The changing role of technology in the mathematics and statistics classroom

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CHAPTER FOUR A framework for types and levels of statistical reasoning with time series

 Introduction

In this chapter the development of statistical reasoning will be examined through a number of different theoretical lenses. The frameworks discussed have slightly different foci but elements from each of them will be utilised to establish a synthesised framework for the types and levels of statistical reasoning with time series. The synthesised framework will be used subsequently to determine the level of statistical reasoning achieved in exemplars of student work completed for the now expired Achievement Standard, AS 3.1, and the current Achievement Standard, AS 3.8. Although a framework for the development of reasoning in time series has not been developed, synthesised frameworks for other areas have been developed. In particular, Mooney, Langrall and Hertel (2014), developed a synthesised framework for probabilistic thinking. It is hoped that the time series framework, developed as part of this research, will not only enable the research question to be answered but may also help to inform instruction techniques and assessment writing. The framework may also be used to examine how current and future curricula reflect the different levels of reasoning described.

 Theoretical perspectives

There are several theoretical lenses that offer a variety of ways to examine the development of mathematical and statistical reasoning. I have categorised these lenses into three distinct groups: Frameworks developed for analysis of levels of reasoning in assessment tasks .Frameworks characterising the development of mathematical thinking. Frameworks characterising the development and dimensions of statistical thinking A full review of frameworks in these categories is outside the scope of this thesis. Each framework selected is briefly described and its key components summarised in Table 3. An examination across the frameworks in all categories provided a synthesis of frameworks which is summarised in Table .
 

Two frameworks developed for analysis of levels of reasoning in assessment tasks

The frameworks selected in this section to inform the development of the time series framework are De Lange’s assessment pyramid and the Structure of the Observed Learning Outcome (SOLO) taxonomy. De Lange’s assessment pyramid was developed within the Realistic Mathematics Education (RME) school of thought. The RME was selected because it focuses on the levels of reasoning which can be instigated through the design of an assessment task and it considers the role of context, crucial in any statistical reasoning, in solving problems. The SOLO taxonomy was selected because it focuses on the levels of student responses to a task. SOLO is also widely used in statistics education research (Watson, 1994) and is also used in New Zealand’s NCEA assessments to describe the different levels of thinking in the mathematics and statistics Achievement Standards. Realistic Mathematics Education RME has been developed over the last 35 years by researchers at the Freudenthal Institute in The Netherlands (Heuvel-Panhuizen, 1996). A strong focus of the curriculum developed by this Institute is the need to provide realistic contexts for students to emphasise the usefulness of mathematics in the real world. This involves developing students’ ability to mathematise real world situations. Students following a realistic mathematics course are also expected to develop a critical stance and evaluate when mathematics has been used to intimidate or even deliberately mislead. In order to assess how well an activity or assessment task reflected the components of an RME style, De Lange developed a pyramidal framework illustrated in Figure 1. (Verhage & De Lange, 1997) There are several components to De Lange’s pyramid, namely 1. Levels of reasoning or thinking. Lower level involving reproduction, a middle level requiring connections to be made and a higher level requiring analytical skills. 2.Context free, camouflage and authentic 3.Ranges from simple to complex.4.Algebra, geometry, number and statistics & probability. A task which requires a lower level of reasoning would be one concerned with basic facts, procedures and recall of knowledge. Questions in such tasks would be similar to ones previously practised and would require little if any new level of reasoning or thinking to achieve. A task which requires a middle level of reasoning would require the student to integrate information from perhaps a number of sources in order to develop a strategy for solution. At the middle level of reasoning a student will need to apply previously learned skills in a new way or to a new and perhaps unfamiliar context. At the highest level of reasoning students would be expected to demonstrate many of the following characteristics – critical attitude, interpretation, reflection, creativity, generalisation and mathematisation. The process of mathematisation is considered to have two components – horizontal and vertical. Horizontal mathematisation requires students to convert a realistic context into a mathematical problem that can be solved mathematically. The processes involved in the mathematical solution are described as the vertical component of mathematisation. A student displaying a higher level of reasoning might either be creating a model to solve a problem or alternatively using a familiar model to generalise about a new problem. Verhage and De Lange (1997), also stress the importance of context if the process of mathematisation is to take place. They state that problems can be divided according to three levels of context. The first level is context-free, which means the task has not been set within any context. In the second level a context may be provided but is regarded as ‘camouflage context’ or where the context is provided to give a relevance to the task but is not an essential element of the task. The third level requires the context to be authentic and an essential and relevant part of the task at hand. Verhage and De Lange maintain that authentic contexts are necessary if students are to engage in higher order thinking. The pyramid structure demonstrates the multi-faceted nature of problems and tasks. For example, a task may require a lower level of reasoning but require complex mathematical skills. Conversely, a task may require a high level of reasoning but require only medium complexity mathematical skills. Often when teachers approach the topic of time series, they initially try to select data sets whose contexts are familiar to students. Within a familiar context the level of reasoning demanded is lower than when a context is unfamiliar. Similarly the complexity of the task will increase if the context is unfamiliar to the student. For time series the NCEA assessment is an extended piece of work which suggests that students will be able to be assessed for higher level thinking. Thus students have the opportunity to demonstrate a critical attitude and an ability to interpret and reason from the data at a high or excellence level. SOLO Taxonomy The Structure of the Observed Learning Outcome (SOLO) taxonomy was developed by Kevin Collis and John Biggs (1982) and has grown in popularity. SOLO taxonomy aims to assess the structural level of a student’s learning and is based on cognitive theory. It allows student learning to be assessed in terms of its complexity and quality rather than how many parts of an assessment they got right or wrong. For teachers it can help them to assess where students are placed in a hierarchical structure of learning based on students’ responses. This in turn can provide guidance as to what future instruction is required in order to assist students in attaining the next level of learning. Creation of the taxonomy began with Piaget’s stages of development in order to establish a categorisation system for student responses. Pegg (2005) gives five modes of functioning with approximate age ranges associated with each one: 1.Sensori-motor (from birth). Physical co-ordination skills often in response to an external. 2.Ikonic (from around 18 months). For example, the development of language. 3.Concrete-symbolic (from around 6 years). Involves use of written language, numbers, symbols, signs and charts. 4.Formal (from around 16 years). The ability to hypothesise and refine structures. 5.Post Formal (from around 20 years). The level usually associated with research as it involves redefining and perhaps extending existing boundaries or ideas.Within each mode identified above learners can operate at one of five different levels. Five structural levels are identified: 1.Pre-structural. A student may focus on one particular aspect, perhaps as a result of some prior knowledge, but effort is often irrelevant to the content of the task. 2.Uni-structural. A student begins to focus on one relevant aspect of the task. 3.Multi-structural. A student is able to work on several aspects of the task, but is unable to relate or integrate them. 4.A student can work on several aspects of the task and integrate them to produce a coherent and meaningful solution. 5. Extended abstract. A student can generalise the task to an, as yet, untaught application. The SOLO taxonomy is illustrated in Figure 2 (Pegg, 1992). The diagram portrays a fairly linear progression from one level to the next which anyone with extensive teaching experience is likely to question. However, Romberg, Zarinnia and Collis (1990, p. 21), describe a process whereby “modes do not successively replace each other…. but as each develops, it is added to its predecessor (which itself continues to develop) and, thus the modal repertoire of the mature adult” is created. Time series analysis, which this research focuses on, is taught in Year 13, and therefore the mode most likely to be involved will be formal, but within that any of the five structural levels might be observed. At the uni-structural level, the student of time series will be capable of carrying out a particular task such as identification of peaks and troughs, but would make no attempt to link this information with the context of the data or to interpret why the peaks or troughs have occurred. At the multi-structural level, the student of time series may be able to identify peaks and troughs, smooth data, identify unusual values and perhaps describe long and short term trends, but they would be unable to relate these properties in order to produce an holistic picture of the time series. At a relational level, not only will the student be able to carry out the tasks mentioned previously but would also be able to relate characteristics of the time series to its context and attempt to interpret those features and offer possible explanations for them. At an extended abstract level, students would be able to analyse time series data from unfamiliar contexts and, through further study and research, be able to provide a coherent interpretation of the time series’ characteristics.

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 Framework characterising the development of mathematical reasoning

The Pirie-Kieren framework (Figure 3) was selected for consideration because it is well-established as a model to describe students’ growth of mathematical reasoning and therefore may provide some insights into the development of the time series framework. The Pirie-Kieren theory of the development of mathematical understanding was first described in 1989 and has since become a well-established model (Martin, 2008; Pirie & Kieren, 1994). A fundamental component of this model is that understanding in mathematics is a dynamic process which involves continual movement between different levels of reasoning and understanding. This dynamic process is not seen as a linear one but one that may involve re-visiting and re-examining previous levels of understanding in order to progress to a higher level of understanding. The process of re-examination is termed folding-back within the Pirie-Kieren theory. The framework of the Pirie-Kieren model comprises eight layers: 1.Primitive knowing. Prior knowledge which can be utilised for the topic currently being It does not necessarily mean low level knowledge. 2.Image making. The student will be involved in activities aimed at assisting them to understand a new mathematical topic or idea. These activities may involve visual images but verbal images or actions may also be indicative of this level. 3.Image having. The student’s understanding allows them to undertake an activity without the visual, verbal or action image for support. A student at this stage has a mental image of what is required. 4.Property noticing. The student is able to investigate and explore the mental images they possess and begin to make connections between them, identify common properties or differences and articulate what they have discovered. The ability to reflect and critique their own work is a characteristic of this level. 5.The student is able to generalise about properties and work with the concept as a formal object. 6.The student identifies connections between formal statements of a concept and defines his/her ideas as algorithms or proofs. 7.The student examines his/her formal observations as a theory and may begin to construct logical arguments in the form of a proof. 8.The student has full understanding of the concept and can pose questions about that concept that may lead to investigation and understanding of new concepts.

CHAPTER ONE 
Introduction 
1.1 Introduction and background
1.2 The need for research
1.3 The research question
1.4 Overview of chapters
CHAPTER TWO A brief history of the secondary school statistics curriculum in New Zealand 
2.1 Introduction
2.2 Brief historical overview of New Zealand statistics curriculum, 1950 – 1980
2.3 1980 – 1992, New Zealand
2.4 1992 – present day, New Zealand
2.5 International influences
2.6 Impact of the reform on the teaching of time series
CHAPTER THREE The changing role of technology in the mathematics and statistics classroom
3.1 Introduction
3.2 Integration of technology
3.3 Technology in the statistics classroom
3.4 Technology in the teaching of time series analysis
3.5 Research on the teaching of time series
3.6 Summary
CHAPTER FOUR  A framework for types and levels of statistical reasoning with time series 
4.1 Introduction
4.2 Theoretical perspectives
4.3 Summary of key components of frameworks
4.4 Framework synthesising characteristics of types and levels of reasoning for time series
4.5 Conclusion
CHAPTER FIVE Research methodology
5.1 Introduction
5.2 Overview of research methods
5.3 Overview of research instruments
5.4 Research procedure
5.5 Quality and validation issues in this research
5.6 Overview of ethical considerations
5.7 Ethical considerations in this research
5.8 Summary
CHAPTER SIX Results
6.1 Introduction
6.2 Student exemplars
6.3 Teacher questionnaires
6.4 Teacher interviews
6.5 Summary of results
CHAPTER SEVEN Discussion and conclusions
7.1 Introduction
7.2 Main research question
7.3 Second research question
7.4 Third research question
7.5 Other issues
7.6 Implications for teaching, assessment and research
7.7 Limitations
7.8 Conclusion

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THE IMPACT OF CURRICULUM CHANGE ON THE TEACHING AND LEARNING OF TIME SERIES

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