Learners’ understanding of mathematics

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Background of the study

The function concept “emerges from the general inclination of humans to connect two or more quantities, which is as ancient as mathematics” (Evangelidou, Spyrou, Elia & Gagatsis, 2004, p. 351) and is “all around us” (Kalchman & Koedinger, 2005, p. 351). This means that the function concept is common among phenomena in everyday life in which people connect quantities to form functional relationships where one quantity completely determines the other. For example, “a functional relationship is at play when we are paying for petrol by the litre or fruit by the gram or kilogram. We can find the amount that we need to pay for the petrol when we know the number of litres that were filled” (Pillay, 2006, p. 4). In addition, phenomena in domains from physics and economics like motion, waves, and electric current and price, demand and rate of inflation respectively, are also modelled by functional relationships (Grinstein & Lipsey, 2001).
Functions are also used “extensively for modeling and interpretation of such phenomena as local and world demographics and population growth, which are critical for economic planning and development” (Kalchman & Koedinger, 2005, p. 351). Surprisingly, in most cases people deal with functional situations like these without being aware of them and use creative or informal ways which may not be well developed and not always consistent when solving functional problems. The function concept “is central to mathematics and its applications” (Evangelidou, Spyrou, Elia & Gagatsis, 2004, p. 351). It is used in every branch of mathematics, such as arithmetic and algebraic operations on numbers, geometric transformations on points in the plane or in space, intersection and union of pairs of sets, some solution sets to equations, formulae used in mensuration (perimeter, area, and volume), and regression functions (Akkoc & Tall, 2005).
In addition, displacement, velocity, and rates of change are typical mathematical topics where functions are applied and learners may learn the mathematical concepts without being aware that they are functions. Apart from their use in calculus and analysis, “functions are also widely used in the comparison of abstract mathematical structures like determining whether two sets have the same cardinality and whether two topologies are homeomorphic. Functions can also be used as elements of abstract mathematical structures such as vector spaces, rings and groups” (Carlson, 1998, p. 115). Without doubt, functions have an important place in the secondary mathematics curriculum (Yerushlamy & Shternberg, 2001).
As early as 1921, “the National Committee on Mathematical Requirements of the Mathematical Association of America recommended that the study of functions be given central focus in secondary school mathematics” (Cooney & Wilson, 1993, p. 17). With specific reference to the South African mathematics curriculum, function-related activities start as early as the fourth grade and continue through to the high school mathematics curriculum. In addition, Froelich, Bartkovich, and Foerester (1991) said “the idea of function is inherent in many parts of today’s algebra and geometry programs” (p. 1), making the concept of function an important part of the school mathematics curriculum. The National Council of Teachers of Mathematics (NCTM) proposed in the Curriculum and Evaluation Standards for School Mathematics that « one of the central themes of school mathematics is the study of patterns and functions » (NCTM, 1989, p. 98).
Statement of the problem Within the function concept in modern mathematics and related fields, problems concerning its teaching and learning are often confronted (Mann, 2000) and internationally its difficulty is acknowledged (Tall & Bakar, 1992). Eisenberg (1992) argues that the function concept is “… one of the most difficult concepts to master in the learning of school mathematics” (p. 140) and has proved to be “subtle and elusive whenever we try to teach it in school” (Tall & Bakar, 1992, p. 1). Despite its importance few teachers know how learners come to understand functions (Yoon, 2007) and how activities can be designed to improve learners’ understanding of the function concept. The understanding of the function concept does not appear to be easy for two main reasons.
Firstly, many learners do not sufficiently understand the abstract but comprehensive meaning of the function concept (Mann, 2000). This may be caused by many different definitions of the function concept that appear in our high school textbooks. As a result teachers are compelled to give simple and understandable definitions of the function concept which do not integrate the core characteristic of the function concept, which sometimes creates misconceptions for learners (Mowahed, 2009). Secondly, because of the diversity of representations associated with the function concept, many learners do not understand the connections between different representations of the same function (Yamada, 2000). As a result, a substantial number of research studies have examined the role of different representations on the understanding and interpretation of functions (for example, Akkoc & Tall, 2005; Thomas, 2003; Zazkis, Liljedahl, & Gadowsky, 2003). However, these studies have done little to demonstrate the connections between the different definitions of the function concept and the different representations.

TABLE OF CONTENTS :

• CHAPTER 1: Introduction
  • 1.1 Background of the study
  • 1.2 Statement of the problem
  • 1.3 Purpose of the study
  • 1.4 Research Questions
  • 1.5 Rationale for the study
  • 1.6 Research design and methodology
  • 1.7 Ethical considerations
  • 1.8 My roles in this research
  • 1.9 Definition of terms
  • 1.10 Layout of the study
  • 1.11 Concluding remarks
• CHAPTER 2: Literature Review
  • 2.1 Introduction
  • 2.2 Learners’ understanding of mathematics
  • 2.3 The teacher’s role in increasing learner understanding of mathematics
  • 2.4 Learning and understanding of the function concept
  • 2.5 Development of the function concept using its definitions
  • 2.6 Characteristics of the function concept
  • 2.7 The role of the definition in teaching the function concept
  • 2.8 Representation of the function concept
  • 2.9 Connections between different representations of the same function
  • 2.10 Inverse function
  • 2.11 Learners’ difficulties with the function concept
  • 2.12 Teaching approaches to the function concept
    • 2.12.1 The situation approach
    • 2.12.2 The example and non-example approach
    • 2.12.3 The pattern approach
    • 2.12.4 The function machine approach
    • 2.12.5 Covariational approach
    • 2.12.6 The word problem approach
    • 2.12.7 Property-oriented approach

• CHAPTER 3: Theoretical Framework
  • 3.1 Introduction
  • 3.2 Constructivist paradigm
  • 3.3 Characteristics of teaching approaches that encourage a constructivist way of learning
  • 3.4 Constructivist mathematics teaching
  • 3.5 APOS theory
    • 3.5.1 Action level of a function
    • 3.5.2 Process level of a function
    • 3.5.3 Object level of a function
    • 3.5.4 Schema level of a function
  • 3.6 Realistic Mathematics Education (RME)
  • 3.6.1 Characteristics of Realistic Mathematics Education
  • 3.6.2 RME’s learning and teaching principles
  • 3.6.3 Using RME principles in designing lessons about functions
  • 3.7 Merging the theories
• CHAPTER 4: Research Design
  • 4.1 Introduction
  • 4.2 Ontological and Epistemological assumptions
  • 4.3 Context
  • 4.4 Sample and sampling techniques
  • 4.5 Research questions
  • 4.6 The research design
  • 4.7 Design research
  • 4.8 Adaptation of my research to the generic design research model
    • 4.8.1 Phase 1: Problem identification
    • 4.8.2 Phase 2: Development of interventions informed by theoretical framework
    • 4.8.3 Phase 3: Using tentative products and theories
    • 4.8.4 Phase 4: Product and theory refinement
    • 4.8.5 Phase 5: Final product and contribution to theory
  • 4.9 Limitations of the design research model
  • 4.10 Rigor and trustworthiness
  • 4.11 Ethical issues
  • 4.12 Summary of chapter
• CHAPTER 5: Presentation and Analysis of Data
  • 5.1 Introduction
  • 5.2 Data analysis for all phases
  • 5.3 Phase 1: Problem identification
  • 5.3.1 The analyses of learners’ initial individual tasks and task-based interviews
  • 5.3.2 Cross case analyses of learners’ calculations and explanations of the critical points
  • 5.3.3 Focus group interviews
  • 5.4 Problem Area 1: Understanding of the function concept
    • 5.4.1 Phase 1: Problem identification
    • 5.4.2 Phase 2: Development of interventions
    • 5.4.2.1 Teaching experiment
    • 5.4.3 Phase 3: Prototype
    • 5.4.3.1 Discussion of learners’ responses on activity
    • 5.4.3.2 Retrospective analysis
    • 5.4.4 Phase 4: Product and theory refinement
    • 5.4.4.1 Teaching experiment 2 (Prototype 2)
    • 5.4.4.2 Retrospective analysis
    • 5.4.4.3 Teaching experiment 3 (Prototype 3)
    • 5.4.4.4 Retrospective analysis
    • 5.4.5 Phase 5: Final product and contribution to theory
    • 5.4.5.1 Activity 4: Deriving the working definition of the function concept
    • 5.4.5.2 Retrospective analysis
  • 5.5 Summary of results on problem area
  • 5.6 Problem area 2: Representations of the function concept
    • 5.6.1 Phase 1: Problem identification
    • 5.6.2 Phase 2: Development of interventions
    • 5.6.3 Phase 3: Tentative products
    • 5.6.3.1 Teaching experiment
    • 5.6.3.2 Retrospective analysis
  • 5.6.4 Phase 4: Product and theory refinement
    • 5.6.4.1 Teaching experiment 5 (Prototype 5)
    • 5.6.4.2 Illustrative example for using the definition-based procedure
    • 5.6.5 Phase 5: Final product and contribution to theory
    • 5.6.5.1 Activity 8: Using definition-based procedures (DBPs) in the
    • translation process
    • 5.6.5.2 Retrospective analysis
    • 5.6.5.3 Summary of results on problem area
  • 5.7 Problem area 3: The inverse of a function
  • 5.7.1 Phase 1: Problem identification
  • 5.7.2 Phase 2: Development of interventions
    • 5.7.2.1 Teaching experiment
  • 5.7.3 Phase 3: Using tentative products and theories
    • 5.7.3.1 Retrospective analysis
    • 5.7.3.2 Phase 4: Product and theory refinement
  • 5.7.4 Interview questions on the inverse of a function (group interview)
  • 5.7.5 Phase 5: Final product and contribution to theory
  • 5.7.6 Summary of problem area
  • 5.8 Summary of chapter
• CHAPTER 6: Summary, conclusions and recommendations
  • 6.1 Introduction
  • 6.2 What is an understanding of functions?
    • 6.2.1 Grade 11 learners’ understandings of the function definition and their weaknesses
    • 6.2.2 Grade 11 learners’ understandings of the function representation and their weaknesses
  • 6.3 Improving understanding of functions
    • 6.3.1 Using design research in the classroom to improve learners’ understanding of functions
    • 6.3.2 A qualitative summary of how learners improved their understanding of functions
  • 6.4 How the material improved learners’ understanding of functions
  • 6.5 Trends of improvement in individual learners’ understanding of the functions
  • 6.6 Discussion of the final products and their contribution to theory
  • 6.7 Reflections on the theoretical framework
  • 6.8 Reflections on my research methodology
  • 6.9 Conclusions
  • 6.10 Recommendations
  • 6.11 Limitations of the study
  • 6.12 Closing remarks
  • List of references

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