Whereas quantitative research is prudently structured with upfront hypotheses, premeditated methods and pursues specific types of measurable data, qualitative research is decidedly semi-structured, open-minded about methods to be used and its data are of unpredictable nature (Maxwell, 2010). The distinction between quantitative and qualitative research is that the former weighs phenomena “in terms of variables and correlations and [the latter] in terms of events and processes”, or put differently,
“variable-oriented” versus “person-oriented” (Maxwell, 2010:476). The aim of qualitative research is not to generalise findings, but rather to gather new insights as interpreted by the researcher. Maxwell (2010) is reluctant to associate the use of numbers exclusively with quantitative research. Indeed, qualitative researchers make use of “quasi-statistical” quantifiers with words like some, few, several, most, frequently, rarely, etc. (ibid.:476). To interpret events and processes qualitatively, it is also necessary to relate the historical progression and conditions under which such events and processes yield a certain outcome. It can therefore be said that even
156 qualitative reasoning has a secondary variable-oriented undertone. For this reason, Maxwell (2010) theorises that in a mixed method study, the quantification of qualitative data can help to answer a research question that probes a relationship between variables. In addition to a pure scientific-based relationship evidenced with quantitative data, the quantification of qualitative data can uncover consistencies or inconsistencies that would otherwise go undetected or refutes the supposed relationship.
The essence of qualitative research is the researcher’s meaning-making and interpretation of data, participants’ understanding of events and the influence of these understandings on the participants’ behaviour (Maxwell, 2008). Consequently, the researcher absorbs the raw data and constructs perspectives by using a specific theoretical lens (Creswell, 2014). Qualitative research may involve a variety of empirical instruments: interviews, participant observations, case studies, discourse analysis and document analysis (Silverman, 2014). Typically, “small, purposeful samples of articulate respondents are used because they can provide important information, not because they are representative of a larger group” (Sale, Lohfeld &
Brazil, 2002:45). The next sections discuss some properties of qualitative methods relevant to this study.
4.4.2 Strengths and weaknesses
The strengths and weaknesses of qualitative research are pointers that should be considered in the design of any study. Table 4.2 outlines certain aspects relevant to the current inquiry (Johnson & Onwuegbuzie, 2004:20):
Table 4.2: Strengths and weaknesses of qualitative research.
Strengths Weaknesses
Useful to describe complex phenomena Results cannot be generalised Useful to small samples Data collection is time consuming Facilitates in-depth probing Data analysis is time consuming Based on participants’ own experiences and
meaning
Subjected to biases Can provide local contexts of phenomena
Can result in grounded theory Data collection in naturalistic setting Determines idiographic causation
157 The weaknesses of qualitative instruments used in this study was offset by the use of quantitative instruments. This is a motivation for mixing methods as discussed in Section 4.9. Silverman (2014:43) summarises the fundamental methods available to the qualitative researcher as follows:
Observation – to understand the natural behaviour of participants
Textual analysis – to understand participants’ activities
Interviews – to understand participants’ views and opinions
Audio and visual materials – to understand how participants work.
The next section describes the qualitative methods used in this study, namely worksheet documents relating to the mathematical modelling tasks, open-ended questionnaires and semi-structured reflective group interviews.
4.4.3 Qualitative methods
There are two options open to researchers who seek further explanation of a phenomenon. Babbie (2010) distinguishes between an idiographic and nomothetic explanation. An idiographic explanation literally applies to a particular participant’s personal or unique empirical experience. The prefix ‘idio’ is the Greek term for individual or personal and ‘graphic’ means to show, illustrate or describe. This type of explanation cannot be generalised as it only intends to detail a singular case in its totality. On the contrary, a nomothetic explanation is a discovery of general experiences or observations. The Greek term ‘nomo’ refers to ‘law’, alluding to a rule that is inclusive rather than exclusive. The Greek term ‘thetikos’ refers to ‘placed’ or ‘to put down’, which relates to a rule that is laid down or pinpointed. A nomothetic explanation aims to lay down the rule between variables that can help to explain a cause-effect relationship (Johnson & Onwuegbuzie, 2004). On the idio-nomo-analysis debate, Hermans (1988) argues for a complementary use of both methods in a single study. “An additional strength of the idiographic approach is that the personal meaning and personal relevance of a more or less general finding [from a nomothetic approach]
can be assessed in intensive idiographic research” (p. 808). In other words, a researcher can use general experiences or observations based on a nomothetic approach to probe in more depth on individualistic experiences with an idiographic approach, even reverting again to a nomothetic approach if necessary. These two
158 approaches to explore participants’ explanations of a situation or experience would generate different types of data. The next two subsections explicate the complementary use of idiographic and nomothetic explanations used as data collection methods for this inquiry.
4.4.3.1 Worksheet documents
While learning in a CAS environment, documents in the form of handwritten text and electronic files are primary sources of data to gain insights in students’ actions, processes, objects, and ultimately their understandings. In this study, worksheet documents refer to hard or soft copies of a document that is used in a computer laboratory. Typically, programming activities result in codes that are typed in a specific software package and then stored as an electronic file; this can then be retrieved and analysed. Hand written documents may complement such electronic documents; for example, to plan a strategy on paper, to draw a rough diagram or graph and to write a report on a certain activity or task. Both hard and soft copy documents are valuable as an additional source to researchers who want to analyse students’ reasoning and thoughts processes.
4.4.3.2 Open-ended questionnaire
The nomothetic approach was applied with an open-ended questionnaire (see Appendix E) during the pilot study to capitalise on the diversity of views relating to students’ experiences and perceptions. Open-ended questions associated with the general experiences of students were collected to formulate a sense of their difficulties, obstacles, the role of CAS and other tools, the use of different representations and the laboratory learning environment. Also, students’ perceptions were analysed for perseverance, motivation, the effect of teamwork on the individual, general feelings after completing the modelling task and the value of the modelling approach. The specific questionnaire items are discussed in Section 4.7.5. Using content analysis, students’ questionnaire responses were searched for a multiplicity of qualities and traits to get a general sense of how a mathematical modelling approach is perceived in a South African university context and culture. This pilot instrument was used in the belief that broader generalisations were first necessary in order to explain the experiences and perceptions of engineering students on a mathematical modelling
159 approach. However, Hermans (1988:790) warns that “broad generalizations are not automatically true of the individual”, therefore, a union with the voice of the individual is also needed. For this purpose, individual perspectives were also gathered through semi-structured reflective group interviews.
4.4.3.3 Semi-structured reflective group interviews
Complementary to the broad interpretations collected via the open-ended questionnaire, the idiographic approach was employed in this inquiry in the form of interviews. This facilitated explanations of circumstantial peculiarities as experienced by individuals. Semi-structured reflective group interviews were used in this study.
During the main study, interview questions were prepared to further probe the responses and outcomes of each group’s modelling documents. Although clear interview mandate and protocols were predetermined, the order of questions and depth of probing were circumstantial and flexible. The interview questions (see Appendix I) aimed to find more clarity on specific shortcomings and/or exemplary responses given in students’ modelling documents. The interviews can therefore be best described as a reflection on group work, navigated with subtle prompts to better understand the rationale behind students’ thinking processes. Consequently, the type of interview will thenceforth be referred to as semi-structured reflective group interviews.
Seeing that certain group dynamics were already in place, a semi-structured reflective group interview was a natural extension of relationships established during the modelling workshops. In the group interviews, it was possible for one student to remember an instance that slipped another’s memory, to delve into their own understanding and to recall the sequence of (often implicit) thought processes that could not be explicitly traced via content analysis. Ultimately, this type of interview provided a platform where students could freely express views on their activities, experiences and perceptions; that is, how, why and when specific actions developed.
Reflective interview questions (see Appendix I) stimulated students’ recounts of their actions, activities, thinking processes and the sequence of forming different objects (Silverman, 2014). To this end, supplying interviewees with copies of their workshop documents served as a visual reminder to assist in recalling processes used.
160 In a group interview, questions like ‘have you considered another strategy’; ‘why did you opt to do it this way’ and ‘can you defend your decision’ would delve deeper into students’ apparent understandings. Through this data source, the evidence collected via worksheet documents was enriched and could explain more comprehensively the complexity of students’ actions, activities, object formulations and processes followed during the modelling tasks. While content analysis of worksheet documents could reveal specific results, it was not suitable to detect reasons why particular decisions were made and what other strategies were contemplated before it had been rejected (Schoenfeld, 1985). The actions, activities and processes to generate objects ultimately carried a substantial amount of information that could be unpacked during semi-structured reflective group interviews. Both introspective and retrospective explanations (Schoenfeld, 1985) were elicited as it was important to know the reasons why certain attempts failed, but also to give students an opportunity to critique their own actions and results. In this sense, content analysis provided macroscopic insights in modelling activities, but during the semi-structured reflective group interviews, stimulated recall (Desoete, 2009) disclosed thinking processes on a microscopic level.
The pairing of content analysis and semi-structured reflective group interviews added diversity of methods and allowed for an exploration into the understandings and meanings of participants. This is the essence of qualitative research (Creswell, 2014).
4.5 THE EMPIRICAL ENVIRONMENT