# Challenges of teaching mathematical symbolisation

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## CHAPTER 3: RESEARCH METHODOLOGY

This chapter discusses the research methodology and design, including sampling, population, establishing rigour during and after data collection, ethical considerations and data analysis. The chapter explains how the research was conducted. A number of measures were taken to ensure that quality data is collected. Ethical considerations and trustworthiness are also discussed.

Research questions

The selection of the methodology for collecting and analysing data was guided by the following research questions:

1. What challenges do secondary school learners encounter when interpreting and using mathematical symbols to understand mathematical concepts and problem solving procedures?
2. What instructional strategies can mathematics teachers use to mitigate the effects of symbolic obstacles?

Research Methodology

Methodology encompasses concepts such as research paradigms, theoretical models and quantitative or qualitative techniques. Burns and Grove (2003) describe methodology as the means or methods of conducting research, which includes the design, setting, sample, methodological imitations, and the data collection and analysis techniques in a study. According to Holloway (2005), methodology means a framework of theories and principles on which methods and procedures are based. In this study, methodology describes how the research was conducted, what data was collected and how it was analysed.
A mixed methods approach was utilised in this study. Mixed methods research refers to quantitative and qualitative procedures of collecting and analysing data in the study (Creswell, 2013). Creswell and Plano-Clark (2007) define mixed methods as a methodology that involves the collection and analysis of qualitative and quantitative data in a single study or series of studies. The main reason for mixing the two research approaches is to obtain better understanding of research problems that either approach cannot achieve alone. The study focused on exploring and describing the experiences of learners as they struggle with the symbolic barrier to understanding mathematical concepts therefore the research approach was dominantly qualitative.

This study is guided by a constructivist paradigm. Creswell and Plano-Clark (2007) defined a paradigm as a worldview. A paradigm is an interpretative framework, which is guided by a set of beliefs and feelings about the world and how it should be understood and studied (Lincoln & Guba, 2000). Constructivism as a paradigm posits that learning is an active, constructive process. The learner is an information constructor. The goals of constructivist research are understanding and structuring, as opposed to prediction. This study explored and described the experiences of FET band learners as they integrate the symbolism in mathematical concepts. The conception of mathematical symbols is constructed through the APOS, Symbol sense, and Procept and Algebraic Insight theories. Different types of data have to be used to construct a complete picture of mathematical symbols.

Qualitative Approach

The dominant research approach for this study is qualitative, since the natural setting is the direct source of the data (Fraenkel & Wallen, 2003). For this study, data was collected from the participants in their natural setting without controlling any aspect of the research situation. Qualitative methodology is interactive and interpretive. In the interaction between the researcher and participants, the researcher discovers the participant’s world and interprets it (De Vos, 2002). This study intended to find out challenges and difficulties learners encounter when dealing with mathematical symbols to develop concepts in the teaching and learning process. The first research question for this study was best answered through a qualitative paradigm. This design allows an in-depth understanding of learners’ challenges about the use of symbols in algebra and in exploring the factors that affect them in learning algebra. In this study, a qualitative method explored and described the challenges teachers and learners encounter when dealing with mathematical symbols, learners’ interpretation of mathematical symbols and instructional strategies to reduce symbolic obstacles.

Quantitative Approach

Quantitative approach measures and analyses the causal relationships between variables. In order to eliminate the weaknesses and limitations of qualitative and quantitative approaches, Laxman (2015) suggests combining them in a mixed methods design. The main weakness of the quantitative paradigm is that the researcher is inseparable from the object of observation (Kura & Sulaiman, 2012). On the other hand, the qualitative research does not generate predictive models that generalise to larger populations. The quantitative paradigm tests and validates existing theories generalising research findings (Johnson & Onwuegbuzie, 2004). Thus, the strengths of both paradigms were combined to offset their mutual limitations.

### Research Design

Research design is the overall plan for obtaining answers to the research questions (Polit Beck, 2004). It is a plan of action that links the philosophical assumptions to specific methods (Creswell, 2013). The research design for this study is in two levels: the logic of the research and at another level, the research design reflects on the purpose of the inquiry, which in this case, is both exploratory and descriptive.
Exploratory research examines a theoretical idea. The researcher has an idea and seeks to understand more about it. This study was informed by the researcher’s observation of learners’ use and manipulation of mathematical symbols without understanding their meanings or concepts they represent. The exploratory research lays the groundwork for future studies on the idea. What is being observed might also be explained by a currently existing theory. Exploratory research identifies the boundaries of the environment in which the problems, opportunities or situations of interest are likely to reside and to elicit the salient factors or variables that might be found there and be of relevance to the research.
On one hand, a descriptive research design provides an accurate and valid representation of the variables that are pertinent and relevant to the research question (van Wyk, 2012).
Methods, on the other hand, refer to specific techniques that are used for data collection and analysis (Creswell, 2003). Kumar (2010) viewed it as a blueprint of how a research study is conducted. It operationalises variables so that they can be measured from a sample and analysis of the data therefrom. This procedure must be carefully adapted by the researcher to answer questions validly, objectively, accurately and economically. Thus, the research design minimizes the chances of drawing incorrect causal inferences from data.

Descriptive Research Design

A descriptive research design was used for the quantitative data collected using the questionnaire survey. Quantitative research designs emphasise objective measurements of data (Babbie, 2010). The study described the status of learners’ understanding of mathematical symbols and their use in conceptual understanding. The dependent or criterion variable is a phenomenon that one is attempting to explain or predict. In this study, the phenomena of interest cover the difficulties that learners and teachers experience due to mathematical symbolisation. Since this study is non-experimental, there are no independent variables that can be manipulated to explain or predict the dependent variable. However, extraneous variables such as demographics of participants need to be controlled in order to obtain meaningful results. Hence, variables such as grade, gender, social economic status, age, home language, geographical location of participants and ethnicity were considered to see the extent to which they influence learners’ understanding of mathematical symbols.

Phenomenological research Design

A phenomenological research study attempts to understand people’s perceptions, perspectives and understandings of a phenomenon (McConnell, Chapman & Francis, 2009). The aim of phenomenological study is to obtain descriptions of experiences from learners who experience problems with mathematical symbols. The aim of the research is to describe the phenomenon of learners’ symbol sense as accurately as possible. Similarly, Sterley (2014) believes that phenomenologists seek to understanding phenomena from the perspectives of the participants. From these descriptions, themes, typologies emerge. It involves interpreting the original descriptions of symbols using reflective analysis and interpretation of the participants’ accounts. Primary methods of data collection are audio-recorded conversations.
A phenomenological methodology was also utilised in this study. Interviews were designed to build a description of the participant’s experiences with symbols. The fundamental assumption made is that the important reality is what people perceive it to be (Alibakhshi, 2015). This perception builds a description of a learner’s conception of mathematical symbols that build mathematical concepts. Thus, the phenomenological interview is a technique ideally suited for data collection in this study.

Intuiting

This process involves thinking through the data in order to obtain a comprehensive and accurate interpretation of what participants mean in a particular description (Leech & Onwuegbuzie, 2007). In order to achieve this, the researcher remains open to the meanings and issues raised by participants in terms of the difficulties they experience with mathematical symbolisation. Intuition leads to a common understanding about the phenomenon that is being studied. It also requires that the researcher creatively analyses the data until such a common understanding emerges. The researcher must be totally immersed in the study of the phenomenon.

Analysing

Analysing involves listening to, comparing and contrasting descriptions of learners’ conceptions of mathematical symbols in to identify the essence of the phenomenon under investigation. Analysis seeks to make sense of the essential meanings of the phenomenon. Common themes emerge as the researcher works with the descriptive data.

Bracketing

Bracketing is a qualitative research technique that suspends assumptions and presuppositions about any knowledge of learners’ difficulties with symbolisation and teachers’ approaches to symbolisation to limit interference with the information given by the participants (Tufford & Newton, 2010). Bracketing improves rigour and reduces bias in research. In this exploration, the researcher suspends his assumptions and preconceptions especially during data analysis. As recommended by Castellan (2010), the researcher remained neutral with respect to belief or disbelief in the existence of the phenomenon. The researcher first identified learners’ preconceptions about mathematical symbolisation. Researcher also had to suspend all prior knowledge about learners’ challenges, to allow the trustworthy “truth” to emerge.

Describing

This is the final step in which the researcher describes distinct, critical elements of the phenomenon. The researcher avoided premature to description of the phenomenon, a common methodological error in this type of research (Vilakati, 2009). In this study, phenomenological describing involved classifying all critical elements common to learners’ challenges in understanding mathematical symbols.
‘Memoing’ was also used in this study. This is recording what the visual, auditory impressions and thoughts of the researcher in the course of collecting and reflecting on the process Groenewald (2004). The researcher complied field notes of what participants were raising during the data-collection process and reflected on the data analysis. As recommended by Ejimabo (2015) the researcher kept updated memos and later correlates them with the data.
In view of the issues discussed above, phenomenology was considered the best method and approach to address the qualitative part of the study.

#### Reflective analysis

Reflexivity is an aspect of a phenomenological research in which researcher assumes the roles of a researcher and the participant at the same time (Finlay, 2012). Researchers continuously reflect on their own preconceived values, participants’ perception of the researcher and reflecting on how it will influence the data collected. In this study, the researcher maintained as self-monitoring stance in order to prevent bias and increase objectivity of the study. As recommended by Holloway and Wheeler (2002) the researcher continuously reflected on his own feelings, actions and conflicts during the research so that they do not affect the credibility of the study.

Mixed Method Approach

Rich and Brown (2014) defined mixed methods as ‘research in which the researcher collects, analyses, mixes, and draws inferences from both quantitative and qualitative data in a single study. Creswell et al (2006:5) define it as “…. a methodology, it involves philosophical assumptions that guide the direction of the collection and analysis of data and the mixture of qualitative and quantitative approaches in the research process”. The researcher selected this approach on the basis that the combined use of quantitative and qualitative approaches provides a better understanding of research problems than either approach alone. Integrating methodological approaches strengthens the research design, as the strength of one approach offsets the weakness of the other (Creswell & Plano-Clark, 2011). The other practical benefit of using a mixed method research is derived from Baran and Jones (2016) who reveal that it encourages interdisciplinary collaboration and use of multiple paradigms in a research.
Although there are on-going debates about whether MMR is a research design or methodology, this study takes a middle ground. MMR is a research design with philosophical assumptions as well as quantitative and qualitative methods. Wilson (2016) describes mixed methods as a research methodology in which data is collected, analysed, and inferences drawn from both quantitative and qualitative data in a study. Qualitative and quantitative designs, methods, data collection and analysis techniques were utilised to provide data that was later mixed to provide a big picture of the findings of this study. The choice of a mixed method approach was derived from the nature of research questions and the kind of instruments used to solicit the data.
The first research question for this study seeks to explore the challenges that learners encounter when interpreting and using mathematical symbols to understand mathematical concepts and problem solving procedures. The second research question is based on instructional strategies that mathematics teachers can use to reduce the effects of mathematical symbolisation obstacles. To address these research questions a survey questionnaire consisting of closed and open-ended questions was used. Quantitative data analysis methods were used to summarise data in the form of descriptive statistics. Open-ended questions were analysed by drawing a list of broad categories that were later qualitatively researched using focus group interviews. Thus, the study utilised qualitative research to gain access to participants’ views about symbolisation while quantitative research allow researcher to make statistical inferences about the phenomenon.

Declaration
Dedication
Abstract
1.0 CHAPTER 1: INTRODUCTION
Introduction
Background of the study
Hypothesis
Definitions of terms
Summary
2.0 CHAPTER 2: LITERATURE REVIEW
Mathematical Symbolisation
Challenges of teaching mathematical symbolisation
Theoretical framework
Justification for combining frameworks
Summary
3.0 CHAPTER 3: RESEARCH METHODOLOGY
Research questions
Research design
Ethical considerations
Summary
4.0 CHAPTER 4: DATA ANALYSIS
Data Analysis
Inductive analysis
Thematic Analysis
Problems encountered in data collection
Summary
5.0 CHAPTER 5: DISCUSSION AND IMPLICATIONS
Discussion
Implications
Summary
6.0 CHAPTER 6: SUMMARY OF THE STUDY
Summary of the study
Conclusions
Recommendations
Limitations of the study
Suggestions for further research
Closing Remarks
7.0 REFERENCES
8.0 APPENDICES
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Mathematical Symbolisation: Challenges and Instructional Strategies for Limpopo Province Secondary School learners