# Heuristics, problem-solving processes and mathematical modelling

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## CHAPTER THREE RESEARCH METHODOLOGY

Research requires decisions with regard to sampling, instrumentation, data-collection and the methods of data analyses (McMillan & Schumacher, 2001). The decisions in respect of method made in this chapter are aimed at establishing the effects of the problem-solving heuristic instructional method designed for this study as will be explained by the preliminary genetic decomposition in section 3.5.3.1. This chapter is organized into 10 main sections. Section 3.1 explains the research paradigm, section 3.2 discusses the research design, section 3.3 discusses the population, sampling and sampling technique, section 3.4 explains the observation of regular mathematics educators during the problem-solving heuristic instructional method, section 3.5 explains the design and implementation of the heuristic teaching design, section 3.6 explains how the effects of the problem-solving heuristic instructional method on learners’ achievements in algebra were measured, section 3.7 discusses the data collection instruments used for this study, section 3.8 discusses the data collection procedure used for this study, section 3.9 explains how data gathered was analysed, and section 3.10 discusses ethical consideration made before data collection.

A research paradigm is “a basic set of beliefs that guide [the] action” of the researcher during research (Guba, 1990, p. 17). A research paradigm explains the researcher’s theoretical lens which influences his or her research methods (Dobson, 2002). A study can be executed with significant achievements if a research paradigm that best suits the study is used (Flowers, 2009). This study is underpinned by the philosophy of pragmatism. Pragmatism, which arose out of the work of William James, John Dewey, and Charles Sanders Peirce (Cherryholmes, 1992), focuses on the research problem and uses pluralistic approaches with the hope of understanding the research problem (see Rossman & Wilson, 1985). The motivation for choosing a research paradigm based on pragmatism was to enable the study to explain fully the effects of a problem-solving heuristic instructional method and how it evolves using both qualitative and quantitative research methods in data collection and analysis. Through this approach the study hoped to report on the true nature of a problem-solving heuristic instructional method on Grade 6 learners’ achievement in algebra. Pragmatism looks at what, and how, to research, and bases decisions on the consequences of the research problem. Hence it makes use of multiple approaches for collecting and analyzing data rather than subscribing to only one method (Cherryholmes, 1992; Creswell, 2009; Morgan, 2007). Pragmatism applies mixed-method research where the researcher draws liberally from both quantitative and qualitative research methods. In this way, the researcher freely chooses methods, techniques and procedures of research that best suit the aims and objectives of the research (Creswell, 2009).

RESEARCH DESIGN

The study followed a mixed-method approach to determine the effects of the problem-solving heuristic instructional method in learners’ achievements in algebra. A mixed-method approach combines both qualitative and quantitative methods of data collection and analysis that can make the study answer the research questions as set out in the study (Cresswell, Klassen, Plano Clark and Smith, 2011). The qualitative component of the research design involved firstly a pre-intervention class observation of Grade 6 mathematics lesson to identify the teaching methods being used by the educators and secondly, the intervention which entailed the design and implementation of the problem-solving heuristic instructional method which helped explain how learners’ knowledge in algebra evolves when they are taught algebra through the problem-solving heuristic instructional method. The quantitative component of the research design involved a non-equivalent control group quasi-experimental design with pre-test and post-test measure and was used to measure the effects of the problem-solving heuristic instructional method on learners’ achievements in algebra.

### Rationale for using a mixed-method approach

A research problem should be investigated holistically, based on the initial premise that human beings who formed the basis of this research are influenced by various factors in several ways. Therefore, one should not only consider the intellectual or psychosocial aspects of a person’s being, but rather all aspects relating to the person-in-the-world (Vrey, 1984). A mixed method approach was chosen since “mixed methods procedures employ aspects of both quantitative methods and qualitative procedures” (Creswell, 2003, p. 17) and the two methods complement one another. In a way, the limitations inherent in either of the methods can be neutralized by combining the effects of both methods (Creswell, 2002). In this way, several challenges that would have arisen if only the quantitative or the qualitative method was used were overcome. Bias that developed in one method neutralizes and cancels bias in the other method, and vice versa. The use of multiple approaches gave deeper insight into the effects of the problem-solving heuristic instructional method. The rationale for the choice of the design was therefore to enable the researcher to determine the impact, if any, of the problem-solving heuristic instructional method on learners’ achievements in algebra and explain how the instruction was used.
Combining both the qualitative and quantitative methods also corroborated the findings of both methods. The quantitative findings informed and supported the richness of the qualitative findings by providing statistical evidence. Hence a more comprehensive investigation of the problem at hand was evident.

Description of the research site and the participants

The research was done in four community quintile 1 schools in the Zululand district of KwaZulu-Natal. The Zululand district has a total land area of 14799km2, and the four schools were at least 50 kilometres away from one another. Two schools represented the control group and the other two schools represented the experimental group. The study was conducted in an environment (classroom) that was familiar to the respondents (learners). This made the implementation of the investigation more convenient and easier to manage. The participants in the research were 198 Grade 6 learners from the four schools, the intermediate phase heads of departments (HODs) in all four schools and the four mathematics educators in all four schools. The two schools in the control group were made up of the two Grade 6 classes with a population of 51 and 55 respectively. The two groups of the experimental group were also made up of the two Grade 6 classes in the respective schools, with populations of 49 and 43 learners respectively. The two Grade 6 classes in school 1 and school 2, the control group, will henceforth be referred to as control group 1 and control group 2, whereas the two Grade 6 classes in school 3 and school 4 will be referred to as experimental group 1 and experimental group 2. Table 3.1 shows the participants at various stages of data collection and Table 3.2 shows the timelines of the various stages of the study.

POPULATION, SAMPLE AND SAMPLING

Population of the study

The population of the study was Grade 6 learners in quintile one schools in the Zululand district of Kwazulu-Natal. Schools from quintile one were chosen as the population for the study because this quintile has the poorest mathematical average in the ANA examination (see section 1.1). Most quintile one schools are in the deep rural areas where there are most commonly inadequate resources in terms of teaching and learning materials, infrastructure and a lack of qualified educators. The study chose Grade 6 learners because the foundational development of algebra in the South African school system begins in the intermediate phase of which Grade 6 is the last grade in that phase. Secondly, the drop in the number of both learners who choose to take mathematics in Grade 10 and the poor quality of mathematics grades in the Grade 12 National Certificate examination have their roots in the teaching of mathematics at the basic level where learners fail to acquire basic mathematical skills (see Campbell Prew, 2014). According to Campbell and Prew (2014), there is an urgent need for attention to pedagogy and content at the upper primary and lower secondary level. Moreover, some studies have revealed that marked changes in learners’ problem-solving skills are observed between the ages of 11 to 14 (see Proctor, 2010; Zhu & Fan, 2006). Yan (2000) also explains that the optimal age at which learners are able to develop their problem-solving skills is from 10 to 16 years. On this basis, the study deemed it fit to develop and test this problem-driven teaching method with Grade 6 learners.

Sample of the study

The sample for the study was intact Grade 6 classes from four schools in the Zululand district of Kwazulu-Natal; one school was selected from each of the four circuits in the district. Although the study took pains to ensure that the schools selected had comparable characteristics in terms of their teaching and learning materials, infrastructure and educators’ qualifications, there were still some inherent differences in these factors among the schools chosen for this study.

#### Sampling techniques used in this study

Purposive sampling was used to sample the four schools used for this study. (Louis, Lawrence, & Keith (2007, pp. 114-115) explains that In purposive sampling, the researcher handpicks the cases to be included in the sample on the basis of their judgement of their typicality or possession of a particular characteristics being sought. In this way, they build up a sample that is satisfactory to their specific needs.
The motivation for choosing a purposeful sampling technique was to ensure that the schools selected were of the same quintile and at the same performance level in terms of their Grade 6 end-of-year results and that these results, taken over a period of time, were comparable. Furthermore, the distance between any two of the four schools measured at least 50 kilometres. This factor ensured that learners in the control and experimental group did not meet each other. Maintaining a long distance between participating schools “prevents diffusion, contamination, rivalry and demoralisation” (Gaigher, 2006, p. 37) as the measure of the true effects of the problem-solving heuristic instructional method may be compromised if the learners in the control and experimental group interact with each other during the intervention stage (Shea, Arnold & Mann, 2004). Contamination also has the propensity to reduce the statistical significance, as well as the observed differences, between the control and experimental groups caused by exposing learners in the control group to the intervention (Howe, Keogh-Brown, Miles & Bachmann, 2007). One study notes that Potential extraneous variables should not prejudice the relationship between the independent and dependent variables” as it may “lead to ambiguous results” of that study (Tierney, 2008, p. 2). This made random sampling basically impractical.

PRE-INTERVENTION CLASS OBSERVATION OF MATHEMATICS LESSONS OF SCHOOLS

Pre-intervention classroom observation was used to investigate the teaching methods adopted in all four schools chosen for this study compared to the problem-solving heuristic instructional method, as well as to examine the similarities and differences in the quality of mathematics teaching among these four schools. This enabled the study to assess whether any of the four schools had an advantage over the others with regard to teaching and learning. Among issues compared were the educators’ lesson plan on the activities they intended to use to develop learners’ understanding in the concept being taught; effective communication between the educator and learners; level of learners’ participation and enthusiasm in the learning process and knowledge creation; integration of authentic real-life problems into the teaching and learning process and whether the specific educator developed a particular mental construction to develop learners’ understanding of the topic being taught. Classroom observation is an excellent instrument to understand the real picture of any social phenomenon (Mulhall, 2003). The behaviour of learners and educators and the interactions between them can best be studied through natural observation of their activities in the classroom (Gay, Mills & Airasian, 2006). Through class observation the study hoped to control some of the variables that could influence the outcome of the problem-solving heuristic instructional method or pose a threat to the internal validity of the study.

PROBLEM-SOLVING HEURISTIC INSTRUCTIONAL METHOD

The teaching treatment was conducted from July to October 2014. The researcher negotiated with the school authorities of the two experimental schools to allocate one hour per week as it was impossible to conduct the study after school hours for operational reasons. In total 9 hours were used in teaching each of the Grade 6 classes in the two experimental schools.

#### Designing the problem-solving heuristic instructional method

The problem-solving heuristic instructional method is underpinned by two theories, namely the APOS theory and the modelling and modelling perspectives approach. Adopting them as guide enabled the researcher to develop a preliminary genetic decomposition, which is a specific mental construction learners may make as they develop a conception in algebra, and this was implemented flexibly. Problem-solving entailing MEAs informed the pedagogical approach used to teach Grade 6 learners algebra. The APOS theory was used as a framework to develop learners’ understanding of algebra. The researcher developed MEAs with which learners are familiar and commonly experience in their daily lives. Most importantly, the modelling-eliciting activities featured components of all six principles of the modelling and modelling activity. The modelling-eliciting activities create the necessary environment for learners to develop a more comprehensive understanding in algebra. When learners create meaning from their own symbolic representation it could be hypothesized that meaningful learning (Ausubel, 1962) is promoted, as opposed to imposing a system of symbols and notations on learners (Chamberlin & Coxbill, 2012). The activity designed had components that guided the development of learners’ conceptual understanding in algebra as it occurs through the mathematization of the modelling-eliciting activity.

Declaration
Abstract
Dedication
Acknowledgements
Abbreviations
chapter One Introduction
1.1 Background to the study
1.2 Problem of the study
1.3 Objectives of the study
1.4 Research questions
1.5 Hypothesis of the study
1.6 Justification for the study
1.7 Definition of key terms
1.8 Structure of the report
1.9 Reflecting on the chapter
Chapter Two: Problem-solving and learning of algebra
2.1 Heuristics, problem-solving processes and mathematical modelling
2.2 Overview of algebra
2.3 THEORETICAL FRAMEWORK
2.4 Reflecting on chapter two
Chapter three: Research methodology
3.2 Research design
3.3 Population, sample and sampling
3.4 Pre-intervention class observation of mathematics lessons of schoolS
3.5 problem-solving heuristic instructional method
3.6 Measuring effects of the problem-solving heuristic instructional method on learners’ achievement in algebra
3.7 Data collection instruments
3.8 Data collection and data collection procedure
3.9 Data Analysis 71
3.9.1 Qualitative data analysis of pre-intervention classroom observation
3.10 Ethical considerations
3.11 Summary and conclusion of chapter THREE
Chapter four: Presentation of the findings
4.1 Pre-intervention Classroom observations
4.2 Development and implementation of a problem-solving heuristic instructional model
4.3 Effects of the problem-solving heuristic instructional method on learners achievements in algebra
4.4 Analysis of sampled learners pre-test and post-test answers
4.5 Concluding remarks on findings of the study
Chapter five: Summary of the study, discussion, conclusions and recommendations
5.1 Summary of the Study
5.2 Discussion of the results in terms of the research questions
5.3 Conclusions
5.4 Recommendations
5.5 Limitations of the study
References
List of appendices
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AN INVESTIGATION INTO THE USE OF PROBLEM-SOLVING HEURISTICS TO IMPROVE THE TEACHING AND LEARNING OF MATHEMATICS