TEACHING AND LEARNING OF FRACTIONS IN 
PRIMARY SCHOOLS IN MASERU 
By 
 
‘Maphole Georgina Marake 
              B. Ed. Honours, BSc. Ed., S.T.C.  
 
Dissertation submitted in fulfillment of the requirement for the degree of 
 
MAGISTER EDUCATIONIS 
 
in the 
 
FACULTY OF EDUCATION 
(SCHOOL FOR MATHEMATICS, NATURAL SCIENCES AND 
TECHNOLOGY EDUCATION) 
 
AT THE UNIVERSITY OF THE FREE STATE 
BLOEMFONTEIN 
 
 
Supervisor:  PROF GF DU TOIT 
 
JANUARY 2013 
 
 
DEDICATION 
 
 
This is dedicated to my mother, ’MaliemisoKuleile, and my late father, Sentebale 
Cletus Kuleile who both instilled in all their children the love of education and 
perseverance in everything they do. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
DECLARATION 
 
I, the undersigned, declare that the thesis hereby submitted by me for the MAGISTER 
EDUCATIONIS (M.Ed.) degree at the University of the Free State is my own 
independent work and that I have not previously submitted the same work for a 
qualification at another university. I further cede copyright of this thesis in favour of the 
University of the Free State. 
 
 
…………………………… 
‘Maphole Georgina Marake 
 
Date: January 2013 
 
 
 
 
 
 
 
 
 
 
 
 
LANGUAGE EDITING 
 
TO WHOM IT MAY CONCERN 
This is to certify that the undersigned has done the language editing for the following 
candidate 
SURNAME AND INITIALS: …………………………………………………………………….. 
 
……………………………………………………………………………………………………… 
 
……………………………………………………………………………….…………………….. 
 
DEGREE: MEd dissertation 
 
 
........................      Date: .................................... 
Name 
 
NOTE WELL:  The language editor does not accept any responsibility for post-
editing, re-typing or re-computerising of the content. 
Residential address: 
 
Postal address: 
 
Tel:
V 
 
ACKNOWLEDGEMENTS 
 
I would like to express my heartfelt thanks and gratitude to the following people for their 
respective contribution during this study: 
• First I would like to thank God almighty who makes all things possible. 
 
• My supervisor, Prof GF Du Toit, for his patience, guidance and encouragement, 
without him this piece of work would have been impossible. 
 
• The principals and teachers of the participating schools who accepted a 
complete stranger in their classrooms.  
 
• The teacher trainers who spared some time in their busy schedule  
 
 
• My husband, Moteaphala, for his support and love throughout my studies and his 
patience with me when the pressure of the study took its toll. 
 
 
• My children who were always there for me and sometimes had to sacrifice their 
time with their mother.  
 
• Lastly I would like to thank all the people who supported me during this period of 
study, my family, more especially ‘Mamasiphole Setoromo, friends and 
colleagues. 
 
 
 
VI 
ABSTRACT 
Throughout the world governments and other education stakeholders advocate quality 
education and education for all. Among others, mathematics education is seen by 
governments as essential in the advancement of the development of countries. Lesotho 
is no exception in this regard hence mathematics is one of the core subjects in 
Lesotho’s education system. Though Mathematics education is seen as pivotal to the 
development of countries, analysis of mathematics Junior Certificate (JC) examination 
results in Lesotho indicates that performance in mathematics is not good. This study 
therefore aspired to investigate teaching strategies predominantly employed by primary 
mathematics teachers and assess their effect on learners’ meaningful learning of 
fractions. In order to meet this aim the study attempted to determine what literature said 
about effective learning and teaching of fractions, the level of training given to 
mathematics teachers and determine whether effective learning and teaching 
materialised in the three classrooms that were studied. 
The existing literature proposed different teaching strategies that resulted in significant 
learning of fractions. To investigate dominant teaching strategies that teachers used in 
the teaching of fractions, class observations of three teachers were conducted. 
Teachers were observed in their classrooms over a period of time and follow-up 
interviews were conducted. Samples of the teachers’ documents and the learners’ work 
were analysed to evaluate the extent to which effective learning and teaching of 
fractions were taking place in these respective classes. Literature indicates that 
effective learning, of fractions, entails meaningful construction of the concept through 
handling of concrete materials and formation of relationship between concepts. Effective 
teaching on the other hand entails the ability to create situations in which learning is 
facilitated. Teachers are said to possess both mathematical knowledge for teaching 
(MKT) and Pedagogical content knowledge (PCK) in order to be able to teach 
effectively. 
In order to fully understand the level of training that the teachers received teacher 
trainers were interviewed. It was found that teachers did not engage learners in high 
order reasoning and problem solving, instead they gave close-ended questions which 
VII 
learners answered by practising rules and procedures that teachers taught. Learners 
therefore did not use their own strategies when writing solutions to questions. It was 
recommended that teachers should use readily available materials like paper and 
papers and when planning lessons they should think of possible errors, misconceptions 
and difficulties that learners were likely to have.
VIII 
KEY WORDS 
 
• Effective Teaching 
 
• Effective Learning 
 
 
• Fractions 
 
 
• Mathematics 
 
 
• Learning Theories 
 
 
• Subconstructs 
 
 
• Problem-Solving 
 
 
 
 
 
 
 
 
 
 
 
 
IX 
ACRONYMS 
 
ECoL             Examinations Council of Lesotho 
 
JC   Junior Certificate 
 
ZPD   Zone of Proximal Development 
 
PCK   Pedagogical Content Knowledge 
 
MKT   Mathematical Knowledge for Teaching 
 
CKM   Common Knowledge of Mathematics 
 
SKM   Specialised Knowledge of Mathematics 
 
 
 
 
 
 
 
 
X 
                                                                  TABLE OF CONTENTS 
TITLE PAGE                                                                                                                                  i                                                                                                          
DEDICATION                                                                                                                                ii                               
DECLARATION                                                                                                                           iii                               
LETTER OF EDITING                                                                                                                 iv                              
ACKNOWLEDGEMENTS                                                                                                            v                                
SUMMARY                                                                                                                                  vi                          
KEY WORDS                                                                                                                              vii                               
ACRONYMS                                                                                                                              viii                            
TABLE OF CONTENTS                                                                                                              ix 
  
CHAPTER 1  
BACKGROUND TO THE STUDY 
 
1.1 INTRODUCTION                                                                                                                   1                                         
1.2 RESEARCH PROBLEM, RESEARCH  QUESTIONS AND AIMS                                        2                   
1.3 THEORETICAL FRAMEWORK                                                                                             4                                     
1.4 RESEARCH DESIGN AND RESEARCH METHODOLOGY                                                 5                                   
1.5 VALUE OF THE RESEARCH                                                                                                6                                 
1.6ETHICAL CONSIDERAIONS                                                                                                  6                            
1.7 RESEARCH LAYOUT                                                                                                            7                            
1.8 CONCLUSION                                                                                                                       7                              
 
CHAPTER 2  
LEARNING OF MATHEMATICS 
 
2.1 INTRODUCTION                                                                                                                   8           
XI 
2.2 MEMORY FORMATION                                                                                                         9                        
2.3 PSYCHOLOGICAL PERSPECTIVES ON LEARNING                                                        10                       
2.3.1 Piaget                                                                                                                     11                         
2.3.1.1 Sensorimotor stage                                                                                12                      
2.3.1.2 Pre-operational stage                                                                             12            
2.3.1.3 Concrete operational stage                                                                    13                        
2.3.1.4 Formal operational stage                                                                       13                   
2.3.2 Bruner                                                                                                                    14                       
2.3.3 Vygotsky                                                                                                                14                       
2.4 CONSTRUCTIVISM                                                                                                   17                       
2.5 LEARNING MATHEMATICS                                                                                               19                          
2.5.1 Learning environment                                                                                          21                    
2.5.2 Effective learning                                                                                                  22                    
2.5.2.1 Dependent, independent and interdependent                                      24       
2.5.2.2 Prior knowledge                                                                                      25                 
2.5.2.3 Experiential learning                                                                               27                  
2.5.3 Conceptual and procedural knowledge                                                              29                 
2.5.3.1 Differentiating between conceptual and procedural knowledge        29          
2.5.3.2 Connecting conceptual and procedural knowledge                            30         
2.6 BARRIERS TO EFFECTIVE LEARNING                                                                             32                          
2.6.1 Lack of motivation                                                                                                 32                        
2.6.2 Learners’ learning styles                                                                                      33       
2.6.3 Fragmented learning.                                                                                    34                  
2.6.4 Language                                                                                                          34    
2.7 CONCLUSION                                                                                                                     38                                  
 
 
XII 
CHAPTER 3    
TEACHING MATHEMATICS 
 
3.1 INTRODUCTION                                                                                                                  39                               
3.2 TEACHING MATHEMATICS                                                                                               39                            
3.3 PEDAGOGICAL CONTENT KNOWLEDGE (PCK) AND 
 MATHEMATICAL KNOWLEDGE FOR TEACHING (MKT)                                                     41                              
3.4 INDUCTIVE AND DEDUCTIVE REASONING                                                              44                              
3.5 PROBLEM SOLVING                                                                                              47                                          
3.5.1 Polya’s four phases of problem solving                                                   48                           
3.5.2  Problem solving strategies                                                                         51     
3.5.3 Factors influencing problem solving                                         
52                                
3.5.4 Nature of problems                                                                                               
54                               
3.6 ASSESSMENT                                                                                                                    55                                  
3.6.1 Types of assessment                                                                                           55                               
3.6.1.1 Formative assessment                                                                           55                           
3.6.1.2 Summative assessment                                                                         56                          
3.6.2 Assessment methods                                                                                          57     
3.6.2.1 Interviewing                                                                                             57                              
3.6.2.2 Observation                                                                                             57                              
3.6.2.3 Portfolio                                                                                                   58                          
3.6.3 Features of assessment                                                                                       58                       
3.6.3.1 Assessment situation, task or question                                               58                      
3.6.3.2 Response                                                                                                 58                                  
3.6.3.3 Assigning some meaning to the 
 interpretation of the student’s responses                                                       59                         
XIII 
3.6.3.4 Recording and reporting                                                                        59                             
3.7 EFFECTIVE TEACHING OF FRACTIONS                                                                         59                                  
3.7.1 Fractions                                                                                                         60                                
3.7.2 Difficulties of teaching fractions                                                                         61                    
3.7.2.1 Handling misconceptions                                                              61                            
3.7.2.2 Difficulties associated with contextual teaching                              63                  
3.7.2.3 Diagnosing and remedying learners’ difficulties                              64                 
3.7.3 Subconstructs                                                                                               65                                          
3.7.3.1 Part-whole subconstruct                                                                        65                         
3.7.3.2 Ratio subconstruct                                                                         67                               
3.7.3.3 Operator subconstruct                                                                         67               
3.7.3.4 Quotient subconstruct                                                                         68                 
3.7.3.5 Measure subconstruct                                                                         68               
3.8 TEACHER DOCUMENTS                                                                                                                       
3.8.1 Lesson plan                                                                                            
69                             
3.8.2 Learners’ mathematics exercise books                                                             71 
3.9 CONCLUSION                                                                                                                    71                    
 
CHAPTER 4      
RESEARCH METHODOLOGY   
 
 4.1 INTRODUCTION                                                                                                          73                                        
4.2 RESEARCH METHODOLOGY                                                                                    73                                      
4.3 QUALITATIVE RESEARCH                                                                                               73                          
4.4 A PHENOMENOLOGICAL STUDY                                                                                    75                                  
4.5 DATA COLLECTION STRATEGIES                                                                                   75                    
XIV 
4.5.1 Observations                                                                                                         75                          
4.5.2 Interviews                                                                                                         77                          
4.5.3 Document analysis                                                                                               79                       
4.5.3.1 Teachers’ lesson plans                                                                         79               
4.5.3.2 Learners’ mathematical exercise books                                         80                   
4.6 POPULATION, SAMPLE AND SAMPLING TECHNIQUES                                         81                                 
4.6.1 The population                                                                                    81                                 
4.6.2 The sample and sampling techniques                                                    81                     
4.7 PILOT STUDY                                                                                                                    83                            
4.8 VALIDITY AND RELIABILITY                                                                                    83                               
4.9 DATA ANALYSIS AND INTERPRETATIONS                                                              86                          
4.10 ETHICAL ISSUES                                                                                                         88                                         
4.11 CONCLUSION                                                                                                                   89                           
 
CHAPTER 5 
ANALYSIS AND INTERPRETATIONS        
 
5.1 INTRODUCTION                                                                                                         90                                          
5.2 SCHOOL A                                                                                                                    90                                               
5.2.1 School’s environment                                                                                    90                              
5.2.2 Teacher’s profile                                                                                               92                        
5.2.3 Class observation                                                                                               92                      
5.2.4 Teacher A’s lesson plan                                                                                   114                  
5.2.5 Learners’ exercise books                                                                                   115                 
 
 
XV 
5.3 SCHOOL B                                                                                                                   118                                          
5.3.1 School’s environment                                                                                   118                            
5.3.2 Teachers’ profiles                                                                                              120                    
5.3.3 Teacher B                                                                                                         120                       
5.3.3.1Class observations                                                                        120                       
5.3.3.2 Teacher B’s lesson plan                                                                        137 
5.3.3.3 Learners’ exercise books                                                             139   
5.3.4 Teacher C                                                                                                              140             
5.3.4.1 Class observations                                                                        140                     
5.3.4.2 Teacher C’s lesson plan                                                                        158      
5.3.4.3 Learners’ exercise books                                                             159             
5.3.5 Reflection on the observation and analysis of the documents                       161   
5.3.5.1 Reflection on the observation of teachers A, B and C                       161   
5.3.5.2 Reflection on the lesson plans used by teachers A, B and C           165    
5.3.5.3 Reflection on the exercise books of learners in 
 schools A, B and C                                                                                           166              
5.3.6 Interviews with teachers                                                                                     167             
5.3.6.1 Planning.                                                                                                 168    
5.3.6.2 Effective learning                                                                                   169               
5.3.6.3 Effective teaching                                                                                  169              
5.3.6.4 Teaching materials                                                                                 170          
5.3.6.5 Teaching strategies                                                                                171     
 
5.4 INTERVIEW OF TEACHER TRAINERS                                                                        172                        
5.4.1 Profiles of  teacher trainers                                                                        172                     
5.4.2 Interviews                                                                                                        172                 
XVI 
5.4.2.1 Methodology                                                                                  173    
5.4.2.1.1 Child development 173                         
5.4.2.1.2 Teaching materials                                                            174             
5.4.2.1.3 Lesson planning                                                            175      
5.4.2.1.4 Teaching method                                                                    176     
5.4.2.2 Content                                                                                            177     
5.4.3 Analysis                                                                                                       177                            
5.5 CONCLUSION                                                                                                                  178                 
 
CHAPTER 6 
FINDINGS, CONCLUSIONS AND RECOMMENDATIONS        
 
6.1 INTRODUCTION                                                                                                       180                
6.2 RESEARCH FINDINGS                                                                                            181             
6.2.1 Effective learning of fractions from literature                                                  181 
6.2.1.1 Findings from literature                                                                       181 
6.2.1.2 Findings from empirical research                                                  182    
6.2.1.3 Conclusion                                                                                             182     
6.2.2 Effective teaching of fractions                                                                                  
6.2.2.1 Findings from literature                                                                       182 
6.2.2.2 Findings from empirical research                                                  184     
6.2.2.3 Conclusion                                                                                             185     
6.2.3 Level of training given to mathematics teachers                                        185 
6.2.3.1Findings from literature                                                                       185 
6.2.3.2 Findings from empirical research                                                 185     
6.2.3.3 Conclusion                                                                                            186      
6.3 RECOMMENDATIONS ON HOW TO TEACH FRACTIONS EFFECTIVELY                 186    
XVII 
6.4 RESEARCHER’S CHALLENGES                                                                                     187               
6.5 SUGGESTIONS FOR FUTURE RESEARCH                                                                    187        
6.6 OVERALL CONCLUSION                                                                                                 188            
LIST OF REFERENCES                                                                                                       189               
 
APPENDICES  
Appendix A: Observation sheet                                                                                            202 
Appendix B: Interview questions for class teachers                                                          204 
Appendix C: Interview question for teacher trainers                                                          205                             
Appendix D: Application letter to the Ministry of Education                                              206 
Appendix E: Permission letter from the Ministry of Education                                          207 
Appendix F: Application letter to the schools                                                                     208 
Appendix G: Letter to teachers                                                                                             210      
Appendix H: Application letter to interview lecturers at LCE                                                             211 
 
LIST OF FIGURES 
Figure 2.1 Fraction as a part-whole                                                                                       10 
Figure 2.2 Fraction as a measure                                                                                          10 
Figure 2.3 Representation of 
                                                                                               24 

Figure 2.4 The connective model of learning mathematics                                                35 
Figure 3.1 Representing addition of fraction                                                                        46 
Figure 3.2 Representing fractions on the number line                                                        49 
XVIII 
Figure 3.3 Fraction as a part-whole                                                                                       61 
Figure 3.4 (a) Fraction misconception                                                                                  62 
Figure 3.4 (b) Fraction misconception                                                                                  63 
Figure 3.5 A whole pizza divided into 4 equal parts                                                            65 
Figure 3.6 A set of 16 balls                                                                                                     66 
Figure 5.1 Floor map of teacher A’s classroom                                                                   91 
Figure 5.2 (a) Fraction board                                                                                                  94 
Figure 5.2 (b) Fraction board                                                                                                  97 
Figure 5.3 Teacher A’s lesson plan                                                                                       114 
Figure 5.4 Learner’s working in teacher A’s class                                                              116 
Figure 5.5(a) Errors made by teacher A’s learners                                                              116 
Figure 5.5(b) Errors made by teacher A’s learners                                                              117 
Figure 5.6 Floor map of teacher B’s classroom                                                                   119 
Figure 5.7 Floor map of teacher C’s classroom                                                                   120 
Figure 5.8 (a) Faulty representations that Teacher B allowed                                            127 
Figure 5.8 (b) Faulty representations that Teacher B allowed                                            129 
Figure 5.9 Teacher B’s lesson plan                                                                                       138 
Figure 5.10 Workings of learners i teacher B’s class                                                          139 
Figure5.11 Errors in teacher B’s class                                                                                  140 
Figure 5.12 (a) Teacher C’s representations of ordering fractions                                    149 
Figure 5.12 (b) Teacher C’s representations of ordering fractions                                    154 
XIX 
Figure 5.13 Teacher C’s lesson plan                                                                                     158 
Figure 5.14 Workings of teacher C’s learners                                                                      160 
 
LIST OF TABLES 
Table 4.1 Coding table                                                                                                             87 
Table 5.1 summary of class observations                                                                            163 
Table 5.2 Summary of teachers’ lesson plans                                                                     166 
Table 5.3 Summary of learners’ exercise books                                                                  167                          
Table 5.4 (a) Planning                                                                                                             168 
Table 5.4 (b) Effective learning                                                                                              169 
Table 5.4 (c) Effective teaching                                                                                             170 
Table 5.4(d) Teaching materials                                                                                            171 
Table 5.4(e) Teaching strategies                                                                                           171 
Table 5.5 Child development                                                                                                 173 
Table 5.6 Teaching materials                                                                                                 175 
Table 5.7 Lesson planning                                                                                                     175 
Table 5.8 Teaching methods                                                                                                  176 
 
 
 
1 
CHAPTER 1  
      BACKGROUND TO THE STUDY 
1.1 INTRODUCTION 
Throughout the world governments and other education stakeholders advocate 
quality education and education for all. Mathematics teachers are no exception in 
this regard, hence so much research is going on regarding contextualising 
mathematics for better understanding and making it realistic (Romberg 2006). 
Despite several research findings of mathematics teaching, learners still perform 
poorly in mathematics. 
Lesotho is no exception in this regard. This is evidenced by the Examinations 
Council of Lesotho’s (ECoL) (2007) comment in the Junior Certificate (JC) pass list, 
“it has observed with great concern the deterioration of candidates’ performance in 
Mathematics over the years.” The situation poses questions such as, “are 
mathematics teachers in Lesotho still employing the drill and practice methods of 
teaching?”  
Goldman (2007:75-76) ascribes poor performance of mathematics to teachers’ 
incompetence and lack of deeper understanding of mathematics concepts. 
Consequently they tell learners what to do and thereafter give them practice 
questions to help them memorize the procedures. 
Alibali (2005) articulates that decimal fraction knowledge is central to mathematics 
understanding, and it is believed that if effectively taught it can lay solid foundations 
to mathematics learning. On the other hand, there are difficulties associated with 
fractions knowledge, some of which are: its multifaceted nature and the language 
used (Charalambos, Charalambous & Pitta-Pantazi 2006:293-316). If this is the 
case, then perhaps the question one can ask is how do teachers teach fractions? 
Are learners given chance to construct their own meaning of fraction?  
As a high school mathematics teacher, the researcher made observations regarding 
students’ misconceptions pertaining to fractions and strategies teachers use to 
teach fractions. When studying the current primary school mathematics syllabus, it 
depicts that fractions are taught from standard 2, known as grade 4 in other 
2 
countries. If fractions are taught so early in primary schools, what is it that makes 
learners demonstrate so much incompetence in fractions when they get to 
secondary schools? Could it perhaps be that primary school teachers contribute 
towards learners’ shallow understanding of fractions?   
1.2 RESEARCH PROBLEMS, RESEARCH QUESTIONS AND AIMS 
Evidence as highlighted by primary literature in the foregoing section demonstrates 
that effective teaching and learning of mathematics is essential. In order to attain this 
goal Yetkiner and Caprano (2009) highlight the importance of effectively teaching 
different mathematical concepts by citing fractions in particular. Yetkiner and 
Caprano (2009) emphasise the significance of effectively teaching fractions by 
stating that, “fractional concepts are important building blocks of elementary and 
middle school mathematics curricula.” They further state that “conceptually based 
instruction of fractions requires teachers to have a complete understanding of the 
subject matter.” This opinion is supported by Shulman, Ball and Bass (as cited by 
Hristovitcn & Mitcheltree 2004) as they state that high quality teacher preparation 
both in content and pedagogy is critical for the improvement of student outcomes in 
mathematics. On this basis teachers should have the skills and competences to 
teach fractions.  
Torrence (2002) testifies that realistic mathematics makes sense for learners. She 
further asserts that if learners are given the opportunity to construct their own 
understanding they learn to think and they learn that they could think. They can also 
learn to see mathematics as creative and pleasurable. Torrence (2002) is also of the 
opinion that if learners are helped to acquire this attitude they could be competent to 
solve problems in any situation. 
The value of realistic mathematics cannot be underestimated as it appears to be a 
promise to the improvement of learners’ mathematical proficiency. It is in this regard 
that Troutman and Lichtenberg (2003:365) underline learners’ difficulties to learn 
mathematics, fractions in particular, to insufficient exposure of manipulating concrete 
materials.  
Research into classroom-based teaching is necessary, as there is literature that 
enunciates the importance and value of realistic mathematics (Romberg 2006). 
3 
There are also suggestions in literature on how to improve mathematics teaching. 
Regardless of these research findings mathematics performance in Lesotho schools 
is persistently low. Learners’ construction of mathematical knowledge and creativity 
in solving novel problems does not seem to improve.  
The study will therefore attempt to answer the following questions: 
• What does literature say on effective learning of fractions? 
• What does literature say on mathematics teaching and effective teaching of 
fractions? 
• What level of training did the teachers get regarding mathematics teaching? 
• Which teaching strategies are predominantly used by mathematics teachers?  
• What do teachers do to verify that students have effectively learned fractions? 
• How can fractions be taught effectively? 
The purpose of the proposed study is to explore and assess how primary school 
mathematics teachers teach fractions and how learners learn fractions. In this 
research study, the general aim is thus to investigate teaching strategies 
predominantly employed by primary mathematics teachers and assess their effect on 
learners’ meaningful learning of fractions. The research will therefore seek to 
address the following objectives: 
• Determine what literature is saying on effective teaching of fractions 
• Determine what literature is saying on effective learning of fractions by 
learners 
• Determine the level of training given to mathematics teachers 
• Determine whether effective teaching is taking place 
• Determine whether effective learning of fractions materialises in selected 
classes 
• Recommend guidelines on how to teach fractions effectively 
4 
1.3 THEORETICAL FRAMEWORK  
Mathematics skills required for people to function at the work place today are 
different from those which were required yesterday, hence the emergence of 
mathematics education reform (Romberg 2006).  Romberg (2006) further articulates 
that mathematics learning is no more regarded as a drill and practice process but 
students actively construct their own mathematics by making connections, building 
mental schemata and developing mathematics based on prior knowledge through 
interactions with others.  
This research intends to construct meaningful reality of the phenomenon under study 
and the way the members understand it. This research will therefore be based on 
both interpretivism and constructivism paradigms. Allan, Carmona, Calvin and Rowe 
(n.d.) state that “... constructivism is synonymous with interpretivism ... and that they 
both share the goal of understanding the complex world of lived experiences from 
the point of view of those who live it.” Higgs (1995:97-98) further explains that people 
understand social reality by constantly interpreting what they see. Constructivism is 
therefore a way of thinking that advocates personal construction of reality by 
interacting with the environment. During interactions people observe, relate and 
connect to what they already know (Troutman & Lichtenberg 2003:18). 
Constructivism argues that learning is meaningful and effective when learners 
generate and construct knowledge for themselves, either individually or within social 
contexts during learning (Muijs & Reynolds 2005:62). Torrence (2002) reveals that in 
realistic mathematics education, students develop mathematics through their social 
interactions in their endeavours to solve problems hence this reform supports 
constructivism.  
The researcher believes that people’s actions are influenced mainly, by their cultural 
beliefs, experiences, values and attitudes. So in order for the researcher to fully 
understand the situation there must be direct interactions between the researcher 
and the respondents. She is also, of the opinion that actions do not speak for 
themselves, instead they need to be interpreted in order to capture the actual 
meaning (Pring 2000: 96).  
 
5 
1.4 RESEARCH DESIGN AND RESEARCH METHODOLOGY 
In order to achieve the stated objectives, a qualitative approach was used in this 
study. The researcher intended to get an in-depth understanding of the participants’ 
world by giving rich descriptions on how they construct their knowledge (Maree 
2007:87). Maree further contends that the truth can be gained through observations 
and interviews.  
The researcher observed lessons where 11 to 12 year old (standard six) children 
were taught fractions. The emphasis of the observation of lessons was primarily to 
determine dominant teaching strategies used when teaching fractions and to find out 
how 11 to 12 year old children conceptualize fraction knowledge.   
In order not to influence the dynamics of the lesson, the observer was part of the 
lesson, but remained uninvolved (de Vos 2002:279; Nieuwenhuis 2009:85). The 
researcher did the study with the aim of understanding the assumptions, values and 
beliefs of the participants. In order to improve the efficiency of the observations, a 
self-designed observation sheet was used.  
At the end of the observation period learners’ mathematics exercise books were 
collected and analysed to determine how they performed calculations of problems on 
fractions.  
At the end of the observation period, that is when the whole topic as planned by the 
teacher is taught, interviews with the observed teachers were conducted using a 
self-designed, semi-structured interview. Interviews were done after school when the 
learners and other teachers had gone home, so that they would not be disturbed.  
The advantages of using semi-structured interviews are that, if questions are 
prepared in advance, the researcher will be able to plan the logic in which questions 
should be asked, identify ambiguous and complex questions and then change them 
before the interview session. In semi-structured interviews the interviewer probes the 
participants so that they elaborate on their responses (de Vos 2002:302; Maree 
2007:87). 
The use of both observations and interviews aimed at ensuring trustworthiness of the 
research findings. This approach, according to de Vos et al. (2002:341) and Shenton 
6 
(2003:65), is called triangulation of measures. According to these authors, a concept 
or phenomenon measured with multiple instruments has greater a chance of being 
valid.   
The sampling of schools was both convenient and purposeful due to restrictions with 
regard to resources such as time and money. Three Government schools formed 
part of the sample but in the end only two participated. In these schools grade 8 
learners and three teachers were observed and the teachers were interviewed.  
Lecturers from the Lesotho College of Education (L.C.E.) were also interviewed. 
Information pertaining to the type of training given to the teachers was sought. Two 
lecturers were included in the study. 
According to de Vos et al. (2002:337), pilot studies are done prior to the actual study 
and few a respondents having the same characteristics as those in the main 
population are used. Pilot study helped the researcher to test the questions, gain 
confidence and improve on their own interview skills. It also helped in building 
relationships and estimating time and costs that would be involved in the main study. 
In this regard the pilot study was done with one school and one teacher.  
1.5 VALUE OF THE RESEARCH 
This research will help to build understanding and enhance both current and further 
literature on the teaching and learning of fractions within the Lesotho context. 
Fractional concepts are some of the practical topics in the mathematics curriculum, 
so this study is intended to uncover the classroom practices of some teachers in 
Lesotho, in order that they can reflect on them and improve on their teaching. 
1.6 ETHICAL CONSIDERATIONS   
Consent from the Ministry of Education was asked for to undertake the study. When 
permission was granted, the schools were asked for consent and the teachers who 
agreed to participate in the study filled in the consent form.  
Teachers voluntarily participated in the research after the researcher had explained 
the purpose of the study to them. To ensure confidentiality, the names of the 
schools, teachers and learners are not used. The schools are referred to as school A 
and school B. Teachers are coded as teacher A, teacher B and teacher C. Fictitious 
7 
names are used to refer to the learners. The researcher ensured that the participants 
will not be harmed emotionally, physically or psychologically. 
1.7 RESEARCH LAYOUT 
In chapter 1, the background to this study, theoretical framework, problem statement, 
research problem, aims and objectives of the research, research design and 
methodology, value of the research and ethical considerations were discussed. 
To really understand the context a literature study on learning Mathematics with 
emphasis on fractions was discussed in chapter 2. 
In Chapter 3, a literature study on teaching Mathematics was discussed. 
In chapter 4, the research methodology was discussed with reference to the 
sampling procedure engaged, the research design and the analysis on the research 
findings. 
In chapter 5, data was analysed and interpreted. 
In chapter 6, research findings, conclusions and recommendations were outlined. 
1.8 CONCLUSION 
The statement of the problem, research questions and aims were established in this 
chapter. Interpretivism which is viewed to be synonymous to constructivsm was 
discussed as the theoretical framework of this study. This study attempts to construct 
the meaning of the phenomenon, teaching and learning of fractions, as interpreted 
by the participants. The research design and methodology were discussed and a 
qualitative approach was used. Class observations, interviews and document 
analysis were utilised as data collecting instruments.  
The value and contribution of this study, the ethical considerations together with the 
chapter layout were also discussed.  
Chapter two reviews literature on effective learning of mathematics with the 
emphasis on fractions.  
 
 
8 
CHAPTER 2 
LEARNING OF MATHEMATICS 
2.1 INTRODUCTION 
“Teaching and learning of fractions has traditionally been one of the most 
problematic areas in primary school mathematics (Charalambos, Charalambous & 
Pitta-Pantazi 2006)”. This has therefore led to poor performance in Mathematics. 
This study focuses on how children learn fractions and how teachers teach them. 
Memory formation will be discussed then learning will be conceptualized from a 
psychological perspective especially with the focus on the work of Piaget, Vygotsky 
and Bruner. The theoretical framework that informs this study is constructivism; 
hence these theorists’ perspectives on learning will be of interest. Their theories on 
learning inform the teaching process especially teaching that enhances construction 
of knowledge.  
After examining in broad terms how children learn, more focus will be on how 
children learn Mathematics. The importance of a learning environment, how it 
promotes or hampers effective learning will all be looked into together with the 
barriers to effective learning. From the constructivist perspective, Mathematics must 
be conceptualized so that it is learned effectively. Learners’ prior knowledge helps in 
making Mathematics meaningful and worth learning as it becomes realistic when 
related to learners’ experiences. Prior knowledge; how it can be used to build 
mathematical concepts and enhance experiential learning will therefore be discussed 
with more emphasis on fractions. The relationship and dependence of both 
procedural and conceptual learning will be clarified. 
Learning can be affected and/or damaged, if barriers to effective learning such as 
learners’ learning styles, language and fragmentation of knowledge are not 
addressed. In this chapter barriers to effective learning will be discussed. 
In the teaching and learning situation teaching and learning are interdependent, that 
is teaching should result in learning and for learning to take place there has to be 
teaching. Since effective teaching should be based on how children learn, the 
approach in this study is to first discuss how children learn mathematics followed by 
discussing effective teaching of mathematics. 
9 
2.2 MEMORY FORMATION 
In our everyday lives we rely on memory for our everyday chores but most 
importantly we need memory to learn. For learning to occur, there has to be the 
processing of information. Information processing involves gathering and 
representing information meaningfully so that it can be stored and retrieved when 
needed (Hamachek 1990:190). This definition brings in an important characteristic of 
learning, which is memory. 
The degree to which one remembers a piece of information depends on where the 
information is stored. Where a piece of information is stored also depends on what is 
done to that information the moment it is received (Entwistle 1981:121-122).  
At times when people receive information they do not pay attention to it, it simply 
gets into the system without being processed. This means if no meaningful 
representations are made, the information will be stored in the short term memory 
where it will be lost within twenty seconds depending on individuals. But if the 
information is dealt with by perhaps associating it with something already known 
then it is likely to be stored in the long term memory (Entwistle 1981:223-224; 
Hamachek 1990:195-199). 
One of the concerns of a teacher should be how to help learners develop long term 
memory. To help teachers address this concern Hamachek (1990:201-202) 
illustrates some of the things learners may do to ensure that what they learn will be 
stored in long term memory. 
Rehearsal strategies: This involves going through the idea again and again. In the 
case of learning fractions, the partitioning exercise could be done again and again 
for different fractions so that the concept and the equality of parts can be stored in 
long term memory. Repetition not only enhances rote learning but it also fosters 
deep rooted understanding. This is because each time an exercise is repeated 
learners are likely to view what they are doing from different perspectives, hence 
getting more insight of the concept they are building.  
Elaboration strategies: New ideas can be elaborated upon by associating them with 
what is already known. In the case of fractions again, learners may associate sharing 
with the actual sharing of items in real life situations.  
10 
Organizational strategies: New information may also be organized in a manner that 


makes sense to the learner. Fraction  may be represented as part of a whole, 

 
Figure 2.1 Fraction as a part-whole 
and as a measure 
 
Figure. 2.2 Fraction as a measure 
Comprehension and monitoring strategies: Include things we do to keep up with our 
learning. This may include note taking and self-questioning to check for 
 

understanding. Learners for example should question themselves why equals  ?  

 
Affective strategies: The way we feel can have a powerful impact on memory. If 
learners feel bored, their motivation level drops and then learning is impeded. 
Affective strategies include time management, establishing and maintaining 
motivation and focusing attention on what is being learned. 
Memory, as one important aspect of learning has been discussed and what teachers 
can do to enhance long term memory. The next section discusses learning from the 
cognitive psychologists’ point of view. 
2.3 PSYCHOLOGICAL PERSPECTIVES ON LEARNING 
Learning is a complex concept defined from different points of view by different 
people. In this section, learning is going to be examined from the psychological point 
of view of three theorists: Piaget, Bruner and Vygotsky. 
 
 
11 
2.3.1 Piaget 
Piaget’s interest in the nature of knowledge led him speculate about the 
development of thinking by children (Biehler, Snowman, D’Amico & Schmid 1999: 
54). He postulates that human beings try to understand their environment by 
organising information into coherent general systems and adapting it to their 
knowledge base (Biehler et.al. 1999:54-56). According to Piaget, organisation occurs 
before adaptation and through it organised patterns of behaviour or thought called 
schemes result. But if new knowledge does not fit into the existing scheme learners 
engage in the adaptation process. In this process learners try to make sense of the 
experience so that new knowledge can be incorporated into their existing scheme, or 
they restructure and modify their schemes which may be faulty, so that they 
accommodate the new knowledge (Biehler et.al. 1999:54-56; Mavugara-Shava 2005: 
41). 
It is mentioned above that two things are likely to happen when one encounters new 
knowledge. That is new knowledge may either fit properly into the existing scheme or 
a misfit may occur. Biehler et al. (1999:56) call these situations equilibration and 
disequilibration. The authors accentuate that equilibration occurs when new 
knowledge fits perfectly with the existing scheme, while disequilibration results when 
there is a discrepancy between the existing scheme and the new knowledge. 
When learning the concept, fractions, learners need to compare them with whole 
numbers which are an existing scheme. For example if they use paper folding to 
investigate fractions, they should first compare it with whole numbers. That is one 
fold gives two pieces each of which is less than one whole (Troutman & Litchtenberg 
2003:342). Ability to form this logic may result in accommodation of the new 
knowledge. The processes of accommodation and assimilation result in mental 
growth because new knowledge gained may result in the change of one’s reasoning, 
thinking and perceptions which is referred to as maturation (Bell 1978:100). 
Through interactions with their environment children gain experiences that Piaget 
calls physical, logico-mathematical and social knowledge (Bell 1978:100; Kamii 
1990:23). Bell (1978:100) defines physical knowledge as the experience gained 
through experiments and observations, such as sharing concrete objects; children 
gain different sets of knowledge such as addition of fractions. Logico-mathematical 
12 
knowledge is the mental actions children perform during accommodation. An 
example of logico-mathematical knowledge is for example where learners give 
meaning to the addition of fractions.  
Social knowledge consists of conventions worked out by society. An example of 
social knowledge is telling learners that the ‘1’ is known as the numerator and the ‘2’ 


is known as the denominator in the fraction ‘ ’. When interacting with their 

environment, children engage in the actual manipulation of concrete materials and 
co-operation with other people. By so doing, children gain knowledge of concepts 
like fractions, their meaning and operation, just to mention a few, hence new 
knowledge is adapted. A typical example of an experience that leads to physical 
knowledge is to engage learners in an activity to cut a pizza into parts, thus working 
with concrete material to assist in conceptualizing fractions. 
As mentioned above Piaget points out that when there is a discrepancy between 
existing schemes and new knowledge, confusion occurs. This confusion if not 
cleared will make it difficult to adapt new knowledge, which will in turn force one to 
find out more about new information so that it can fit the present cognitive structure, 
or for it to be discarded (Fisher 1995:12). This results in cognitive growth. If the 
confusion is left unattended it becomes difficult to learn because of the gap that 
results from the “disequilibration”. Fisher (1995:12) further posits that teachers 
should look for signs of cognitive conflict and challenge the learners ideas so that 
they attain higher levels of thinking.  
Piaget insists that schemes undergo systematic changes at particular points in time 
from birth to adulthood; hence he theorizes that there are distinct stages of cognitive 
development through which children transcend.  
2.3.1.1 Sensorimotor stage 
This is the first stage at which infants to two year old develop schemes by exploring 
their own bodies, senses and later explore external objects and situations.  
2.3.1.2 Preoperational stage 
The two to seven or eight year old thinking centres on mastery of symbols. Children 
cannot think of conservation problems as those needing logical thinking. Piaget (as 
13 
cited by Biehler et al. 1999:58), clarifies this by citing this example; a teacher pours 
fruit juice into two identical short glasses until the child agrees that each contains an 
equal amount,  and may be half full. Then the juice is poured from one of the glasses 
into a taller thin glass. The child is asked; if there is more juice in the tall thin glass or 
the short one?   
Children at this stage will think there is more juice in the tall thin glass than in short 
squat glass (Biehler et al. 1999:58). 
They cannot think of more than one quantity at a time, mentally reverse the situation 
and take another person’s point of view.   
2.3.1.3 Concrete operational 
This is the third stage at which the eight to 11 or 12 year old begin to develop 
schemes that allow understanding of logic based tasks. But their thinking is limited to 
concrete objects. Piaget postulates that they are not able to solve abstract problems 
by engaging in mental explorations; usually they need to manipulate concrete 
objects physically. For instance, when teaching fractions learners should be 
physically engaged like; cutting of real objects so that they can develop schemes 
such as; ordering, fair share and equivalence. Piaget in Biehler et al. (1999:59-60) 
encourages manipulation of concrete objects because children at this stage cannot 
generalise from one situation to a similar one with any degree of consistency. 
Repeated manipulation of real objects helps to develop schemes which will later 
enable children to generalize. 
2.3.1.4 Formal operational stage 
Children at this stage are able to generalize and engage in mental processes by 
thinking up hypotheses and testing them in their head (Biehler et. al.1999:60-61). 
Knowledge gained through the formation of relationships created by an individual is 
called logico-Mathematical knowledge (cf. 2.3.1; Kamii 1990:23). They can list 
equivalent fractions without necessarily sharing concrete objects. So children who 
have reached this stage are able to create relationships, hypothesize and test them. 
This study focuses on how the 11 to 12 year old learn fractions, so the last two 
stages are of importance as far as this study is concerned. 
14 
On social interactions, Piaget believed that peer interactions do more to spur 
cognitive development than do interactions with adults. His argument is that children 
are able to discuss, analyze and debate merits of another child’s point of view. On 
social interaction Piaget comments that, no amount of teaching could accelerate the 
rate at which children could progress through stages. By this Piaget perceives that 
human development precedes learning (Bobis et al. 2004:7). 
2.3.2 Bruner 
In his study on the cognitive development of children, Bruner (n.d.) came up with 
three stages, which are not necessarily hierarchical in nature. The first one is an 
enactive (action-based) representation, in which learners handle concrete objects in 
order to understand their environment (Biehler et al. 1999:140). They do so without 
the use of words. The second stage is the iconic (image-based) representation in 
which learning depends on visual or sensory organization. In this stage learners use 
diagrammatic sharing to represent concepts like fractions. The third stage is 
symbolic (language-based) representation. In order to understand the environment, 
learners use language, logic and mathematics. The symbolic representation enables 
learners to arrange ideas and store them so that they can be retrieved when needed. 
Bruner maintains that these stages are evident in both young and adult learners 
when they are faced with new situations (Biehler et al. 1999:140).  
Based on the stages of cognitive development discussed above, Bruner views 
learning as an active process in which learners construct and discover things for 
themselves. When solving problems, learners bring in their informal, current and/or 
prior knowledge to solve the problems. In doing so, they discover new truths about 
what they are learning and they also develop their own strategies to solve problems 
(Bruner n.d.; Orten 2004:194-195).  
During knowledge construction, learners may use one or all of the three stages of 
learning, select and transform information, formulate hypotheses and make 
decisions (Biehler et al. 1999:140). As learners categorize their knowledge to try and 
understand whatever they are learning they may gain more understanding, find other 
ways of explaining and in so doing they discover new knowledge hence Bruner 
views learning as the process of discovery and enquiry. In order to enhance 
discovery learners are given a variety of examples, facts and information and they 
15 
are encouraged to find answers or underlying rules or principles (Hamachek 1990: 
222). 
Another critical feature of discovery learning is that learners make errors in drawing 
conclusions. Instead of discarding them they should serve as points for instruction in 
which the teacher may ask some learners to explain why the answer is wrong. This 
can be afforded in the classroom climate that tolerates wrong hypotheses and 
incorrect answers (Hamachek 1990:224). 
For effective knowledge construction, activities should be of the difficulty that is not 
too complex for learners to handle; hence Bruner (n.d.) advocates a spiral approach 
to curriculum design at classroom level. It is the teacher’s task to design curriculum 
at classroom level because they know the learners’ context and needs better hence 
they can make informed decisions regarding what to teach and what not to teach at 
different stages. He posits that spiralling benefits learners in that they continually 
build on what they have already learned. For example learners first learn addition 
with whole numbers and they learn addition with fractions. Hence spiralling promotes 
the use of one’s prior knowledge. 
Learners could be allowed to work in small groups so that they can share their 
experiences (Orton 2004:198-199). In these cooperative learning groups as they are 
referred to by Bruner (n.d.) teachers and learners engage in active dialogue. They 
analyse and reflect one another’s experiences and hence construct their own 
understanding. During dialogue, teachers or learners may explain and ask questions. 
Teachers do so to direct the learners’ focus to key concepts of what they are 
learning whereas learners may do so while sharing their experiences with their peers 
or trying to make their point heard (Fisher 1995:12). This process Bruner refers to it 
as scaffolding. During scaffolding the mediator provides help and suggestions, but 
gradually withdraws as learners reach a level of constructing their own internalized 
understanding (Donald, Lazarus & Lolwana 2002:04-105). 
Unlike Piaget, Brunner theorizes that teachers should engage in spiral curriculum 
organization because learners can learn anything as long as it is simple enough for 
them to understand (Bruner n.d.; Fisher 1995:12). In relation to fractions, learners 
should first be helped to build the concept of fractions before ordering fractions 
16 
because having conceptualized fractions it is only then that learners will be able to 
determine their sizes. 
2.3.3 Vygotsky 
Like Bruner, Vygotsky believes that cognitive development is largely due to social 
processes particularly interactions with others who are more knowledgeable and 
competent (Biehler et al. 1999:69-71). He assures that children are first introduced to 
a culture’s major psychological tool (speech, writing and numbers) through social 
interactions with parents and later with teachers. 
He argues that children can learn anything not very far from what they already know, 
as long as they are exposed to it through collaborations with experts such as 
teachers, parents and/or experienced learners. While Piaget’s theory encourages 
teachers to plan their teaching basing themselves on their learners’ cognitive 
developmental stage, Vygotsky advises teachers to expose and guide their learners 
through challenging situations. He believes that well designed instruction aimed 
slightly higher than what children know and can do at the present time will pull them 
along, helping them master things they cannot learn on their own, hence the zone of 
proximal development (ZPD) (Biehler et al. 1999:69-71). 
Vygotsky in Biehler et al. (1999:69-71) defines the zone of proximal development 
(ZPD) as “the difference between problem solving ability that a child has learned and 
the potential that the child can achieve from collaboration with a more advanced peer 
or expert, such as a teacher.” Teachers are cautioned though that the tasks they 
give to learners should not be at a level where learners can work without guidance 
because they will practise previously learned material and result in no new learning 
(Biehler et al. 1999:69-71). They should also not be too far above learners’ current 
level of competence because the result is that the learners will be off task and no 
learning takes place. So tasks should be pitched between these two extremes as it is 
the learning zone (ZPD) in which learners can operate only with some form of help 
(Mavugara-Shava 2005: 45). 
The three theorists agree on the issue that learning is an active process in which 
knowledge is constructed. Piaget focuses on individual construction of knowledge 
through the use of one’s’ cognitive structures while Bruner and Vygotsky focus on 
17 
knowledge construction in social groups. The next section focuses on constructivism 
and how it relates to teaching and learning. 
2.4 CONSTRUCTIVISM 
The way people understand the world around them is determined by their 
epistemological underpinnings. It is mentioned in chapter one that the 
epistemological basis of this study is constructivism. Troutman and Lichtenberg 
(2003:14-18) give account of the origin of constructivism from the pioneering work of 
Piaget and Vygotsky.  
Constructivism focuses on how knowledge is acquired and emphasizes knowledge 
construction not knowledge transmission (Mudork-Steward 2005:14). Constructivism 
also views human beings as agents of their own development because they actively 
shape their development by interacting with their environment which enhances the 
development of skills essential in their every day functioning. Through interactions 
with the environment learners make hypotheses, test them and draw conclusions. In 
so doing they create knowledge, adapt and assimilate it to create new internal 
representations (Troutman & Lichtenberg 2003:18). Constructivism therefore views 
learning as an active process. 
As learners explore their environment, they sometimes find things they do not 
understand and hence fail to be accommodated in their existing schemes. In an 
attempt to understand this new information learners engage in the process of 
eliminating fallible information and adjusting their schemes. This process is referred 
to by Piagetian psychologists and educators as constructivism (Biehler et al. 
1999:54-56). Hence Piaget is also seen as a constructivist. 
This in turn creates what Piaget refers to, as cognitive conflict (cf. 2.3.1).  As Bruner 
mentioned, teachers should now play the role of the facilitator and mediator by either 
asking questions or explaining to learners so that they help them construct 
meaningful knowledge. This process is referred to by Bruner as scaffolding (cf. 2.3.2; 
Donald et al. 2002:111). Scaffolding as one of the teaching strategies of 
constructivism should be done intensively during the beginning of instruction. Then 
withdrawn gradually so that learners become independent (Donald et al. 2002:111). 
Constructivism does not stipulate what should be done in the teaching and learning 
18 
situation, it guides teachers as to how they can maximise learning by giving 
suggestions of what could be done such as pitching activities to be at the learners’ 
level (cf. 2.3.3; Mudork-Steward  2005:15). 
Since learning is viewed from a constructivist perspective as an active construction 
of knowledge, before going to class, teachers should prepare activities that will 
enable learners to construct knowledge. In order to do that teachers may structure 
the teaching and learning situation that will enable the formation of co-operative 
groups in which learners will share information as they engage in class discussions. 
As Bruner indicated, (cf. 2.3.2) co-operative learning is characteristic to 
constructivism, because in these groups learners share ideas and then alter their 
schemes as they use their peers’ ideas to build on new knowledge. Through 
discussions, learners organize their experiences and prior knowledge and then adapt 
it in their schemes as indicated by Piaget (Orton 2004:197). It is also through co-
operative groups that learners develop communication and interpersonal skills, 
which is a basic skill needed for effective functioning in social groups (Donald et al 
2002:108). 
The role of the teachers is to create opportunities for action in which they give 
learners tasks which will engage them in action (Biehler et al. 1999:396; Donald et 
al. 2002:108). Tasks should be grounded in learners’ contexts so that learning 
becomes realistic for them to make connections easily. Learners may be asked to 
work in groups and share a pizza among six people and write the fraction of the 
pizza each person will get. Through this co-operative exercise, learners share their 
experiences and strategies, they engage in arguments and give each other support 
and assistance (cf. 2.3.3). Hence, learning is a social activity and knowledge is 
socially constructed. 
Learners have different experiences and background knowledge, so teachers in 
constructivist learning should vary activities and provide many experiences that 
require manipulation of materials, so that all learners are offered equal chances to 
construct meaningful knowledge (Bobis et al. 2004:7-8). Learners should also be a 
given chance to make decisions regarding how and what they want to learn and 
teachers should create and structure learning environments that promote effective 
learning. It is true that it may not be possible to let every learner decide on what and 
19 
how to learn, but giving everybody a chance to participate and come up with own 
their own strategies and views during learning may in some way give learners a 
sense of ownership and control, hence, effective learning may result. For retention 
and long term acquisition of the new knowledge, Bruner and Vygotsky suggest that 
some form of apprenticeship should be given (cf. 2.3.2 & 2.3.3). The authors further 
suggest that learners must take greater responsibility for their own learning; hence 
learning must be learner centred and personal. 
From the above discussion about constructivism one can deduce that Piaget, Bruner 
and Vygotsky are constructivists. Piaget advocates individual knowledge 
construction. He indicates that human beings always try to make sense of their 
environment by organising their knowledge, comparing what they know to new 
information therefore adjusting their schemes. This is an individual activity hence 
Piaget talks of individual knowledge construction.  
Bruner and Vygotsky add that to effectively construct new knowledge learners need 
other people. When working in groups, as Bruner indicated, learners may exchange 
ideas and understandings of what they are learning (cf. 2.3.2). Vygotsky on the other 
hand adds that to construct knowledge, learners need somebody who knows more 
than them to guide them. This more knowledgeable other could be a teacher or other 
learners (cf. 2.3.3). 
2.5 LEARNING MATHEMATICS   
Learning, as defined from the cognitive psychology point of view is the acquisition of 
knowledge. It comprises all activities that increase the individual’s knowledge, skills 
and understanding of the world so that they can interact effectively and successfully 
with their environment (Steffens 2001). From the works of Piaget, Bruner and 
Vygotsky, learning is an individual construction and accumulation of knowledge 
through collaborations with other people like peers and/or the more knowledgeable 
other who could be a teacher, parent or other children (cf. 2.3.3; Bruner n.d.; Biehler 
et al. 1999). 
Learning is also perceived as a personal activity which  is determined by a variety of 
factors such as; learning environment, individual preferences, context in which the 
content is presented and the quality of instruction (Hierbert et al. 1997:5-6;Fennema 
20 
& Romberg 1999: 5). Knowledge is constructed as learners make sense of the new 
information and connecting it to their experiences so that it makes more sense, 
hence, it is cumulative. Some people learn best when they collaborate with others 
and share ideas, argue their strategies or findings and explain what they learned to 
others, hence, learning is a social process (Mueller, Yankelewitz & Maher 2010). 
Russell (1999:1) as well as Haylock and Cockburn (2008:226) view Mathematics as 
a discipline which deals with abstract entities, as a collection of ways of thinking and 
reasoning and ways of organizing and internalizing information received from the 
external world. This conceptualization of mathematics accords with Piaget’s theory of 
accommodation and assimilation (cf. 2.3.1), in which learners need to think about the 
new knowledge, compare it with what they already know and organize it in such a 
way that it makes more sense to them. Mudork-Steward (2005:13) adds that 
mathematics is a language that is used to describe the physical and non-physical 
aspects of the world we are living in. For example learners may be asked what the 
fraction of girls in their classroom is. Therefore learners need to have a good 
command of this language so that they can communicate effectively. 
It is mentioned earlier in the works of Bruner and Vygotsky that learning is socially 
constructed. Through social learning learners become mature in terms of knowledge 
possession and reasoning capability. This indicates that social learning is vital when 
learning Mathematics as it is a language which needs to be communicated. So 
through learning in a social setting, learners have the chance to share their 
understanding, difficulties and observe one another. Through these sharing 
sessions, learners have the chance to modify their understandings, rethink their 
conceptions and do things they were not able to do before engaging in group work. 
One of the research questions is to determine what literature says on effective 
learning of fractions, so this section will reveal what a learning environment is, how it 
enhances the learning of mathematics; what is effective learning, what factors 
contribute to effective learning and what are its indicators? Lastly the barriers to 
effective learning will be discussed. 
 
 
21 
2.5.1 Learning environment  
A learning environment is viewed in relation to the factors in the teaching and 
learning situation which affect learning positively. Bobis et al. (2004:302) refer to the 
learning environment conducive to the development of mathematical power for all 
learners as those that:  
• encourage students to explore 
• help students verbalize their mathematical ideas 
• show students that many mathematical questions have more than one right 
answer 
• teach students the importance of careful reasoning and disciplined 
understanding through experience 
• build confidence in all students so they can learn mathematics 
Troutman and Lichtenberg (2003:561-562) suggest that teachers must create an 
environment that promotes respect and empathy. When learners know that they are 
respected, they are free to explore their environment and participate in class 
activities knowing that their contributions will be appreciated. Teachers may allow 
learners to express what they like, value and believe. Troutman and Lichtenberg 
(2003:561-562) presume that, these interactions will provide richer content for 
problem solving tasks as teachers will formulate tasks around the learners’ context.  
A learning environment is more than a designated room or physical space in which 
instruction occurs; it is created by the hidden messages conveyed about what is 
important in learning and doing mathematics (Bobis et al. 2004:304). It is therefore 
socially, emotionally and physically created. If teachers hold the belief that 
mathematics is a set of rules and procedures to be transferred from one person to 
the other, this will be depicted in the classroom setting in which there will be silence 
and the teacher will be the only one responsible and talking (Troutman & Lichtenberg 
2003:304). Learners will not be given chance to verbalize their mathematical ideas.  
In the learning process which supports knowledge construction by learners, mistakes 
are inevitable (Ding 2007). In an attempt to construct knowledge learners are likely to 
22 
make wrong connections and generalizations. So the situation should afford learners 
to ask questions to seek for clarification and guidance. Teachers should also ask 
learners questions which will direct their focus to important issues hence result in a 
healthy classroom discourse (Huerta 2009). In rich learning environments learners’ 
mistakes are used as opportunities for learning. The whole class analyzes mistakes 
in order to find why they are mistakes instead of rejecting them without giving 
justification. Teachers and learners in healthy and rich learning environments are 
free to take risks and deal with difficulties (Ding 2007; Hiebert et al. 1997). This helps 
learners to develop confidence and hence face real life problems and challenges and 
solve them confidently (Abramovich & Brouwer 2007). 
2.5.2 Effective learning 
Effective learning is an ultimate goal of knowledge formation and does not occur in a 
vacuum but in a learning environment that supports it. Some learners are naturally 
motivated and have high self-confidence but if this is not nurtured they end up de-
motivated. In learning environments that support effective learning, most learners 
become free to experiment and hence discover things for themselves (Brown & 
Quinn 2007). The knowledge base learners get from experimentation (physical 
knowledge) enables the process of accommodation since what is learned makes 
more sense (Kamii 1990:22). 
So this new knowledge is meaningful and effective if it benefits the learner (Beihler, 
et al.1999). It is essential that learners are helped to learn effectively by exposing 
them to situations that will facilitate knowledge construction through cooperative 
groups, stipulating the goals, employing spiral approach to curriculum and 
encouraging independence and autonomy (Bruner n.d.). Beihler, et al. (1999:387) 
emphasize that effective learning results when: 
• Learners understand the fundamental ideas of a subject and how they are 


related. That is, understanding as a number which can be marked on a 

number line 
• Learners construct their own knowledge   
23 
• When constructed knowledge can be applied to different situations. That is, 


understanding   as the ratio between two numbers like 1 and 2 and dividing 

sweets amongst children in the ratio 1 to 2 
Employing a spiral approach to mathematics teaching makes use of learners’ prior 
knowledge. That is when learning fraction addition learners make use of the 
knowledge of addition of whole numbers. Hence spiralling results in the 
accumulation of knowledge. Learning is cumulative and effective because 
relationships are established between concepts. 
Brown and Quinn (2007) suggest that learners should be given freedom to 
experiment and come up with their own strategies when answering questions and 
solving mathematical problems. Mathematical problems should be rooted in learners’ 
real life so they can make use of their prior knowledge (Troutman & Lichtenberg 
2003: 561-562). In this way learners will have effective learning because they gained 
skills and insight into the problem. Thus learning should be contextualised and 
situated. So when faced with the same problem they will use the acquired skills. This 
suggestion is supported by Bruner (in Beihler et al. 1999:389) as he emphasized that 
learning should be “discovery because conceptions that children arrive at on their 
own are usually more meaningful than those imposed by others.” Therefore learning 
is personal and discovery learning results in effective learning. 
Learners who have effectively learned a concept are able to communicate it 
effectively to others as it makes more sense. Communication is regarded as another 
tool for effective learning (Mueller et al. 2010). Through communication learners 
share ideas and help each other view the same situation or problem from different 
perspectives. This could be achieved through collaborative groups.  
During collaborative encounters, learners ask questions to seek for clarifications and 
they explain their thinking. As they communicate they rethink their strategies and 
hence use their peers’ reasoning to improve on their own (Vygotsky in Dahl, n.d.). 
Through these social encounters learners can learn and acquire skills which are 
thought to be beyond their level of comprehension (cf. 2.3.3). 
Mueller et al. (2010) underline that understanding, which is an indication of effective 
learning, is meaningless without a serious emphasis on reasoning. One who 
24 
understands has the capacity to explain, justify and think critically. As a result 
learning is facilitated because they are now able to learn quickly and are able to use 
their knowledge flexibly in novel situations (Newton 2000:7-8). When learning about 
sharing, learners should be able to give reasons behind their solutions such as why 
 
the shaded part is  not  (Charalambos et al. 2007).  
 
 

Figure 2.3 Representation of  


If they are able to argue that; the fraction is  because there are two shaded parts out 

of five parts that the whole is divided into. This will indicate that learners use 
reasoning to understand relationships and make connections to new ideas. 
The definition of effective learning of De Corte (1996:34-37) summarises the 
previous discussions on effective learning and is further more based on international 
research on learning. In this definition effective learning is considered a constructive, 
cumulative, goal-directed, situated, collaborative, self-regulated and individually 
different process of knowledge-building through meaningful construction. 
Having discussed what effective learning is, it is worth explicating the character traits 
needed to enhance effective learning. In the next section, the three levels of maturity 
one needs to have to participate effectively in the teaching and learning situation, will 
be discussed. 
2.5.2.1 Dependent, independent and interdependent 
From the preceding discussions it is mentioned that learning is acquired through 
interactions with one’s environment including the people around. People interact 
effectively when they have attained higher levels of maturity, being independent and 
interdependent. For clarity the discussion will commence by discussing the lowest 
level of maturity which is dependence. 
 
25 
• Dependent  
Children at Piaget’s first stage of development (sensorimotor stage) (cf. 2.3.1.1) are 
entirely dependent on other people for everything. But as they develop to the second 
stage they begin to have some physical independence. 
• Independent  
As children advance to Piaget’s second stage of cognitive development they begin to 
have some independence. That is, they decide for themselves when they want to 
walk or play. But they still lack emotional and intellectual independence. In the 
teaching and learning situation, learners should be helped to be intellectually and 
emotionally independent. Covey (1989: 50) explains that people are said to be 
emotionally and intellectually independent when they do not need other people’s 
opinion to feel good and are able to make their own decisions, think and reach their 
own conclusions. 
Since learning is defined among others as collaborative, then for effective learning to 
take place learners should be emotionally and intellectually independent. If not, they 
will be unable to participate in collaborative groups because of the fear of threatening 
their self-worth.  
There is a higher level of development, one in which learners are aware that they 
need other people’s thinking to complement their own. This is interdependence. 
• Interdependent 
This is the highest level of maturity in which one accepts that they are capable to 
produce, but they are more capable and can accomplish more when working with 
others (Covey 1989:50-51). These people participate in co-operative groups 
effectively as they take criticism and disagreements positively. They acknowledge 
that they need other people’s thinking to complement their own and they are ready to 
change their point of view after analysing other peoples thinking. 
2.5.2.2 Prior knowledge 
It is mentioned in 2.3.1 that people learn effectively when they are able to assimilate 
and accommodate new knowledge to what is already known, otherwise confusion 
26 
results and learning becomes minimal. Bruner on the other hand in 2.3.2 advocates 
spiralling of curriculum design. With regard to prior knowledge, spiralling of 
curriculum design makes it possible for learners to revisit the basics of a concept- 
fraction in this case-at different levels of schools. When learning fraction addition 
spiralling will call for reviewing the meaning of the concept, fraction. In this way 
learners’ prior knowledge is tapped so that it assists in the effective learning of 
fraction addition. 
Prior knowledge comprises all the formal and informal knowledge together with 
experiences learners bring into the learning environment which needs to be 
confirmed or refuted (Mueller et al. 2010). If no match exists between the learners’ 
prior knowledge and the new knowledge, effective learning does not occur (cf. 2.3.1; 
Bobis et al. 2004:13). As Piaget indicated earlier, that a misfit of old and new 
knowledge leads to learners being confused (Fisher 1995:12; Murdock-Steward 
2005:14-17). 
The use of learners’ prior knowledge as reference for learning increases their 
curiosity and they get deeply involved in learning. This enables learners to create 
their own ways of understanding and build representations and understanding based 
on their previous knowledge and experiences (Theunessen 2003; Mueller et al. 
2010).  
Mueller et al. (2010) contend that learners make maximum use of their prior 
knowledge when the learning materials used in the lesson, provide them with 
opportunities to have full experience of the content. When adding mixed fractions 
learners may be asked to write how much chocolate do the girls have if Neo has one 

 

bar and a half (1 ) and Lipalesa has one and a quarter (1 ). With real chocolate or 
 
slabs made from clay, learners may use them to do addition.  
The use of learning materials provides, what Piaget calls, physical experience. 
Hence, it is imperative for teachers, to have a thorough knowledge of the learners’ 
socio-cultural background from which they would formulate problems and activities 
that make sense to all learners, hence, situated learning is enhanced (Bobis et al. 
2004:302). If the concept fraction is introduced through sharing, learners should then 
engage in the actual sharing of concrete materials they share in their everyday life, 
27 
or what Dewey (1969) calls experiential learning. In that way learners will be able to 
compare old and new knowledge, contrast what is and what is not, separate out 
patterns with respect to fixed conditions, make generalizations after seeing the same 
phenomenon under different tool usages, and simultaneously bring together different 
and varying aspects to construct new knowledge (Mueller et al. 2010). 
The use of learners’ prior knowledge seems to be crucial as it appears to bridge the 
gap between informal and formal knowledge. The fact that it encompasses 
experiential learning and enhances critical thinking also seems to be an answer to 
the educational needs of all nations (Freire 1973:36). 
2.5.2.3 Experiential learning 
Dewey (1969:118) argues that experience is an active-passive process in which “We 
do something to the thing and it does something to us in return”, that is, the 
experience should result in the change. The change might be the growth of 
knowledge or an insight into whatever was experienced. 
With regard to learning, it is indicated above that incorporating learners’ prior 
knowledge enhances effective learning and that learning should begin with the actual 
manipulation of concrete objects. During manipulations of real objects learners use 
in their real-life, learners should move backward and forward connecting what they 
do, what they know and the outcomes of their actions (Dewey 1969:118). In the case 
of sharing as they learn fractions, learners should experience the feelings and 
emotions of unfair share. By this it means the pain or the joy of the sharing 
experiences. In this way Brookfield (1996:750), Beard and Wilson (2006:2) and 
Dewey (1969:119) say that it is learning from experience. Having experienced 
sharing, learners may be in a position to form relationships between fractions and 
equal sharing. 
Unlike traditional learning which starts with abstractions which may not bear any 
relationship to learners’ experiences, experiential learning emphasises starting with 
concrete experiences. Teachers should therefore design activities which will help 
learners have meaningful experience of what they learn. When learners have 
learned from experience and they have established the connections and 
relationships between activity and the consequence, it then becomes possible to 
28 
make accurate and comprehensive foresight (Dewey 1969:123). Establishment of 
connections and formation of relationships result in the cumulative growth of 
knowledge hence experiential learning is continuous. 
In order to maximize learning, activities should demand the use of multiple senses. It 
is indicated in previous paragraphs that, in the sharing exercise to build the fraction 
concept, learners do not only share but they use their hands to touch as they share, 
their eyes to compare the size of the pieces and their brains to give reasons to their 
observations and discuss their observations amongst themselves. This is supported 
by Dewey (1969:120), Beard and Wilson (2006:6-7) when they contend that 
experiential learning results in deeper learning when the experience requires the use 
of most of the senses, mind and body and when the activities provide challenge to 
engage and test mental and physical endurance. 
Dewey (1969) referred to traditional classrooms as those in which, learners are 
expected to sit passively in class as the teacher talks, moves and demonstrates. 
Dewey (1969:120) contends that this is divorcing the bodily activity from the 
perception and meaning. When this is done, mechanical use of bodily actions, or 
rote learning, is encouraged. In this manner learners are denied the opportunity to 
learn using their senses, namely seeing, hearing, feeling and tasting. Even today 
that arrangement is still referred to as a traditional classroom. 
Caution should be taken though, that experiential learning could result in gaps that 
may hinder future learning (Beard & Wilson 2006: 21). The authors explain that 
during experiences two situations may arise. The first one may be where learners 
have established connections and relationships. That is, they have developed 
methods and strategies to act in a certain situation. They are also able to discern 
why they work and why they do not work. 
The second situation according to Beard and Wilson (2006:21) is where learners do 
not develop connections and relationships but act in a trial and error fashion. In this 
situation they have not developed strategies and cannot discern why this works and 
that does not work.  As a result they are not in a situation to decide with confidence 
how to act in the future. 
29 
The discussion indicates therefore that teachers should be careful they facilitate 
learning so that learners make sense of the experiments they engage with. This is to 
ensure that experiential learning results in effective learning. Conceptual 
understanding of the concept learned is contributing towards effective learning. 
2.5.3 Conceptual and procedural knowledge 
2.5.3.1 Differentiating between conceptual and procedural knowledge 
In every day endeavours when people encounter problems they try different 
approaches to solve such problems. When doing thus they often search for 
relationships and connections to what they already know and then determine the 
strategy they think could work (cf. 2.3.1 & 2.5.2.2). At other times relationships are 
not established and no connections are made hence disequilibration occurs. So 
when one is faced with this situation and cannot confidently establish connections 
and relationships one tends to search more into the problem and use a trial and 
improvement approach to solve the problem (cf. 2.5.2.2).  
In the first situation where relationships and connections are established it is said 
that conceptual knowledge is gained. Conceptual knowledge is therefore defined as 
knowledge of facts and properties of mathematics that are recognised as being 
related in some way (Owen & Super 1995:138; Effandi 2009). The authors further 
explain that the interconnections between ideas explain and give meaning to 
mathematical procedures and algorithms hence conceptual knowledge is gained. 
Owen and Super (1995:138) maintain that a piece of information becomes 
conceptual knowledge only when it is integrated into a larger network that is already 
in place. This once again links to Piaget’s theory of accommodation.  
Building conceptual knowledge also makes use of one’s prior knowledge. With 
regard to fractions, learners are said to have conceptual knowledge if, for example, 
they learn fraction position values by relating them to whole numbers and/or fractions 
(Owen & Super 1995:138). Since learners already know about the whole number 
system and their position on the number line, then using this knowledge will facilitate 
fraction knowledge construction. 
In order to enhance conceptual understanding learners should be encouraged to 
handle and experiment with physical concrete objects, as it is through manipulation 
30 
of concrete objects that they gain physical and logico-mathematical knowledge (cf. 
2.3.1). These are believed to facilitate construction, understanding and retrieval of 
mathematical concepts (Mudorck-Steward 2005: 35). 
In most cases people are satisfied with the knowledge of facts and procedures to 
follow when doing certain tasks. Most people do not ask the why of what they are 
doing. This type of knowledge is called procedural knowledge. It is defined by 
(Barker & Czarnocha 1995; Effandi 2009: 202) as the mastery of computational skills 
and familiarity with procedures for identifying mathematical components, algorithms 
and definitions. It is characterised by lack of relationships and of connections to prior 
knowledge (Owen & Super 1995:138). A learner who can only read 14 as one 
quarter without realising that it is a number greater than zero but less than one has 
procedural not conceptual knowledge (Owen & Super 1995:138). 
Learners who have conceptual knowledge are able to tackle a variety of problems in 
different contexts whereas those with procedural knowledge are able to compute 
procedures but fail to solve problems in different contexts from those they were 
drilled on. 
2.5.3.2 Connecting conceptual and procedural knowledge 
Barker and Czarnocha (1995) point out that learning begins with actions on existing 
conceptual knowledge and with increasing knowledge, learners acquire procedural 
knowledge. When learning fractions learners have some informal knowledge 
regarding sharing but in most cases they are not aware of the importance of equal 
sharing. So when teachers emphasize equal sharing their knowledge becomes 
modified and increases. Learners then gain procedures regarding how to share 
objects of different shapes in order to come up with different fractions. 
Fennema and Romberg (1999:4) and Meagher (2002) caution that conceptual 
knowledge should precede procedural knowledge so that effective learning can 
occur. They concur that learners who possess procedural knowledge only, simply 
practise procedures and algorithms without understanding what they are doing. 
When faced with a novel problem such learners do not have alternative methods 
should they forget the algorithm or the learned procedure (Carruthers & Worthington 
2006:75; Meagher 2002). 
31 
On the other hand learners could be helped to develop conceptual knowledge by 
means of procedures. As they execute procedures learners could be encouraged to 
embark on the cognitive process of reflecting on the procedure by looking for 
patterns and relationships (Hiebert 1990:34). When adding fractions by embarking 
on the algorithm of finding the lowest common multiple, learners could be asked to 
find how and why the algorithm works. If, through procedures learners could develop 
conceptual understanding, they will have a cushion to fall back on should they 
encounter difficulties when confronted with novel problems.  
Hiebert as (cited by Owen and Super 1995:139-140) describes three types of 
connections between conceptual and procedural knowledge. The first being 

 
attaching meaning to mathematical symbols, that is, when computing 2  x 6  =, a 
 

 
learner must have an understanding of 2 ; x; 6 ; =; and ___. Without conceptual 
 
knowledge, gained through social, physical and/or logico-mathematical knowledge of 
these symbols, learners will not be able to make sense of the multiplication of 
fractions.  
The second connection is with regard to the internal steps of the procedure and their 
conceptual underpinnings. Learners are expected to understand that the product 
would have the denominator as a multiple of 5 (Hiebert in Owen and Super 
1995:139-140). 

 
The last connection would be to realize that 2 x 6  = ___ must be somewhere 
 


between 12 and 21. Twelve is 2 x 6 while 21 is obtained from 3 x 7 since 2  is 


between two and three, while 6  is between six and seven. Owen and Super 

(1995:139-140) cite Hiebert as explaining that interpreting a computed answer with 
meaning is highly dependent on conceptual underpinning of the number and 
operation symbols. 
Possession of conceptual understanding of mathematical concepts enables learners 
to reason from different perspectives because conceptual learning is believed to 
enhance more mathematical content (Mudork-Steward 2005:23). 
 
32 
2.6 BARRIERS TO EFFECTIVE LEARNING  
As discussed, learning is very complex and depends on a number of factors: there 
are some that facilitate learning and some that impede learning. This section 
discusses the factors that impede learning. 
2.6.1 Lack of motivation  
Some people are naturally motivated while others are naturally demotivated. 
Motivation can be viewed as “the engine that powers and directs behaviour” 
(Entwistle 1981:193; Hamachek 1990:262). Some people have the inner natural 
powers that direct their behaviours while others need the external drive to behave. 
Such people are said to have intrinsic and extrinsic motivation respectively. 
Learners may enter the learning situation with intrinsic motivation but this may 
diminish due to circumstances beyond the individual, such as persistent failure no 
matter how hard the learner tries to succeed; lack of opportunities for enhancing 
personal adequacy; and lack of self-worth (Hamachek 1990:263). To help them feel 
fulfilled they should be afforded the opportunity to participate in cooperative learning 
groups. In such groups learners may have the platform to demonstrate their abilities 
which may boost their motivation level. 
Learners with high self-esteem perceive themselves as achievers and in most cases 
strive to maintain high standard, whereas those with low self-esteem view 
themselves as failures and also maintain that standard (Hamacheck 1990:282-283). 
Learning environments should therefore provide extrinsic motivation to those 
learners with low self-esteem so that their perception of themselves may change for 
the better and hence boost their motivation level.  
Learning situations should be structured with learners’ characteristics in mind, that is, 
learners’ intelligence, motivation, personality and prior knowledge (Entwistle 
1981:83). For low achievers, they may be given tasks of low cognitive order so that 
their chances to succeed are increased. On the other hand, high achievers may be 
given demanding task which will challenge them so they stay motivated (Hamachek 
1990:283). 
33 
For any activity that learners do they should be made aware of its relevance and be 
helped to see how it relates with their prior knowledge. If no relationship is 
established then some learners are likely to be demotivated. Learners whose level of 
motivation is low do not put in any effort to understand or learn. They are, in most 
cases pushed from behind in order to learn something. 
2.6.2 Learners learning styles 
Learning has been defined, among other things as a personal process (cf. 2.5). It 
was mentioned earlier that learners receive information, assimilate, accommodate 
and construct knowledge differently due to their different characteristics. Hence, 
people adopt different strategies to learning. 
When confronted with a situation, in order to understand it, others prefer to take bits 
and pieces of information and link them logically to build a global understanding or 
generalisation. Others prefer to begin with a big idea and then break it into smaller 
pieces of information, hence they develop deep understanding. The former group is 
referred to as inductive or serialist learners while the latter is the deductive or holistic 
learners (Entwistle 1981:90-92; Marwaha 2011). The general tendency that one 
adopts when learning is referred to as learning style (Entwistle 1981: 93). 
Within the two broad learning styles discussed above, there are those who 
understand best when they write down the steps. That is, talk about what they do 
and demonstrate it in different forms, like drawing concepts maps, diagrammatically 
or otherwise. There are those who understand when they start with a bigger picture 
like a concept map or a diagram and then discern why and how, concepts relate. 
The implication to education is that learning situations in classrooms should promote 
the use of multiple senses (cf. 2.5.2.3). If the classroom situation favours one type of 
learning style, say inductive learners, those who are deductive are disadvantaged 
hence, minimal learning results. In the classroom learners should be encouraged to 
use all their senses; otherwise they will experience learning difficulties (cf. 2.5.2.3).  
 
 
 
34 
2.6.3 Fragmented learning  
Street or informal mathematics is the knowledge base acquired out of school and it is 
mainly oral arithmetic. School or formal mathematics on the other hand is the 
knowledge base acquired in schools (Nunes & Bryant 1996:104).  
Informal mathematics is mostly physical knowledge gained through experimentation 
(cf.2.3.1). The authors explicate that representation of numbers in oral arithmetic 
allow learners to calculate and think of values they are working with at the same 
time. Whereas at school, the meaning of numbers or logico-mathematical knowledge 
is not thought of during calculations, instead numbers are operated with and spoken 
about outside their context (Nunes & Bryant 1996:104-108). The question can be 
asked whether this is still the situation now in 2012.  Kamii (1990:22-23) states that 
this happens especially when learners are not helped to identify relationships. This 
approach to learning results in fragmented learning. 
Fragmented learning is manifested by the lack of solving problems similar to those 
that learners encounter in real life. With reference to fractions, fragmented learning is 
explicated by incidences such as: two pizzas of the same size cut into different 
pieces. One is cut into six pieces while the other one is cut into eight pieces, if one 
person gets a piece from each pizza, from which pizza does this person get more? 
The authors indicate that if learning is fragmented, learners will say the person gets 
more pizza from one that is cut into eight pieces. The reason being that, learners 
know that eight is greater than six. The question can be asked whether the fraction 
concept was developed from only one representation? 
2.6.4 Language  
Learning mathematics requires understanding of carefully worded expositions as 
well as appreciation of carefully drawn distinctions. Mathematics is a language; 
therefore to understand it teachers should create appropriate experiences in which 
symbols, pictures, context and language are meaningfully connected (Association of 
Teachers of Mathematics 2010:1). This accords with Bruner’s cooperative groups in 
which learners work together during which they explain and ask questions (cf. 2.3.2). 
During these collaborative encounters learners use language and hence may 
become conversant with the appropriate language.  
35 
 
     
Figure 2.4 The connective model of learning mathematics (taken from Association of 
Teachers of Mathematics 2010) 
Regarding fractions, learners are not immersed in fractions as they are in numbers. 
They are not exposed to situations in which they actively engage with them. 
Therefore learners do not become familiar with the language and as a result they do 
not work and move confidently from context to symbols and images (Association of 
Teachers of Mathematics 2010:1). 
Learners could be immersed in a situation where they share a chocolate bar or any 
concrete material among four people (concrete experience). To represent the 
chocolate bar pictorially, a rectangle cut into four equal parts (pictures) may be 
drawn. When giving each of the four people their share, that is, one quarter or one 
part out of the four parts then the appropriate language is used. The (symbol) 


representing one part out of four  can now be written (Association of Teachers of 

Mathematics 2010:1).  
With this type of representation, learners would have got a complete picture of the 
concept and would have gained a complete meaning of fractions. In this regard if this 
holistic approach to fractions is not presented, learners will encounter difficulties in 
the use of language of fraction and fragmented learning will result. 
Other difficulties that language impose on the learning of mathematics bear 
reference to the restricted and imprecise use of language in describing fractions 
such as representing fractions by shapes. Karslake (1991:86-87) exemplifies this by 
giving a situation in class where a circle is divided into two parts to give two 
36 


semicircles. Each semicircle is labelled one-half or  but a semicircle is a whole in its 


 
own right. Another example is taking an example of  an apple and  of a bun. 
 
Karslake (1991:86-87) explain that if learners are exposed to fractions as parts of a 
whole and they are to operate them, then problems of meaning arise as there is no 
possible meaning that can be attached to adding one-half of an apple to three-
quarters of a bun. This is because it is numbers that are added not pieces of food. It 
is even more inexplicable to interpret multiplication of 12 of an apple and 34 of a bun. 
This implies therefore that teachers should first help learners conceptualise fractions 
as numbers and represent them as different constructs (part-whole, measure, ratio, 
quotient and operator) as presented by Charalambos, Charalambous and Pitta-
Pantazi (2007). When learners have fully conceptualised fractions as numbers, now 
they can be taught how to operate them. Based on Karslake’s (1991:86-87) example 
it is also advisable that learners, especially in primary schools should operate 
fractions with meaning. One example could be to find the amount of fruit juice John 


would have drank if Sana gives him one quarter ( ) of a litre and Lebo gives him two 


fifths ( ). 

Classroom teaching is full of jargon Karslake (1991:88). Some of the phrases and 
words that teachers use, intentionally or unintentionally have some impact on the 
learning process. Some of this jargon, just to mention a few examples, are ‘to share 
by’ and ‘cancelling’. If there are sweets to be shared among a certain number of 
children, teachers or learners use the phrase ‘share by’. To share sweets by children 
is bizarre. The author attributes this confusion to stem from the comparison with 
‘dividing by’, which applies strictly to numbers only. 
The word ‘cancelling’ also has evidenced to cause difficulties in the learning process. 
In general speech ‘cancel’ has the idea of: to remove, to undo and to annul and this 
may explain the way many children appeal to the notation of cancelling to get them 
out of the difficulty (Karslake 1991:90). It is true that some learners do not 
understand that cancelling in mathematics is used when dividing. As a result they 
use it as a convenient technique mostly when simplifying fractions but they do not 
relate it clearly to the concept involved. This incorrect use of cancelling Barmby, 
Bilsborough, Harries and Higgins (2009:81- 82) attribute it to often teaching through 
37 
memorization. That is if teachers do not use symbols, pictures, context and language 
learners will tend to misuse cancelling as they often lack understanding of the 
concept. 
When learning new abstract concepts, learners are often helped to make sense of 
this new knowledge by relating it to their prior knowledge (cf.2.5.2.2). In order to do 
this, teachers often use metaphors. Nolder (1991:105) mentions that “metaphors 
enable people to deal with novel experiences whether describing something which 
they have never previously encountered, or seeking to comprehend a new idea.” 
Nolder (1991:106) explains that teachers use metaphors with the purpose of  
• offering something concrete and familiar to help learners understand an 
unfamiliar abstract idea 
• linking a new concept to the learners’ experiences 
The author points out that, teachers do this with the intention of making the concept 
more secure and hope that the metaphor by its novelty may enhance memorability. 
He points out that research has indicated that metaphors can create difficulties to 
learn mathematics. 
When learning number concept and thus multiplication, learners are often told that 
multiplication makes bigger, but when applying multiplication to  fractions, the 

 
 
 
 
 

generalization no more holds because ×  =  and  is less than both  and so 
     
learners become confused and encounter difficulties when multiplying fractions. 
This possible difficulty implies that teachers should use metaphors, but with great 
care, so that the learning should not be polluted. That is teachers should first 
examine the metaphors for their danger of causing confusion and misunderstanding. 
In class communication teachers often ask questions and learners give responses. 
Depending on the learners’ response, teacher gives feedback. Cangelosi (2003:63) 
distinguishes between two types of language teachers often use in classroom 
communication. These are descriptive and judgmental languages.  
In classrooms teachers often label learners as good or bad depending on their 
responses. Cangelosi (2003:64) explains that the use of judgmental language may 
38 
have negative impact on learning. This is because learners may refrain from 
responding to questions with the fear of being labelled as either good or bad. 
Instead of using labels the author advocates the use of language that provides rich 
information which provides learners with specific feedback. When learners explain 

 

correctly why  is greater than  , descriptive feedback could be; ok I understand that 
 

 

half ( ) a loaf is bigger than a quarter ( ) loaf. Judgmental language such as ‘correct, 
 
not quite, wrong, no’ is often used and that does not provide learners with important 
information. 
Cangelosi (2003:63-64) therefore encourages teachers to use descriptive language 
as learners are more likely to listen and pay attention to what their teachers say if the 
teachers’ comments are full of helpful information. 
2.7 CONCLUSION 
This chapter has elucidated that reform in mathematics teaching requires an 
understanding of how memory is formed and how children learn. Therefore learning 
was discussed from the psychological point of view of Piaget, Bruner and Vygotsky.  
The aspects of effective learning of mathematics such as ensuring a favourable 
learning environment that support cognitive growth, using learners’ prior knowledge 
and engrossing learners in manipulations of concrete materials were discussed. It 
was also indicated that helping learners develop conceptual knowledge before 
procedural knowledge facilitates effective learning. 
Since learning could be facilitated, it can also be impeded. It is in this regard that 
barriers to effective learning were also discussed. These are lack of motivation, 
learning styles, fragmented learning and language. De-motivated learners find it 
difficult to learn so do learners whose learning styles are not incorporated in the 
learning encounter. It was also indicated that learners who are not exposed to 
different representations of a concept may not develop a holistic understanding of 
the concept. Inappropriate language use may also be a barrier to learning. 
 
39 
CHAPTER 3 
TEACHING MATHEMATICS 
3.1 INTRODUCTION 
Teaching is a very complex activity which goes beyond the management of 
classroom discipline, motivation and imparting of knowledge from the teacher to the 
learners. It is dependent on a number of things such as the learners’ characteristics; 
context and teachers’ beliefs about their role in the teaching situation and their 
beliefs about the subject they are teaching (Kroll & Miller:1993).  
3.2 TEACHING MATHEMATICS 
Mathematics has been defined in section 2.5 as a discipline that deals with abstract 
entities, a collection of ways of thinking and reasoning and of organising and 
internalising information received from the external world. Because of the abstract 
nature of Mathematics, one of the tasks of teachers is to try to make it concrete. This 
is because learners of concern in this study understand best when they work with 
concrete materials, since they are at the beginning of the concrete operational stage 
as discussed in section 2.3.1.3.  
It will be discussed later in this chapter that one way of teaching learners to think, 
reason and organise mathematical information is through problem solving. It is in this 
regard that Strutchens (2007) as well as Fajemidagba and Olawoye (n.d.) state that, 
teachers’ beliefs determine how they plan their teaching activities. Teachers who 
hold the belief that mathematics is a way of thinking and reasoning will use teaching 
strategies which align with this belief. But those who believe that it is a set of isolated 
facts and skills or, as rules to be memorised and practised, then they will not teach it 
for understanding (Hiebert et al. 1997 xvi). Such teachers employ traditional methods 
of teaching in which the main emphasis of the lesson is to practise procedures rather 
than develop conceptual knowledge (Fennema & Romberg 1999:4). This implies that 
teachers should work on their beliefs so that their perception of mathematics may 
also change as will their teaching strategies. 
Teaching is about assisting others develop intellectual growth. Shulman (1987:7) 
states that; 
40 
A teacher knows something not understood by others, presumably the 
students. The teacher can transform understanding, performance skills, or 
desired attitudes or values into pedagogical representations and actions. 
Teachers therefore need to understand what is to be learned and how it is to be 
taught as this is the basis for teaching (Ball n.d.:1). Shulman (1987) adds that it is 
also imperative for teachers to have multifaceted understanding of the concept to be 
taught so that they can guide learners, to a better understanding. By the multifaceted 
nature of the concept, the author means, its representation-either pictorial or 
otherwise-then how the concept relates with other concepts, within and outside the 
subject. 
In addition to the knowledge of the subject matter, Shulman (1987:14) cite the 
following pedagogical reasoning and actions teachers need to undergo. 
• Comprehension of subject matter within and outside the discipline 
• Transformation of comprehended ideas by critically preparing the given text. 
Then representing the ideas in the form of analogies and metaphors by 
selecting from a variety of teaching methods. Since learners are different and 
have different needs, teachers may adapt the representations to learners’ 
characteristics 
• Instruction, which includes, inter alia, organizing and managing the classroom, 
presenting clear explanations and descriptions; assigning and checking work 
and interacting effectively with students through questioning and probes; 
answers and reactions and praise and criticism 
• Assessment which involves checking for learners understanding during 
interactive teaching and at the end of a lesson and evaluating the teacher’s 
own performance and making adjustments 
• Reflection involves reviewing and critically analyzing teacher’s own 
performance and that of the learners 
In order to achieve this goal of teaching, teachers should have a good command of 
the subject matter because if they are largely ignorant or uniformed they can do 
much harm by passing on inaccurate bodies of knowledge to their students. Such 
41 
teachers may even fail to realize and correct students' misconceptions (Ball & 
McDiarmid n.d.; Haylock & Cockburn 2008:6-7; Zakaria 2009). Because children 
understand structures of mathematical ideas differently from adults, teachers should 
find what understanding learners have regarding a concept because Haylock and 
Cockburn (2008:7) believe that this will enhance and strengthen the teachers’ own 
understanding of mathematics and help teachers with identifying learners’ 
misconceptions. 
Murdock-Steward (2005:21) communicates that mental activities which contribute to 
the development of mathematical understanding are; (a) constructing relationships, 
(b) extending and applying mathematical knowledge, (c) reflecting about 
experiences, (d) articulating what one knows and (e) making mathematical 
knowledge one’s own. Teachers should go through these activities so that they are 
able to help learners do so (Haylock & Cockburn 2008).  
In the above discussion there has been mention of teachers’ knowledge of the 
subject matter and the reasoning and actions teachers need to undergo for them to 
teach. The next section will discuss the knowledge base teachers need to have that 
will help them be effective teachers. 
3.3  PEDAGOGICAL CONTENT KNOWLEDGE (PCK) AND MATHEMATICAL 
KNOWLEDGE FOR TEACHING (MKT) 
The knowledge base that teachers should possess must be different from the one 
possessed by any educated adult. This section unfolds the knowledge base that is 
essential for teachers, especially mathematics teachers. 
Ball (n.d.:3) suggests that teachers should have subject matter content knowledge 
which goes beyond knowing topics on the curriculum. The author defines subject 
content knowledge as the knowledge of the topic, concepts and procedure. Teachers 
must be able to explain why concepts are correct, why it is worth knowing and how 
they relate to other concepts.  
Shulman (1987:7) and Ball (n.d.:3) corroborate that knowledge of content alone is 
not enough; in addition, teachers must also possess knowledge of pedagogy. Apart 
from the general and broad principles and strategies of classroom management and 
organization, teachers should know how to help learners access the subject matter 
42 
easily. Shulman (1987) therefore conceptualises pedagogical content knowledge 
(PCK) as the knowledge a teacher has regarding how to represent and formulate the 
subject matter so that it becomes accessible to learners. PCK is essential for 
planning and executing lessons that facilitate effective learning. This is because PCK 
enables teachers to predetermine possible misconceptions and difficulties learners 
are likely to have. 
Mathematical knowledge needed by a mathematics teacher is different from the one 
needed by a mathematics specialist. Hence Hill, Schilling and Ball as cited by 
Research Summary (2009:1 of 8) explain that a mathematics teacher needs to 
possess both common knowledge of mathematics (CKM) and specialized knowledge 
of mathematics (SKM). They explain that common knowledge is mathematics 
possessed by any mathematically educated adult while specialized knowledge of 
mathematics is the possession of mathematical knowledge and skills such as 
explaining why an algorithm works or being able to provide learners with multiple 
representations to address their diverse learning styles. SKM in relation to fractions 
include being able to explain fractions in different ways inter alia: as a number on the 
number line or a relation between two quantities.  
In addition to CKM and SKM, Hill, Schilling, and Ball (Research summary 2009:2 of 
8) further explain that teachers need to possess knowledge of learners and their 
ways of thinking about the content. Like SKM knowledge of learners and their ways 
of thinking about the content are unique to teachers. This body of knowledge 
specifically relevant to mathematics teachers is an extension of PCK and it is 
theorized as the Mathematical Knowledge for Teaching (MKT). 
Both PCK and MKT enable teachers to know and understand what makes learning 
of certain topics easy or difficult and hence represent and formulate them in a way 
that they become easily conceptualized by the learners (Izs´ak 2008). PCK also 
enables teachers to predetermine learners’ misconceptions and difficulties. MKT on 
the other hand goes further by equipping teachers with skills to analyse and evaluate 
errors and determine their source (Ding 2007:20). Therefore Ding concurs that MKT 
is an extension of PCK and it enables effective teaching of mathematics. 
It is mentioned in section 2.5.1 that when learning fractions learners make mistakes. 
Apart from mistakes they ask questions, make representations and rethink their 
43 
strategies when solving problems. When teachers attempt to respond to learners’ 
difficulties and mistakes, they make representations. What remains crucial is 
deciding on the representation that effectively explains the concept; connects 
alternative representations; explains the mathematics underlying each 
representation and deciding on the representation which will be understood easily by 
most learners. So teachers need to possess MKT in order to make effective 
pedagogical decisions (Meixia 2007). 
What teachers do in mathematics classrooms should be determined by their MKT. 
Regarding the teaching of fractions teachers should undergo the process of 
pedagogical reasoning, as discussed in the second paragraph above. That is, they 
should comprehend the concept fraction, transform the content by sequencing the 
activities in a way that will make the fraction concept easir to understand, for 
example to build conceptual understanding of equivalence before adding fractions. 
This is because equivalence serves as the basis for addition, so when the concept is 
clearly developed fraction addition becomes easy to understand (Charalambos et al. 
2006).  
Apart from the reasoning, teachers should engage in actions that help learners learn 
that a fraction can be presented differently, that is, half should be presented, among 
others as  
a part-whole  
 
a measure.  
 
 
Teachers who are able to employ multiple representations of a concept are said to 
posses MKT.  
44 
Teachers should also anticipate difficulties learners are likely to encounter so that 
they plan strategies to help learners overcome them. Difficulties in ordering fractions 
are associated with inadequate exposure of learners to manipulate concrete 
materials (Troutman & Lichtenberg 2003:365). In this way teachers will plan in 
advance how to help learners to avoid such difficulties (Meixia 2007). MKT also 
enables teachers to interpret and evaluate learners’ diverse solutions to a problem 
and respond to them in a way that will challenge them. Prospective teachers should 
also make connections by moving flexibly among subtopics so that learners can see 
relationships in them (Izs’ak 2008; Meixia 2007).  
It  was mentioned in section 2.4 that mathematics is viewed as a way of thinking and 
reasoning, so the next section discusses types of mathematical thinking and how it 
enhance the teaching of mathematics. 
3.4 INDUCTIVE AND DEDUCTIVE REASONING 
The importance of the manipulation of concrete objects in developing mathematical 
conceptions has already been emphasized. This experience results in the 
development of physical knowledge as indicated by Piaget (cf. 2.3.1).  
Kamii and Warrington (1999:83) emphasize that, physical, social and logico-
mathematical knowledge are important to learners, therefore teachers should avoid 
teaching only the physical and social knowledge at the expense of logico-
mathematical knowledge.  
Cooney, Davis and Henderson (1975:152) suggest a teaching strategy in which 
teachers give mathematical instances to learners and guide them with questions, to 
the objective of the teaching. An example could be to ask learners to share equally a 
whole, like a pizza, between two, four and six people. Then ask them to compare the 
size of the piece each person gets. From this they will see that when the number of 
people sharing one whole increases, the resulting pieces become small. They would 

 
 

have learned that > >  and so on. 
  
This type of reasoning is called inductive reasoning (Marwaha 2011) whereas 
Cooney et al. (1975:152) call it an inductive discovery teaching strategy. Marwaha 
45 
(2011) and Cooney et al. (1975:152) agree that the strategy moves from the specific 
idea to the general idea. 
In the traditional methods of teaching, teachers give learners algorithms to use when 
solving mathematics problems. Instead of letting learners use these algorithms 
without understanding them, Cooney et al. (1975:153-158) suggest teachers to 

 
  
 
employ deductive discovery strategy. If learners work out  +  as  + = , then 
    
teachers can design the learning experience in which learners will discover how the 
answer comes about.  
The situation could be; two sisters are each given a piece of pizza, one is given one 

 

half ( ) and the other one quarter ( ), so what fraction of the whole pizza do the girls 
 
get? 
The teacher may ask learners to draw diagrams to represent this situation or the 
teacher may do it on the board. But eventually they should see one half given to one 
girl. 
 
and one quarter given to the second girl.  
 
Then to find the fraction of the pizza they got, the pizza divided into two parts will 
now be divided into four parts  
46 
 
Figure 3.1Representing addition of fraction 
and the shaded quarters now added equal three quarters.  
From this and similar situations, learners are expected to deduce the generalization 
that when adding fractions they should have the same denominator or they should 
find the lowest common multiple. 
Cooney et al. (1975:157) highlight that “the key success in a deductive discovery 
strategy is the teacher’s ability to ask a sequence of leading questions that will guide 
the students to deduce the generalization the teacher wishes to teach.”  
Inductive and deductive reasoning can therefore be regarded as the stepping stone 
to problem solving as a mathematical teaching strategy (Marwaha 2011). Cooney et 
al. (1975:157) also link with Piaget’s theory of learning, discussed earlier in this 
chapter, in that in the processes of accommodation and assimilation one uses both 
inductive and deductive reasoning (cf. 2.3.1). Take the example of sharing a pizza, 
learners assimilate the new knowledge and then accommodate it into their schemes.  
Both inductive and deductive reasoning are therefore pivotal in the learning of 
Mathematics, but at different stages as indicated by Piaget who points out that 
children begin to think inductively and deductively when they are at the formal 
operational stage (cf. 2.3.1.4). Both deductive and inductive reasoning provide one 
with conceptual knowledge as decisions they make may not be based on guess work 
but on Mathematical facts. Da Ponte (2001:53) adds that inductive reasoning is 
essential in creating new knowledge while deductive reasoning is necessary for 
organizing new knowledge and deciding what is valid and what is not. 
 
 
47 
3.5 PROBLEM SOLVING  
As mentioned earlier, learning is a social activity, so is mathematics (Fajemidagba & 
Olawoye n.d.; Hiebert 1997:5-6). Mathematicians engage in mathematical activities 
using systematic steps to observe, study and experiment to come up with 
generalizations and new rules.  Quinones (2005: 8-10) as well as Brown and Quinn 
(2007) suggest that one way of helping learners think and behave like real 
mathematicians is to use problem solving as a teaching strategy. 
Through problem solving learners learn to think critically about what to do, how to do 
it and then monitor their progress. The National Council of Teachers of Mathematics 
(NCTM) in Ding (2007) state that “effective mathematics teaching requires 
understanding what students know and need to learn and then challenging and 
supporting it well.” Zakaria (2009) supports this as he states that students learn 
Mathematics through classroom experiences and problems given by teachers. The 
actions teachers take in class are determined by both their PCK and MKT.  
Kroll and Miller (1993:59-60) as well as Bobis et al. (2004:40-41) define a problem as 
a situation in which an individual sees no apparent means to obtaining a solution. 
They further point out that a mathematical task is a problem only if the problem 
solver reaches a point where he or she does not know how to proceed. Problem 
solving as a teaching strategy, engages learners in the thinking process as they 
attempt to come up with appropriate solutions. 
A problem solving program should include instruction to help learners know how to 
approach unfamiliar problems (Bobis et al. 2004:46). One way would be to 
familiarize learners with a variety of problem solving strategies or heuristics like 
drawing a diagram, completing a table or making a list (Kroll & Miller 1993:64-68; 
Quinones 2005:9). These are methods and rules of mathematical inventions and 
discoveries used by mathematicians to represent problems in different forms that 
make more sense to them. Kroll and Miller (1993:64-68) as well as Quinones 
(2005:9) also indicate that experience with using different heuristics gives learners a 
better understanding of the problem solving process and of possible strategy 
choices. 
48 
Being able to interpret a problem and present it in a different form is motivating to 
learners and it aids as an encouragement to continue with the attempt to solve the 
problem. 
Polya (in Kroll & Miller 1993:61) suggests a problem solving approach in which 
learners are expected to read and understand the problem, devise a plan, carry out 
the plan on how to solve the problem and look back to verify the effectiveness of 
their plan of action. 
3.5.1 Polya’s four phases of problem solving 
Polya indicates that the purpose of a teacher is to help students gain much 
experience and independent work. He warns teachers though, that the help they give 
should not be too much or too little because if it is too much it deprives learners of 
the chance to develop cognitively. So learners should be left with a reasonable 
amount of work that will result in some learning when the task is completed. This 
notion is in accordance with Vygotsky’s ZPD and Bruner’s process of scaffolding as 
discussed in sections 2.3.2 and 2.3.3 respectively. Bruner indicated that teachers 
should provide assistance to a certain degree and then withdraw gradually for 
learners to gain independence and hence grow cognitively. Vygotsky on the other 
hand warned that tasks given to learners should be just above their zone of proximal 
development so that they remain engaged and challenged. 
In order to help learners, teachers should ask learners questions and make 
recommendations that will make learners think and engage in mental operations. 
Learners should be made aware that mathematics problems require the use of 
common sense and ask questions they would ask themselves in real life situations 
(Polya 1985:2). Teachers can facilitate and enhance learners’ problem solving skills 
by creating situations in which learners can imitate and practise how teachers solve 
problems. 
Polya (1985) maintains that these stages can be followed to solve any problem from 
any discipline because, as mentioned, they are more of common sense than 
academic. These stages are: 
 
49 
• understanding the problem 
In order to understand the problem, (Polya 1985:6; Troutman & Lichtenberg 
2003:57-58) suggest that problem solvers should ask questions such as; Can I 
identify needed information? What are the given data? Can I generate all 
possibilities? These questions are essential to ask because, according to Troutman 
and Lichtenberg (2003:43) as well as Kroll and Miller (1993:66), problems may 
sometimes contain extraneous information or may not contain all the necessary 
information. So it is the problem solvers’ task to identify these. This indicates that 
learners should possess deductive reasoning skills as discussed in section 3.4, in 
order to be able to solve mathematical problems.  
Care should be taken when using problem solving to teach mathematical 
generalization that, problems do not contain information that may result in no 
learning. When building the concept fraction, from the problems given learners 
should learn that a whole is shared equally in order to be able to say this is a certain 
fraction.   
Teachers should ask learners questions that will focus their attention to the 
unknown.  
Example: on the number line what fraction is represented by each division? 
 
Figure 3.2 Representing fractions on the number line 
The teacher and learners can now engage in a dialogue that will elicit understanding. 
Learners should indicate that the unknown is the fraction that represents each 
division on the scale. The given data are zero and one, which is one whole. They 
should also be able to tell whether the problem is reasonable or not. 
• Devising a plan 
With the necessary knowledge of mathematics to solve the problem, learners should 
be given the chance to come up with their own plan (Polya 1985:9). Polya suggests 
that this can be achieved by engaging in the dialogue which will provoke the idea of 
50 
how to solve the problem. Learners’ attention should once again be focused to the 
unknown. They may also be asked to come up with a familiar problem having the 
same or similar unknown. 
It is also essential to devise more than one plan so that one can compare the 
effectiveness of each plan (Troutman & Lichtenberg 2003:57-58). They further 
suggest that when carrying out the plan and it seems not to be giving positive 
results, problem solvers should not be afraid to start all over again as one of the 
qualities of problem solvers is perseverance. 
In the example explicated above, a familiar problem would be the one in which 
learners are to find the fraction represented by each division when all the divisions 
are of equal width.  
 
If they are able to come up with a similar problem this may suggest that they have a 
plan.  
• Carrying out a plan 
Coming up with a plan is not very easy since it requires one to engage in deep 
mathematical thinking (Polya 1985:12). Once the plan is conceived, to implement it, 
is not as challenging as coming up with it. Now that learners have realized that the 
divisions in the problem are not even then they should be urged to carry out the plan 
of making divisions of equal width and then find the solution, which is one eighths in 
this problem. 
• Looking back  
This is the last of Polya’s stages of problem solving. This calls learners to check the 
effectiveness and accuracy of their plans and the reasonableness of their solution. 
Also to reconstruct and re-examine the result and the path that led to the solution 
(Polya1985:14). 
If problem solvers feel that the solution is not reasonable enough then they can go 
through the steps to find where they might have gone wrong. When looking back to 
51 
how the problem is solved, learners are likely to identify their mistakes and then see 
why they are mistakes. As another strategy to verify one’s solution, learners should 
be given the chance to discuss each other’s plan so that they could criticize them 
and hence build on their own and attain cognitive growth (Leung, 2010; Mueller et al. 
2010). 
Polya’s model gives learners the freedom to solve problems the way they make 
sense to them. If problems are rooted in learners’ real life situations they will 
understand and interpret the problem the way they exist in their context and hence 
employ strategies they find suitable (Fennema & Romberg, 1999:5; Hiebert et al. 
1997:4). The model also aligns with the constructivist philosophy in which learners 
construct knowledge through social interactions and by making connections to their 
own experiences (cf. 2.3). 
3.5.2 Problem solving strategies  
Ding (2007) indicates that trial and improvement is a problem solving strategy that 
can be used to solve mathematical problems. Through it, learners struggle, make 
mistakes and then see why they are mistakes. Trial and improvement as a problem 
solving strategy could be done in small groups or through whole class discussions in 
which learners criticize each other’s problem solving strategies and hence build on 
their own strategies (cf. 2.3.2; Leung 2010; Mueller et al. 2010).  
The above discussion indicates that emphasis should not be put on correct answers 
but on the process of attaining that answer (Kroll & Miller 1993: 61).The authors posit 
that one of the hindrances to effectively learning of mathematics is emphasizing the 
attainment of correct answer. Problem solving, as a teaching strategy, therefore can 
contribute to effective learning of mathematics. 
Problem solving can be used in mathematics as a teaching strategy, that is, learners 
may be given a problem to work on individually or in groups, and later the whole 
class engage in the whole-class discussion of their solutions. Teachers may 
structure the discussion such that learners come up with their own conclusions. 
When learners have thorough understanding of what they did, the teacher may now 
introduce the new mathematical concept learned through that problem. Carpenter 
(1989:191) calls this teaching through problem solving.  
52 
The author defines teaching through problem solving as giving learners a problem to 
solve using their own strategies and through the problem solving process they learn 
new mathematical concepts.  
As discussed teachers at times find it worth teaching learners the strategies they 
could use when solving mathematical problems, like Polya’s model of problem 
solving. This is called teaching problem solving by Carpenter (1989:199). In this way 
learners are taught the problem solving strategies to follow when faced with a 
problem. In this type the prime focus is not to learn new mathematics but to acquire 
the problem solving process. Like in the teaching through problem solving, learners 
need to use all the mathematics they know to come up with the solution to the 
problem. 
Meaningful learning is the outcome if teaching is effective. As argued above teaching 
through problem solving enhances deep and meaningful understanding of 
mathematics.  
3.5.3 Factors Influencing problem solving 
Kroll and Miller (1993:61-65) suggest that teachers should consider the following 
factors which may affect problem solving; knowledge factors; beliefs and affective 
factors.  
• Knowledge factors 
In order that learners can solve problems Schoenfeld in Thompson (1989) cite that 
they should have mathematical resources, which he describes as the knowledge of 
concepts, facts and procedures. Learners often encounter difficulties when solving 
problems if they lack the necessary resources which Kroll and Miller (1993:61-65) 
call algorithmic or computational skills. 
It was discussed that procedural knowledge should follow conceptual knowledge (cf. 
2.5.3.1). In this regard teachers should help learners develop procedural knowledge, 
it should not be compromised. For learners who did not develop procedural 
knowledge it sometimes becomes difficult for them to decide on the operation to use. 
This difficulty Kroll and Miller (1993) attribute to the development of procedural 
knowledge, that is, the ability to calculate without paying attention to developing 
53 
understanding. Teachers are encouraged not to spend time teaching learners to rely 
on key words to decide what operation to use as this strategy has been observed to 
break down quickly as problems become more complex. 
• Beliefs and Affective factors 
At times learners may hold believe that mathematical problems should be able to be 
solved quickly. They may also belief that there is just one right way to solving a 
mathematical problem. These beliefs may have a negative impact on the learners’ 
motivation and hence imply that when a mathematical problem does not have a 
straight forward solution it is not solvable and hence be poor problem solvers. 
When using problem solving as a teaching strategy, teachers should help learners 
develop positive beliefs about problem solving. One way could be by introducing 
problem solving by giving learners easy tasks which have low cognitive demand as 
this will help them improve their motivation level. Again teachers should allow 
learners see them as they struggle with a problem (Owen 1993: 66).  
Teachers often undermine the learners’ capability and hence give them problems 
with low cognitive demand. Teachers who do this are said to have low expectations 
for their learners and such learners often do not perform well. For learners to do well 
in mathematics, teachers should give problems, within the learners’ cognitive 
capability, but which are more demanding. This is what Spady (1994:180) calls high 
expectations for the learners. Spady (1994:16) accentuates that most learners, 
whose teachers have high expectations of them, will raise their performance in more 
challenging levels of learning. 
Kroll and Miller (1993) point out that some researchers have linked work on meta-
cognition with work on problem solving. In the words of Schoenfeld in (Lesh & 
Zawojewski 2007:771) metacognition is described as “self regulation, or monitoring 
and control … [of] … resource allocation during cognitive activity and problem 
solving.” Problem solvers or learners should be able to decide; on what they know, 
what they do not know and what they need to know; on the mathematical knowledge 
needed to solve the problem; what to do to solve the problem. 
Since learning can also be socially constructed (cf. 2.3.2 & 2.3.3), metacognition can 
be enhanced through group work. In those groups, learners can provide to each 
54 
other, explanations to each other regarding their understanding of the problem, the 
method they use and others may ask questions seeking for clarification or indicating 
a mistake.  
That is, learners should be helped to be aware of their own knowledge and thinking 
process and monitor their thinking and progress when involved in a problem 
situation. This ability, of learners, to regulate and monitor their own problem solving 
is what is called metacognition (Thompson 1989:234). 
3.5.4 Nature of problems 
The nature of problems given to learners has direct impact on the learners’ 
motivation and perception as problem solvers. Teachers should try their best to 
compose meaningful and solvable problems (Kroll & Miller 1993:66). For a start 
teachers should give problems which are solvable and require low order cognitive 
reasoning (cf. 3.5.3) so that learners’ confidence is increased. When they have 
gained confidence, then learners could be exposed to problems that require high 
order cognitive reasoning which are within their cognitive development stage. 
Real-life application problems should be used as they often seem more meaningful 
and interesting to students. This could be done by adapting problem details from 
learners’ interests such as using a popular context (Kroll & Miller 1993:66). Learners 


could be asked to find how many apples a friend will have after he is given half ( ) a 



box of apples by one friend and a quarter ( ) box of apples by the other. Giving 

learners this type of problems may indicate to them that mathematics is real and 
hence it may become meaningful to them. 
While introducing learners to problem solving, care should be taken when 
formulating problems. Kroll and Miller (1993: 66) warn teachers that problems should 
not contain extraneous information or incomplete or contradictory information. This is 
likely to confuse learners and hence they perceive that problem solving is difficult 
and not possible to do. It is true that in Polya’s first stage of understanding the 
problem, learners should be able to identify what belongs and what does not. But 
this could well be done by learners who are at the advanced stages of problem 
solving. For young learners teachers should avoid confusing them with a lot of 
unnecessary or wrong information. Instead they should begin with simple problems 
55 
with no extra unnecessary information, then gradually introduce them to extra 
information. 
In order to monitor learners’ progress and/or understanding of what they are doing in 
the classroom when solving problems, teachers should continually assess the 
learners’ understanding. The next section discusses assessment and how it 
enhances teaching. 
3.6 ASSESSMENT  
The major goal of mathematics teaching is to develop an understanding of 
mathematical concepts so that learners will have the capacity to integrate, apply and 
communicate their mathematical understanding in and out of school. To achieve this 
learners must be confident, interested, curious and inventive (Lambdin 1993:10). 
One of the tasks of teachers is to design instructional activities that will enhance the 
achievement of mathematical teaching. Teachers give instructional activities that do 
not only facilitate learning but give them information regarding the learners’ 
competence. Webb (1993: 1) indicates that it is the teachers’ tasks to engage in an 
on-going assessment that will help learners attain the goals of curriculum. 
“Assessment is a comprehensive accounting of a student or a group of student’s 
knowledge (Webb 1993:1).” Engelbrecht, Green, Naicker & Engelbrecht (1999:117) 
on the other hand define assessment as:  
The process of identifying, gathering and interpreting information about 
learner’s learning. The central purpose of assessment is to provide 
information on learner achievement and progress and set the direction for 
ongoing teaching and learning. 
This definition brings us to the discussion of two types of assessment; formative and 
summative assessment. 
3.6.1 Types of assessment 
3.6.1.1 Formative assessment 
It has been discussed in section 3.3 above that during the teaching and learning, 
teachers should identify learners’ difficulties so that they can assist them. This could 
56 
be done through observing and questioning learners as they engage in classroom 
activities. The information that teachers get from such encounters should guide their 
decision-making and actions to facilitate learning. 
This continuous monitoring of learners’ strengths and needs facilitates and informs 
teachers of the actions that will help them pace learners accordingly. This form of 
information gathering is called formative assessment and it is referred to as the 
assessment for learning (Cangelosi 2003:147; Engelbrecht et. al. 1999:110).  
Cangelosi (2003:147) and Chambers (1993:17) warn that assessment must be an 
interaction between teachers and learners with the teacher continually seeking to 
understand what a learner can do and how a learner is able to do it and then using 
this information to guide instruction. Cangelosi (2003:147) call this formative 
assessment. So assessment must be part of every mathematics lesson so that 
effective teaching can occur. 
Since the purpose of assessment for learning is to facilitate learning, learners should 
be given specific and detailed feedback immediately (Earl & Kartz 2006:33). 
Descriptive feedback enhances learning as is the case with descriptive language (cf. 
2.6.4). Descriptive feedback gives learners information regarding what they know, 
what they do not know and what they should do in order to learn effectively. Coupling 
assessment with descriptive feedback therefore facilitates effective learning.  
3.6.1.2 Summative assessment 
Other than gathering information that will help teachers make informed teaching 
actions, information may also be gathered at the end of a learning period. That is at 
the end of a learning unit, a term, semester or year. The intention being to provide 
other stake holders about the teaching and learning that has taken place 
(Engelbrecht et al. 1999:111).  
Effective assessments require teachers’ to have thorough knowledge of the 
mathematical domain under investigation and the nature of the learners’ thinking 
within that domain or MKT (cf. 2.5.3; Lambdin 1993:10).  
Both formative and summative assessments are important in teaching and learning 
and more specifically in the teaching and learning of fractions. Since this study is 
57 
concerned about effective teaching, this study will focus mainly on formative 
assessment as one of the aims of the study is to determine whether effective 
teaching is taking place (cf. 1.2). As discussed in 2.6.1, through formative 
assessment teachers should be able to identify learners’ errors, misconceptions and 
difficulties and hence effective learning would take place.  
3.6.2 Assessment methods 
Different assessment methods that teachers could use include; interviewing, 
observations, portfolio, performance, essay writing and objective tests. 
3.6.2.1 Interviewing 
This is not necessarily a separate technique; it can be part of the observation and 
can be done during the normal teaching. As learners participate in classroom 
activities, teachers may notice certain behaviour. In order to make sure that they 
understand what that behaviour means, teachers may have to interview or question 
the concerned learner (Engelbrecht et al. 1999:117).  
From interviews, teachers may get a lot of information pertaining to the learners’ 
feelings and beliefs about themselves and about mathematics (Engelbrecht et al. 
1999:117). They may also gather information about the thinking behind their actions 
as they perform mathematics tasks. Information obtained can inform teachers about 
the learners’ level of understanding of the subject, expectations and/or beliefs. 
Hence the information obtained may be of use to the teacher when designing further 
instruction. 
3.6.2.2 Observation 
Through observations of learners as they participate in class, teachers can also 
judge learners’ level of motivation and confidence when they engage in a 
mathematics task and when they interact with other learners. However as said in the 
above paragraph teachers should make sure that they do not misjudge learners. 
They could do this by verifying an observation with an interview (Engelbrecht et al. 
1999:117; McMillan 1997:52). 
Engelbrecht et al. (1999:117) warn teachers that observations should not be done by 
chance but should be a planned assessment strategy. For example if a teacher 
58 
wants to determine whether learners can competently demonstrate addition of 
fractions, then they should plan activities from which they can competently draw a 
conclusion. 
3.6.2.3 Portfolio  
A collection of learners’ work over an extended period may also serve as an 
assessment strategy. From this collection of the learner’s work, teachers may make 
informed decisions about the learner’s progress. It can be deduced whether there is 
improvement or not. Based on this evidence, teachers may discuss the learners’ 
work with the concerned learner or other parties, like parents and/or the principal 
(Engelbrecht et al. 1999: 120). 
The next section discusses features of assessment as identified by Webb (1993:3-
4). 
3.6.3 Features of assessment 
3.6.3.1 Assessment situation, task or question 
This could be a problem presented to learners, a classroom discussion or activity or 
any other action that elicits learners’ responses. 
3.6.3.2 Response 
In order for teachers to be able to have some information regarding learners’ 
knowledge, teachers should ask learners to display this knowledge either by 
• Writing numerical answers 
• Writing a paragraph explaining the thinking process behind the solution 
• Performing oral presentation 
• Performing journal entry 
• Keeping a portfolio of their work over an extended period of time 
• Lastly teachers should Interpret student’s responses 
59 
This can be attained by comparing responses to defined goals or expectations. This 
could also involve understanding what the response reveals about the learners’ 
structure of mathematical knowledge. It requires making deductions on the basis of 
an understanding of mathematics, learning mathematics and knowledge of the 
students. 
3.6.3.3 Assigning some meaning to the interpretation of the student’s 
responses 
This could be demonstrated by locating learners’ responses on a scale that 
represents the range of all possible responses. 
3.6.3.4 Recording and reporting  
Recording can be done by writing ‘good’ on the margin of the book or by writing the 
marks in the teachers record book. Reporting can be by verbally talking to the 
learner to explain how they have performed, what problems are identified and what 
improvements need to be done and then recording the findings in a book for 
reference when designing future instruction. This form of communication gives 
learners feedback (Earl & Kartz 2006:33) on their work and performance as a whole 
and therefore facilitates learning. 
Learners who have effectively learned a concept should have the capacity for 
integrating, applying and communicating their mathematical understanding (cf. 
2.5.2), and this ability is referred to by Lambdin (1993:10) as mathematical power. 
The author further explicates that learners should show confidence, interest, curiosity 
and inventiveness in working with mathematics ideas. This he calls mathematical 
disposition. If learners demonstrate mathematical power through multiple 
behaviours, then teachers must also use a variety of methods for assessment. That 
is, teachers should assess whether learners have all the aspects that culminate to 
the ultimate goal of mathematical learning (Lambdin 1993:11). 
3.7 EFFECTIVE TEACHING OF FRACTIONS 
It has already been discussed that effective learning requires assimilation and 
accommodation of one’s experiences into the already existing schemes. Based on 
what effective learning in mathematics is, effective teaching in mathematics, more 
60 
particularly fractions, will be discussed from that angle. The discussion will begin by 
focusing on what effective teachers do and what effective teaching is. 
Children learn by manipulating concrete materials, making sense of what they are 
learning by relating it to what they already know and then articulating it within 
collaborative groups (cf. 2.5.2). 
Ball and McDiamid (n.d.:4) conceptualise effective teaching as a result of what 
teachers do, how they understand their work and decide what to do. The author 
explicates that effective teaching depends on what the teachers’ believe about their 
teaching and about the subject matter. 
Teachers who perceive mathematics as the set of rules to be learned through drill 
and practice teach learners rules and then give them exercises to practise them. 
Those who believe mathematics is learned by constructing their own understanding 
and meaning, either individually or in groups, create situations which will enable 
them to construct meaning and understanding. Ball and McDiamid (n.d.:36) 
elucidates that what teachers decide to do depends on their subject matter 
knowledge. For teachers who give learners opportunities to construct meaning and 
understanding, it is because they have the knowledge that enables them to reason 
and relate concepts. So they are able to explain why learners’ reasoning is 
mathematically wrong or correct. That is, prospective teachers have MKT (cf. 3.3). 
So effective teaching should include giving learners activities which are rooted in 
their contexts with the attempt to extend, answer their questions and give meaning to 
what they already know (Donald et al. 2002:109).This enables learners to construct 
their own meaning and understanding (cf. 2.5.3.1). 
3.7.1 Fractions 
Fractions can be understood from different angles of their usage. They can be 
understood as numbers, positions on the number line or ruler and as the ratio 
between a part and a whole (Haylock & Cockburn 2008:49-50). They are elements of 
the set of rational numbers and it is defined as an integer divided by an integer not 
equal to zero. 
 
61 
3.7.2 Difficulties of teaching fractions 
As stated earlier, Charalambos et al. (2006) and Leung (2009:3) emphasize that 
teaching and learning of fractions is one of the most problematic areas in primary 
school mathematics. Haylock and Cockburn (2008:49) and Barmby et al. (2009:61) 
corroborate with them and then posit that the problems relate to the way fractions 
are represented as numbers and the standard ways that we use to represent 
fractions in diagrams and pictures. They suggest that the number concept should 
first be established in learners before they can be introduced to fractions. This is 
because the concept of a fraction as a number can be overwhelming to learners 
because initially fractions were rejected as numbers.  
Some of the causes of these difficulties arise from the misconceptions associated 
with fractions that learners bring along and the multifaceted nature of fractions 
(Bezuk & Bieck 1993:119; Charalambos et al. 2006).  
3.7.2.1 Handling misconceptions  
The resistance to accept fractions as numbers can lead to the misconception that a 
fraction is two different whole numbers, that is, to find the fraction of the shaded part 
in the diagram below  
 
Figure 3.3 Fraction as a part-whole 

They count the number of shaded sectors as 3 and write the fraction as . To get the 

fraction of the unshaded part, instead of subtracting fraction of the shaded part from 
one whole the learners would count again (Barmby et al. 2009:62).  That is, they 

count the unshaded sectors as 5 and then write the fraction as . The authors explain 

that this could be that learners do not see the connection between the fraction of 
  
shaded and unshaded or complimenting addition, that is,  +  =  = 1. 
  
62 
Troutman and Lichtenberg (2003:290) advise teachers that if they realise that their 
learners have this difficulty they should engage them in the hands on manipulation of 
concrete materials. They contend that this will help learners realise that they do not 
have to count every time they encounter this situation. 
In order to handle this difficulty, teachers could engage learners in the counting 
activities when finding fractions. When they are satisfied that every learner can do 
this exercise, then they can ask learners to deduce how to find one fraction when 
they know the other (cf. 3.4). 
Another misconception as cited by Nunes, Bryant, Hurry, and Pretzlik (2006) 
together with Murdock-Steward (2005:6) is equating the number of shaded parts to 

the numerator and the number of un-shaded parts to the denominator, which is , 


instead of . 

 
Figure 3.4 (a) Fraction misconception  
  
and hence result in computational errors when operating fractions like   	  = . As 
  
discussed earlier, teachers are advised to engage learners in manipulating concrete 
materials. They should also help learners conceptualise fraction as a number 
(Charalambos et. al. 2006; Haylock & Cockburn 2008: 45-52). 
So effective teaching and learning will take place if teachers can assist learners not 
to develop misconceptions or to clear them if they already exist. This could be 
achieved by creating learning situations in which learners interdependently construct 
understanding of all the constructs (cf. 2.5.2). Misconceptions such as equating the 


shaded part as  

63 
 
Figure 3.4 (b) Fraction misconception 

  
and calculating  +  =  are attributed to inappropriate exposure of learners to 
  

manipulate real life concrete materials and hence acquire physical knowledge (cf. 
2.3.1; Murdock-Steward 2005: 6; Troutman & Lichtenberg 2003: 365). 
3.7.2.2 Difficulties associated with contextual teaching  
Difficulties associated with learning fractions among others are that fractions are not 
related to the learners’ real-world and fraction learning experiences are not 
sequenced in a cohesive way(Murdock-Steward 2005:4-5). If this is the case, 
teachers should, as mentioned, use examples from the learners’ context and 
approach to the topic should be spiral (cf. 2.3.2). Charalambos et al. (2006); Leung 
(2009) as well as Pearn (2007) do not agree with Murdock-Steward that fractions are 
not related to the learners’ real-world. They argue that when teaching fractions, 
teachers use language that is used in learners’ real-life. The example being when 
talking about sharing, teachers refer to concrete objects that are familiar to learners, 
like sharing a pizza, sweets and many more (Charalambos et al. 2006; Leung 2009; 
Pearn 2007). 
On the other hand Kerslake (1991:86-87) argue that fractions are mostly represented 
as parts of things learners use in their everyday life, such as an apple or a bun (cf. 
2.5.2). So according to Kerslake, the language learners use in their every-day life 
with an attempt to contextualise their teaching imposes difficulties especially when 
operating fractions as the author indicated where no possible meaning could be 
attached to adding one half of an apple to three quarters of a bun (cf. 2.5.2). So 
contextualising could also result in learners experiencing difficulties.  
64 
In order for teachers to help learners not to experience difficulties, especially when 
using their context for teaching, they should invest more time in helping learners 
develop conceptual understanding by engaging them in hands-on activities using 
concrete objects. This is because in this study learners are at both Piaget’s concrete 
operational and formal operational stages; the entry point of cognitive development 
(cf. 2.3.1.3 & 2.3.1.4). They should also help learners build the concept of a fraction 
as a number and they should build it from different perspectives (Caswell 2007; 
Haylock & Cockburn 2008: 6-7; Kerslake 1991: 88).  
3.7.2.3 Diagnosing and remedying learners’ difficulties 
It is inevitable that, in the teaching and learning environment, learners will 
experience difficulties. So, it is indicated in section 2.6.3 that teachers should posses 
the skills to identify and remedy learners’ difficulties. Cooney, Davis and Henderson 
(1975:202-239) suggest that teachers should approach diagnosis and remedying of 
learners’ difficulties as problem solving. They should go through the four stages of 
problem solving as discussed by Polya (cf. 2.5.2.1).  
Firstly teachers should be aware of the problem. For example a learner cannot add 
fractions. Then they should define it. By this he means that teachers should state the 
kind of errors learners make, what misunderstanding they manifest and what it is that 
they cannot do (Troutman & Lichtenburg 2003:287; Cooney et al. 1975:203).  
Having defined the problem, teachers may now collect data that will enable them to 
solve the problem. If the problem is: learners make mistakes when adding fractions, 

 
 
that is, they write  	  = . Teachers then have to decide on the cause of this error.  
  
They may conjecture that learners do not have conceptual knowledge regarding 
sharing. To test the conjecture teachers may now engage in formative assessment 
by asking learners questions that need them to have the possession of conceptual 
knowledge of sharing. Data will now be the answers that learners. Once the correct 
conjecture has been determined, teachers may now decide on what kind of remedial 
teaching to do (Cooney et al. 1975:202-214).  
Point of clarification is given to teachers though that, they should remember that 
learning difficulties are not attributed to learners’ intelligence only. There are other 
factors such as; physiological, social, emotional and pedagogical factors. As it is 
65 
easier to correct one’s own problems than others, teachers should start by doing 
introspection before looking at other factors (Cooney 1975: 214). That is, teachers 
should assess their teaching. Teachers should ask themselves the following 
questions; “Have I used the learners’ prior knowledge to assist in the development of 
conceptual knowledge? What is it that I have done wrong? What should I do to 
improve my teaching” (Cooney et al. 1975:214)?  
The next section discusses the five subconstructs teachers are advised to consider 
when teaching fractions. 
3.7.3 Subconstructs 
Haylock and Cockburn (2008:6-7) indicated that teachers should engage with 
structures of mathematical ideas in terms of how children come to understand them 
so that they can teach them effectively. So in this section the five subconstructs 
pertaining to fractions as identified by Behr et al. (1983) and Charalambos et al. 
(2006) will be discussed. These are part-whole, measure, operator, quotient and 
ratio subconstructs.  
3.7.3.1 Part-whole subconstruct 
This is defined as the situation in which a whole or a set of discrete objects is divided 
into equal parts (Charalambos et al. 2006; Leung 2009). 
 
Figure 3.5 A whole pizza divided into 4 equal parts.  


The shaded part represents one quarter ( ) of a pizza 

66 
 
Figure 3.6 A set of 16 balls 


The shaded balls represent one quarter ( ) of the whole set of balls. In both cases 



the fraction,  represents the comparison between the number of parts of the 

partitioned unit to the total number of parts in which the unit is partitioned 
(Charalambos et al. 2006; Leung 2009:4). The authors indicate that to understand 
the part-whole subconstruct, learners should develop the concepts that; a) the parts 
into which the whole is divided into must be equal; b) the more parts the whole is 
divided into, the smaller the produced parts become; c) the parts taken together 
must be equal to the whole; d) the relationship between the parts and the whole is 
conserved regardless of the size, shape and orientation of the equivalent parts. 
Learners should also be made aware that they should count the parts twice, once for 
the numerator and then for the denominator (Charalambos et al. 2006; Nunes & 
Bryant 1996:203). Learners who have not developed this concept will write the 

 

shaded part as , instead of  as discussed above. 
 
 
Learners should further develop the concept of repartitioning already partitioned 
units. For example, from the whole partitioned into fourths, learners should construct 
 

 or any fraction equivalent to . 
 
 
67 
3.7.3.2 Ratio subconstruct 
Fraction as a ratio conveys the notion of comparison between two quantities. 
Carraher (in Charalambos et al. 2006) posits that a fraction is therefore a 
comparative index not a number. Teachers should assist learners to understand that 
for a fraction to be a comparative index, similar quantities should be compared. Two 
girls share a pizza and three boys share a pizza, which one gets more pizza, a boy 
or a girl? To answer this question so that the comparative notion is employed, 
learners should compare the same thing that is the amount of pizza each girl gets to 
the amount of pizza each boy gets. 
Since this study is based on age group 11 to 12, it may not be important to let them 
know the difference between rate and ratio. However problems and activities that are 
given should develop what they are intended to develop, either rate or ratio 
(Charalambos et al. 2006). 
The authors further make teachers aware that, for learners to fully develop the 
subconstruct, they need to construct the idea of relative amounts. Learners need 
also to understand what it means to say that there is a relationship between two 
quantities and that when quantities in the ratio relationship change together the 
relationship between them remains invariant (Charalambos et al. 2006). 
3.7.3.3 Operator subconstruct 
A fraction as an operator is viewed as a function applied to some number, an object 
or a set. Lamon (in Charalambos et al. 2006) says that a fraction as an operator 
could be viewed as a composite function which results from two multiplicative 

operations or as two discrete but related functions applied consecutively, that is,  of 

		
8 could be viewed as  or as 3		2. An operator is also defined as a transformer 

that lengthens or shortens the length of a line segment, increases or decreases the 
number of tasks in a set of discrete objects and takes a figure in the geometric plane 
and maps it onto a larger or smaller figure of the same shape. 
From the above analysis, Lamon (in Charalambos et al. 2006) further decomposes 
the operator subconstruct into two subconstructs: stretcher/shrinker and as 
68 
duplicator/partition reducer. But Behr (in Isz’ak 2008) have the third subconstruct, 
namely multiplier/divisor. This then brings in the other subconstruct, the quotient. 
3.7.3.4 Quotient subconstruct   
A fraction can be viewed as a result of a division situation (Charalambos et al. 2006; 

Iza’ks 2008; Leung 2009). That is  is a numerical value obtained when a whole 

number x is divided by another whole number y (y ≠ 0). Situations usually used to 
help learners construct the quotient subconstruct involve finding fair shares of 
continuous quantities. Because of its numerical value, a fraction as a quotient can be 
of any size; less than, equal to, or greater than a unit. A problem situation in which 
three pizzas are shared by four children depicts two different measure spaces; 
pizzas and children. The pizzas are shared into four equal parts, the dividends, 
which is the number of parts in each share, and each child gets three quarters of a 
pizza, the divisor, which is the fraction name of each share. 
3.7.3.5 Measure subconstruct   
Fraction as a measure is connected to two inter-related notions. The first one is how 
big the fraction is, and the second one is associated with the measure assigned to 
some interval (Charalambos et al. 2006; Leung 2009:5). A unit fraction is defined as 


 so when used repeatedly from the starting point then the distance can be 

 

determined. For example,  corresponds to the distance 3 (  unit) from a given point. 
 
Hence to build the measure subconstruct, number lines and measuring devices like 
scales and rules are usually used. 
3.8 TEACHER DOCUMENTS 
Curriculum design on a micro level can enhance effective teaching of mathematics, 
and more specifically effective teaching of fractions. Lesson plans, work schedules, 
assessment activities, grading documents are all examples of teacher documents.  
In this study focus will be on lesson plans and learners’ mathematical exercise books 
or assessment products. 
 
69 
3.8.1 Lesson plan 
The effective ways of teaching mathematics, fractions in particular, were reflected on 
in the previous paragraphs. Before the actual teaching of their classes, teachers 
have other responsibilities (Cangelosi 2003:17). Cangelosi (2003:20-22) name 
these, among others, as planning and organising courses by writing syllabi; 
arranging and organising the classroom. 
A lesson plan, according to the Department of Education (2003:2) “describes 
concretely and in detail teaching, learning and assessment activities that are to be 
implemented...’.  It could be a daily, weekly or quarterly lesson plan. 
A lesson plan prepared by an individual teacher and its contents are inter alia: topic, 
date, objectives, list of materials needed, teaching approach, method, procedures, 
potential problems and pitfalls and methods of evaluation (Cangelosi 2003:54; 
Department of Education 2003:4; Wagner 2004). 
• Objectives 
Objectives or learning objectives are short and measurable statements that describe 
what learners will be able to do as a result of learning. Objectives should begin with 
action an action verb and when possible they should include performance standards 
on how the learners will be evaluated  
• List of materials 
List of materials needed. When thought of well before the lesson they save the 
teacher much frustration and time.  
• Procedures   
Procedure is a list of all the steps the teacher intends to follow in the lesson. This 
may be how to introduce the lesson, how to activate prior knowledge and so on. 
• Potential problems and pit falls 
Potential problems and pit falls that may occur are that learners do not have the 
requisite knowledge necessary for the lesson, so teachers should have a plan if this 
be the case (Wagner 2004). 
70 
• Assessment tasks 
Assessment task is a question, exercise or problem that a teacher intends to give to 
the learners during or at the end of the lesson. The purpose of which is to determine 
whether learners have learned what was taught. Learners’ performance on the 
assessment task provides teachers with information pertaining to the achievement of 
the lesson objective/s.   
Wagner (2004) further indicates that there are different types of lesson plans. The 
type of lesson plan each teacher uses depends on individual circumstances, lesson 
being taught, cognitive readiness and social stance of learners involved. 
Cangelosi (2003:194) agrees with Wagner (2004) about different types of lesson 
plans.  But Cangelosi (2003:194) describes a four stage lesson plan that can be 
used by mathematics teachers regardless of the intention of the lesson. The four 
stages as described by Cangelosi (2003:182-194) are:  
• Experimenting 
It is discussed in section 2.5.2.3 that mathematics can be learned effectively through 
experiments. Hence teachers should plan in advance experiments they intend to 
give to learners.  
• Reflecting and explaining 
In this section of the lesson, teachers should plan how learners will be helped to 
reflect on the experiment. Learners may be asked to explain the problems and 
difficulties they encountered and how they overcame the difficulties. As learners 
reflect on their experiences, they get used to using the appropriate mathematical 
language and also improve on reasoning skills (cf. 2.3.2 & 2.5).  
• Hypothesising and articulating  
At the end of an experiment, learners should be encouraged to discuss the findings 
and come up with their own generalizations. This makes learning personal and 
hence learners gain conceptual knowledge (cf. 2.5.3.2). 
 
71 
• Verifying and refining 
Teachers may now conclude the lesson by helping learners to verify their findings 
and make corrections where needed.   
3.8.2 Learners’ mathematics exercise books 
The researcher is of the opinion that exercise books of learners can provide valuable 
information to the researcher regarding the performance of learners. This information 
may include, inter alia, the type of mathematics questions given to learners, the 
strategies learners use to perform calculations and the feedback that teachers give 
to learners. 
The researcher believes that by studying learners’ mathematics exercise books, 
information can be gained regarding performance, errors and misconceptions that 
learners may have. 
3.9 CONCLUSION 
Chapter 2 elucidated what effective learning is and how learners could be helped to 
achieve effective learning. This is because the teaching of mathematics is informed 
by the knowledge of how children learn and what effective learning is. 
This chapter discussed effective teaching strategies that teachers can use to assist 
learners to learn effectively. These teaching strategies are the use of problem 
solving as Quinones (2005:8-10) suggests. It is one way of helping learners think 
and behave like real mathematicians. Teachers could use this strategy to teach 
mathematical concepts. Using problem solving as a teaching strategy could help 
learners develop both inductive and deductive reasoning, which are high order 
reasoning skills essential in solving every-day real-life problems.  
In order to be effective in this regard mathematics teachers should have both PCK 
and MKT. This is because these bodies of knowledge help teachers make informed 
decision such as: sequencing of the subject matter and teaching activities such that 
learning becomes accessible; the teachability of the topic; identifying and rectifying 
learners’ difficulties and misconceptions; and selecting appropriate assessment 
strategies that will facilitate learning. 
72 
Teachers are also advised on how to teach fractions, that is the five subconstructs 
pertaining to fractions were discussed. It is said that teachers who teach all these 
constructs and help learners identify the relationships are embarking on effective 
teaching. 
Lesson plan and learners’ mathematical exercise books are discussed as important 
documents in this study. Lesson plan books indicate the type of preparation that 
teachers do. That is, type of activities and questions they intend to give their 
learners. Learners’ mathematics exercise books on the other hand indicate the 
layout and strategies that learners use when solving problems.  
In the next chapter describes the research methodology that will be used in this 
study. 
 
 
73 
CHAPTER 4 
RESEARCH METHODOLOGY 
4.1 INTRODUCTION  
The purpose of this study as was stated in chapter 1 is to explore and assess how 
primary school mathematics teachers teach fractions and how students learn 
fractions. In Chapters 2 and 3 literature on the effective teaching and learning of 
mathematics with emphasis on fractions was reviewed. This chapter, on 
methodology, will explicate how empirical data was collected that helped fulfil the 
purpose of this study. 
4.2 RESEARCH METHODOLOGY 
Methodology is defined by Silverman (2000:13) as “a general approach to studying 
research topics.” Wagner and Okeke (2009:69) indicate that research methodology 
is determined by the epistemological underpinning of the researcher. In chapter 1 it 
was communicated that constructivism, a branch of the interpretive approach 
(Nieuwenhuis 2009:58), is the epistemological underpinning of this study in which 
the phenomenological approach will be employed. The phenomenon of this study is 
teaching and learning fractions. The research intends to give a descriptive report 
regarding these phenomena. Since phenomenologists believe that phenomenology 
is the study that, “wants us to see clearly and to describe accurately what we see, 
before we start explaining in a scientific manner what we have seen” (Higgs 1995:9). 
This study is therefore qualitative in nature.  
4.3 QUALITATIVE RESEARCH 
The theoretical perspective of interpretivism sees the world as too complex to reduce 
it to observable laws (Gray 2004:31). This is because the behaviour of human beings 
consists of the physical actions that we see and the reasons or the meanings 
attached to such actions. Hence to understand these behaviours fully the observer 
should understand the reasoning that human beings attach to their actions (Carr & 
Kemmis 1986:88). The observer can understand by doing some form of 
interpretation.  
74 
It was mentioned in chapter 1 that this research study is intended to get insight and 
understanding of the strategies teachers use to teach fractions and how learners 
learn fractions. In order to achieve this, the researcher will observe and interpret both 
the teachers and learners actions. Nieuwenhuis (2007:50) together with Wagner and 
Okeke (2009:63) describe qualitative studies as those that attempt to get rich 
descriptive data regarding a phenomenon under study. So this study intends to do 
that hence it is qualitative.  What teachers and learners do in their classrooms is 
sometimes determined by other factors, so the researcher entered the situation 
without predetermined knowledge but tried to understand what teachers and learners 
do from their point of view. 
Invankova, Creswell and Clark (2007:257) state the following characteristics of 
qualitative studies: 
• the research questions are broad and general 
• seek to understand participants’ experiences with the central phenomenon 
The selected sample is small and purposeful. 
With regard to the above characteristics of qualitative research, Gray (2004:320) 
adds that qualitative research:  
• be conducted through intense contact within a “field” or real life setting 
• main focus is to understand the ways in which participants (teachers) act 
(teach) and account for these actions 
• supports the researcher’s role of getting a “holistic” or integrated overview of 
the study, including the perceptions of the teachers 
• provides  data  open to multiple interpretations 
• allows participants (teachers) to review the themes emerging from the data 
• allows for the simultaneous data collection and analysis but not necessarily 
analysis occurring sequentially after data collection 
With regard to the above characteristics, this is a phenomenological study because 
the phenomenon under study is studied intensely in its real setting. 
75 
4.4 A PHENOMENOLOGICAL STUDY 
The phenomenon in this study is teaching and learning fractions. Phenomenological 
research is a complicated process since the researcher has her own conceptions 
which she brings to the study (Smith & Eatough 2007:38). This is true because in 
this case the researcher is a Mathematics teacher and she has her own 
understanding and experiences regarding the phenomenon under study. The 
authors purport this as they state that the researcher’s conceptions help her to make 
sense of the participants’ worlds’ view through a process of interpretative activity.  
The purpose of phenomenology is not to theorise but to describe what all 
participants have in common as they experience the phenomenon. In this research 
therefore the researcher works from the participants’ specific statements and 
experiences rather than abstracting from their statements to construct a model 
(Lichtman 2011:243). In the words of Berglund (2007:76) “the goal of 
phenomenological methods is to study the meaning of phenomena, and human 
experiences in specific situations…” 
In this study the researcher, aspires to determine a view into the teachers and 
learners’ life worlds, which is their teaching and learning of fractions respectively, 
and how learners learn fractions (cf.1.3; Johnson & Christensen 2012:383). 
Therefore this is a phenomenological study. 
In order to obtain this view the researcher laid aside her understanding of the 
phenomena and opened up for new meanings by focusing on what she is 
experiencing at that moment. This she did so that she could get the subjective 
experiences of the teachers and learners (Gray 2004:21-22; Johnson & Christensen 
2012:383-384).  
4.5 DATA COLLECTING STRATEGIES 
4.5.1 Observations  
Observation as a qualitative data gathering technique is defined as “a systematic 
process of recording the behavioural patterns of participants, objects and 
occurrences without necessarily questioning or communicating with them” 
(Nieuwenhuis 2007:83). Observations can be done by watching and/or listening 
(Thomas 1998:136). There are three types of observation namely, casual, participant 
76 
and systematic (Dyer 2006:80-81). Dyer (2006:80-81) indicates that in systematic 
observations the researcher is detached from the subjects and data collection is 
systematically organized, whereas in casual and participant observation, the 
researcher is fully involved in the social activities and data collection is not 
organized, the observer acts like one of the subjects. This study therefore followed 
the systematic observation as the researcher took the role of the detached observer 
(Nieuwenhuis 2007:83-87). 
The researcher sat in front of the class, facing in such a way that most of the 
learners’ faces, together with the teacher, could be seen throughout the lesson.  
A voice recorder was used to record the classroom discourse. Together with the 
voice recorder an observation guide was used to supplement the recorded 
information by noting non-verbal cues such as facial expressions of both teachers 
and learners, or any other behaviour that may be of value during data analysis and 
interpretation (Nieuwenhuis 2007:86). The observation sheet used is as shown in 
appendix B. 
The observation guide defines the behaviours of interest. Dyer (2006:89) warned 
that pre-determined behaviours should be defined in such a way that they can be 
easily identified and not confused with other behaviours. Since in a normal 
classroom even those behaviours which are of no concern of the study will occur, the 
observation form was as exhaustive as possible. Maree (2007:86) warns that 
excluding other behaviours which might be of value may create problems during data 
interpretations. He maintains that it is better to have excess observation records than 
to have inadequate. In addition to the predetermined observations, other behaviours 
that occurred during the observations, like late coming, were also recorded as they 
may add value during interpretations. Some of the behaviours observed were: 
• how the lesson was introduced, (do teachers remind the learners about the 
previous lesson, if they do how: do they give questions, ask learners to 
explain what they learned etc) 
• teaching strategies used during the lesson, (questioning, demonstration, 
problem solving, discussions etc) 
• types of problems given to learners (real life, abstract non-relevant) 
77 
• how teachers respond to learners difficulties; and  
• the time given to learners to execute the given task 
One of the disadvantages of observations is that, if people who understand what is 
happening are observed, like the teachers in this study, they may change their 
behaviours and behave in a way they perceive desirable (Dyer 2006:94-95). As 
mentioned a voice recorder was used to help minimize the cues that may signal that 
observations are in progress (Dyer 2006:4-95).  
In school A, observations were done over a period of eight days while in school B 
they were done for five days in one class and for 11 days in the other class. 
The main focus of this study is to get a deep understanding of how teachers teach 
fractions and how learners learn fractions, the researcher felt it is not enough to just 
observe classrooms. She went further to interview teachers as an attempt to get 
teachers’ perspectives regarding the actions that transpired in their respective 
classrooms. She also collected the learners’ mathematics exercise books in order to 
determine the types of questions given to them, how they answered them and the 
type of feedback that teachers gave to learners. 
The next section explains how interviews were conducted. 
4.5.2 Interviews 
Interview as a qualitative data collecting technique is defined as “attempts to 
understand the world from the participant’s point of view, to unfold the meaning of 
people’s experiences and to uncover their lived world prior to scientific explanations” 
(Greeff 2002:292). It is discussed in 4.3 that the distinct feature of human beings’ 
actions is the motive and intention in performing the action. Such actions become 
intelligible to others when they understand the meaning attached. The purpose of 
supplementing observations with interviews is therefore to find out what is in 
someone else’s mind in order to minimize subjectivity in the study (Best & Kahn 
2003: 255). Greeff (2002:292) and Lankshear and Knobel (2004:198) warn that if 
observations are not followed by interviews there is a danger of making assumptions 
regarding people’s behaviours. 
78 
Each of the three teachers was interviewed and all of their lessons, on fractions, 
were observed. Each teacher was given chance to determine the time that would be 
convenient to have an interview. They all chose to be interviewed after school when 
everybody had gone home so that we would not be disturbed. The interviews were 
semi-structured in that, there were predetermined themes the interviewer intended to 
cover (Dyer 2006:32; Wellington 2000:71-80). The topics mapped the interview but 
as the interview proceeded the questioning were guided by the content of the 
respondent’s answers (Dyer 2006:32). The researcher was therefore able to probe 
and hence the interviewees were able to clarify themselves and make it possible for 
the researcher to get a deeper understanding of their view on teaching fractions. 
This is because one of the characteristics of qualitative research is that the 
researcher should neither be objective nor detached, but should be engaged (Greeff 
2002:299). Interview questions for class teachers and teacher trainers are attached 
as appendix C and D respectively. 
Interview sessions were voice recorded and supplemented with notes on non-verbal 
cues. The use of a recorder helped the researcher to maintain eye contact and show 
interest in the person being interviewed (Dyer 2006:40; Lankshear & Knobel 
2004:199; Nieuwenhuis 2007:89). The authors indicate that if eye contact is 
maintained the researcher can get the respondent’s feelings and emotions even 
without them being said. 
Some of the precautions the researcher took were to open up and show interest in 
what the interviewees say and asked more questions to elicit more information. This 
helped in determining if the interviewees were just trying to be smart or they are 
being authentic (Lankshear & Knobel 2004:198).  
Teachers’ choice of pedagogical actions is influenced by their beliefs, attitudes and 
the type of training they got while being trained as teachers. Some of them model 
themselves on their lecturers at teacher training institutions while others teach the 
way they were taught while they were in primary and high school. So to get more 
insight into their choice of pedagogical practices, Mathematics lecturers from the 
Lesotho College of Education (LEC) were interviewed. 
In this regard lecturers were asked to explain to the researcher the level of training 
given to these teachers that would enable them to teach Mathematics. That is, were 
79 
they ever given a chance to demonstrate their understanding of mathematical 
concepts they would teach at the end of training or to demonstrate how they would  
use different teaching strategies, such as problem solving and/or group work to 
teach mathematics and fractions in particular. 
The interviews were also voice recorded and supplemented with notes to capture 
those cues which could not be recorded. 
4.5.3 Document analysis  
Observations and interviews such as data gathering techniques are classified as 
interactive or obtrusive measures (Gray 2004:263). Document analysis on the other 
hand is a non-obtrusive strategy for obtaining qualitative data with little or no 
reciprocity between the researcher and the participant (McMillan & Schumacher 
2001:451). Documents, among others, are organizational records, personal records 
and school records. The documents of concern in this study were teachers’ lesson 
plans and learners’ mathematics exercise books. These documents were viewed as 
important in this study because, as discussed in section 3.8, lesson plans and 
learners exercise books enhance effective learning.  
Punch (2009:158) maintains that documents are rich sources of data and may be 
collected in conjunction with observations and interviews. 
4.5.3.1 Teachers’ lesson plans 
From these documents the researcher determined how teachers prepared for their 
mathematics lessons. Among other things the researcher wanted to find out whether 
teachers wrote lesson plans before teaching. If they did, then what is it that they 
write in their lesson plans?  The contents of the teachers’ lesson plans will be 
studied in relation to the contents stipulated in section 3.8 as: 
• topic 
• date 
• objectives 
• methods 
80 
• procedure 
• list of materials needed 
• potential problems and pitfalls   
• methods of evaluation 
Apart from the contents of a lesson plan effective ways of teaching mathematics 
were discussed in chapter 3, so from lesson plans the following information will be 
sought: 
• teachers’ anticipation of learners’ mistakes and difficulties regarding the 
concept to be taught 
• pre-determination of problems to be solved to elicit learners understanding 
• whether the types of problems used are derived from learners’ contexts or not  
4.5.3.2 Learners mathematics exercise books 
From these documents, information regarding effective teaching of mathematics will 
be depicted. That include: 
• how learners layout their work, that is, whether their work is neat and easy to 
read 
• the use of learners’ own strategies when solving problems enhances 
conceptual knowledge hence effective learning (cf. 2.5.3.1)  
• whether learners use algorithms all the time when they solve problems. When 
this is done learners acquire procedural knowledge which promotes 
procedural knowledge (cf. 2.5.3.1) 
• whether they use their own methods. It is discussed in section 2.5.2.3 that 
through experiential learning, learners come up with their own methods of 
solving mathematics problems 
 
 
81 
 
4.6 POPULATION, SAMPLE AND SAMPLING TECHNIQUES 
4.6.1 The population 
Population is defined as the total set of individuals or target group from which the 
individuals of the study are chosen and on which to generalize and apply the 
research findings (Strydom & Venter 2002:198). In this study the population refers to 
all 11 to 12 year old pupils in primary schools in Morija (in Maseru district), primary 
Mathematics teachers in the district of Maseru Lesotho and Mathematics teacher 
trainers at the Lesotho College of Education (LCE).  
Maseru is a large district of 4279 square kilometres, so it is not feasible to study the 
entire population. The researcher therefore selected a small proportion of the 
population especially as the study is qualitative in nature and the aim of this study is 
not to generalize but to get an understanding of how fractions are taught and how 
learners construct understanding.  
The next section describes the sampling techniques used in this study. 
4.6.2 The sample and sampling techniques  
Sampling is defined by Nieuwenhuis (2007:79) as a small proportion of the 
population of study which has all the characteristics of the population. The 
researcher did this study and at the same time she was a fulltime teacher, so 
studying the entire population is not feasible due to the constraints of time, 
accessibility and finances (Nieuwenhuis 2007:80). 
Purposive sampling of the participants was employed. The majority of Primary 
school teachers in Lesotho are trained at the Lesotho College of Education and 
those who further their studies often do so with the National University of Lesotho 
(NUL). In the recent years some cross the borders of Lesotho and train with the 
University of Free State (UFS).  
Initially the researcher intended to study three teachers who trained with the above 
institutions so that comparison could be made regarding how they taught fractions. 
82 
However this was not possible because standard six teachers in the schools which 
the researcher was able to observe trained at the LCE.  
In this regard the participants comprise three primary Mathematics teachers teaching 
standard six (known to be grade eight in other countries such as South Africa). The 
researcher’s place of residence is Morija, therefore primary schools around Morija 
participated in this study. All the learners in these three classrooms and two 
mathematics teacher trainers both from LCE formed part of this study. I did not find it 
relevant anymore to interview lecturers from NUL because none of the observed 
teachers studied at NUL.  
Silverman (2000:105) purports that purposive and theoretical sampling are often 
treated as synonyms. The author cites three features of theoretical sampling as: 
• choosing cases in terms of our theory 
• choosing deviant cases  
• changing the size of our sample during the research 
The researcher’s decision to study learners in standard six was based on the theory 
that they were at the end of the concrete operational stage and were moving towards 
the beginning of the formal operational stage of cognitive development (cf. 2.3.1.3 
and 2.3.1.4). Children at this stage are expected to begin to reason inductively and 
deductively.   
The investigations, in the form of fieldwork, took place in three classrooms in two 
schools. Two of the classrooms are in the same school but they were observed at 
different times. All three were standard six classes.  These three cases provide an 
opportunity to examine and understand in depth the phenomenon in a classroom 
setting and determine how feasible information about teaching and learning of 
fractions from the literature translates into existing classroom practice (McMillan & 
Schumagher 2001:398).   
In school A the researcher took eight days without interruptions. In the first class in 
school B she also took five days without interruptions. In the second classroom she 
took overall 11 days. After seven days we were interrupted for one day by the 
debating activity which was held one Friday. The last four days went without 
83 
interruptions. Details of the context in which the teaching took place were recorded, 
including information about the physical environment and social factors that were 
relevant to the situation (Punch 2009:119). 
Interviewing mathematics teacher trainers from LCE provided information on whether 
teachers practiced the training they got or whether their teaching was mainly 
influenced by their beliefs, environment, and culture of the schools or other factors. 
From these interviews the researcher gained a deeper understanding of the type of 
content matter, pedagogical content knowledge and the attitudes of teachers which 
helped them develop into becoming effective teachers. 
4.7 PILOT STUDY 
Pilot studies are referred to as either small scale feasibility studies done to prepare 
for the major study or a study done to test a particular research instrument (Gall, 
Borg & Gall 1996:65; Van Teijlingen & Hundley 2001). In this study piloting was done 
with both aims of testing the feasibility of the study and testing the research 
instruments.  
The pilot study was done in a school with the same environment and socio cultural 
background as those two included in this research study. This helped the researcher 
improve on the skills she needed to conduct the major study; these include 
observation and interviewing skills together with document analysis. In so doing the 
researcher gained confidence and hence embarked on the major study with a 
positive attitude and higher level of competence. These views are supported by Van 
Teijlingen and Hundley (2001) as they state that one advantage of pilot studies is to 
train researchers about different elements of the research process. 
After conducting the preliminary study and having identified problems, adjustments 
were done accordingly. Some of them included the focus of observations and 
structuring of the interview questions.  
4.8 VALIDITY AND RELIABILITY  
Creswell (2008:266) as well as Whittemore, Chase and Mandle (2001: 528-529) 
warn qualitative researchers that throughout the entire research process they should 
make sure that the findings and interpretations are accurate.  Whittemore et al.  
(2001:528-529) state the criteria that should be used by qualitative researchers to 
84 
maximise quality of their research studies. These are credibility, authenticity, 
sensitivity and creativity, just to mention a few.  
Care was therefore taken in this study to be accurate while reporting on the findings 
and interpretations. The researcher supplemented the findings and interpretations 
with verbatim statements of the participants. The researcher was also conscious of 
the conceptions she brought to the study and that her involvement in the research 
process could hinder the participants to speak authentically for themselves. The 
precautions that the researcher took were according to Whittemore et al. (2001:529) 
are important to increase credibility and authenticity of qualitative research. 
Criticality and integrity of qualitative research as discussed by Whittemore et al. 
(2001:531) refers to whether the researcher worked repeatedly with data and every 
time tried to find different and/or multiple understanding. The researcher therefore 
followed this procedure because she engaged with data continuously. She engaged 
with it individually and collaboratively. At the end of the observation she asked 
observed teachers to each check on the transcriptions to verify their accuracy. She 
also asked her study leader to reflect with her on the raw and coded data. This is 
referred to by (Creswell 2008:267) as member checking and it is another strategy 
used to ensure validity of qualitative studies.  
Gray (2004:90) on the other hand states that to ensure validity, research instruments 
must measure what they are intended to measure. In this regard Hopkins (1980:45) 
defines validity as the degree to which a research instrument accurately describes 
the attribute being observed. It also ensures that collected data is relevant to 
research (Birly & Moreland 1998:41). 
Observations pose the danger of producing invalid results (Gray 2004:256). This is 
because when observing one event, two people may come up with different 
interpretations. So this indicates that observations, as data gathering instruments, 
are subjective. To reduce this error and hence increase the credibility and 
authenticity of data and findings, triangulation of data collecting instruments was 
used (Creswell 2008:266; Golafshani 2003:603; Nieuwenhuys 2007:113). That is, 
observed data was supplemented with interviews with the teachers who presented 
the authenticity of their behaviour and actions in their respective classrooms. 
85 
Triangulation of data collecting instruments can be upheld as a claim for validity. The 
reason being that data and research finding becomes credible if data collected by 
one instrument is confirmed by data collected from another instrument. 
When designing interview schedules, the researcher ensured that the content of the 
interview questions directly concentrates on the research objectives (Gray 2004:218- 
219). Gray further suggests to researchers to do the following in order to increase 
validity of interviews: 
• using interview techniques that build rapport and trust, thus giving informants 
the scope to express themselves 
• prompting interviewees to illustrate and expand on their initial responses 
• ensuring that the interview process is sufficiently long for subjects to be 
explored in depth 
• constructing interview schedules that contain questions drawn from the 
literature and from pilot work with respondents 
All the points suggested by Gray were taken care of because all teachers observed 
are known by the researcher as they work in the neighbouring schools and one of 
them is the researchers’ former student. So a good rapport was established as all 
seemed to be at ease during observations and interviews. Interview questions were 
also drawn from the literature. 
It is mentioned above that the researcher was in direct contact with data throughout 
the process. So during data analysis the researcher constantly inspected and 
compared all data fragments that arose in a single case. The data from one case 
was further compared with data from the other two cases. This validating method is 
referred to as constant comparative method by Silverman (2000:179). Apart from this 
the number of instances a category emerged from data was recorded as this was 
believed to provide grounds for generating conclusions. This is also as depicted by 
Silverman (2000:184-185) 
The steps taken to ensure validity of the research study according to Lincoln and 
Guba in Golafshani (2003:601-602) also maintains reliability as they state that "Since 
86 
there can be no validity without reliability, a demonstration of the former [validity] is 
sufficient to establish the latter [reliability]".  
4.9 DATA ANALYSIS AND INTERPRETATIONS  
Notes on teachers’ planning and learners’ workbooks; transcriptions of observations 
and interviews form data to be analysed. The number of learners in each class, their 
age distribution and number of repeaters; age of the teacher, highest qualifications, 
years of teaching experience, their qualifications and the set-up of the class formed 
the basis for the description (Gray 2004). In qualitative research data collection and 
analysis are done simultaneously. At the research site, the researcher took notes of 
the events that happened as she thought some might be helpful during data 
analysis.  
One teacher was observed at a time, so this gave the researcher enough time to 
work with one set of data. Data from each teacher was filed and put aside while the 
next teacher was observed. The data that is referred to here are; field notes, lesson 
and interview transcripts, teachers’ and learners’ documents from the same school.  
The next phase was to use a word processor to transcribe the audio recorded 
lessons (de Vos 2002:347; Gray 2004:331; Nieuwenhuis 2007:108). When typing 
even the non-verbal cues like silence, laughter and words such as “er”… “well”… 
were included as they portray, for instance, uncertainties and will add value during 
data interpretation (Nieuwenhuis 2007:108-110).  
After transcribing the observations and interviews, the researcher then read the field 
notes, listened to the tape and read the transcripts over and over so that she 
became familiar with the data. She further wrote impressions she had as she went 
through the data which Nieuwenhuis (2007:104-105) referred to as a reflective 
journal.  
The process of analysis and interpretation was done based on the six steps as 
depicted by Radnor (2002:71). These wre: 
• topic ordering; which is listing topics that appear on reading the whole text 
• constructing categories; some of which were derived from literature and some 
as explicated by data 
87 
• reading for content; highlighting and attaching codes in the text 
• completing the coding sheet; under each category, the codes from all 
teachers were written in order to give the researcher an idea of how much 
data there was for each category 
• generating coded transcripts; copied and pasted chunks of data to appropriate 
categories in the coding sheet 
• analysis and interpretation; written statements that supported and 
summarised the findings within each category 
To address steps one and two the researcher drew a table, as shown in table 4.1, in 
which she filled in the topics, codes and categories (Nieuwenhuis 2004:108; Radnor 
2002:85) as derived from literature discussed in chapters two and three.  
Table 4.1 Coding table 
Topics  Code  Category    
Effective learning  EL Manipulation of concrete materials 
   Using learners’ prior knowledge 
Conceptual knowledge before procedural 
knowledge 
Learning environment 
Group work  
Effective teaching  ET Problem solving 
Inductive reasoning 
Deductive reasoning 
Teachers’’ PCK and TK Analysing learners’ mistakes 
MKT Sequencing of subtopics 
Multiple representations 
Traditional teaching TTS Rules before concept 
strategies Drill and practice of rules 
Lecture  
 
 
88 
The researcher was flexible enough to remain open and include other themes as 
they emerged in data during data analysis (Mertens 2010:428). She did not only 
focus on identifying themes from literature. 
She once again read the transcripts for content, by highlighting and attaching codes 
to chunks of data. The convention used here was such that EL 2a represents the 
topic effective learning (EL), category number two (using prior knowledge) appeared 
for the first time (a) (Radnor 2002:71-90). These were then filled in the coding sheet 
so as to give the researcher an idea of the amount of data there was for each 
category. 
Then coded data, for each topic, was copied and pasted on a master assessment file 
(Radnor 2002:80). This enabled the researcher to have access of data from all three 
teachers at the same time so that analysis could be done with ease. 
After arranging data in this manner, the researcher wrote statements that summarise 
the findings within each category as interpreted by the researcher. It is worth noting 
that phenomenologist do not assume that individuals are completely unique, in this 
regard the researcher not only focused on the differences between the teachers’ 
teaching approaches but also looked for the commonality of experience (Johnson & 
Christensen 2012:385). 
4.10 ETHICAL ISSUES 
The researcher had the responsibility to adhere to the ethical issues. Before doing 
the study, first the researcher sought permission to do the study in the schools from 
the Ministry of Education and Training (MoET). When she had permission she 
approached the principals of the schools and informed them of the purpose of the 
study. The concerned teachers were approached, told the purpose of the study and 
they willingly agreed to participate in this study. Teachers who willingly accepted to 
participate in this study also signed a consent form which stated clearly the purpose 
of the investigation (Kvale 1996:112). 
Confidentiality of the identity of teachers, learners and schools and results was 
observed. This was done by labelling schools by letters A, B and C and identifying 
learners as learner 1, learner 2, teachers as ‘Teacher A, Teacher B and Teacher C’. 
89 
Where names were mentioned the researcher used fictitious names (Kvale 
1996:112; Radnor 2002:34-35). 
While doing class observations, the researcher did not interfere with the class 
timetables, the normal running of the schools or the personal activities of the 
teachers. She did class observations at the timetabled time and did not ask for 
favours that would suit her schedule. 
During class observations the researcher sat at the front of the class where she 
could not interfere with the smooth running of the class but where she had a good 
view of both the teacher and learners. When learners were given class work, the 
researcher went around with the purpose of looking at how the learners’ did their 
work. 
4.11 CONCLUSION 
This chapter dealt with the research design followed in this study which helped to 
answer the research questions as stated in chapter one. In this chapter the research 
methodology that was followed while conducting the study was discussed together 
with the rationale for following the phenomenology as the research approach.  
The study followed a qualitative approach in which class observations, teachers and 
teacher trainers’ were interviewed. Interviews were conducted in order to supplement 
information gathered through class observation teachers’ documents and learners’, 
mathematics books.  
The population and the sampling techniques employed were discussed. The data 
collecting strategies used in this study, which were observations, interviews and 
document analysis together with their validity, reliability and rationale for using them, 
were discussed. Data analysis and interpretations together with ethical issues were 
dealt with.  
In the next chapter collected data will be analysed, interpreted and discussed. 
 
 
 
90 
CHAPTER 5 
ANALYSIS AND INTERPRETATIONS 
5.1 INTRODUCTION 
In chapter four the empirical research was described as a phenomenological study. It 
was discussed that teachers were observed in their natural setting as they teach 
fractions to standard six learners. The purpose of this study was stated in chapter 
one to assess and explore how mathematics teachers teach fractions and how 
learners learn fractions. 
In chapters three and four of this study, effective methods of learning and teaching 
mathematics, with emphasis on teaching and learning fractions, were discussed. 
Expectations regarding what teachers should do to help learners learn effectively 
were elucidated together with the knowledge base teachers need to possess in order 
to be effective in this regard. 
This chapter therefore, interprets and analyses data collected through class 
observations, teachers’ lesson plans, learners’ exercise books, interviews of class 
teachers and also teacher trainers.  
5.2 SCHOOL A  
5.2.1 Schools’ environment  
School A was situated at about 8 kilometres South of where the researcher resided. 
The school was in a rural village in which the community mainly relied on 
subsistence farming and the rearing of cattle.  
The school was a three block building where each block comprised of three rooms. 
There was no fence around the school hence the villagers pass freely next to the 
classes. The first block comprised of standard five, the principal’s office, standards 
six and seven.  
The inside walls of the classroom were not clean. They were painted with white and 
blue paint, but most of the paint on the lower parts of the walls was removed possibly 
by the rubbing of desks or hard objects. There were neither posters nor charts on the 
walls except one hand written chart of multiplication tables on the wall.  
91 
The area of the classroom was five by four metres and there were 50 students in all. 
There were two chalk boards, one on the front wall and the other one on the back 
wall. There were two teachers responsible for standard six, but they taught different 
subjects, and their tables were placed on either side of the class. The teacher who 
participated in this study taught mathematics, science and social studies while the 
other one taught the remaining subjects. 
The floor map below represents the sitting arrangement of both teachers and the 
learners 
 
Figure 5.1 Floor map of teacher A’s classroom 
T1 and T2 refer to the teachers’ tables while the small rectangles refer to the 
learners’ desks with chairs. The small circles represent stools made of aloe that were 
used by some of the learners who did not have desks. 
The observer sat at T1’s table. This was the table of the mathematics teacher who 
participated in this study.  
 
 
 
92 
5.2.2 Teacher’s profile   
Teacher A is a male teacher in the late twenties. He has been teaching for five 
years. His highest qualification is a three year Diploma in Education Primary. 
5.2.3 Class observation 
In this section the researcher describes the observations as she conducted them. 
There were 50 learners in teacher A’s class. There were pieces of dry aloe at the 
back of the class and some of the pieces were sculptured such that they served as 
stools on which some learners sat. The teacher was involved in a project which 
involved integrating technology in teaching, and he used aloe in this project. 
As mentioned in section 5.2.1 there were two teachers assigned in this class but 
they taught different subjects. During mathematics lesson the other teacher sat at 
her table doing other business such as marking or preparing for her next lesson.  
There was one incidence in which she helped the mathematics teacher to mark class 
work but most of the time she would be busy with her work. 
At the beginning of every lesson of teacher A one learner took the mathematics text 
books from the cupboard situated behind the door and gave each learner a copy of 
this book. 
Primary school education is free and compulsory in Lesotho and learners do not buy 
their own stationery and text books. Instead the government supplies schools with 
stationery and text books. In this school text books are kept at school and they are 
given to the learners only when they are given work from the books as homework. 
Schools are not supplied with text books every year, so it is their responsibility to 
take care of them. Hence, they keep them at school.  
As the learner distributed the books the teacher would be writing the date, subject 
and class on the chalk board.  
All the mathematics lessons were in the morning hours, that is, from eight to eleven. 
A single lesson took forty minutes and a double lesson eighty minutes. There were 
two single lessons and three double lessons. During each of these morning lessons 
the teacher would be interrupted by a learner who was late. He never seemed 
bothered as he often allowed them in without a comment.  
93 
The series of his lessons were such that he taught fraction concepts, comparing 
fractions, equivalent fractions, and addition of both proper and mixed fractions, 
multiplication and division by a whole number and making fractions. These were 
taught in eight consecutive lessons.  
When the teacher reviewed fraction concepts, he introduced the lesson by telling a 
story about a mother who shared an orange among her four children. He had an 
orange and was holding it up for all the learners to see. He did not share the actual 
orange instead he had a chart on which an orange was divided into four parts. 
Holding this chart he asked learners to name the fraction of an orange that each 
child would get.  
All learners answered one quarter. He also asked how many parts would be left after 
the mother has given one part to one of the children. The learners all said three 
quarters. 
The teacher wrote both fractions on the chalkboard and then told learners that one of 
the quarter is the numerator and the four is the denominator. He then wrote the 


fraction  and asked learners to name the numerator and denominator.  

The teacher then drew rectangles on the board and shaded part of each rectangle. 
He asked the learners to name the fraction of the shaded parts. Learners were able 
to name the fractions and name numerators and denominators. 
Towards the end of the lesson the teacher drew this shape 
 
He asked one learner to name the fraction of the shaded part. 
The learner said  
One third  
The teacher  
94 
Is it one third? 
He called another learner who gave the correct answer which was one quarter. He 
ended the lesson by giving a written exercise in which the learners repeated the 
same exercise of writing fractions of shaded parts. 
The given diagrams were a partitioned circle, a rectangle and a group of twelve 
eggs, three of which were coloured black. 
The second lesson was about comparing fractions, the teacher introduced the lesson 
by saying: 
Today we will compare fractions and before we can do that I want us to look 
back and remind ourselves about what we did yesterday, we were using the 
fraction wall or the fraction tables to identify the fractions, so I want someone 
to remind me these fractions that I’m putting on the board here. 
After doing the oral exercise with four fractions the teacher drew a fraction wall like 
the one below.  
 
Figure 5.2 (a) Fraction board 
After drawing it he informed learners about what they were going to use the fraction 
wall for. 
We are going to use the fraction wall and the number line to compare 

 

fractions. Do you think these fractions  and  are the same? 
 
95 
Learners: (Together) No sir 
Teacher A: You are correct. That means the other fraction is bigger than the 
other, isn’t it? 
Learner: (Together). It is so. 
The teacher asked learners to name the fractions of different sections of the fraction 
wall, may be to make sure that they understood it.  
Teacher A: What do we have on the first bar of the fraction wall? (Students 
shout sir, sir!) 
Tumi: One whole 
Teacher A: We have one whole right? One whole, that’s correct. On the 
second bar we have? 
Hloni: Half 
Teacher A: We have half; I’m not going to ask further questions, I want to ask 
you this, how many halves make up a whole? That’s the question I want to 
ask. How many halves make up one whole? 
The teacher straight away asked learners to name the number of halves in one 
whole. The learner who was chosen was able to say  
Two halves in one whole. 
This is what the teacher said after the response 
Teacher A: That means we must have 2 halves and you remember that when 
we add fractions we only have to add them when the denominators are? 
Learners: (Together) Same 
Teacher A: When denominators are the same.  Are the denominators the 
same? 
Learners: (Together) Yes 
Teacher A: Are they the same? 
96 
Learners: (Together) Yes 
Teacher A: Alright let’s get right away, how do we do it? That is… yes 
Thabisang! 
Thabisang: 1+1 is 2 
Teacher A: One plus one is two next, yes ‘Malebo 
‘Malebo: Over 2 
Teacher A: Over the denominator 2. Do not add the denominators. So we are 
saying that is 2, let us try to simplify this fraction that is 2 over 2 simplified, yes 
Katji 
Katji: 1 whole 
Teacher A: One whole correct that’s correct. So this is how we can compare 
fractions. So you will realize that um… 2 out of 2 is equivalent to 1 that is 1. 
Learners were then told to look at the fraction wall in their text books and find the 
number of quarters that make up one whole. In order to find the number of quarters 
in one whole the teacher encouraged them to add the fractions as above. 
The teacher gave four sets of fraction slabs to four desks and asked those who did 
not have the slabs to move to the desks where the fraction slabs were placed. He 
asked the groups to compare the halves, thirds, quarters, fifths sixths and eighths to 
one whole. The teacher went from group to group and looked at what they were 
doing. 
After this exercise, the teacher wrote <, >and= and asked learners to name them. 
One learner said; It means greatest.  
The teacher said greater than and then showed half and one fifth slab and asked;  
Which is bigger than the other? 
‘Malebo: One fifth. 
Teacher A: I would advise you to answer the question looking at the fraction 
wall but not on this. Between half and one fifth which is bigger? Yes ‘Mamo. 
97 
‘Mamo: One fifth. 
Teacher A: One fifth is bigger! Why? Look at the fraction wall not this. 
Between half and one fifth which one is bigger? 
Bongi: One fifth. 
Teacher A 
 
: One fifth is bigger? He moves the chalk around  and and then 
 
asks which is bigger? I think if I can rub that people can now get to where I 
want them to, to drive them to. Half and one fifth which one is bigger? 
Katji: Half 
Teacher A 
 
: Half is bigger. We say  is greater than . (The teacher moved the 
 
chalk around the two fractions) Please ensure that there are spaces before 
and after fractions (and formulas).  

 
 
 

After comparing these set of fractions,  and ,   and , the teacher asked learners 
   

to come up with the generalisation regarding comparing fractions. He asked: 
So what does this tell you? What does this tell you as you compare fractions? 
Say something about the denominator. 
The teacher accepted the answer  
The fraction that comes after half is smaller than half. 
After this answer he asked: 
If you are to choose which orange to take between one eighth and half, which 
one would you choose? And why would you choose that one? 
The learner who responded to the question was able to say half because it is bigger 
than one eighth. 
The teacher assessed the lesson by giving learners an exercise from the text book. 
The question was to compare two pairs of fractions by writing the appropriate sign 
between<,  >and=. 
98 
When teaching equivalent fractions, which was the third lesson, the teacher 
introduced the lesson by drawing a fraction wall on the chalk board and pointed at 
the whole slab, half, one third up to the eighths and asked. Each time he pointed, the 
learners named the fraction in unison. 
He then shaded the fraction board as shown below and pointed to the line that 
divided a whole into two halves.  
 
Figure 5.2 (b) Fraction board 
He then told the class that  
This line divides the fraction bar into two halves. 
The learners then named the fractions whose bars shared the same half line (drawn 

 
 

in bold in the diagram above). The learners stated that they are , , and  
  
Having listed the above fractions the teacher then asked learners the following.  
Teacher A 
: I would like us to compare these 2 fractions, writes  =  on the 
 
board. But this fraction is incomplete. We have half here and I want us to have 
99 
a fraction that we can say it is equal to this one, and for us to come and find 
the missing numerator here. I would like you to tell us what we need to do 
alright? Suppose we want to find the fraction that is equal er… equivalent to 
half, what do we do? We only have this part here. Ntate (sir) 
Makaja: We take the half over 6 
Teacher A: Let me get this clear what did we do to get 6 from 2, the 
denominator 6 from 2 We only have denominators right? We only have one 
denominator I mean numerator rather and we have another one which is 
missing so we only have this two which we know so what did we do to 2 so 
that we get 6?  
‘Malebo: We multiply by 3 
Teacher A: We multiply 2 by 3 and that will give us 6. Is that true? 
Learners together: That is true 
From this point the teacher helped learners practise the rule by calculating 
equivalent fractions and every time that they did it the teacher would be saying:  
What we do to the numerator we also do it to the denominator  
Having drilled learners on how to calculate equivalent fractions, he asked learners to 

 

find the smallest fraction which was equivalent to  and . These are the 
 
responses: 
Lala: 2 
Teacher A: The fraction, 2 is the whole number 
Lala: Half 
To check whether it is half the teacher divided the numerator 15 with 15 and the 
denominator 30 with 15. He ended the lesson by asking learners to complete the 
statements. He then reminded learners that they should remember the rule of 
equivalent fractions. 
100 
3
=  
9 3
5
=  
20 3
1
=  
4 12
 
 
Lesson four was on addition of fractions. This time the teacher did not review the 
work done in the previous lesson. Instead he asked learners to give him two 
fractions.   
Can you, can you just give me any, any fraction, yes  
The learners shouted half, One quarter  
The teacher told the class that they are going to add fractions and reminded them to 
remember the rule that they talked about yesterday. 
Teacher A: I would like us to come and add the two fractions, now. (Pauses). 
Before we add there is one rule of fractions that we talked of yesterday. What 
is that rule? A few hands were up and he repeats, what rule of fractions that 
we talked of? Yes girl.  
‘Mato: … (Talks with a very low voice such that the teacher did not hear and 
he said) 
Teacher A: Don’t talk to the window but to me 
The student did not seem confident so the teacher called on another to give the 
answer. 
TK: What you do to the numerator you also do to the denominator 
Teacher A: That is the rule for the fractions. What we are going to do here 
which is addition we are going to use that rule, especially when we add 
fractions that have different denominators, alright? 
101 
Learners: (Together) Yes sir 
From this point the teacher asked open questions, explained and demonstrated to 
the class how they to add fractions and emphasized rules they should use. 
Teacher A: You cannot add fractions if they have different denominators and 
that means we have to make those denominators. …That means you have to 
make the denominators the same. What are the denominators of the fractions 
that we are adding now? What are the denominators of the two fractions that 
we are adding now? (Pause as he waits for the hands and call! Yes boy). 
Tboss:  2 and 4 
Teacher A: Repeats; 2 and 4. You cannot add fractions that have different 
denominators. …So like when we are, when we are dealing with the 
equivalent fractions we had to make the two denominators the same, right?  
Learners: (In chorus) Yes sir 
Teacher A: We had to make them the same. Which is the bigger 
denominator? Which is the bigger denominator? (Yes girl, Likeleli) 
Likeleli: The bigger denominator is 4 
Teacher A: The bigger denominator is 4. Now if the bigger denominator is 4 
we have to look for means of making the smaller denominator the same as 
the bigger denominator, alright? 
Learners: (Together) Yes sir 
Teacher A: And in this case I must explain that for making the smaller 
denominator being equal to the bigger one we are to multiply the smaller 
denominator by a certain number or…r see if the smaller denominator can 
divide the bigger one without having a remainder, ok? 
There are 2 methods; multiply the smaller denominator by a certain number to 
get the bigger denominator or see if the smaller denominator can divide the 
bigger one without getting a remainder. Those are the two methods. But I 
102 
would like us to go for this one of multiplying. Which number can we multiply 2 
so that it becomes 4? 
Learners: (Shout) sir, sir 
Teacher A: Calls a student to give the answer 
TK: We multiply by 2 
Teacher A: 2 multiplied by 2 equal 4. Correct 2 by 2 correct. Tell me the rule 
now 
Sir, sir, sir 
‘Malebo: 4 
Teacher A: The rule sir, not the answer. The rule that we must apply, (Calls 
another student ntate! (sir))  
Tboss: We add. (Others shout sir, indicating that they disagree with him) 
Teacher A: Do we add? 
Lala: Whatever we do on the denominator we also do on the numerator 
Teacher A: Repeats and then asks, what do we do on the denominator? So 
what do we do to the denominator or what should we do to the denominator 
and what should we do to the numerator? 
TK: Multiply 2 by 1 
Teacher A: We multiply 2 by 1, he writes it 
TK: And we multiply 2 by 2 
Teacher A: This is the rule that we have to apply. What ever that we do on 
the numerators we do to the denominators. As simple as that, and now it goes 
something like that, so let us try to simplify this, what is 1 by 2? 
Learner 10: 1 plus 2 
103 
Teacher A: It is not plus but it’s… (somebody shout times). Teacher then 
says the answer is? 
Learners: (Together) 2 times 1 
Teacher A: The answer is 2, right 
Learners: (Together) Two times two equals four (2 x 2 is 4) 
Teacher A: Two times 2 is 4, now here we go, here we go, (as he writes on 
the chalkboard) do you see what we mean? 
Learners: (Together) Yes sir 
Teacher A: Now we have this now we have this as he writes  

		 
  

 +  =  + .Now the denominators are 
		   
Learners: (Together) The same 
Teacher A: Ke eona ntho e re e batlang (that is what we were looking for). 
That’s a rule that is a rule, you cannot add the fractions that have different 
denominators why? So now we can go on adding. What do we say? 
Teacher A: Another rule, if the denominators are not the same, you don’t add 
 
 #

them. Some make the mistake of saying + = . We don’t do that ok? We 
  #
only add the numerators, the top numbers, that is, what we only do. So this is 
how fractions that have different denominators are being added. Look at the 
picture, look at the pictures 
Learners: (Together) Yes sir 
Teacher A: Ah… ah, I would like to put the question here on the board and 
 
 
 
someone should come and solve it. (He writes +  and  + ) 
 
  
After demonstrating, he called upon two learners to come to the board and add 
fractions. They worked out the sums at the same time. They worked quietly and 
applied the rule they had just learned. They worked them like this; 
	$	 
  
 
+  = +  =  
	$	 
 
 
 

104 
	
	$	    
 +  =  +  =    
	$	    
When they finished working out these two questions he showed the learners how to 
add fractions with denominators that are not multiples. Like in the first case, he 
taught the rule. 
Teacher A: Here is another instance I want you to look at. Let us look at this 

 

one now, writes + . The denominators are 3 and 4 is there any number you 
	 
can multiply 3 by in order to make it 4 
Learners: (Together) No 
Teacher A: The second option; can 3 get into 4 without leaving a remainder? 
Learners: (Together) No sir 
Teacher A: So what is 3 into 4? It goes?  
Learners: (In chorus) It goes once and leaves remainder 1 
Teacher A: You are right, and the other one is the? That’s the remainder I’m 
talking about. In this case, look at this carefully please; in this case if we try to 
balance the 2 fractions, since we don’t have the number that can multiply 3 to 
be 4. We try to balance the fractions. Why? By exchanging, exchanging the 
denominators, you see? (multiplies one third by 4 and one quarter by three.) 
Learners: (In chorus) Yes sir. 
Teacher A: Through multiplying. That means what we do is this. Here we 
have the denominator 4. To balance it we bring that denominator to this side.  
He demonstrates on the board 

 

 +  
	 

		 
		
=  +  
		 		
 
=  +  

 

105 

= . 


So we apply the rule of fractions, whatever that is done on the numerator we 
do on the denominator. The bottom line is to get the denominator the same. 
I know some will have problems especially (ha se li le peli) when they are two. 
You see here we are working with one side, mona re sebetsa (here we work) 
with two sides. Do you all see? (re a bona). So I want you to help each other 
in realising how we should treat the sides. That is, under which circumstances 
do we work with one side? (Na ha re treata side le le leng ke ha denominator 
li le joang) and under which circumstances do we work with two side? (le hore 
na ha re treata two sides ke ha re reng). So I want us to work in groups so 
that we help each other and every one, everybody is going to participate ok? 
After teaching addition of simple fractions, the fifth lesson was on addition of mixed 
fractions.  
This was how the lesson transpired:  
Teacher A: (raises up 3 pieces of chalk and says to the class) I have this in 
my hand, now ah… how many pieces are here? Yes boy! 
Sako: There are 3 pieces  
Teacher A: Of what? 
Sako: There are 3 pieces of chalk. 
Teacher A: … of chalks. (Breaks the other piece and asks) how many pieces 
are here? Yes girl! 
Lebo: There are 2 and half pieces 
Teacher A: Can somebody come and write 2 and half on the board. Boy  
Sups: 
 Writes 2  

Teacher A: She has done this, pointing to the fraction written on the board, is 
he right? 
106 
Learners in chorus: Yes sir 
Teacher A: Now how many wholes are there in this number that Sups has 
written? Yees (Did he stretch out the word as he spoke? If so – keep it as 
such) 
Learners in chorus: 2 wholes 
Teacher A: 2 wholes. So which is the whole number there? Ntate TK (sir) 
TK- 2 is a whole number 
Teacher A: 2 is a whole number, 2 is aaa… whole number. Now here 
(pointing to the fraction on the board) 2 doesn’t go alone, what else goes with 
2? What else goes together with 2? 
TK: Fraction 
Teacher A: What is that fraction? Yes man! 
Katso: Half 
Teacher A: Half is that fraction. So the whole number goes together with a 
fraction. So this whole number together with the fraction, what do we call them 
… together? What do you call a fraction and a whole number together? Lebo 
(girl) 
Lebo: Improper fraction 
Teacher A: I don’t think so 
Lebo: With a low voice, a mixed number 
Teacher A: Did you hear that over the corner? 
Learners at the back of class together: No sir 
Teacher A: Speak aloud 
Lebo: A mixed number 
107 
Teacher A: That is a mixed number. That means the numbers are being 
mixed, a whole number and a…  
Learners: (Together) fraction 
Teacher A: As you have seen there were 2 whole pieces of chalks right? And 
they went together with 
Learners together: Half 
Teacher A: (Repeats) half. That means 2 chalks are whole that means they 
are complete. Which is why we have the 2 here right? 
Learners together: Yes sir 
Teacher: And we also have the half part of a piece of chalk and we have here 
next to the whole number 2 so we say that it is a mixed number. The numbers 
have been brought together, so we say they are mixed numbers, so now we 
are going to change this mixed number into another type of fraction which is 
going to… which is not going to be a mixed number any more. So we have 
212 as a mixed number, as a mixed number. Remember, remember here in 
fractions we work with rules. How many rules do you remember now? Yes 
man! 
Thisa: One 
Teacher A: One! Thisa only remembers one rule. Which is that rule Thisa? 
Thisa: What you do (teacher interrupts) 
Teacher A: I can’t hear you 
Thisa: Whatever you do on the denominator you must also do on the 
numerator 
Teacher A: (Clears his voice) we are saying that we are now changing the, 
the mixed number to something else, when we change um… the mixed 
number to a fraction we multiply the denominator with the whole number and 
add the whole number, oh… the product with the numerator like this; as he 
writes what he’s saying on the board. A ke re? 
108 
Learners together: Yes sir 
Teacher A: And the total that you’ll get you put it over the denominator (writes 

) the very same denominator. Somebody come to the board and do just that 

Learners shout- sir, sir sir 
Teacher A: TK 
TK: (As he writes teacher asked him to talk to the class) talk to us 
TK: 2 times 2 is 4 plus 1is 5 (writes 5) 
Teacher A: There is TK is he right? 
Some learners said yes some said no 
Teacher A: Yes? 
Learners together- no, yes  
Teacher A: Those who say TK is not correct, he is not correct, raise up your 
hands 
(Learners who said he is not correct raised their hands) 
Teacher A: It is obvious that those who did not raise their hands say he is 
correct. (Learners raise their hands shouting sir, sir.) 
Teacher A: Yes Mamo (girl) 
Mamo : 2 times 2 plus 1 is 5. (She writes  and says 2 into 5 goes 2 and 



leaves remainder 1 (writes 2 ) 

Teacher A: Repeats, 2 by 2 is 
Learners in chorus: Four  
Teacher A: And 4 plus 1 that is  
Learners together: Five 
109 
Teacher A: And she puts 5 over here (writes 5) and she said 5 out of 2 (writes 

). So she puts it over 2, and now Mamo has further simplified this er… 

fraction. She said 2 into 5, she said it goes 2 times (writes it), as a whole 
number and it leaves 1 out, out of  
Learners in chorus: Two (says it with the learners. which is the same as the 
first one. You see? (Directs them to the first one with an arrow?) We are 
saying… what is the name of this type of fraction?  
(No response and the teacher called) every body… over here… what is the 

name of this type of fraction (taps on  on the chalkboard) (girl) 

Lebo: Improper fraction 
Teacher A: We call it improper fraction (learners said it with them). You will 
see that improper fraction has denominator… a…a… it has its numerator 
being bigger than the denominator. A ke re?(isn’t it?) 
Learners together: Yes 
Teacher A: Yes. An improper fraction has its numerator being bigger than its 
denominator. That’ an imp… 
Learners finish off the sentence: ...roper fraction 
Teacher A: Now I would like us to go further. But before we go further, err… 
we are going to do a very quick piece of work from our text books.  
Very quickly lets do this in your exercise books and I’m going to mark it right 
now. You change this thing into improper fraction. Change (pauses as he 
looks around to check whether they are getting started). 
The fractions are  


a) 2  



b) 1  



c) 3  

110 
Teacher A: After a while he asks one learner, ha eka ha uso qale tje u etsa 
joang? U etsa joang ha eka hau so qale tje (you have not yet started, why 
have you not started?) 
When the denominators are the same, a ke re. 
While other learners worked on the given exercise, others moved up and down the 
class making a noise. The teacher did not try to stop them, he marked a few learners 
then stopped for learners to collect the books and put them on his table he would 
mark them later. 
At the beginning of the class, the teacher told the learners that today they were going 
to add fractions. But they only changed mixed fractions to improper fractions and 
they did not add them. 
Lesson six was on multiplication of fractions. The teacher used the same approach 
that he used for addition. He began the lesson by asking learners to give a fraction 
they would multiply by a whole number. 
Teacher A: I would like you to give me just any fraction that, er… you will 
have to multiply with a whole number, a fraction multiplied with a whole 
number. Can you just give me that? (Girl) 
Lebo: Two times one half. 
Teacher A 
: Writes 2 x . (boy) 

Sako: One times one quarter. 
Teacher A 
: Something like this (writes) ×1 



He then asked one learner to work out 2×  on the chalkboard” 

Mamo-  
	 		
	 	She writes  x  =  =  = 1 repeating everything she writes (2 

  
		 
times 1 is 2, and 1 times 2 is 2, 2 into 2 goes once). 
Teacher A: Asks the class why did she write 2 whole as 2 over 1? (that is 2 = 

) 


111 
Learners: (In unison) No sir. 
Teacher A: Do you want to know. 
Learners: (In unison) yes sir. 
Teacher A: The thing is, let’s get this clear; the thing is a whole number can 
be changed to a fraction, alright? If you have a whole number the thing that 
you must do is that you must put it over 1 because when you divide that whole 
number by 1, you are going to get the same whole number, right? 
Learners: Together they say, Yes 
Teacher A: For instance we have 2 out of 1( ). What is 1 into 2? 


Learners: Some say one, some say two. 
Teacher A: It goes two times, right. So it takes you back to that whole 
number. That is why we are putting it over 1. Now somebody come and do 
number. (As a student goes to the board he said) I think you must look at the 
differences when you are adding and when you are multiplying, alright? You 
are going to tell the differences. So be very observant as you are watching 
here. (yes boy) 
Phisa: 
 
 
 
×
 
× 1 = × = =  
  
 ×
	 
The product to the fractions they were multiplying were all proper fractions in their 
simplified form. He told learners to simplify the answers if they found that they were 
not in their simplified form.  
He performed fraction multiplications whose answer was an improper fraction. Here 
again he told learners what to do in order to change improper fractions to mixed 

fractions. When he finished demonstrating, he asked one learner to work out 3 ×	  

  		  

on the chalk board and this is how he worked it x =  =   = 2 .	 Some of the 

  
		  
learners were able to see that he made a mistake when changing it to a mixed 
fraction and raised their hands with an attempt to be given the chance to correct it. 
So this was what they did 
112 
Lebo 		  :  =  = 1  (repeats every thing she does) 

		  

She quickly realised the mistake and wrote the correct answer 1  

Towards the end of the lesson the teacher used the diagram taken from the text 
book and asked learners the following question  
 
Teacher A: How many shapes are there which are whole numbers? How 
many shapes are there that are whole numbers that are coloured? In the first 
example, look in the text books you’ll see what I’m talking about. How many 
coloured shapes are there?  
Lebo: Two coloured 
Teacher A: Where are they? Show me this is one this is two. (Still Lebo did 
not know what to say).  
Learner: There are 5 shapes that are coloured. 
Teacher A: There are 5 shapes that are coloured is that so? 
Learners: (Together) It is so 
Teacher A: So the whole number will be? The whole number will be…? 
Learners: (Together) The whole number will be five. 
Teacher A: The whole number will be five. So we are going to, we are going 
to multiply that by half, right? 
Learners: (Together) Yes sir.  
At the end of the lesson learners were given class work to do which required the 
application of the rule that the teacher had given them. In most cases learners did 
the work individually except in one instance in which the teacher encouraged 
learners to help each other understand the rule used to add fractions. This was what 
he said 
113 
So I want you to help each other in realising how we should treat the sides ... I 
want us to work in groups so that we help each other and every one, 
everybody is going to participate ok? 
When he gave class work the teacher used questions in the text book and he went 
round, from desk to desk checking whether all were doing the work. After making 
that round he would stand in front of the class and learners one by one would go to 
him so that he can mark their work. He was not able to mark the work of all learners, 
so he collected their books and marked them as the other teacher taught.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
114 
5.2.4 Teacher A’s lesson plan  
It was discussed in section 3.8 that there were documents that teachers kept that 
helped them organise their teaching. Teacher A’s lesson plans had the same format 
as indicated below.   
 
Figure 5.3 Teacher A’s lesson plan 
The structure of his lesson plans was the same for all lessons. The format was such 
that he wrote the class, date, duration and topic. He then listed the objectives, wrote 
the materials that would be used and the teaching methods. Under the procedure, he 
wrote the steps he would follow when teaching, together with the content he would 
be teaching. In the assessment he wrote a list of questions he would give to the 
learners at the end of the lesson.  
115 
As discussed in section 3.8.1 teacher A wrote short and measurable objectives that 
could be attained within the duration of the lesson. In the example lesson plan above 
there was one objective and the duration of the lesson was 40 minutes. He also 
wrote a list of materials he would use where applicable. Regarding the procedure, he 
wrote what he would do such as asking questions and/or demonstrating on the 
chalkboard, but did not indicate potential problems and/or misconceptions that 
learners were likely to have. He always concluded his lessons by giving learners an 
assessment task, in the form of closed ended questions which were done 
individually.  
The teacher did not keep the lesson plan book next to him when teaching. As a 
result he did not use the examples that he had written in the lesson plan but he used 
similar ones. On the plan he indicated that he would ask learners to identify the 
 
numerator and denominator in the fraction , in the actual teaching he wrote . 
 
5.2.5 Learners’ exercise books 
Learners divided each page into two columns, in this way the layout of their work 
was neat and easy to read.  
It was indicated in the discussion above that learners used procedures provided by 
the teacher when answering questions. They were not given a chance to come up 
with their own strategies. The teacher always provided them with procedures that 
they should follow when answering questions. The example below indicates what 
procedure they were given when subtracting fractions and all learners used it. 
116 
 
Figure 5.4 Learner’s working in teacher A’s class 
When learners had forgotten the procedure they did not know what to do. There was 
one incidence in which the learner multiplied like this 
 
Figure 5.5(a) Errors made by teacher A’s learners 
117 
When the teacher asked the learner to explain where one, the denominator of six, 
comes from, he said;  
It is this one (pointing to the denominator of three)  
Another learner had multiplied like this 
 
Figure 5.5 (b) Errors made by teacher A’s learners 
When the teacher asked where the eight comes from, she said: 
6 into 6 goes 1 and that 1 times 8 is 8. 
From the learner’s explanation one deduced that this learner had not fully 
conceptualised the concept of simplifying fractions and fraction multiplication. The 
learner was practising the procedure which she did not conceptualize. This could 
also be that the learner had fragmented learning because the teacher did not help 
learners understand and relate multiplication of fractions to multiplication of whole 
numbers. May be this difficulty arose from the fact that throughout the whole 
observation period, the teacher used was the part-whole representation only. In no 
single case did he engage learners in the other four subconstructs, namely; ratio, 
operator, quotient and measure (cf. 3.7.3). 
Other than these spotted difficulties those who were said to have understood were 
those who were able to repeat the procedures.  
During the lesson not all learners’ work were marked, so the teacher would normally 
collect the books he had not marked and then mark them as the other teacher taught 
118 
her subjects. In general the amount of work given to learners was little. During the 
lessons they were given not more than four questions, and for assignments, which 
were not always given, learners were given again not more than four questions. 
Learners were not given constructive feedback. The teacher assigned a tick and said 
good to learners who got the questions correct. To those who experienced 
difficulties, he would say things like, not quite or no we do not do it like that go and 
correct it.  
5.3 SCHOOL B 
5.3.1 School’s environment 
School B is situated about two kilometres from the researcher’s place of residence. 
The school is surrounded by a number of institutions, hence the socio-economic 
status of the learners’ families is diverse as most parents are employed in these 
institutions.  
Unlike in school A, school B has a fence around and villagers do not pass through 
the school grounds. There is a senior and junior block. The senior block is comprised 
of standard six and seven. There are three standard six and three standard seven 
classrooms.  
The observed two standard six teachers in this school, but did it in two successive 
years. The classes were both painted white though the paint was worn out and the 
walls appeared to have been exposed to rubbing off by hard objects like desks 
and/or chairs. There were no posters or pictures of any form on the walls.  
The first teacher the researcher observed in school B is referred to as teacher B and 
the second one as teacher C. There were 40 learners in teacher B’s class and 39 
learners in teacher C’s class. 
Both classrooms are of the area 5 by 3 meters. The floor map of each of these 
classrooms, the arrangement of both the teachers’ and learners’ desks and the 
position where the observer sat are shown below. 
 
 
119 
 
 
Teacher B’s classroom 
 
Figure 5.6 Floor map of teacher B’s classroom 
The large rectangles are the learners’ desks while the small rectangles represent the 
chairs. There are two teacher’s desks so the observer used the one which was in 
front of the class so that she could have a clear view of both the teacher and 
learners during the lesson. 
 
 
 
 
120 
 
 
Teacher C’s classroom 
 
Figure 5.7 Floor map of teacher C’s classroom 
Large rectangles represent the learners’ desks and small rectangles represent the 
learners’ chairs. The observer used the teacher’s table during observations. 
5.3.2 Teachers’ profiles  
Teacher B is a female teacher in her early forties and has been teaching for 21 
years. Her highest teaching qualification is a three year Diploma in Education 
Primary.  
Teacher C is a female teacher in her early fifties who has been teaching for 17 
years. Her highest qualification is also a three year Diploma in Education Primary. 
5.3.3 Teacher B  
5.3.3.1 Class observations 
On the first day of the observation, the mathematics lesson was at eight o’clock in 
the morning. The researcher arrived just after the morning assembly when learners 
were getting into their respective classrooms. The teacher greeted the researcher 
121 
and they entered the classroom before all learners could enter. The researcher had 
already been to the school before the observation in which she and the class teacher 
made arrangements as to how they were going to work. The researcher took her 
place as the learners entered the class. 
When they were all seated the teacher introduced the researcher and they greeted 
her enthusiastically.  
There were three class six classrooms and this one that the researcher observed 
was standard 6B. There were no repeaters this class. They were all doing standard 
five the previous year. 
Unlike teacher A, who taught fractions in eight lessons, teacher B taught fractions in 
five lessons. The duration was 80 minutes for two lessons and 40 minutes for the 
other three lessons. The sequence in which she taught fractions was sharing, 
denominators and numerators, comparing, ordering and equivalent fractions.  
Like teacher A all lessons were in the morning hours. The class activities were not 
very different from those of teacher A. Introductions to the lessons were the same for 
all lessons. She asked learners to state what was done in the previous lesson and 
then reviewed the content by asking questions like: 
Teacher B: Who can remind us what we have already done yesterday in 
fractions? Who can tell us what we have already done yesterday in fractions, 
(Calls a learner by his name) 
Lele: We have done numerators and denominators 
Teacher B: We have already done the numerators and the denominators. 
What did we say the numerators are and what did we say the denominators 
are? Who can tell us what numerators stand for and what denominators stand 
for? 
The teacher then linked the previous lesson to the present lesson. The first lesson 
that I observed was about sharing and its duration was 40 minutes. The teacher 
used fraction slices (ready-made fraction pieces) to demonstrate sharing.  Before 
that she told a short story about a woman who stayed with her children.  
122 
Teacher B: Ok, there was an old woman who was just staying in her hut with 
her children and you know what!  The lady was alone in the house so 
everything she had she shared. She used to share everything she had with 
her… children. 
She used two learners to show how they would share a fraction slab between two 
people. This was what transpired in this class. 
Teacher B: So in life, in real life, we are just going to see how we share 
things and in maths we also learn to… we are going to see how we… share… 
(Pause then takes out fraction slabs and asks one student to come to the front 
of the class) 
The teacher gives one student a square fraction slab and then asks  
 What do we have in here? 
Learners: (Together) One whole.  
Teacher B: We take this as one whole. Should any one come and hold this 
(fraction slabs representing 1 whole). Can I have two learners to come in 
front, so the other learner should try to share this (whole slab) into two people 
but we consider this, to share this whole number into 2 people. 


(The student took two  slabs and covered the whole slab with them, then he 

gave each of the half slabs to the students he is sharing to). 
Teacher B: So are you aware from this whole number how many parts do we 
have so far? 
Learners: (Together) 2 parts 
Teacher B: Let us see when we want to divide this one, suppose we want to 
share this half to two people. Two people to come and share this one and the 


other one to come and share that half (two more students came in front took  


 

slabs covers  slab with 2 of the   slabs. 
 
What can you see and what can you say about this exactly? 
123 
Seth: In half there are 2 quarters. 
Teacher B: In half we have got these two quarters meaning that in a half we 
have  
 
Learners: (Together) 2 quarters 
Teacher B: 2 quarters. May I have another 2 learners to come (they 
volunteered). So let us divide the second half [the second half] that is held by 
(calls her by name). 
Seth &: Takes 2 quarters and place them over the  slab 
'
 
 
Teacher B: How many parts are there? Before you go 
Seth: Two parts 
124 
Teacher B: Two parts. So let us just take this one half and put them here so 
that you could see how many parts are there in here  
How many parts do we have (calls him by name)?  
Lele: 4 parts 
 
Teacher B: Why do we say it’s a quarter? 
Tebo: (Shouts) we say it is a quarter because there are 4 parts in a whole. 
This whole has been divided into how many parts? 
Teacher B: We say it is a quarter because there are 4 parts in a whole. This 
whole has been divided into how many parts? 
Learners: (Together shout) four  
Teacher B: It has been shared to 4 … 
Learners: (Together shout) 4 parts 
Teacher B: 4 parts. This is why we are saying that it’s one quarter, it’s one 
over 4, and this one over four it’s a quarter. It has been divided into 4, ok? 
Learners: (Together) yes  
Teacher B: Thanks go back  
(Learners who were sharing went back to their seats) pauses 
Teacher B: Let us just go to our exercises and in your exercise books, just 
draw a whole number and then divide the whole number into 2 equal parts 
125 
then from that one you also divide the same whole number into 4 parts. You 
do it quickly. 
At the end of the lesson learners were asked to draw a rectangular bar three times. 
The first one was to be labelled one whole, and then the second bar was to be 
divided into two equal parts and the third into four equal parts. 
Teacher B: (Calls a learner by his name) Seth go to the chalkboard and show 
us what the teacher is saying. 
Seth drew the fraction wall from one whole to quarters. 
Teacher B: Let us look at what he is showing and you will see that you can do 
it within a very short time. 
When he finished Seth turned back trying to go back to his seat but the 
teacher stops him. 
Teacher B: Go and explain what you have done. 
Seth: These two parts give the whole of this fraction. When we divide this 
fraction into 2 parts is going to give a half and when you divide the half to 2 
equal parts again it’s going to be quarters and a half, a whole is equal to 2 
halves and 2 halves are equal to 4 quarters. 
Teacher B: So what do we mean by a fraction? We mean what? We mean 
parts. In order to make this whole we must have at least 2 parts and we call 
these 2 equal parts half ok? 
Learners together: Yes  
In the second lesson the teacher asked what was done in the previous lesson, 
having stated what they did the teacher then drew a rectangle, divided it into four 
equal parts and shade three of those. She then asked learners to name the fraction 
of the shaded part. Learners did not give the fraction straight away, they struggled a 
little. The teacher then became irritated and raised her voice as she repeats and 
directs learners to what the question wants. She straight away provided learners with 
the meaning of the denominator and numerator. 
126 
Teacher: Yesterday we have seen a whole then we divided a whole into two 
equal parts and then we also divided it into four equal parts. Therefore today 
let us try to… let us try to look at this diagram on the board 
 
By looking at this diagram on the chalkboard, who can tell us what fraction 
err… what part is shaded. What fraction is shaded on this diagram on the 
chalkboard?  
Nka: 3 is shaded 
Teacher B: Ah… 
Nka: 3 is shaded 
Teacher B: Who can correct him, who can add something on what he has 
already said?  
Tebo: Three quarters 
Teacher B: Three quarters, listen to the question, the question says what 
(raises voice) fraction is shaded, what fraction is shaded? Tebo. 
Teacher B: This numerator tells us about the parts which are shaded. What 
about the denominator? Who can say something about the denominator? If 
the numerator tells us about the number which are shaded what about the 
denominator? 
Tebo: The denominator tells us how many of them are there. The total parts 
Fractions were mainly represented by partitioning shapes and mainly rectangles. 
There was one instance though where she drew a circle and partitioned it unequally. 
This is the discourse that transpired  
Teacher B: The denominator is dealing with the total parts of the  
127 
Learners: (Together) Fraction 
Teacher B: Fraction. So from now you should know that the top numbers are 
called the numerators and the bottom numbers are called the denominators 
only in fractions.  
Let us have another diagram. 
 
Figure 5.8 (a) Faulty representations that Teacher B allowed 
Teacher B: How many parts are shaded in this diagram? How many parts are 
shaded, what fraction is shaded in the second diagram? 
You may even come closer and count, but don’t tell others. 
Lele: Four parts are shaded out of nine 
Teacher B: Is he correct? 
Learners: (Together) Yes madam 
Teacher B : 4 parts are shaded out of nine . When looking at these two 

numbers 4 and 9 which one is the numerator and which one is the 
denominator? 
Luke: 4 is the numerator and 9 is the denominator 
Teacher B: Ok good. Four is the numerator whereas nine is the denominator 
(writes the words denominator and numerator). So let me just quickly write 
these numbers for you so that you show me the numerators and the 
denominators. 
128 
The teacher concluded the lesson by asking learners to write a list of numerators 
and denominators from a set of fractions she wrote on the chalk board. 
& ( *
Writes on the board , ,  
' ) +
i.  write down all the numerators 
ii. Write down all the denominators  
The teacher marked this work in class. She went around from desk to desk. When 
she came across one girl, Thato, she found that she had exchanged the numerators 
with denominators and this is what transpired. 
Teacher B: (Shouting) You must have copied; you wrote them right here and 
because you copy, you erased your correct answers and wrote the wrong 
one. Stop copying from others. 
The third lesson was comparison of fractions. In this lesson she asked questions that 
reviewed the previous day’s lesson: 
Teacher B: Who can remind us what we have already done yesterday in 
fraction? Who can tell us what we have already done yesterday in fractions? 
Tumi! 
Tumi: We have done numerators and denominators. 
Teacher B: We have already done the numerators and denominators. What 
did we say the numerators are and what did we say the denominators are? 
Who can tell us what numerators stand for and what denominators stand for? 
Famo! 
Famo: Denominators mean the bottom number. 
Teacher B: But what do they mean? (The tone of her voice increases) Fine 
numerators are the top numbers. The top number we call it the numerator and 
the bottom number we call it the denominator but what does numerator tells 
us? Luke, what does numerator tells us? 
Luke: Numerator stands for the shaded parts. 
129 
After reviewing denominators and numerators the teacher stated what they were 
going to do today. 
Teacher B: For now we are just going to count or compare, who can just go 
to the fraction board and just show us where half should be, where is half? 
The teacher drew the big rectangle and then divided it into four rectangles. She then 
asked learners to indicate where one whole was, another one was to divide the 
second rectangle into halves, quarter and eighths.  
 
Figure 5.8 (b) Faulty representations that Teacher B allowed 
The teacher accepted the partitioning of quarters and eighths as they are 
demonstrated above. This is how the discussions went. 
Teacher B: Pointing to the first bar of the fraction board, what do we have 
here? 
Lele: Whole 
Teacher B: This one is a whole, she writes whole next to the bar 
What about here? Pointing to the second bar 
Famo: Half 
Teacher B: How many halves are there? 
Seth: Two 
Teacher B: Who can come up here and show us quarters on the fraction 
board 
One learner goes to the board and divides the third bar into 4 parts 
130 
 
Teacher B: What can you say about these? 
Lele: Madamm… they are not equal  
Teacher B: They are not equal? 
Lele: Yes madam. 
Teacher B: But let us talk about the number. Even though they are not equal 
are they not the quarters? 
Lele: Yes 
Teacher B: Well even though they are not exactly equal are they not the 
quarters? 
Learners: (Together) they are quarters. 
Teacher B: Uh… so in this part this part (pointing at the bar of quarter) has 
been divided into? 
Learners: (Together) 4 parts. 
The teacher asked learners to compare fractions using the fraction board which was 
not partitioned equally. She asked them to count the number of quarters that make 
up one half and the number of eighths that made three quarters. This was an oral 
exercise and those learners who answered the questions answered them correctly.  
After this oral exercise, the teacher gave an individual exercise in which learners 
were to complete the following statements to make them true using the fraction wall 
on the chalk board. The statements were 


 =  
	 


 =  
	 
  = 1 

131 
These are solutions from some of the learners 
 

 =  
	 
 

 =  
	 


  = 1 


 = 1 
	
 

This is what the teacher said to the learner who have written ( = ) 
	 
Teacher B: ‘Mantho Let us just come to the fraction board just count five 
eighths. 
‘Mantho: Counts 1, 2, 3, 4, 5 
Teacher B: Are they equal to eight? Where is half? Show me half, are these 
equal to half? 
‘Mantho: No madam [with a shaky voice as if she is afraid] 
Teacher B: So you said five eighths are equal to half? Mm… use the fraction 
board [throwing the students books on the floor. 
The teacher became frustrated and lost her temper. She shouted at learners and 
threw their books down. While this happened learners laughed at those whom the 
teacher was shouting at. This is how the teacher wrapped up the activity and the 
lesson. 
There are people who did not use the fraction board therefore they come up 
with the wrong answers. You may find that if ever you use this you could have 
got the answers correct. Someone said 3 eighths are equal to half, yet the 
fraction board is already there, people you must always think and use the 
teaching material which are already on the chalkboard. So we shall see each 
other tomorrow.  
The fourth lesson was ordering fractions. In this lesson again the teacher drew a 
fraction board and this time it was partitioned equally.  
132 
She directed the learners’ attention to half, one third and one quarter parts on the 
fraction board. She then asked which part they would prefer to have if those were the 
slice of bread.  
Teacher B: So if they are the slice of bread, which one would you like to take 
and why? Seli! 
Seli: Half, one half because it is bigger than one third. 
Some of the learners struggled to decide which slice they would prefer. One of the 
learners said he would take half because it is bigger than one whole. 
The teacher ran the chalk around one whole and around half and asked  
Is this half bigger than this one whole? 
The fraction board was extended to the eighths and the teacher asked learners to 
compare fractions orally. 
A written exercise was to order two sets of fractions starting with the smallest. The 
fractions were. 
 
  
1. , , ,  
   
 
  
2. , , ,  
	 	 	 	
 
When they finished they formed a queue in front of their teacher in order for her to 
mark their work. 
After marking most of the learners work she gave them an extra exercise. This was 

 
 

, , , 1 
 	 	
While learners tried to order the fractions, she took out fraction pieces and showed 
them to the whole class, each time she raised a piece she asked: What is this? And 
learners shouted the fraction. 
With these fraction pieces arranged on the teacher’s table she asked  
Teacher B: So which is the smallest?  
133 
Learners together: (Shouted) One sixth.  
Teacher B: Followed by? 
Learners: (All shouted) One fifth,  
The teacher listed the fractions as 

 
 

, , , 1 
	 	 
After this exercise the teacher concluded the lesson by asking learners to say 
something about the denominator. She then directed their attention to each of the 
slabs and named its denominator. 
Teacher B: Say something about the denominator. Is there anyone who can 
say something about the denominator? And tell us something about the 
denominator and the size of the fraction. Do you want to say something? 
Lele: When the denominator is big the fraction is small. 
The lesson that followed ordering fractions was equivalent fractions. As usual the 
teacher reviewed the previous lesson and then informed learners of what they are 
going to do today.  
Teacher B: When you just want to make the order be simple, you have to 
change and make the denominators the same, if the denominators are the 
same is now that you will be able to arrange the fractions in order of their 
sizes and the size will be easy for you. Do you get me? 
Learners: (Together) Yes madam 
The teacher straight away tells learners what to do 
Teacher B: If you can change the denominators and make the denominators 
to be the same ordering the fractions will be easier, ok? 
Learners: (Together) Yes madam 
Teacher B: Now who can tell me what is this? Lifts the fraction board 
Learners: (Together) Fraction board 
134 
Teacher B: (Raising fraction slices and asks) What is this?  
Learners: (Together) Fraction slice 
Teacher B: So for today we are just going to divide ourselves into 3 groups 
because we are just going to use these 2 fraction boards and fraction slices, 
therefore we are just going to divide ourselves into 3 groups. I think it should 
be groups of 8; we should be 8 in each group. Let us just be 8 in each group 
and one member to come and get the fraction board and fraction slices  
When you are 8 come and take the fraction boards and fraction slices. 
Learners formed groups but they were far more than eight in each group. In two 
groups they were 13 and in the other group they were 12. Some moved in front and 
took the fraction board and fraction slices. The learners made a lot of noise as they 
moved into groups. 
Teacher B: (Talking to one of the groups that have already taken the 
material): you are going to use this fraction board and you are just going to 
use this desk to place them, ok? 
Learners: (Together) Yes madam 
Teacher B: Just place them on a desk so that they end up making what you 
really see how these things are done. Now that you have fraction board don’t 
just take out the parts before, don’t just take out the parts before. 
Gave instructions and moved around to check whether they were following 
Try to show me the whole number, where is the whole number? 
Learners show a whole on the fraction board and fraction slices 
Teacher B: Are you sure this is a whole number? 
Learners: (Together) Yes madam 
Teacher B: Ok fine if this one is a whole number, where is the whole number 
in this? 
135 
The teacher continued going from one group to the next asking them to show 
 
fractions that were equal to one whole. She got answers like = 1, = 1 
 	
After finding fractions that were equivalent to one whole the teacher used the fraction 
pieces from the fraction stacks and asked: 
Teacher B: What does this red (pieces are coloured differently) stand for?  
Learners: (Together) One quarter. 
Teacher B: One quarter, so we say that these half and two quarters what do 
they mean? 
Learners: (Together) Half 
Teacher B: They are equal, you know what even though they are not exactly 
but the value is the same, ok?  
Learners: (Together) Yes madam 
Teacher B: The value of two quarters and a half is equal, even if when we 
write them they are not the same, we may say that half is equal to two 
quarters, ok?  
Learners: (Together) Yes madam  
Teacher B: What we are trying to say is that this half is equal to two quarters, 
so we are saying that half is equivalent to two quarters, ok? The value of half 
and this two quarters is the same, ok?  Mm… so I want you to take out your 
math books so that you see what I mean by saying, talking about equivalent 
fractions. Let us quickly go back to our seats [students take the board and 
slices in front, on the teacher’s table as they move to their seats]  
Teacher B: The equivalent fractions are equal in what? 
Learners: (Together) Value 
136 
Teacher B: In value, so to make fractions to be equivalent, suppose that we 

have may be this  so we want another equivalent fraction so we can multiply 
		
		 
this by two, so we are going to have   = therefore  
			 			
 
is equivalent to . Normally when we are just doing equivalent we do 
				 				
fractions in pairs, ok?  
Learners: (Together) Yes 
Teacher B: We do them in pairs three quarters is equivalent to six eighths, 
ok? 
Learners: (Together) Yes  
Teacher B: This is the same as when we are saying err… we are just doing 
this, we are trying to multiply this to help you to work out the fractions when 
the fraction boards are not available, you should also be able to … work out 
the equivalent fractions. Do you get me? Mm… so this is how we can get the 
equivalent fractions by multiplying both the numerators and denominators so 
 
when you say   is equivalent to , ok? 
				 				
Learners: (Together) yes 
Teacher B: Mm … so you write all the fractions in twelve fraction board at the 
moment I’m going to work out this one with you, the other ones you shall work 
it alone. 


Let us have;  =   who can just tell us how we work out this one? 
				
Lele: Madam it’s two eighths  
Teacher B: How did you get that? Come and show us on the chalkboard 
Lele: Madam I multiplied by 2; one times 2 is equal to two and four times two 

		 
is equal to eight =  
					 
Teacher B: So let us go back to the fraction board and try to prove what he is 
saying. Look at one quarter; do you all see one quarter? 
137 
Learners: (Together) (some say) Yes (some say) No 
Teacher B: So let’s go back to one eighth and see 
&
The teacher used the fraction slices to verify the answer. She lifted the  slice and 
)
' & 'the slice. She then put  slice over  slice and they exactly covered each other 
+ ) +
Teacher B: Is this one quarter equal to two eighths. 
Learners: (Together) Yes madam 
The lesson was ended with an exercise in which learners changed fractions to their 
equivalence. 
5.3.3.2 Teacher B’s lesson plan  
The lesson plans of teacher B that the researcher looked at all had the same format. 
As indicated in the interview which will be discussed in section 5.4. She stated the 
date, duration and the number of learners, topic, objectives, teaching materials, 
introduction, development and the assessment (3.8.1). The format was as indicated 
below. 
138 
 
Figure 5.9 Teacher B’s lesson plan 
Objectives were short, clear and measurable. She made a list of the teaching 
methods and teaching materials she intended to use. The teacher also wrote the 
procedure she intended to follow and the possible responses she expected to get 
from the learners. Regarding potential problems and misconceptions learners were 
likely to have, the teacher did not indicate them. She ended her lessons by giving 
learners closed-ended questions which required learners to remember and use 
procedures. 
139 
It was discussed in section 3.8.1 that teachers could follow a four stage lesson in 
which learners experimented, reflected and/or explained, hypothesised and verified 
their findings. There was no lesson in which teacher B employed this procedure. She 
also did not engage learners in high order thinking and/or problem solving.  
5.3.3.3 Learner’s exercise books 
Like teacher A’s learners, the learners drew a line in the middle of the page to divide 
the pages into two. In this way the layout of their work was neat and easy to read. As 
they answered questions, they wrote the steps going down the page. 
There was not much information that the researcher got from the learners’ exercise 
books regarding the conceptual learning. When learners answered the questions 
they used procedures that their teacher told them in class. None of the learners had 
done anything different from what the teacher did, except those who forgot the 
procedure and/or rule and then got everything wrong.  
This is how the learners worked out questions given as class work. 
 
Figure 5.10 Workings of learners i teacher B’s class 


In this case learners were given this statement, = , to complete. This was what 
 
one of the learners did  
 
140 
 
Figure 5.11 Errors in teacher B’s class 
As indicated in section 5.3.3.1, teacher B was not patient and hence did not assess 
formatively. She lost her temper and threw down learners’ books when she was 
supposed to give them feedback that would enhance understanding and learning. In 
this particular case, the teacher did not write any comment on the learner’s exercise 
book. This meant that this particular learner did not know why she got the question 
wrong. 
The teacher gave two or three close ended questions as an assignment. She often 
did not mark all the learners’ class work, but marked all of their assignments. 
5.3.4 Teacher C 
5.3.4.1 Class observations 
The first day of observation of teacher C the researcher arrived while the learners 
were still outside for a short break. The teacher let me in and we looked for a place 
to sit which was where the researcher would be able to see most of the learners. It 
was agreed that her table was the best option.  
There were 39 learners altogether in this class. They were all doing standard six for 
the first time. There were three standard six classes; these are 6A, 6B and 6C. All 
repeaters are in class 6C. The class that the researcher observed was standard 6B. 
When the learners arrived the researcher was already seated, so they looked at her 
and looked around as if they were wondering who she was. The teacher introduced 
me to the class and their faces loosened and they greeted me enthusiastically.  
All lessons were in the morning hours. On Mondays, Tuesdays and Fridays the 
lessons were at eight o’clock in the morning and they were 80 minutes long. On 
141 
Wednesdays and Thursdays they were 40 minutes long. On Wednesday the lesson 
was at eleven o’clock and Thursday it was at 0950 hours. 
Like the other two teachers, the teacher wrote the date and subject at the top of the 
chalk board while one of the learners was distributing textbooks to the rest of the 
learners.  
The teacher taught fractions in nine consecutive lessons. The first lesson was a 
review of fractions, such as, what is a fraction, distinguishing between denominators 
and numerators and so on.  
The teacher introduced the lesson by relating a short story about sharing. 
‘M’e ‘Malipuo has one loaf of bread. She wants to share the loaf equally 
among her 4 children. How much will each child get? 
Learners wrote the answers and the teacher marked ten learners who were the first 
to finish writing. After marking she drew a rectangle and divided it into four equal 
parts and provided this explanation 
This is ‘m’e Malipuo’s one loaf of bread. (She drew a rectangle on the board). 
This loaf has to be given to 4 children, so all you have to do is to see to it that 
the pieces that you give to each child are the same size. There shouldn’t be a 
child who gets a bigger share, shares or parts should be the same size. So 
this is one, one, one (as she writes on each part). She then writes each piece 


as … it has been cut into 4 parts. One child will get one out of 4, the other 1 

over 4 and so on and so forth.  
 
What do we call this type of numbers? What do we call this type of numbers 
where we have a number on top of another, what do we call this type of 
numbers?    
The teacher drew a square, divided it into four equal parts and shaded three of them. 
She asked learners to name the fraction of the shaded part. Learners gave the 
fraction as three quarters.  
142 
 
The teacher wrote the fraction on the chalkboard and told learners which were 
denominators and numerators 
Teacher C: 3 parts are shaded (writes it on the board). What i would like you 
to learn now is that the number of divisions in which a whole has been divided 
or shared is called a denominator. Denominator is the equal part of a whole 
and they are the ones at the bottom. And the ones that are the shaded ones, 
we said there are 3 shaded parts. The three shaded parts are called the  
Learners: (Together with the teacher said) Numerators 
The teacher gave other examples for learners to name the fractions they 
represented and to state numerators and denominators. 
Teacher C: We can also have the whole group, like you are a group of 
standard 6B, let’s say there are 40 of you and there are 19 boys, so the 


fraction of boys will be . From our book there are some oranges, the basket 

has some oranges. How many oranges are in the basket? Yes girl, Keny 
Keny: There are 12 oranges 
Teacher C: There are 12 oranges then how many oranges are shaded 
Lefa: 5 orange are shaded 
Teacher C : Which is the denominator when we have ? Yes Mary 


Mary: 12 
Teacher C: 12 is the denominator and what is the numerator? Yes boy! 
Bobo: 5 is the numerator 
143 
This oral exercise was followed with a short written exercise in which learners were 
to identify numerators and denominators of given fractions. These were 

 
 and  
 
In the same lesson fractions were compared. Like the other two teachers, she drew 
the fraction board and partitioned it up to the eighths. She asked learners to name 
the fraction of each strip as she pointed it. Answers were given in chorus. 
She asked how many quarters would give one whole when brought together. 
Learners answered in chorus. She directed the learners’ attention to the sizes of 
different fractions on the fraction chart and said: 
Looking at this it is smaller than this (pointing at the halves, quarter, thirds and 
so on) now what we have to do is try to compare them. We use this sign <, 
what sign is it? 
Learners responded together. She gave an oral exercise in which she asked 
learners to use the fraction board and put the correct sign between these fractions to 
make the statement true. 
1 1
∗  
2 3
She took out half and put one third over it and showed them to the learners and 
asked; what do you see? Learners together said half is greater than one third. She 
then called one learner to insert the symbol on the chalk board. 
Individual written work that she gave towards the end of the lesson was to insert 
   

either <, >or= in the place of * in order to make the following true: ∗  and ∗ . 
   
The second lesson was on equivalent fractions. The teacher began the lesson by 
reviewing denominators and comparing fractions by using inequality signs. 
After marking the first ten learners who had finished the review questions she asked 
    
 
learners to complete this statement by writing the correct sign.  ∗ , ∗  and ∗ . 
     
These were answered orally. 
144 
Teacher C: Now we are going to fill in any of the three signs to the different 
statements that we have. She writes  
 
a) ∗  
 
Nt’sala:  8 over 8 equals 4 over 4  
Teacher C:  Ee! Do you agree with him? 
Learners: (Together) Yes madam. 
Teacher C    : Writes = . Number b) (writes ∗ ). Yes boy, Lebo 
   
Lebo: 6 eighths is greater than 3 quarters 
Teacher C: Ee! Is he correct? 
Learners: (Together) No  
Teacher C: Sefiso is trying, yes 
Sefiso: 6 eighths equals 3 quarters 
Teacher C:     (Repeats and writes) ∗ , = . We are coming to verify this so that 
   
e! People who might have missed the point here can you understand. I hope you 
still have the fraction chart and you are using them. 

 
Then number c) ∗ , 
 
Lillo: 1 quarter is less than 3 eighths 
Teacher C: Is it less than 3 eighths? Mm! is that so? 
Learners: (Together) Yes.    
Teacher C: I want everybody to say yes or no. I want to be given answers by all 
of you because I have allowed you to use... what do you say class? 
Learners: (Together) 1 quarter is less than 3 eighths 
Teacher C 
 : Repeats as she writes <  
 
145 
   
The teacher then demonstrated on the fraction wall that 8 eighths  =  ,  =  and that  
   

 
< . 
 
Then she drew a fraction wall on the chalkboard, she made sure that the partitions 
were equal because she used the fraction pieces from the fraction chart. That is, 
when drawing a whole she used the piece which was a whole from the fraction 
pieces, for half she used the piece which has a half and so forth. 
Teacher c: How many halves are equal to 1 whole? Let’s observe the fraction 
wall. Lebo  
Lebo: 2 halves 
Teacher C: Aha! We have this 1 half and this 1 half. (Pointing at the halves on 
the fraction wall). These 2 halves are equal to 
Learners: (Together) 1 whole 
Teacher C: 
How many quarters are equal to half? Writes  =  as she repeats 3 
 
times. Yes girl! 
Puseletso: 2 quarters. 
Teacher C: 2 quarters. I don’t think we need more exercise, all we can do is to 
try on our own. 
Learners: (Together) Yes madam.  
Teacher C: Can we? Or are there some people who still have some difficulty? 
Learners: (Together) No madam. 
Teacher C: Do you see? You people. This halves are equal (writes = on the 
chalkboard) to 1 whole, pointing to 2 halves and then 1 whole, akere? 
Learners: (Together) Yes madam. 
Teacher C: Counts the halves 1, 2 is equal (writes = on the board) to 1 whole. 
Alright now let’s try to do this in our exercise books. 
146 
Having done this oral exercise she gave class work, with this she ended the lesson, 
in which they were to fill in the missing numbers in the given fractions 
1.  = 1 

2.  = 1 



3.  =  
 
In the third lesson they ordered fractions. She did this by engaging learners in the 
paper cutting exercise. 
On each desk the teacher put three pieces of paper cut in circular form and told them 
to imagine those papers as cakes. She instructed each group to cut one of the cakes 
into quarters, halves and eighths and gave them a hint on how to share the cakes 
You share the cakes in different fractions, the first one you make it in quarters, 
the second one in eighths and the third one in halves. 
 It is easy if you fold your cake (demonstrates by folding her paper such that the 2 
parts are equal). 
Having finished sharing all the three cakes, the teacher asked learners to count the 
number of divisions they have on each cake. When counting on the cake cut in to 
eighths this is how they did it.  
Learners: (Together) Count together 1, 2, 3, 4, 5, 6, 7, 8 
The teacher then asked learners to take out one piece from each cake and then 
arrange them in order so that they can compare their sizes.  
 
After this exercise learners were given a set of fractions to arrange in order of 
   

size starting with the smallest. The fractions were  , , ,  
   
147 
When learners were trying to settle and arrange the fractions the teacher directed 
their attention to the fraction slabs 
Teacher C: Looking at the column of eighths where is 8 eighths. Let’s have 
somebody to come and count for us. Yes girl! Puseletso 
Puseletso: (Counts the eighths) 1, 2, 3, 4, 5, 6, 7, 8 
Teacher C: These are 8 eighths a ke re? (Isn’t it?) 
Puseletso: Yes madam.  
Teacher C: What’s the other number? 5 eighths, let’s have another one to come 
and show us 5 eighths. 
Bobo: (Went to the board and counts the eighths) 1, 2, 3, 4, 5. Learners counted 
the 5 sixths 
Teacher C: Are they the 5 eighths? 
Learners: (Together) Yes  
Teacher C: Ee? 
Learners: (In chorus) No madam 
Teacher C: 5 eighths 
Sefiso: (Counted the eighths) 1, 2, 3, 4, 5 
Teacher C: Now from here to here (indicating on the fraction wall where 5 
eighths are) 
Then 1 eighth 
Lebo: Points where 1 eighth is 
Teacher C: Ok then 3 eighths 
Learners shouted “Madam” so that they can be given chance to go to the board and 
show where 3 eighths are.  
Nka: Counts 1, 2, 3 
148 
Teacher used different colours to indicate where each of the four fractions were. 
She then directed the learners’ attention to the marked fractions by saying as she 
points 
This is 3 eighths, 1 eighth, 5 eighths and 8 eighths 
Teacher C: Where do we start, from the biggest or from the smallest? 
Lefa: Smallest 
Teacher C: From the smallest 
Mary: Yes 
Teacher C: Ok somebody to come and write the one that is the smallest 
One boy volunteered and the teachers asked 
Teacher C: Are you not coming for the second time? 
Nt’sala: No. Went to the board and struggled to write. The teacher reminds him.  
The smallest, you write the smallest 
Learner: Writes 1 eighth 
Teacher C: What comes next? 
Molemo: Writes 3 eighths.  
He did not write it next to 1 eighth 
Teacher C: Do you see that you disturb the arrangement write it next to the first 
one. Next, yes boy! Thabiso. 
Thabiso: (Writes 5 eighths and 8 eighths) 
The final arrangement was; 

   
 , , ,  
   
Teacher C: Do you all agree that 1 eighth is the smallest? 
149 
Learners: (The whole class) Yes madam. 
Teacher C: Ok. With the help of what we have been doing here try to locate this 
err on our chart we do see that 1 eighth is smaller than other fractions and then 
comes one err 3 eighths, 5 eighths and lastly 8 eighths which is a whole a ke re? 
Learners: (Together) Yes madam. 
The teacher then gave class work.  
 
  
a) Arrange the fractions from the smallest to the greatest , , ,  
   

 
 
 

 b) Arrange from greatest to smallest , , ,  

   
Most learners were able to order fractions. The teacher wrapped up the lesson by 
demonstrating with fractions stacks that one fifth is the smallest and four fifths is the 
largest 
 
Figure 5.12 (a) Teacher C’s representations of ordering fractions 
In the fourth lesson the teacher wrote two sets of fractions for the learners to arrange 
and she marked the first 10 learners as usual. As she marked she made comments 
Teacher C: Yesterday you were able to order fractions, but now there are only 
two people who got these ones correct. Now if we have these eight, ten and five 
parts, (showing eighths, tenths and fifths on the fraction wall), really which is 
smaller 
Thabiso: 10 
Teacher C: Then? 
Bobo: Eight 
150 
Teacher C: And then 5. So do you see that this shows that you did not work with 
understanding yesterday, you were just excited. She wrote another set of 
fractions with different denominators and they also order it orally. The fractions 

  
were; , ,  

  
After ordering the above fractions correctly, some learners shout no, no, while others 
complained, Teacher you cheat me. When the teacher checks the fractions again 
she found that she had ordered them incorrectly and hence marked learners wrong 
yet they were correct. This is what she said; Okay thank you. So you see a teacher 
can also go wrong. Ok that is good of you. I nearly get you confused yet you 
understood what you were doing.  
The teacher then proceeded with the lesson of the day which was equivalent 
fractions. She drew a fraction wall on the chalkboard. She covered one half of the 
fraction wall and gave the following instruction 
Teacher C: You look through this half and wherever there is a line cutting 
through you should say the fractions. This is half, where do we find another line 
that goes straight from half, which is that other line? 
The lesson proceeded as follows 
Learners:  (Together) 2 quarters 
Teacher C: 2 quarters, which one else? Yes boy, Sello 
Sello: 4 eighths 
Teacher C: Counts 1, 2, 3, 4 eighths. Now try to do that in your text books. Cover 
1 half of your fraction table then say the fractions we have which are equal to 
half. Ok Mary! 
Mary: 1 sixth 
Teacher C: 1 sixth why are you skipping others? Yes another one. 
Mary: 2 quarters 
Teacher C: 2 quarters 
151 
Learners: (Together) 3 sixths, 4 eighths, 5 tenth 
Teacher C: Alright there is something that is a bit new today. You have half 
equals 
Learners did not wait for the teacher to finish they answered: 2 quarters 
Teacher C: Another half is equal to...  
Learners: (Together) 4 eighths 
Teacher C: Do you have another one in your head? 
Learners:  (Together) Half equal to 6 twelfths 
Teacher C: Ok thanks, you can go on and on to find fractions that are equal to 
half. So in mathematics when we have this kind of err fractions we say they are 
equivalent, that is, they are equal in value. Equi means equal. Let’s try to find a 
fraction that is equivalent to 1 quarter. Let’s try that from our err... fraction chart. 
Try to find the fraction that is equivalent to a quarter. That is you have to cover 
the fraction chart.  
Learners shout madam 
Teacher C: Yes  
Keny: 2 sixth 
Others shout no madam, madam 
Teacher C: Yes calling him by the name. Lefa! 
Lefa: 2 eighths 
Teacher C: Two?  
Lefa: Eighths 
Teacher C: Another one yes 
Learner: 4 twelfths 
152 
Teacher C: Alright we need to know how we can find this without the help of the 
fraction chart. We have to learn this. Let us suppose you are in the exam room 
and you are to write an equivalent fraction to a quarter or a half. You won’t have a 
fraction chart in front of you; all you have to do is to find a way out to find this.  
Instead of waiting for the learners to come up with their way of finding equivalent 
fractions without using the fraction wall, the teacher told them what to do. This is 
what she said;  
Teacher C: So you will have to find one number that you are going to multiply 
both the numerator and the denominator with.  
For example looking at this number we will say half times something (writes 

		×	
). This number should be the same. A ke re? (Isn’t it?) 
	×	
If it is 2 it is going to be 2, if you decide to make it 3, we will have 3 on both sides. 
So 1 times 2 
Learners: (Together) 2 
Teacher C: 2 times 2 
Learners:  (Together) 4  
Teacher C 
		×  : So you see = .. Half is equals to 2 quarters 
	×  
The lesson that followed this one was about simplifying fractions. This was the fifth 

lesson. After reviewing equivalent fractions, she asked, what is this number  ? 


(Writes  on the chalkboard) 

Learners: (Together) 2 quarter 
Teacher C: What?   
Learners:  (Together) 2 quarters 
Teacher C: Just think and think, how can you make this fraction the smallest or 
smaller than it is now yet it doesn’t change the value (pause) how can you make 
this (points at the fraction) 
153 
Learners: Divide 
Teacher C: Divide, do you agree with him? 
What followed after was that the teacher demonstrated how to divide the fraction by 
the same number to simplify it. The learners then simplified on the chalkboard and 
later into their exercise books. 
As the teacher marked she came across the situation where one learner did not fully 
	 
 	 
simplify. Instead of having = , one learner had written = . Another one had 

  
 


	 
written = . 
 

The teacher helped the first learner to identify that he had not fully simplified and 
asked him to think of a number that could divide both five and 10. The boy was able 
to find that number and further divided by five to get half. 
The second learner was not marked, and hence his misconception was left 
unattended. 
The sixth lesson was comparing fractions. She compared fractions in the first lesson 
when reviewing fractions, but this time she engaged learners in activity. She 
demonstrated with fraction slabs and drawings.  
The teacher asked, orally, learners to write each of these signs <,	 =	 or	 >	 to make 
each of the statement true 
 
1.  *  

	 

 
2.  *  
	 
Learners gave the correct answers 
 
1. >  

	 

 
2. <  
	 
 
The teacher then asked learners to explain how they got <  
	 
The responses were as follows;  
154 
Moshe: Madam 7 eighths is nearer to a whole 
Teacher C: What about if I add 1 eighths here (to 7 eighths) and I add a quarter 
 
here (to 3 quarters), wont they make a whole? So how do you get that < ? 
	 
Err? Talk to each other, I agree with you 3 quarters are less than 7 eighths. Yes 
boy, Malinga 
Malinga: Err... madam it’s because the eighths are smaller than the quarters and 
the eighths are close to being a whole there  
Teacher C: Laughs and repeats, the eighths are close to being a whole and add 
the 3 quarters are close to being a whole  
The teacher then drew this on the chalkboard in order to clarify what Malinga said. 
 
Figure 5.12 (b) Teacher C’s representations of ordering fractions 
After the teacher finished drawing this rectangle Malinga raised his hand and said: 
Madam when you times 3 quarters by 2 you get 6 eighths 
The teacher was very happy and asked everybody to clap for him and then said:  
Yes all we do is to make denominators the same, common denominators, that is 
the fact. Ke eona taba a ke re? (That is the fact, isn’t it?) 
		 
She then demonstrated how to make the denominators the same =	 . She 
			 
gave learners the chance to practise the procedure on the chalkboard and later in 
their exercise books as the evaluation of the lesson. 
The seventh lesson was on mixed numbers, the teacher drew this on the chalkboard 
 
155 
She asked learners whether they remember this. They struggled for some time but 
quickly they said 
Mixed number 
The teacher together with the learners together counted the parts and got seven. 
She then asked seven over what? 
Learners: (Together) four 
Teacher C:  Writes  and said: Somebody might not know why we say over 4 yet 

here we have only 3. That person who does not understand, should know that 
there was a part here, it is only that it has not been included. It might happen that 
we have all these parts and only these are shaded, so don’t think it is not there, 
right? 

From this point the teacher asked learners what type of fraction  is. The teacher 

then showed learners how to change mixed fractions to improper fractions and 
improper fractions to mixed fractions. This was how she proceeded 
 
×#
1 =  
 
As usual she gave learners two mixed fractions and asked them to change to 
improper fractions. These were 

3. 1  


4.  

Learners used the procedure their teacher has shown them. That is; 
 
×# 
1. 1 = =  
  

2. = 

 
156 
 

  = 6  
 
Teacher C taught addition and subtraction of mixed numbers in the last two lessons, 
lesson eight and nine. She drew two rectangles and partitioned them as shown 
below. 
 
 
She then asked learners what fraction of each rectangle is shaded and the teacher 
wrote the fractions below each rectangle. She also asked 
Teacher C: Now here how many parts are shaded in the 2 wholes? How many 
are shaded altogether? Yes boy! Mosa. 
The learner struggled but she probed him until he gave the correct answer. 
Mosa: 6 tenths. 
Teacher C: 6 tenths? E? How much is this whole? 
Malinga: 1 whole 
Teacher C: Ee? 
Puseletso: 1 whole 
Teacher C: Ee! How much is it? 
Lefa: 5 fifths 
Teacher C: and this one? 
Lefa: 5 fifths 
Teacher C: And how many fifths are shaded altogether? 
Nt’sala: 6 
157 
Teacher C : Do you agree with him? (She writes ). 

From this the teacher demonstrated how to add mixed fractions. This was what she 
did:  

  
  
1  + 2  = (1+2)  +  = 3  
    
The learners practiced the same procedure and towards the end of the lesson they 
were given two sums to perform in their exercise books. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
158 
5.3.4.2 Teacher C’s lesson plan 
Below is an example of a lesson plan of teacher C. 
 
Figure 5.13 Teacher C’s lesson plan 
Like the other two teachers, teacher C wrote short clear objectives, listed materials 
that would be used and wrote the procedure she intended to follow in the lesson. 
She did not use teaching materials in all lessons, so in that case she did not write 
them in her lesson plan. 
159 
There is no evidence, in her lesson plan, of predetermined problems, mistakes 
and/or difficulties learners were likely to make. In the lesson on simplifying fractions 

 
one learner simplified like this = . The teacher did not address this error, may be 
 

because she did not anticipate this error so she did not know how to address it. She 
also did not use a four stage lesson plan in any of her lessons. Learners were not 
given chance to work with concrete materials and then come to their own 
conclusions or discoveries. 
During the lesson, the teacher always had her lesson plan book around and referred 
to it whenever need arouse. The examples she used were those that are in her plan 
book even the class work questions she copied them from her book.  
It was true that she sometimes thought of fractions on the spot, but in most of her 
lessons she used what she had planned to use. 
The questions that she gave to learners were abstract and closed. She did not give 
questions that were extracted from the learners’ context. Learners were therefore not 
challenged and always used the rules and algorithms that the teacher gave in class.  
5.3.4.3 Learners’ exercise books 
Learners’ exercise books were also divided into two sections by drawing a vertical 
line in the middle of each page. In this regard learners’ work was neat and easy to 
read. 
The teacher even emphasised that learners should write the steps to a question 
vertically, such that the answer should be written at the end with an arrow pointing to 
the answer as this particular learner did in his working. 
The teacher gave closed ended questions similar to the ones she used in class. All 
learners were therefore obliged to use the procedures and algorithms that the 
teacher gave in class. No single student used a different method, may be because 
the questions did not call for high order thinking. 
An example of their working is shown below. 
160 
 
Figure 5.14 Workings of teacher C’s learners 
When marking the learners’ work, the teacher either wrote a tick or a cross. 
Sometimes for wrong answers she drew a circle around the mistake. She did not 
give feedback on what learners should have done. She often marked the first ten 
learners to finish the class work. The rest of the class would know whether they got 
them correct or not when the questions were done on the chalkboard. 
During each lesson the teacher gave introductory review questions and assessment 
tasks.  In both of these exercises, the teacher marked the first ten learners to finish 
the work. For both exercises the teacher gave learners chance to work the questions 
on the chalkboard. This is where learners whose work was not marked would see 
whether they got the questions correct or wrong. The last exercise that she gave was 
at the end of the lesson. In this exercise the teacher marked most of the learners’ 
work but often questions were not solved on the chalkboard. 
 
 
 
 
161 
5.3.5 Reflection on the observation and analysis of the documents 
5.3.5.1 Reflection on the observation of teachers A, B and C  
When teaching fractions all three teachers did not use concrete teaching materials 
that are found from the learners’ context. They used ready-made fraction boards and 
fraction charts. Even when using these ready-made materials, learners did not 
manipulate them; instead teachers demonstrated with them and drew 
representations on the chalkboard. Learners then answered questions referring to 
the fraction board drawn on the chalkboard. 
In the earlier lessons, teacher C made an attempt to elicit reasoning from the 
learners before giving them answers. She also tried to engage learners by giving 
them paper cutting exercise through which learners were able to order fractions in 
order of size. This experience provided learners with an opportunity to handle 
concrete materials visualise fraction parts and hence gain conceptual knowledge (cf. 
2.5.3.2). 
Teachers asked closed-ended questions in which learners did not need to think but 
recall what was done earlier. These were either oral or written. In the case of written 
work learners worked individually. These types of questions did not give learners the 
opportunity to explain or reason, hence no high order thinking skills such as inductive 
and deductive reasoning (cf. 3.4) were incorporated in these teachers’ classrooms. 
On few occasions teachers B and C asked learners to give reasons for their 
answers. 
It was discussed in chapter two that concepts should be related since learning is 
cumulative (cf. 2.5.2). Regarding learners’ prior knowledge teachers did not use it to 
its greatest advantage. The prior knowledge that teachers used was with regard to 
what was done in previous lessons. They would ask learners what they did in the 
previous lesson but did not use all the knowledge of mathematics that they knew. 
For example, when adding fractions, teachers A and C did not bring in addition of 
whole numbers and compare it with addition of fractions. Literature recommends that 
teachers use learners’ out of school mathematics so that learners could understand 
that mathematics is not abstract and meaningless (cf. 2.5). 
162 
It was mentioned, above that learners did not handle and manipulate materials, and 
that the questions given did not allow them to explore and experiment. Teachers 
therefore provided rules and procedures to use when working with fractions. The 
class room environment did not give learners the opportunity to construct their “own” 
rules and hence answer questions using their own strategies (cf. 2.3.1). When 
teaching about ordering of fractions, teacher C tried to help learners find the pattern 
by themselves. In as far as other concepts were concerned she also provided 
learners with rules and procedures to use when working with fractions. 
Teacher C taught equivalent fractions in two separate lessons, the second and the 
fourth lesson. The first time she used fraction board and diagrams to help learners 
identify that different fraction pieces were not equal. The second time, equivalent 
fractions were calculated. When asked why she did not teach in two consecutive 
lessons she indicated that she remembered that she did not teach how to find 
equivalent fractions by calculation.  
Learners experience different barriers to learning (cf. 2.6). One of the teachers’ tasks 
is to be aware of these barriers to effective learning so that they could help learners 
overcome them. The teachers did not seem to consider learners’ barriers because 
they did not engage different teaching methods in one lesson so that most of the 
learners’ learning styles could be catered for (2.6.2). Learners who learn best by 
doing were not catered for because teachers did not design their lessons in such a 
way that learners could handle and experiment with real objects. This approach 
could lead to learners experiencing very limited level of motivation and lack of 
experiential leaning (cf. 2.5.2.3 & 2.6.1). 
It is indicated above that learners’ prior knowledge was not used to its maximum, 
hence learners were not helped to identify relationships. This may result in 
fragmented learning as discussed in section 2.6.3. Fragmentation result when 
teachers do not relate fraction addition and subtraction to equivalent fractions. 
Inappropriate language may result in a learning difficulty; hence teachers are 
encouraged to use appropriate language. When building fraction concepts, all three 
teachers moved from context, pictures, symbols and language.   
163 
Teacher B talked of sharing a number. This might result in learners encountering 
difficulties as it was discussed that inappropriate language use could result in 
learning difficulties (cf. 2.6.4). When calculating equivalent fractions, teacher C 
referred to numerators and denominators as sides so this is a language issue that 
could result in learners experiencing difficulties. 
Incorporating assessment, when teaching, has been found to maximise teaching and 
learning (cf. 3.6). Through formative assessment, teachers engage in the diagnosis 
of learning, such as asking questions to determine understanding and/or difficulty 
and hence providing learners with feedback to assist learning. Teachers did not 
make an attempt to find causes of difficulties and errors that learners made. They 
either endorsed a cross next to the learners’ answer or just made a verbal comment 
that they were wrong.  
All three teachers did not give learners novel mathematics problems to challenge 
them. Learners were given questions through which they executed rules and 
procedures they learned in class. Learners were neither taught problem solving or 
were taught through problem solving as discussed in literature (cf. 3.5.2). 
Section 3.7.3 discussed the recommendations that teachers should follow in order to 
teach fractions effectively. It was suggested that the five subconstructs of fractions 
should be taught because this provides a holistic approach to fractions. All three 
teachers only taught the part-whole subconstruct. Teachers did not even incorporate 
the two aspects of part-whole subconstruct, which are, partitioning the whole object 
and/or a set of similar objects. Teachers mainly partitioned the whole object and 
engaged learners in finding fractions of each piece. As discussed this approach 
could result in fragmented knowledge of fractions and hence learners incur learning 
difficulties in higher mathematics (cf. 2.6.3).    
The three teachers are compared in table 5.1 below. 
 
 
 
 
164 
Table 5.1 Summary of class observations 
Effective Teacher A Teacher B Teacher C 
learning 
EL 
Manipulation None None Paper cutting 
of concrete once 
materials 
Prior No out of school No out of school Prior mathematics 
knowledge mathematics mathematics learned was used. 
used No out of school 
mathematics 
Conceptual Taught Taught Taught 
knowledge procedures. procedures. procedures. 
before No relationships No relationships No relationships 
procedural formed formed formed 
knowledge 
Learning Did not enhance Did not enhance Did not enhance 
environment discovery discovery discovery through 
 through through experimentation 
experimentation experimentation 
Group work None None  None  
Formative Very little Very little Very little 
assessment  
Handling of Not well done Not well done Not well done 
learning 
barriers 
Inductive Not done Very limited Very limited 
and 
deductive  
reasoning 
 
 
165 
5.3.5.2 Reflection on the lesson plans used by teachers A, B and C 
All three teachers wrote the topic, date and objectives in all their lesson plans but did 
not write teaching methods and teaching materials in all lesson plans. Regarding 
procedures to follow during the lesson, teachers wrote them clearly. They indicated 
the teacher’s and learners’ activities. For all three teachers under the teacher’s 
activities, they gave sample questions that they would ask learners. In some cases 
teachers wrote the answers they expected learners to give. 
None of the teachers wrote problems, errors and/or difficulties they expected 
learners to have. When such errors occurred, teachers were unable to help learners, 
hence did not give learners constructive feedback. 
Lessons were assessed by giving closed questions extracted from the prescribed 
text book. Such questions were not rooted in the learners’ context. 
In most of the lessons, teachers asked oral questions, demonstrated with ready-
made materials and/or one learner would write answers on the chalkboard. Learners 
were not engaged in the manipulation of concrete materials hence teachers did not 
use experimentation in their lessons.  
 
 
 
 
 
 
 
 
 
 
166 
Table 5.2 Summary of teachers’ lesson plans 
Contents  Teacher A Teacher B Teacher C 
Topic, date, objectives Done  Done  Done  
 
Teaching methods and Not always Not always  Not always 
teaching materials 
 
Procedure Clearly stated Clearly stated Clearly stated 
 
Potential problems and Not indicated Not indicated Not indicated 
pitfalls 
 
Methods of assessment Written Written Written 
answers to answers to answers to 
closed closed closed 
questions questions questions 
Experimentation  Not done Not done Done once in 
the series of 
lessons 
 
5.3.5.3 Reflection on the exercise books of learners in schools A, B and C 
Learners in all three classes partitioned the pages of their mathematics books in two 
and this made the presentation of their work neat and easy to read. When answering 
questions, they wrote the steps going down the page such that the final answer 
appeared at the bottom. Teacher C even asked learners to draw an arrow pointing to 
the final answer. She emphasised this because she did not mark any learners’ work 
that had not drawn an arrow.  
All three teachers did not employ formative assessment because on the learners’ 
exercise books, teachers did not write comments that would guide learners. They 
simply wrote a tick for correct answers or a cross for wrong answers.  
167 
The amount of work given to learners was also little because teacher A only gave 
assessment tasks and sometimes did not give assignments (cf. 5.2.5) while teacher 
B gave two or three assignment questions (cf. 5.3.3.3). Teacher C gave more work 
than the other two teachers because it was indicated that she gave learners three 
different sets of tasks in one lesson (cf. 5.3.4.3).   
All teachers gave learners rules and algorithms, so all learners used rules and 
algorithms given by the teachers when answering questions. Table 5.3 summarises 
the findings from learners’ exercise books. 
Table 5.3 Summary of learners’ exercise books 
Contents Teacher A Teacher B Teacher C 
Layout and • Page • Page • Page 
neatness divided into divided into divided into 
two  two  two  
• Writing • Writing • Writing 
steps going steps going steps going 
downwards downwards downwards 
• Neat  • Neat  • Neat  
Use of own Not  Not  Not  
strategies 
Use of algorithms Every time Every time Every time 
Marking  Marked all Did not mark all Did not mark all 
learners  learners learners 
    
 
5.3.6 Interviews with teachers  
It was indicated in chapter one and four that after conducting class observations the 
researcher would interview each teacher in order to understand each teacher’s view 
regarding the teaching of fractions. From the three interviews five themes were 
identified and they were: planning, effective teaching, effective learning, teaching 
materials and choice of teaching strategies.  
168 
Tables 5.4 (a), (b) (c) (d) and (e) below summarise the analyses of the interviews 
with the three teachers. 
5.3.6.1 Planning 
Planning is an essential element of teaching. It was discussed in chapter two that 
children learn differently at different stages of development, hence Piaget’s stages of 
cognitive development (cf. 2.3.1). It was discussed that teachers should know the 
learners context before teaching. It is in this regard that planning is an essential 
aspect of teaching.  
From interviews, teachers indicated that they planned before teaching. Teacher A 
said that while planning he decided on the teaching materials that would help him 
deliver the content he intended to teach. 
Teacher B on the other hand said that when planning she bore in mind the age of 
learners, their prior knowledge and then she wrote a lesson plan. 
Teacher C also considered learners’ age, prior knowledge, teaching materials and 
methods.  
Table 5.4(a) Planning 
Themes  Teacher A Teacher B Teacher C 
Planning I firstly look at I consider the The things that I 
the topic that age of the group consider are the; class, 
I want to treat I am planning for, topic, prerequisite 
and I would the content they knowledge that the 
go for those have. ...I write a learners in that class 
teaching aids lesson plan are supposed to have, 
which I think then the materials, and 
and believe the methods 
that they 
would err… 
reinforce the 
content 
 
169 
5.3.6.2 Effective learning 
Chapter two discussed features of effective learning which teachers should be aware 
of when teaching. The table below displays teachers’ views regarding effective 
learning. 
From interviews teacher A indicated that when teaching he considered learners’ prior 
knowledge as it helps build new knowledge on it and hence effective learning results.  
Teacher B on the other hand indicates that when learners are able to answer 
questions it indicated that learners have acquired effective learning. 
Teacher C considered using concrete materials to enhance effective learning. 
Table 5.4(b) Effective learning 
Theme  Teacher A Teacher B Teacher C 
Effective I first started If they are able to When you take an 
learning with the topic answer questions orange and share it 
which I think and when I give among say 3 
they were them an exercise children equally they 
familiar to. they get them right see that the pieces 
Their prior I move on to are equal 
knowledge that something else 
would help me because that 
build up on shows that they 
that knowledge have understood 
for this new 
one that I want 
to teach 
 
5.3.6.3 Effective teaching  
Teacher A said he ensured effective teaching by connecting learners’ prior 
knowledge to the new content that was being taught.  
170 
Teacher B on the other hand said that she emphasised rules because that was what 
mathematics is. 
Teacher C said that she engaged her learners in the partitioning of whole objects 
when teaching fractions. When the partitioning was done she then asked learners 
questions to ensure that they have understood what they were doing. 
Table 5.4(c) Effective teaching 
Theme  Teacher A Teacher B Teacher C 
Effective Normally Mathematics is full We only partition 
teaching emm… take of rules so I make wholes in class. I do 
them from sure that they are not use other 
what they able to use those methods I only give 
already know rules because they them questions 
to this thing need them when when I come across 
that I want to they write them in the text book 
introduce examinations 
 
5.3.6.4 Teaching materials 
Teacher A said he used materials that are easy to get. Teachers B and C said that 
they were aware that they should use concrete materials to help learners understand 
what is taught. But teacher B added that concrete materials like food items were not 
easy to get as they were expensive   
 
 
 
 
 
 
 
171 
Table 5.4(d) Teaching materials 
Theme  Teacher A Teacher B Teacher C 
Teaching I choose I know that they So I have learned that 
materials materials should use real working with tangible 
firstly by things like paper, things help them to 
looking at oranges, sweets understand 
how available and you name 
they might them. ...And if you 
be, I don’t buy them yourself 
look for those it becomes 
abstract and expensive 
are… hard to 
get 
 
5.3.6.5 Teaching strategies  
Teachers A and C did not say anything about how they decided on teaching 
strategies. Teacher B specified that she chose teaching strategies depending on the 
number of learners in the class, the topic and the availability of teaching materials 
that learners could use. 
Table 5.4(e) Teaching strategies 
Theme  Teacher A Teacher B Teacher C 
Choice of  I use teaching  
teaching strategies 
strategies depending on the 
topic, availability of 
teaching materials 
and the number of 
learners in the 
class. 
 
 
172 
5.4 INTERVIEWS WITH TEACHER TRAINERS 
5.4.1 Profiles of teacher trainers 
Lecturer 1 
Lecturer 1 is a male in the early thirties. He has been training mathematics teachers 
for six years. His highest qualification is Master of Education in mathematics.  
Lecturer 2  
Lecturer 2 is a female in the late forties. She has been training mathematics teachers 
for 15 years. Her highest academic qualification is Master of Education in curriculum 
studies. 
5.4.2 Interviews 
For both lecturers interview sessions were conducted in the morning in their offices. 
The interview session with lecturer 1 was scheduled at eight o’clock in the morning. 
With lecturer 2 it was on the same day at ten o’clock in the morning. 
The interview with lecturer 1 began at quarter past eight and it took about one and a 
half hours. With lecturer 2 we began at ten as we planned. 
The lecturers used different offices. During the interview sessions they were the only 
one present. I did not ask about the whereabouts of the other office occupants.  
I asked the lecturer to explain in detail the type of training they offered mathematics 
teachers especially those who would be teaching in primary schools. 
The lecturers indicated that the college offers a three year teachers’ diploma. 
Student teachers spend two years, first and third year in the college while they spend 
the second year in schools doing teaching practice. During the first year they are 
offered two courses, namely methodology and content.  
The analysis of the interviews of teacher trainers was done according to the themes 
identified from literature discussed in chapters two and three. 
 
 
173 
5.4.2.1 Methodology  
The lecturers indicated from the interviews that, in the methodology course, mostly 
done in first year, student teachers were taught the essentials of teaching such as: 
the psychology of child development and learning; designing and selection of 
appropriate teaching materials; different teaching methods; and designing lesson 
plans. 
5.4.2.1.1 Child development  
Effective teaching encompasses, among others, teachers’ knowledge of the 
children’s level of development. This knowledge helps teachers to decide on 
appropriate classroom discourse (cf. 2.3 & 2.5.2.2). 
Table 5.5 Child development  
Theme  Lecturer 1 Lecturer 2 
Child We train student teachers First of all I think the most 
development  about the child development. important thing that we expect our 
You know children learn student teachers to have is to 
differently at different stages have the knowledge of a child, as 
of development. So since child as in the um… psychology 
primary teachers are going to that surrounds the child as a 
work with children at different person, that is, how the child 
stages we train them in that develops so that they can 
regard. When they teach approach the children knowing 
fractions in class one and exactly what they expect out of 
class five or six, they should them, secondly when they have 
know what activities they learned about the child then we 
should give them depending expect them to know how the 
on what they are capable to child learns for example using 
learn. concrete materials, using pictures 
using all those things, we expect 
them to know those things and 
how to use them at different 
stages 
174 
 
Unlike Lecturer 1, Lecturer 2 explained that in standard 1 learners are supposed to 
manipulate concrete materials whereas in higher classes they begin to learn by 
using pictures and symbols.  
Lecturer 2: In standard one we expect our students to be using concrete 
materials most of the time, but when you go further you find that when you go to 
upper classes they learn by using pictures and then from there we can look at 
abstract symbols that we use in mathematics, we can go to symbolism now. 
5.4.2.1.2 Teaching materials  
Regarding teaching materials, both lecturers explained that they helped student 
teachers to develop teaching materials for different topics. This is what they said 
regarding teaching aids: 
 
 
 
 
 
 
 
 
 
 
 
 
 
175 
Table 5.6 Teaching materials 
Theme Lecturer 1 Lecturer 2 
Teaching ok one of the So when we have addressed the teaching 
materials things that we help methods which we think are more 
them do here is to effective in the teaching of mathematics 
construct teaching then we help them to because we 
materials understand that if the student can be able 
to understand the learning aids, the 
teaching aids then they are going to own 
the concept and it also makes 
mathematics easier to teach 
 
 
5.4.2.1.3 Lesson planning 
The lecturers further indicated that they teach student teachers how to write a lesson 
plan and the importance of having one before one enters the classroom.   
Table 5.7 Lesson planning 
Theme  Lecturer 1 Lecturer 2 
Lesson We train them on how to So if they have the teaching 
planning design lesson plans, to methods, they understand the child, 
do this they should know they have teaching aids, and then 
how to construct we plan our lesson, because it’s 
objectives, decide on through planning the lesson that you 
teaching materials and can deliver. That is what we have 
teaching methods that discussed. So we help them to make 
will help them achieve the lesson plan and to make sure 
the objectives that they really understand how a 
 lesson has to be taught we have 
We give them chance to what we call peer teaching, where 
do presentations in class. they present in the class. 
 
176 
5.4.2.1.4 Teaching methods 
Regarding teaching methods, both lecturers explained that they help student 
teachers to use teaching methods that help learners discover rules for themselves. 
The choice of appropriate teaching methods seems to correlate with the teachers’ 
PCK and MKT. This is because in section 3.3 it is mentioned that both PCK and 
MKT enable teachers to know and understand what makes learning of certain topics 
easy or difficult. This is what they said regarding teaching methods: 
Table 5.8 Teaching methods 
Theme Lecturer 1 Lecturer 2 
Teaching  We actually discourage We also have those teaching 
methods them from giving rules aids that we say if you use this 
because mathematics is one then you ask students 
not about rules; rules different questions based on 
only help them when this teaching aid then they will 
they have understood be able to say, at the end of the 
the concept then they day, oh… this is the formula for 
can use the algorithm finding this and if they have 
just for efficiency gone through that process of 
knowing exactly how we got the 
formula then they will not have 
the problem of forgetting and 
mixing things up. 
 
Lecturer 1 explained in detail how they train student teachers to teach fractions 
constructively. 
Lecturer 1: In the methodology part we have a section where we teach fractions. 
First we unpack the knowledge of what is a fraction, we start with a whole and 
then we start dissecting some pieces. Recently we have introduced something 
very interesting, the concept of slicers where we show them how to add fractions 
practically or by means of drawing, they can draw some sketches to add, subtract 
multiply and divide fractions. 
177 
Lecturer 1 raised his concern that, regardless of what they were taught, student 
teachers teach rules when they get to schools.   
Lecturer 2 explained that they teach teaching strategies appropriate for teaching 
mathematics. She said these are: discussions, discoveries and investigations. 
5.4.2.2 Content 
Section 3.3 indicated that teachers needed to possess different bodies of 
mathematical knowledge, CKM, SKM and MKT. So the content that student teachers 
were taught was both CKM and SKM because they needed both knowledge base as 
mathematics professionals and mathematics teachers. The teachers needed CKM 
not only for their day-to-day teaching but also when they furthered their studies. 
During peer teaching the lecturers have indicated that student teachers use the 
mathematics they used from high school. But they selected the content from the 
primary mathematic syllabus.  
Lecturer 2: We give them content that will help them teach effectively but which 
will also help them further their studies. In other words we don’t say, since you 
are going to teach primary students we are going to give you primary content 
ah… ah… (laughs) we go further and say even if you want to go further with your 
studies we have to give you enough content that will enable you to further your 
studies. 
Lecturer 1 did not say anything about the type of content they taught. He paid more 
attention on the methodology. 
5.4.3 Analysis 
Both lecturers have indicated that during the first year of training, student teachers 
are taught both content and methodology. By methodology they meant the child 
development in relation to learning, selection and design of teaching materials, types 
of teaching strategies, lesson preparation and actual teaching. They indicated that 
under the methodology they also unpacked the knowledge base of different topics 
such as fractions. 
178 
The lecturers have indicated that student teachers selected content from the primary 
school mathematics syllabus. Since most of the student teachers have not taught 
before, they worked in groups when preparing for lessons. They brought their lesson 
plans to the lecturers and they discussed it together. 
Then one of the group members would teach the lesson to the whole class after 
which the whole class would analyse the lesson.  
During year two, student teachers go to schools to practise teaching for the whole 
year. During this period, lecturers visit student teachers at their respective schools to 
observe them teach.    
Regarding the content, the lecturers indicated that the type of content they teach is 
not what the teachers will teach when they get to schools. This is the mathematics 
content that will help when they further their studies. This is CKM that will help 
teachers gain higher reasoning skills and participation in higher institutes of learning 
(cf. 3.3). 
Teachers A, B and C teaching strategies seem to accord with lecturer 1’s concern 
that teachers taught rules when they got to schools (cf. 5.2.7) regardless of how they 
were trained to teach. Again the teachers presented fractions by partitioning and by 
symbols as said by lecturer 2 that in lower classes learners must handle concrete 
materials but in higher classes teachers should teach by partitioning and symbols. 
5.5 CONCLUSION 
In this chapter, data collected from all the three schools was analysed and 
interpreted. The biographical data of each teacher, the context of each school and 
each class room were discussed. Class observations were discussed and compared 
in order to identify the similarities and differences in the teachers’ classroom 
activities. Class observations were then followed by interviewing teachers. As 
discussed in chapter four, this was done to get deep understanding of each teacher’s 
view on the teaching of fractions.  
The teachers’ lesson plans were evaluated in order to determine how teachers plan 
for their lessons and to see the type of questions they gave to the learners. Learners’ 
179 
mathematics exercise books were also analysed with the purpose of determining 
whether learners used their own strategies or rules when doing mathematics.   
Teacher trainers were also interviewed with the intention of determining the level of 
training that teachers received. These interviews were discussed and analysed. 
In the analysis the teachers’ actual teaching were compared with their responses to 
the interview questions, the learners’ workings to given questions, and the teacher 
trainers’ responses to interview questions. 
180 
CHAPTER 6   
FINDINGS, CONCLUSIONS AND RECOMMENDATIONS 
6.1 INTRODUCTION 
Chapter one introduced the study by presenting the problem, the purpose and the 
significance of the study. The research methodology and the procedure that were 
applied in the study were also outlined. Chapters two and three reported on the 
literature in which the aspects that determine effective learning and teaching of 
mathematics with emphasis on fractions were discussed. Chapter four then 
discussed in detail the empirical research. In chapter five interpretations of class 
observations and interviews, teachers’ lesson plans’ and learners’ mathematics 
exercise books were done. From this analysis and interpretation, the objectives of 
the research were met. In this chapter therefore the research findings regarding the 
research objectives will be discussed. These objectives are: 
• Determine what literature is saying on effective learning of fractions by 
learners 
• Determine what literature is saying on effective teaching of fractions 
• Determine the level of training given to mathematics teachers 
• Determine whether effective teaching is taking place 
• Determine whether effective learning of fractions materializes in selected 
classes 
• Recommended guidelines on how to teach fractions effectively 
 
 
 
 
 
181 
6.2 RESEARCH FINDINGS 
6.2.1 Effective learning of fractions  
The first and fifth objectives of this study are addressed in this section. 
6.2.1.1 Findings from literature 
From literature it is found that effective learning is a constructive, cumulative, goal 
oriented, collaborative, situated, self-regulated and individually different process of 
knowledge building through meaningful construction (cf. 2.5.2). Literature therefore 
indicates that, in order for teachers to enhance effective learning, especially of 
fractions, in this instance they should help learners manipulate concrete materials 
which learners use in their real life.  
Section 2.3.2 suggests that learners could be allowed to work in small groups as 
they experiment with concrete materials. In those cooperative groups learners 
should communicate among themselves. Through communication and 
experimentation learners may connect new knowledge to their prior knowledge and 
come up with their own strategies to solve problems. If learners manage to 
accommodate and assimilate new knowledge to the existing schema, they will 
develop conceptual knowledge and hence develop effective learning results (cf. 
2.3.1 & 2.5.3.1).  
Literature also depicts that learners who have conceptual knowledge are able to 
solve unfamiliar problems as they do not have fragmented knowledge (2.6.3). 
Fragmented knowledge is said to result when learners are not helped to establish 
relationships among different concepts such as, equivalent fractions and addition of 
fractions.   
It is also said that for effective learning to take place, the learning environments 
should permit learners to explore and work cooperatively with others (cf. 2.5.1). 
When learners explore they use most of their senses and in this regard effective 
learning is said to take place. If learners are not allowed to employ multiple senses 
when learning, this may result in a learning barrier as learners are individuals and 
learn differently.  
182 
Motivation is also found to be an important aspect of learning. Lack of or very low 
motivation may serve as a learning barrier (cf. 2.6.1). Learners who are motivated 
are able to learn even under difficult conditions but those with very little or no 
motivation give up when they face challenges. It is in this regard that learners’ 
motivation level should be kept high so that learning can occur. 
6.2.1.2 Findings from empirical research  
From the empirical research it was found that learners’ context and hence their out of 
school prior knowledge was not used when learning mathematics, fractions. Even 
when they did, they used the knowledge of mathematics that learners learned in 
school not the informal knowledge gained out of school.  
Learners memorised procedures and reproduced them when answering questions. 
They did not handle concrete materials, instead, teachers demonstrate with ready-
made teaching materials. The most dominant ready-made teaching materials in all 
three classrooms were fraction boards and fraction stacks. 
6.2.1.3 Conclusion 
The findings regarding effective learning indicated that learning environments in the 
three classrooms did not facilitate effective learning. Learners memorised 
procedures and rules and reproduced them when answering questions.  
The teaching and learning discourse that transpired in the classrooms may result in 
fragmented knowledge as learners were not exposed to experimentation and 
discovery of relationships and rules. 
6.2.2 Effective teaching of fractions 
This section presents findings regarding the second and fourth objectives:  
6.2.2.1 Findings from literature 
It was found from literature that effective teaching begins with thorough planning. 
Teachers should know, inter alia, the age group of the learners in class and their 
socio-economic status. This information helps teachers select appropriate content, 
teaching materials and activities that will facilitate teaching and learning.  
183 
It was found from literature that from age 11 years, learners are at the beginning of 
the concrete operational stage (cf. 2.3.1.3). Learners at this age and class have 
learned basic concepts of fractions. It was explained that for effective teaching of 
fractions, teachers should help learners gain conceptual knowledge before 
procedural knowledge. In order to do this, teachers are advised to create 
opportunities for learners to experiment by handling concrete materials. During 
experimentation, learners may come up with their own methods and hence gain 
conceptual knowledge. As learners experiment they may work in groups in which 
they share ideas and help each other. Through collaborative groups learners are 
said to gain essential skills like communications.  
Teachers are also encouraged to teach fractions through problem solving. It is said 
in literature that through problem solving learners can gain skills like critical thinking, 
communication and many more essential skills for a mathematician (cf.3.5). 
Teachers are again advised that during problem solving, teachers should ask 
learners a series of questions that will help them engage intensively in analysis and 
critically think of what they do (cf. 3.5.1). If teachers are able to create situations in 
which learners are able to solve and deduce solutions then effective teaching would 
have occur. 
Literature also depicts that for effective teaching to take place, teachers should 
engage in a continuous assessment of learning and provide learners with 
constructive feedback. That is teachers should engage in formative assessment (cf. 
3.6.1.1). Assessment for learning is said to enhance learning because learners are 
provided with useful information that helps them know what they should do. 
Embarking continuously on assessing learners, teachers may even identify learners’ 
difficulties and/or errors and hence help learners overcome them.  
Teachers are also encouraged to engage in multiple representations of fractions. 
That is they should help learners conceptualise fractions as a part-whole, quotient, 
ratio, measure and operator (cf. 3.7.3). Literature indicates that learners, who have a 
holistic understanding of the concept fraction, will not have problems handling 
fractions in different situations.  Then learners who are able to work with fractions in 
different contexts are said to have effectively learned the concept (cf. 2.5.3.1). 
184 
It is worth noting that literature explains that in order for teachers to be able to 
achieve all of the above, they should have two bodies of knowledge, PCK and MKT 
(cf. 3.3). Both PCK and MKT help teachers to plan their teaching and assess both 
teaching and learning (cf. 3.3). 
6.2.2.2 Findings from empirical research 
It seemed that the teachers had a grasp of the psychology of learning hence they all 
used concrete materials but not really relevant to the context of the learners. They 
did not though create learning environments in which learners handled and 
experimented with concrete materials. Instead teachers demonstrated with these 
materials and provided learners with rules.  
Teachers did not help learners develop conceptual knowledge. The cumulative and 
collaborative aspects of learning were not considered as teachers did not consider 
learners’ prior knowledge and they did not employ group work in their classes. 
Learners’ out of school knowledge of mathematics and real life concrete materials 
were not used. In this regard teachers did not give learners opportunities to solve 
mathematical problems. Learners did not engage in inductive and deductive 
reasoning. They were taught rules and procedures which they memorised and 
retrieved when needed, and were not given time to do mathematics their own way.   
Continuous formative assessment and more specifically constructive feedback was 
not used. This was evident because in their lesson plans teachers did not identify 
and/or specify difficulties that learners may experience so that they could prepare 
strategies to address them should they occur during their lessons. The importance of 
predetermining possible errors and difficulties was evidenced in teacher B’s lesson 
 

when she was not able to help a learner who made an error when simplifying 	to	  
	 
(cf. 5.3.3.3). So teachers did not handle learners’ difficulties and provide them with 
feedback that facilitated learning. 
All three teachers taught the part-whole sub-construct only. They did not represent 
fractions as measure, ratio, quotient and operator. This may result in fragmented 
learning and hence impede further learning of fractions or mathematics in general. 
 
185 
6.2.2.3 Conclusion  
Teachers who participated in this study seemed not to indulge in effective teaching. 
The type of learning environments they created in their respective classes did not 
support knowledge construction. They did not engross learners in tasks which 
demanded high order thinking, formation and discovery of relationships among 
mathematical concepts. Learners learned and practised rules and procedures used 
to operate fractions.  
Formative assessment, and specifically feedback, was used to the minimum 
because teachers did not try to identify difficulties that learners experienced. Even 
when they identified that certain learners experience difficulties they did not help 
such learners overcome them. 
The part-whole sub-construct was the dominant sub-construct in these three 
classrooms. The other four were not used hence effective teaching was very minimal 
in these classrooms. 
6.2.3 Level of training given to mathematics teachers 
In this section findings regarding the third objective are presented.  
6.2.3.1 Findings from literature 
It is found from literature that teachers should possess different bodies of knowledge. 
These are: the psychology of child development and learning (cf. 2.3) factors that 
determine effective learning and how to enhance effective learning (cf. Chapter 2) 
common and specialised knowledge of mathematics (cf. 3.3) MKT and PCK (cf. 3.3). 
All aspects of assessment, ranging from assessing learners’ socio-economic status 
of the learners to assessing learning (cf. 3.6). Designing and planning of the 
curriculum through scheming and lesson planning. 
6.2.3.2 Findings from empirical research 
From interviews with teacher trainers it was found that mathematics teachers are 
taught the psychology of child development and learning, designing teaching 
materials and lesson plans. Teacher trainers indicated that designing teaching 
materials help teachers to understand the structures of mathematics that are to be 
186 
delivered through those materials. This is what is called specialised knowledge of 
mathematics (cf. 3.3). Teachers are also taught common knowledge of mathematics. 
Teacher trainers also indicated that teachers are trained on how to plan for teaching. 
By this they referred to designing a lesson. 
6.2.3.3 Conclusion 
What teachers did in class seems to correlate with what teacher trainers indicated 
that they have taught. All three teachers wrote a lesson plan before going to class. 
The lesson plan of all three teachers contained the same contents. This is supportive 
to the information obtained from the teachers, namely that that they were trained at 
the same institution.  
When teaching fractions all three teachers used concrete materials, although not 
contextualised, indicating that they knew how learners learn at this age. 
Based on the conclusions of the first two objectives the effectiveness of the teacher 
training can be questioned. 
Teacher trainers did not say anything about teaching teachers to use assessment 
and how to incorporate assessment when teaching. It was evident in the classrooms 
that teachers did not have appropriate training on how to continuously assess 
learning through implementing formative assessment. 
6.3 RECOMMENDATIONS ON HOW TO TEACH FRACTIONS EFFECTIVELY 
The following are some of the recommendations that teachers may follow when 
teaching fractions in order to enhance effective learning: 
• Teachers should use readily available materials such as; paper, small stones 
and the learners themselves. They are readily available and teachers do not 
need money to have them 
• Learners must also work collaboratively in groups where they share ideas 
• Teachers should facilitate learning by helping learners identify relationships 
between and among different concepts 
187 
• When planning for a topic, teachers should identify misconceptions that 
learners may have, errors they may make and difficulties they may encounter 
and plan accordingly to address these barriers to learning 
• Teachers should do thorough lesson planning and address all aspects of 
teaching that will facilitate effective learning 
• Teachers should read research on mathematics learning and teaching so that 
they stay up to date with recent developments in the field 
• Teachers should have forums in which they discuss their teaching and help 
each other improve their teaching 
• The number of learners in class should be reduced to between 20 and 30. A 
smaller teacher learner ratio helps teachers identify possible barriers to 
learning such as different learning styles, cultural background and other 
aspects. Teachers can then adapt their teaching to ensure effective learning 
for all 
6.4 RESEARCHER’S CHALLENGES 
The researcher is a full time teacher so she faced a lot of challenges when 
conducting this study. These challenges were with regard to 
• Time: The researcher struggled to keep a balance between work, family and 
her studies. 
• Participation of teachers: Teachers did not like to be observed when teaching. 
The researcher wanted to include teachers from other schools as well but 
those teachers refused to participate in this study because they did not want 
to be observed. 
6.5 SUGGESTIONS FOR FUTURE RESEARCH 
The following areas on researching classroom teaching and learning need to be 
investigated: 
• The implementation of formative assessment to improve teaching and 
learning of mathematics 
188 
• How to engage learners in the manipulation of concrete materials that are 
contextualised so that they can construct their own knowledge 
• How to create powerful learning environments that will enhance critical 
thinking and discovery of procedures 
• How to identify barriers to learning and how to address them so that effective 
learning could be enhanced 
6.6 OVERALL CONCLUSION 
This study was aimed at understanding how mathematics teachers in Lesotho 
primary schools teach fractions. In order to execute this study, objectives were 
formulated as stated in chapter one. 
The research process as described in chapter one, was such that class observations 
and analysis of teachers’ lesson plans and learners’ exercise books were conducted. 
This was done to determine the type of contents the teachers included in their 
preparation and also to find out the type of questions they posed to learners. At the 
end of the whole class observations, the respective teachers were interviewed 
together with teacher trainers. 
The aim of the research was met as the researcher determined that teachers 
predominantly use lecture and demonstration methods when teaching fractions. It is 
also found that learners in these classes did not meaningfully learn fractions. This 
was because of insufficient manipulation of concrete objects by learners and also 
that learners are not helped to connect concepts and formulate relationships and 
generalisations.  
  
189 
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202 
APPENDICES 
 
Appendix A: OBSERVATION SHEET 
Date ____________________________                      
Class___________________________ 
Topic ___________________________                   
 Duration ________________________ 
 
 
Behaviours  How behaviours Other behaviours 
are demonstrated  
1. Teaching strategies:   
 
 
 
2. Nature of problems:    
• recall questions  
• Application questions 
 
3.  Individual,  Group work or   
whole class discussions 
 
4. Are rules given and practiced?   
 
 
5. Do learners manipulate   
concrete materials to develop 
conceptual knowledge before 
they are given rules 
203 
6. Are students given   
opportunities to communicate 
their understanding with their 
classmates 
7. Identifying learners’   
misconceptions and difficulties 
8. Addressing and rectifying   
learners’ difficulties and 
misconceptions  
 
9. Alerting learners of mistakes   
they are likely to make  
 
10. Use of concrete materials   
 
 
 
 
 
 
 
 
 
 
 
 
 
 
204 
Appendix B: Interview questions for class teachers 
1. When planning to teach a topic, which things do you consider, that will help 
enhance effective learning and teaching of that particular topic? 
2. Can you also explain what influences you when you decide on the teaching 
strategy?  
3. Which teaching strategy do you often use and why? 
4. In your own opinion does it work for you?  
5. What makes you think so? 
6. When teaching the first lesson on fraction what did the learners know about 
fraction?  
7. How do you perceive the importance of prior knowledge in your teaching? 
8.  What is your understanding of a mathematics problem? 
9.  How do you use problem solving as a teaching strategy? 
10.  How do you think problem solving as a teaching strategy enhances effective 
learning of mathematics?  
 
 
 
 
 
 
 
 
 
 
 
 
205 
Appendix C: Interview question for teacher trainers 
Can you please tell me what knowledge base you expect teachers to have at 
the end of training that will help them teach mathematics effectively? 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
206 
Appendix D: Application letter to the Ministry of Education 
 
P.O. Box368 
Morija 190 
16 May 2011 
The Principal Secretary 
Ministry of Education and Training 
Maseru 100     
Dear Sir 
RE: Application for conducting research in Morija Primary Schools in the 
district of Maseru 
I wish to apply for permission to conduct research in primary schools in Morija.  
The title of my research dissertation is:  
Teaching and learning of Fractions in schools in Maseru 
The research is part of my studies I am doing with the University of Free State. 
The research is intended to inform myself and other teachers and empower us in 
our teaching career. 
The study will not interfere with the normal running of the school and all the 
ethical considerations, such as confidentiality of the results and findings of the 
study and the protection of participants’ identities will be observed.  
I am thanking you in advance for your attention. 
Yours sincerely 
      ‘Maphole Marake 
 
207 
Appendix E: Permission letter from the Ministry  
 
 
208 
Appendix F: Application letter to the schools 
P.O. Box 368 
Morija 190 
01 August 2011 
The Principal 
Dear Madam 
RE: Permission to do research at your school 
I am a registered student for the Masters’ degree at the University of the Free 
State. As part of my degree, I need permission to conduct a research project at 
your school. 
The title of my research is 
Teaching and learning of Fractions in primary schools in Maseru 
Three of my research aims are to:  
Reflect on the teaching strategies predominantly used by teachers.  
Reflect on what teachers do to elucidate that learners have meaningfully learned 
fractions. 
Reflect on the students’ explanations to the strategies they use to solve problems 
regarding fraction. 
I am therefore asking for permission to observe all the lessons on fractions in 
standard six. 
The study will not interfere with the normal running of the school and all the 
ethical considerations, such as confidentiality of the results and findings of the 
study and the protection of participants’ identities will be observed.  
Attached is a copy of the letter of approval from the Ministry of Education and 
Training. 
209 
I am thanking you for your attention. 
Yours faithfully, 
‘Maphole Marake (Mrs) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
210 
Appendix G: Letter to teachers 
 
P.O. Box 368 
Morija 190 
01 August 2011 
 
 
Dear teacher 
I am a registered student for the Masters’ degree at the University of the Free 
State. As part of my degree, I am cordially asking for your permission to observe 
your lessons on fractions.  
The title of my research is: 
Teaching and learning of Fractions in primary schools in Maseru 
 
I am thanking you in advance for your participation. 
 
Yours faithfully 
 
‘Maphole Marake (Mrs)                                               Teacher’s signature 
          
………………………………………………… 
 
 
 
211 
Appendix H: Application letter to interview lecturers at LCE  
P.O. Box 368 
Morija 190 
12 August 2011  
 
The Director  
Lesotho college of Education 
Maseru 100 
Dear Sir 
RE: Application to interview mathematics teacher trainers 
I am a registered Masters student at the University of Free State and I wish to 
apply for permission to conduct research in your institution. The title of my 
dissertation is  
Teaching and learning of Fractions in primary schools in Maseru 
One of my research aims is to determine the level of training given to teachers to 
enable them to teach mathematics effectively. I am therefore asking for 
permission to interview two lecturers from your institution. 
The ethical considerations, such as confidentiality of the results and findings of 
the study and the protection of participants’ identities will be observed.  
Attached is a copy of the letter from my Supervisor. 
I am thanking you for your attention. 
Yours faithfully, 
 
‘Maphole Marake (Mrs) 
212 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
213