85 | P a g e
Table 5.3: Final themes generated from interview data FINAL THEMES Theme 1: Usefulness of teachers’ content knowledge Theme 2: Teachers’ conception of remedial teaching Theme 3: Strategies for dealing with learners’ errors
86 | P a g e
Figure 5.3 Sample of learner H’s work in the class activity
5.4.2 Observation of Anele’s lesson
Anele started the lesson by first writing the topic, which was algebraic equations, on the chalk board. The teacher then used questions and answers to discuss relevant previous knowledge that the learners had on the topic; for example, he used a question like “If you subtract your age from your mother’s age, how old was your mother when she gave birth to you?” to check learners’ previous knowledge which was relevant to the topic of algebraic equations. The teacher gave learners a few minutes to give their answers to the question. He then started to write some algebraic equations on the board. He first worked out some on the board and explained to learners how to solve algebraic equations, and later gave the learners an opportunity to ask questions. Figure 5.4 shows an example of an algebraic equation the teacher worked out on the chalk board.
Activity 1.1.4
Solve for x in the equation: 2𝑥𝑥 − 3(3 + 𝑥𝑥) = 5𝑥𝑥 + 9 Solution:
2𝑥𝑥 − 3(3 + 𝑥𝑥) = 5𝑥𝑥 + 9 2𝑥𝑥 − 9 − 3𝑥𝑥 = 5𝑥𝑥 + 9
2𝑥𝑥 − 3𝑥𝑥 − 5𝑥𝑥 = 9 + 9 −6𝑥𝑥 = 18
−6x−6 = −186 x = -3
Figure 5.4 Example of Anele’s lesson on algebraic equation in class
87 | P a g e
After explaining the activity in Figure 5.4, he then gave a learner exercise which was marked in class for learners to do their corrections.
Anele later introduced to learners the topic of exponential equations. He further reminded learners that they were still doing algebraic equations, but this time they were algebraic equations involving exponents. He gave the learners a worksheet which contained the rules or laws of exponents, which learners had already learned in the previous term. The teacher then told learners that the rules they had learnt in the previous lessons would be applied in the new topic which is algebraic equations involving exponents. Anele then wrote three questions on the board and asked learners to do them on their own. After few minutes he requested three learners to present their answers on the board. As the learners presented their work on the board and mistakes were identified, he then started engaging with the errors with the whole class. In some cases, Anele asked the learners to explain to the class how they arrived at the correct answers.
5.4.3 Analysis of data from lesson observation
Data from classroom observation were deductively analysed using the themes generated from the framework as shown and discussed in Chapter Three. The researcher consulted literature and the framework to develop the initial strategies which teachers employ in analysing learners’
errors by identifying 1) the forms of error analysis, and 2) the strands of teachers’ mathematical knowledge. To ensure alignment with the framework, participant practices were categorised using the strands of mathematical knowledge (Ball et al., 2008), and categorised for analysing learners’ errors as mentioned by Sapire et al. (2014).
Table 5.4: Generating codes and themes from observation data.
Formulation of categories Theme Rationale
Recognition of mistakes or errors in the learners’ steps.
For example, Zafira recognised mistakes made by the learners.
Anele worked on solutions with learners
Teacher knowledge and using procedural knowledge to address errors
To identify the types of error strategies teachers’ use
Using explanation to address the error. For example, Zafira highlighted errors made and provided explanation
Knowledge teachers need to teach mathematics and understanding of conceptual errors
To establish the kind of mathematical knowledge teachers have
88 | P a g e
Drawing from one’s
knowledge to address the error.
For example, Anele used his mathematical knowledge to identify learners’ errors
Knowledge of students and the content in relation to teachers’
awareness of commonly identified learners’ errors
To establish the knowledge teachers have to identify learners’ errors
Table 5.4 above shows how codes were generated from the lesson observations and how these were formulated into categories. Themes were then generated from the categories to help the researcher to answer the research questions using classroom observation. The rationale from the themes which were created support the already existing themes from the framework of the study.
Sapire et al. (2014) developed six criteria for error analysis, as follows: conceptual understanding, procedural understanding, awareness of errors, multiple explanation of errors, use of everyday knowledge, and diagnostic reasoning. However, the data from this study could be grouped into three, that is, teacher knowledge and procedural knowledge, teacher knowledge and understanding of conceptual errors, and awareness of errors. For effective engagement of learners’ errors, it is important to note that teachers are supposed to have their mathematical knowledge in place in order to be able to deal with learners’ errors. According to Ball et al.
(2008) mathematical knowledge is the knowledge that teachers need in teaching mathematics.
Shulman (1986), as cited in Ball et al. (2008), argued that knowing a subject for teaching requires more than facts and concepts. In other words, mathematics teachers must understand the principles and structures as well as the rules for establishing what is legitimate to do and say in the classroom.