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CHAPTER TWO THEORETICAL FRAMEWORK AND LITERATURE REVIEW
This chapter introduces the theoretical perspectives used in the study and reviews the literature concerning imagery and visualisation in the learning of mathematical analysis by undergraduate students in particular.
THEORETICAL FRAMEWORK
The researcher’s experience in teaching mathematics in general made him aware of the well-known epistemological problem, i.e. that it is impossible to see or know what anyone is thinking. The only way to gain access to the students’ thinking processes was through their utterances, algebraic symbols, numbers, diagrams or graphs. Accordingly, the idea of using a semiotic perspective, which looks at the production of signs, became an attractive possibility as a means to an understanding of the students’ mathematical activities. The theoretical framework for this study draws from the Theory of Registers of Semiotic Representations (TRSR) (Duval, 1999, 2006, 2017).
The study also incorporates other perspectives of analysis that go beyond the semiotic approach but are relevant to the role of imagery and the visualisation of students’ thinking processes in mathematical analysis. Among them, the researcher highlights the visual-analytic coordination strategy (Zazkis, Dubinsky & Dautermann, 1996) as an analytical tool to explain the thinking processes of undergraduate students when solving mathematical analysis tasks. The researcher also recognizes other important frameworks such as the APOS theory (Dubinsky et al., 1991) as potential tools to analyse advanced mathematical thinking processes.
Theory of Registers of Semiotic Representations (TRSR)
Duval (2017) provides a useful formulation of the learning of mathematics using a semiotic perspective. The researcher agrees with Duval’s (1999) views that the only possible access to mathematical objects is through their representations in their different semiotic registers. Duval (1999) argues that semiotic representations play several fundamental roles in mathematics. As an example, semiotic representations refer to mathematical objects; they allow one to communicate about mathematics and they are necessary for mathematical processing. A mathematical activity can use a variety of semiotic representation systems, each with its own possibilities. The registers of semiotic representations comprise natural language (as used in definitions and proofs), numeric, algebraic and symbolic notations, geometrical figures and Cartesian graphs. Duval (2006) further argues that mathematical activity is a transformation of one semiotic representation into another in the same or different register.
In the discipline of mathematics, a representation is a symbolic, graphic or verbal notation to express concepts and procedures of the discipline, as well as their more relevant characteristics and properties. Representations can be classified in registers of representations (Duval, 1999). In the context of cognitive psychology, the notion of representation plays an important role in the acquisition and the treatment of an individual’s knowledge. Duval (1999, p. 1) points out that “representation and visualisation are at the core of understanding in mathematics”. According to him, …representation refers to a large range of meaning activities: steady and holistic beliefs about something, various ways to evoke and denote objects, how information is coded. On the contrary, visualisation seems to emphasize images, and empirical intuition of physical objects and actions (Duval, 1999, p. 1).
Duval (2017) further points out that no knowledge can be mobilised by an individual without a representative activity. This fact makes the study of representations very important in order to explain the understanding of the concepts and the learning of mathematics.
Comprehension of the theory on registers of semiotic representation requires consideration of three key characteristics (Pino-Fan et al.., 2017):
i. There are as many different semiotic representations of the same mathematical object as semiotic registers utilised in mathematics.
ii. Each different semiotic representation of the same mathematical object does not explicitly state the same properties of the object being represented; what is being explicitly stated is the content of the representation.
iii. The content of semiotic representations must never be confused with the mathematical objects that these represent.
It is important to point out that there are two fundamental cognitive activities within the TRSR: treatment and conversion. The activity of treatment on the one hand consists of a transformation carried out in the same register. In other words, only one register is mobilised. The activity of conversion, on the other hand, consists of the mobilisation from one register into another, where the articulation of representation becomes fundamental. According to Duval (1999), the study of the activity of conversion makes it possible to comprehend the close relation between “noesis” and “semiosis,” a relation which is essential in intellectual learning. Semiosis is the mobilisation and creation of mathematical signs while noesis is the action and effect of understanding (Duval, 1999). Semiosis is necessary for noesis. However, it must be taken into account that the operation of conversion brings some difficulties, including the fact that the representation of the source register does not have the same content as the destination register. Another difficulty lies in the treatment, which becomes complex because of the use of the register of natural language and those registers that allow ‘visualising’ (graphs, geometrical shapes, etc.). Figure 2.1 illustrates various representations in algebraic register of a function in space to its graphical representations in space and plane.
There is a need to observe that not all semiotic systems are registers of semiotic representations. Duval (1995) defined registers of semiotic representations as all semiotic systems that allow the construction of the mathematical concepts through the following cognitive activities: (i) representation of concepts in a given register; (ii) treatment of these representations within the same register; (iii) conversion of these representations from a given register to another. For instance, with the concept of function, there are graphic, algebraic, numerical and verbal registers. There might be others, but these are the most used in teaching. It is possible to carry out processing into each register, that is, transformations of the representations in the same register in which they were created. It is also possible to realise conversions between different registers of representations that are transformations of one representation made in a register into another representation in another register. In the instance of a function, a conversion can be a translation of the function’s tabular information into a graphic representation. Figure 2.2 provides a summary of how the transformations of treatment and conversion work.
CHAPTER ONE: INTRODUCTION
1.1 BACKGROUND TO THE STUDY
1.2 STATEMENT OF THE PROBLEM
1.3 RESEARCH QUESTIONS
1.4 RATIONALE FOR THE STUDY
1.5 SIGNIFICANCE OF THE STUDY
1.6 SCOPE AND DELIMITATION OF THE STUDY
1.7OPERATIONAL DEFINITION OF TERMS
1.8 CONCLUDING REMARKS
CHAPTER TWO: THEORETICAL FRAMEWORK AND LITERATURE REVIEW
2.1 THEORETICAL FRAMEWORK
2.2 REVIEW OF LITERATURE
2.3 CONCLUDING REMARKS
CHAPTER THREE: RESEARCH METHODOLOGY
3.1 RESEARCH PARADIGM
3.2 POPULATION AND SAMPLE
3.3 INSTRUMENTATION
3.4 VALIDITY OF COGNITIVE TESTS
3.5 PILOT STUDY
3.6 VALIDITY OF THE CLINICAL INTERVIEW
3.7 RELIABILITY OF THE CLINICAL INTERVIEW
3.8 DATA ANALYSIS
3.9 ETHICAL CONSIDERATIONS
3.10 CONCLUDING REMARKS
CHAPTER FOUR: FINDINGS
4.1 RESULTS OF THE COGNITIVE TEST
4.2 STUDENTS’ THINKING PROCESSES IN LEARNING OF MATHEMATICAL ANALYSIS ASSESSED FROM FOLLOW-UP INTERVIEWS
4.3 SUMMARY OF FINDINGS
CHAPTER FIVE: DISCUSSION OF FINDINGS
5.0 INTRODUCTION
5.1 THE NATURE OF IMAGERY AND VISUALISATIONS IN THE LEARNING OF MATHEMATICAL ANALYSIS CONCEPTS
5.2 CLASSIFICATION OF VISUAL IMAGERY OF MATHEMATICAL ANALYSIS CONCEPTS
5.3 STUDENTS’ VISUAL THINKING IN SOLVING MATHEMATICAL ANALYSIS TASKS
5.4 THE ROLE OF IMAGERY AND VISUALISATIONS IN PROVING THEOREMS IN MATHEMATICAL ANALYSIS
5.5 CONCLUDING REMARKS
CHAPTER SIX: SUMMARY, CONCLUSIONS AND RECOMMENDATIONS
6.1 SUMMARY OF FINDINGS
6.2 CONCLUSIONS
6.3 RECOMMENDATIONS
6.4 LIMITATIONS OF STUDY
6.5 SUGGESTIONS FOR FUTURE RESEARCH
6.6 CONCLUDING REMARKS
REFERENCES
APPENDICES
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