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Additional comments on Ben’s teaching

These comments are meant to give an overview of how Ben teaches algebra with an eye to identifying how he creates OTL. The literature reviewed pointed to the fact that OTL is concerned with the conditions under which learners have to learn and is positively associated with achievement. In this study OTL is considered in the context of the learning of algebra at grade ten level. OTL is determined largely by the way the teacher instructs the learners because what is learnt depends on what is taught (Kilpatrick et al., 2001). The way in which OTL are generated in the classroom differs from class to class. In this section I consider the way Ben affords his learners
the opportunity to learn grade ten algebra by considering his choices of registers of representations. I will use the examples described above to highlight the way Ben attempts to give his learners OTL.
Mathematical knowledge is constructed in the mind of the individual. Students’ construction of mathematical knowledge is greatly influenced by the experiences they gain through interaction with the teacher (Cobb and Steffe, 1983). Teachers decide upon the strategies to engage students. They create the opportunities for students to learn the knowledge and skills required in society and also in order to pass examinations so that they have something to show for their effort.
In his teaching Ben emphasises procedures. All teachers when they attend to a class of learners want to give them an opportunity to learn, each in his or her own way. For Ben learning is accomplished when learners complete given sentences using short phrases or single words. He starts the sentence spelling out a procedure and the learners complete it. For example he says, ‘Here we are supposed to make ‘h’ the subject of the what? The learners respond, ‘Formula.’ Most of the questions were of this nature. Questions should be constructed to stimulate different forms of thinking.
Ben’s questions did not stimulate high order thinking and problem solving. A teacher’s questions can shape a learner’s learning as the type of questions asked place emphasis on the process strands that are valued in the learning of mathematics (Posamentier and Jaye, 2006).
Communication with his learners was largely one sided, the teacher did most of the talking and all of the chalkboard working. Ben followed the examples as set in the Classroom Mathematics textbook. He worked out the examples in order of difficulty as arranged in the text book. However there are dangers associated with textbook driven instruction in that the questions might not apply to the context familiar to the learners. Learners naturally relate favourably to questions that mean something to them than to those that are far removed from their own experiences. Ben always repeated the answers that the learners gave him.

Classroom C: Mt Carmel Catholic High School

This section of the report describes the second case of this study. It reports on lessons observed at Mt Carmel Catholic all girls’ school. Mt Carmel is about eight minutes drive from the city centre. I observed eight mathematics lessons there.
The mathematics classroom is on the first floor. Adjacent to it is a small mathematics room where the mathematics teachers meet to share ideas or to wait to enter their classroom. It is in this room that we would talk about the lessons before and after they had taken place. As you enter the classroom you face the teacher’s table which is aligned with the front desks of the learners.
There are thirty-two learners desks arranged in a column leaving a space allowing passage in any direction. There were seventeen learners doing grade ten mathematics. The empty places were not arranged in any specific order because the occupants of those seats, who are also in grade ten, moved to another venue because they do mathematical literacy.

READ CONCEPTIONS OF EXCELLENT TEACHING AND SELFREPORTED TEACHING PRACTICES IN CHINESE MIDDLE SCHOOLS

Chapter One: Background to the study
1.1 Introduction
1.2 Motivation
1.3 Background
1.4 Statement of the problem
1.5 Research Questions
1.6 Objectives of the study:
1.7 Delimitations of the study
1.8 Limitations of the study
1.9 Feasibility of the Study
1.10 Organization of the study
1.11 Conclusion
Chapter Two: Literature Review
2.1 Introduction
2.2 The Curriculum in South Africa
2.3 Education Structures in South Africa
2.4 Constructivism
2.5 Opportunity to learn
2.5.1 The History of OTL
2.5.2 Applications of OTL
2.5.2.1 OTL as a standard to measure school effectiveness
2.5.2.2 OTL as a research concept
2.6 How to study OTL
2.6.1 Some example of OTL Studies
2.7 The Teaching and learning of mathematics.
2.7.1 Learning and teaching of school algebra
2.8 OTL Conceptual Framework
2.9 The Theoretical Framework
2.10 Conclusion
Chapter Three Methodology
3.1 Introduction
3.2 Review of Research Questions
3.3 Research Design
3.3.1 Qualitative Research
3.3.2 The Design –Case Study
3.4 Qualitative Methods of Data Collection
3.4.1 The Interview
3.4.2 Observation
3.5 Procedures
3.6 Data Analysis
3.6.1 Observation Data
3.6.2 Interview Data
3.7 Reliability and Validity
3.7.1 Reliability
3.7.2 Validity
3.8 Research Ethics
3.9 Challenges
3.10 Conclusion
Chapter Four Data Presentation, Analysis And
Interpretation
4.1 Introduction
4.2 Reports On The Classroom Observations
4.2 CASE 1:
4.2.1 St Bernard Catholic High School.
4.2.2 The teacher (Ben)
4.2.3 Classroom B
4.2.4 Lesson 1
4.2.5 Lesson 3
4.2.6 Additional Comments On Ben’s Teaching
4.3 Case 2
4.3.1 Classroom C: Mt Carmel Catholic High School
4.3.2 The teacher: Cherry
4.3.3 The lesson
4.3.4 Topic of Lesson: Making y or any selected variable the subject of the
equation
4.3.5 Comments on Cherry’s teaching
4.4 Case 3
4.4.1.St Anne Catholic High School
4.4.2 The teacher: Ann
4.4.3 The first Lesson/ generalising from number patterns
4.4.4.St Anne High School / Lesson 2
4.4.5 The lesson
4.4.6 Comments on Ann’s teaching.
4.5 Conclusion
Chapter Five :Data Analysi
5.1 Introduction
5.2 Commonalities
5.3. How The Teacher Provided Their Learners With Opportunities To Learn
Algebra.
5.3.1 Ann’s teaching approach.
5.3.1.1 Ann’s posing of questions
5.3.1.2 Uses of terminology
5.3.1.3 The types of tasks
5.3.1.4 How Ann used different registers of representation
5.3.2 Ben’s teaching approach
5.3.2.1 Ben’s posing of questions
5.3.2.2 Uses of terminology
5.3.2.3 Types of tasks given by Ben
5.3.2.4 Use of different registers of representation
5.3.3 Cherry’s approach to teaching
5.3.3.1Cherry’s posing of questions
5.3.3.2 Uses of terminology
5.3.3.3 Type of tasks
5.3.3.4 Use of different registers of representation
5.4 Conclusion
Chapter Six General: General Discussion, Synthesis
And Conclusio
6.1 Introductio
6.2 Defining Opportunities to Learn
6.2.1 Assessing content coverage
6.2.2 Assessing content exposure
6.2.3 Assessing content emphasis
6.2.4 Assessing the quality of instructional delivery
6.3 Synthesis
6.3.1 Considering formation
6.3.2 Considering treatmen
6.3.3 Considering conversion
6.4 Conclusions
6.5 Recommendations
6.6 General Conclusion