4.10 Data analysis procedures
4.10.3 Pattern-matching analysis
4.10.3.2 Analysis of interviews data
Research methodology Data analysis procedures
Also, I attached memos to these text segments to record the ideas, insights, interpretations or understanding that may arise from the data. Then, I displayed tree nodes to see how the participant talked about, for example, “semantic contamination”. Notably, tree nodes are categories organised hierarchically into trees. Thus, I used single item search to ensure that every mention of the word
“textbook”, for example, was coded under the “factors” tree node.
The final stage involved recording of insights gained into a memo from the display. This memo contained my commentary on text from the document to use in the interpretation stage of the project. Each node on the tree accommodated similar data and allowed storage of the comments I made. Then, I searched the indexing system to retrieve data according to themes identified in literature. The text was rechecked for the occurrence of these categories to seek patterns so as to determine relationships. I explained the relationships between the categories to seek patterns to interpret the data from the standpoint of participant’s perspectives, in their own voice (McMillan
& Schumacher, 2010). As I transcribed the interview verbatim, I demonstrated that the analysis is a nonlinear but recursive process involving a search for themes to categorise.
Research methodology Data analysis procedures
with the approach of thematic analysis in which data are coded to fit into a preexisting coding frame, the purpose of the qualitative phase was to provide a less detailed description of the data overall, and a more richer analysis of some aspects of the data (Braun & Clarke, 2007). The need to provide such a description was driven by what I as researcher and teacher educator have experienced as a current gap: the absence of South African learners’ views on functional understanding of proof. This purpose was realised through consideration of the frequency of occurrence and defining each theme sufficiently so that it is clear what it represents. The existence and prevalence of themes is not represented by quantitative methods but by the use of the qualitative phrase “Most issues raised by Presh N indicated that …”
All text in the transcript were coded and collated without trying to answer the research question (Kawulich & Holland, 2012). Thus, a long list of the different codes identified across the entire dataset was produced. In the end, the result of this coding process was a coding scheme that listed all the codes thus identified in this way providing an overall insight into the codes. For instance, the inteview subquestions and probes used to obtain answers to code around in order to obtain answers to the overriding research question included “What are the functions you believe proof performs in mathematics?”, “Why do you think so?”, respectively. Finally, I considered how these themes interrelated and divided into different levels. A detailed analysis for each individual theme was conducted to identify the story that each theme told and how the theme related to others in answering the qualitative research question (Braun & Clarke, 2007). In other words, this analysis phase essentially involved sorting the different codes into potential themes, considering how different codes may combine to form an overarching theme. I selected compelling extract examples to demonstrate the prevalence of the themes.
The coding of participants’ answers to questions “What, in your view, is proof in mathematics?” or “Do you think that proving propositions in mathematics is necessary?” involved checking whether the responses demonstrated (1) the definition of mathematical proof as a product of an argumentation process based on a finite sequence of axioms to reach a conclusion, (2) an appreciation of the generality of proof, (3) de Villiers’ (1990) categorisation of the functions of proof: truth (verification), explanation, communication, discovery, and systematisation. Further,
Research methodology Data analysis procedures
in response to the question, “How do you learn your theorems?”, the participant may respond,
“Our teacher starts by giving us the theorem and follows with problems where we apply the theorems”. This statement would then be coded and categorised as “Teacher influence”.
I performed constant comparison method in which each new piece of data is compared with the previously coded ones thus giving rise to new codes if none of the previous ones provided adequate description (Kawulich & Holland, 2012). These codes were kept brief and succinct. The next step involved grouping these codes into exhaustive and mutually exclusive categories and that their labels not only reflected the purpose of this study but were germane to the research question (Cohen, Manion, & Morrison, 2011). These categories were mutually exclusive in that no data were assigned to more than one category. An example of the coding and categorising process is shown in Figure 4—9.
Research methodology Data analysis procedures
Interview question Aim of interview question Data extract Code Category
(a) What, in your view, is proof? To obtain insights into whether learner’s definition of proof is in terms of a particular role of proof;
verification.
“It’s like something that convinces you hmmm [inaudible] yes hmmm. It’s like it helps you to see that what you are testing is hmmm [hesitation]
true. (Thandi B07a)
Proof convinces
Semantic contamination
(b) Do you agree with the learner’s thinking that finding the same answer after trying many cases proves?
(c) How can the learner be sure that the statement that “The sum of the angles of a triangle sum up to 1800” always works?
To check if learners conceive of empirical argument as proof.
Empirical argument
Figure 4—7. Sample analysis process of semistructured questions in Interview Schedule
I constructed different triangles and measured their
angles with a protractor.
This is proof that “The sum of the angles of a triangle
sum up to 1800.
Research methodology Data analysis procedures
As the participant noted astutely, “All statements that require proof must be proven in ways that show evidence”. Although these descriptions point to a linear analytical process, the analysis involved a back-and-forth movement between the whole and parts of the text. Any text that could not be categorised with the initial coding scheme was allocated a new code.