METHODOLOGICAL APPROACH

According to Creswell (2014:5), the broad research approach is the “plan or proposal to conduct research”. The purpose of the research aids in deciding the methodology and the design of the research and the methods are the range of approaches used to gather data (Cohen, Mannion and Marrison, 2011). As a researcher it is very important for me to design the research method. The aim of a research methodology is to assist me to develop a research strategy, understand the process of data collection and to be able to analyse the data.

All aspects of this research study are interrelated as it determines why the qualitative, quantitative or mixed approaches were used. According to Creswell (2014:4) qualitative research is used to explore and comprehend the meaning that individuals ascribe to a human problem and quantitative research tests the “objective theories by relationships variables”. In the qualitative research the researcher is the primary tool to conduct the research. Qualitative research allows for the phenomena under investigation to be explored and is designed to support and understand people in the environment they live or work in. It presents data of a descriptive nature, in this research observing the pre-service teachers teaching mathematical problem solving in the classroom. In summation, the methodological approach is reference to the broad research methodologies or paradigms that guide the study.

The data was collected more or less simultaneously and this is known as convergent parallel mixed methods. Convergent parallel mixed methods are a “form of mixed methods” (Creswell,

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2014:15) as it allows the researcher an opportunity to amalgamate both qualitative and quantitative data in order to present a “comprehensive analysis” of the research problem (Creswell, 2014:15).

I engaged in multiple methods to acquire evidence and knowledge on the parallel phenomenon using different research measures. Multiple methods involve mixed methods, i.e., when two or more methods are used to collect data, in this instance the qualitative and quantitative methods (Cohen et al, 2011). Since this study will follow a mixed methodological approach by conducting a qualitative and quantitative study, it will result in the amalgamation of data and this is called triangulation. Triangulation is when qualitative and quantitative data (methodological triangulation) is obtained from several sources to strengthen the research design (Kothari, 2014) and is used to evaluate and to ascertain if it corroborates and validate the research findings (Creswell, 2014). If one method shows weakness during the study, there is an opportunity through triangulation to verify the facts and findings (Kothari, 2014; Denizen and Lincoln, 2011). Triangulation improves trustworthiness and uncovers biasness.

Denizen and Lincoln (2011) described triangulation as crystallization which is more than the concept of triangulation. In this process a story is told through data gathered from a range of sources. The crystallization process involved the temporary suspension of examining data in order to reflect on the analysis experience and an attempt to identify and articulate patterns and themes during the immersion process. The immersion is the process when the researcher immerses himself in the data he has collected by reading and examining the data in detail and the dual processes continue until all the data have been examined and substantiated in detail (Denizen and Lincoln, 2011; Cohen et al, 2011).

Cohen et al (2011) and Lincoln, Lynham and Guba ( )*stated that qualitative research is interested how individuals understand themselves and construct meanings of their lives. The qualitative research is designed to enlighten the researcher (how) and the why (meanings) things ensue in order to get an in-depth understanding of the situation. Cohen et al (2011), Denizen and Lincoln (2011) and Lincoln, Lynham, and Guba ( )* described that the quantitative research method as being able to generate data that would be justifiable and unbiased, as they produce knowledge which de-personalizes the obtained data. The use of both quantitative and qualitative methods is therefore necessitated by the need to verify and understand the perspectives of the participants.

121 4.3 RESEARCH DESIGN

According to Creswell (2014) and Denizen and Lincoln (2011) a research design comprises an investigation with qualitative and quantitative and mixed method approach. This study will follow a mixed methodological approach by conducting both a quantitative and qualitative study. For the purpose of this study I used a combination of research instruments. The instruments used were a questionnaire for pre-service teachers (Annexure 1), observation of pre-service teachers in the classrooms (Annexure 3), semi –structured interviews with pre- service teachers (Annexure 2), evaluation worksheets for both pre-service teachers (Annexures 4 and 5) and learners (Annexures 6,7,8 and 9) and examination of learner‟s books to garner evidence. I used these instruments as it would assist me in garnering the necessary data for this study.

These instruments are discussed below. 4.4 QUESTIONNAIRE

A questionnaire is a paper and pen method of collecting information from participants. It is a popular and advantageous way to gather data from a wide range of issues as it is through this medium that data is collected. As a data collecting instrument, a questionnaire is often used in a wide range of settings to gather a variety of information. The questionnaire for this study (Annexure 1) comprehensively covered the relevant areas that I prosed to investigate. The said questionnaire was divided into three sections. The first section focused on the pre-service teacher‟s knowledge of the Curriculum Statement and Assessment Policy (CAPS) and their knowledge of problem solving; the second section examined their classroom observation and teaching experience knowledge relating to strategies used to teach problem solving and the third part focused on the pre-service teacher‟s knowledge on visualization and their ability to use this to teach and solve problems in the classrooms. The questionnaire comprised multiple choice questions, dichotomous questions, open and closed ended questions and Likert scale ratings was used to obtain an in-depth understanding on the pre-service teacher‟s knowledge and utilization of visualization skills in teaching and solving mathematical problems.

A questionnaire is also practical in that some will find it easier to write than to talk and much more information can be garnered from participants via a questionnaire than they will contribute to through an interview (Cohen et al, 2011). The questionnaires were handed out to the pre-service teachers during their lectures and they were requested to return it within the agreed time frame, that being a week. I allowed them this period as they had lectures and assignments to contend with. It also gave them an opportunity to complete it on their terms and

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in an independent environment. The stress free environment allowed the pre-service teachers to think over their responses before responding to the questions. Removing myself from the equation (environment) as their lecturer alleviated the possibility of biasness as they did not have to provide responses to satisfy me. Furthermore, the neutral environment also assured their confidentiality and anonymity.

In order to obtain the appropriate data the questions should be relevant and carefully constructed. Wherever the alternatives were being given the „answers‟ should not be obvious to the respondent. Explanatory type of questions must be free from ambiguity. If this exists then the respondents will have limited opportunities for elaboration. Furthermore they can leave questions unanswered or provide superficial responses. In order to collect authentic data, the set questions must give all respondents an opportunity to provide an answer to all the questions. The respondents must be competent to provide the responses. All of these factors were considered when designing the questionnaire. As this study focused on the responses of the pre-service teachers it qualified their responses to the study.

When conducting a research study using a questionnaire the following factors, namely, informed consent, confidentiality, and the right to withdraw without prejudice have to be considered. A covering letter (Annexure 12) outlining the rationale of the study and afore mentioned factors were explained and attached to the questionnaire. Another letter (informed consent) from the institution these pre-service teachers attend was attached as well. These letters gave an overview of my intention as the researcher and purpose of the study. I allowed the pre-service teachers an opportunity to peruse both the attachments. I drew their attention to their rights to confidentiality and the right to withdraw from the study. I placed a lot of emphasis on both these aspects as I didn‟t want the pre-service teachers to feel threatened and prejudice the study by not responding objectively.

Whilst the questionnaire can be considered as the prime method of collecting data in a research study, it has its limitations. The response rate is low from individuals; there is a refusal to co- operate; providing brief explanations or ambiguous and incomplete answers to questions (Kothari, 2014; Cohen et al, 2011).

4.5 OBSERVATION

Observations require a researcher to develop an observation schedule that will direct what needs to be observed by the researcher. According to Cohen et al (2011) observation is the gathering of information as they occur and at the site it occurs as it allows for obtaining „live‟ data from „live‟ situations. The lesson observations allowed for the direct collection of data from an

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authentic source. The goal of the lesson observations was to understand the approaches and specific methods used by the pre-service teachers in the classroom. According to Lester (1985) classroom observation is the most appropriate medium to obtain first-hand information. Information can be gathered from extensive observation of actual teaching of authentic mathematics in the classroom. For the purpose of this study, the information is sought by the researcher in his own direct surveillance as it is currently happening.

Lester (1985) advocated qualitative methods for conducting research in problem solving. He stated that “adopting a holistic view of problem solving and problem solving instructions necessitates the use of naturalistic (inquiry) rather than traditional scientific research paradigms” (Lester, 1985:52). Naturalistic inquiry also known as constructivist inquiry (Lincoln and Guba, 1985) is the reference to qualitative research been done in a natural setting to examine a social or human issue. In this study the natural setting is the classroom and the area of investigation is how pre-service teachers teach problem solving. The pre-service teachers were observed teaching mathematics during their teaching practice at their allocated schools. I needed to get a sense of their teaching practices, procedural knowledge and how they taught problem solving. Observing the pre-service teachers in different geographical settings allowed for the testing of various teaching practices in the classroom.

I used a pre-determined observation checklist (Annexure 3) when observing the lessons in the classrooms. The reason, using reliability as a factor, was to ensure that all pre-service teachers were observed accordingly so that the data collected can be considered valid and free from bias. 4.6 SEMI-STRUCTURED INTERVIEW

Interviews are a valuable source of information. As interviews are face to face interviews it can be classified as a two directional conversation (Cohen et al, 2011; Maree, 2007) as the interviewees are able to express themselves by providing their own explanations and interpretation of the questions. Semi-structured interviews allow for leading questions to be asked of all participants and also allows for the possibility of asking unplanned questions based on the responses of the participants. The responses to the questions are direct and given without assistance or prompts. The interview, in my opinion, provides more reliable data than a questionnaire or the evaluation worksheet. The answers to the questionnaire can be obtained from the internet or copied from colleagues whilst in the interview process you have to give your own true response to the questions asked.

A semi-structured interview was used. I intended discovering first-hand the thoughts of the participants. A list of standardised open ended questions was asked (Annexure 2) and the

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responses were recorded. The standardized framing of the questions allows the interviewer to ask more than one interviewee the same question. This reduces researcher bias and allows for an increased intensity of comparability (Cohen et al, 2011).

The questions were based on visualization skills and problem solving strategies and how it was used in the classroom. Open ended questions also formed part of this interview. The use of open ended questions in these face to face interviews reveals the thoughts of the interviewees. This secures a more spontaneous and factual response. In an interview the interviewees reveal their true thoughts and feelings without inhibitions. These kinds of interviews allow for additional questions to be asked in the event of obtaining further clarity to a response. As the interviewer I am in a position to ask the interviewee additional probing questions in order to delve further or to ask them to elaborate their responses (Cohen et al, 2011; du Plooy-Cilliers et al, 2014). The asking of the additional questions also avoids misinterpretations of questions. The interview allowed me an opportunity to get a sense of their beliefs on problem solving and the teaching strategies or practices they employed. I was very much interested on how problem solving was taught using visualization.

According to Maree (2007:87) a semi structured interview can be used to “to corroborate data emerging from other sources”. The data obtained from the interview was used to corroborate data obtained from the questionnaire, lesson observation, evaluation worksheet and learner‟s books.

Scheduling the interviews with the pre-service teachers was problematic. The participants in the interview process all live in different areas from their institution and logistically it was difficult to get them together for the interview process. This was further compounded as they attended lectures at different times. As time was a determining factor, all the interviews were scheduled for approximately thirty minutes. This allowed me access to the pre-service teachers during their non-lecture periods. To ensure reliability I formulated the questions as clearly as possible for all pre-service teachers. The interviews were audio recorded, transcribed, coded and categorised to suit the research questions.

The interview as a data collecting process has its disadvantages. It is a very time consuming and an arduous task to transcribe and compile the data. It also brings the problem of the interviewee to be punctual for the interview. With the interviewer been part of the interview it can lead to the contamination of data. The possibility may arise that the interviewer will ask additional questions to obtain data to suit the research study.

125 4.7 EVALUATION WORKSHEETS

Two evaluation worksheets (Annexures 4 and 5) comprising mathematical terminology and non-routine mathematical questions were given to the pre-service teachers. The reasoning behind using mathematical terminology and concepts in the evaluation worksheet was to gauge their pedagogical knowledge. The non-routine mathematical questions were taken from previous Association for Mathematics Educators of South Africa (AMESA) and Mathemagica Plus mathematics problem solving competitions. The purpose of administering this evaluation worksheet was to determine the methods or strategies used by the pre-service teachers when problem solving. This evaluation worksheet was given to the pre-service teachers as an activity during a routine lecture period and collected thereafter.

An evaluation worksheet comprising five non routine problems (with/without diagrams) was given to learners in grades four to seven (Annexures 6, 7, 8 and 9). These evaluation worksheets were given to the pre-service teachers to administer to their classes during their teaching practice sessions. The purpose of the worksheet was to give the learners an opportunity to show how they solved these problems using mathematical strategies and visual skills. The evaluation worksheet was within the ability levels of these learners and was sourced from AMESA and Mathemagica Plus problem solving competitions. Every endeavour was made to ensure that the learners had no come across these questions.