introduced later).

▪ 64 + 14 = [ ] + 64

▪ 64 + [ ] = 18 + 64

▪ 64 + [ ] = 18 + 62

Extend to sums like: 86 + 57 = 143 -> 88 + 55 = same. Learners need to understand that this sum

should give the same result.

Commutativity

Commutativity means that when we are adding, the order of the numbers does not matter. For

young learners, the word is not important, but the idea. Using diagrams or blocks to introduce

commutativity and show that the order in the addition does not matter. For children 99 + 3 is much

easier than 3 + 99. Often children learn to put the first number in their head and count to the next

number. We want children to recognize when it is easier to change the order of numbers in the sum.

Repeating and Growing Patterns

The main element to look out for in patterns is to expose learners to a variety of patterns (repeating

and growing, arithmetic and geometric) and visualisations. Familiarize learners with the key elements

for repeating and growing patterns:

Repeating patterns: The ‘unit’ that repeats

How many elements in this unit?

Growing patterns:

How it starts

How it grows

Learners need experience with growing patterns in both geometric and arithmetic (number) formats:


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▪ 5, 6, __ , __ , 9 , __

▪ __, 9, 12, __ , 18, __

▪ 70, __, 50, __ , __ , 20

▪

Some children have a limited understanding of patterns as only repeating. Children can extend

patterns, but have trouble describing and generalizing patterns. Use many exercises where learners

need to find elements far down the sequence (Moss et al., 2006). Mason (2008) suggests visualization

and manipulation of geometric patterns as a step towards construction of the rule.

Gradually, teachers should move from word descriptions to numerical and algebraic descriptions.

These allow to find out how the pattern will evolve without having to draw it.

For example, study the pattern, made of matchsticks, below.

How many squares and matchsticks would the:

▪ 10th picture have?

▪ 75th picture have?


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Can you write a general rule? How many squares and matchsticks would the n-th picture have? Use

the table below. Notice that with this question, learners gradually move to more abstract problem

solving.

Activities

1. True or False (relational reasoning)

Let the learners find out, discuss and explain why:

▪ 37 + 56 = 56 + 37

▪ 37 + 56 = 38 + 59

▪ 37 + 56 = 38 + 57

▪ 37 + 56 – 56 = 37

▪ 458 + 347 – 347 = 458

▪ 56 – 38 = 56 – 37 – 1

▪ 56 – 38 = 56 – 39 + 1

▪ 56 – 38 = 56 – 36 – 2

▪ 3 x 5 = 3 x 4 + 5

▪ 3 x 5 = 3 x 4 + 3

▪ 64 ÷ 14 = 32 ÷ 28

▪ 64 ÷ 14 = 32 ÷ 7

▪ 42 ÷ 16 = 84 ÷ 32

Next, move to open number sentences (using quasi-variables, represented by open brackets):

▪ 64 + 14 = [ ] + 64

▪ 64 + [ ] = 18 + 64

▪ 64 + [ ] = 18 + 62

▪ 647 – 285 = [ ] – 300

▪ 671 – 285 = 640 – [ ]

▪ [ ] – 285 = 640 – 285

▪ 3 x 5 = 5 x [ ]

▪ 3 x 5 = 3 x 4 + [ ]

▪ 3 x [ ] = 3 x 5 + 3


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2. Repeating Patterns

What is the repeating unit? How many elements does the repeating unit contain? Create a repeating

pattern with a repeating unit with 4 elements. Can you continue your repeating pattern? What would

go in 84th position? The 407th position? Ask a partner to answer these questions for your pattern.

Create a repeating pattern with a repeating unit with four elements using only ‘0’ and ‘1’

3. Growing Patterns

Make the matchstick pattern in the figure below.

How many squares are there in the 5th and 6th patterns?

How many matchsticks are there in the 5th and 6th patterns?

How many squares/matchsticks are there in the 12th/13th/23rd /407th positions?

How can we work out how many squares and matchsticks there will be (general rule):

▫ in the 18th picture

▫ In the 97th picture


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Other growing patterns:

76, 73, 70, 67, 64, …

▪▪ Describe the pattern precisely in words? Can you describe how it grows?

▪▪ Now express the pattern with numbers.

▪▪ Work out the next few numbers.

▪▪ Work out the number in the 114th position.

Make a repeating pattern and then a growing pattern with your matchsticks.

Describe your pattern in words using the critical features mentioned above

Give your description to a partner on another table. Can they re-create your pattern?

Let learners analyse the pattern below. Is it a growing or a repeating pattern? Have them make

a table (see below). How many blocks are there at the 6th position (Note: common mistake = 20,

versus 19). How many blocks at the nth position?


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4. Distinguishing repeating and growing patterns

Can you describe these patterns? What kinds of questions can we ask about these patterns?

In what ways are these two patterns similar? In what ways are they different from each other?

Continue the pattern. Describe the pattern. In what ways are these four patterns similar to each

other? How are they different from each other? What would be in the 12th position in each pattern?

The 13th position? The 23rd position? The 108th position?

5. Using pattern cards or counters to let learners explore patterns

Use counters and number cards to let learners construct their own pattern. Next, you can let learners

try and recognize each other’s patterns. They must also be able to describe the pattern (repeating

or growing, unit…).


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Use group work and class discussion to construct understanding on patterns:

1. What makes some patterns easy for others to recognize the rule?

2. Think of one of the patterns around the room that might have been more difficult for you

to figure out.

Sources: https://buildingmathematicians.wordpress.com/2016/08/27/how-do-you-give-feedback/

http://www.nelson.com/linearrelationships/From%20Patterns%20to%20Algebra%20Sampler%202012.pdf

6. Word problem: Frog activity

This problem introduces algebraic thinking, without already using symbolic language.

Francine the frog is a champion precision jumper. All her jumps are the same size (as are her steps).

Francine makes 4 jumps and 8 steps. For her that is exactly the same as 52 steps. How many steps

is a jump?

A good strategy to deal with this kind of word problems is “specialize, then generalize”. First, explore

the specific problem with drawings, tables etc. Then, use other numbers and try to find a general

rule. You can find an example of a drawing below.


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7. Cube stickers (Moss et al., 2006)

This is another word problem that introduces algebraic thinking.

A company makes coloured rods by joining cubes in a row and using a sticker machine to put “smiley”

stickers on the rods. The machine places exactly 1 sticker on each exposed face of each cube. Every

exposed face of each cube has to have a sticker. This rod of length 2 (2 cubes) would need 10 stickers.

How many stickers would you need for:

▪▪ A rod of 3 cubes





▪▪ What is the general rule?

8. Trapezoid Tables (Moss et al., 2006)

Nicolette decided she would place the chairs around each table so that 2 chairs will go on the long

side of the trapezoid and one chair on every other side of the table. In this way, 5 students can sit

around 1 table. Then, she found that she could join 2 tables like in the figure below, so that now 8

students can sit around 2 tables.


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How many students can sit around 3 tables joined this way?

How many students can sit around 56 tables?

What is the rule? How did you figure it out?

9. Perimeter Problem (Moss et al., 2006)

This is a 3x3 grid of squares with only the squares at the outside edge shaded. If you had a 5x5 grid

of squares where only the outside edge of squares is shaded, how many squares would be shaded?

If you had a grid of 100 number of squares, how many would be shaded? Is there a rule? How did

you figure it out?

10. Handshake Problem (Moss et al., 2006)

Imagine that you are at a huge party. Everyone starts to shake hands with all the other people who

are there. The problem can be represented by a table or by a drawing (see table and figure below).

If 2 people shake hands, there is 1 handshake.

If 3 people are in a group and they each shake hands with the other people in the group,

there are 3 handshakes.


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How many handshakes if there are 4 people? 10 people? Can you use a rule to help you

figure this out?

11. Linking Patterns to Generalization

An important step to move from arithmetic to algebra is to recognize and describe patterns. You can

use exercises such as this one to let learners generalize patterns.

Choose an even number

Choose another even number

Add them together

What kind of number do we get?

Choose another pair of even numbers

Is the result the same kind of number?

Will the result always be the same kind of number?

Use a diagram or a word explanation to show why your result is true.

What about even + odd?

What about odd + even?


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What about odd + odd?

Try this:

▫ 1 + 3

▫ 1 + 3 + 5

▫ 1 + 3 + 5 + 7

▫ …

▫ What can you say about the results?

12. Word problems of type “Think-of-a-Number (TOAN)

If you know the sum and the difference of two numbers, can you figure out what the two numbers

are? Example: A + B = 13 and A – B = 5. List all sums and differences and look for pattern. Learners

should eventually find out that A can be found by taking (sum + difference)/2.

A variation of this

Example: TOAN, add 5, double, add 2, half the answer and subtract your original number.

The result is always six!

Play the game a few times with different numbers.

Try and find the explanation why the result is always six. Come to a generalized statement

(using A and B).

You can challenge learners to make their own TOAN activity.

13. Word problem: Buying T-Shirts

Concord Trading sells T-shirts for 3000 Frw each, but adds a delivery charge of 5000 Frw regardless

of how many T-shirts you order. True Sports sells the same T-shirt for 4000 Frw each without any

delivery charge. Better still, for every order, True Sports gives a discount of 2000 Frw on the entire

bill.


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What is the cost of buying 5 T-shirts from each store? Of buying 10 T-shirts? For which number of

T-shirts will the price be the same in both shops?

Source: https://elsdunbar.wordpress.com/2016/05/27/learning-from-a-5th-grade-math-team/


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Section 4: Probability and Statistics

A key learning outcome in probability and statistics is to make learners familiar with the concept of

probability. Through a variety of daily life situations and using concrete materials, learners explore

questions like:

How likely is something going to happen?

What are the chances of an event happening?

Examples of concrete materials and situations are:

counters in different colours: “how likely am I going to pick a blue counter?”

dice: “what is the chance of throwing a 5?”

spinners: see below

learners themselves:

▫ how likely is it that a learner’s birthday falls in November?

▫ how likely to pick a girl if teacher picks a name at random?

Below we provide some ideas on how you can use self-made spinners in your lessons on probability.

1. Using spinners

Figure 39: Example of a spinner


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Let learners draw spinners based on guidelines that you provide. Probabilities in different

mathematical notations are introduced and practiced, as well as terminology such as likely, unlikely

and certain.

Table 10: Colour in the spinners to show the different probabilities.

50:50 chance of red or white

¾ chance that you will get blue

More likely than you will get red than green and less likely that you

will yellow than red

Certain that you will get a yellow

Unlikely that you will get red

Likely that you will get yellow

Not impossible to get green

Where it is impossible to get red but likely to get white.

3 in 8 chance that you will get red

2 in 8 chance that you will get blue

Impossible to get yellow


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In a subsequent activity, learners can design their own spinners and explain the probability of landing

on different colours, using a worksheet like the one below.

Table 11: Colour in the spinners to show the probabilities that you define

2. Investigation activity to introduce probability

To introduce the concept of probability, you can use the following investigation activity with the

learners.

1. Let learners in pairs toss a coin in the air for a total of 30 times.

Let them predict how many times they will have head or tail.

Every time the coin lands they record whether they get a ‘head’ or a ‘tail’.

They write H for ‘head’ and T for ‘tail’ in a table.

How many times did you get a ‘head’ (H)? _______________

How many times did you get a ‘tail’ (T)? _______________

2. Secondly, learners roll a dice for a total of 30 times.

Let them predict how many times they will throw a 1, 2…

Every time they roll, they record the score on the dice and write it in a table.


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Number of 1s: __________ Number of 4s: ________



Further questions you can ask for discussion:

If we rolled 2 coins what possible outcomes could we get?

If we rolled more than 1 dice what possible outcomes could we get?

What would our chances of getting 2 heads or a 6 be like then?


182

MODULE REFERENCES

Aikman, S., & Underhalter, E. (Eds.). (2007). Practising Gender Equality in Education. Oxford: Oxfam.

Retrieved from http://policy-practice.oxfam.org.uk/publications/practising-gender-equality-

in-education-115528

Ainscow, M. (2005). Understanding the development of inclusive education system. Electronic

Journal of Research in Educational Psychology, 3(7).

Altinyelken, H. K. (2010). Curriculum change in Uganda: Teacher perspectives on the new thematic

curriculum. International Journal of Educational Development, 30(2), 151–161.

Askew, M., & Wiliam, D. (1995). Recent research in mathematics education 5-16 (Vol. 53). London:

Ofsted Reviews of Research, HMSO.

Ball, D. L. (1990). The mathematical understandings that prospective teachers bring to teacher

education. The Elementary School Journal, 90(4), 449–466.

Ball, D. L. (1997). From the general to the particular: Knowing our own students as learners of

mathematics. The Mathematics Teacher, 90(9), 732.

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content Knowledge for Teaching What Makes It

Special? Journal of Teacher Education, 59(5), 389–407.

Baroody, A. J. (1990). How and when should place-value concepts and skills be taught? Journal for

Research in Mathematics Education, 21(4), 281–286.

Beckmann, S. (2013). Mathematics for elementary teachers with activity manual. Pearson.

Beilock, S. L., Gunderson, E. A., Ramirez, G., & Levine, S. C. (2010). Female teachers’ math anxiety

affects girls’ math achievement. Proceedings of the National Academy of Sciences, 107(5),

1860–1863. https://doi.org/10.1073/pnas.0910967107

Benson, S., Addington, S., Arshavsky, N., Al Cuoco, E., Goldenberg, P., & Karnowski, E. (2004). Ways to

Think About Mathematics: Activities and Investigations for Grade 6-12 Teachers (1st Edition).

Corwin Press.

Black, P., & Wiliam, D. (2001). Inside the black box: Raising standards through classroom assessment

(p. 14). British Educational Research Association. Retrieved from http://weaeducation.

typepad.co.uk/files/blackbox-1.pdf


183

Black, Paul, & Wiliam, D. (1998). Assessment and classroom learning. Assessment in Education, 5(1),

7–74.

Bloom, B. S. (1968). Learning for Mastery. Instruction and Curriculum. Regional Education Laboratory

for the Carolinas and Virginia, Topical Papers and Reprints, Number 1. Evaluation Comment,

1(2), n2.

Bloom, B. S., Hastings, J. T., & Madaus, G. F. (1971). Handbook on formative and summative

assessment of student learning. New York: McGraw Hill.

Brodie, K. (2014). Learning about learner errors in professional learning communities. Educational

Studies in Mathematics, 85(2), 221–239.

Burns, M. (2015). About teaching mathematics: A K-8 resource (4th edition). Houghton Mifflin

Harcourt.

Bushart, B. (2016, February 4). Writing Numberless Word Problems. Retrieved from https://bstockus.

wordpress.com/2016/02/04/writing-numberless-word-problems/

Carbonneau, K. J., Marley, S. C., & Selig, J. P. (2013). A meta-analysis of the efficacy of teaching

mathematics with concrete manipulatives. Journal of Educational Psychology, 105(2), 380.

Carroll, W. M., & Porter, D. (1997). Invented strategies can develop meaningful mathematical

procedures. Teaching Children Mathematics, 3(7), 370–375.

Chapin, S. H., O’Connor, C., O’Connor, M. C., & Anderson, N. C. (2009). Classroom discussions: Using

math talk to help students learn, Grades K-6. Math Solutions.

Chappell, M. F., & Strutchens, M. E. (2001). Creating connections: Promoting algebraic thinking with

concrete models. Mathematics Teaching in the Middle School, 7(1), 20.

Chick, H. L., & Harris, K. (2007). Pedagogical content knowledge and the use of examples for teaching

ratio. In Proceedings of the 2007 annual conference of the Australian Association for Research

in Education (pp. 25–28).

Clements, D. H. (2000). ‘Concrete’ manipulatives, concrete ideas. Contemporary Issues in Early

Childhood, 1(1), 45–60.

Consuegra, E. (2015). Gendered teacher-student classroom interactions in secondary education:

perception, reality and professionalism. Vrije Universiteit Brussel.


184

DfES. (2004). Pedagogy and Practice: Teaching and Learning in Secondary Schools - Unit 10: Group

Work (No. 0433–2004).

Fujii, T., & Stephens, M. (2008). Using number sentences to introduce the idea of variable. Algebra

and Algebraic Thinking in School Mathematics: Seventieth Yearbook, 127–149.

Hattie, J. (2009). Visible Learning: A Synthesis of over 800 Meta-Analyses Relating to Achievement.

Routledge.

Hattie, J., & Timperley, H. (2007). The power of feedback. Review of Educational Research, 77(1),

81–112.

Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Unpacking Pedagogical Content Knowledge:

Conceptualizing and Measuring Teachers’ Topic-Specific Knowledge of Students. Journal for

Research in Mathematics Education, (4), 372.

Hyde, J. S., Fennema, E., & Lamon, S. J. (1990). Gender differences in mathematics performance: a

meta-analysis. American Psychological Association.

Jordan, L., Miller, M. D., & Mercer, C. D. (1999). The Effects of Concrete to Semiconcrete to Abstract

Instruction in the Acquisition and Retention of Fraction Concepts and Skills. Learning

Disabilities: A Multidisciplinary Journal, 9(3), 115–22.

Kainulainen, M., McMullen, J., & Lehtinen, E. (2017). Early Developmental Trajectories Toward

Concepts of Rational Numbers. Cognition and Instruction, 35(1), 4–19. https://doi.org/10.1

080/07370008.2016.1251287

Klibanoff, R. S., Levine, S. C., Huttenlocher, J., Vasilyeva, M., & Hedges, L. V. (2006). Preschool children’s

mathematical knowledge: The effect of teacher” math talk.”. Developmental Psychology,

42(1), 59.

Krach, M. (1998). Teaching Fractions Using Manipulatives. Ohio Journal of School Mathematics, 37,

16–23.

Krathwohl, D. R. (2002). A revision of Bloom’s taxonomy: An overview. Theory into Practice, 41(4),

212–218.

Lemov, D. (2015). Teach like a champion 2.0. San Francisco, CA: Josey-Bass.


185

Litwiller, B., & Bright, G. (2002). Making Sense of Fractions, Ratios, and Proportions. 2002 Yearbook.

ERIC.

Martino, A. M., & Maher, C. A. (1999). Teacher questioning to promote justification and generalization

in mathematics: What research practice has taught us. The Journal of Mathematical Behavior,

18(1), 53–78.

Mason, J. (2008). Making use of children’s powers to produce algebraic thinking. Algebra in the Early

Grades, 57–94.

Mercer, N., & Sams, C. (2006). Teaching Children How to Use Language to Solve Maths Problems.

Language and Education, 20(6), 507–528. https://doi.org/10.2167/le678.0

MINEDUC. (2015). Education For All 2015 National Review Report: Rwanda. Kigali, Rwanda: MINEDUC.

Mlama, P. M. (2005). Gender Responsive Pedagogy: A Teacher’s Handbook. Nairobi, Kenya: Forum

for African Women Educationalists. Retrieved from http://www.ungei.org/files/FAWE_GRP_

ENGLISH_VERSION.pdf

Moss, J., Beatty, R., McNab, S. L., & Eisenband, J. (2006). The potential of geometric sequences to

foster young students’ ability to generalize in mathematics. In The annual meeting of the

American Educational Research Association.

Moss, J., & Case, R. (1999). Developing children’s understanding of the rational numbers: A new

model and an experimental curriculum. Journal for Research in Mathematics Education,

122–147.

National Research Council. (2002). Helping Children Learn Mathematics. Washington, DC: The

National Academies Press.

Nicol, D. (2007). Principles of good assessment and feedback: theory and practice. In from the REAP

International Online Conference on Assessment Design for Learner Responsibility, 29-31 May

2007. Retrieved from http://www.reap.ac.uk/reap/public/papers//Principles_of_good_

assessment_and_feedback.pdf

NISR. (2015). Integrated Household Living Conditions Survey (EICV) (EICV No. 4). Kigali, Rwanda:

National Institute of Statistics in Rwanda. Retrieved from http://www.statistics.gov.rw/

survey/integrated-household-living-conditions-survey-eicv


186

Nsengimana, T., Habimana, S., & Mutarutinya, V. (2017). Mathematics and science teachers’

understanding and practices of learner-centred education in nine secondary schools from

three districts in Rwanda. Rwandan Journal of Education, 4(1), 55–68.

Nsengimana, T., Ozawa, H., & Chikamori, K. (2014). The implementation of the new lower secondary

science curriculum in three schools in Rwanda. African Journal of Research in Mathematics,

Science and Technology Education, 18(1), 75–86.

Nunes, T., & Bryant, P. (1996). Children doing mathematics. Wiley-Blackwell.

OECD. (2006). Assessing Scientific, Reading and Mathematical Literacy: A Framework

for PISA 2006 (p. 187). OECD. Retrieved from http://www.oecd.org/edu/school/

assessingscientificreadingandmathematicalliteracyaframeworkforpisa2006.htm

OECD. (2015). The ABC of Gender Equality in Education. Paris: Organisation for Economic Co-

operation and Development. Retrieved from http://www.oecd-ilibrary.org/content/

book/9789264229945-en

Page, A. (1994). Helping students understand subtraction. Teaching Children Mathematics, 1(3),

140–144.

Polya, G. (1945). How to solve it: A new aspect of mathematical method. Princeton university press.

REB. (2015). Competence-Based Curriculum Pre-Primary to Upper Secondary. Kigali, Rwanda:

MINEDUC. Retrieved from http://reb.rw/fileadmin/competence_based_curriculum/syllabi/

CURRICULUM_FRAMEWORK__FINAL_VERSION_PRINTED.compressed.pdf

REB. (2017). Literacy, Language and Learning Initiative (L3): National Fluency And Mathematics

Assessment Of Rwandan Schools: Endline Report. Kigali, Rwanda: USAID, EDC.

Richardson, K. (2012). How Children Learn Number Concepts: A Guide to the Critical Learning Phases.

Bellingham, Wash.: Math Perspectives Teacher Development Center.

Rwanda Education Board. (2015). Summary of Curriculum Framework Pre-Primary to Upper

Secondary. Kigali, Rwanda: Rwanda Education Board. Retrieved from http://reb.rw/fileadmin/

competence_based_curriculum/syllabi/CURRICULUM_FRAMEWORK_FINAL_PRINTED.

compressed.pdf

Save the Children, Mureke Dusome project. (2017). Parent-School Partnerships for Education Toolkit:

Training Modules for Sector Education Officers. Save The Children, REB, USAID.


187

Schweisfurth, M. (2011). Learner-centred education in developing country contexts: From solution

to problem? International Journal of Educational Development, 31(5), 419–426.

Shulman, L. S. (1986). Those Who Understand: A Conception of Teacher Knowledge. American

Educator, 10(1).

Smith III, J. P., Disessa, A. A., & Roschelle, J. (1994). Misconceptions reconceived: A constructivist

analysis of knowledge in transition. The Journal of the Learning Sciences, 3(2), 115–163.

Stacey, K., Burton, L., & Mason, J. (1982). Thinking mathematically. Addison-Wesley.

Stewart, G. (2014). Promoting & Managing Effective Collaborative Group Work. Belfast: Belfast

Education and Library Board. Retrieved from http://www.belb.org.uk/Downloads/i_epd_

promoting_and_managing_collaborative_group_work_may14.pdf

Subrahmanian, R. (2005). Gender equality in education: Definitions and measurements. International

Journal of Educational Development, 25(4), 395–407.

Swan, M. (2005). Standards Unit-Improving learning in mathematics: challenges and strategies.

DfES, http://www. ncetm. org. uk/files/224/improving_learning_in_mathematicsi. pdf.

Sweller, J., & Cooper, G. A. (1985). The Use of Worked Examples as a Substitute for Problem Solving

in Learning Algebra. Cognition and Instruction, 2(1), 59–89. https://doi.org/10.1207/

s1532690xci0201_3

The National Council of Teachers of Mathematics (NCTM). (2007). What Do We Know about the

Teaching and Learning of Algebra in the Elementary Grades? (Research Brief). The National

Council of Teachers of Mathematics (NCTM).

UNICEF. (2017). UNICEF Gender Action Plan, 2018–2021. New York: UNICEF. Retrieved from https://

www.unicef.org/gender/files/2018-2021-Gender_Action_Plan-Rev.1.pdf

Vamvakoussi, X., Van Dooren, W., & Verschaffel, L. (2012). Naturally biased? In search for reaction

time evidence for a natural number bias in adults. The Journal of Mathematical Behavior,

31(3), 344–355.

Van de Walle, J. A., Karp, K. S., & Williams, J. M. B. (2015). Elementary and middle school mathematics.

Teaching development (9th Edition). Boston: Pearson.


188

Van Den Heuvel-Panhuizen, M. (2003). The didactical use of models in realistic mathematics

education: An example from a longitudinal trajectory on percentage. Educational Studies in

Mathematics, 54(1), 9–35.

Vavrus, F., Thomas, M., & Bartlett, L. (2011). Ensuring quality by attending to inquiry: Learner-

centered pedagogy in sub-Saharan Africa. UNESCO-IICBA Addis Ababa, Éthiopie.

Venkat, H., & Askew, M. (2012). Mediating early number learning: specialising across teacher talk

and tools? Journal of Education [P], 56, 67–89.

Weimer, M. (2012). Five characteristics of learner-centered teaching. Faculty Focus.

Wiliam, D. (2016, September 2). The 9 things every teacher should know. Retrieved 8 March 2017,

from https://www.tes.com/us/news/breaking-views/9-things-every-teacher-should-know

Zuze, T. L., & Lee, V. E. (2007). Gender Equity in Mathematics Achievement in East African Primary

Schools: Context Counts (Working Paper Series 15 No. 15). Cape Town, South Africa: Southern

Africa Labour and Development Research Unit.


189

APPENDIX

Appendix 1: Self-Evaluation for Primary Mathematics Teachers

How confident are you to apply appropriately following techniques for mathematics teaching?te

chni

ques

for

mat

hem

atics

teac

hing

Not

con

fiden

t

A bi

t con

fiden

t

Qui

te c

onfid

ent

Very

con

fiden

t

Questioning

Use open questions to challenge pupils and

encourage them to think

Use voting to involve all learners in

questioning

Use questioning techniques that stimulate

interactions between learners and not only

between the teacher and learners

Let learners formulate mathematical

questions

Mathematics Conversations

Use techniques that stimulate learners to

express their mathematical ideas

Summarise and review the learning points in

a lesson or sequence of lessons

Developing problem solving skills

Use activities that stimulate the

development of problem-solving skills


190

Use word problems that stimulate learner’s

thinking and understanding of mathematical

concepts.

Learner errors and misconceptions

Be familiar with common mathematical

misconceptions with learners

Use techniques to expose and change

learner misconceptions about mathematics

Connecting concrete, pictorial and abstract

representations of mathematical concepts

Introduce a mathematical concept with

concrete materials or experiences, and

gradually move to pictorial and abstract

representations of the concept.

Use low-cost materials to teach and learn

mathematics

Mathematical games

Use games to increase understanding about

mathematical concepts

Use games to practice basic mathematical

skills

Gender and Inclusivity in mathematics

Address gender stereotypes about

mathematics

Make sure that boys and girls have equal

opportunities to take part in lessons and

achieve learning outcomes in mathematics

Practice differentiation to make learning

mathematics more inclusive

Assessment


191

Use formative assessment techniques to

inform yourself and learners about their

learning.

Use the results from formative assessment

to change your teaching.

Based on your self-evaluation above, formulate 3 priorities for yourself in this CPD Programme.

1.

2.

3.


192

Appe

ndix

2: S

ampl

e Le

sson

Pla

n of

prim

ary

mat

hem

atics

less

on

IMBA

TA Y

’ISO

MO

RY’

IMIB

ARE

– U

MW

AKA

WA

KABI

RI (R

EB, P

lay-

Base

d Le

arni

ng Te

ache

r gui

de fo

r prim

ary

scho

ol, 2

016

in p

rogr

ess)

Izin

a ry

’ishu

ri: G

S AK

ANZU

A

maz

ina

y’um

war

imu:

M

USA

BYIM

ANA

Dani

el

Igih

embw

e:Ita

riki:

Inyi

gish

oU

mw

aka

wa

Um

utw

e w

aIs

omo

rya

Igih

e is

omo

rimar

a

Um

ubar

e w

’aba

nyes

huri

Cya

112

Gas

hyan

tare

20

16Im

ibar

eKa

biri

2 5

/24

Imin

ota

4046

Abafi

te ib

yo b

agen

erw

a by

ihar

iye

mu

myi

gire

no

mu

myi

gish

irize

n’u

mub

are

wab

oU

mw

ana

umw

e ufi

te u

bum

uga

bw’in

ging

o (A

gend

era

ku m

bago

imw

e) a

ricar

a ha

fi y’

ikib

aho.

Um

utw

e Im

ibar

e ku

va k

uri 0

kug

era

500

Ubu

shob

ozi b

w’in

genz

i bu

gam

ijwe

- Kub

ara,

gus

oma,

kw

andi

ka, g

uton

deka

, kug

erer

anya

, gut

eran

ya g

ukub

a no

kug

aban

ya n

eza

imib

are

ishyi

tse

kuva

kur

i 0 k

uger

a ku

ri 50

0

Isom

oM

ara

ya 4

Inte

go n

gena

muk

oro

Hifa

shish

ijwe

udup

apur

o tw

andi

tseh

o im

ibar

e 1

kuge

za 1

0 n’

agak

ino

kitw

a “D

ukin

e du

kuba

na

4” b

uri

mun

yesh

uri a

raba

ash

obor

a gu

kuba

nez

a um

ubar

e afi

te n

a ka

ne k

andi

aka

bivu

ga m

u ijw

i rira

ngur

uye

mu

min

ota

5

Imite

rere

y’a

ho is

omo

riber

aM

u ish

uri,

Aban

yesh

uri b

aric

ara

mu

ishus

ho y

a U.

Hag

ati y

’ikib

aho

n’ab

anye

shur

i har

asig

ara

nibu

ra m

eter

o 3

Imfa

shan

yigi

sho

- ud

ukar

ito 1

0 tu

rimo

uduf

uka,

bur

i gaf

uka

karim

o am

abuy

e an

e -

Udu

papu

ro tw

andi

tseh

o im

ibar

e ku

va k

uri 1

kug

era

ku 1

0, ik

ibah

o, in

gwa,

ikar

ito ir

imo

ibik

ores

ho 4

0.

(am

abuy

e, u

dufu

niko

tw’a

mac

upa)

Imya

ndik

o n’

ibita

bo

byifa

shis

hijw

e In

tega

nyan

yigi

sho

y’im

ibar

e ic

yici

ro c

ya m

bere

cy’

amas

huri

aban

za p

ge 1

6,

Num

erac

y le

arni

ng th

roug

h pl

ay

Ibic

e by

’isom

o +

igih

e

Gus

oban

ura

mur

i mak

e ig

ikor

wa

umw

arim

u n’

umun

yesh

uri b

asab

wa

guko

ra

Ubu

shob

ozi

n’in

ging

o

nsan

gany

amas

omo

(an

dika

ubus

hobo

zi

+ ig

isob

anur

o

kigu

fi ki

gara

gaza

Hifa

shish

ijwe

uduf

uka

turim

o ut

ubuy

e tu

netu

ne. M

u m

atsin

da m

ato,

aba

nyes

huri

bash

aka

umub

are

w’u

tubu

ye t

wos

e tu

ri m

u ga

karit

o. B

afas

hijw

e na

mw

arim

u, b

atah

ura

ko iy

o

uter

anyi

je ib

intu

bin

gana

ku

bury

o bw

isubi

ram

o bi

ngan

a no

kub

ikub

a izo

nsh

uro.

Ibik

orw

a by

’um

war

imu

Ibik

orw

a by

’um

unye

shur

i

Inta

ngiri

ro:

Imin

ota

5

Gu

saba

aba

nyes

huri

kuvu

ga m

ara

ya k

abiri

m

u nj

yana

.

Kuvu

ga m

u nj

yana

mar

a ya

2

Isom

o ny

irizi

na:

Imin

ota

25

a) ig

ikor

wa

k’iv

umbu

ra

Gu

fash

a ab

anye

shur

i gu

kora

am

atsin

da

atan

datu

Guta

nga

amab

wiri

za y

’um

ukin

o

Kw

erek

a ab

anye

shur

i ib

ikor

esho

bi

tand

ukan

ye b

iri b

wifa

shish

we

-Du

fite

uduk

arito

10.

-Bu

ri ga

karit

o ka

rimo

uduf

uka

turim

o am

abuy

e 4

mur

i kam

we

-Bu

ri ts

inda

riro

here

za u

mun

tu u

mw

e uz

a gu

tom

bora

-M

urat

angu

ranw

a ku

bwira

um

ubar

e w

’am

abuy

e yo

se

ar

i m

u ga

karit

o m

wat

ombo

ye

-U

tsin

da n

i ur

avug

a um

ubar

e ny

awo

kand

i ak

abya

ndik

a

ku

bury

o bw

oros

hye

kubi

som

a.

Gu

kora

am

atsin

da

baku

rikije

am

abw

iriza

y’u

mw

arim

u.

M

u m

atsin

da a

bany

eshu

ri ba

raki

na

agak

ino

buba

hiriz

a am

abw

iriza

ya

tanz

we

ku b

uryo

buk

urik

ira:

-Bu

ri ts

inda

rir

oher

eza

umun

yesh

uri

umw

e gu

tom

bora

ag

akar

ito k

amw

e

-Ab

agize

its

inda

bar

abar

a by

ihus

e ut

ubuy

e tw

ose

turi

mur

i bu

ri ga

fuka

, bab

ashe

kub

ona

utub

uye

twos

e tu

ri m

uri b

uri g

akar

ito

-Bu

ri ts

inda

rir

andi

ka

umub

are

w’u

tubu

ye

babo

nye

baga

raga

za

n’uk

o ba

bige

zeho

.

Ubu

rezi

bu

dahe

za:

umw

arim

u yi

ta k

u m

wan

a

ufite

ub

umug

a bw

’ingi

ngo

kand

i aka

ba h

amw

e n’

aban

di

amw

icaz

a im

bere

.

Ubu

fata

nye,

im

iban

ire

ikw

iye

n’ab

andi

n’u

bum

enyi

ngiro

mu

buzi

ma

bwa

buri

mun

si: m

u gi

he a

bany

eshu

ri

bako

rera

ha

mw

e m

u

mat

sinda

bas

haka

ibisu

bizo


193

Ibic

e by

’isom

o +

igih

e

Gus

oban

ura

mur

i mak

e ig

ikor

wa

umw

arim

u n’

umun

yesh

uri b

asab

wa

guko

ra

Ubu

shob

ozi

n’in

ging

o

nsan

gany

amas

omo

(an

dika

ubus

hobo

zi

+ ig

isob

anur

o

kigu

fi ki

gara

gaza

Hifa

shish

ijwe

uduf

uka

turim

o ut

ubuy

e tu

netu

ne. M

u m

atsin

da m

ato,

aba

nyes

huri

bash

aka

umub

are

w’u

tubu

ye t

wos

e tu

ri m

u ga

karit

o. B

afas

hijw

e na

mw

arim

u, b

atah

ura

ko iy

o

uter

anyi

je ib

intu

bin

gana

ku

bury

o bw

isubi

ram

o bi

ngan

a no

kub

ikub

a izo

nsh

uro.

Ibik

orw

a by

’um

war

imu

Ibik

orw

a by

’um

unye

shur

i

Inta

ngiri

ro:

Imin

ota

5

Gu

saba

aba

nyes

huri

kuvu

ga m

ara

ya k

abiri

m

u nj

yana

.

Kuvu

ga m

u nj

yana

mar

a ya

2

Isom

o ny

irizi

na:

Imin

ota

25

a) ig

ikor

wa

k’iv

umbu

ra

Gu

fash

a ab

anye

shur

i gu

kora

am

atsin

da

atan

datu

Guta

nga

amab

wiri

za y

’um

ukin

o

Kw

erek

a ab

anye

shur

i ib

ikor

esho

bi

tand

ukan

ye b

iri b

wifa

shish

we

-Du

fite

uduk

arito

10.

-Bu

ri ga

karit

o ka

rimo

uduf

uka

turim

o am

abuy

e 4

mur

i kam

we

-Bu

ri ts

inda

riro

here

za u

mun

tu u

mw

e uz

a gu

tom

bora

-M

urat

angu

ranw

a ku

bwira

um

ubar

e w

’am

abuy

e yo

se

ar

i m

u ga

karit

o m

wat

ombo

ye

-U

tsin

da n

i ur

avug

a um

ubar

e ny

awo

kand

i ak

abya

ndik

a

ku

bury

o bw

oros

hye

kubi

som

a.

Gu

kora

am

atsin

da

baku

rikije

am

abw

iriza

y’u

mw

arim

u.

M

u m

atsin

da a

bany

eshu

ri ba

raki

na

agak

ino

buba

hiriz

a am

abw

iriza

ya

tanz

we

ku b

uryo

buk

urik

ira:

-Bu

ri ts

inda

rir

oher

eza

umun

yesh

uri

umw

e gu

tom

bora

ag

akar

ito k

amw

e

-Ab

agize

its

inda

bar

abar

a by

ihus

e ut

ubuy

e tw

ose

turi

mur

i bu

ri ga

fuka

, bab

ashe

kub

ona

utub

uye

twos

e tu

ri m

uri b

uri g

akar

ito

-Bu

ri ts

inda

rir

andi

ka

umub

are

w’u

tubu

ye

babo

nye

baga

raga

za

n’uk

o ba

bige

zeho

.

Ubu

rezi

bu

dahe

za:

umw

arim

u yi

ta k

u m

wan

a

ufite

ub

umug

a bw

’ingi

ngo

kand

i aka

ba h

amw

e n’

aban

di

amw

icaz

a im

bere

.

Ubu

fata

nye,

im

iban

ire

ikw

iye

n’ab

andi

n’u

bum

enyi

ngiro

mu

buzi

ma

bwa

buri

mun

si: m

u gi

he a

bany

eshu

ri

bako

rera

ha

mw

e m

u

mat

sinda

bas

haka

ibisu

bizo


194

b) Is

esen

gura

Ku

baza

ibib

azo

bire

bana

n’u

ko b

agez

e ku

gi

subi

zo

-N

i iki

cyab

ashi

mish

ije m

uri a

ka g

akin

o ?

-N

i gut

e m

wag

eze

ku g

isubi

zo?

-

N’ik

i cya

bago

ye m

uri a

ka g

akin

o

Ku

reba

ab

akor

eshe

je

gute

rany

a no

ku

baba

za in

shur

o ba

tera

nyije

no

kuba

baza

ik

imen

yets

o gi

kore

shw

a m

u m

ibar

e iy

o us

haka

kug

arag

aza

insh

uro

wak

oze

ikin

tu.

Ku

reba

kub

akor

eshe

je u

bury

o bw

o gu

kuba

ak

aber

eka

ko

=

4x1

= 4

4

+4 =

4x2

= 8

4+

4+4

= 4x

3 =

12

4+

4+4+

4 =

4x4

= 16

4+4+

4+4+

4 =

4x5

= 20

…

.

guko

mez

a gu

tya

kuge

za k

u gi

kubo

‘40’

-Bu

ri ts

inda

rira

gend

a ris

oban

ura

uko

bage

ze k

u gi

subi

zo

-Bu

ri m

unye

shur

i ara

vuga

ibya

giye

bi

muk

omer

era

-Bu

ri m

unye

shur

i are

gend

a ya

ndik

a m

u ik

ayi

isano

rir

i ha

gati

yo

guku

ba n

o gu

tera

nya

4 =

4

x1 =

4

4+

4 =

4x2

= 8

4+4+

4 =

4x

3 =

12

4+

4+4+

4 =

4

x4 =

16

4+

4+4+

4+4

= 4x

5 =

20

Ubu

shis

hozi

no

gu

shak

ira

ibib

azo

ibis

ubiz

o: M

u gi

he

aban

yesh

uri

baso

banu

ra

inzir

a ba

ciye

mo

kugi

ra n

go

bage

re k

u gi

subi

zo c

yabo

Ubu

fata

nye,

im

iban

ire

ikw

iye

n’ab

andi

: m

u gi

he

aban

yesh

uri

bako

rera

ham

we

mu

mat

sinda

bash

aka

ibisu

bizo

by’

ibib

azo

baha

we.

c) Ik

omat

anya

Kwan

dika

mar

a ya

kane

kuva

kuri

rimw

e ku

geza

ku ic

umi

O O

O

O

O

O

O O

O

O

O O


195

b) Is

esen

gura

Ku

baza

ibib

azo

bire

bana

n’u

ko b

agez

e ku

gi

subi

zo

-N

i iki

cyab

ashi

mish

ije m

uri a

ka g

akin

o ?

-N

i gut

e m

wag

eze

ku g

isubi

zo?

-

N’ik

i cya

bago

ye m

uri a

ka g

akin

o

Ku

reba

ab

akor

eshe

je

gute

rany

a no

ku

baba

za in

shur

o ba

tera

nyije

no

kuba

baza

ik

imen

yets

o gi

kore

shw

a m

u m

ibar

e iy

o us

haka

kug

arag

aza

insh

uro

wak

oze

ikin

tu.

Ku

reba

kub

akor

eshe

je u

bury

o bw

o gu

kuba

ak

aber

eka

ko

=

4x1

= 4

4

+4 =

4x2

= 8

4+

4+4

= 4x

3 =

12

4+

4+4+

4 =

4x4

= 16

4+4+

4+4+

4 =

4x5

= 20

…

.

guko

mez

a gu

tya

kuge

za k

u gi

kubo

‘40’

-Bu

ri ts

inda

rira

gend

a ris

oban

ura

uko

bage

ze k

u gi

subi

zo

-Bu

ri m

unye

shur

i ara

vuga

ibya

giye

bi

muk

omer

era

-Bu

ri m

unye

shur

i are

gend

a ya

ndik

a m

u ik

ayi

isano

rir

i ha

gati

yo

guku

ba n

o gu

tera

nya

4 =

4

x1 =

4

4+

4 =

4x2

= 8

4+4+

4 =

4x

3 =

12

4+

4+4+

4 =

4

x4 =

16

4+

4+4+

4+4

= 4x

5 =

20

Ubu

shis

hozi

no

gu

shak

ira

ibib

azo

ibis

ubiz

o: M

u gi

he

aban

yesh

uri

baso

banu

ra

inzir

a ba

ciye

mo

kugi

ra n

go

bage

re k

u gi

subi

zo c

yabo

Ubu

fata

nye,

im

iban

ire

ikw

iye

n’ab

andi

: m

u gi

he

aban

yesh

uri

bako

rera

ham

we

mu

mat

sinda

bash

aka

ibisu

bizo

by’

ibib

azo

baha

we.

c) Ik

omat

anya

Kwan

dika

mar

a ya

kane

kuva

kuri

rimw

e ku

geza

ku ic

umi

O O

O

O

O

O

O O

O

O

O O

4x1=

4

4x2=

8

…

4x10

=40

Guko

resh

a um

wito

zo w

o gu

fata

mu

mut

we

mar

a ya

kan

e

Aban

yesh

uri

bara

subi

ram

o in

shur

o

nyin

shi m

ara

ya k

ane

kugi

ra n

go b

ayifa

te

mu

mut

we.

Um

usoz

o w

’isom

o:

(Isuz

uma

)

Imin

ota

10

Gu

tang

a am

abw

iriza

y’a

gaki

no k

’isuz

uma

ko g

ukin

a ba

kuba

na

kane

an

erek

ana

ibik

ores

ho (u

dupa

puro

) biri

bw

ifash

ishw

e

Gu

kurik

irana

im

igen

deke

re y

’aga

kino

mu

mat

sinda

ata

nduk

anye

Gu

saba

am

atsin

da a

mw

e ku

jya

imbe

re n

o gu

kina

aga

kino

aba

ndi b

ose

baku

rikiy

e

Mu

mat

sinda

ya

babi

ribab

iri, a

bany

eshu

ri

bara

kina

um

ukin

o w

o gu

kuba

;

-um

unye

shur

i w

a m

bere

ar

aato

mbo

ra a

gapa

puro

kand

itseh

o um

ubar

e (1

kug

era

10).

-Ah

isha

agap

apur

o m

u ki

ganz

a ki

mw

e

-as

aba

mug

enzi

we

gufin

dura

ik

igan

za g

ihish

emo

agap

apur

o

-m

ugen

zi w

e af

indu

ra

ikig

anza

ki

rimo

agap

apur

o

-iy

o ag

asan

zem

o, a

som

a um

ubar

e ur

imo

-ak

uba

uwo

mub

are

na k

ane

maz

e ak

avug

a ig

isubi

zo

-M

ugen

zi w

e na

we

ahita

ato

mbo

ra

Gus

aban

a m

u nd

imi

zem

ewe

guko

resh

wa

mu

gihu

gu :

buz

agar

agar

ira m

u

kuga

nira

ha

gati

y’ab

anye

shur

i na

mw

arim

u

ndet

se

n’ab

anye

shur

i

ubw

abo.


196

ku

yobo

ra ik

igan

iro g

isoza

-U

yu m

unsi

tw

ize ik

i?

-N

iba

usha

ka k

ugira

am

akay

e ab

iri m

uri

buri

som

o m

u m

asom

o an

e ak

urik

ira:

Ikin

yarw

anda

, Im

ibar

e Ic

yong

erez

a n’

imbo

neza

mub

ano,

ubw

o m

u by

’uku

ri am

akay

e yo

se u

kene

ye n

i ang

ahe?

-M

u ru

go

mur

i ab

ana

bane

. M

ama

agus

abye

kug

urira

bur

i w

ese

bom

bo

enye

, U

bwo

uzag

ura

bom

bo z

inga

he

kuri

butik

e?

-um

ukin

o uk

omez

a gu

tyo

mu

mat

sinda

ya

babi

ribab

iri.

Gu

subi

za

utub

azo

tuga

raga

ra

mu

kiga

niro

bag

irana

na

mw

arim

u.

-U

yu m

unsi

twize

mar

a ya

kan

e…

-W

aba

uken

eye

amak

aye

8

-U

zagu

ra b

ombo

16

Kwis

uzum

a (u

mur

ezi)

Mui

r iri

som

o ab

anye

shur

i bish

imiy

e um

ukin

o bi

ze k

andi

bize

guk

uba

/ m

ara

ya k

ane

biny

uze

mu

muk

ino.

Bav

umbu

ye

isono

riri

haga

ti yo

guk

uba

no g

uter

anya

Date post:	12-Jul-2020
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