ANNEXURE A A scenario of a Grade 8 Mathematics lesson presented by Sesedi, a novice mathematics teacher. During the lesson I (Lekwa) was offering Sesedi classroom support, which was part of my roles as the Department Head for Mathematics, Sciences and Technology at school. Topic: Number Patterns Background The lesson took place in a grade 8 class that had learners who come from different cultural backgrounds. This was an overcrowded classroom with 110 learners. Learners were seated in groups of 5s and 6s. There were six columns with four rows of groups. The space inbetween the rows allowed little room for movement to check on how individual members in groups were working. The noise in the class was loud and Sesedi tried harder to let them keep quiet. The learners were uncontrollable at the start as some struggled to settle down from the previous technology lesson presented by another teacher. It was easy to notice that some learners playing and not concentrating on the given work. But, it was hard for Sesedi to see the boys who were playing at the back and not doing the activity. The learning episode The lesson itself focused on number patterns. The activity that was given on the board had eight items:

1. 5 ; 9 ; 13 ; __ ; __ ; __

2. 14 ; 11 ; 8 ; __ ; __ ; __

3. 3 ; 4 3 ; 16 3 ; __ ; __ ; __

4. 1 ; 4 ; 9 ; __ ; __ ; __

5. 3 2 ; 9 6 ; 27 8 ; __ ; __ ; __

6. 2:5 ; 2:5:5 ; 2:5:5:5 ; __ ; __ ; _

7. A ; C ; E ; G ; __ ; __ ; __ 8. 5 ; 15 ; 25 ; __ ; __ ; __ Before they could engage with the activity, textbooks were issued to all the groups. The following example was then presented on the board as rehearsal of what was done previously. 1 ; 4 ; 7 ; __ ; __ ; __ ; __ ; __ The learners were asked to give subsequent numbers with Sesedi completing them on the chalkboard as presented. So they raised their hands and only one by one responded: 10 ; 13 ; 16. Realising that this could go on and on, I then changed then changed the question and said If the number was 25, what was the previous number? Following a moment of silence the hands started going up slowly as Sesedi kept on repeating the question. When we ultimately asked for responses we had some learners saying it is 23, some 22, and even 27. With the latter response the class reminded the learner that we wanted the number before. Following short mumblings 22 was accepted as the appropriate answer. It was at this stage that we asked learners to explain how they got 22. This evoked different answers among which were: Group C4-1: We add four numbers Group C5-4: We skip two numbers Group C2-2: We skip two numbers Whilst a clear majority were agreeing that they skip two numbers, group C4-1 maintained their stand and said that they add four numbers. We became interested into what the learners meant by adding four numbers and which were those. So we asked them to explain and it was only after repeating what they were saying that we saw how that was done. It went like; Group C4-1: 22 plus 1 is 23, 23 plus 1 is 24, 24 plus 1 is 25. [as they said that they also demonstrated using their fingers. As they say 22, they had the first finger raised, 23 their second finger raised, 24 their third finger raised, and 25, their fourth finger raised. The other groups also felt like explaining their system and they indicated how by skipping two numbers resulted in a right answer. Sesedi was a little worried at this stage that the learners were not seeing the addition of three. So he went to the board and wrote: +3 +3 +3 1 ; 4 ; 7 ; 10 ; 13 ; 16 ; __ ; __ ; __ Before he could go further and before the learners could pick up what he was trying to explain I intervened. I was not convinced that the groups were really counting forward as they had explained, whether adding or skipping. Hence I asked, If the number is 77, what would be the number before that? This took a few seconds and group C4-1 said 71. No, 74, 75 were responses that were now coming from the floor. Group C4-1 changed again before settling for 74. The interactions continued until there was a consensus that if 3 is subtracted from 77 then the answer 74 will be obtained. That is, if we move forward we add 3 and if we move backward we subtract 3 from the given number. The learners were then asked to work on the activity on the board. As they worked on that we moved from group to group trying to observe how they were going about it. In some cases we observed group members working as groups whilst in some we noted those members who preferred to work on their own. In group C3-3 for example we had this girl who continued to work on the activities on her own even during whole class discussions. Group C1-1 took sometime on the first item and even attempted to introduce n as a way of generalising their pattern. However, both of us could not spend sufficient time with the group to understand their argument. After a while we noticed that with item 2, most groups stopped at 2. We wanted to find why this was the case and therefore asked what number should be after 2? Almost all the groups were saying is nothing after 2 with the exception of three groups: C4-1, C3-2 and C5-4. Groups C4-1 and C3-2 were both having 1 whilst group C5- 4 had 0. We became more interested in the 0 and asked the group to explain. That took us a while as we both could not follow the argument. It was after some time that one learner from the group pointed to the board and said: look where 2 is when you start from the beginning, it is 14 ; 11 ; 8 ; 5 ; 2 ; 0 ; 2. It was at this stage that we realised that the 2 that we wrote on the board was like we left three after 8. In a way the learners thought we were asking them for the number to fill up that space. So at least this problem was resolved when we clarified our question. So we turned our attention to other groups. Sesedi introduced a number line on the board with the hope that learners will use it to get the expected answer. That did not prove useful as only the other two groups could follow. So at the end only those groups could get to 1. In the process of working on these items, a learner from group C5-4 kept on attempting to give answers in the process changing mind as he continued to work. It looked like he really enjoyed making attempts. Unfortunately the period was over before we could see how learners attempted other items

4. The lesson narrated in Annexure A, was jointly facilitated by two teachers; Sesedi the subject teacher and Lekwa the departmental head. Refer to Paul Cobbs (2007) work titled Coping with multiple theoretical perspectives to compare and contrast the two teachers philosophical commitments. In responding to this question focus should be on:

4.1 The theoretical perspectives recommended by Cobb (2007), which were adopted by each teacher. This should include how the teachers navigated through the issues raised by Cobb for teachers to consider in choosing theoretical perspectives. (15)

4.2 How each teachers classroom practice reflected success or failure in taking caution of aspects identified by Cobb (2007) for teachers to take caution on when choosing a theoretical perspective to adopt for teaching mathematics? (5)