FIRST SET OF INTERVIEWS

Interviews were conducted with the same eight learners discussed in the pre-test results above. The purpose of the interviews was to gauge learners’ understanding of the problems when they verbally explained their strategies. Verbal explanations were considered against recordings of their solutions in the pre-test. Interview questions were structured in order to establish how learners were able to explain their solutions based on their writing when solving problems of the pre-test. Interviews helped to explore learners’ thinking and understand what they were doing. There was evidence of scaffolding in some interviews: learners needed prompts to explain their strategies. Most of the selected learners found

strategies did not always reflect what they had written on paper. Some learners, particularly from the average and below average groups, seemed to lack the mathematical vocabulary to explain what they had done.

Learner 1 (AA) was able to explain his strategies verbally even though this learner sometimes had the incorrect solution. At this stage of the data collection (pre-test), he had not written an explanation of his thinking when solving the problem because learners had not yet encountered the use of writing tasks. His strategies showed that he could use his conceptual understanding to represent his thinking. He was able to combine more than one method in certain strategies to reach his solution (Figure 4.1).

Learner 2 (AA) was able to explain her strategies verbally according to what she had done. This ability allowed an understanding of the problems of the pre-test compared to the writing she used in her strategies. In her writing she sometimes used tallies to represent her strategy. At the time of the pre-test, some mathematical problems had a low enough number range for tallies to be used as a strategy. Below is an excerpt from the interview where Learner 2 explained how she used tallies as a strategy.

Researcher: And the one with the cricket team?

Learner 2: I put, I had 90, I put 94 circles then I crossed out 47 of them and so I counted the rest of them and it gave me 47.

Learner 2 (AA) Pre Interview

When solving the first four problems of the pre-test, Learner 3 (A) used strategies that reflected the problem types represented. For example, the third problem required use of addition as a strategy which was reflected in his writing. However, he came to the incorrect solutions for these problems. As a result of this pre-test, it appeared that Learner 3 (A) was able to determine the underlying mathematical concepts needed to solve the problem but could not solve the problem. He needed many prompts during the interview to help explain or justify his strategies. He seemed to find difficulty applying the strategy to his writing in order to reach the solutions successfully. The following excerpt is from the pre-test interview conducted with Learner 3 (A).

Researcher: Let’s look at the problem with Rodney and the doughnuts. It says that Rodney’s putting 42 doughnuts on platters for his party. He places 7

doughnuts on each platter. How many platters will he have? Now what...how did you solve this problem?

Learner 3: I did put 7 platters then I put 7 on each platter. Researcher: Ok, how did you know that you had to do that? Learner 3: Because there’s 7 of each.

Researcher: And when you put the 7 out, what did you come up with? Learner 3: 48

Researcher: 48...but how many doughnuts does Rodney have? Learner 3: 42.

Researcher: 42...ok, so how come you ended up with more doughnuts? Learner 3: Because I add more.

Learner 3 (A) Pre Interview

His explanation of the strategy made sense according to the multiplication/division problem type (problem 2 of the pre-test). The recording of this strategy and his verbal explanation showed that he solved the problem at level 1 (initial grouping) of early multiplication and division. The above excerpt displays an understanding of the required operation or strategy to solve this problem. However, he did not follow through with this strategy and came to an incorrect answer.

During Learner 4’s (A) interview, it appeared that there was an understanding of the mathematical concepts required by each problem. The learner generally represented her thinking by using drawings or tallies. This was often her strategy when recording her thinking while she solved mathematical problems. However, drawings were not used when solving the third problem about the school sports team. In the excerpt below, Learner 4 (A) explained that drawing would be time-consuming when solving this problem since the numbers were too high. In this case, she was able to adapt her strategy and change her usual method of representation to suit the needs of the problem. In the following excerpt she explains why drawings were not used as a strategy for this particular problem.

Researcher: Let’s look at the school sports team problem. What did you do here? Learner 4: Well, I usually... when it’s a long, what I do is I just make a sum and

then I use my brain to, um, add 16. Researcher: Ok, so you added that without drawing? Learner 4: Yes.

Learner 4: Yes.

Researcher: Wow, that’s good. Ok, because I see you didn’t do a drawing there. I see you started and then you erased it.

Learner 4: Yes because I knew it would take too long. Researcher: Is it? Why did you think it would take so long? Learner 4: Because the numbers are high.

Learner 4 (A) Pre Interview

In the pre-test interviews with the selected learners from the above average and average ability groups, it seemed that they had the necessary conceptual understanding to solve mathematical problems according to the problem type as mentioned in chapter 2. Conversely, their solutions were not always correct: they had either misread on misinterpreted the problem. Some learners needed more prompting than others when verbally explaining their problems. Only one learner wrote a brief explanation of her strategy during the pre-test (see Figure 4.7) which was significant because learners had not been exposed to the various writing tasks at this stage of the data collection. They were not expected to write explanations of their problem-solving strategies but, in a few instances, learners wrote statements of their solutions without explanations of the strategies they used when solving the problems. During the pre-test, learners were asked to solve the problems showing their strategies and solutions. They were not asked to write explanations of their strategies.

The three learners from the below average ability group did not use strategies appropriate to the problem types presented in the pre-test. This failure showed their lack of conceptual understanding related to the mathematical problems. This lacking could be linked to their language ability: two of these learners (Learner 7 and Learner 8) had below average reading and comprehension abilities. When these learners were interviewed, they had difficulty explaining what they had done. Below is an excerpt from the interview conducted with Learner 7 (BA) which displays his difficulty in explaining his strategy.

Researcher: Let’s look at the first problem that you did. Do you want to explain to me how you solved this problem?

Learner 7: I...I did dots there and I carried on. Researcher: Ok, how did you carry on?

Researcher: But how did you know in the problem that you had to carry on. When you read the problem, what made you think that you had to carry on adding?

Learner 7: I don’t know. Learner 7 (BA) Pre Interview

It was evident in their strategies that the three below average learners comprehended the third problem about the school sports team as an addition problem type even though their solutions or answers were incorrect: displayed in the interview with Learner 8 (excerpt below).

Researcher: Ok, and then let’s look at the sports team. I see you just did a sum here. You started doing a drawing and then you erased it. So explain to me what were you thinking here.

Learner 8: I done a sum and then I made circles and then I erased it. Then I added the circles together and I kind of plussed it so I made 48 plus 10 plus 16 and it was equal to 73.

Researcher: Ok, so did you add this up in your head or did you use the drawing to help you?

Learner 8: I used my drawings to help me. Learner 8 (BA) Pre Interview

Learner 8 did not elaborate upon her strategy for this problem. At first, she explained that she had erased her drawings, in which she used the tally method, and solved the problem using a number sentence. When asked how she added the numbers, she said that she used drawings. This explanation did not make sense since she had erased her drawing. It is possible that she may have erased the drawing after she arrived at the answer. Figure 4.14 shows some evidence of her erased drawings and the number sentence that she wrote to solve this problem.

Figure 4.14: Learner 8 (BA) Pre Q3

During the pre-test interviews, learners often had difficulty giving verbal explanations of their problem-solving strategies. A possible factor in their inability to do so could have been a lack of appropriate mathematical vocabulary to clarify their thoughts. Another factor could have been that they had not previously explained their strategies verbally in the way that was expected during this study. Learners used limited details and explanation in their writing which may have led to the difficulty in their verbal explanations: they were not expected to use writing in this manner. Learners had not yet been exposed to using writing in mathematics through various writing tasks.

Once all the pre-test interviews were concluded, the various types of writing tasks, as modelled by Burns (1995a), were implemented in the selected Grade 3 class. Later, a post- test was conducted and the same learners were interviewed to compare their use of strategies and how the writing tasks supported them in reaching solutions. The findings of the post-test and interviews are elaborated later in this chapter.