Formas de operación del SPAV-UN para la identificación del sistema

Themba: How did your lesson go? Alfred : Well, I can say it went well.

Themba: What do you think you have achieved by presenting this lesson?

Alfred: Now learners know types of linear graphs and different methods of determining it.

Themba: Your lesson focused on the content of mathematics with no reference to

real life. Did you have similar lessons in the past?

Alfred: Yes I did. You need content knowledge before the contexts but not at the large extent.

Themba: Is this the way you approach the subject?

Alfred : Yes. I teach them content and relate it to the real life situations Themba: So, in the examinations you set mathematics content as well?

Alfred : Most assessment tasks are based on application of content in real life. I have a very limited content knowledge in assessment.

Themba: How did you see the participation from your learners?

Alfred: They always participate. I always create an environment for flow of communication.

When I present the analysis of lessons I will refer to this excerpt.

180 Having presented the description of the two lessons, and Alfred‟s reflections, I now present the analysis of these two lessons. Here again, I will pay more attention to the key aspects identified in the introduction of this chapter. These are related to curriculum and pedagogy, resources and learner participation.

Curriculum and pedagogy

The two lessons described above were very different in most aspects e.g. content and focus. There are, however, similar teaching styles used in both lessons. Both types of curricula, as described by Bernstein, were represented in the two lessons. Lesson 1 represented an integrated type, while lesson 2 represented a collection type. Lesson 1 was driven by both content and contexts (Pedagogic Agendas 1 & 2) while Lesson 2 was mainly content driven (Pedagogic Agendas 3 & 4) with no application or reference to real life contexts. The content that was presented in lesson one was weakly classified, while the content presented in lesson 2 was strongly classified. This is consistent with Alfred‟s interview presented in Chapter 6.

It was shown in Part 1 of Chapter 5 that 68.3% of the 60 teachers agree with the statement that learners must be taught content then contexts. Furthermore, 58.4% of the 60 teachers agreed with the statement that sometimes in Mathematical Literacy it is important to teach only mathematics content. In Chapter 6 Alfred argued that basic mathematics concepts must be taught without necessarily relating them to real life contexts. The analysis of lesson 2 concurs with Alfred‟s stated Pedagogic Agendas and Orientations in the previous chapters. In excerpt 7, Alfred explains why he taught content only without any reference to context. His explanation is consistent with the responses he gave during the interviews in chapter 6.

Resources

In both lessons there were no evident problems associated with resources. All learners were provided with copies of worksheets. These worksheets were compiled from different textbooks. While in the previous chapter resources were identified as a challenge in Mathematical Literacy, in these two lessons this problem did not emerge.

181 Learner participation

In both lessons learners were actively involved, irrespective of the nature of the content they were dealing with. Neither mathematics content nor context seemed to negatively impact on learner participation. This finding provides a different view from the general view shared by other teachers which assumes that more content knowledge in Mathematical Literacy would affect learners‟ active participation.

Summary of findings of lesson observations

In this phase I have presented an analysis of four lessons, across two teachers in different schools. Analysis of these lessons has revealed that the two teachers have mixed approaches to teaching Mathematical Literacy. There was consistency in the lessons presented by Mr Khumalo. It is possible to explain his implementation of the Mathematical Literacy curriculum as in Pedagogic Agendas 1 and 2. On the other hand, Alfred had different approaches, particularly when coming to the issue of contexts and content. His first lesson resembled Agendas 1 and 2 and his second lesson resembled Pedagogic Agendas 3 and 4. Pedagogic Agendas observed were largely consistent with what these teachers had stated in earlier interviews about the purpose and teaching of Mathematical Literacy. Table 45 below presents a summary of classroom observation notes:

Table 45: Summary of classroom observation notes Aspect of the

lesson

Khumalo Alfred

Introduction of the lessons

In both lessons Khumalo introduced the lesson by indicating to learners the purpose of the lesson. He continued to guide the learners for further classroom discussions. He always linked each lesson to the previous lesson(s).

Alfred introduced the lessons- He allowed learners to continue with classroom discussions.

182 Teaching

strategy

In all lessons observed a teacher- centred approach and activity- based teaching strategies were used with a lot of guidance.

A learner-centred approach was used with activity-based teaching in both lessons. Alfred mentioned cooperative teaching as a strategy.

Focus and emphasis

The lessons focused on contexts and put more emphasis on real life contexts with little emphasis on mathematics content knowledge. Khumalo emphasised that learners need to engage with real life contexts.

Varies lesson by lesson. Lesson 1 was context and content focused while lesson 2 was strictly content based. Alfred emphasised that learners need some basic mathematics first, before they are exposed to contexts.

Resources Khumalo relied on specific textbooks - for the lessons observed; only one kind textbook was used by both teacher and learners.

He uses a variety of textbooks that are rich in basic mathematics content knowledge. He developed worksheets from various textbooks for all learners. Driving agenda In both lessons Pedagogic

Agendas 1 and 2 were foregrounded. There was more emphasis on real life contexts in the lessons observed.

All agendas visible (Pedagogic Agendas 1 - 4). Lesson 1 forgrounded agendas 1 and 2 and lesson 2 forgrounded agendas 3 and 4. Lesson 1 addressed both content and contexts while lesson 2 focused on content only.

Orientations In both lessons only Orientations 1 and 2 were foregrounded. More emphasis was on contexts.

All orientations were visible (Orientations 1-4). Lesson 1 was dominated with Orientations 1 and 2 and lesson 2 with Orientations 3 and 4.

Assessment Assessment was based on the textbook provided. All activities were drawn directly from the textbook.

Assessment was based on the worksheets provided drawn from various sources.

Feedback Feedback was given to individual groups; Khumalo was actively involved in giving solutions.

Feedback was given to the whole class.

Pedagogic models

In all lessons which were analysed, competence models were more foregrounded.

The lessons were moving between performance and competence models, e.g. lesson 1 (on business finance) was more aligned to competence model and lesson 2 (on linear equation) was more aligned to performance model.

183 Summary of the key findings from the analysis of lesson observations

Chapter 7 presented the case study of two Mathematical Literacy teachers‟ lessons. Data was collected through the classroom lesson observations presented in Chapter 7. The focus question was on how teachers implement or teach Mathematical Literacy. This focus question was addressing critical question 2 of the study. The four lessons that were analysed provided insight into how the two teachers teach Mathematical Literacy. Besides the nature of the content that was presented by the two teachers, it was evident how the teachers‟ Pedagogic Agendas and understanding of effective teaching and learning of ML play out in teaching.

Connections in the three phases of this study

Analysis of the three phases revealed some connections across phases. At this stage it is relevant to present visible connections amongst the three phases of data analysis. In doing so, I pay special attention to findings related to the three aspects, namely; main idea, common idea and contradicting ideas. These aspects are presented in table 46 below:

Table 46: The connections and common findings in the three phases of data analysis Phase1 questionnaire Phase 2 Interviews Phase 3 Lesson observations Main idea and key findings

General view about what Mathematical Literacy is, and how it should be implemented. ML is a real life subject Experiences in teaching Mathematical Literacy, successes and challenges. The idea that ML is a real life subject was shared

How Mathematical Literacy is implemented at classroom level.