Data analysis procedures

After collecting the data from observations and interviews, the analysis began. There were two types of data, one from the interviews and the other from the observations. Data from the observations focuses on the teachers’ actions and their knowledge while the data from the interviews focuses on the reasons given by the teachers for their actions and it feeds to some extent into teachers’ knowledge as well. Next, I describe the procedures of data analysis in each case then explain how the cross- case analysis was done.

4.7.4.1. Observation data analysis procedures

When I came back to the UK after doing the fieldwork, I started to explore the entire data with my supervisors to assess its richness. This task required me to transcribe all the important incidents from the observations and all the data from the interviews. I created a word table (see below Table ‎4.7 for an example) with two columns in which I wrote the lesson’s incidents of the first teacher on the left and the related chunks of the interview on the right using number codes. These codes, like (417) mean the seventh question in the first lesson of the fourth teacher, in the example below are used in the interview and the incident to link them together. The number in the incident column means that I asked the trainee about the response before the number and his answer in the interview is the one next to the same number.

Time Incident Interview

30:00 – 32:00

Fahad says ‘If you go to the shop to buy something, what I am going to buy?’ A student says ‘Milk or juice’. ‘No. These were in the last lesson when we learnt capacity, (417) now we are studying mass. If you buy 24 apples, how many kilo will

(417) Fahad said ‘I want to tell the student that there are differences between millilitres and litres, and grams and kilograms, as this might be a common misconception. ‘I

you ask the cashier for?’ Some students say ‘2 kilos’ and some say ‘2.5’. While Fahad is standing still near the board, different answers are reached such as 3 kilos, 3.5 kilos and 4 kilos. (418) Fahad asks ‘Why did you say 4 kilos?’ Then the student says ‘Because 6 plus 6 equals 12 and another 6 plus 6 equals 24’ Fahad says ‘We said one kilo equals 6 apples so four times 6 equals 24 apples’.

never like to ignore a student, even when I am sure he will answer incorrectly, because if I ignore him many times he will lose confidence and might become shy and not participate in the class’

Table ‎4.7: A sample of one incident from one lesson

After writing all the lessons’ incidents and the related interviews for all the teachers I started to analyse the observation. At first, I analysed the first lesson for the first teacher with Sfard’s model as I mentioned earlier in section (4.5.2) and I found that the analysis was very long. Therefore, after changing the framework to the KQ I started with the same lesson and found it was a manageable size.

Next, I recognised that even analysing eight lessons of the three teachers would be very long for the thesis so I decided to include only three lessons from each teacher in the analysis but bearing in mind the danger of missing some important data. I started exploring Abdulla’s lessons by looking at all his lessons to find the desired incidents in which he interacted with the students. For example, I made a table to summarise the frequency of these incidents in every lesson as in Table ‎4.8 below.

Teacher Observed Lessons Grade Incidents per lesson

Abdullah 1 6 4 2 6 5 3 5 2 4 5 3 5 6 3 6 6 1 7 5 5

Table ‎4.8: A summary of Abdullah lessons

After identifying the incidents in which Abdullah responded to students’ contributions, I started to order the lessons decreasingly in terms of the number of

incidents in every lesson. I assumed that the higher the number of incidents the more responses occurred which would give more diversity in these responses. I chose the first three lessons with the highest number of incidents to include in the analysis. Before starting the analysis of the first lesson I searched the excluded lessons for any important aspects of responses that needed to be included in the analysis. I had in mind the idea of possibly changing the chosen lessons if I found a lesson with less incidents but richer in content. I found that the response patterns are almost the same among all the lessons. I did the same procedure to Fahad and Onizan’s lessons and came back with nine lessons in total.

The first lesson was analysed by writing a short synopsis (about 500-600 words) of the lesson then choosing one incident to be described in detail after the synopsis as an example of interaction. This work was followed by the actual analysis which was done by applying the KQ’s codes through MAXQDA software (see http://www.maxqda.com) to the extended version of the lesson, not the short description, in order to give a more detailed picture of the types of knowledge and different responses. The code ‘responding to children’s idea’ was applied to all the sequences of teacher-student- teacher interaction. Then these sequences were categorised into different codes of teachers’ responses to answers, suggestions, questions and comments. I re-coded all the lessons of one teacher on this basis.

Next, I noticed that the interaction in the form of question-answer was the most observed sequence while the other sequences were rare. As a result I thought of focusing the analysis more on dealing with students’ answers as it was the most frequent in the data. I then subdivided the code ‘teacher response to answer’ into teacher response to correct and incorrect answer. The latter refers to an incomplete answer or a completely incorrect answer. After identifying the sequence of each response to correct and incorrect answers I investigated the teacher third move (follow up move) which follows the students’ answer, and checked the type of these moves (described in section 3.2.2) and the type of exchange according to the Sinclair and Coulthard’s (1992) model discussed earlier in the literature review chapter. For example, the teacher’s move could be a comment move in which he comments on the answer by expanding it or it could be asking for justification (press move) and so on. By that time, I had, to some extent,

some information about teacher knowledge and their ways of responding but I still needed to know their justification of why they responded in that way. This information came from the interview data. Next, I explain how the interview data was analysed.

4.7.4.2. Interview data analysis procedures

The interview data was divided into two types of data: one was generally related to the teachers’ background of beliefs of mathematics teaching and the other more relevant to certain parts of the observed lessons. Both of this data provided extra information about teacher knowledge but the latter provided more information about the reasons behind the ways of response. I started to code the data of the first lesson with an open-minded view in which I let the data speak. I paid more attention to parts dealing with teacher knowledge and justification as I coded them according to the aspect of teacher knowledge and response. Then the response code was subdivided into reasons of response to correct and incorrect answers. Next, all the different given reasons for one response were grouped together to form a pattern. For example, in the example above in Table ‎4.8 the teacher rejected the answer verbally and he reasoned that he was treating a common misconception, so the act of rejecting verbally was associated with treating common misconceptions and other reasons given in different incidents in the lesson. These reasons were grouped together to see if the similarity between them would suggest an overall reason that would join them together. After finding all the responses and their reasons and enriching the teacher knowledge data from the KQ by adding some aspect of teacher knowledge from the interview, I was ready to write the first lesson account.

I started writing the analysis of the first lesson by focusing on the codes of the foundation, transformation, less on connection but heavily on the contingency dimensions. After writing the first lesson I repeated the process for the subsequent lessons. I wrote about teacher knowledge and responses to students’ answers in detail in two lessons and in less detail in the third, if it looked like the other two, to avoid repetition. After writing all three lessons analyses I wrote an overview of the teacher knowledge and response patterns among the lessons which was achieved by searching for patterns emerging from the codes of the teacher responses to students’ answers. Then I moved to another teacher and checked if he fit with the existing coding or

needed re-coding again. After writing all the case reports I moved to the cross-case analysis which is described below.

4.7.4.3 Cross-case analysis procedures

When doing cross-case analysis, researchers look for similarities and differences across the cases in order to fully understand the phenomena under study (Yin, 2009; Stake, 2006). After writing each case report of this research, the three cases were looked at together to see if they had common themes and to highlight the most important findings.

To do this, I re-read all the cases again mainly focusing on the overview in each case. I looked to the common response patterns among the participants and their reasons. Then I grouped the response patterns into two groups of actions and the reasons into two groups of reasons. I investigated the relationships between these two elements and teacher knowledge. Next, I moved to the difference between responses in which I found a variety of responses within the cases and I compared these regarding the various themes. This work was followed by a summary of the findings.