4.8 Instrumentation
4.8.3 The Interview Schedule
Research methodology Instrumentation
My statement is that ………
………
My reason is that ………...
………
Arguments against my idea might be that ………...
………
Figure 4—4. The argumentation frame in Euclidean geometry
As can be seen in the figures above, I used Toulmin’s (2003) argument structure and adapted Osborne et al.’s (2004) tool to develop an argumentation framework as a data collection instrument. In light of the fact that English was the second or third language for most of the participants, basic everyday English language was used to create this tool. This approach was also used to improve the reliability of findings. Next I turn attention to the qualitative data collection tool, the Interview Schedule.
Research methodology Instrumentation
The Interview Schedule was pilot tested on one participant (not Presh N), purposefully selected from those who had completed the survey in the first, quantitative phase of the study. On the basis of the results obtained from the pilot interview analysis, the order of the Interview Schedule questions was slightly revised and additional probing questions were developed. For example, the question, “What do you think is your role in the proofs in textbooks?” was regarded as too broad and consequently revised to read “If the textbook has verified the truth of a theorem, what do you think is your role, next?”
The schedule comprised the participant’s demographic information, interviewer’s, transcriber’s, and data capturer’s names, time, date, and questions and probes. The participant’s responses were probed with the aim of encouraging elaboration on her responses so that a better understanding of the thinking behind her ideas could be gained and also to seek clarity. These questions were organised around the following themes:
1. Establishing whether Euclidean proof was covered in Grade 11, the second term as scheduled in CAPS.
2. Obtaining insights into learner’s definition of proof whether it is in terms of a particular role of proof. Also this question will elicit the role of semantic contamination.
3. Understanding learners’ views about whether understanding proof is innate or takes effort and practice. This will help see if they resort to memorisation.
4. Understanding whether the learner appreciate the need to read and understand a theorem for themselves rather than rely on the authority of the teacher.
5. Understanding the extent to which the textbook influences her belief about proving.
6. Checking if learner conceives of an empirical argument as a means to convince herself that a proposition holds true and/or regards deductive arguments as a means to explain and/or communicate to others why the proposition is always true.
7. Eliciting whether the type of proof presented to a learner influences their understanding of the functions of proof.
8. Determining what the learner attributes lack of success to and how they think proof learning can best take place in the classroom to elicit the influence of collectivist culture.
The semistructured interview also contained task-based interview, a particular form of clinical interview, as the secondary data gathering strategy for this study. According to Maher and Sigley (2014), this type of interview can be traced to Piaget in the early 1960s who pioneered clinical interviews in his quest to gain deeper understanding of children’s development. I follow the
Research methodology Instrumentation
definition of task-based interviews as ‘interviews in which a subject or group of subjects talk while working on a mathematical task or set of tasks (Maher & Sigley, 2014, p. 579). To Maher and Sigley (2014), task-based interviews make provision for more open-ended questions requiring qualitative analysis wherein participants interact not only with the interviewer but also with the task environment.
The choice of the strategy was informed by the suggestion that task-based interviews provide the best context for assessing and probing for the roots of participants’ beliefs (Hurst, 2008). Thus, the interview was designed to elicit participant’s perspectives on the functions of proof through integrating ordinary text with a cartoon. The reason for using a cartoon in the Interview Schedule was that, as Stephenson and Warwick (2002) point out, they allow for the disassociation of the ideas from those of particular participants so that it is not they who may be proved to be “wrong” but, the cartoon character.
The questions that incorporated diagrams (Figure 4—5) were presented on paper for respondents to use at any time. A cartoon depicting a learner reaching a conclusion about “the sum of angles in a triangle” on the basis of construction and measurement, was used in the Interview Schedule. Ibrahim, Buffler, and Lubben (2009) suggest that the use of real-life figures and names can lead to prejudice towards the making of a decision. As a consequence, the cartoon was used not only because it did not refer to gender, race or culture but also to improve construct validity of the responses (Ibrahim et al., 2009). Generous wait time (a definite pause between asking a question and requiring answers from respondents) of 5 seconds was allowed.
Research methodology Instrumentation
Main questions Probes (Follow up questions)
Please, consider the following cartoon and its proposition. Given that the proposition has worked in every case that the teacher has tried so far, how can we be sure that the method always works?
Why do you think so?
Figure 4—5. Sample main question and possible probes in the Interview Schedule
Qualitative data were collected through semistructured task-based interview to understand the factors influencing understanding of Euclidean proof in mathematics and thus answer research question, Why does Presh N hold informed beliefs about the functions of proof? This interview was considered as the best method to gain insights into Presh N’s reasons for holding the understanding of functions of proof she did. The questions in the Interview Schedule (Appendix B4) were organised according to the theoretical perspectives on factors influencing proof understanding. The first question posed was intended to understand whether Presh N has had experiences of proof, at all, in the present and previous mathematics classes. Thus, the Interview Schedule included questions such as: “Since this term began, have you done a proof?” and “Tell me about one theorem you just did in class”. These questions and their accompanying probes were important in the context of the South African classroom given findings that teachers tend to leave out some topics citing time constraint as a reason (Mji & Makgato, 2006).
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 Instrumentation
Probing as a technique in interviews enables delving more deeply into learners’ hidden interpretations of their functional understanding of proof and thus developing deeper insight into how and why they conceive of proof as they do. For example, Figure 4—4 is a sample core question aimed at elicitation of the effect of empirical argument on functional understanding of proof. This question followed the notion that learners inappropriately use 300 + 1500 = 1800 to
“prove” that the angles on a straight line are supplementary. Also, Kunimune, Fujita, and Jones (2009), report that even learners who can construct deductive proof do not understand why such proofs are necessary in geometry. It is reasonable to suggest that such learners lack an appreciation of the function that proof performs in mathematics.
Another source of qualitative data that was useful in the analysis stage was field notes.
These are notes I created during fieldwork to recall and record the behaviours, events, and other features to supplement interview data. A better understanding of what is said in an interview comes from its context, including a range of cues that are simply not captured on the audiorecorder. In addition, the notes were used to reflect on the research settings, difficulties encountered, procedures followed, distractions and nonverbal cues during interview, and as a backup in the event that recording equipment malfunctioned. The benefit that accrues with keeping field notes is that an interrater can use it to understand the coding followed in this study (Saunders, Lewis, &
Thornhill, 2012). As with interviews, field notes were transcribed for analysis purposes. Further, the Interview Schedule consisted of spaces between the questions to write these notes.