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Thus, an analysis of interview data from different students was done, based on different questions, and yielded similar explanations for the error made, leading to the formulation of a common theme.

Comparison of data analysis within individual cases, on the other hand, involved themes generated from, for example, a student’s error as depicted by scribblings in the MC pre-test or explanations in the discussion pre-test, and the same student’s (faulty) verbal explanation during interviews. An example is given below, in which a snippet shows a student’s erroneous

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calculation (even though the student ended up choosing the correct option) and, interestingly, another error of associating “2 units to the right” with “-2”.

Figure 4.16: Snippet showing an error in calculation, identified during pre-test analysis and later confirmed through analysis of interview

When the researcher interviewed the student, she established that the student indeed had the misconception: “When two signs are the same, the answer is positive”. That is the student firstly did not make a careless mistake when she wrote “-2 - 2” instead of “-2 +2”. Second, the student did not realise that the “same signs” rule did not apply to addition and subtraction. Hence, as shown in Table 4.8, the analysis of data provided by the student in the above snippet (in both her test and her interview) resulted in the formulation of the code and theme as “Algebraic

signs” and “Errors involving basic operations” respectively.

As mentioned before, data from both the pre-tests and the interviews were used to answer the same research question. Therefore, the thematic map already designed following the review, defining and naming of themes (Phases 4 and 5 of thematic analysis) during analysis of pre- test data, also included data from the analysis of interviews. The analysis of interview data, however, had the added advantage over that of pre-tests in that the researcher was able to, later on, undertake “member checking” with the interviewees to determine the accuracy of themes against the sentiments expressed by the interviewees (Ngulube, 2015). Similarly to pre-test analysis, the themes derived from the analysis of interview data found their way into the final report, produced in the next chapter (Phase 6 of thematic analysis).

4.8.3. Stage 3: Identification of (emergent) themes from the intervention programme

As mentioned earlier in the section on data collection methods, data for the third stage were collected through participant observation, informal conversations with students, researcher’s field notes and students’ written work. Therefore, the analysis involved firstly reading and re-

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reading notes and information (both descriptive and reflective, as mentioned earlier on) from the researcher’s notebook and from students’ written work (Phase 1 of thematic analysis). In addition, the researcher analysed relevant images from video clips taken during lesson presentations by, for example, organising such images under relevant themes or codes. Analysis of data from lessons presented during the intervention programme was done to answer mainly the second research question. However, there was an anticipation that data obtained from lesson presentations might augment that which were obtained from pre-tests and interviews. See section 4.7.3 under Stage 3 of data collection. In order to answer the second research question, data analysis from lesson presentations resulted in the researcher addressing errors involving aspects such as the use of (inappropriate) vocabulary (mis)application of rules of transformation, performance of procedures for transforming shapes (mis)interpretation of instructions, and so on.

Analysis of data from lesson presentations also occurred through coding, searching for, reviewing, defining and naming themes (Phases 2, 3, 4 and 5 of thematic analysis) obtained from the data sources used during lesson presentations. Ngulube (2015, p. 140) maintained that a characteristic of thematic analysis is that “open and selective coding may be used to analyse the first case, and the themes that emerge are used as a basis for comparison with further cases”. This characteristic was relevant here because themes that emerged from the analysis of the first case, namely, pre-tests (as well as the second case of interviews), were used to analyse and compare data from lesson presentations. Examples of coding and theme generation during lesson presentations as part of the intervention programme appear in the table below. A similar more-comprehensive table appears as part of the data analysis report (Stage 6 of thematic analysis) in the next chapter.

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Table 4.9: Example of how codes and themes were generated from an analysis of data during lesson presentations

Data obtained during lesson presentations

Source of data Initial Coding

Theme/Sub-theme

During Information Phase – Reflection

Question: If the vertical line in the middle is the line of reflection, draw the image of the given figure

Students’ (erroneous) written responses and a portion of researcher conversations with students:

Sample 1

Student: “I know the arrows will point in opposite

direction”

Researcher: “What about the distance of the

image from the line of reflection…. the mirror line?”

Sample 2

Student: “I copied the figure on the other side of

the line”

Researcher: “Did you consider what the direction

of arrows will look like in the image….If you were to “flip” the image along the line of reflection, onto the original figure, would it fall exactly on top of it?” Students’ written work on their notebooks (plus) Researcher’s field notes following informal conversations with students Properties of transformations Incomplete descriptions

Addressing errors involving application of incorrect or incomplete rules and properties of changes resulting from specific transformations

It should be noted from the above table that there was a link between themes created from analysis of data during lesson presentations and those generated from the other stages of data analysis. The reason for this is that the intervention programme lessons were based on and designed so as to address the errors and misconceptions that already emerged from pre-tests and interviews. Therefore, in addition to analysing lesson presentation data that contained errors, the researcher also analysed information regarding the extent to which the intervention programme addressed these errors. She did this in two ways. Firstly, she analysed students’ responses to her guidance and probing at the particular moment when she was addressing the error (for example, do they realise there are instances in which it would be easier to do

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calculations rather than drawings, or vice versa), to see if the student understands the error and reason for it to be labelled as such. Second, the researcher continuously checked if students remembered and applied the aspects involving their corrected errors, as they continued with similar questions during different phases of Van Hiele phase-based instruction. She then recorded her impressions in her notebook. For example, when an error involving students misreading or not following instructions was corrected, the researcher checked if the students correctly read and followed instructions in subsequent phases of the instruction. She then wrote down this information as field notes resulting from participant observation. The information was then analysed and reported on, as part of answering the second research question.

4.8.4. Stage 4: Analysis of students’ responses from the post-test

Data involving students’ performance in, and responses to the post-test were analysed in order to answer the second research question. The question was: “To what extent can Van Hiele phase-based instruction be used to address students’ errors and misconceptions involving transformation geometry?” The researcher had already started to answer this question before, at the first level, when students’ errors and misconceptions were addressed by the researcher as they emerged during the lessons presented in the intervention programme (see discussion in the previous Section 4.8.3). The analysis of students’ responses to the post-test was the second level of analysis to answer the same question. Data analysis at this level sought to find out if students’ responses showed improvement in terms of errors committed.

4.9. Trustworthiness of the study

Qualitative research is often criticised for lack of scientific rigour (Ngulube, 2015), and is labelled as “subjective, anecdotal, subject to researcher bias, and lacking generalizability by producing large quantities of detailed information about a single, unique phenomenon or setting” (Cope, 2014, p. 89). Therefore, the challenge for researchers would be to convince the reader that their qualitative studies, despite being characterised as subjective, are trustworthy. Hence the researcher in the current study had to ensure that this study does not carry with it notions of untrustworthiness that will expose the study to such criticism. The researcher had to ensure the soundness of the research conducted, in terms of how appropriate the methods used are, and the trustworthiness of the conclusions reached (Noble & Smith, 2015).

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Trustworthiness refers the extent to which those who assess the research study are confident of the study and its conclusions (Robson, 2011).

Lincoln and Guba (1985) initially suggested four criteria that can be used to develop trustworthiness in qualitative research, namely, credibility, dependability, confirmability and transferability. These terms, in essence, “replace the more conventional views of reliability and validity” (Cohen, Manion & Morrison, 2007, p. 158) and should be applied to both the research process and the results so as to ensure high rigour and robustness (Leung, 2015). In the current study, various strategies were used to enhance the trustworthiness of the study. These strategies are discussed, in the context of the four criteria, below.

4.9.1. Credibility

The credibility of research describes the extent to which the researcher truthfully represents and interprets data (Noble & Smith, 2015), so that the findings and conclusions reached are believable. In the current study, the researcher used several strategies to enhance the credibility of the study. For example, the researcher had a prolonged engagement with participants (students) with the aim of building trust with them (Petty, Thomson & Stew, 2012). The researcher’s success in creating a relationship of trust with students in the current research resulted in students scribbling freely (on encouragement by the researcher) explanations that accompanied their answers, for both the tests and the interviews. This enabled her to gain a deeper and enriched understanding of the phenomenon being studied, from their rich and thick descriptions, thus reducing the possibility of researcher bias resulting from her own opinions. “Rich data are a hallmark of qualitative inquiry and are frequently regarded as key to lending credibility and persuasive strength to qualitative studies” (Schultze & Avital, 2011, p. 1).

To further offset the researcher’s bias, the researcher captured accurate observations of participants’ actions, using a video camera during both interviews and intervention programme lesson presentations. The use of video camera enabled the researcher to truthfully represent, analyse and interpret participants’ opinions, as opposed to possibilities of reporting inaccurate data due to her limited memory. She was then able to provide rich descriptions of participants’ ideas, when writing and communicating the report of data analysis. The report includes quotations of students’ own words as well as researchers’ interpretations which were confirmed and verified by students themselves, through member checking (Cope, 2014; Leung, 2015).

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The reader of this current research can therefore provide a meaningful critique of the credibility of this study and authenticate the interpretations of the researcher.

Triangulation, that is “the process of using multiple sources to draw conclusions” (Cope, 2014, p. 90), was used to increase the credibility of the study. Tests, interviews, participant observation and field notes were all used to produce more comprehensive findings of the research. These sources or methods of collecting data supported each other to produce consistent findings. In addition, the researcher recorded details from sources such as interviews and field notes immediately after analysis, while the details were fresh in her mind, so as to avoid inconsistencies between observed (or spoken) data and recorded data. The literature supports the idea that, when information from data sources is recorded and reflected soon after observation, this carries more credibility (Walliman, 2011; Schwandt, 2015). Walliman (2011) further argued that immediate recording and reflection allows the researcher to identify gaps in the data and be more receptive to new ideas that challenge the researcher’s assumptions and biases.

As mentioned before, as part of data analysis in the current study, the researcher had to immerse herself in the data. However, she also had to “conversely distance herself” from the data for some time, so as to do the analysis by assessing and examining the accuracy of the coding and theme development processes from an “outsider perspective” (Vaismoradi, Jones, Turunen & Snelgrove 2016, p. 106). This allowed her to “maintain a sense of self-criticism” (Vaismoradi, Jones, Turunen & Snelgrove 2016, p. 106). Particularly because this is an action research study, in which the researcher had a vested interest in the outcomes of the research; she had to critically reflect on her biases. For example, during data analysis, she had to sometimes reject certain themes which were initially developed, when new data emerged, even though she had her own pre-conceived and convincing ideas about the relevance of such themes. Petty, Thomson & Stew (2012) are of the opinion that, when a researcher acknowledges his or her own biases and does ongoing critical reflection on sampling methods, data collection techniques and analysis, this contributes towards the credibility of the qualitative study. In the current study, reflection by the researcher involved, for example, her decision to combine the questions of the two pre-tests into one post-test during the final stage of data collection. She did this to avoid students discussing the next test before they wrote it (if the two were not combined), and thus compromising the level of credibility of the test results.

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4.9.2. Dependability

“Dependability refers to the logical, traceable, and carefully documented research process” (Kihn & Ihantola, 2015, pp. 235-236). It is important to corroborate the findings of a research study in terms of consistency and the extent to which they can be repeated. This would allow someone else who was not part of the original research to “follow, audit, and critique the research process” (Moon, Brewer, Januchowski-Hartley, Adams & Blackman, 2016, p. 18). In the current study, the researcher applied certain strategies to ensure the dependability of the study. These strategies are discussed below.

The researcher maintained an “audit trail” which documents the procedures and processes followed while doing the research (Ngulube, 2015, p. 149). The audit trail would then allow the readers of the research to track these processes in order to understand how decisions were made by the researcher. Cope (2014) listed examples of documents and materials that can be used to check the trail, involving decisions and assumptions made by the researcher. These are interview transcripts, data analysis and process notes and drafts of the final report. In addition, Ngulube (2015) suggested that one of the elements that help maintain an audit trail is a record of how codes were identified and themes developed. For the current study, detailed documents such as interview transcripts and data analysis process notes were used or reviewed by other researchers to draw similar conclusions to those of the current study. Even if there were to be cases of variations in findings and conclusions (Petty, Thomson & Stew, 2012), such variations are “essential and inevitable” as they “add extra dimensions and colours to enrich the corpus of findings” (Leung, 2015, p. 324).

While extracting data from original sources, the researcher should validate the accuracy of the data by doing a constant comparison and comprehensive data analysis that includes references to quantitative aspects where possible (Leung, 2015). In the current research, the researcher provides a comprehensive analysis of data that includes quantitative data in tables. Any outside researcher can then refer to this analysis to explore and possibly challenge data analysis processes and procedures, as well as to understand the interpretation of findings that were made. In this way, they would be able to establish whether the study is dependable or not.