Primary Students’ Interpretation of Maps: Gesture Use and ... · more detailed analysis of students’ interpretation of map tasks. Video data from an existing data set was reorganised

Primary Students’ Interpretation of Maps: Gesture Use and Mapping Knowledge

Tracy Michelle Logan (BEd)

Centre for Learning Innovation

Faculty of Education

Queensland University of Technology

MEd (Research)

2010

Prof Carmel Diezmann (QUT, Principal Supervisor)

Prof Tom Lowrie (CSU, Associate Supervisor)

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Abstract

Maps are used to represent three-dimensional space and are integral to a range of

everyday experiences. They are increasingly used in mathematics, being prominent

both in school curricula and as a form of assessing students understanding of

mathematics ideas. In order to successfully interpret maps, students need to be able to

understand that maps: represent space, have their own perspective and scale, and their

own set of symbols and texts. Despite the fact that maps have an increased prevalence

in society and school, there is evidence to suggest that students have difficulty

interpreting maps.

This study investigated 43 primary-aged students’ (aged 9-12 years) verbal and

gestural behaviours as they engaged with and solved map tasks. Within a

multiliteracies framework that focuses on spatial, visual, linguistic, and gestural

elements, the study investigated how students interpret map tasks. Specifically, the

study sought to understand students’ skills and approaches used to solving map tasks

and the gestural behaviours they utilised as they engaged with map tasks.

The investigation was undertaken using the Knowledge Discovery in Data (KDD)

design. The design of this study capitalised on existing research data to carry out a

more detailed analysis of students’ interpretation of map tasks. Video data from an

existing data set was reorganised according to two distinct episodes—Task Solution

and Task Explanation—and analysed within the multiliteracies framework. Content

Analysis was used with these data and through anticipatory data reduction techniques,

patterns of behaviour were identified in relation to each specific map task by looking

at task solution, task correctness and gesture use.

The findings of this study revealed that students had a relatively sound understanding

of general mapping knowledge such as identifying landmarks, using keys, compass

points and coordinates. However, their understanding of mathematical concepts

pertinent to map tasks including location, direction, and movement were less

developed. Successful students were able to interpret the map tasks and apply

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relevant mathematical understanding to navigate the spatial demands of the map tasks

while the unsuccessful students were only able to interpret and understand basic map

conventions. In terms of their gesture use, the more difficult the task, the more likely

students were to exhibit gestural behaviours to solve the task. The most common

form of gestural behaviour was deictic, that is a pointing gesture. Deictic gestures not

only aided the students capacity to explain how they solved the map tasks but they

were also a tool which assisted them to navigate and monitor their spatial movements

when solving the tasks.

There were a number of implications for theory, learning and teaching, and test and

curriculum design arising from the study. From a theoretical perspective, the findings

of the study suggest that gesturing is an important element of multimodal engagement

in mapping tasks. In terms of teaching and learning, implications include the need for

students to utilise gesturing techniques when first faced with new or novel map tasks.

As students become more proficient in solving such tasks, they should be encouraged

to move beyond a reliance on such gesture use in order to progress to more

sophisticated understandings of map tasks. Additionally, teachers need to provide

students with opportunities to interpret and attend to multiple modes of information

when interpreting map tasks.

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Keywords

Mathematics education, maps, map reading, graphics tasks, gestural behaviours,

assessment, spatial reasoning

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Table of Contents

Abstract ......................................................................................................... ii 

Keywords ...................................................................................................... iv 

Table of Contents .......................................................................................... v 

List of Tables ................................................................................................. x 

List of Figures ............................................................................................. xii 

Statement of Original Authorship ............................................................. xiv 

Acknowledgments ...................................................................................... xv 

Chapter 1. Introduction ................................................................................ 1 1.1. Preamable ........................................................................................................... 1 

1.2. Overview of the Chapter .................................................................................... 1 

1.3. Mathematics Education in the 21st Century ....................................................... 1 

1.4. The Research Investigation ................................................................................ 3 

1.5. Significance and Innovation of the Investigation............................................... 4 

1.5.1. Significance ................................................................................................. 4 

1.5.2. Innovation ................................................................................................... 5 

1.6. Overview of the Thesis ...................................................................................... 5 

1.7. Chapter Summary .............................................................................................. 6 

Chapter 2. Literature Review ........................................................................ 7 2.1. Introduction ........................................................................................................ 7 

2.2. Conceptual Framework ...................................................................................... 7 

2.2.1. The Multiliteracies Framework ................................................................... 7 

2.2.2. Multimodality .............................................................................................. 9 

2.2.3. Linking the Theory and the Literature ...................................................... 10 

2.3. Visual and Spatial Meaning in Mathematics ................................................... 10 

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2.3.1. Representation in Mathematics ................................................................. 12 

2.3.2. Graphicacy ................................................................................................ 13 

2.3.3. Graphics in Mathematics .......................................................................... 14 

2.3.4. Graphical Languages ................................................................................. 15 

2.3.5. Understanding Maps ................................................................................. 18 

2.3.6. The Use of Maps in Primary School Curricula ......................................... 20 

2.3.7. Content Knowledge of Maps .................................................................... 22 

2.3.8. Map tasks and Gender ............................................................................... 24 

2.3.9. Summary of Visual and Spatial Meaning in Mathematics........................ 26 

2.4. Gestural Meaning ............................................................................................. 26 

2.4.1. Gesture and Mathematics .......................................................................... 27 

2.4.2. Hand Gestures ........................................................................................... 27 

2.5. Linguistic Meaning .......................................................................................... 31 

2.6. Conclusion ....................................................................................................... 32 

2.7. Chapter Summary ............................................................................................ 32 

Chapter 3. Context for the Study ............................................................... 34 3.1. Introduction ...................................................................................................... 34 

3.2. Setting the Scene .............................................................................................. 34 

3.3. Overview of the Graphical Language in Mathematics Project ........................ 35 

3.3.1. Phase One: The Mass Testing ................................................................... 36 

3.3.1.1. The Instrument ................................................................................... 36 

3.3.1.2. The Participants .................................................................................. 37 

3.3.1.3. Data Collection .................................................................................. 37 

3.3.1.4. Findings from the Mass Testing in the GLIM Project ....................... 38 

3.3.2. Phase Two: Interviews .............................................................................. 38 

3.3.2.1. The Instrument ................................................................................... 38 

3.3.2.2. The Participants .................................................................................. 39 

3.3.2.3. Data Collection and Interview Protocol ............................................. 39 

3.3.2.4. Summary of Findings from the Interview Component of the GLIM

Project ............................................................................................................. 40 

3.4. Relationship between the ARC Project and the Masters Project ..................... 41 

3.4.1. Duties Associated with Mass Testing in Phase One ................................. 42 

3.4.2. Duties Associated with the Interviews in Phase Two ............................... 42 

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3.4.3. Publications from the Original Study ........................................................ 43 

3.5. The Masters Study ........................................................................................... 43 

3.6. Chapter Summary ............................................................................................ 43 

Chapter 4. Design and Methodology ......................................................... 45 4.1. Introduction ...................................................................................................... 45 

4.2. Research Questions .......................................................................................... 45 

4.3. The Research Design –Knowledge Discovery in Data .................................... 45 

4.4. Selection of the Data ........................................................................................ 47 

4.4.1. Video Data ................................................................................................ 47 

4.4.2. The Map Tasks .......................................................................................... 48 

4.4.2.1. The Picnic Park Task ......................................................................... 50 

4.4.2.2. The Playground Task ......................................................................... 51 

4.4.2.3. The Street Map Task .......................................................................... 52 

4.4.3. Participants ................................................................................................ 53 

4.5. Preprocessing the Data ..................................................................................... 54 

4.5.1. Organising the Data .................................................................................. 54 

4.5.2. Analysing Video Data with Studiocode .................................................... 56 

4.6. Transformation of the Data .............................................................................. 58 

4.6.1. Technique of Content Analysis ................................................................. 58 

4.6.2. Process of Content Analysis ..................................................................... 59 

4.6.2.1. Coding Procedure for Task Solution .................................................. 59 

4.6.2.2. Coding Procedure for Task Explanation ............................................ 61 

4.7. Data Mining ..................................................................................................... 64 

4.8. Interpretation/Evaluation of the Analysis of Data ........................................... 65 

4.9. Overview of the Design ................................................................................... 66 

4.10. Quality and Rigour of the Study .................................................................... 68 

4.10.1. Credibility ............................................................................................... 68 

4.10.2. Dependability and Transferability........................................................... 69 

4.10.3. Ethics ....................................................................................................... 69 

4.11. Chapter Summary........................................................................................... 70 

Chapter 5. Results and Discussion ........................................................... 72 5.1. Introduction ...................................................................................................... 72 

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5.2. Introduction to the Three Tasks ....................................................................... 72 

5.3. The Picnic Park Task ....................................................................................... 74 

5.3.1. Task Solution and Relationship Between Correctness and Purposeful

Gesture Use ......................................................................................................... 75 

5.3.2. Mapping Skills and Solution Approaches Utilised During Task

Explanation ......................................................................................................... 76 

5.3.3. Types of Gesture Utilised During Task explanation ................................. 80 

5.4. The Playground Task ....................................................................................... 81 

5.4.1. Task Solution and Relationship Between Correctness and Purposeful

Gesture Use ......................................................................................................... 82 

5.4.2. Mapping Skills and Solution Approaches Utilised During Task

Explanation ......................................................................................................... 83 

5.4.3. Types of Gesture Utilised During Task Explanation ................................ 86 

5.5. The Street Map Task ........................................................................................ 87 

5.5.1. Task Solution and Relationship Between Correctness and Purposeful

Gesture Use ......................................................................................................... 88 

5.5.2. Mapping Skills and Solution Approaches Utilised During Task

Explanation ......................................................................................................... 89 

5.5.3. Types of Gesture Utilised During Task Explanation ................................ 93 

5.6. Understanding Students’ Performance and Behaviour on Map Tasks............. 94 

5.7. Patterns Across Map Tasks .............................................................................. 96 

5.7.1. Students’ Performance and Use of Gesture During Task Solution ........... 96 

5.7.1.1. Correctness and Purposeful Gesture Use on Map Tasks ................... 97 

5.7.1.2. Analysing Individual Tasks’ Correctness and Purposeful Gesture Use

......................................................................................................................... 99 

5.7.1.3. The Impact of Gender on Correctness and Purposeful Gesture Use for

Individual Tasks ............................................................................................ 101 

5.7.2. Students’ Mapping Skills and Solution Approaches During Task

Explanation ....................................................................................................... 104 

5.8. Understanding the Patterns Among Students’ Behaviour on Map Tasks ...... 107 

5.9. Profiles of map tasks ...................................................................................... 110 

5.9.1. The Picnic Park ....................................................................................... 114 

5.9.2. Example of One of the Most Common Pathways for The Picnic Park ... 115 

5.9.3. The Playground ....................................................................................... 117 

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5.9.4. Example of the Most Common Pathway for The Playground ................ 120 

5.9.5. The Street Map ........................................................................................ 122 

5.9.6. Example of the Most Common Pathway for The Street Map ................. 125 

5.10. Understanding Task Profiles ........................................................................ 127 

5.11. Chapter Summary......................................................................................... 129 

Chapter 6. Conclusions ................................................................................ 131 6.1. Introduction .................................................................................................... 131 

6.2. Summary of Findings for Each Research Question ....................................... 131 

6.2.1. What Mathematical Understandings do Primary-Aged Students Require to

Interpret Map Tasks?......................................................................................... 132 

6.2.2. What Patterns of Behaviour do These Students Exhibit When Solving

Map Tasks? ....................................................................................................... 133 

6.2.3. What Profiles of Behaviour do Successful and Unsuccessful Students

Exhibit on Map Tasks? ..................................................................................... 135 

6.3. Limitations of the Study ................................................................................. 136 

6.4. Implications of the Study ............................................................................... 138 

6.4.1. Implications for Theory........................................................................... 138 

6.4.2. Implications for Learning and Teaching ................................................. 138 

6.4.3. Implications for Test Designs and Curriculum Design ........................... 140 

6.5. Avenues for Further Research ........................................................................ 141 

6.6. Chapter Summary .......................................................................................... 143 

References................................................................................................. 144 

Appendices ................................................................................................ 157 Appendix A. Mass Testing Protocol ................................................................. 157 

Appendix B. Interview Protocol for GLIM ...................................................... 158 

Appendix C. Information Package for Parents ................................................. 163 

Appendix D. The Use of Chi Square Procedures .............................................. 166 

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List of Tables

Table 2.1 The Six Design Elements of the Multiliteracies Framework ....................... 8 

Table 2.2 An Overview of Spatial Thinking .............................................................. 11 

Table 2.3 Graphical Languages in Mathematics ........................................................ 16 

Table 2.4 New South Wales Mathematics Syllabus Continuum for Position ........... 22 

Table 2.5 Overview of the Content Knowledge of Maps .......................................... 24 

Table 2.6 McNeill’s Four Major Types of Gesture.................................................... 28 

Table 3.1 Phases of the Original Graphics Study and this Masters Project ............... 36 

Table 4.1 Overview of the Three Map Tasks ............................................................. 49 

Table 4.2 Composition of the Schools and the Participants....................................... 53 

Table 4.3 Organisation of the Video Data ................................................................. 54 

Table 4.4 Codes for Mapping Skills .......................................................................... 62 

Table 4.5 Codes for Solution Approaches ................................................................. 63 

Table 4.6 Codes for the Types of Gestures used during Task Explanation ............... 63 

Table 4.7 Symbiosis of Research Questions and Research Design ........................... 67 

Table 5.1 Contingency Table for Cross Tab Analysis for Task Solution on The Picnic

Park ..................................................................................................................... 76 

Table 5.2 Mapping Skills for Picnic Park Task by Correctness ................................ 77 

Table 5.3 Solution Approaches for Picnic Park Task by Correctness ....................... 78 

Table 5.4 Types of Gestures for Picnic Park Task by Correctness ............................ 81 

Table 5.5 Contingency Table for Cross Tab Analysis for Task Solution on The

Playground .......................................................................................................... 83 

Table 5.6 Mapping Skill for The Playground Task by Correctness ........................... 83 

Table 5.7 Solution Approaches for The Playground Task by Correctness ................ 84 

Table 5.8 Types of Gestures for Playground Task by Correctness ............................ 87 

Table 5.9 Contingency Table for Cross Tab Analysis for Task Solution on The Street

Map ..................................................................................................................... 89 

Table 5.10 Mapping Skills for The Street Map Task by Correctness ........................ 90 

Table 5.11 Solution Approaches for The Street Map Task by Correctness ............... 91 

Table 5.12 Types of Gestures for Street Map Task by Correctness........................... 94 

Table 5.13 Proportion of Students Achieving a Correct Solution on the Three Map

Tasks ................................................................................................................... 97 

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Table 5.14 Proportion of Students Using Purposeful Gesture across the Three Map

Tasks ................................................................................................................... 98 

Table 5.15 Frequency Distribution by Correctness and Purposeful Gesture Use ...... 98 

Table 5.16 Frequency Counts for Task and Gestural Use by Success ....................... 99 

Table 5.17 Means and (Standard Deviations) for Task Correctness by Gender Across

Tasks ................................................................................................................. 101 

Table 5.18 Means and (Standard Deviations) for Purposeful Gesture Use by Gender

across Tasks ...................................................................................................... 103 

Table 5.19 Mapping Skills Used Across the Three Tasks ....................................... 104 

Table 5.20 Solution Approaches Employed Across the Three Tasks ...................... 106 

Table 5.21 The Five Aspects of Task Profiles ......................................................... 111 

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List of Figures

Figure 2.1. Five multiliteracy elements as they relate to student engagement with map

tasks. ...................................................................................................................... 9 

Figure 3.1. Development of the current research project. ........................................... 42 

Figure 4.1. An overview of the KDD process ............................................................. 47 

Figure 4.2. The Picnic Park Task. ............................................................................... 50 

Figure 4.3. The Playground Task. ............................................................................... 51 

Figure 4.4. The Street Map Task. ................................................................................ 52 

Figure 4.5. A description and representation of Episodes 1 and 2. ............................. 55 

Figure 4.6. The Studiocode coding window. .............................................................. 57 

Figure 4.7. Purposeful gesture type one. ..................................................................... 60 

Figure 4.8. Purposeful gesture type two...................................................................... 60 

Figure 4.9. Purposeful gesture type three.................................................................... 60 

Figure 4.10. Non-purposeful gesture type one. ........................................................... 61 

Figure 4.11. Non-purposeful gesture type two ............................................................ 61 

Figure 4.12. The Data Mining analysis process. ......................................................... 65 

Figure 5.1. The Picnic Park Task. ............................................................................... 75 

Figure 5.2. The Playground Task. ............................................................................... 82 

Figure 5.3. The Street Map task. ................................................................................. 88 

Figure 5.4. The inverse relationship between task correctness and gesture. ............. 100 

Figure 5.5. The distribution of students across the three tasks who exhibited certain

behaviour characteristics. .................................................................................. 100 

Figure 5.6. The proportion of boys’ and girls’ correct responses across task. .......... 102 

Figure 5.7. The proportion of boys’ and girls’ purposeful gesture use across task. . 103 

Figure 5.8. Profile of The Picnic Park Task. ............................................................. 113 

Figure 5.9. Purposeful gestures used during Task Solution on The Picnic Park and

tracked on the map. ........................................................................................... 116 

Figure 5.10. Transcript of The Picnic Park explanation, cross referenced with deictic

gesture use. ........................................................................................................ 117 

Figure 5.11. Profile of The Playground Task. ........................................................... 118 

Figure 5.12. Sequence and transcript of a student demonstrating the most common

pathway for The Playground task. .................................................................... 121 

Figure 5.13. Profile of The Street Map Task. ........................................................... 123 

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Figure 5.14. Sequence and transcript of a student demonstrating the most common

pathway for The Street Map task. ..................................................................... 126 

xiii

Statement of Original Authorship

The work contained in this thesis has not been previously submitted to meet

requirements for an award at this or any other higher education institution. To the best

of my knowledge and belief, the thesis contains no material previously published or

written by another person except where due reference is made.

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Acknowledgments

The support, encouragement and assistance I have received along this journey have

been greatly appreciated. To my partner Mathew, and my two children, Lachlan and

Alexander, thank you for your unconditional love and support. The time and space

you provided allowed me to undertake this study, and for that I am sincerely grateful.

Heartfelt thanks are extended to my two supervisors Carmel Diezmann and Tom

Lowrie. In tandem, your capacity to provide me with structure, depth, technical detail

and the freedom to explore my ideas have taught me much about the research process.

I also appreciate your trust and confidence in my ability by allowing me to reanalyse

work from your project.

I would also like to acknowledge the support of my family, friends, and colleagues

who allowed me to verbalise my thinking and gave me the encouragement and

support to keep going.

Chapter 1. Introduction

1.1. Preamable

In the past decade, maps have become much more prevalent in our society. Maps have

increased sophistication and detail (e.g., Google Earth) and there are increased opportunities

for people to engage with maps. Maps are readily available in cars (through GPS systems)

and even on mobile phones. As a consequence, maps have become more likely to be used by

people in everyday situations (Godlewska, 2001). In the past, many maps were used for

relatively specialised purposes, by professionals such as geographers, whereas today most

citizens engage with them on a regular (and even daily) basis (Godlewska, 2001). Maps

convey information graphically through coordinates, landmarks, simple icons, different

perspectives, and common grids. However, even though maps in some form are embedded in

the curriculum from the first years of school (e.g., Board of Studies NSW, 2002; National

Council of Teachers of Mathematics [NCTM], 2000), many primary students experience

difficulty interpreting relatively simple maps. Diezmann and Lowrie (2008b) found that

students had difficulty with some vocabulary presented in maps; that students were distracted

by different foci on the map; and that information critical to understanding was often

overlooked. Knowing how students solve tasks and the difficulties they experience in solving

the tasks are important components of pedagogical content knowledge (Carpenter, Fennema,

& Franke, 1996). Thus, an investigation that provides a focused, in-depth look at how

children comprehend and interpret map tasks was required.

1.2. Overview of the Chapter

The purpose of this chapter is to orientate the reader to the study. It contains three further

parts. The first part discusses the position of mathematics in the 21st century and identifies

some of the current thinking about what constitutes mathematics (Section 1.3). The next part

identifies the purpose of the research investigation (Section 1.4) and highlights the

significance and innovation of this study (Section 1.5). The final part presents an overview of

the document (Section 1.6) and a summary of the chapter (Section 1.7).

1.3. Mathematics Education in the 21st Century

The 21st century is marked by three converging trends in mathematics education. First,

mathematics is fundamental to an individual’s capacity to function effectively in everyday

life. As the 21st century proceeds, the demand for mathematical proficiency is increasing due

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to technological advances. For example, bookkeepers and administrative workers are now

expected to be proficient with spreadsheet and accounting software. However, mathematical

proficiency extends beyond arithmetic proficiency. As described earlier, mapping requires

increased attention. The widespread availability and use of maps requires the development of

skills and processes which include visualising and manipulating mental representations of

objects, perceiving an object from different perspectives, and interpreting and describing the

represented physical environments (Lowrie & Logan, 2007). Not surprisingly, the acquisition

of mapping knowledge has had increased attention in school mathematics (NCTM, 2000).

The NCTM recommends fostering spatial reasoning by creating and reading maps, planning

routes and designing floor plans. Therefore, it is evident that mapping knowledge and skills

are a fundamental aspect of mathematics education in today’s society.

Ball (2003) acknowledged that while number sense and computational skills will always be a

necessary component of mathematics, other aspects of mathematics such as knowledge and

the associated skills of statistics, probability, and quantitative information will play an

essential role in students’ lives. In today’s society, mathematics is concerned with a broad

range of skills that require students to understand and interpret mathematics principles that

require reasoning in various quantitative situations (Ball, 2003). Maps are one aspect of

mathematics where quantitative information is embedded in the representation and the ability

to decode maps is critical to engaging in society.

There is also a trend in mathematics education where graphics have assumed an important

role in organising and representing mathematics. Graphics are seen as visual representations

for “storing, understanding and communicating essential information” (Bertin, 1967/1983, p.

2) and include graphs, maps, diagrams and networks. Graphics have increased in popularity

with the technological advances in computer software. Given the abundance of graphics in

schools and society (i.e., a diagram of a train system, or a weather map on the news), notions

of mathematics are often experienced by students in a wide variety of ways beyond

traditional mathematics. Thus, graphics are becoming increasingly important for

communicating and representing mathematical ideas.

Lastly, the increased demand for mathematics proficiency and the use of graphics in

mathematics has resulted in an elevation of the importance of graphics in mathematics

assessment tasks. Mathematics assessment has changed significantly in the last ten years and

2

the representation of mathematics has become increasingly visual with the use of graphics

utilised to convey both mathematics meaning and content (Lowrie & Diezmann, 2009). For

example, in two national-based primary mathematics assessment booklets, Lowrie and

Diezmann (2009) found 85% of the items contained a graphical representation. The use of

graphics is particularly useful because they are not reliant on language, and hence, help to

overcome the language barriers as societies become more internationalised. However, the use

of graphics in test items places a large emphasis on students’ ability to correctly interpret and

utilise them. Lowrie and Diezmann (2007a) have reported that students have difficulty with a

range of graphical representations, including maps and are not necessarily proficient at

interpreting the graphics they encounter in assessment items. Additionally, there are gender

differences in performance on map graphics. Thus, it is important to understand students’

knowledge of maps, which is the focus of this study

1.4. The Research Investigation

This study aims to investigate the behaviours that students exhibit as they interpret map tasks. The research questions are:

1. What mathematical understandings do primary-aged students require to interpret Map tasks?

2. What patterns of behaviour do these students exhibit when solving Map tasks?

3. What profiles of behaviour do successful and unsuccessful students exhibit on Map tasks?

To respond to these questions, this investigation focussed on students’ verbal and non verbal

behaviours while completing multiple choice assessment tasks that incorporated maps. The

aspect of gender differences is explored through the second research question. Differences in

performance between boys and girls will be analysed within this research question. The

investigation was undertaken using a data mining design in which existing video data of

children’s performance on map tasks was examined (see Section 4.3). This design allowed

the identification of patterns within the data with the aim of understanding student behaviours

in interpreting maps in assessment tasks.

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1.5. Significance and Innovation of the Investigation

1.5.1. Significance

This project is significant in three ways. First, it investigates issues in mathematics education

which are both contemporary and problematic, namely students’ capacity to interpret maps in

assessment tasks. Specifically, the investigation documents how students solve a wide variety

of map tasks which require both knowledge of maps (e.g., knowledge of keys) and

mathematical understandings of spatial concepts (e.g., location and position) associated with

maps. The analysis examines verbal and non verbal responses from students and considers

their patterns of behaviour in relation to specific map tasks (see Section 2.3.7). In addition,

this study builds upon an extensive number of recent studies (Aberg-Bengtsson & Ottosson,

2006; Diezmann & Lowrie, 2006; 2007; 2008a; Lowrie, 2008; Lowrie & Diezmann, 2007a)

which specifically focus on students’ sense making when solving tasks rich in graphics.

Second, this investigation embraced the notion that communication is multifaceted in nature

and reflects current thinking on the role of non verbal communication in mathematics

education (Edwards, 2009; Kaput, 2009; Williams, 2009) and education in general (Babad,

2005). By analysing students’ verbal and gestural behaviours the current project considers the

collective role that these perspectives have on students’ ability to communicate their

understandings of specific tasks.

Third, this investigation uses standard assessment tasks from state and national tests.

Assessment tasks are of particular interest in mathematics education because there is an

increased international and national prevalence in the use standardised tests. There is also a

reliance on high stakes testing and the subsequent accountability attached to such measures

(Coyne & Harn, 2006). Thus, this study will provide new insights into not only student

performance but a range of behaviours that influence student performance on test items. To

date, the reporting of most high stakes testing has been limited to task correctness. As McNeil

(2000) warned, over emphasising students’ performance on high stakes tests has led to an

over reliance on test preparation, which limits the range of educational experiences enjoyed

by students. By moving beyond descriptive statistics, as is typical in the reporting of student

performance on these tests, and examining student behaviour from a holistic perspective, this

study will provide insights into how to support and guide students in interpreting maps. In

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turn, this knowledge can be used to inform and ultimately enrich student learning experiences

in mathematics.

1.5.2. Innovation

This study is innovative in two ways. The first innovative aspect is the use of a data mining

design (Section 4.3). The Knowledge Discovery in Data (KDD) design utilises existing data

sets to systematically process the data in order to discover patterns which will lead to new

knowledge. The use of this design within education is quite innovative since it is

predominately used in the fields of Artificial Intelligence and statistics with large numerical

data sets. Within this study, the design was applied as a sequential model to enable the

analysis of existing interview data which was video-taped for a previous project. This

innovative design narrowed the focus of the investigation and provided the opportunity to re-

analyse data using an additional conceptual framework; namely, the multiliteracies

framework.

The second innovative aspect of this study is the use of Studiocode (Studiocode Business

Group, n.d.) research tool. Studiocode enables the user to capture, compact, classify, observe,

and search video and audio very easily (Section 4.5.2). This software allowed for the

simultaneous consideration of the verbal and non verbal behaviours associated with task

completion. Thus, the use of Studiocode ensured that the design elements of the

multiliteracies framework could be analysed whilst students solved the tasks.

1.6. Overview of the Thesis

This document is organised into six chapters. Chapter 1 has provided an orientation to the

broad notions of mathematics in today’s society and provides a brief overview, outlining the

significance and innovation of the study. Chapter 2 presents an overview of the theoretical

underpinnings of this study and a review of the literature on representation, graphics and

maps in mathematics. It also provides an overview of gestural research in mathematics and

the use of student explanations. Chapter 3 explains the context for this study and how it

evolved from a larger project undertaken by Diezmann and Lowrie. It also outlines my

involvement with that project. Chapter 4 describes the design and methodology of this study.

It outlines each stage of the design process and describes the selected data and the data

analysis processes. Chapter 5 presents the results of the study with a discussion about those

5

results. Chapter 6 outlines the findings in relation to the research questions and discusses the

implications and limitations of the study and suggests avenues for further research.

1.7. Chapter Summary

Graphics, especially maps, are an important aspect of mathematics and they are becoming

common place in schools. Maps represent information both visually and graphically through

conventions such as coordinates, landmarks, simple icons, different perspectives, and

common grids. It is essential for students to have an understanding of graphics within

mathematics and proficiency in this area is increasingly important given extensive use of

graphics in modern society. Within the domain of graphics, maps have an important role to

play because of their widespread use and their availability for students to access. This study

goes beyond previous work in the field by considering the notion that communication is

multifaceted by investigating students’ verbal as well as non verbal behaviour. Its

significance is heightened by the use of map tasks from state assessments. This study utilised

an innovative design in the Knowledge Discovery in Data, which allowed for the

interrogation of pre-existing video data across a multimodal conceptual design. The use of

computer-based research tool Studiocode ensured that the relevant design elements of

students solving map tasks could be analysed with the multiliteracies framework. The notions

of multiliteracies and multimodality are discussed further in Chapter 2 which presents the

conceptual framework of this study and reviews the associated literature related to map tasks.

6

Chapter 2. Literature Review

2.1. Introduction

This chapter provides an overview of the theoretical stance on the investigation of students’

interpretation of map tasks from two complementary perspectives and explores the literature

associated with each component of the theoretical framework

This chapter contains four parts. The first part identifies the theoretical underpinnings of the

study (Section 2.2). The second part discusses the use of visual and spatial meaning in

mathematics by looking at representations, graphics and maps in mathematics (Section 2.3).

The third part focuses on visual and verbal behaviour of students during task completion. The

use of gestures in mathematics is discussed and a categorisation of hand gestures is identified

(Section 2.4). This is followed by a discussion of student explanations in educational contexts

(Section 2.5). The final part presents a conclusion, linking the theoretical underpinnings of

this study to the research literature (Section 2.6) and a chapter summary (Section 2.7).

2.2. Conceptual Framework

This study was investigated using a combination of two complementary perspectives. The

multiliteracies framework provides an opportunity to investigate students’ holistic

understanding about map tasks (Section 2.2.1). In tandem, the notion of multimodality

provides an opportunity to consider the interconnectivity of students’ verbal and non verbal

behaviours (Section 2.2.2). These frameworks are used as a background to the literature

related to students’ understanding of maps in mathematics (Section 2.2.3).

2.2.1. The Multiliteracies Framework

A significant shift in thinking occurred over the past 20 years with the coining of the term

multiliteracies by the The New London Group (2000). There are two main points associated

with this term. First, the variability of sense-making in different cultural, social or domain-

specific contexts results in differences that affect our everyday communication. Second,

sense-making is made in ways that are increasingly multimodal—where written or linguistic

forms of meaning interact with oral, visual, audio, gestural, and spatial patterns of meaning.

This project is concerned with the second point, in particular, how gestures (non verbal

behaviours) are involved in interpreting and completing a map task. Using the underpinnings

of the multiliteracies framework, student behaviours can be analysed taking into account the

7

design elements in tasks. Within the notion of multiliteracies, six major design elements that

describe and explain patterns of meaning-making have been identified (see Table 2.1). Each

design element is not isolated or discrete, but intertwined, with characteristics overlapping.

Table 2.1

The Six Design Elements of the Multiliteracies Framework.

Element Description Linguistic design elements of linguistic meaning including delivery and vocabulary

Visual design elements of visual meaning including colours and perspectives

Audio design elements that constitute music and sound effects

Gestural design elements that constitute behaviour, gesture, and feeling and affect

Spatial design elements that constitute geographic meanings

Multimodal design this element represents the dynamic interconnection between and

among the other areas

(Adapted from The New London Group, 2000, p. 26).

Five of the six multiliteracies designs are particularly appropriate to investigating a holistic

view of students’ behaviour on map tasks. These are the visual, spatial, linguistic, gestural,

and multimodal meanings (The New London Group, 2000). An adapted model (see Figure

2.1) provides an overview of how the five elements combine when students engage in

mathematics tasks, such as interpreting maps. The elements that make up the visual and

spatial meaning relate to the actual map tasks, such as the symbols used in the maps and the

arrangement of elements on the maps. The linguistic meaning relates to the verbal

explanations of students. The gestural meaning relates to the hand gestures students use when

solving the map tasks, in conjunction with their verbal explanations. The multimodal

meaning draws on the combination of different modes of communication within a map task

(e.g., text and visual spatial representations) and in their response to the task (e.g., verbal

language and gestures).

8

Figure 2.1. Five multiliteracy elements as they relate to student engagement with map tasks (adapted from The New London Group, 2000).

2.2.2. Multimodality

A consideration of multimodality is of importance in this study which investigated the

different behaviours that students employed whilst solving map tasks. In recent years,

multimodality has developed in and across many various fields of study, from cognitive

neuroscience to communication to teaching and learning (Granström, House, & Kralsson,

2002). Multimodality refers to the connection between all our sensory modes which at any

one time can be providing a person with differential information (Gallese & Lakoff, 2005):

“Sensory modalities like vision, touch, hearing, and so on are actually integrated with each

other and with motor control and planning” (p. 459). This stance is drawn from the

9

neuroscience perspective, which suggests that the sensory-motor system of the human brain

is multimodal in function rather than modular. In this sense, multimodality can be seen as

“the multiplicity of modes of communication”, in which modes are represented by meaning-

making systems that are organised and shaped to articulate and communicate meaning

(Kress, Charalampos, & Ogborn, 2006, p. 1, italics in original). Further, Nemirovsky and

Ferrara (2009) identified the modes of communication as “all types of body activity that play

a part in a given conversational turn or transaction … that includes multimodal aspects such

as: facial expression, gesture, tone of voice, sound production, eye motion, body poise, gaze,

and so forth” (p. 162). These aspects of communication are not only interrelated but

according to Sfard (2009), “there is an intimate relationship, indeed symbiosis, between

gestures and language” (p. 192). Thus, to fully understand students’ sense making on map

tasks, this investigation needs to be conducted within a framework that considers the

multimodal aspects of task completion in tandem with the multiliteracies that are involved in

this process. Collectively, multimodality provides a comprehensive view of children’s

communication and meaning-making systems. This view takes into account their language

and gestures as well as their understanding of the visual and spatial aspects of map tasks and

how they connect in a multimodal manner.

2.2.3. Linking the Theory and the Literature

The theoretical perspectives of multiliteracies (Section 2.2.1) and multimodalities (Section

2.2.2) guide the review of literature. Visual and spatial meaning is primarily concerned with

the map tasks themselves, and provides a background to the recent research undertaken in

this area (Section 2.3). Gestural meaning is addressed through a discussion of the recent

increase of attention the study of gesture has been receiving in the mathematics community

(Section 2.4). It also details four specific hand gestures which may relate to this study.

Linguistic meaning is investigated through the benefits of using student explanation after task

completion (Section 2.5).

2.3. Visual and Spatial Meaning in Mathematics

Visual and spatial awareness is highly important in mathematics (NCTM, 2000) and across

the curriculum (National Research Council, 2006). In the 2006 report by the National

Academy of Sciences (National Research Council, 2006), spatial thinking was seen to

encompass three major aspects namely, representation, space, and reasoning. Table 2.2

provides a summary of these aspects with examples related to mapping. These three aspects

10

of spatial thinking (i.e., representation, space, and reasoning) all have an affinity with

graphics in mathematics including maps, and how they are understood. For example, Liben

(2008) advocated learning and instruction about maps as a powerful way to develop spatial

thinking in children.

Table 2.2

An Overview of Spatial Thinking.

Aspect of spatial thinking Abstract concept Example Representation the relationships among views plans versus elevations of

buildings, or orthogonal versus perspective maps

the effect of projections Mercator versus equal-area map projections

the principles of graphic design

the roles of legibility, visual contrast, and figure-ground organisation in the readability of graphs and maps

Space the relationships among units of measurement

kilometres versus miles

different ways of calculating distance

miles, travel time, travel cost

the basis of coordinate systems

Cartesian versus polar coordinates

the nature of spaces number of dimensions [two- versus three-dimensional]

Reasoning the different ways of thinking about shortest distances

as the crow flies versus route distance in a rectangular street grid

the ability to extrapolate and interpolate

estimating the slope of a hillside from a map of contour lines

making decisions given traffic reports on a radio, selecting an alternative detour

(National Research Council, 2006, pp. 12-13)

11

To provide a further background to visual spatial meaning in mathematics focusing on maps,

a brief background on the use of representations in mathematics is presented (Section 2.3.1),

followed by the identification of graphics as a means of visual communication (Section

2.3.2). Then, the use of graphics in mathematics is outlined (Section 2.3.3) and a structural

framework for identifying maps is provided (Section 2.3.4). This framework highlights what

is required to understand maps (Section 2.3.5) and how maps feature in mathematics

curriculum (Section 2.3.6). This section also summarises the generic content knowledge

needed to interpret maps (Section 2.3.7) and recognises the issue of gender differences in

map tasks (Section 2.3.8).

2.3.1. Representation in Mathematics

Representations of mathematical ideas and concepts are of particular importance because

they provide a basis from which people understand and use mathematical ideas (NCTM,

2000). Representations tend to fall under two systems, namely internal and external

representations. Internal systems deal with how students’ personal ideas (based on the notion

of affect), constructs (natural language, problem solving processes) and images they create in

their own mind (visual and spatial imagery) are associated with their understanding of

mathematical objects and processes (Goldin & Shteingold, 2001, Presmeg, 1986). External

systems deal with the written aspects (numeration, algebra, calculus) and the visual and

spatial aspects (diagrams, graphs, representations of geometric shapes) of how mathematics is

presented and communicated to others (Goldin & Shteingold, 2001). These two systems

cannot and do not exist as separate entities. Instead they are seen as, “a two-sided process, an

interaction of internalization of external representations and externalization of mental

images” (Pape & Tchoshanov, 2001, p. 119). This study investigated maps as part of the

external system of representation and students’ understanding of maps as part of the internal

representation system.

External representations such as maps, diagrams, graphs and charts, are expressions of

mathematical concepts that “act as stimuli on the senses” to help people comprehend

complex ideas (Janvier, Girardon, & Morand, 1993, p. 81). However, representations are not

always understood by primary-aged students (Diezmann & Lowrie, 2006). One plausible

reason for this difficulty is that the creator and interpreter of the representation are different

people. von Glasersfeld (1987) emphasises that the interpreter of a representation plays a key

role: “(an external representation) does not represent by itself—it needs interpreting and, to

12

be interpreted, it needs an interpreter” (p. 216). Pape and Tchoshanov (2001) add that a

further complexity in the interpretation of a graphic is that the relative expertise of the creator

of a representation can be far below the level of expertise of its interpreter: “(the

representations of) mathematical concepts are developed by experts, embody experts'

conceptions of mathematical ideas, and may not be readily available or understandable to the

novice” (p. 124). This study focuses on students’ interpretation of map items that have been

prepared by mathematics assessment item writers.

Internal representations are generally considered pictures “in the mind’s eye” (Kosslyn, 1983)

and include various forms of concrete, pictorial and dynamic imagery (Presmeg, 1986)

associated with personalised ideas, images and processes. This study considered the

approaches students undertook to solve the map tasks by observing particular internal

representations. These internal representations were based on the students’ solution strategy

and the processes they undertook to solve the tasks such as using gesture or visualising. Thus,

this study details students’ performance and behaviours when engaging with map tasks from

both an external and internal perspective. It also contributes to knowledge about what

students know and understand about interpreting map tasks and any difficulties they might

experience in interpreting an external graphic created by an expert.

2.3.2. Graphicacy

In order to consider the collective and individual components that make up various types of

representations, a broad background needs to be acknowledged and theories outside the usual

domain of mathematics education are a useful starting point. Not surprisingly, the discipline

areas of cartography and geography provide structural frameworks which consider a range of

graphical representations collectively. Over 40 years ago, Balchin and Coleman (1966)

coined the term “graphicacy” within such a discipline framework as a means of highlighting

the influence graphical representations have on communication as both an aid and a way of

making meaning from visual representations. Balchin and Coleman (1966) defined

graphicacy as:

the intellectual skill necessary for the communication of relationships which cannot be

successfully communicated by words or mathematical notation alone; it is a skill to be

possessed by both those wishing to communicate and those attempting to understand;

visual aids, especially maps, photographs, charts and graphs, are the media of

communication. (p. 1)

13

As Balchin and Coleman note, spoken language, written language, and computation are not

necessarily interchangeable. There are times when mathematical notation is the best way to

represent mathematical information, such as a formula. However in other instances

presenting information in written or verbal form or graphically is a more effective

representation, for example, describing (verbal) and representing (graphical) climate patterns

on a weather map. Consequently, one form of representation is not ultimately superior to

another, rather sometimes particular representations are more or less appropriate for

representing different types of information. At times, information can be presented by more

than one representation, for example, “a half” can be represented graphically and also

described verbally. Using complementary representations is a very effective means of

communication because there are two different ways that the interpreter can access

information about the mathematical idea or situation.

2.3.3. Graphics in Mathematics

The representation of mathematical ideas, concepts and relationships in graphical form is not

new to mathematics. Graphics include graphs, maps, diagrams, and networks and according

to Bertin (1967/1983) are visual representations for “storing, understanding and

communicating essential information” (p. 2). Recent research on graphics in mathematics

education has focused on specific types of graphics. For example, there has been extensive

research done on number lines (Bobis, 2007; Bobis & Bobis, 2005; Diezmann & Lowrie,

2007), graphs (delMas, Garfield, & Ooms, 2005; Friel, Curico, & Bright, 2001; Roth &

Bowen, 2001) and maps (Lowrie, Francis, & Rogers, 2000). Other research has considered

relationships among and between graphics representations. Novick (2004), for example,

investigated diagrams, networks, and hierarchies. Lowrie (1996) and colleagues (Lowrie &

Kay, 2001) found that diagrams and other graphical representations particularly assist

students when solving complex and novel problems. On a different but aligned note,

Abergberg-Bengtsson and Ottossons’ (2006) work has reported that students’ graphic

abilities are closely aligned to their general knowledge and language skills. Until recently,

however, few studies have actually examined the particular nature and elements of a range of

graphics on students’ sense making and understanding on mathematics assessment tasks. In

the last five years, Diezmann and Lowrie (e.g., Diezmann & Lowrie, 2009; Lowrie &

Diezmann, 2005, 2009) have investigated the influence a broad range of graphical

representations have on students’ understanding on these tasks. Specifically, their work

focuses on how students interpret the mathematical aspects of various graphics, all with their

14

own structure and spatial arrangement. As previously discussed, maps are important everyday

graphics, and hence worthy of further exploration (Section 1.1). Because Diezmann and

Lowrie (2008b) identified some gender differences in students’ performance on maps, gender

differences are discussed shortly (Section 2.3.8).

2.3.4. Graphical Languages

There are many unique graphics in use in mathematics. These include maps, number lines,

bar graphs, Venn diagrams, and pie charts. Within the mathematics domain however, there

has been limited research into categorising these graphical representations into a

comprehensive system. However, work in information communication by Mackinlay (1999)

is applicable to mathematics. Mackinlay provides a specific structure in which graphics that

convey information can be represented and he refers to these representations as information

graphics. He categorises information graphics according to six types of “graphical

languages”. A graphical language is a collection of graphics which are grouped according to

specific criteria related to the perceptual elements they use and how these elements are

encoded. The six graphical languages are Axis, Apposed-position, Retinal-list, Map,

Connection and Miscellaneous. As Mackinlay (1999) explains:

Single position languages encode information by the position of a mark set on one axis.

Apposed-position languages encode information by a mark set that is positioned between

two axes. Retinal-list languages use one of the six retinal properties of the marks in a

mark set to encode the information. Since the positions of the marks do not encode

anything, the marks can be moved when retinal list designs are composed with other

designs. Map languages, which have fixed positions, encode information with graphical

techniques that are specific to maps. Connection languages encode information by

connecting a set of node objects with a set of link objects. Miscellaneous languages

encode information with a variety of additional graphical techniques. (p. 75)

The graphical languages are variously represented by sets of perceptual elements which

include position, length, angle, slope, area, volume, density, colour saturation, colour hue,

texture, connection, containment, and shape (Cleveland & McGill, 1984). For example, Map

languages typically have fixed positions and use a variety of visual elements such as symbols,

area, density, colour saturation and texture. Table 2.3 provides a brief description of each of

the graphical languages together with examples and the encoding technique that each one

uses.

15

Table 2.3

Graphical Languages in Mathematics.

Graphical

Languages

Examples Encoding Technique

Axis Languages Number line, scale A single-position encodes information

by the placement of a mark on an axis.

Apposed-position

Languages

Line chart, bar chart, plot

chart

Information is encoded by a marked

set that is positioned between two

axes.

Retinal-list

Languages

Graphics featuring colour,

shape, size, saturation,

texture, orientation

Retinal properties are used to encode

information. These marks are not

dependent on position.

Map Languages Road map, topographic map Information is encoded through the

spatial location of the marks.

Connection

Languages

Tree, acyclic graph, network Information is encoded by a set of

node objects with a set of link objects.

Miscellaneous

Languages

Pie chart, Venn diagram Information is encoded with additional

graphical techniques (e.g., angle,

containment).

(Lowrie & Diezmann, 2005, p. 266)

Mackinlay’s (1999) framework of graphical languages is relevant in considering the manner

in which mathematics graphics are presented in classroom situations, in text books, and

indeed, in assessment and mass testing situations. Work conducted to date in Australian

settings (e.g., Diezmann & Lowrie, 2008a; Logan & Greenlees, 2008) demonstrates the

extent to which graphics can be classified under Mackinlay’s (1999) framework are actually

used within testing situations. Categorising graphics in such a manner allows researchers to

better understand how assessment items are structured and the various types used within

mathematics assessment. For example, work conducted by Lowrie and Diezmann (2009) has

been useful in identifying the types of information graphics being used in the National

Assessment Plan for Literacy and Numeracy (NAPLAN) testing in Australia. This work

revealed that particular types of graphics are represented more frequently on these tests than

16

other types of graphics. This over-representation of some graphics in particular Year Level

tests suggests that at certain ages some types of graphics are considered more relevant than

others. Typically, maps are represented in national tests in the primary years (e.g., Australian

Curriculum, Assessment and Reporting Authority [ACARA], 2009).

Over the past six years, there has been considerable Australian research conducted on student

sense making when decoding mathematics assessment tasks with high graphical content by

Diezmann and Lowrie (e.g., Diezmann & Lowrie, 2009; Lowrie & Diezmann, 2007a). They

have examined student decoding ability within and between graphical languages in a

longitudinal study of students aged 9-12 years. Some of their work has been devoted to

specific languages including Axis (Diezmann & Lowrie, 2006; 2007), Maps (Diezmann &

Lowrie, 2008b) and Apposed-position (Lowrie & Diezmann, 2007b). Other studies have

considered the relationship between performance on each of the six graphical languages

(Lowrie, 2008; Lowrie & Diezmann, 2005). This concentrated body of work provided

insights into students’ performance and conceptual development over time. In each year of

the study, when students were between 9 and 12 years old, students’ performance increased

in each of the six languages at a statistically significant level (Lowrie, 2008). Additionally,

there were significant correlations across the six languages, although it needs to be noted that

the relationships were only moderate, with the strongest correlations commonly within the

range of r = 0.3 to r = 0.4 (Lowrie & Diezmann, 2005). This result is worth noting since

students did not develop generic decoding skills that could be used to interpret the variety of

graphics in the assessment items. In other words, students who were successful at solving

Apposed-position tasks were not necessarily successful at solving Map tasks. The relationship

between these two languages for 9-10 year-olds, for example, was quite weak with

Spearman’s coefficient measuring only r = 0.21 (Lowrie & Diezmann, 2007a). Consequently,

further work needs to be undertaken to examine individual students’ performance on

particular graphical languages such as Maps.

Diezmann and Lowrie’s (Diezmann & Lowrie, 2007, 2008b; Lowrie & Diezmann, 2007b)

work also found variations between students’ performance within specific languages, and

therefore, there is scope for more detailed work around the different types of information

graphics within a particular graphical language. Diezmann and Lowrie (2008b) reported that

in some Map language items, students encountered some graphics from a birds-eye

perspective while others were from a front-view perspective. Another major finding was the

17

emergence of gender differences within the graphical languages (Lowrie, Diezmann, &

Logan, 2009) in which boys outperformed girls in all of the six graphical languages.

Statistically significant differences were found in Map languages at each grade level. This

finding further supports the investigation of students’ performance on Maps. Gender

differences are discussed further shortly (see Section 2.3.8).

Given the above findings reported by Diezmann and Lowrie, a number of issues emerged

which warrant further investigation. More attention needs to be devoted to student

performance in specific languages. This focus examines the processing and behaviours

employed to solve the graphics tasks and any inconsistencies regarding the performance of

students within each language. Consistent with the needs identified in recent research and due

to the importance of maps, this investigation focuses specifically on the Map language items.

Recall, maps are graphics where information is encoded through the spatial location of fixed

position marks (Mackinlay, 1999). Clarke (2003) also acknowledges the limited research on

map reading.

2.3.5. Understanding Maps

Information in maps is encoded through the spatial location of fixed marks and symbols

(Mackinlay, 1999). Clarke (2003) argues that in order to be map literate, the user is required

to develop and access a variety of spatial information. He states that spatial applications are

used for the simple location of places, right through to discerning complex spatial patterns.

Irrespective of the complexity of a given task, sound spatial reasoning skills are needed in

order to decode maps because the spatial relationship amongst visual elements is of particular

importance. As Godlewska (2001) argued, maps “express facts or concepts that derive a large

part of their significance from their spatial relationships” (p. 18). Maps are reliant on their

overall effect, but also specifically on locational features, intricate symbols, and the

relationship between locations and symbols. They also serve a communicative function

insomuch as they are designed with a specific purpose in mind—namely to represent the

spatial character of any given environment. Muehrcke (1978) broadly suggests that the main

purpose of a map is for recording and storing information, assisting data exploration, and

visualising the world around us. He also identifies the need for map interpretation,

highlighting that many people have not been taught specifically how to read maps. According

to Liben (2008), two main areas of children’s cognitive skills which relate specifically to

maps are representation (the content, the what and how of maps) and space (spatial

18

information such as scale, direction, and angle). Thus, developing these cognitive skills in

children is of particular importance to enhance their ability to interpret maps.

Although maps provide an authentic context for learning mathematics and assessing

mathematical knowledge, students do not always find their interpretation straightforward. For

example, Diezmann and Lowrie (2008b) reported that 10- to 13-year-olds experienced

difficulty with some of the vocabulary presented in maps; that students were distracted by

different foci on the map; and that information critical to understanding was often

overlooked. The ability to interpret or decode maps involves the student analysing: locations

(through position and placement) and attributes (what is actually represented); and

understanding that the map representation is presented within some form of scale and as a

result are smaller depictions of real world place or spaces (Wiegand, 2006). Although young

children are not often required to interpret scale or ratio on a map, they do need to appreciate

that a map is a reduced representation of something that is real and three dimensional. Indeed,

Liben (2008) suggests that young children have an elementary understanding that maps

represent and depict places, but interpret the scale incorrectly. Other difficulties identified in

Liben’s research relate to children misinterpreting the representation of symbols (for

example, believing that the symbol represented on the map has the same attributes in the real

world), and confusion over perspectives and different angles used to represent different maps

(for example, elevation view and birds-eye-view). Hence, reading and understanding a map is

a skill in itself, with certain fundamental features that need to be taken into consideration. In

order to fully understand maps, Wiegand (2006) identified five types of essential knowledge:

(a) understanding that maps represent space; (b) understanding a map’s alignment and angle

(perspective); (c) understanding scale; (d) understanding symbols and texts; and (e) using

maps to find the way. These understandings provide a basis from which students’

comprehension of map tasks can evolve. However, it is not just the features of a map that

need to be addressed. The perceptual and cognitive processing required to read a map are also

fundamental to proficiency with map tasks.

Map reading occurs at three levels of sophistication (Muehrcke, 1978; Wiegand, 2006). The

initial stage involves extracting information from a map and generally reading names and

attributes. In this phase, the user records or recognises visual stimuli and is able to recognise

and identify specific elements (or icons) that are contained within maps. The subsequent

phase involves ordering and sequencing information. This could include measuring,

19

calculating, comparing, and even manipulating information or data. Finally, interpretation

requires higher levels of mathematical thinking and decision making involving the

application of information. In this phase, the user is required to draw on prior knowledge and

experience in order to fully interpret information. These fundamental understandings are

evident in school syllabi around Australia (e.g., Board of Studies, 2002; Department of

Education and Training, Western Australia, 2005). Given the sophistication involved in map

understanding and reading, it is likely there will be differences in students’ ability to

understand and comprehend maps at various ages, it is vital that children have opportunities

to develop the necessary skill set to understand maps.

2.3.6. The Use of Maps in Primary School Curricula

Maps are typically part of the mathematics curriculum nationally and internationally for

primary-aged students. This study was conducted in New South Wales (NSW). Hence, an

overview of mapping in the curricula for this state is presented here. In addition, a national

perspective is presented because it informs the NSW curriculum and that of other Australian

states.

Within NSW, the teaching and use of maps in primary schools falls within both the

Mathematics syllabus (Board of Studies New South Wales [NSW], 2002) and the Human

Society and its Environment (HSIE) syllabus (Board of Studies NSW, 1998). Inside the

mathematics syllabus, map knowledge is located in the Space and Geometry strand under the

sub strand of position. Within the HSIE syllabus, maps are located in the environments strand

under the sub strand of location, position, and direction. Both syllabi provide specific

references to maps. However the mathematics syllabus provides much more detail on the

exact nature of maps and the type of spatial thinking advocated by the National Research

Council (2006) (see Section 2.3). The position sub strand content within the mathematics

syllabus is now examined because the focus of the study is on map items in mathematical

assessment tasks.

From the first year of schooling in NSW, maps are supposed to be prominent in the

curriculum, with students expected to “develop their representation of position through

precise language and the use of common grids and compass directions” (Board of Studies

NSW, 2002, p. 23). Students commence schooling in Kindergarten at 5 years of age. The

continuum for Position provides an overview of the expectations of students in NSW schools

20

for maps and their relationship to the world. Table 2.4 highlights the continuum of mapping

knowledge outlined in the mathematics syllabus, as students’ progress through the four stages

of primary schooling in New South Wales.

From a national perspective, the Curriculum Corporation (2006) developed the Statements of

Learning for Mathematics which guide the various state syllabi around Australia and provide

an outline of the skills, knowledge, and understandings that all Australian students are

expected to learn. Maps are addressed through the topic of Space. From these national

statements, expectations of children’s understandings about maps at Grade 3 and Grade 5

follow. These ages were selected because the study uses data from students in the middle to

upper primary years (See Section 3.3.2).

The expectations for Grade 3 in the Statements of Learning for Mathematics (Curriculum

Corporation, 2006) are.

Students interpret simple maps and plans and identify the most obvious features that have

been marked. They make reasonable sketches of familiar local environments such as the

school grounds or a particular room. They interpret the language of turns (half, full,

quarter, three-quarter) as they follow and give directions for moving around these

environments or for locating specific features. (p. 7)

The expectations of students at Grade 5 in the Statements of Learning for Mathematics

(Curriculum Corporation, 2006) are:

Students recognise and interpret the symbols and conventions used on different maps,

plans and grids to locate key features and landmarks. They use the North symbol, the

symbols within the legend and alpha-numeric grids to plan movement around those

environments. They understand the relationship between the four major compass points

and the amount of turn (quarter, half, three-quarter and full turns) and how these can be

used when giving directions. They use simple scales to estimate distances on maps and

plans. (p. 10)

While some of the language used in the Statements of Learning for Mathematics (Curriculum

Corporation, 2006) is different to the NSW mathematics syllabus (Board of Studies NSW,

2002), the key themes running through both the documents are similar. Moreover, within a

number of the Australian state syllabi, there are specific references to these same themes—

21

space, position, location, movement, direction, and arrangement (see for example Department

of Education and Training, Western Australia, 2005; Queensland Studies Authority, 2007a;

2007b). However, while maps are recognised throughout syllabi documents in New South

Wales and other states, some Australian students still struggle to interpret many different

types of maps (Diezmann & Lowrie, 2008b).

Table 2.4

New South Wales Mathematics Syllabus Continuum for Position.

Stage Expected outcomes

Early Stage 1

(Kindergarten)

Give and follow

simple directions

Use everyday language

to describe position

Stage 1

(Grades 1-2)

Represent the

position of

objects using

models and

drawings

Describe the position of

objects using everyday

language, including left

and right

Stage 2

(Grades 3-4)

Use simple maps

and grids to

represent position

and follow routes

Determine the directions

N, S, E and W; NE,

NW, SE and SW, given

one of the directions

Describe the location

of an object on a

simple map using

coordinates or

directions

Stage 3

(Grades 5-6)

Interpret scales on

maps and plans

Make simple

calculations using scale

(Board of Studies NSW, 2002, p. 36)

2.3.7. Content Knowledge of Maps

An overview of the content knowledge that students may require when decoding map tasks is

necessary in order to comprehend the skills and knowledge needed to interpret maps. Based

on research on mapping and the content of the NSW mathematics syllabus (Board of Studies

NSW, 2002), three key areas emerged in relation to mapping: (a) the language; (b) the

mapping knowledge; and (c) the mathematics concepts.

22

First, the language associated with maps is everyday language (e.g., left and right) and more

specific mapping terminology (e.g., North). The NSW mathematics syllabus (Board of

Studies NSW, 2002) identifies the need for younger children to be utilising everyday

language to describe position and direction. However, they advocate that students from Grade

3 onwards should be utilising mathematical terminology based on compass directions.

Second, mapping knowledge is based around the themes of children using a key or legend,

the landmarks and symbols and understanding the perspective and arrangement of map tasks.

Third, the mathematical concepts relate to children being able to move around the space

finding locations, considering direction, and working mathematically through different

processes.

Content knowledge for maps is extensive as shown on Table 2.5. Thus, children need an

understanding of the relevant language, mapping knowledge, and mathematics concepts to

successfully interpret a particular map. To investigate students’ understanding of maps

adequately, there is a need to use a variety of mapping tasks in which students are required to

utilise a range of mapping skills and mathematical concepts. These types of maps include

simple iconic maps, maps displayed from different perspectives, common grid maps, and

maps that include coordinates and landmark features. Given the difficulties some students

have with applying this content knowledge to maps (Diezmann & Lowrie, 2008b; Liben,

2008) a study that looks at specific knowledge relating to map tasks is timely.

23

Table 2.5

Overview of the Content Knowledge of Maps.

Language Mapping Knowledge Mathematical Concepts

Everyday language

• next to

• above

• behind

• near

• between

• left and right

Key/legend

• Uses key

• Compass points

Location and movement

• Position

• Orientation

• Navigating space

Landmarks/symbols

• Features on the map

• Symbolic representations

Direction

• Follow routes

• Search for and identify

destinations

Mathematical language

• North, east, south, west

Arrangement

• Co ordinates

• Scale

Measurement/Processes

• Search and Identify

• Count, compare and

contrast

• Estimate

Perspective

• Birds-eye-view

2-D/3-D representations

2.3.8. Map tasks and Gender

Gender differences have been the focus of spatial ability studies for many years. From a

broad perspective, a body of literature has examined the differences between males and

females on spatial tasks and identified gender differences in favour of boys (e.g., Halpern,

2000; Linn & Peterson, 1985; Voyer, Voyer, & Bryden, 1995). According to Boardman

(1990):

spatial visualisation is an ability to manipulate or rotate two-and three-

dimensional pictorially-presented visual stimuli. An example of the relationship

between spatial ability and map tasks is recognising and reading the signs and

symbols on a map when it is not held the right way up (p. 61).

24

In a review of recent literature on gender differences in mathematics, Spelke, (2005)

suggested that boys tend to perform better than similarly aged girls on tasks that require

mental rotations, or when tasks encourage the manipulation of objects in the mind or required

higher degrees of spatial reasoning. Although gender differences have been found in

numerous studies (e.g., Linn & Petersen, 1985), differences are typically reported on spatial

orientation tasks (e.g., Coluccia & Louse, 2004).

Boardman (1990) identifies spatial orientation as “an ability to remain unconfused by the

changing orientation in which a spatial pattern of visual stimuli may be presented. It requires

comprehension of the arrangement of elements within the pattern and an aptitude for

comprehending the pattern in relation to the orientation of the observer” (p. 61). In a

comprehensive review, Coluccia and Louse (2004) discussed the wide variations of findings

in relation to spatial orientation tasks, which include specific map tasks. Their review covered

many areas such as research conducted in real environments, simulated environments, and

with maps. Coluccia and Louse found that when a spatial environment task was represented

in a map form, in over 42% of the reported studies, males outperformed females, while in

over 39% of studies, no gender differences emerged. However, they also found that in over

18% of studies, females performed better than males. Lowrie and Diezmann (in press) found

that boys outperformed girls on map tasks when the tasks required high levels of dynamic

imagery and the interpretation of directional processing. Their study revealed that boys

performed better on tasks that required them to interpret both two-dimensional and three-

dimensional representations. They also suggested that boys performed better on tasks where

the processing of directional information was required. To fully substantiate the potential

existence of gender differences, studies need to focus on specific aspects of spatial tasks

within extensive longitudinal studies given irregularities in the research literature.

Generally, the extent to which gender differences are identified, the age when these

differences occur (and/or diminish), and the nature of these differences have raised

considerable debate. To move forward from this debate, Fennema and Leder (1990)

highlighted the need for a more focused and strategic examination of possible differences

between males and females within mathematics. Hence, this study aimed to examine any

patterns of behaviour that emerged (which included gender differences) when primary

students engaged with map tasks.

25

2.3.9. Summary of Visual and Spatial Meaning in Mathematics

The notions of spatial thinking identified by the National Academy of Sciences (National

Research Council, 2006) (Section 2.3) suggest the need for specific research on areas that

will assist students to thinking spatially. Graphics such as maps are becoming more prevalent

within both curricula and assessment. By identifying a specific graphical framework

(Mackinlay, 1999), information graphics in mathematics can be categorised, providing the

opportunity to look at individual graphical languages in more depth. Investigating maps

provides an opportunity to examine students’ use of maps in mathematics generally, and to

consider more specifically how mapping content knowledge is utilised by students as they

engage with mapping tasks. These tasks also provide the opportunity to explore any patterns

of behaviour among students.

2.4. Gestural Meaning

The underpinnings of the multimodal (Section 2.2.2) and multiliteracy conceptual framework

(Section 2.2.1) of this study highlight the importance of also investigating students’ non

verbal behaviour on map tasks. In line with the broader notion of communication, gestural

signs displayed during learning or task engagement is considered to be a useful mechanism in

ascertaining student sense making. The capacity to consider a range of behaviours that go

beyond written text and verbal accounts provides a useful insight into how students are

thinking about mathematical tasks. Thus, gestural language can provide additional evidence

regarding the strategies and approaches students employ to solve tasks. As Edwards (2009)

explains, “spontaneous gesture produced in conjunction with speech is considered as both a

source of data about mathematical thinking, and as an integral modality in communication

and cognition” (p. 127). Gestures might include a variety of personal communication

approaches which include facial expressions, hand and body movements, as well as

expressions of engagement and excitement. As Garber and Goldin-Meadow (2002) indicated,

“the gestures speakers spontaneously produce when they talk can reflect substantive ideas

relevant to the task at hand” (p.118). In some ways, the notion of gesturing provides explicit

observable details of some affective aspects of learning which were once considered internal.

Thus, studying gestures allows another meaning-making system to be investigated which

might provide an additional view of students’ concepts and understandings and which is

complementary to spoken language. It should also be acknowledged that gesture use can be

affected by different cultural perspectives since some cultural groups utilise gesturing in

different ways to others (Tversky, 2007). This section describes how gestures can contribute

26

to understanding student responses to mathematics tasks (Section 2.4.1). It also considers

more specifically how hand gestures can be interpreted to assist in understanding student

sense-making when engaging with map tasks (Section 2.4.2).

2.4.1. Gesture and Mathematics

In recent years within mathematics education, gesture has become a much researched

phenomenon. Volume 70 of a 2009 issue of the well respected journal Educational Studies in

Mathematics was dedicated entirely to embodiment and gesture within mathematical

contexts, highlighting the growing importance and relevance of gesture to the mathematics

community. Specific papers concentrated on gestures as semiotic resources (Azarello, Paola,

Robutti, & Sebena, 2009); gestures and conceptual integration in mathematical talk

(Edwards, 2009); bodily experience and mathematics conceptions (Roth & Thom, 2009);

embodied multi modal communication (Williams, 2009); and gestures, drawings and speech

in the construction of the mathematical meaning (Maschietto & Bartolini Bussi, 2009). These

papers bring to the fore the notion that gesture and mathematics are intrinsically linked. More

specifically, work undertaken by Tversky (2007) and Heiser, Tversky and Silverman (2004)

makes connections between gesture and maps. Tversky and colleagues’ findings suggested

that certain types of gestures were used for different purposes when navigating maps and that

movement of the gestures were confined to the map itself, with pointing and tracing gestures

prominent. While Tversky and colleagues’ research does not detail the exact nature of all

gestures, it certainly highlights that spatial arrangements such as maps are highly conducive

for gestural use when explaining and reasoning. This study will concentrate on the students’

use of hand gestures as they solved the map tasks. The focus on hand gestures is necessary

because the available data set had limited vision of students’ faces and the most prominent

gesture exhibited as they engaged with the map tasks was with their hands. Thus, a study that

includes attention to gestures, and more specifically hand gestures, provides a promising

avenue to explore students’ interpretation of map tasks.

2.4.2. Hand Gestures

The most prominent theory on gestures and their related meaning has come from the work of

McNeill (1992). He defined gestures as the “spontaneous and idiosyncratic movements of the

hands and arms accompanying speech” (p. 37) and suggested that they are a person’s inner

thoughts rendered visible which relate to “memory, thought, and mental images” (p. 12). He

advocates that gestures are symbols which display meaning designated by the speaker and

27

their coexistence with words and speech offer an insight into the process of sense making.

McNeil identified four major types of gestures people use when they talk, namely iconic,

metaphoric, beat and deictic. Table 2.6 outlines these gestures and provides an overview of

each type of gesture.

Table 2.6

McNeill’s Four Major Types of Gesture.

Name of Gesture General Overview

Iconic Bears a close relationship to the semantic content of speech

Metaphoric Similar to Iconic, but represents an abstract idea

Beat Rhythmical movements in time with the pulsation of speech

Deictic Pointing relating to objects and events in the concrete world

Iconic gestures are movements which relate directly to the words being spoken. That is, the

gesture represents aspects of the same scene being presented by speech. It is not so much the

words themselves but the nature of the scene described by the speech that the gesture relates

to. To illustrate an iconic gesture, McNeill (1992) uses the following example (the spoken

words are underlined, the meaningful part of the gesture is in brackets, and a description of

the gesture is italicised):

and he [bends it way back] (emphasis in original)

Iconic: right hand appears to grip something and pull it back from front to own

shoulder.

As the speaker described this scene he appeared to grip something in his own

hand and pull it back toward his shoulder. The grip shape of the hand and the

backward trajectory displayed aspects of the scene that speech was also

presenting. (McNeill, 1992, pp. 78-79)

The above example highlights the connection between speech and gesture and how a gesture

or gestures can be used to help elicit and understand a person’s complete thought process.

Iconic gestures are performed typically in three phases—preparation, stroke, and retraction.

The preparation refers to the speaker initiating a movement, for example lifting an arm or a

hand. The stroke is the main source of the gesture and refers to the action taken by the arm or

28

the hand. The retraction refers to the movement of the arm or hand back to a resting position.

These three phases are also applicable to metaphoric gestures.

Metaphoric gestures are similar to iconic gestures in that they are representations of the

images in a person’s head but they differ in the content of the gesture. A metaphoric gesture

will refer to an abstract concept such as genre, meaning, knowledge, or language itself

through a physical representation. The following is an example from McNeill (1992) of a

metaphoric gesture:

it [was a Sylves]ter “sic” and Tweety cartoon (emphasis in original)

Metaphoric: hands rise up and offer listener an “object.”

A particular cartoon event is concrete, but the speaker here is not referring to a

particular event: he is referring to the genre of the cartoon. This concept is

abstract. Yet he makes it concrete in the form of an image of a bounded object

supported in the hands and presented to the listener...this is the metaphor: the

concept of a genre of a certain kind (the Topic) is presented as a bounded,

supportable, spatially localisable physical object (the Vehicle). (McNeill, 1992,

pp.14-15)

Metaphoric gestures are fundamentally more involved than iconic gestures. In addition to the

phases of preparation, stroke, and retraction, there are two aspects of metaphoric gestures.

The Base is the action that the gesture is presenting, and the Referent is the concept the

gesture is presenting.

Beat gestures are when a finger, hand, or an arm moves along with the rhythm of speech,

with these gestures tending not to rely on the content of speech to produce a certain form.

These types of movements do not have any apparent meaning associated with them and are

generally slight, fast flicks of fingers or hands that require minimal energy and take up little

space. Moreover, they occur where ever the hands are at the time, including in the person’s

lap, on the armrest of the chair or next to the face. McNeill (1992) offers the following

example:

29

when[ever she] looks at him he tries to make monkey noises (emphasis in

original)

Beat: hand rises short way up from lap and drops back down.

...a beat that accompanied a reference to the theme of an episode. The spoken

utterance does not refer to a particular incident but characterises a class of

incidents, and the beat marked the word (whenever) that signalled this reference

to the discourse as a whole rather than a specific event. (McNeill, 1992, p. 16)

While beat gestures often look insignificant compared to other gestures, they reveal the

speakers’ impression and perception of the topic as a whole. Beat gestures differ from both

iconic and metaphoric gestures in that they are usually only presented in two phases as

opposed to three. The two phases being a movement such as in/out or up/down and hence can

be recognised by their repetitive movements.

Deictic gestures indicate objects and events in the real world or in the immediate

environment. Generally these types of gestures are pointing movements with fingers, hands

or with some extension of these such as a pencil. A gesture can also be classified as deictic

even if there is nothing concrete to point at. According to McNeill (1992), these abstract

deictic gestures occur most often in narrative dialogue. He offers this example.

[where did you] come from before? (emphasis in original)

Deictic: points to space between self and interlocutor.

The gesture is aimed not at an existing physical place where the interlocutor had

been previously, but at an abstract concept of where he had been before.

...although the space may seem empty, it was full to the speaker. It was a palpable

space in which a concept could be located as if it were a substance. (McNeill,

1992, p. 18)

According to Haviland (2000) pointing gestures are attached to speech through “pronouns,

tenses, demonstratives, and so on” (p. 18). That is, vocabulary such as that or there is often

combined with a deictic gesture aimed at what it is they are talking about. Where the target of

the gesture is immediately perceivable, its location and other features may be taken for

30

granted in the conversation. Where the target is not present, the deictic gesture becomes more

abstract and its actual location may be irrelevant to the conversation.

The mapping study utilised McNeill’s (1992) four categories of hand gestures (iconic,

metaphoric, beat and deictic) to explore students’ understandings and reasoning about maps

in conjunction with their explanations. The students’ use of hands when solving map tasks

provides an additional avenue through which their meaning can be identified. Understanding

students’ gestures coupled with their verbal explanations as to how they solved each map

tasks provided a comprehensive view of their knowledge of how to interpret maps. It also

links with the gestural meaning element of the multiliteracies framework which highlights the

connection between gesture use and the communication of meaning.

2.5. Linguistic Meaning

Language and linguistic communication is seen as “the prime way of exchanging meaning

between human beings who have acquired spoken language” (Lloyd, 1990, p. 51). While it is

not the only way (e.g., gestures), it is the most fundamental of all human communication. The

ability to give verbal explanations is considered highly important in educational contexts

(Donaldson & Elliot, 1990). Students’ explanations and verbal reports of their own thinking

provide an opportunity for the listener to gain insight into the knowledge they bring to the

situation and their cognitive processes (Chi, de Leeuw, Chiu, & LaVancher, 1994).

In order to know what a student is thinking, they generally have to verbalise and explain what

is in their thoughts. Enabling students to produce and convey their own explanations of their

reasoning offers the opportunity for deep learning on behalf of the speaker (Kastens & Liben,

2007). Much research in the field of education to date has focused on the use of self-

explanations as a way to improve student learning using worked-out examples (Chi, Bassok,

Lewis, Reinmann, & Glaser, 1989), expository texts (Chi, de Leeuw, Chiu, & LaVancher,

1994), and word problems (Mwangi & Sweller, 1998). However, these types of self-

explanations tend to occur during problem solution in the form of think-aloud strategies as

opposed to interview situations, where the explanations are elicited post solution. The

advantage of interviews is that they allow researchers to access personal thoughts and

processes in a controlled manner, with the emphasis on capturing the lived experiences and

related meanings of the students in their own words (Kvale, 2007). The controlled nature of

interview explanations allows for information that is specifically related to a certain topic, to

31

be gathered from a number of sources, such as different aged children, while assuring the

comparability of the explanations (Kumar, 1996). From this perspective, students’

explanations can be seen as reflections on their own knowledge, with the intent of

demonstrating what the student knows. Donaldson and Elliot (1990) argue that explanations

“demonstrate not only that you know, but you know how you know” (p. 48). These types of

explanations provide the opportunity to delve into the actual conceptual and cognitive

understandings the student holds as opposed to them just giving a correct answer (Donaldson

& Elliot, 1990). In this study, student explanations are used to gather information about their

content knowledge and understandings about maps. Information about the type of language,

the mapping skills, and the mathematical concepts that students use serve as a basis for

extracting students’ understandings about map tasks based on their verbal explanations

(Section 2.3.7). These explanations are considered in conjunction with student gestures to

connect the linguistic and gestural meaning elements of the multiliteracies framework. This

multimodal relationship contributes to a comprehensive view of students’ interpretation of

map tasks.

2.6. Conclusion

Communication is much more than verbal dialogue. It includes non verbal communication

such as hand gestures. Thus, the multimodal viewpoint considers a range of communicative

modes to understand the behaviours students are displaying. The multiliteracies framework

promotes the connection between the linguistic (verbal) and gestural (non verbal) aspects of

the data with the opportunities to interpret students’ understanding of visual elements and

spatial design presented within the Map tasks and how these aspects interconnect through

multimodality, therefore utilising five of the six multiliteracy elements within the project.

This comprehensive perspective allowed the researcher to interrogate different types of data

(e.g., explanations, gestures) in different ways. Furthermore, the Map tasks which are the

focus of this current investigation fit the multiliteracies framework particularly well, given

the fact that their high visual- spatial design is so critical to the way the tasks are presented

and interpreted.

2.7. Chapter Summary

The multiliteracies framework (i.e., the spatial, visual, linguistic, and gestural elements) and

the connectivity of these elements (i.e., multimodal forms of communication) provide a basis

32

from which the investigation was conducted. The visual and spatial meaning aspects of the

multiliteracies framework relate directly to the use of map tasks in this study.

Graphics, such as maps, are increasingly being used in mathematics in both curricula and

assessment. Graphicacy is a way of representing and communicating meaning from graphics.

The Graphical Languages framework (Mackinlay, 1999), categorises different types of

graphics that all have their own structures and purposes. These languages are Axis, Apposed-

position, Retinal-list, Map, Connection and Miscellaneous. This study focuses on the Map

language, in which information is encoded through the spatial location of fixed marks and

symbols. In order to fully understand maps, students need to be able to appreciate that maps

represent space and have their own perspective and scale and their own set of symbols and

texts. School curricula documents highlight that map knowledge is an important aspect of

mathematics knowledge, with maps having explicit content, namely the type of language

used, the map knowledge, and the mathematical concepts. Gender may be a performance

variable because some studies have identified gender differences in favour of males on some

mapping tasks.

Students’ understanding of maps can be investigated through gestures and their explanations.

The gestural meaning element of the multiliteracies framework identifies the importance of

students’ gestures in the field of mathematics. Hand gestures are an increasing area of

research within mathematics education. These gestures can be identified as Iconic, Abstract,

Beat or Deictic. Students’ explanations of their solution strategies give linguistic meaning to

the task. Their explanations provide an opportunity to gain insight into the knowledge a

student brings to a mathematical situation. Following this background of the visual and

spatial elements of map tasks and the multimodality of communication, the context for this

study is presented in Chapter 3.

33

Chapter 3. Context for the Study

3.1. Introduction

The preceding chapter identified how a multiliteracies framework has guided this

investigation, and outlined the literature on the visual and spatial, gestural, linguistic, and

multimodal elements related to this framework. The purpose of this chapter is to explain the

source of the data to be used in this study. The chapter has four parts. The first part provides a

context to this study and an introduction to the original project (Section 3.2). It also provides

an overview of the Graphical Languages in Mathematics project, detailing the distinct parts

and the relevant design and methodological issues (Section 3.2). The next part discusses the

relationship between the larger project and the development of this study (Section 3.4). The

third part pertains directly to this study and what it aims to achieve (Section 3.5). Finally, a

chapter summary is presented (Section 3.6).

3.2. Setting the Scene

The current research project is an extension and elaboration of one aspect of an Australian

Research Council (ARC) Discovery grant (# DP 0453366) titled “How primary school

students become code-breakers of information graphics in mathematics”, in which I provided

support for 3 years as a Research Assistant. The large four-year project was undertaken by

Professor Carmel Diezmann (Queensland University of Technology) and Professor Tom

Lowrie (Charles Sturt University) and funded between 2004 and 2007. The broad aim of the

ARC project was to increase fundamental knowledge about primary students’ decoding of

information graphics in mathematics. The ARC project was longitudinal, used a multi method

approach, and had two phases. Phase One comprised the annual mass testing of primary

students on a Graphical Languages in Mathematics (GLIM) instrument (Section 3.3.1). This

instrument is described shortly (Section 3.3.1.1). Phase Two consisted of individual

interviews with a different cohort of primary students based on the items from the GLIM test

over a 3 year period (Section 3.3.2). These phases are described in more detail shortly. This

current study on map tasks represents an extension to the ARC study of primary students’

interpretation of information graphics. Prior to explaining the current study and its

relationship to the ARC study (Section 3.4), the original study is outlined (Section 3.3)

34

3.3. Overview of the Graphical Language in Mathematics Project

To provide a background to the current investigation, a brief overview of the design of the

ARC study is first presented.

The original 4-year project employed a multi method longitudinal design (Willett, Singer, &

Martin, 1998) to investigate how primary-school aged students decoded information graphics

such as diagrams, charts, graphs, and maps presented in mathematical tests. The research

aims were to:

• Determine students performance on the graphical languages;

• Understand the knowledge that students utilise when decoding graphical languages in

mathematics;

• Discover any difficulties that students encounter as they decode graphical languages;

and

• To ascertain whether there is a hierarchy of complexity of the graphical languages.

The original study was divided into two phases over the 4 year period, with further analyses

and publications continuing on. The current investigation provides an additional phase

following on from the original study (Table 3.1). The phases and relevant findings of the

original study are now described.

35

Table 3.1

Phases of the Original Graphics Study and this Masters Project.

1st year 2nd year 3rd year 4th year 5th year 6th year

Phase One:

ARC Mass

Testing

X X X

Phase Two:

ARC

Interviews

X X X

Phase Three:

Analysis and

Publications

X X X X

Phase Four:

Current

Study

X

3.3.1. Phase One: The Mass Testing

Phase One investigated students’ performance on mathematical tasks with embedded

graphics over a 3-year period (Table 3.1, Years 1-3). This phase of the study sought to

determine students’ performance on mathematics tasks often found in national testing

instruments that were rich in information graphics such as number lines, bar graphs, and

maps. Students’ performance was assessed in a mass testing situation using the Graphical

Languages in Mathematics (GLIM) instrument.

3.3.1.1. The Instrument

The GLIM instrument was designed by Diezmann and Lowrie (2009) in order to better

understand students’ ability to interpret and decode tasks with high graphical content. The

instrument was a central component to both phases of the project. The instrument was

extensively trialled before development (see Diezmann & Lowrie, 2009). As Diezmann and

Lowrie (2009) explain:

The Graphical Languages in Mathematics [GLIM] Test is a 36-item multiple choice test

that was designed to investigate students’ performance on each of the six graphical

36

languages. This instrument comprises six items that are graduated in difficulty for each

of the six graphical languages. The test was developed from a bank of 58 graphically-

oriented items. These items were selected from published state, national and international

tests that have been administered to students in their final three years of primary school

or to similarly aged students … due to the limited Connection items in existing

mathematics tests, content free Connection items were sought from science tests … the

tasks in the item bank were variously trialled with primary-aged children (N = 796) in

order to select items that: (a) required substantial levels of graphical interpretation, (b)

required minimal mathematics knowledge, (c) had low linguistic demand, (d) conformed

to reliability and validity measures, and (e) varied in complexity. Our selection of items

was also validated by two primary teachers. (p. 136)

In its complete form, (the 36 item instrument was used for mass testing of students in each of

three years of the study (Phase One). It was also used during interviews (Phase Two) see

Section 3.3.2).

3.3.1.2. The Participants

In the mass testing phase, a total of 327 students (Female = 154, Male = 173) completed the

GLIM test annually for three years. These students commenced in the study as 9/10-year-olds

in Grade 4, and were tested again as 10/11-year-olds in Grade 5, and as 11/12-year-olds in

Grade 6. Participants were sourced from six schools in a large regional city and two schools

in a metropolitan city across two states in Australia. The participants’ socioeconomic and

academic background varied from school to school and across state, with the sample being

representative of the general population. Less than 10% of participants had English as their

second language.

3.3.1.3. Data Collection

The GLIM test was administered three times annually approximately 12 months apart in

whole-class situations in the presence of the classroom teacher. The researchers administered

the instruction protocol verbally and explained the nature of the study (see Appendix A for

the mass testing protocol). Participants were given up to one hour to complete the instrument,

with all students completing the test within this time frame. The tests were marked by hand

and cross-checked independently by two research assistants for accuracy. Data were entered

into SPSS (SPSS, 1990) to undertake statistical analysis.

37

3.3.1.4. Findings from the Mass Testing in the GLIM Project

Over the 3-year period, the mass testing (Phase One) provided insight into students’

developmental performance on tasks that are rich in graphics. The students’ performance on

the GLIM instrument increased significantly in each year of the 3-year study. Moreover, in

each year of the study, students’ performance significantly increased across each of the six

graphical languages represented in the instrument (Lowrie & Diezmann, 2007a; Lowrie,

2008). This finding is interesting given that students are not typically taught about many of

the graphical languages in primary school (Diezmann & Lowrie, 2008b).

Particularly noteworthy was the fact that boys outperformed girls in each year of the study

and in each of the graphical languages (Lowrie, 2008). Boys’ performance was significantly

higher than that of girls on two of the six graphical languages, namely Map and Axis

languages. Consequently, there is scope for analysis of the structure and nature of different

Map tasks in relation to students’ processes and performance which is the central concern of

the current investigation. Thus, the opportunity to investigate students’ understandings of

map tasks will build on and complement the previous work by Diezmann and Lowrie. The

current study will focus on specific types of map structures and how students decode these

graphics (Section 2.3.7).

3.3.2. Phase Two: Interviews

The second phase of the study commenced in Year 2 of the study (See Table 3.1) and

involved an in depth examination of 98 students’ sense making as they described the way in

which they solved the mathematics tasks from the GLIM instrument. The students were

interviewed on 12 different tasks in each of the three years (Table 3.1, Years 2-4) from the

GLIM test. Their performance was video-and audio-taped for analysis purposes.

3.3.2.1. The Instrument

The tasks for the Interview component comprised the 36 items from Phase One, organised

into three 12-item booklets. Items within each language were categorised as “easy” for

Booklet One, “moderate” for Booklet Two and “difficult” for Booklet Three according to the

results of the Grade 4 data in the mass testing phase. The booklets in the interview

component of the GLIM test comprised two items of each graphical language category within

each difficulty level (2 items x 6 graphical languages x 3 booklets in levels of difficulty = 36

items).

38

3.3.2.2. The Participants

The 98 interview participants (44 Males, 54 Females) were sourced from three schools in a

large regional city and two schools in a metropolitan city across two states. The interview

cohort differed from the mass testing cohort. The participants commenced in the project when

they were aged 9/10 (Grade 4 in 1NSW, Grade 5 in Qld) and they were interviewed again

when they were aged 10/11 (Grade 5) and aged 11/12 (Grade 6). The students’

socioeconomic, cultural and academic backgrounds varied, with less than 5% of the students

speaking English as their second language. This reflected the composition of the local

community.

3.3.2.3. Data Collection and Interview Protocol

In the first year of the interview study, the students were interviewed on the two easiest tasks

from each graphical language (Booklet One). In the following two years of the study the

same students were presented with the moderate (Booklet Two) and difficult (Booklet Three)

tasks respectively. Interviews took place approximately 12 months apart. Thus, the students

had been interviewed on the 36 tasks of the GLIM instrument by the last year of their primary

schooling. In each year of the interviews, the students completed one pair of tasks (e.g., Map

tasks), explained their responses, and were then encouraged to justify their thinking. It was

important to note that no scaffolding was provided by the researcher. Participants were video-

and audio-taped in each of the three years of the project.

For each year of the study, the interview process was replicated (see Appendix B for the

interview protocol format). Participants were briefed about the project and what their role

would be. Additionally, both students and their parents signed consent forms to allow

students to participate in the project. Each participant’s interview took place over two days,

with students completing six tasks each day. The timing was to minimise any effects of

fatigue on students. In the interviews, two tasks from each of the six graphical languages

were presented to the students in turn. Thus, in the first interview, students were asked to

solve two Axis, two Apposed-position and two Retinal-list tasks. In the second interview,

students were presented with two Map, two Connection and two Miscellaneous tasks. Thus,

over the two days they had responded to all twelve tasks of the booklet over the two

interviews. The identical format occurred in subsequent interviews.

1 Henceforth, the NSW grade levels are used because NSW data are used in the current study.

39

In each interview, after participants had completed two tasks from each of the languages,

semi-structured interview questions were posed. These questions were intended to evoke their

understanding and sense making for the respective tasks. The semi-structured questions

consisted of both general open-ended questions and more focused questions that related to

specific tasks. Some of the general questions are stated below:

Can you tell me how you worked out the answer?

What information was there on the diagram that helped you work out the answer?

How does it tell you that information?

Tell me what you did to work out the answer.

Please tell me a bit more about that.

These questions were designed to support students to explain their thinking and the strategies

that they had used to solve the tasks. After the students had explained their thinking on both

tasks within a graphical language, they were asked to compare the two items from the same

graphical language using the following questions:

Which of the two tasks did you find the harder?

What made it harder?

The purpose of these questions was to ascertain how the students perceived each task in

relation to another one from the same language.

The interview data were analysed to identify the strategies that students used in solution and

the difficulties they encountered with particular reference to the graphic in the task. The data

from the interview phase of the study provides a rich data set from which further

investigations are possible. With the endorsement of the ARC chief investigators, the data

from the Map items in the interview phase of the ARC project (Phase Two) was used in the

current study.

3.3.2.4. Summary of Findings from the Interview Component of the GLIM Project

The longitudinal interview data revealed three key issues of interest namely: (a) some

students had inappropriate conceptions of basic graphics; (b) students’ conception of a

graphic often determines the approach they take to solve a task; and (c) student capability in

retrieving information from graphical representations was a major factor in the approaches

40

taken to decode the graphics tasks (Diezmann & Lowrie, 2008a). For example, on an Axis

item, Diezmann and Lowrie (2006) explained that students tend to interpret specific graphics

(e.g., a number line) by either using a counting model or a measurement model. For those

who use a counting model, the predominant strategy was to count forward or backward along

the axis, whereas students who utilised the more sophisticated measurement model were able

to proportionalise space between numbers, locate points of reference, and had an

understanding of directionality. Additionally, the high performing students used multiple cues

within the graphic to ascertain meaning (Diezmann, Lowrie, & Kozak, 2007). Each of these

findings can inform instruction. These findings also highlight the insight gained by

investigating individual graphical languages in detail. Aligned with the findings from the

mass testing, which identified gender differences on map tasks, an investigation of map tasks

which concentrates on what students know about maps would be of value. One aspect of the

interview component of the GLIM project which can be interrogated further is the video of

the children engaging with the GLIM items. Through the video data, aspects of students’

engagement with and understanding about maps tasks can be gathered from considering

students’ gestures and verbal explanations as they complete various map tasks. Hence, this

study utilised the video of the participants’ interviews on the map tasks from the GLIM study

to investigate students’ understanding of these tasks in detail.

3.4. Relationship between the ARC Project and the Masters Project

The following graphic provides an overview of the relationship between the ARC project and

the current study. My involvement with the ARC project was as a Research Assistant in

Phases One and Two the final 3 years of the study (Years 2-4), undertaking a varied role

within the team relating to mass testing (Section 3.4.1); interviews (Section 3.4.2), and

publication (Section 3.4.3). As shown on Figure 3.1, the current study extends the original

study by focussing on one particular language in ways not outlined in the aims of the original

study.

41

Figure 3.1. Development of the current research project.

3.4.1. Duties Associated with Mass Testing in Phase One

My main duty associated with the mass testing in the ARC project was data collection and

analysis. With other research assistants, I undertook the second and third year mass testing

data collection and contributed to analysing the data within the frameworks developed by the

chief investigators. Administrative duties, such as data storage, as required by the chief

investigators were also undertaken. Thus, I have a sound knowledge of how the mass testing

data were collected, stored and analysed statistically.

3.4.2. Duties Associated with the Interviews in Phase Two

My main duties associated with the interview phase of the ARC project were to carry out the

first and second year interviews and to assist in analysing these data within the protocols

developed by the chief investigators. In each year after data collection, I was responsible for

coding of students’ verbal responses from the interviews. I was also involved with the

administrative side of the project, coordinating the data collection process, data entry, and

maintaining and updating project records. Thus, I was involved in the collection, coding and

storage of the interview data.

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3.4.3. Publications from the Original Study

I have contributed to publications by the chief investigators and co-written papers with them

for peer reviewed journals and conference publications (Diezmann, Lowrie, Sugars, &

Logan, 2009; Lowrie, Diezmann & Logan, 2009). I have also undertaken an additional small

study associated with the original study and published independently (Logan & Greenlees,

2008). The additional small study considered the effect that modifying assessment items had

on students’ performance and strategy use.

3.5. The Masters Study

Due to my extensive involvement in the ARC project as described above and with Ethics

approval (see Section 4. 10.3), the chief investigators authorised me to undertake further

analysis on a specific section of the interview data for this Masters Research project. The two

ARC chief investigators are the supervisors of this Masters project. This study aimed to

describe a range of student behaviours as they solve map tasks and the processes they used to

solve the tasks through the interrogation of existing video data. Thus, this project adopted a

multimodal theoretical framework to describe and analyse students’ behaviour by

investigating both their verbal and non verbal behaviours as they complete a set of map tasks.

This project focused on understanding students’ sense-making on map tasks using existing

video data collected during interviews. The aim was to describe the behaviours that students

exhibited and used to solve these tasks and to make sense of the verbal and non verbal

behaviours. The investigation also documented the extent to which the task type and gesture

use influence these behaviours. A data mining and knowledge discovery design (described in

Section 4.3) is used to reanalyse data sourced from the ARC project, using retrospective

techniques to explore data from a different perspective.

3.6. Chapter Summary

The current study is an elaboration and extension of one aspect of a larger ARC funded

project undertaken by Diezmann and Lowrie. That project used a longitudinal multi method

design incorporating both a mass testing phase and an interview phase. A key finding from

the mass testing was the discovery of gender differences in favour of boys on the Map

languages. Findings from the interviews revealed that students had inappropriate conceptions

of basic graphics including Maps. Both the findings of the ARC project on the interpretation

of maps and the availability of extensive video data generated in the interview phase of that

project were factors in the identification of the topic for the current study. Also contributing

43

to the context of the current study was my involvement with the original ARC project, having

a thorough knowledge of both the mass testing and interview phases, and contributing to

publications. This study utilised the existing video data from the interview phase of the ARC

project in order to describe a range of student behaviours as they solved map tasks. In

particular, students’ gestures and explanations were examined. The design and methodology

of the Master’s study are presented in Chapter 4.

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Chapter 4. Design and Methodology

4.1. Introduction

The purpose of this chapter is to describe and justify the design and methodology of this

investigation. The chapter has five parts. The first part outlines the investigation’s three

research questions (Section 4.2). The second part discusses the research design by

introducing the Knowledge Discovery in Data (KDD) process (Section 4.3). The third part

describes each step of the KDD process in more detail by outlining the Selection (Section

4.4), Preprocessing (Section 4.5), Transformation (Section 4.6), Data Mining (Section 4.7),

and Interpretation/Evaluation (Section 4.8) steps respectively. The fourth part presents an

overview of the design which highlights the connectivity of the design to the research

questions (Section 4.9), and then provides a justification for the quality and rigor of the study,

including ethical considerations (Section 4.10). Finally, a summary of the chapter is

presented (Section 4.11).

4.2. Research Questions

The following research questions concerning Map tasks emerged from the literature review

(Chapter 2) and the preceding investigations of graphics (Chapter 3).

1. What mathematical understandings do primary-aged students require to interpret Map tasks?

2. What patterns of behaviour do these students exhibit when solving Map tasks?

3. What profiles of behaviour do successful and unsuccessful students exhibit on Map tasks?

4.3. The Research Design –Knowledge Discovery in Data

Due to ready access to a set of quality interview data (Chapter 3), this investigation of

students’ performance on Map tasks was undertaken using a Data Mining approach. Hand

(1998) defined data mining as “the process of secondary analysis of large databases aimed at

finding unsuspecting relationships which are of interest or value” (p. 112). While data mining

is generally the domain of large, numerical datasets, Hand (1998) suggests that increasingly

data mining is being used with other non-numerical data sources such as audio, image and

text data. From this perspective, an analysis of video interview data is within the scope of

data mining. Data mining can be a standalone process, as suggested by Fayyad, Piatetsky-

Shapiro, and Smyth (1996). It is also acknowledged as one step in a larger process called

45

Knowledge Discovery in Data (KDD). KDD is an area of research which aims to address the

rapidly growing proliferation of digital data by extracting useable knowledge from a

collection of data. Data mining has been defined as “the nontrivial process of identifying

valid, novel, potentially useful, and ultimately understandable patterns in data” (Fayyad et al.,

1996, pp. 40-41). For the purposes of this project, KDD provides a model for the design of

the study. While traditionally KDD stems from fields of research such as artificial

intelligence, machine learning, and statistics, the process is interdisciplinary in nature and is

applicable in many research contexts. The process is both interactive and generative and

involves a series of sequential steps and corresponding decision making processes (Fayyad et

al., 1996).

According to Fayyad et al. (1996), there are five steps in the KDD process (Figure 4.1). (a)

The Selection step involves selecting data from the larger database to create a target data set.

The target data set is based on the goals of the project and the relevant prior knowledge of the

data, i.e., focusing on a subset or a sample of data. (b) Preprocessing involves reducing the

target data set to the useful features which represent the goals of the project, essentially

sorting and organising the data. Preprocessing requires the researcher to look at the data in a

manner which allows them to make decisions about the exact nature of analysis. (c)

Transformation of the data requires a suitable analysis technique to be identified based on the

goals of the project and the type of data being utilised. Data can be transformed through any

analysis technique, with the aim to classify, cluster and summarise the data. (d) Data mining

is seen as searching for and “determining patterns from observed data” (Fayyad et al., 1996,

p. 43). This step can often involve a form of visual representation of the extracted patterns.

(e) The Interpretation/Evaluation step consists of interpreting any patterns and themes

identified in the data mining step in relation to the project goals and evaluating their

usefulness and potential interest to others. This study followed these steps.

46

Figure 4.1. An overview of the KDD process (Fayyad et al., 1996, p. 41).

4.4. Selection of the Data

The selection step requires the researcher to identify the required data from the original data

set. This initial step for this project consisted of identifying a subset of the data from the ARC

project (Section 3.3.2.3) to suit the research questions (Section 4.2). The subset of data was

identified in three ways. First, the data was restricted to only the video data from the

interview phase of the original study (Section 4.4.1). Second, only selected three tasks from

the GLIM instrument which had typical map structure (road map, pictorial map, coordinate

map) were used (Section 4.4.2). Third, the data was limited to 43 participants from one state

who completed each of the three map tasks in the interview phase of the original study.

Because a detailed analysis was undertaken, an analysis of the full cohort of 98 from both

states was beyond the scope of this project. Explanation of the video data (Section 4.4.1), the

map tasks (Section 4.4.2), and the participants (Section 4.4.3) follows.

4.4.1. Video Data

Video has numerous strengths as a qualitative research tool (Penn-Edwards, 2004; Ratcliff,

2003). In particular, video can assist in capturing subtleties and easily overlooked details,

such as body language and facial expressions; enable data to be viewed repeatedly; facilitate

the re-coding of data from different perspectives; and support in depth, fine-grained data

analysis. These features are especially advantageous in situations where the focus of inquiry

is working with children in one-on-one situations, where verbal explanations may not always

be adequate. The use of video data in this study provided access to gestures, which is an area

47

48

of investigation not previously addressed in the original project, with the video data capturing

children’s engagement with map tasks with respect to their language and gesture use. In

particular, the video data allowed the researcher to see how the participants reacted to

particular map tasks and their reaction as they explained their solutions.

In this project, the interviews were analysed from a visual as well as verbal perspective.

Visual data included students’ gestural hand movements. These data were taken into

consideration along with their verbal explanations in investigating how students interpreted

map tasks. Thus, this study attempted to delve further into the nature of students’ engagement

with and solution of, Map tasks by going beyond their verbal explanations and solution

pathways.

The video data was edited using the Studiocode (Studiocode Business Group, n.d.) software

in order to narrow the data set from the entire interview to only the section where the

participants were engaging with and solving the Map tasks (see Section 4.5.2). The video

excerpt for each task was approximately 4 minutes in duration. Hence, there was a total of 12

minutes of video analysed per student (3 tasks x 4 minutes) which equates to an overall total

of 516 minutes (12 minutes per student x 43 students) or 8.6 hours of video analysed in

depth.

4.4.2. The Map Tasks

The tasks utilised in this study were three typical map tasks that students undertook in the

interview phase of the ARC project (see Section 3.3.2). Originally, these tasks were selected

from state-based published numeracy tests and were designed for students in the upper

primary grade levels (see Section 3.3.2). Hence, these tasks are representative of the types of

tasks that students in the primary years are expected to be able to complete. The three map

tasks have a number of elements related to the language, mapping knowledge, and

mathematics concepts that students are required to utilise and process in order to interpret the

tasks. These are outlined on Table 4.1.

Table 4.1

Overview of the Three Map Tasks.

Map Task Language Mapping knowledge Mathematics concepts

The Picnic Park task Everyday: rides, bike, what part. Uses a Key

Landmarks (labelled points)

Birds-eye-view perspective

Location (position)

Arrangement (co ordinates)

The Playground task Everyday: gate, tap, shed, how many. Landmarks (symbolic)

Birds-eye-view perspective

Direction (following routes)

The Street Map task Everyday: left, right.

Mathematical: north, first, second.

Uses a Key

Landmarks (road names, symbols and labelled points)

Birds-eye-view perspective

Arrangement (co ordinates and scale)

Direction (search for and identify destinations)

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4.4.2.1. The Picnic Park Task

The Picnic Park Task (Queensland School Curriculum Council, 2001) (Figure 4.2) was

represented as a co ordinate map, with positional points used to indicate landmarks (as

opposed to pictorial representations of specific landmarks, like those illustrated in The

Playground Task in Section 4.4.2.2). A key is also provided to indicate a specific landmark

and a compass bearing given to indicate North. The task also required an understanding of co

ordinates.

Figure 4.2. The Picnic Park Task.

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4.4.2.2. The Playground Task

The Playground Task (Queensland Studies Authority, 2002a) (Figure 4.3) was a pictorial

representation of a playground. Locational features include easily identifiable objects

depicted in pictorial form (e.g., a pool), and a representation not to scale. Locational skills

need to be employed to find the track, while directional skills are needed in order to locate

given landmarks and navigate a directional pathway to identify a solution.

Figure 4.3. The Playground Task.

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4.4.2.3. The Street Map Task

The Street Map Task (Queensland Studies Authority, 2002b) (Figure 4.4) is a traditional

street directory representation with some features represented from a birds-eye-view

perspective (e.g., the netball courts). Other features are represented in the key and depicted

pictorially on the map (e.g., the post office). The map includes co ordinates, a scale and a

compass bearing. Specific mathematics understandings (i.e., North) and everyday language

(i.e., first right, second left) are required to navigate a route along streets to complete a

journey and then identify a landmark (i.e., a street name).

Figure 4.4. The Street Map Task.

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4.4.3. Participants

The original data on the map tasks were collected from forty-three students (22 girls, 21

boys) at three Australian primary schools in the state of NSW between 2005 and 2007 (3

years) (See Section 3.3.2.3). The participants commenced in the study when they were aged

9-10 (Grade 4) and were interviewed annually for three years. The three schools involved

consisted of one government, one Catholic and one independent school and they all catered

for children aged 5-12 years (Kindergarten to Grade 6). Situated in a regional city with a

population of over 50 000, these medium-sized schools all had enrolments of over 200

students. Given the diversity of the school environments, the participants were from varying

socioeconomic and academic backgrounds, and reflected the ethnic and cultural composition

of the local community, with less than 5% of the students speaking English as their second

language. The participants could be described as relatively monocultural with students

typically from a white, Anglo Saxon background. An overview of the information about

participants is presented in Table 4.2.

Table 4.2

Composition of the Schools and the Participants.

School School Type Total Students Male Female

Northern Government 13 6 7

Western Catholic 15 8 7

Southern Independent 15 7 8

The participants were not subject to any treatment programs in the study and they continued

with the mandatory syllabus of the state. None of the participants had received overt

instruction about how to interpret these types of graphics because neither the state

mathematics syllabus nor the school mathematics programs included a specific focus on

learning about graphics. However, the participants might have encountered these types of

graphics previously during instruction, in the use of textbooks, or in other assessment tasks.

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4.5. Preprocessing the Data

The preprocessing step is the second of five steps in KDD (Fayyad et al., 1996) and involved

organising the data for analysis. This was using data reduction techniques and “finding useful

features to represent the data depending on the goals of the task” (Fayyad et al., 1996, p. 42).

In this project, the video data was separated to provide the distinction between Task Solution

and Task Explanation. These episodes are described shortly (See Section 4.5.1). The

preprocessing step also identified the tool with which the analysis was undertaken, in this

case Studiocode (Studiocode Business Group, n.d.) software (Section 4.5.2).

4.5.1. Organising the Data

The target video data were organised into two episodes. Each video segment was separated

into two distinct episodes which occurred in sequential order. Episode 1 focussed on the Task

Solution and consisted of the video excerpt that involved the participants’ solving the

question. Episode 2 was the Task Explanation and consisted of the video excerpt of

participants’ explaining their solution. These two video episodes allowed data to be analysed

at a detailed level. Table 4.3 outlines how each video segment was organised for each

participant to aid in transforming the data.

Table 4.3

Organisation of the Video Data.

Episode What part of the video? Coding

1. Task

Solution

Participants solving the

task

Dichotomous coding with students

either using a purposeful gesture or

non-purposeful gesture

2. Task

Explanation

Participants’ explanation

of their solution

Coded according to the participants’

verbal explanations and gestural

behaviour

Task Solution (Episode 1) was measured from when the student started to work out the task

to when they had written their answer (generated a solution). The two categories in this first

episode are identified as internal and external behaviours. Internal behaviours were when the

students worked out the solution in their heads without any sort of gestural movement.

External behaviours were when students used purposeful gestural movement to navigate

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around the map task. Thus, this episode was dichotomous in nature from the view that

students were deemed to be making or not making gestures.

Task Explanation (Episode 2) commenced from when the interviewer asks the student to read

out the question and explain how they answered the task to when they move on to the next

task. This episode involved both verbal and non verbal observations of student behaviour and

the analysis consist of a number of different categories. Data consisted of students’ verbal

and gestural explanations of their strategies and solution of each Map task. Participants’

verbal explanations were categorised according to: (a) their knowledge of specific mapping

terms and functions; and (b) the mathematical concepts and understandings they possess (see

Section 2.3.7 for a detailed description of these categories). The mapping aspect of students’

explanations examined the specific indicators of mapping knowledge. This knowledge

included reference to the conventions of map tasks, such as using keys and legends (Section

2.3.7), and whether students use these in their explanations. Similarly, students’ knowledge

of mathematical concepts was examined through their language and gestures with particular

attention to specific mathematical ideas such as location, movement and direction (Section

2.3.7). Throughout the Task Explanation (Episode 2), students’ non verbal responses were

categorised according to the hand gestures they were exhibiting during their explanation.

These were classified as iconic, metaphoric, deictic, or beat (McNeill, 1992, see Section 2.4

for a detailed description of McNeill’s categorisation). The following graphic shows an

overview of the focus in each Episode (Figure 4.5).

Figure 4.5. A description and representation of Episodes 1 and 2.

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4.5.2. Analysing Video Data with Studiocode

Analysis of the video data was undertaken with a computer assisted software package called

Studiocode (Studiocode Business Group, n.d.). Studiocode is a video editing and analysis tool

that allows users to annotate video and record the frequency and times of specific events that

occur throughout the video. Studiocode enables the user to capture, compact, classify,

observe, and search video and audio very easily (Gyorke, 2006). Users generate codes and

code sets which are created in a code window, where they build a personalised set of buttons

and labels based for specific analysis (see Figure 4.6). Other features associated with codes

include the ability to develop relationships among codes via linking, the option of lead and

lag times for code buttons, and colour coding. Code templates can be edited as often as

necessary, and do not have predefined structures, making them very flexible. These codes can

then be applied to video segments which will produce a timeline. Each code in the timeline is

linked to an individual movie segment highlighting that particular code. These coded movie

segments can be viewed repeatedly and modifications can be made to the codes during

subsequent viewing. According to Rich and Hannafin (2009), Studiocode “provides simple

quantitative analysis for codes and transcripts, which can be exported for detailed data

analysis” (p. 60). The code matrix function in Studiocode allows the user to view a summary

of the number of occurrences of each code, which can be exported into other programs if

further statistical analysis is required. It also allows the user to click on a specific code and all

instances of video data with that code are collated and available to view. These data can also

be searched using Boolean searches.

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Figure 4.6. The Studiocode coding window.

Studiocode (Studiocode Business Group, n.d.) has been used very effectively to gain insight

into educational issues. For example, Clarke and associates (Clarke, Emanuelsson, Jablonka,

& Mok, 2006; Clarke, Keitel, & Shimizu, 2006) used Studiocode software to perform

detailed analysis in their international research projects, The Negotiation of Meaning Project,

The Learner’s Perspective Study and the Casual Connections in Science Classrooms Project.

Clarke and Xu (2008) reported that Studiocode provided “basic descriptive coding statistics

with a capacity to reveal temporal patterns in a highly visual form ... and supports coding of

either events in the video record or the occurrences of specific terms in the transcript” (p.

967). In another study using the software, Armstrong and Curran (2006) reported that

teachers using Studiocode found “this piece of software useful because it provided a visual

representation of patterns of similarity and difference across the sequence of lessons video

recorded” (p. 342). Although the software is relatively new in the field of Education, Rich

and Hannafin (2008) argue that “(it will) provide significant data mining capabilities,

management, and fine grained analysis and reporting” (p. 66). Thus, Studiocode is a highly

57

appropriate tool for investigating understanding of children’s engagement with Map tasks and

their associated mathematical sense making.

This study utilised Studiocode in two ways. First, the software was used to edit the video data

from the original study. The video excerpts were partitioned and classified as Task Solution

and Task Explanation for each task in order for analysis to take place. The 43 video excerpts

for Task Solution from an individual task were stacked together to form one continuous video

excerpt of the cohorts’ Task solution episode. The same procedure was repeated for Task

Explanation. For example on The Picnic Park, there were 43 video excerpts that made up the

Task solution video and 43 video excerpts that made up the Task Explanation video. This

provided the opportunity to use a single coding window to analyse each episode for the entire

cohort.

Second, various codes were developed (Figure 4.6) to enable Content Analysis to take place.

These codes were developed using set procedures which are described shortly (Sections

4.6.2.1 & 4.6.2.2). The coding window was developed to reflect the different episodes and

the different codes within each episode.

4.6. Transformation of the Data

Transformation of the data is the third step in KDD (Fayyad et al., 1996) and involves

identifying an analysis technique that will classify, categorise and summarise the target data

with respect to the aims of the project (Fayyad et al., 1996). This investigation utilised

Content Analysis to undertake the transformation step of the target video data. Initially, the

technique of content analysis is explored (Section 4.6.1), and then the process of content

analysis as it applies to the study is outlined (Section 4.6.2).

4.6.1. Technique of Content Analysis

Content Analysis was utilised to classify the data during the Transformation step of analysis.

According to Payne and Payne (2004) “Content Analysis seeks to demonstrate the meaning

of written or visual sources ... by systematically allocating their content to pre-determined,

detailed categories, and then both quantifying and interpreting the outcomes” (p. 51). This

qualitative identification of meaning from both written and visual sources links closely to the

multiliteracies (Section 2.2.1) and multimodal underpinnings (Section 2.2.2) of the study.

Best and Khan (2006) maintain that Content Analysis is concerned with the explanation of

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particular phenomena at a particular time and is useful in adding knowledge to specific fields

of inquiry and in explaining particular contexts and events. They also suggest it aids “in

yielding information helpful in evaluating or explaining social or educational practices” (p.

258). Content Analysis enables data to be collated through frequencies and counts. This

utility allows some aspects of the data to be analysed using quantitative measures. As

Krippendorff (2004) argued, there is no dichotomous distinction between quantitative and

qualitative approaches in content analysis which suggests that both approaches can be utilised

in analysis. Therefore, content analysis is a useful tool to investigate the verbal and non

verbal behaviours of students as they engage with map tasks.

4.6.2. Process of Content Analysis

Students’ engagement with the Map tasks during both Task Solution (Episode 1) and Task

Explanation (Episode 2) were analysed using content analysis to observe the frequency of

each occurrence of students’: (a) internal or external behaviour when solving the task; (b)

verbal references to the conventions of Map reading; (c) verbal references to specific

mathematics concepts; and (d) non verbal hand gestures (See Sections 5.3, 5.4, & 5.5). This

counting procedure provided a quantifiable ‘what’s there’ reference (Miles & Huberman,

1984) to make further decisions about the data. Descriptive statistics were used with the

outcomes of the content analysis from each Episode and the subsequent categories of

purposeful gesture use or non-purposeful gesture use and mapping skills, mathematics

concepts and gestural behaviours. These statistics were used to outline student performance

and observable behaviours. These counts were also used to produce matrices which helped to

identify any general patterns of behaviour in relation to Map tasks. A description of the

coding for the Task Solution (Section 4.6.2.1) and Task Explanation (Section 4.6.2.2) follow.

4.6.2.1. Coding Procedure for Task Solution

This coding procedure is for Task Solution (Episode 1). Recall, Task Solution consisted of

the video excerpt of the participant solving the map tasks (Section 4.5.1). The first level of

analysis was to undertake a content analysis on Episode 1. This described participants’

performance and the behaviours they exhibited whilst solving the tasks. In this instance, the

focus is presented in a dichotomous form. That is, participants were deemed to have

answered the task as correct or incorrect and be using purposeful gesture or non-purposeful

gesture when solving the task.

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When participants were coded as using purposeful gesture to engage with the task, three

types of gestural behaviours was observed. The first gestural behaviour was classified as

direct touching of the page with a finger or a pen. The majority of these gestures were

directed specifically towards the map on the page. The second type of gestural behaviour was

classified as hovering over the page, without direct contact on the page. The majority of these

gestures were directed towards the space above the map, and between the map and the

answers. The third type of gestural behaviour was classified as a place keeper. This gesture

was used to track their thinking through the task such as counting on fingers or drawing on

the page. Figures 4.7, 4.8 and 4.9 are pictorial examples of the three types of behaviour that

were coded as purposeful gesture.

Figure 4.7. Purposeful gesture

type one: The participant using

a finger or a pen to touch the

page as he solves the task.

Figure 4.8. Purposeful

gesture type two: The

participant using his pen to

hover over the page as he

engages with the task.

Figure 4.9. Purposeful

gesture type three: The

participant counting on her

fingers as a place keeper, to

keep track of thinking.

When participants were coded as using non-purposeful gesture to engage with the task, two

types of behaviours were counted. The first behaviour was classified as non-hand movement.

This type of behaviour was identified when participants either clasped their hands together, or

put their hands under the table. The second type of behaviour was classified as nervous

movements or twitching. This type of behaviour was identified when participants were

“playing” with their hands or their pen, but no movement was directed towards the task.

Figures 4.10 and 4.11 provide pictorial examples of the types of behaviours which were

coded as non-purposeful gesture.

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Figure 4.10. Non-purposeful gesture type

one: The participant sits with his hands

under the desk, making no movement toward

the task with his hands.

Figure 4.11. Non-purposeful gesture type

two: The participant plays with her pen, but

makes no movement toward the task.

Based on the above classifications, participants were coded according to which type of

behaviour they exhibited during Episode 1. These classifications are coded in relation to the

type of engagement the students had with the task as they went through the process of solving

the task. Thus, habitual tendencies were not considered to be gestures for the purposes of this

study. This coding did not reflect the number of times a student gestured, but rather whether

they exhibited this behaviour at least once, or not, during Task Solution. Hence the content

analysis of Episode 1 meant that students were coded as either using purposeful gesture or

non-purposeful gesture to engage with and solve the task.

4.6.2.2. Coding Procedure for Task Explanation

This section considers the particular approaches and behaviours employed by the students as

they solved the three map tasks. These data are drawn from student explanations of how they

solved the respective tasks and consequently go beyond the dichotomous analysis (gesture-

non gesture) of Task Solution (Episode 1)

In order to identify the approaches and behaviours that were both common and distinctive

across the tasks, detailed analysis of individuals’ task solution and explanation were

undertaken. This pilot coding involved selecting a purposeful stratified sample of students

based on performance on the three items. In this sample, one third of the 21 students who

correctly solved all three map tasks (n=7) and one third of the nine students who correctly

solved one of the three tasks (n=3) were selected. The collective groups’ (n=10) video

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excerpts were used to construct pilot codes (Miles & Huberman, 1994) in order to develop a

set of behaviour categories which could be adapted for further analysis. This process of

content analysis (See Section 4.6.2) provided a quantifiable ‘what’s there’ reference (Miles &

Huberman, 1984) to make decisions about student approaches and behaviours in relation to

their mapping and mathematics understandings.

The coding of the verbal explanations was based around the extent to which the students

utilised mapping skills and the solution approaches they used on the map tasks. From the

pilot coding of highly successful students and less successful students, a set of mapping skills

was identified (Table 4.4). These mapping skills identified elements on the maps that students

recalled in their explanation of how they solved the tasks.

Table 4.4

Codes for Mapping Skills.

Mapping skill Code

Identify and use landmarks LM

Identify and use co ordinates CO

Identify and use compass point CP

Identify and apply key KE

The pilot coding also identified the approaches utilised by students to solve the map tasks

(Table 4.5). These solution approaches were coded directly from the explanations students

gave in regard to the way they worked out each task.

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Table 4.5

Codes for Solution Approaches.

Solution approaches Code

Describe movement DM

Understood relationship between landmarks and movement RLM

Followed a set of directions (route) FD

Misunderstood ordinal sequence OS

Process of elimination PE

Immediately accessed positional information with the key API

Fixated on reference point RP

Other OH

With regard to the gestural behaviour exhibited by students as they explained their solution,

McNeill’s (1992) classification of hand gestures was utilised (Section 2.4.2). Hence, students

were coded as using an iconic, metaphoric, deictic or beat gesture (Table 4.6). Also, those

students who did not exhibit any gesture were coded appropriately.

Table 4.6

Codes for the Types of Gestures used during Task Explanation.

Type of gesture Code

Iconic IGE

Metaphoric MGE

Deictic DGE

Beat BGE

No gesture NGE

The codes presented in Tables 4.4, 4.5, and 4.6 were the final codes used during the content

analysis step of the design. Students were assigned a code according to the explanation they

gave and the type of behaviour they exhibited during that explanation.

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4.7. Data Mining

Data mining is the fourth step in KDD and is seen as “searching for and determining patterns

from observed data” (Fayyad et al., 1996, p. 43). The observed data in this study are taken

from the content analysis during the Transformation step (Section 4.6). The transformed data

provided descriptive counts in relation to student performance and observable behaviours on

the Map tasks. This information was used to find patterns that emerge among groups of

students with similar verbal and non verbal behaviours. These patterns were considered with

particular reference to the (a) respective Map tasks; (b) students’ performance on these tasks;

(c) students’ behaviour; and (d) students’ explanations. The process for developing a

framework to identify commonality traits among students was generated by Anticipatory

Data Reduction (Miles & Huberman, 1984). This reduction occurs by focusing and bounding

the data. In this analysis, the bounding was undertaken, in the first instance, by analysing

each Map task separately. This provided a form of data reduction which allows for an

analysis of task type. Visual data displays—using both Studiocode (Studiocode Business

Group, n.d.) and matrices—were generated for each task. These data displays were analysed

in relation to student performance (task correctness) and potential patterns that emerge with

specific reference to gesture use, mapping skills and solution approaches. This process was

replicated for the three tasks. Further analysis drew meaning from these data by seeking

patterns, explanations, causal flow, and irregularities (Miles & Huberman, 1984). This

component of the analysis created reasoning paths that represent various types of solutions.

Reasoning paths are a visual representation of the flow of students’ behaviours and reasoning

(Diezmann, 2004). Representing solutions on these pathways enables the researcher to

identify commonalities and differences among students. They also allow the researcher to

present a holistic view of a particular type of reasoning during the solution of a task.

An overview of the analysis process for Data Mining based on Miles and Huberman’s (1984)

model for data analysis follows (Figure 4.12). As shown, the process moves through a

process of transforming and reducing data to a set of reasoning paths which can be displayed.

This data mining step isolated each individual Map task and attempted to pass the

transformed data through two separate lenses, namely task correctness and purposeful gesture

use. Some points of interest which arose from this viewpoint were: Do students who answer

this task correctly behave in a similar manner? and Are the students who answer incorrectly

behaving differently to those who answered correctly? The two lenses worked together in

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order to identify profiles of groups of students who behaved in similar ways.

Reasoning/behaviour paths were produced to show the flow of students’ understanding on

each task and on Map tasks in general (See Section 5.9).

Figure 4.12. The Data Mining analysis process.

4.8. Interpretation/Evaluation of the Analysis of Data

Interpretation/Evaluation is the final step of the KDD process (Fayyad et al., 1996) and

provides an opportunity to present findings about discovered knowledge by documenting and

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66

reporting on the analysis of data and establishing recommendations for relevant stakeholders.

This step also allows for the resolving of any potential discrepancies in data analysed in the

Data Mining step. In the current study, this section summarises findings based on the

interactive and generative interpretations of the Transformation analysis (Section 4.6) and the

reasoning/behavioural profiles generated in the Data Mining step (Section 4.7). The

information for this step is represented using a combination of linguistic, numerical and

graphical formats. Hence, multiliteracies (Section 2.2.1) and multimodalities (Section 2.2.2)

inform the use of tools and techniques for analysis and also provide the medium for

representing the outcomes of these analyses.

4.9. Overview of the Design

The KDD research design is appropriate for this analysis because it provides a mechanism for

systematically transforming and mining existing data in order to discover new knowledge.

Specifically, it provides scope for video data to be re-analysed using both quantitative and

qualitative techniques. Table 4.7 highlights how the structure of this design allowed for the

research questions to be investigated.

Table 4.7

Symbiosis of Research Questions and Research Design.

KDD Design step Question Intended outcomes

1. Selection Selection of target data.

2. Preprocessing Organisation of data in preparation for analysis.

3. Transformation What mathematical understandings do primary-aged students require to interpret Map tasks?

Descriptive counts from each episode with relevance to how the participants solved the map tasks in relation to their mathematics knowledge, mapping knowledge and their gestural behaviours.

4. Transformed Data What patterns of behaviour do these students exhibit when solving Map tasks?

Using the descriptive counts, matrices can be produced to identify any patterns of similarities on the Map tasks generally.

5. Data Mining What patterns of behaviour do these students exhibit when solving Map tasks?

Patterns and themes discovered in the transformed data will attempt to uncover and explain some common behavioural and reasoning traits among students on each map task.

6. Data Mining and Interpretation/ Evaluation

What profiles of behaviour do successful and unsuccessful students exhibit on Map tasks?

This project will use success as a lens through which the data is viewed to determine if successful and unsuccessful students are attempting these types of tasks in a similar manner.

7. Interpretation/ Evaluation

Summary of findings, implications and recommendations.

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4.10. Quality and Rigour of the Study

In order to enhance the methodological rigor of the study, the research findings need to be

presented in a manner which achieve trustworthiness. Graneheim and Lundman (2004) maintain

that notions of credibility (Section 4.10.1), dependability, and transferability (Section 4.10.2) are

interrelated aspects of trustworthiness which contribute to the rigor of a study. In particular, they

argue that these three aspects of trustworthiness are necessary for studies that utilise content

analysis. The rigour of a study is also related to the ethical conduct of the study, with particular

consideration of the rights of the participants and the responsibilities of the researcher (Section

4.10.3).

4.10.1. Credibility

Credibility refers to confidence in how data and processes address the research questions.

Graneheim and Lundman (2004) suggest that credibility is enhanced when participants in the

study are of various ages and include both males and females. They also suggest that

observations should be considered from different perspectives. In this study, the 43 participants

chosen were a mix of boys and girls, were from varying academic backgrounds, and reflected the

socioeconomic make up of the large rural city in one state of Australia (Section 4.4.3). This

make up of participants helped to provide diversity to this study. The one-on-one semi-structured

interviews from the original study provided opportunities for participants to express their

understanding about various Map tasks (Section 3.3.2.3). The interviews were conducted over a

three-year period and thus captured students engaging with various maps tasks at different ages.

The use of video in the interviews helped to capture participants’ gestures whilst they were

solving the Map tasks. This allowed for the investigation of participants’ gestures along with

their verbal explanations. The use of video also allowed for repeated viewing of the video data.

This helped to maximise the accuracy of the content analysis process because it provided

opportunities to analyse the data from different perspectives, and thus, enhance the credibility of

the study. Additionally, Graneheim and Lundman (2004) suggest that another critical issue for

achieving credible results within the content analysis process comes from selecting the most

appropriate and meaningful codes which reflect the context of the situation. Initial coding of data

was taken from the content knowledge of maps such as use of keys and use of co ordinates

(Section 2.3.7), and the interpretation processes using interactive and generative processes within

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the KDD design (Fayyad et al., 1996) (Section 4.3). Content analysis and interpretation were

checked by my supervisors, who as chief investigators on the original study were familiar with

these data sets.

4.10.2. Dependability and Transferability

The notions of dependability and transferability relate to the possible replication of the study and

the consistency of the procedures for keeping thorough notes and records of activities (Hittleman

& Simon, 2006). Graneheim and Lundman (2004) suggest that clear descriptions of culture,

context, and a thorough detailing of the design provide an opportunity for replication. In this

study, dependability was enhanced through verbatim accounts of student voice to clearly present

the link between the theoretical underpinnings and the analysis process. The video data was

viewed in its original form, and thus, precise descriptions of events were analysed. In addition, I

was a research assistant in the data collection for the original study for a considerable period of

time and observed a full range of interview sessions within the original study (see Section 3.4.2).

Hence, I am very familiar with the context of the original project and the interview data

collection. To facilitate transferability, specific features of the participants, the data collection

and the analysis process were explicitly detailed outlining the design of the original study

(Chapter 3) and the current investigation (Chapter 4).

4.10.3. Ethics

A number of ethical issues need to be considered in this investigation, specifically, ethical issues

of confidentiality, use of pseudonyms in reporting, storage of data, and ethical issues related to

the conduct of the researcher (Kumar, 1996).

With regards to the current investigation, considerable attention was paid to adhering to the

rights of the participants involved in the original study including how the data were reported in

this follow up study. As Kumar (1996) stipulated “it is important to ensure that research is not

affected by the self-interest of any party and is not carried out in a way that harms any party” (p.

192). This project aims not to harm any participants through this research process and will

address the ethical issues in the following way. In terms of confidentiality, all participants were

provided with full anonymity. All participants were assigned a four-digit code to replace their

name on data records and pseudonyms were used in qualitative findings when student voice was

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required. Pseudonyms were assigned based on the first letter of the participant’s name. However,

care was taken that these pseudonyms were reflective of participant’s gender.

With respect to the original ARC project (Section 3.3), ethical clearance was obtained through

QUT Human Research Ethics Committee (No. 3728H). The current project is within the broad

aims of the ARC project, as highlighted by the following research foci addressed in the original

study: (a) to determine students’ performance on the graphical languages; and (b) to understand

the knowledge students utilise when decoding graphical languages in mathematics. Hence this

study falls under the ethical consent obtained for the original study (QUT Human Research

Ethics Committee No. 3728H). Parental permission was obtained to use photographs/video stills

in reporting. The data were not used in this project in a manner that contradicts the initial consent

agreement between students, parents/caregivers and the research team (see Appendix C for the

original information package provided to parents). In accordance with the Australian code for the

responsible conduct of research (Australian Government, 2007), the target data subset will be

stored on hard drives which are password protected and stored in locked in cabinets for 5 years. All

other data will also be stored on password protected hard drives. At all times, I conducted this

research in an ethical manner based on the principles of the Australian Association for Research in

Education’s (1997) Code of Ethics.

4.11. Chapter Summary

The design of this study capitalises on an existing data set to undertake more detailed analysis of

students’ performance on Map items. The Knowledge Discovery in Databases (KDD) design

(Fayyad et al., 1996) utilises existing data to explore patterns and relationships within that data.

The KDD design is made up of five sequential steps. The first step involved selecting the target

data to be utilised in the process. In this study, the target data was identified by restricting the

original data to only the interview video data, then more specifically to only three map tasks

which had typical map structures from within the interview data. It was also restricted to 43

participants to enable detailed analysis of students’ performance. The second step involved

preprocessing the data so that it is organised in a way to facilitate analysis. In this study, the

target data was organised into two distinct episodes—(a) when the student was solving the task

(Episode 1), and (b) when the student was explaining their solution (Episode 2). This step also

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identified Studiocode (Studiocode Business Group, n.d.) as a useful cognitive tool that was used

to analyse the data. The third step involved transforming the data to classify, categorise, and

summarise it. In this study, the organised data set was analysed using content analysis to count

frequencies of observed behaviour. The fourth step was data mining. This step aimed to find

patterns and relationships between the transformed data. This study attempted to find patterns of

behaviour in relation to each specific map task by looking at task correctness and gesture use

through anticipatory data reduction techniques. Profiles of student behaviours were also created

using reasoning/behaviour paths. The fifth step in the process related to interpreting and

evaluating the analysis. This study interpreted any new knowledge about students’ engagement

with and performance on map tasks and provided a discussion about the results with particular

reference to any gesture use (Chapter 5).

The quality and rigor of this study were achieved by addressing issues of credibility,

dependability and transferability. This study conducted the research in an ethically sound

manner, taking into consideration issues such as confidentiality, data storage, and use of

pseudonyms in reporting. Ethical approval was obtained to undertake this study using data from

the original ARC study. The results and discussion of the data mining and analysis process

described in this chapter are presented in Chapter 5.

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Chapter 5. Results and Discussion

5.1. Introduction

This chapter presents the results of analysis undertaken for the three map tasks and has three

main parts. The first part presents an introduction to the analysis (Section 5.2) and details how

the students solved the three tasks: The Picnic Park (Section 5.3), The Playground (Section 5.4)

and The Street Map (Section 5.5). It presents data from Task Solution (Episode 1) and Task

Explanation (Episode 2) for each of the tasks. With respect to Task Solution these sections

examine the relationship between correctness and purposeful gesture use. For Task Explanation,

these sections examine the mapping skills, solution approaches and types of gestures used as

students explained their solutions. The results of this section are then discussed (Section 5.6).

The second part of the chapter (Section 5.7) identifies patterns of student behaviour across the

three tasks in relation to Task Solution (Section 5.7.1) and Task Explanation (Section 5.7.2). For

Task Solution, this section takes into account correctness, purposeful gesture use and considers

the impact of gender on both of these variables. For Task Explanation, this section explores

commonalities in mapping skills and solution approaches across the three tasks. A discussion of

these results is then presented (Section 5.8).

The final part of the chapter presents profiles of student behaviour on the map tasks (Section

5.9). The results of both Task Solution and Task Explanation are combined to create profiles of

groups of students who exhibited certain behaviour pathways. These are presented in order of

difficulty: (a) The Picnic Park (Section 5.9.1), with an example of the most common pathway for

this task (Section 5.9.2); (b) The Playground (Section 5.9.3), with an example of the most

common pathway (Section 5.9.4); and (c) The Street Map (Section 5.9.5) also with an example

(Section 5.9.6). A discussion of the task profiles is then presented (Section 5.10) followed by a

summary of the chapter (Section 5.11).

5.2. Introduction to the Three Tasks

Prior to examining the three map tasks in depth, an overview of the analysis process is presented.

(For details of the analysis see Sections 4.5, 4.6 & 4.7). These analyses will draw on data from

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both the Task Solution (Episode 1) and the Task Explanation (Episode 2) in order to identify the

behaviours exhibited by students for each task.

In considering Task Solution, the results of analysis of student success and gesture use during

task solution (Episode 1 data) are presented for each task. The participants’ data was coded

according to a two-step coding procedure: (a) Being correct or incorrect in the answer, where

students were scored as 1 for correct response or 0 for incorrect response; and (b) using either a

purposeful or non-purposeful gesture, where gestures were scored as 1 for a purposeful gesture

or 0 for a non-purposeful gesture. With regard to gestures, the participants were in the process of

working out a solution to the task, and therefore no verbal exchanges between the student and

interviewer took place. Hence, gestures were classified only as purposeful or non-purposeful.

Recall, a purposeful gesture included movements of the hands which reflected engagement with

the task such as direct touching of the page with a finger or a pen, hovering over the page

without direct contact on the page and counting on fingers or drawing on the page. While a non-

purposeful gesture included no hand movements at all or nervous, habitual movements like

twitching (see Section 4.6.2.1 for coding procedure). Chi square analyses was undertaken in

order to ascertain the relationship between correctness and purposeful gesture use (See Appendix

D). In each of the three tasks, the analysis sought to determine whether there was a relationship

between students’ use of purposeful gesture and task success on individual tasks.

With regard to Task Explanation, these data are drawn from student explanations of how they

solved the respective tasks. Codes developed for this analysis were associated with the students

mapping skills and their solution approaches (See Section 4.6.2.2 for the coding procedure).

These skills included identifying and using landmarks, coordinates, keys, and compass points on

the maps. The approaches to solution included describing movement, understanding the

relationship between landmarks and movement, following a route, and using a process of

elimination. The different types of gestures used during Task Explanation were also examined.

However, gesture use in the Task Explanation was analysed differently to Task Solution

(purposeful, non-purposeful). Unlike the Task Solution where there was no verbalisation to

accompany the gestures, the participants’ verbalisations as they explained their solution enabled

a more sophisticated analysis of their gestures. Because these gestures were observed

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concurrently with speech, McNeill’s (1992) classification of gestures could be employed: Iconic

(e.g., movements that resemble word spoken); Metaphoric (e.g., movements for an abstract idea);

Deictic (e.g., pointing); and Beat (e.g., tapping) (Section 2.4.2). An individual participant may,

over the course of their explanation, use more than one type of gesture. However, this study will

concentrate on the type of gesture most often observed as students explained their solutions. If

students did not exhibit any type of gesture during Task Explanation, they were coded as “No

gesture”. With regard to the four gesture types, analysis of data revealed that students only

exhibited two of the four types of gesture, namely Deictic and Iconic. Therefore, results for

Metaphoric and Beat gestures are not displayed in the data as their counts were zero for all three

tasks.

The tasks are presented in order from easiest to the most difficult task based on the proportion of

students who responded to each task correctly: (a) The Picnic Park (88% success) (Section 5.4);

(b) The Playground (72% success) (Section 5.5); and (c) The Street Map (65% success) (Section

5.6).

5.3. The Picnic Park Task

The Picnic Park task was a co ordinate map, with positional points used to indicate landmarks

(Figure 5.1). A key was also provided to indicate a specific landmark and a compass bearing

given to indicate north. The task required an understanding of locating landmarks, co ordinate

position and compass direction. The task was sourced from the Queensland Year 5 test: Aspects

of numeracy (Queensland School Curriculum Council, 2001, p. 2) and hence, was suitable for

the age of participants in this study.

Figure 5.1. The Picnic Park Task.

5.3.1. Task Solution and Relationship Between Correctness and Purposeful Gesture Use

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The Picnic Park was the easiest of the three tasks with 38 of the 43 students (88%) correct, while

19 of the 43 students (44%) exhibited a purposeful gesture. An examination of task correctness

and purposeful gesture use across Task Solution was undertaken using a Chi square procedure.

This procedure considered the degree of the relationship between task correctness and purposeful

gesture use. The Pearson’s Chi-square value X2(1, 43) = 4.48, p = .04 was statistically significant

at a p = .05 level. Therefore on the Picnic Park, there was a significant relationship between

students’ purposeful gesture use and task correctness. The contingency table (Table 5.1) shows

that 88% of students answered this task correctly, with an even distribution of those students who

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used (50%) and did not use (50%) purposeful gestures. Thus, students who used a purposeful

gesture were more likely to solve the task correctly than those students who did not use a

purposeful gesture. The significant probability level was due, in part, to the fact that all five

students who answered incorrectly, did not exhibit a purposeful gesture. Therefore, on The

Picnic Park, all students who gestured correctly solved the task. Nevertheless, it should be noted

that a purposeful gesture was not essential in order to answer the task correctly.

Table 5.1

Contingency Table for Cross Tab Analysis for Task Solution on The Picnic Park.

Correct Total No (% of total

correct) Yes

Used Gesture No (% of total correct) 5 (100) 19 (50) 24 (56) Yes 0 (0) 19 (50) 19 (44) Total 5 (100) 38 (100) 43 (100)

5.3.2. Mapping Skills and Solution Approaches Utilised During Task Explanation

This section reports on the participants’ mapping skills and solution approaches as they

explained their solutions to the Picnic Park task. Various mapping skills and solution approaches

were identified in relation to this task

The majority of the students, both correct and incorrect, demonstrated their mapping skills with

all 43 students able to identify and use the landmarks (100%) and 42 students able to identify and

use the co ordinates (98%). Over half of the students, 27 of the 43 (63%) were able to identify

and apply the key (Table 5.2).

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Table 5.2

Mapping Skills for Picnic Park Task by Correctness.

Type of behaviour

Correct responses(N=38)

(%)

Incorrect responses

(N=5) (%)

Total

(N=43) (%)

Identify and use landmarks 38

(100) 5

(100) 43

(100)

Identify and use coordinates 38

(100) 4

(80) 42

(98)

Identify and apply key 25

(66) 2

(40)

27 (63)

A total of 38 students (88%) were successful and five students (12%) were unsuccessful on this

task. The successful students used one of three approaches: (a) process of elimination; (b)

immediately accessed positional information; and (c) indicated that the black line did not go

through the B4 square. The majority of successful students employed two main solution

approaches with 16 (42%) using a process of elimination and 18 (47%) accessing positional

information. The remaining four successful students (11%) were coded as Other because their

explanations lacked enough detail to be classified as either of the previous codes. All

unsuccessful students used a fourth approach, namely they fixated on a reference point (Table

5.3). An overview of the approaches used by successful students is presented followed by that of

unsuccessful students.

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Table 5.3

Solution Approaches for Picnic Park Task by Correctness.

Type of behaviour

Correct responses(N=38)

(%)

Incorrect responses

(N=5) (%)

Total (N=43)

Employed process of elimination 16

(42) 0 16

(37)

Immediately accessed positional information (with key)

18 (47)

0 18 (44)

Fixated on reference point 0 5

(100) 5

(12) Other: Indicated the black line did not go through B4 square (specific to this task)

4 (11)

0 4 (9)

A process of elimination was employed by 16 successful students (42%). They worked through

all the multiple choice answers until they found the co ordinate cell that did not have part of the

bike track running through it. For example, Kayla identified her answer as B4 and then explained

her reasoning for eliminating other possible answers.

Kayla: I chose B4 because I had a look at all the answers and B4 was the one spot that

she didn’t ride through, so that must be the answer. I had a look at A5, but she

went (sic) through A5 because the bike track is in it [eliminate answer option], so

she went through it and B5 she went through [eliminate answer option], and A4

[eliminate answer option], and so I chose B4.

Although Kayla’s approach was effective, it was time consuming.

Accessing positional information was employed by 18 successful students (47%) who

immediately used the key to identify the respective landmark on the map. For example, Bella

looked at the key and then explained that the bike track did not go through B4.

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Bella: I chose B4 because there was a little sign that says key on it and there wasn’t a

track through it [pointing to B4] and so I chose that one and if she did ride

through it she would probably get wet [emphasis added].

Students who used this strategy considered if the black line of the bike track ran through that co

ordinate square or applied prior knowledge about the context of the task (i.e., they just knew that

she wouldn’t ride through a pond). Students who were able to solve this task correctly and

explain their solution considered all relevant aspects of the map, along with the information

contained in the question in their solution.

The four successful students coded as Other simply indicated that the black line did not go

through that co ordinate square, without any elaboration as to how they worked that out. For

example, Jackson was able to identify that the bike track did not go through the B4 square and he

identified the key, however, he did not elaborate on his answer. Further follow up questions may

have encouraged these students to more fully explain their solution. However, given the study is

re-analysing already collected data, this was not possible.

Jackson: I looked at the circle and I looked at B and it wasn’t in there and I went up to the

4 and yeah, I went up to the 4. There’s that, right there [pointing to the pond in the

B4 square], that little island thing, and it said the key and so I though the key

would be the little pond.

In sum, the successful students used either a process of elimination, immediately accessed

positional information or indicated that the black line did not go through B4 square as their

solution approach. Although each of these strategies proved effective, both using a process of

elimination and immediately accessing positional information were the more efficient

approaches because they are sound strategies to work out multiple choice questions for map

tasks.

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The five unsuccessful students (12%) all focussed on the bike track as a reference point. They

were looking for the co ordinate square with the least amount of line through it, indicating they

may have misinterpreted the intent of the question. Thus, these students appeared to be overly

influenced by reference to the bike track in the question and the representation of the bike track

on the map and could not look outside these parameters or past the black line on the map. For

example, Sean did not indicate which elements of the task he relied on during his solution.

However, he focussed his answer based on the coordinate square with the least amount of track

running through it.

Sean: I chose B5 because it is the one with the least track through it.

The five unsuccessful students were unable to look past the key word in the task and became

overly concerned with the bike track. Had they considered the map in its entirety, they may

noticed other elements on the map and been able to use those to answer correctly.

The analyses of successful and unsuccessful students above demonstrate the importance of

solution approach in relation to correctness. Hence, using a process of elimination, or

immediately accessing positional information provided on the page seems to improve students’

likelihood of success. In contrast, becoming fixated on a certain element of the task like the bike

track seems unlikely to lead to a successful solution.

5.3.3. Types of Gesture Utilised During Task explanation

This section presents the type of gesture participants exhibited as they explained their solutions

to the Picnic Park task. An examination of the types of gestures used during Task Explanation

revealed that 36 students (84%) exhibited a deictic gesture, with three students (7%) using iconic

and four students (9%) not gesturing at all during their explanation (Table 5.4). Thirty-two of the

students (84%) who used a deictic gesture answered correctly, as did the four students who did

not gesture during task solution. The high proportion of students who used deictic gestures to

explain their solution could indicate that the structure of the task required students to point to the

page to indicate their position within the map. Thus, the students’ spatial understanding of the

task was reinforced and communicated as they pointed and explained.

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Table 5.4

Types of Gestures for Picnic Park Task by Correctness.

Gestures Correct responses

N=38 (%) Incorrect Responses

N=5 (%) Total

N=43 (%) Deictic 32 (84) 4 (80) 36 (84) Iconic 2 (5) 1 (20) 3 (7) No gesture 4 (11) 0 4 (9)

5.4. The Playground Task

The Playground task required students to interpret a birds-eye-view representation of a

playground (Figure 5.2). Location skills were required to identify specific landmarks (e.g., the

track) while directional skills (e.g., movement) were needed to navigate from specific landmarks

through a given route. The task was sourced from the Queensland Year 3 test: Aspects of

numeracy (Queensland Studies Authority, 2002a, p.11) and hence, was suitable for the

participants in this study.

Figure 5.2. The Playground Task.

5.4.1. Task Solution and Relationship Between Correctness and Purposeful Gesture Use

The Playground was of moderate difficultly in relation to the three tasks with 31 students correct.

The potential statistical relationship between task correctness and purposeful gesture use across

Task Solution (Episode 1) was examined using a Chi square analysis. The Pearson’s Chi-square

value X2(1, 43) = 1.03, p = .26 was not statistically significant at a p = .05 level, and therefore,

there was no relationship between students’ purposeful gestural behaviour and task correctness.

The contingency table (Table 5.5) shows that 74% of students solved the task correctly and used

purposeful gesture whereas 26% of students who solved the task correctly did not use purposeful

gesture. Of the 12 students who answered incorrectly, seven students (58%) used a purposeful

gesture and five students (42%) did not gesture. There was also a consistent pattern in terms of

the proportion of total students who correctly solved the task (31 of 43 students) and the number

of students who used purposeful gesture (30 of 43 students). That is, the total correct is almost

equal to the number of purposeful gestures. Consequently, there was no relationship between

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task correctness and gesture use since this task had an almost even balance of correctness and

gesture use.

Table 5.5

Contingency Table for Cross Tab Analysis for Task Solution on The Playground.

Correct Total No (% of

total correct) Yes

Used Gesture No (% of total correct) 5 (42) 8 (26) 13 (30) Yes 7 (58) 23 (74) 30 (70) Total 12 (100) 31 (100) 43 (100) Note: due to rounding, totals may not equal 100.

5.4.2. Mapping Skills and Solution Approaches Utilised During Task Explanation

This section draws on the participants’ mapping skills and solution approaches as they explained

their solutions. One mapping skill and various solution approaches were identified in relation to

this task. All of the students, both correct and incorrect, were able to identify and use landmarks

(100%) on the map (Table 5.6). This was the only mapping skill demonstrated by the students on

this task. A compass point was located in the bottom left corner however, none of the students

referred to it at any stage.

Table 5.6

Mapping Skill for The Playground Task by Correctness.

Type of behaviour

Correct responses N=31 (%)

Incorrect responses N=12 (%)

Total N=43

Identify and use landmarks

31 (100) 12 (100) 43 (100)

A total of 31 students were successful and 12 students were unsuccessful on this task. The

successful students were able to follow a set of directions through a route and monitor the

sequence of events that occurred, while the unsuccessful students were not able to fulfil all these

requirements (Table 5.7).

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Table 5.7

Solution Approaches for The Playground Task by Correctness.

Type of behaviour Correct responses N=31 (%)

Incorrect responses N=12 (%)

Total N=43

Described movement

30 (97) 8 (67) 38 (88)

Understanding of relationship between location of landmarks and movement

30 (97) 2 (17) 32 (75)

Followed a set of directions (route)

28 (90) 0 28 (65)

Other: Counted number of landmarks and vague response (both specific to this task )

1 (3) 4 (33) 5 (12)

In order to answer the task correctly students had to follow a set of directions (route) and

remember the number of times the track had been crossed. For example, Henry was able to

navigate the route whilst counting the number of times he crossed the track.

Henry: I went from the gate over to the tap [followed directions], so that’s once

[counted]. Then from the tap to the shed [followed directions], that is twice

[counted] and then to the rubbish bins it doesn’t go anymore.

The ability to follow the set directions through to a conclusion was required in order to be

successful. The ability to count the number of times the track was crossed on this route

demonstrated competent mathematics understandings.

The students coded as understanding relationship between location of landmarks and movement

were able to identify the relevant landmarks and in addition, had some sense of the movement

required to navigate between the landmarks. Therefore, these students were able to demonstrate

some understanding of sequenced movement within the scope of the task. For example, Sian

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answered incorrectly, however she was still able to demonstrate that she moved between the

landmarks, but was unable to fully complete the instructions outlined in the task.

Sian: I chose three because he went to the tap then the shed then the rubbish

bins [Understanding of relationship between location of landmarks and

movement]

Int: So he crossed the track three times?

Sian: Yes

Students such as Sian tended to count the number of movements between the landmarks as

opposed to the number of times the route crossed the track.

A majority of the students (88%) were able to describe movement. Many of the students were

able to express that some form of movement around the map was required. This was classified as

the most basic of approaches and a student was given this code if they were able to indicate that

movement was required on the map.

Many of the successful students were given three codes because they were able to: (a) describe

movement, (b) understand the relationship between landmarks and movement, and (c) were able

to follow the set directions. The approaches listed are of a hierarchical nature, in that most of the

students were able to describe movement, but only the students who answered correctly were

able to follow a set of directions. Hence, describing movement could be classified as the most

basic of solution approaches, while being able to follow a set of directions through to a

conclusion maybe seen as a skilful approach for competent students.

Counting the number of landmarks results in students obtaining an incorrect solution. These

three students misunderstood the final part of the question asking how many times the track was

crossed. For example, Jackson did not interpret the question properly and got confused by what

he was meant to do.

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Jackson: my answer was four. I wrote down all the things he went past and it came

up as 4 (counted the number of landmarks)

Jackson’s response is indicative of students who misread the intent of the task and concentrated

on the landmarks as opposed the sequential movement between them.

Two students, one successful and one unsuccessful, coded as having a vague response were

unable to verbalise their approach. For example, Adele answered correctly and was able to

identify the landmarks; however, her response gave no real insight into her thinking and how she

approached the task. Again, further follow up questions may have encouraged these students to

more fully explain their solution but as this study is re-analysing existing data, this was not

possible.

Adele: I thought it was 2. The track is there, and the shed and the gate are there. It was

kind of 2.

It could be the case that these two students were not ready to verbalise such metacognitive

thinking, and hence, were only able to give a vague indication of how they solved the task.

The unsuccessful students on The Playground task misinterpreted the intent of the question by:

(a) counting landmarks, (b) not fulfilling all the sequential elements of the task or (c) being

unable to verbalise their solution clearly. The unsuccessful students were ineffective at applying

all of the information in the question to the map.

5.4.3. Types of Gesture Utilised During Task Explanation

An examination of the types of gestures exhibited during Task Explanation revealed deictic

gesture was the predominate gesture displayed with 40 of the 43 students (91%) exhibiting this

type of gesture. Only two participants (5%) exhibited iconic gestures and one student (2%) did

not gesture at all (Table 5.8). With respect to the use of deictic gestures, there was no distinction

between students who were correct or incorrect, with Allen (2003) indicating that deictic

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gestures were common for this type of route finding task. This suggests that this type of gesture

does not necessarily lead to a correct solution or a self correction on an incorrect solution.

Table 5.8

Types of Gestures for Playground Task by Correctness.

Gestures Correct responses

N=31 (%) Incorrect Responses

N=12 (%) Total

N=43 (%) Deictic 30 (97) 10 (83) 40 (91) Iconic 1 (3) 1 (8) 2 (5) No gesture 0 1 (8) 1 (2)

5.5. The Street Map Task

This task has a traditional street directory representation with some features represented from a

birds-eye-view perspective (e.g., the netball courts) (Figure 5.3). Information represented in the

key is depicted pictorially on the map (e.g., the post office). Other features on the map include a

scale and a compass bearing within a co ordinate arrangement. Specific mathematics

understandings (i.e., North and ordinal numbers) and everyday language (i.e., right, left) are

required to navigate a route along streets to complete a journey and then identify a landmark

(i.e., a street name). The task was sourced from the Queensland Year 5 test: Aspects of numeracy

(Queensland Studies Authority, 2002b, p.7) and hence, was suitable for the age of participants in

this study.

Figure 5.3. The Street Map task.

5.5.1. Task Solution and Relationship Between Correctness and Purposeful Gesture Use

In order to examine the degree of the relationship between task correctness and purposeful

gesture use, a Chi square procedure was undertaken. The Pearson’s Chi-square value X2(1, 43) =

1.82, p = .18 was not statistically significant at a p = .05 level and therefore there was no

relationship between task correctness and purposeful gestural behaviour. The contingency table

(Table 5.9) shows that of the 28 students who answered correctly, 25 (89%) exhibited a

purposeful gesture. This was the predominant behaviour for this task. However, 11 of the 15

students (73%) who answered incorrectly also exhibited a purposeful gesture as they solved the

task. Hence, the majority of students (84%) exhibited a purposeful gesture on this task

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suggesting that students were more inclined to use gesture in order to help them solve the task—

and this was the case irrespective of task success.

Table 5.9

Contingency Table for Cross Tab Analysis for Task Solution on The Street Map.

Correct Total No (% of total

correct) Yes

Used Gesture No (% of total correct) 4 (27) 3 (11) 7 (16) Yes 11 (73) 25 (89) 36 (84) Total 15 (100) 28 (100) 43 (100)

5.5.2. Mapping Skills and Solution Approaches Utilised During Task Explanation

This section draws on the participants’ mapping skills and solution approaches as they explained

their solutions to The Street Map. Three key mapping skills were identified in relation to this

task (Table 5.10), namely (a) identify and use landmarks, with 100% of students able to

demonstrate this skill; (b) identify and use compass point, with 77% of students able to utilise

this skill; and (c) identify and apply key, with only 12% of students using this element of the

map. This task was designed with a lot of visual cues and elements to help children answer the

question such as a scale, coordinates, and key. However it seems that not all elements were

utilised by all the students.

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Table 5.10

Mapping Skills for The Street Map Task by Correctness.

Type of behaviour

Correct responses (N=28) (%)

Incorrect responses (N=15) (%)

Total (N=43)

Identify and use landmarks

28 (100) 15 (100) 43 (100)

Identify and use compass point

24 (86) 9 (32) 33 (77)

Identify and apply key

3 (11) 2 (13) 5 (12)

A total of 28 students were successful and 15 students were unsuccessful on this task. The

approaches taken to solve this task were similar to The Playground. The successful students were

able to follow a set of directions through a route and monitor the sequence of events to find the

unknown location, while the unsuccessful students were not able to fulfil all these requirements,

using a number of different approaches to solve the task (Table 5.11).

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Table 5.11

Solution Approaches for The Street Map Task by Correctness.

Type of behaviour Correct responses

(N=28) (%) Incorrect responses

(N=15) (%) Total

(N=43) Described movement

28 (100) 15 (100) 43 (100)

Understanding of relationship between location of landmarks and movement

28 (100) 14 (93) 42 (98)

Followed a set of directions (route)

28 (100) 0 28 (65)

Misunderstood ordinal sequence

0 12 (80) 12 (28)

Fixated on a reference point

0 1 (7) 1 (2)

Other: left and right confusion (specific to this task)

0 2 (13) 2 (4)

The successful students on this task were all able to monitor the sequence of events and follow

the set directions (100%). For example, Pippa demonstrates how she monitored which road was

the first and second on the left.

Pippa: I went from the pool and drove North [followed directions] using the compass.

He turns right and passes the first road on the left [followed directions] and goes

to the second road on the left, which is School Road.

The ability to navigate such ordinal directions resulted in students answering correctly. The

majority of students were able to understand the relationship between landmarks and movement

indicating that were able to identify the relevant landmarks and were able to demonstrate some

understanding of sequenced movement within the scope of the task. For example, Kai answered

incorrectly however he was able to show that movement between the landmarks was required.

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Kai: I looked for the pool and went up and then turned right and then left and that’s

Post Road there [relationship between landmarks and movement]

The ability to understand the relationship between landmarks and movement, on its own, was not

an adequate enough concept to fully answer this task correctly. This was a ‘stepping stone’ to

answering correctly, but further mathematical concepts needed to be applied in order to answer

correctly.

The entire cohort, regardless of correctness, was able to describe movement around the map. This

suggests that these students were familiar with the purpose of a street map as a navigational tool.

Nevertheless, the successful students were able to handle the various elements of the task and the

multiple directions given in the text. That is, they were able to distinguish between just turning

right and left, by making the connection that it had to be the first right and the second left.

With respect to those students who answered incorrectly, three different approaches were

employed. Of the 15 students who answered incorrectly, 12 (80%) misunderstood the ordinal

sequence, meaning they chose the first road to the left instead of the second. These students

failed to retain all the information in the question when applying it to the map. Some students

counted the road they were in as the first road on the left, while others had a different

interpretation again. For example, Sian was interpreting the second road on the left to mean the

second turn that Bill makes. So she was able to navigate past the first turn on the right, however,

she then turned left straight away as it was the second turn that Bill could make.

Sian: He drives North and then takes the first road which is this one [pointing]. Then it

says the second road on the left which is that one [pointing] and it says the Post

Road.

The ordinal sequence is a key element of the task and those students who misunderstood this

element were not able to answer correctly.

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Two of the students (13%) were confused by their left and right. This resulted in the students

being on the opposite side of the map than what they needed to be. For example, Shane turned

left first and then right, ending up in Jones St.

Shane: I went from the pool and that’s the North [pointing to compass], so then went up

North and took my first right [indicating left] and my second left [indicating

right].

What may seem like a simple mistake of confusing left and right, can often lead students

completely astray on map tasks.

One student (7%) got fixated on a reference point, that being Bill’s house. This student

attempted to navigate their way to Bill’s house via a series of left and right turns.

Ava: I went the pool, then I went down Green Road, then I went up Jones Road and

then I went to Beef Road and then I went to Bill’s house.

This student seemed to misunderstand the intent of the task, focussing on an aspect of the map

that was not even mentioned in the question, namely Bill’s house.

Given The Street Map was the most difficult task of the three, it is not surprising that such a

variety of approaches were employed to solve the task as the students came to grips with

mathematical language (north), ordinal numbers (first, second) and everyday language (left,

right). Thus the cognitive demands of the task were high with students having to deal with a

number of interrelated mathematical ideas.

5.5.3. Types of Gesture Utilised During Task Explanation

An examination of the types of gestures used during Task Explanation revealed that 39 of the 43

students (90%) exhibited a deictic gesture, two students (5%) using iconic gesture and two

students (5%) not gesturing at all during their explanation (Table 5.12). The highest proportion

of students used deictic gestures to explain their solution. This result is consistent with The

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Playground task. On both tasks, students were required to follow a set of directions. Allen (2003)

suggested that pointing gestures (deictic) were the most common type of gesture for route

finding tasks.

Table 5.12

Types of Gestures for Street Map Task by Correctness.

Gestures Correct responses

N=28 (%) Incorrect Responses

N=15 (%) Total

N=43 (%) Deictic

25 (89) 14 (93) 39 (90)

Iconic

2 (7) 0 2 (5)

No gesture

1 (4) 1 (7) 2 (5)

5.6. Understanding Students’ Performance and Behaviour on Map Tasks

Students’ behaviour can be understood by considering their responses to each of the map tasks.

The first aspect of this analysis focussed on task correctness. The map tasks investigated in this

study are representative of tasks presented to students in standardised assessment situations and

of the type of map tasks specifically taught in school curricula (Section 2.3.6). Individually, each

task had a fairly high correct response rate with a range from 88% to 65%. This indicates that the

tasks were appropriate for the grade levels and supports Lowrie and Diezmann’s (2007a) finding

that map tasks were one of the easier graphical languages to solve at these grades. The second

aspect of analysis looked at student behaviours in relation to gesture use. Many of the students

exhibited a purposeful gesture during task solution and those students who did gesture, generally

answered correctly. The analysis of gestures is considered later in further detail after additional

results on students’ gesture use have been presented (see Section 5.8)

Students’ mapping skills and approaches to solving the tasks exhibited an understanding about

representation, space and reasoning with spatial tasks (Section 2.3, Table 2.2). Students were

able to comprehend the principles of graphic design by being able to locate landmarks and

labelled points (representation) and they had a clear understanding of the basis of the coordinate

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system (space). Many students were able to appreciate different ways of thinking about position

and distance, however fewer students were able to make informed decisions about the pathways

they took (reasoning). These kinds of approaches are aligned to the same principles of the

National Research Council’s (2006) aspects of spatial thinking (Section 2.3). The students’

mapping skills indicated that they grasped the basic elements of map reading and their solution

approaches suggested that they were not as adept in the mathematical concepts required to solve

the tasks. Thus, the identification of different levels of competence is consistent with Clarke’s

(2003) suggestion that spatial applications are used for simple location of places but these

applications become increasingly complex when you have to follow several patterns or

relationships (i.e., having multiple directions).

In relation to student behaviours and gesture use, almost all students used a gesture as they were

explaining their solution. The majority used deictic gestures which specifically included

gesturing where participants pointed towards the page or touched the page. Some students used

iconic gestures, especially for The Picnic Park. These gestures commonly provided a figurative

representation of parts of the task. For example, participants circled with their finger in the air to

represent the circular bike track. As noted previously (Section 5.2), no observations of

metaphoric or beat gestures were recorded for the three tasks within this study. It is not

surprising that students exhibited deictic gesture through task explanation because they were

encouraged to verbalise their thinking in relation to tasks that required high levels of spatial

reasoning. The use of this gesture allowed students to demonstrate the approaches they

undertook to navigate the spatial arrangement and locate specific location of landmarks on the

maps. The high proportion of participants using deictic gestures supports previous research by

Heiser, Tversky, and Silverman (2004) and Tversky (2007), who found a high proportion of

deictic gestures associated with map tasks. Therefore, it seems that the deictic gestures observed

in this study were used as a communication tool to aid explanation. In relation to the lack of

metaphoric and beat gestures used by students in this study, Allen (2003) indicated that such

gesturing was rarely used in explanations associated with spatial representations. Since the tasks

in this study were concrete pen and paper map problems, there was less of a need for metaphoric

and beat gestures. A low number of students did not gesture at all.

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The next section of analysis goes beyond looking at each task individually by considering task

correctness within and across the respective map tasks and student gesture use across these tasks

with particular attention draw to the difficulty of the three tasks. These forms of analysis provide

scope for more detailed descriptions of students’ performance in relation to different types of

maps and the role that gesture plays in assisting students to navigate the task. This examination

of gesture use responds to Radford’s (2009) call for research which investigates “how gestures

relate to learning and thinking” (p. 112). In addition, further analysis is presented on the mapping

skills and solution approaches utilised by the students in order to identify similarities and

differences across the three tasks.

5.7. Patterns Across Map Tasks

This section describes common patterns across correctness, gesture use and the mapping skills

and solution approaches that emerged across the three tasks. Previous analysis (Sections 5.2, 5.3

and 5.4) has considered data within task, whereas this section identifies patterns across Task

Solution (Section 5.7.1) and then Task Explanation (Section 5.7.2).

5.7.1. Students’ Performance and Use of Gesture During Task Solution

In looking at the overall performance and participants’ gesture use, the three map tasks were

initially considered as a set of tasks, so patterns of performance and gesture use over the set

could be identified (Section 1.7.1.1). Tasks were scored as 1 for correct response or 0 for

incorrect response. The overall maximum score for task correctness was 129 if every participant

(n=43) was to answer each of the three tasks correctly (43 participants x 3 tasks). At this overall

level, 97 out of 129 (75%) of responses were coded as correct. Gestures were scored as 1 for a

purposeful gesture or 0 for a non-purposeful gesture. Hence, similar to task correctness, the

maximum score for using a purposeful gesture is 129, that is, if every student used a purposeful

gesture on every task. Across the set of tasks, 85 out of 129 (66%) of responses were coded as

using purposeful gesture. These results indicate that there is sufficient variance with regard to the

spread of counts between (a) tasks that are correct and incorrect, and (b) use of purposeful and

non-purposeful gesture to warrant further analysis (Burns, 2000).

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5.7.1.1. Correctness and Purposeful Gesture Use on Map Tasks

Participants differed in their ability to correctly solve the map tasks. Twenty-one of the 43

participants (49%) correctly solved all three tasks, while 21% solved one or no tasks correctly

(Table 5.13). The results of nil, one, two or three tasks correct, present a relatively even spread of

student performance across the three tasks. Only one student was unable to solve at least one task

correctly. Thus, almost all of the students were at a level of understanding where they could

engage in the task solution with some level of confidence and indicates that the tasks were grade

appropriate.

Table 5.13

Proportion of Students Achieving a Correct Solution on the Three Map Tasks.

No. correct 0 1 2 3 Frequencies (n=43) (%)

1 (2)

8 (19)

13 (30)

21 (49)

With regard to gesture use, a high proportion of participants utilised a purposeful gesture across

at least one of the three tasks. Fourteen of the 43 participants (33%) used purposeful gestures for

each of the three tasks, while four participants (9%) did not use a purposeful gesture on any of

the tasks (Table 5.14). With 25 participants (58%) purposefully gesturing on only one or two

tasks, it seemed that the behaviours exhibited by participants were aligned to the task rather than

the individual student. These results could indicate that the participants were selective in

deciding whether a purposeful gesture was required to solve certain tasks as opposed to students

using a gesture all the time because it is in their nature. Therefore, a closer look at the interaction

between task correctness and purposeful gesture use was undertaken to identify any patterns

between task correctness and purposeful gesture use.

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Table 5.14

Proportion of Students Using Purposeful Gesture across the Three Map Tasks.

Gestures on the three tasks

0 1 2 3

Frequencies (N=43) (%)

4 (9)

7 (16)

18 (42)

14 (33)

A frequency table (Table 5.15) was produced to gauge the interaction between correctness and

purposeful gesture use by the participants. There was a loading of counts within particular cells.

The cells with the highest number of frequencies are loaded towards the cells that represent a

higher success rate and higher gestural use. For example, in Table 5.15, the ten participants1 (see

superscript 1 on Table 5.15) who correctly solved all three tasks also used a purposeful gesture

on each task. The seven participants2 who correctly solved each task used gesture for only two of

these tasks. The participants who gestured in each of the three tasks were likely to answer two or

more of the tasks correctly. Whereas, the success rates for participants who gestured on only two

of the tasks were spread relatively evenly across one, two or three tasks correct. Hence, although

the use of a purposeful gesture appeared advantageous for a correct solution it did not guarantee

a correct solution. Because, the cell1 with the highest frequency showed that students who

correctly solved the three tasks and gestured on each of these tasks, those who used gestures

appeared more likely to be successful.

Table 5.15

Frequency Distribution by Correctness and Purposeful Gesture Use.

No. purposeful gestures used

No. correct responses Total

0 1 2 3

0 0 2 0 2 4

1 1 1 3 2 7

2 0 5 6 72 18

3 0 0 4 101 14

Total 1 8 13 21 43

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5.7.1.2. Analysing Individual Tasks’ Correctness and Purposeful Gesture Use

The individual tasks were analysed with respect to task correctness and purposeful gesture use in

order to ascertain the interaction between correctness and purposeful gesture use of each

particular task during Task solution (Table 5.16). In terms of task correctness, The Picnic Park

was the easiest of the three tasks with 88% of participants solving this task correctly; while 72%

and 65% of participants correctly solved The Playground and The Street Map respectively. With

regard to purposeful gesture use, gesture use varied from 84% of participants using a purposeful

gesture on The Street Map, to 70% on The Playground and 44% on The Picnic Park. Thus, there

was a spread of 40 percentage points across purposeful gesture use (an increase of 91%) and a

spread of 23 percentage points across task success (an increase of 35%) with these measures

indicating variations in both performance and preferences for gesture use. This result indicates a

linear relationship in gesture use, with more students exhibiting a purposeful gesture as the tasks

became more difficult (Figure 5.4).

Table 5.16

Frequency Counts for Task and Gestural Use by Success.

Task N Correct frequency

(%)

Gesture frequency

(%) The Picnic Park 43 38 (88) 19 (44) The Playground 43 31 (72) 30 (70) The Street Map 43 28 (65) 36 (84)

These data on correctness and purposeful gesture use reveal an inverse relationship between task

difficulty and the proportion of participants who purposefully gestured within each task (Figure

5.4). That is, the proportion of participants who evoked purposeful gestures for the easiest task

(The Picnic Park) was relatively low (44%) given the high success rate (88%) whereas on the

most difficult task (The Street Map) the converse occurred with gesture use quite high (84%)

while the success rate was lower (65%). This inverse pattern suggests that the use of a purposeful

gesture was a tool that participants employed as they navigated their way through each task.

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Figure 5.4. The inverse relationship between task correctness and gesture.

Similarly, the distribution of students who did not exhibit a purposeful gesture decreased as the

task got more difficult (Figure 5.5). The 19 students who did not purposefully gesture to solve

The Picnic Park correctly, halved to eight students for The Playground and then more than

halved again to three students for the hardest task, The Street Map. This shows that those

students who were not gesturing on The Picnic Park, began to gesture and answered correctly or

began to gesture and answered incorrectly on The Playground and The Street Map.

The Picnic Park The Playground The Street Map

no gesture/correct 19 8 3

gesture/correct 19 23 25

gesture/incorrect 0 7 11

no gesture/incorrect 5 5 4

0

5

10

15

20

25

30

35

40

The Picnic Park The Playground The Street Map

No. correct

No. gestures

Figure 5.5. The distribution of students across the three tasks who exhibited certain behaviour characteristics.

Figure 5.5 shows the flow of student behaviour from no gesturing with a correct response to

gesturing with a correct response (represented with arrows from the top left hand corner to the

middle and bottom). However, the number of students who did not gesture and answered

incorrectly stayed relatively constant, with either five (The Picnic Park and The Playground) or

four students (The Street Map) across the three tasks. The finding of a shift in students’ gesture

use across the tasks highlights a change in behaviour with students using more purposeful

gestures as the tasks became more difficult, even for those students who initially did not require

the use of a gesture.

5.7.1.3. The Impact of Gender on Correctness and Purposeful Gesture Use for Individual

Tasks

This section examines the potential performance differences of boys and girls on the three map

tasks. A gender comparison was undertaken because Diezmann and Lowrie (2008b) found

differences in the performance of boys and girls on map tasks and there is ongoing debate as to

whether gender difference occurs in mathematics (Section 2.3.8). Means and standard deviations

for task correctness by gender are presented in Table 5.17. For each of the three tasks, boys had

higher mean scores than the girls suggesting that the boys found the tasks easier to solve than the

girls (Figure 5.6). Nevertheless, t-tests revealed no statistically significant gender performance

differences on each of the three tasks. These statistical results could be due in part to the fact that

the sample size of 43 is relatively low. Burns (2000) argues that with less than 30 subjects per

variable (in this case less than 30 boys and 30 girls), the statistical power of the t-test procedure

is reduced and therefore it is likely that Type II error will occur. A Type II error is manifested by

saying there is no gender differences when in fact there may well be. Thus, there were consistent

performance differences for boys and girls on these tasks however the fact that boys

outperformed girls cannot be justified statistically.

Table 5.17

Means and (Standard Deviations) for Task Correctness by Gender Across Tasks.

The Picnic Park

X (SD)

The Playground

X (SD)

The Street Map

X (SD)

Gender Boys .95 (.23) .81 (.40) .71 (.46) Girls .82 (.39) .64 (.49) .59 (.50)

t-tests 1.3 (ns) 1.4 (ns) .83 (ns)

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0

10

20

30

40

50

60

70

80

90

100

The Picnic Park The Playground The Street Map

Boys

Girls

Figure 5.6. The proportion of boys’ and girls’ correct responses across task.

The potential differences between boys and girls use of purposeful gesture on Task Solution was

also investigated. Means and standard deviations for purposeful gesture use by gender are

presented in Table 5.18. Both boys and girls exhibited more purposeful gestures in The

Playground and The Street Map than in The Picnic Park. On the two easiest tasks, boys’ gesture

use was higher than the girls, with the difference being 12 percentage points on The Playground

(a 19% difference) and 7 percentage points on The Picnic Park (a 17% difference). However, on

the most difficult task, girls’ exhibited more purposeful gestures than the boys with the

difference being 15 percentage points (a 20% difference). The statistical significance of these

apparent differences was investigated using t-test procedures. The t-tests for each of the three

tasks revealed there were no statistical differences on purposeful gesture use between boys and

girls (Table 5.18). Nevertheless, a pattern of gesture use across the three tasks was apparent (see

Figure 5.7) with the boys’ gestural use remaining constant across the more difficult tasks, while

the girls’ gesture use proportionally increased across the three tasks. This suggests that girls

employ the use of gesture more often than boys as tasks become more difficult. Although there

were no statistically significant differences between the boys’ and girls’ use of gesture during

task solution, girls used more gestures on the hardest task.

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Table 5.18

Means and (Standard Deviations) for Purposeful Gesture Use by Gender across Tasks.

The Picnic Park

X (SD)

The Playground

X (SD)

The Street Map

X (SD)

Gender Boys .48 (.51) .76 (.44) .76 (.44) Girls .41 (.50) .64 (.49) .91 (.29)

t-test .88 (ns) .43 (ns) -1.3 (ns)1

Note1: The negative t-test indicates the direction of the mean difference. Thus, girls had a higher mean than boys on The Street

Map.

0

10

20

30

40

50

60

70

80

90

100

The Picnic Park The Playground The Street Map

Boys

Girls

Figure 5.7. The proportion of boys’ and girls’ purposeful gesture use across task.

Since there were no statistically significant differences between boys and girls on performance or

gesture use, issues related to gender are not explored further in this study.

103

104

5.7.2. Students’ Mapping Skills and Solution Approaches During Task Explanation

The analysis of students’ Task Explanations identified the mapping skills and solution

approaches that students employed as they solved the three map tasks. The frequencies and

percentages of the types of skills and approaches employed by the students across the three tasks

are presented (Table 5.19). Note, these counts do not relate to task success. With respect to the

mapping skills, the entire cohort (100%) was able to identify and use the landmarks on all of the

maps. In addition, 98% of the students were able to identify and use the coordinates on The

Picnic Park. However, not one student used the coordinates on The Street Map despite

coordinates being present. Similarly, both The Picnic Park and The Street Map had keys, with

63% and 12% of the students able to identify and apply the key on those tasks respectively. Also,

78% of students were able to identify and use the compass point on The Street Map but not on

the other two tasks, despite the compass point being present on all three tasks.

Table 5.19

Mapping Skills Used Across the Three Tasks.

Type of skill Proportion across tasks

The Picnic Park (%)

The Playground

(%)

The Street Map (%)

Identify and use landmarks 43 (100)

43 (100)

43 (100)

Identify and use coordinates 42 (98)

— 0

Identify and use compass point 0 0 33 (77)

Identify and apply key 27 (63)

— 5 (12)

The different influences of the map features in the tasks can be seen through two of John’s

transcripts. John, who answered all three tasks correctly, explains how he identified and used, or

did not use, the various features of The Picnic Park and The Street Map.

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The Picnic Park

John: There is a key and it is a picture of something and then it says pond next to it

(emphasis added).

The Street Map

John: I did look at the arrow to know to go north. I didn’t look at the key because it

didn’t really have much to do with the question (emphasis added).

While John used the key on The Picnic Park, he chose not to use it on The Street Map, instead

focussing on the compass point. This example does not necessarily indicate that students did not

observe these features (e.g., a key) as they solved the respective tasks but it does suggest that

only certain features and attributes were influential in their solutions to the tasks. For example,

the elements (i.e., north) pertinent to the written information were important for John.

A major pattern to emerge from the solution approaches is that the students employed similar

approaches to solve both The Playground and The Street Map tasks (Table 5.20). For both tasks,

the most common approach was to follow a set of directions, even though the two tasks required

students to achieve different outcomes. For example, The Playground asked students to count the

number of times the route crossed the track, while The Street Map asked them to locate the street

that the route finished on in. The following two transcripts highlight the similarities between

solution approaches with each student following the directions (route) on two different tasks.

The Playground

Ivy: [he] went from the gate to the tap and crossed it once, then from the tap to the

shed and crossed it again, then from the shed to the bins and didn’t cross the track

again, so 2. [Followed a set of directions]

The Street Map

Tim: I chose school road. I looked at the street map and looked at the compass with

north facing straight up, then the first right. I skipped the next left then I went the

second left which was school road. [Followed a set of directions]

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Table 5.20

Solution Approaches Employed Across the Three Tasks.

Type of approach Proportion across tasks

The Picnic Park (%)

The Playground

(%)

The Street Map (%)

Described movement 0 38

(84) 43

(100) Understanding of relationship between location of landmarks and movement

0 32 (75)

42 (98)

Followed a set of directions (route) 0 28

(65) 28

(65)

Employed process of elimination 16

(42) 0 0

Immediately accessed positional information (with key)

18 (44)

— 0

Fixated on reference point 5

(12) 0 1

(2)

Misunderstanding of ordinal sequence

— — 13

(30)

Other 4

(9) 5

(12) 2

(4)

One possible explanation for the occurrence of common solution approaches across The

Playground and The Street Map tasks was the requirement that students were expected to follow

a set route from a given starting point. Whereas students approached The Picnic Park completely

differently than the other two tasks, focussing on using a process of elimination or immediately

accessing positional information with the key to find a solution. The following transcripts

highlight two different approaches to The Picnic Park.

Ivy: I checked the coordinates and B4 was the only one that didn’t have part of the

bike track in it. [Process of elimination]

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Tim: I looked at the graph and it was a pond and there was no bike track going through

it. I looked on the graph and I saw the pond there. [Immediately accessed

positional information with key]

Although the three maps have very different graphic representations (pictorial, coordinate and

street map) and each task was unique, students employed similar solution approaches to The

Playground and The Street Map tasks and different approaches for The Picnic Park. The

employment of similar approaches could be a result of the spatial arrangement of the respective

tasks, particularly in relation to navigation and movement within a confined space.

The three maps utilised in this study were unique tasks that required different sets of skills to

solve correctly. However, patterns emerged from the students’ performance, gesture use,

mapping skills and solution approaches that could indicate a close connection between the three

tasks. Students required the use of gesture more often as the tasks increased in difficulty and less

than half were able to solve all three tasks correctly. While students’ mapping skills on the three

tasks seemed well developed, their approaches to the tasks differed according the type of

questions associated with the map. Students employed similar approaches to a very simple

pictorial map (The Playground) and a complex street map with a variety of elements contained

within (The Street Map), while employing two separate approaches to solve The Picnic Park. An

expected pattern in gender difference did not emerge from the data suggesting that boys and girls

were similar in performance and gesture use across the three tasks. A discussion of these findings

is presented next (Section 5.8).

5.8. Understanding the Patterns Among Students’ Behaviour on Map Tasks

This section presents a discussion of the patterns that emerged from an analysis of the three tasks

in relation to performance, purposeful gesture use, mapping skills and solution approaches. The

analysis of patterns among the tasks provided the opportunity to find out how students’

performance and gesture use were linked across the three tasks. It also highlighted patterns of

student mapping skills and solution approaches across the three tasks.

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Generally, the students performed well on each of the three tasks, however, it was interesting to

note that less than half the students in this study were able to answer all three tasks correctly.

Given the presence of mapping skills in the NSW state mathematics syllabus (Board of Studies

NSW, 2002) and indeed the HSIE syllabus (Board of Studies NSW, 1998), it could be assumed

that the students had previous exposure to such tasks, and the concepts within the tasks.

However, Lowrie and Diezmann (2005) found that students’ performance on map tasks

correlated moderately with other graphical languages and that these correlations were sometimes

higher across other languages than within the map languages. In other words, students’

performance within map tasks was not always consistent. The low number of students answering

all three tasks correctly could indicate that explicit teaching about map tasks is not widely

undertaken in schools and could be an area considered for future teacher education and research.

The identification of an inverse relationship between correctness and gesture use across the three

tasks suggests that gesture use was a tool students utilised to make connections between

mathematical ideas. Generally, students used purposeful gesture to help make sense of the spatial

challenges among the respective tasks. It provided a concrete tool to help students track their

thinking on the maps, and this was especially true when students were required to follow a route

on the map. Those students who did not gesture either did not require such support or were

unable to engage with the task with enough understanding to utilise such a tool. The utilisation of

this “concrete” tool draws attention to Pirie and Kieran’s (1994) theory of mathematical growth

where students “fold back” to more concrete forms of image making in order to fully engage

with the task. This process provides students with support to more easily understand multiple

representations as they are solving the task. In fact, such behaviour is aligned closely to theories

of multimodal learning where the student is required to simultaneously process information in

multiple ways and within multiple forms of visual and spatial representation given each of the

map tasks in this study was unique in layout and form (The New London Group, 2000).

Map tasks have their own set of specific elements where information is encoded through the

spatial location of marks and symbols (Mackinlay. 1999). The three tasks presented in this study

differed in structure and were encoded with various elements. For example, The Playground was

a basic pictorial map with little other information than labelled pictures and a compass point. By

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contrast, The Street Map was a complex map with scale, coordinates, landmarks, a compass

point and a key. The Picnic Park fell in between these two tasks, with labelled points, a

coordinate grid and a key all utilised. Many of the students were able to identify these elements

and most were able to distinguish between which elements were pertinent to answer the

questions. This relates to the simplest level of map reading where students are able to extract

information from a map by reading names and attributes, and recognising visual stimuli and

specific elements (or icons) on the map (Muehrcke, 1978; Wiegand, 2006). The majority of

students, both correct and incorrect, were able to achieve this first level of map reading.

Although the structure of The Playground and The Street Map were quite distinct, students

utilised similar approaches to solve these tasks. As a result students were required to undertake

the task in similar ways even though the tasks were represented using different encoding

techniques. The mean scores of 72% and 65% for The Playground and The Street Map

respectively provided similar levels of difficulty and required similar approaches to solve the

tasks. These tasks were of similar complexity with similar approaches adopted and yet the task

structures were very different. For example, on The Playground students only needed to interpret

labelled pictures and a compass point on the map itself. While on The Street Map, students

needed to interpret street names, labelled points, a key, coordinates, scale and a compass point.

The Street Map was a much more dense graphical representation than The Playground. Thus, the

route finding nature of the tasks seems to be the determinant for complexity rather than the

structure of the task. Hence, some students struggled with the cognitive demands of the tasks in

relation to following set directions and monitoring the sub-components of the tasks. Students’

difficulty with following sequential directions resonates with Wiegand’s (2006) second level of

map reading which involves both ordering and sequencing information and for these specific

tasks included counting, ordering and comparing information and data.

The following section describes the interpretation and evaluation of the data, through task

profiling, in order to develop new knowledge bringing together each section of analysis in order

to be able to make informed judgments about students’ sense making and the behavioural

characteristics they exhibit.

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5.9. Profiles of map tasks

This section combines data analyses from both the Task Solution (Episode 1) and the Task

Explanation (Episode 2) for the three tasks in order to develop a complete representation and

understanding of performance and behaviours of all students on each of the three tasks. The

combination of both episodes enables the creation of profiles of groups of students who exhibited

certain solution pathways. These pathways show the flow of students’ behaviour and

explanations on each task. The profiling process involved tracking five aspects of students’

performance and behaviour in the following sequence: (a) task correctness, (b) purposeful or

non-purposeful gesture use, (c) mapping skills, (d) solution approaches, and (e) type of gesture

used during Task Explanation. These aspects are detailed in Table 5.21. The vertical columns

contain the data for the students’ collective performance and behaviour on each task. These

profiles go beyond initial analyses by making connections between the five aspects in terms of

various solution pathways undertaken by students.

Table 5.21

The Five Aspects of Task Profiles.

Task Solution Task Explanation Correctness Gesture use Mapping skills Solution approaches Type of gesture

during explanation

Correct solution (CS) Incorrect solution (IS)

Purposeful gesture (PG) Non-purposeful gesture (NG)

Identify and use: Landmarks (LM) Coordinates (CO) Compass point (CP) Identify and apply Key (KE)

Describe movement (DM) Understood relationship between landmarks and movement (RLM) Followed a set of directions (route) (FD) Misunderstood ordinal sequence (OS) Process of elimination (PE) Immediately accessed positional information with the key (API) Fixated on reference point (RP) Other (OH)

Deictic (DGE) Iconic (IGE) No gesture (NGE)

111

112

There were four possible pathway approaches for each of the tasks. These pathways were: (a)

Pathway 1 - correct solution with purposeful gesture (CS,PG); (b) Pathway 2 - correct solution

with non-purposeful gesture (CS,NG); (c) Pathway 3 - incorrect solution with purposeful gesture

(IS,PG); and (d) Pathway 4 - incorrect solution with non-purposeful gesture (IS,NG). For

example, Pathway 1 for The Picnic Park (Figure 5.8) represents the 19 students who correctly

solved the task and used purposeful gesture (CS,PG). The proportion of students who utilised

specific mapping skills is first displayed on Figure 5.8. All of these 19 students were able to

identify and use landmarks and identify and use coordinates while nine (47%) identified and

applied the key (KE). The next aspect along the pathway identified the proportion of students

who employed a particular approach to solve this task. For example, 10 of the students (53%)

used a process of elimination (PE). The final aspect along the pathway indicated the type of

gesture students used during task explanation. For students in Pathway 1 this was predominately

deictic pointing gesture (DGE) (90%). Thus, only two of these 19 students did not use deictic

gestures (IGE or NGE).

For Pathway 2, the representation is restricted to those students who answered the task correctly

without using purposeful gesture (CS,NG). Pathway 3 described those students who answered

incorrectly and used a purposeful gesture (IS,PG). For The Playground task, no students

exhibited such behaviours and as a result no further data are represented on this pathway.

Pathway 4 includes data concerning those students who incorrectly solved the task without using

purposeful gesture (IS,NG). Thus, the 43 students are represented on three of the possible four

pathways for The Picnic Park (Figure 5.8).

Correctness Mapping skills Gesture use Types of Gestures Solution approaches

Figure 5.8. Profile of The Picnic Park Task.

IGE – 1 (20%)

DGE – 4 (80%) Fixated on reference point – RP

– 5 (100%)

KE - 2 (40%)

CO - 4 (80%)

LM - 5 (100%)Non purposeful gesture NG 5 (100%)

Purposeful gesture PG

0

Pathway 4

Incorrect IS

5 (12%)

Pathway 3

NGE – 3 (16%)

IGE – 1 (5%)

DGE – 15 (79%)

Other – OH - 1 (5%)

Process Elimination – PE - 6 (32%)

Accessed positional information – API – 12 (63%)

KE - 16 (84%)

CO - 19 (100%)

LM - 19 (100%)Non purposeful gesture NG 19 (50%)

NGE – 1 (5%)

IGE – 1 (5%)

DGE – 17 (90%)

Other – OH - 3 (18%)

Process Elimination – PE - 10 (53%)

Accessed positional information – API – 6 (32%)

KE - 9 (47%)

LM - 19 (100%)Purposeful gesture

PG Pathway 1

CO - 19 (100%)

19 (50%)

Pathway 2

Correct CS

38 (88%)

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5.9.1. The Picnic Park

The Picnic Park revealed three distinct pathways (Figure 5.8) with Pathway 3 not exhibited by

students. Pathway 1 and Pathway 2 contained students who were successful on the task (88%).

These students’ correct responses were evenly distributed between those students who exhibited

a purposeful gesture (50%, Pathway 1) and those who did not (50%, Pathway 2). There were

distinctions between the skills and approaches the successful students employed to solve the task.

Although all of the students, irrespective of whether or not they gestured, were able to identify

and use landmarks and identify and use co ordinates, there were marked differences in whether

or not the students identified and apply the key as part of their solution approach. For those

students in Pathway 1, only 47% identified the key, with the most common approach being a

process of elimination (53%). This Pathway suggests that although many of the students knew

the key was there, they chose to work out the task by using the answers as an initial starting point

and finding each set of co ordinates by using one hand or one finger to move along the x axis and

the other hand or another finger to move along the y axis until they met. The students on

Pathway 1 also predominately used deictic gestures during their explanation. As mentioned

earlier (Section 5.6), this type of gesture is common when engaging with spatial tasks. An

example of a student solution in Pathway 1 is presented shortly (Section 5.9.2).

By contrast, for those students in Pathway 2, 84% identified the key, with a majority of the

students (63%) immediately accessing positional information with the key and using it as the

basis for solving the task. This Pathway suggests that the students who did not gesture were more

likely to be immediately drawn to the key as their initial starting point. These students were able

to visually assess where the key was located on the map, and identify if the bike track was

located in that co ordinate square. They were able to navigate the map and the information it

contained without the need for a concrete tool to locate a point. One interesting component of

Pathway 2 is the three students who did not exhibit any gesture during task explanation. These

students answered correctly without using a purposeful gesture and were then able to explain

their solution without using any type of gesture to aid communication. Therefore, across the

entire task, these three students did not require the use of gesture in any way, either as a tool to

help solve the task, or as an aid to explain their thinking. Success despite the lack of gesture

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suggests that these students had the mathematical proficiency and confidence to explain their

thinking without the need to draw attention to specific parts of the map. Hence, students might or

might not need to gesture, with an omission of gesture suggesting a higher level of proficiency.

Pathway 4 presents the five unsuccessful students with all of these students not exhibiting a

purposeful gesture as they solved the task. All of these five students were able to identify and use

landmarks, with fewer able to identify and use coordinates and identify and apply the key (80%

and 40% respectively). In terms of solution approach, these five students all became fixated on a

reference point, namely the bike track. This approach suggested they did not take into account all

the necessary information and hence became fixated on the main component of the question and

the map (i.e., the actual bike track). Given that the use of gesture was often a tool to aid students

in navigating the spatial requirements of the map, the lack of gesturing during Task Solution may

have affected their capacity to answer correctly. It could be suggested that had the students in

Pathway 4 employed a purposeful gesture to solve the task, it may have aided them in their

solution.

5.9.2. Example of One of the Most Common Pathways for The Picnic Park

Pathway 1 was one of the most common pathways for The Picnic Park in which students used a

purposeful and a deictic gesture, and a process of elimination in their solution approach. Laura

used Pathway 1 and her pathway is described below. Laura reached a correct solution, by

identifying and using the landmarks and the coordinates. She utilised a process of elimination

solution approach and explained her answer with a deictic gesture (Figures 5.9 & 5.10). The

sequence of numbered images (Figure 5.9) depicts Laura using purposeful gestures as she solved

the task. This numbered sequence is tracked on the map, showing where Laura’s finger

movements were in relation to the task on the page. Image 1 depicts Laura locating the co

ordinate square A5, which is the first option in the answers. Image 2 depicts Laura locating the

co ordinate B5, the second option in the answers. Image 3 depicts Laura locating A4 and Image 4

depicts her locating B4 on the map, the last option in the answers. From here she chose her

answer, B4.

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1 2

3 4

2

Figure 5.9. Purposeful gestures used during Task Solution on The Picnic Park and tracked on the map.

Laura’s solution approach indicates where her hands were moving on the map as she was

explaining her solution (Figure 5.10). The deictic gestures used during Task Explanation were

pointing to the specific aspects of the task that she was explaining. The first image shows Laura

pointing to the answer box, explaining that she “went through all the answers”. The second

image demonstrates how Laura moved her finger around the bike track in a clockwise direction

as she talked about the answer options “all had a little bit of the bike track on them”. The last

image shows Laura pointing to the co ordinate square B4 that she chose as her answer explaining

“B4 didn’t have any part of it [the bike track] on it”. This sequence highlights the process of

elimination solution approach and the type of deictic gestures used during Task Explanation.

1

3 4

1 2

3 4

1 2 3 Laura: I picked B4. I went through all the answers (1, she points to the answer box) and

they all had little bit of the bike track on them (2, she moves her finger around

the bike track) and one, the B4 (3, pointing to the B4 square), didn’t have any

part of it [the bike track] on it. (emphasis added in brackets)

Figure 5.10. Transcript of The Picnic Park explanation, cross referenced with deictic gesture use.

Laura’s example is representative of those students in Pathway 1 who utilised a purposeful

gesture during Task Solution, used a process of elimination as their approach and used deictic

gestures during Task Explanation.

5.9.3. The Playground

The profile of The Playground task shows four pathways with connections between the Task

Solution (Episode 1) and Task Explanation (Episode 2) (Figure 5.11). Thirty-one of the 43

students (72%) were successful on this task. Pathway 1 highlights the students who used a

purposeful gesture to successfully complete the task. The majority of these 23 students identified

and used the landmarks (100%), described direction (96%), knew the relationship between

landmarks and movement (96%) and were able to follow set directions (route) (91%). Almost all

of these students were able to engage with the higher cognitive demands of the task by

considering the sequence of directions in the correct order and navigating their way to the

unknown location. Students in Pathway 1 utilised deictic gestures during Task Explanation with

96% of them exhibiting such behaviour.

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Correctness Mapping skills Gesture use Solution approaches Types of Gestures

Figure 5.11. Profile of The Playground Task.

NGE – 1 (20%)

DGE – 4 (80%)

Other - vague response – OH - 1 (20%)

Relationship landmarks & movement – RLM - 1 (20%)

Described movement – DM - 4 (80%)

LM - 5 (100%)

Non purposeful gesture NG

5 (42%)

IGE – 1 (14%)

DGE – 6 (86%)

Other – counted landmarks – OH - 3 (43%)

Relationship landmarks & movement – RLM - 1 (14%)

Described movement – DM - 4 (57%)

LM - 7 (100%)Purposeful gesture PG

7 (58%)

Pathway 4

Incorrect IS

12 (28%)

Pathway 3

DGE – 8 (100%)

Followed set directions (route) – FD - 7 (86%)

Relationship landmarks & movement – RLM - 8 (100%)

Described movement – DM - 8 (100%)

LM - 8 (100%)

Non purposeful gesture NG

8 (23%)

IGE – 1 (4%)

DGE – 22 (96%)

Other - vague response – OH - 1 (4%)

Followed set directions (route) – FD - 21 (91%)

Relationship landmarks & movement – RLM - 22 (96%)

Described movement – DM - 22 (96%)

Purposeful gesture

PG 23 (77%)

Pathway 1 LM - 23 (100%)

Pathway 2

Correct CS

31 (72%)

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119

In relation to the eight students (23%) in Pathway 2 who were successful but did not use a

purposeful gesture, these students’ skills and solution approaches were predominantly the same

as those students who did gesture. The majority of Pathway 2 students were able to use a skilful

approach to solution by following a set of directions in relation to finding the route. These

students, who apparently were able to navigate the movement of the task “in their mind’s eye”

by using internal representations, could do so as effectively as those students who gestured to

scaffold their movements around the task. Consequently, a sizable proportion of students (23%)

were able to maintain the cognitive demands of following set directions and making decisions

regarding movement between landmarks without having to “trace” these movements with the use

of an external gesture. The complexity of the task did not prompt these students to use more

“concrete” procedures to solve the task as their internal representations were sufficiently

adequate. Thus, not all successful students required the use of gestures. It might be that when a

student has a good understanding of the task, they no longer needed to use gestures when solving

map tasks. All eight of the Pathway 2 students used deictic gestures to explain their solutions.

These gestures helped communicate their explanations to the interviewer, even though they did

not require a purposeful gesture to solve the task.

By contrast, of the 12 students (28%) who were unsuccessful, there was a more even distribution

among those students who utilised a purposeful gesture on Pathway 3 (58%) and those who did

not use a gesture on Pathway 4 (48%). The 12 students who had an incorrect response were less

likely to be able to describe movement within the task or appreciate the relationship between

landmarks in the map and movement between the landmarks. These students understood many

aspects of the task however the incorrect response typically occurred at the point when higher

levels of reasoning were required. Thus, when students were asked to keep track of the

landmarks, follow them in order and monitor how many times they crossed the track, the

cognitive load was too demanding for them. In these situations the use of a gesture in Pathway 3

was not a sufficient tool to help them navigate the requirements of the task. Again, deictic

gestures were the most prominent of gestures exhibited for Pathway 3 and Pathway 4. Only one

student who answered incorrectly, did not gesture at all as he/she engaged with the task. This

student did not use a purposeful gesture or any type of gesture during their explanation. It could

be that this student might have needed the scaffold of a gesture to help him/her solve the task in

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order and keep track of his/her movements on the page. However, regardless of success or

gesture use on this task, most students appeared to find it helpful to use deictic gestures during

their explanation.

5.9.4. Example of the Most Common Pathway for The Playground

Pathway 1 was the most common pathway for The Playground with students using a purposeful

and a deictic gesture, identifying the landmarks and following set directions in their solution

approach. Jeremy’s solution and explanation for Pathway 1 is described below (Figure 5.12). His

response is typical of the students who followed this pathway. The sequence of numbered photos

depicts Jeremy using purposeful gestures as he solved the task. The deictic gestures used during

Task Explanation were similar to that shown during Task Solution, where Jeremy tracked the

route from landmark to landmark with his hand. Hence, one set of images is used to illustrate

both the purposeful gestures and the deictic gestures for this task. The numbered sequence is

tracked on the map, showing where Jeremy’s hand movements were in relation to the task on the

page. With respect to Task Solution, Jeremy immediately found the first landmark in the bottom

right of the map (image 1, the gate). From there he moved to the tap, which is located in the

middle right of the map (image 2, the tap). He then went to the shed located on the bottom left

side of the map (image 3, the shed). Lastly he moved to the top left of the map to where the

rubbish bins were located (image 4, the rubbish bins).

The transcript provided outlines Jeremy’s solution approach and indicates where his hands were

moving on the map as he was explaining his solution, cross referenced to his gesture use. In a

similar manner to his Task solution, the deictic gestures Jeremy used during Task explanation

tracked the movement of the route on the map. Hence, as Jeremy explained “because when he

goes from the gate to the tap, he crosses”, his pen moves from 1 (the gate) to 2 (the tap). As

Jeremy continues “and all the way to the shed, he crosses again”, he moves his pen from 2 (the

tap) down to 3 (the shed) on the left side of the map. Finally Jeremy explains “but when he gets

to the rubbish bins, he doesn’t have to cross over again”, moving his pen in an arced manner

from 3 (the shed) to 4 (the rubbish bins) in order to take the shortest route to the last landmark,

meaning he did not cross the track again in that movement.

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1 2

3 4 Jeremy: I chose two because when he goes from the gate to the tap [1 2] he crosses and

all the way to the shed [2 3] he crosses again, but when he gets to the rubbish

bins [3 4] he doesn’t have to cross over again.

Interviewer: So what did you have to look at to work that out?

Jeremy: The names of everything...every time I crossed the track I would have 1 in my

head and get ready to put another one if I crossed again.

4 2

3

1

Figure 5.12. Sequence and transcript of a student demonstrating the most common pathway for The Playground task.

Within his explanation, Jeremy also stated that he kept count in his head of how many times he

crossed the track. This could indicate that he split the demand of keeping track of information

between his head and his hands. Keeping track of the route with his hands allowed him to

concentrate on the question, that is, how many times the track was crossed during that route. This

coordinated approach to keeping track of his movements was a practical way to distribute the

cognitive load of the task between his hands and his mind. This example provides a snapshot of

the types of behaviours exhibited by the students in Pathway 1 as they engaged with The

Playground task.

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5.9.5. The Street Map

This section combines data from both the Task Solution (Episode 1) and the Task Explanation

(Episode 2) analyses for The Street Map task. The profile of The Street Map (Figure 5.13)

revealed four pathways. Pathways 1 and 2 highlight the 28 students (65%) who were successful

on this task. On Pathway 1, 25 of these students (89%) used a purposeful gesture to solve the

task, while on Pathway 2 three students (11%) did not use a purposeful gesture. Irrespective of

gesture use, students on Pathways 1and 2 employed the same solution approaches to solve the

task. All of the students identified the landmarks, described movement, understood the

relationship between landmarks and movement and followed the set directions in order to find

the solution. These students were able to understand the complex combination of language and

apply those directions to the map. Students who used Pathways 3 or 4 were unsuccessful.

With respect to Pathway 1, 25 of the 28 successful students (89%) used a purposeful gesture

when solving this task. This suggests that using gesture to help navigate the route on the map

was a tool students utilised to track their thinking. The use of a tracking tool allowed these

students to monitor the sequence of movements and directions as the navigated the map. In

relation to the types of gestures exhibited during their explanation, 23 (92%) of the students in

Pathway 1 used deictic gesture. Two students (8%) in Pathway 1 used iconic gestures during task

explanation. As these two students were explaining their solution, they were showing how they

positioned themselves in the map and how they would move around the map by using iconic

gestures as they spoke. Unlike the students who used deictic gestures, these two students did not

point to the map at any stage, with most of their hand movements being in the space in front of

them, showing the direction they were facing and the route they took.

Correctness Gesture use Mapping skills Types of Gestures Solution approaches

Figure 5.13. Profile of The Street Map Task.

DGE – 4 (100%)

Misunderstood ordinal sequence – OS - 4 (100%)

Relationship landmarks & movement – RLM - 4 (100%)

Described movement – DM - 4 (100%)

CP – 3 (75%)

LM - 4 (100%)Non

purposeful gesture NG

4 (27%)

NGE – 1 (9%)

DGE – 10 (91%)

Fixated on reference point – RP – 1 (9%)

Other – left/right confusion – OH - 2 (18%)

Misunderstood ordinal sequence – OS - 8 (73%)

Relationship landmarks & movement – RLM - 10 (91%)

Described movement – DM – 11 (100%)

KE - 2 (18%)

CP – 6 (55%)

LM - 11 (100%)Purposeful gesture PG 11 (73%)

Pathway 4

Incorrect IS

15 (35%)

Pathway 3

NGE – 1 (33%)

DGE – 2 (67%)

Followed set directions (route) – FD - 3 (100%)

Relationship landmarks & movement – RLM - 3 (100%)

Described movement – DM - 3 (100%)

CP – 2 (67%)

LM - 3 (100%)Non

purposeful gesture NG

3 (11%)

IGE – 2 (8%)

DGE – 23 (92%)

Followed set directions (route) – FD - 25 (100%)

Relationship landmarks & movement – RLM - 25 (100%)

Described movement – DM - 25 (100%)

KE - 3 (12%)

Pathway 1 LM - 25 (100%)Purposeful gesture

PG CP – 22 (88%)

25 (89%)

Pathway 2

Correct CS

28 (65%)

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The three successful students on Pathway 2 utilised the same solution approaches as those

students on Pathway 1. It seems they were able to navigate the task without the use of a concrete

tool to track their movements as they were able to monitor sequence in their “mind’s eye”. One

student who answered correctly but did not used purposeful gestures in task solution, did not

exhibit any type of gesture as they explained their solution. Therefore, across the entire task, this

student did not use of gesture in any way, either as a tool to help solve the task, or as a way of

explaining his/her thinking. This suggests that this student possibly had the spatial ability and

confidence to undertake the task using mental imagery and the verbal ability to explain their

thinking without using any type of gesture to aid communication.

With respect to the 15 unsuccessful students (35%), 11 of these students (73%) were in Pathway

3 and used a purposeful gesture during task solution. The 11 students in Pathway 3 employed a

number of different solution approaches in order to solve this task. All 11 students were able to

identify the landmarks and describe movement on the map. However, eight of these students

(73%) misunderstood the ordinal sequence in the task and chose the first road to the right as

opposed to the second. Two of the students (18%) were confused by their left and right meaning

they went left initially and then right, when the task required the students to go right then left.

One student became fixated on Bill’s house and followed a route of left and right turns in order

to arrive at Bill’s house. All of these students used a purposeful gesture to help them solve the

task however the cognitive demands of the task were too great for that tool alone to be

supportive. Thus, having to combine different types of mathematical language and follow these

through a sequence required more than the use of gesture to scaffold understanding. The majority

of Pathway 3 students used deictic gestures during their explanation. One student however used

no gesture to help communicate his solution.

Four of the 15 unsuccessful students (27%) were on Pathway 4 and did not utilise a purposeful

gesture. All of these students identified and used the landmarks, described movement and

understood that movement was required between the landmarks. However, all four of these

students misunderstood the ordinal sequence of first and second. These students were able to

negotiate the majority of the required sequence except for the ordinal numbers and how they

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were applied to the route. That is, they took the first road to the left instead of the second. All

four students in Pathway 4 used deictic gestures during their explanation.

5.9.6. Example of the Most Common Pathway for The Street Map

Pathway 1 was the most common pathway for The Street Map in which students used a

purposeful and a deictic gesture, identified the landmarks and followed set directions in their

solution approach (Figure 5.14). Lachlan use of this pathway is described below. His response is

typical of the students who followed this pathway. The sequence of numbered photos depicts

Lachlan using deictic gestures as he explained how he solved the task. The purposeful gestures

used during Task Solution were similar to that shown below, where the student tracked the route

from landmark to landmark with his hand. Hence, one set of images is used to illustrate both the

purposeful gestures and the deictic gestures for this task. The numbered sequence is tracked on

the map, showing where Lachlan’s hand movements were in relation to the task on the page.

With respect to Task Solution, Lachlan immediately found the first landmark in the bottom

middle of the map (image 1, the pool). From there he moved north along Stoney Road (Blue

arrow on the map and image 3) to the intersection of Stoney Road and Wattle Road. Continuing

to follow the directions, Lachlan moved toward his right and stopped at the first road, Post Road

(image 4) then continued moving right to the second road, which is School Road (image 5). As

Lachlan read the task, he moved his pen to follow the directions given in the written information.

The transcript provided outlines Lachlan’s solution approach and indicates where his hands were

moving on the map as he was explaining his solution, cross referenced to his gesture use. In a

similar manner to his Task Solution, the deictic gestures Lachlan used during Task Explanation

tracked the movement of the route on the map. Hence, when Lachlan explained “First I had a

look where the pool was, found it”, his pen locates the pool (image 1). He then explained that he

“used the north” pointing to the compass point (image 2) to indicate that he used this to help him

work out which way was north. As Lachlan continued his explanation “he drives north and takes

the first right, which is Wattle Road”, he pointed his pen to the first intersection with a right hand

turn (image 3). From there, Lachlan knew that he was looking for the second turn on the left and

so he moved past the first left—“that is the first” (image 4, pointing to Post Road)—and

continued on to the second road on the left—“that is the second, which is School Road” (image

5).

1 2

3

4 5

Lachlan: First I had a look where the pool was, found it (1, pointing to the pool) then he

leaves the pool and I used North (2, pointing to the compass). So he goes, he

drives north and takes the first right, which is Wattle Road (3, pointing to the

intersection of Stoney and Wattle Roads), and takes the second road on the left,

that is the first (4, pointing to Post Road) and that is the second (5, pointing to

School Road), which is School Road. (emphasis added in brackets)

5 4

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Figure 5.14. Sequence and transcript of a student demonstrating the most common pathway for The Street Map task.

3

1

2

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This example highlights how students such as Lachlan used Pathway 1 for The Street Map task.

The successful students in this pathway used a gesture to navigate the map and followed the

directions provided. They aided their explanation by using deictic gestures to communicate and

show how they worked out the task.

5.10. Understanding Task Profiles

The construction of task profiles assigns specific performance and behaviour to students’ skills

and approaches, presenting a full representation of the pathways students undertook in order to

solve the respective map tasks. These pathways represent the collective forms of analysis

undertaken on students’ Task Solution and Task Explanation and provide an interpretative

mechanism which highlights patterns in ways previously not addressed.

An analysis of the profiles revealed that task complexity had a substantial impact on not only the

approaches students used to complete the task, but also the extent to which they exhibited

gestural behaviour. For The Picnic Park, the easiest of the three tasks, the proportioning of

gesture use was similar on Pathways 1 and 2, that is, a 50% split between those who used and

those who did not use a purposeful gesture. For this task, those students in Pathway 1 who

employed a purposeful gesture generally employed different approaches to solve the task than

students in Pathway 2 who did not use gesture to solve the task. Pathway 1 students tended to use

a process of elimination to complete the task using pointing gestures to identify specific locations

or objects on the map. Pathway 2 students, who did not gesture, immediately located the relevant

key and generated a solution. These students used visual imagery to locate relevant coordinates

without the aid of gesture. Presmeg (1986) recognised that the use of imagery is not always the

most effective way to solve tasks however such processing is particularly useful when students

experience novel or relatively complex tasks. In this study however, this assertion is not

necessarily helpful because the tasks had high spatial demands. Thus, for the easiest of the three

tasks, students were able to choose between utilising gesture or not in their approach to solving

the task. On more complex tasks, the proportion of students using gestures increased and hence

visual imagery possibly declined.

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In relation to both The Playground and The Street Map, almost all of the students who

successfully completed the tasks in Pathways 1 and 2 were able to engage with the higher

cognitive demands of the tasks, irrespective of gesture use. In other words, approaches to solving

the tasks were similar irrespective of gesture use. These tasks were more difficult than The

Picnic Park and more students found it necessary to employ gestural behaviours. Despite the

increase in gesture use in Pathway 1 (from 50% of successful responses on The Picnic Park to

77% and 89% of successful responses on The Playground and The Street Map respectively)

Pathway 2 students on The Playground (23%) and The Street Map (11%) were still able to solve

the tasks correctly without gesturing. Thus, Pathway 2 students were able to navigate the spatial

demands of the tasks without the support mechanism that assisted in reducing the cognitive

demands of the tasks. Lowrie and Diezmann (2007a) found that spatial reasoning ability was a

strong predictor of success on graphics tasks which helps to explain why these students were

able to complete these tasks without gesturing. Given these tasks required spatial thinking, these

students were probably exhibiting characteristics of internal representation as they solved the

task. In these instances, these students processed the information in their “mind’s eye” rather

than using gestural movement to navigate the problem context. In terms of The Playground,

Pathway 2 students probably imagined moving from the gate across the track to the tap, then

crossing the track to reach the shed, before moving on the rubbish bins. For The Street Map,

Pathway 2 students could have imagined a similar navigational pathway but this time, imagined

moving within a road map context such as imagining moving in a car on the road (see Figure

5.14). Kosslyn (1983) regarded this type of visual reasoning as “inside space” visualisation, that

is, visualisation that places the subject into the spatial representation. While gesture use

increased as the tasks became more difficult, it was evident that some students were able to

employ visual processing to successfully solve these map tasks without the aid of gesture.

With respect to those students who were not able to solve the tasks correctly (28% and 25% of

the cohort respectively), it was evident that they were unable to fulfil all the requirements of the

tasks. Many of the students on Pathways 3 and 4 did not possess the appropriate understandings

to follow the set directions through to generating a correct solution. Although they were able to

identify most relevant aspects of the tasks, they could not follow the sequence of events, monitor

essential counts (The Playground) or interpret directions (The Street Map) as they completed the

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tasks. For The Playground task, most incorrect responses in Pathways 3 and 4 were a result of

only a partial understanding of the essential aspects of the task related to monitoring the travel

path of the movements and where it crossed the track. For The Street Map, most incorrect

responses in Pathways 3 and 4 were a result of students not being able to interpret all of the

directions given in the task or apply them in the correct sequence. In both cases, it was an

inability to monitor and act on two sets of information. Hence, these students were unable to

engage with these tasks at Wiegand’s (2006) second level of map reading because they struggled

to analyse the tasks in relation to ordering and sequencing information. This level of map reading

occurred whether or not the students gestured, and hence the gesturing was an ineffective support

for problem navigation. If students are unable to readily access appropriate information to solve

a task, it seems gesturing can only be a supportive mechanism if students have the capacity to

monitor the sequence of events.

5.11. Chapter Summary

An analysis of existing video data using the KDD design provided scope to assess how students

solved map tasks commonly found in national assessment instruments. There were three main

components of this analysis. The first component of the analysis described how students solved

map tasks. The second component identified patterns across the map tasks and the final

component presented profiles of performance and behaviour on each task.

This first level of analysis examined the way students solved map tasks, from Task Solution

through to Task Explanation. This analysis revealed that students had a relatively sound

understanding of general mapping knowledge however their understanding of mathematical

concepts pertinent to map tasks were less developed. In terms of their behaviour, typically

students utilised deictic gestures as they explained their solutions. These types of gestures aided

the students’ communication of their explanations.

The second level of analysis considered the patterns that emerged across the three map tasks.

One finding considered the influence gestural behaviours had on students’ performance when

solving map tasks that required considerable levels of spatial thinking. That is, successful

students who utilised gesture tended to approach the task in the same manner as students who did

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not gesture. An inverse relationship was found between task difficulty and purposeful gesture

use, where more students’ used purposeful gestures as the tasks increased in difficulty. Students

generally solved the two most difficult tasks in a similar manner due to the fact that they were

both route finding tasks even though the graphical structure of the tasks was different.

The final level of analysis described profiles of students’ performance and behaviour on each of

the three map tasks. Four pathways were identified for each map task based on correctness and

gesture use. The profiles offered the opportunity to examine the differences between successful

and unsuccessful students. Both successful and unsuccessful students were able to read the

symbolic features of the maps. However, successful students were able to interpret the map tasks

with higher levels of mathematical understanding than the unsuccessful students. Drawing on the

results and discussion of this chapter, Chapter 6 addresses the research questions and provides

concluding comments on the study.

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Chapter 6. Conclusions

6.1. Introduction

This chapter presents conclusions to the study and has four main parts. The first part presents a

summary of findings with a view to answer the three questions (Section 6.2). The research

questions were:

1. What mathematical understandings do primary-aged students require to interpret map

tasks? (Section 6.2.1)

2. What patterns of behaviour do these students exhibit when solving Map tasks? (Section

6.2.2)

3. What profiles of behaviour do successful and unsuccessful students exhibit on Map

tasks? (Section 6.2.3)

The second part of the chapter identifies the limitations of the study by acknowledging issues

which arose from the trustworthiness of the design and methods (Section 6.3). The third part of

the chapter considers the implications of the research (Section 6.4). Implications are presented

for practice – learning and teaching (Section 6.4.1) and for theory – test designers and policy

makers (Section 6.4.2). The final part of the chapter identifies avenues for further research

arising from the study (Section 6.5) and provides a chapter summary on how primary-aged

students interpret map tasks (Section 6.6).

6.2. Summary of Findings for Each Research Question

This study investigated how primary-aged students interpreted and understood map tasks. In

particular, the study considered how students solved tasks and the patterns of behaviour they

exhibited as they solved the tasks. The research findings consider data from transformation, data

mining and interpretation/evaluation stages of the KDD research design. The three research

questions are addressed in turn (Sections 6.2.1, 6.2.2, and 6.2.3).

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6.2.1. What Mathematical Understandings do Primary-Aged Students Require to Interpret

Map Tasks?

The first research question considered the types of knowledge that students exhibited as they

solved map tasks. With almost half the cohort (49%) able to solve all three map tasks, the

students demonstrated a number of mapping skills and mathematical concepts related to map

tasks. These students demonstrated generic mapping knowledge including an understanding of

the use of a key and the ability to locate landmarks and symbols. By contrast, approximately one-

third of the students were only able to solve two of the three tasks and therefore it could be

proposed that some forms of specific mathematical concepts, including an understanding of

arrangement and direction, were required to complete the task successfully.

The three tasks required various forms of content knowledge and the necessity to decode

information that was represented in different ways. The three tasks were different in

representation; one being a pictorial representation (The Playground, Figure 5.2), the second

using a coordinate structure (The Picnic Park, Figure 5.1), and the third using a combination of

both pictorial and coordinate structure in a complex street map representation (The Street Map,

Figure 5.3). Generally, the students were familiar and unperturbed with these different

representations in terms of their ability to decode the graphical structure. Therefore, from a

decoding perspective, the results showed that students were able to interpret information in

different types of map tasks with particular graphic representations. These findings align to the

simplest level of map reading where students are able to extract information from a map by

reading names and attributes, and recognising visual stimuli and specific elements (or icons) on

the map (Wiegand, 2006).

The relative difficulty of the respective tasks was predominately associated with the application

of the sequence of events required to complete the task. In particular, the requirement to keep

track of sequential movement challenged more than half the students (51%). In these situations

the students were required to interpret several aspects of the map tasks in order make

navigational decisions, especially for The Playground and The Street Map tasks. With these two

tasks, task complexity was due to the requirement to navigate and monitor the process of finding

routes. This process mainly required an understanding of mathematics concepts including

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location, movement and direction rather than the mapping knowledge of locating landmarks and

symbols. The ability to apply these mathematical concepts resonates with Wiegand’s (2006)

second level of map reading which refers to the ability to process and sequence information. This

study found that the students’ conceptual understanding for this second level of map reading

ability was somewhat limited when students were required to make sense of tasks such as The

Playground and The Street Map, which required counting, ordering and comparing information.

6.2.2. What Patterns of Behaviour do These Students Exhibit When Solving Map Tasks?

The second research question moved beyond an interpretation of students’ mathematical

understandings to consider patterns of behaviour exhibited by students as they solved the tasks.

The theoretical framework of the study considered the multimodal nature of engagement with,

and reasoning on, highly spatial tasks. As a result it was necessary to look at different types of

behaviour in order to gain a better understanding of how students solved these tasks. These

behaviours included the gesturing students used to solve the tasks, their utilisation of gesture to

explain their reasoning, and the relationship between task complexity and gesturing. The analysis

also considered the role of gender in relation to these gestural behaviours.

Gesturing was certainly a prominent feature of students’ behaviour. This study analysed

students’ gesturing whilst solving map tasks and as they explained their solutions. One of the

main findings of this study with regard to gesture use occurred during Task Solution. As students

solved the tasks, purposeful gesture was exhibited by all but four of the participants in the study.

By contrast, one third of the participants (33%) used purposeful gestures for each of the three

tasks. These gestures were commonly concentrated on the map itself, with students using their

fingers or a pen to track their progress on their task. With regard to gesture use during Task

Explanation, deictic gestures were the most prominent gestures used as students explained their

solutions. Deictic gestures are pointing type gestures. Allen (2003) suggested that these pointing

gestures act as an aid to communication and are especially common when spatial thinking is

involved.

There is a view that gesturing influences the representations and processes that take place in

students’ minds as they engage with spatial tasks and that such behaviour can influence the

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pathways of spatial reasoning (Alibali, 2005). Thus, the students who gestured for each of the

three tasks did so as a component of their information interpretation and cognitive processing.

These behaviours appeared to help their performance because they provided a support

mechanism for monitoring spatial information. Despite 33% of students utilising gesture to solve

all three tasks, 58% of students gestured only on one or two tasks. This selective use of gestures

could indicate that the behaviours were aligned to the actual task rather than the students’

individual processing needs. In relation to the particular task, the structure and design of certain

tasks appeared to prompt students to gesture in order to navigate the spatial challenges of the

task. Knowing that the students were able to decode the mapping elements of the tasks (as

explained in Section 6.2.1), gesturing appeared to be a tool used to support their navigation

around the respective tasks. The types of gesture exhibited during task solution all pertained to

location, movement and arrangement and certainly involved mathematical concepts rather than

the identification of landmarks and symbols. To this point, fewer students utilised gesture on The

Picnic Park which required less demanding sequencing steps than the other two tasks.

In terms of individual students’ behaviours, more students gestured as the spatial demands on the

mapping task associated with location, movement and arrangement increased. In such situations,

gesturing appeared to become a support mechanism for monitoring progress and determining the

ordering and sequencing of directions. The strongest pattern to emerge from the study in relation

to gesture use was the inverse relationship between it and success. Students’ use of gesture as a

support tool may have allowed them to revert back to less abstract approaches by using

purposeful gestures in order to fully engage with the tasks (Pirie & Kieran, 1994). Furthermore,

the necessity to consider multiple forms of representation, sometimes simultaneously, could also

be supported through the use of gesture. As The New London Group (2000) maintained, gesture

is an important element of meaning making. The finding that gesture was related to task

correctness, to the best of my knowledge, has not been reported in the literature. It also makes a

contribution to research required “to determine how gestures relate to learning and thinking”

(Radford, 2009, p. 112).

A final pattern of behaviour that was investigated in this study related to the extent to which

gender differences affected performance and gesture use. A growing body of literature has found

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that boys tend to outperform girls on a range of graphics tasks and particularly on mapping tasks

(Lowrie & Diezmann, 2005; 2009). In the current study, boys scored higher than girls on each of

the three map tasks, however differences were not statistically significant. As tasks became more

difficult, girls’ gesture use increased at a higher rate than that of boys, but once again, the

differences were not statistically significant. Although these findings suggest a conflict with

respect to recent findings on mapping tasks, the small sample size could have been a major factor

in determining probability values for statistical significance (Burns, 2000).

6.2.3. What Profiles of Behaviour do Successful and Unsuccessful Students Exhibit on Map

Tasks?

The third research question considered the behaviours of successful and unsuccessful students on

the three map tasks. With respect to successful students, the approaches they employed to solve

the tasks, and particularly the two more difficult tasks (The Playground and The Street Map),

were similar irrespective of whether they gestured or not. Thus, the act of gesturing did not

appear to alter or impact on solution approach. This finding suggests that the students who did

not gesture were likely using imagery to solve the task. The rationale for assuming that non-

gesturing students were visualising was due to the high spatial nature of these tasks (see

Presmeg, 1997 for a discussion of visualisation and spatial tasks). The successful non-gesturing

students used similar strategies to those students who gestured, but instead of using their hands to

navigate the spatial demands of the task they apparently did so “in their mind’s eye” (Kosslyn,

1983).

Visualisation is often regarded as a more powerful form of reasoning than gesturing. As Pirie and

Kieran (1994) explained Image Having is the capacity to carry a mental plan of the particular

mathematical concepts, and is at a higher level of understanding than Image Making where the

reliance is on a tool to aid concept development. Thus, non-gesturing students might have had an

image, whereas gesturing students were making an image with their gestures. Martin (2008)

argued that a student would intuitively return to a more primitive way of knowing when faced

with a task that is challenging. In this study, those students who did not gesture could be

considered at a higher level of understanding on a particular map task than those students who

required gestures to help them solve the task. This assertion is reinforced for The Street Map task

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where only three students (7%) successfully completed the task without gesturing, and hence,

were able to visualise the navigational space of the most difficult task.

The unsuccessful students were, at times, able to use some mapping knowledge and some

mathematical concepts to solve the tasks, however limited understandings resulted in ineffective

solution approaches. As was the case with the successful students, the act of gesturing did not

appear to affect the approach that these students took to solve the task since the approaches were

similar regardless of gesture use or not. Nevertheless, those students who gestured did so in a

purposeful manner as they tried to use gesturing to help them navigate spatial pathways and

arrangements. Like the non-gesturing students, however, they were unable to combine the

necessary mapping knowledge and mathematics concepts to achieve a successful solution.

6.3. Limitations of the Study

The limitations of the study are expressed in terms of the study’s trustworthiness. When

evaluating the trustworthiness of the study, it is necessary to consider how the (a) data analysis;

(b) researcher; and (c) setting have influenced the results and their interpretation. The limitations

of this study are outlined in terms of these three essential criteria in order to present a clearer

description of the results and how they are addressed in the research questions.

One limitation of this study concerned the data mining techniques used to interpret the data

sources. Although data mining is a well recognised process of analysing existing data, the static

nature of the available data set needs to be acknowledged. That is, the interview protocols,

camera angles, and the actual tasks presented to the participants during the data collection

process were predetermined by specific goals and research questions of the original project. As

mentioned earlier, one of the advantages of data mining is that data can be reanalysed with a

different set of research questions to extend existing knowledge (Section 4.3). However, one of

the limitations of reanalysing data concerns the inability to manipulate the data collection

process (Kelder, 2005; van den Berg, 2005) and specifically in this study, the design of the

interviews, the tasks and the way the video data were presented. In a differently designed study,

the researcher would have been able to pose open-ended questions that encouraged the students

to explain their internal thoughts more clearly. This elicitation of students’ thinking would have

137

provided a more flexible data set with the researcher able to ascertain some of the internal

representations the students evoked as they solved the tasks. To this point, all observations on

behalf of the researcher needed to be of the external verbal and non verbal behaviour.

Consequently, the researcher was able to hypothesise what the students might have been

thinking, but could not triangulate these data with confirmation from the students.

A related limitation was associated with the role of the researcher in this study and in particular,

the categorisation of a students’ use of gesture to communicate meaning. This attribute is

personalised and is potentially egocentric in nature. In this study, decisions about students’ use of

gesture were categorised and analysed in terms of observations of hand movements from the

video data. There was no possibility for further probing into reasons why students exhibited such

behaviour due to the fact that gesture use was not part of the original research project.

Consequently, analysis of the gesturing data needed to be clearly defined through traditional

observational techniques specifically in relation to McNeill’s (1992) categorisation of hand

gestures. Therefore, the trustworthiness of the data coding was reliant on the interpretations of

the researcher. Nevertheless, this limitation was reduced, to some extent, since the researcher had

some understanding of these students’ responses to the tasks given the fact that she had worked

with them over extended periods conducting interviews as a research assistant on the original

project.

The final limitation of the study relates to the setting of the study, specifically the sample size of

the cohort. Although the sample size would be considered adequate for the qualitative

component of the study, a sample size of 43 reduced the statistical power of the quantitative

aspects of analysis. This limitation did not impact on the analysis of dichotomous data or

descriptive coding but did impact on the analysis of variance procedures (see Section 5.7.1.3).

The small sample size is a result of analysing only one state cohort from the larger study. To use

data from the interview cohorts in both states (n=93) would have been impractical because the

size of the qualitative data set would have become unmanageable. As a result of the small sample

size, there were performance and behaviour trends in relation to students’ gender but these

results were not statistically significant due in part to the power of the design. Nevertheless, the

multi method approach to analysing the data within the Knowledge Discovery in Data design

138

was both innovative and combined sound features of both quantitative and qualitative

methodologies.

6.4. Implications of the Study

A number of implications for theory and practice emerged from this study. These implications

related to theory (Section 6.4.1), learning and teaching (Section 6.4.2), and test design and

curriculum design (Section 6.4.3).

6.4.1. Implications for Theory

From a theoretical perspective, the findings of the study suggest that gesturing is an important

element of multimodal engagement in mapping tasks. Although gesturing has long been

recognised as an important component of communication (Radford, 2009; The New London

Group, 2000), the findings reveal the effectiveness of gesturing when students are solving

mapping tasks that require spatial processing. Therefore, this study highlights the utility of

multimodal frameworks in ways that go beyond the nature of communication. The study also has

implications for theoretical models which describe students’ sense making and reasoning when

solving mathematics tasks. Taking notice of when students use gesture to solve tasks can help

researchers identify the cognitive challenges students are experiencing since gesturing is

particularly effective when students need concrete support to solve tasks. Just as it is important to

recognise the importance of visualisation in processing, so too is the need consider the influence

of gesturing. Thus, this study can inform theoretical frameworks which aim to describe students’

thinking and processing of information.

6.4.2. Implications for Learning and Teaching

The research findings revealed that students generally had a well informed understanding of the

skills associated with mapping knowledge. The generic mapping skills of using keys, compass

points, coordinates and landmarks seem well established with students of this age, and therefore

the foundations are present. By contrast, students had less knowledge of the specific mathematics

concepts associated with location, movement and direction. More challenging map interpretation

is required in order to better establish these important mathematics concepts. Thus, it is

important that teachers explicitly address these mathematics understandings in relation to map

139

tasks in order for more students to move toward the second level of Wiegend’s (2006) map

reading ability, namely processing and sequencing information.

A related implication concerns the state syllabus document. It is interesting to note that in the

NSW mathematics syllabus (Board of Studies NSW, 2002), Grades 3-4 (Stage 2) students are

expected to interpret and decode maps at Wiegand’s (2006) first level of map reading ability

however scant attention is given to more complex forms of map reading in Grades 5-6 (Stage 3).

Thus, the curriculum addresses the general mapping skills but not the more complex

mathematics concepts associated with multiple directions, locations and movements. Such an

instructional framework is concerning given the level of complexity required for students to

interpret mathematics mapping tasks within national assessment programs. In other words, the

curriculum requires one level of analysis and yet assessment practices require a higher level.

Therefore, teachers need to challenge students to develop skills that allow them to interpret and

sequence multiple forms of information. One way to do this would be to encourage students to

create their own maps and pose problems for others to solve (Silver, 1994). These problem

posing situations allow for open-ended task development and are more likely to introduce and

present directions and representations in multiple forms. These maps could include “treasure

maps” where students follow compass direction in conjunction with an ordinal sequence. For

example, the instructions for the map could be “Go North for 7 steps and turn right at the third

tree”. Such multiple processing allows students to combine general mapping knowledge with the

more specific mathematics knowledge associated with maps. Such engagement can promote a

variety of meaningful learning situations and certainly has the potential of improving primary-

aged students’ ability to interpret maps.

Classroom teachers should also be encouraged to watch how students solve map tasks. Although

teachers should be well equipped at assessing students work samples, assessments tasks and

many other holistic dimensions of their learning, this study has revealed the importance of

noticing when students use gesture to solve map tasks. While successful students tend to employ

the same approaches irrespective of whether they gesture or not, the act of gesturing does

highlight particular aspects of learning and spatial development. In their theory on dynamical

growth in mathematical understanding, Pirie and Kieren (1994) maintained that students’

140

understandings become more sophisticated when they become less reliant on gesturing.

Gesturing is an important tool for navigating spatial relationships however at some stage these

concrete supports need to become more abstract. As an example, for a student to remember “how

to turn left” a teacher may encourage the student to “shake their wrist which has their watch on

it” as a cue to distinguish left from right. Sooner or later the student needs to recognise direction

without this gestural (concrete) support. Nevertheless, this study has shown how useful gesturing

can be for interpreting spatial information. Hence, teachers should encourage students to utilise

gesturing but to also appreciate that students need to move beyond reliance on such behaviours

in order to move toward more sophisticated understandings of map tasks.

6.4.3. Implications for Test Designs and Curriculum Design

The findings of the study also present implications for test design. It was evident that the

students in the study were at ease decoding the different types of maps which included a pictorial

map, a coordinate map and a street map. These distinct representations of map graphics had quite

different map features, and various types of objects, images and elements to decode. The

majority of students were able to navigate around these representations and locate specific

objects without difficulty. Thus, the different forms of map representation did not appear to

influence whether or not students were able to successfully solve the tasks. The most influential

aspects of task success seemed to involve students attempting to follow multiple instructions in a

task and the requirement to navigate space by processing and sequencing information. Therefore,

the degree of difficulty for each question was more to do with the number of instructions the

student had to follow than anything else. It would be worthwhile for test designers to construct

assessment items which actually required specific decoding skills across the respective type of

map representations as a way of assessing the capacity to interpret maps.

In terms of implications for curriculum design, there needs to be an increased concentration of

mapping activities in the last two years of primary school (Board of Studies NSW, 2002 [Stage

3]). This is necessary for two reasons since: (1) assessment practices require this level of

analysis; and (2) the interpretation of maps is increasingly necessary in out-of-school contexts.

Curriculum documents should extend the complexity of mapping activities in these grades so

that levels of understanding required in Wiegand’s (2006) second level of map reading ability

141

can be developed. At present, the NSW syllabi (Board of Studies NSW, 2002) plateaus at the

first level of map reading ability with students not required to make connections between general

map knowledge and relevant concepts associated with location and direction. Students of this age

are more likely to engage with maps than ever before especially when you consider the number

of maps embedded in computer games, on the internet and the general exposure students receive

from GPS devices. Curriculum designers have the opportunity to ensure that authentic map

activities are presented to students through learning activities which challenge them to order and

sequence information in map tasks.

6.5. Avenues for Further Research

Four research issues arose from the study which warrants further exploration. First, one of the

major findings of the study, that students’ gesture use increased as tasks became more complex,

should be explored in more detail. To this point in time, there has not been detailed research on

the relationship between gesture use and students’ performance in mathematics (Radford, 2009),

with most research on gesture devoted to the connection between language (communication) and

gesturing. For example, Goldin-Meadow and colleagues (Goldin-Meadow, 2000; Goldin-

Meadow, Nasbaum, Kelly & Wagner, 2001) have focused on the mismatch between the meaning

expressed from language and the meaning exhibited through gesture. This Master’s study

concentrates on the influence of gesture on students’ sense making and consequently adds to the

research literature. Given the obvious link between gesture use and task complexity, further

research should focus on the connection between how students engage with spatial mathematics

tasks and the extent to which they make sense of tasks as they become more complex or novel. It

is important that further research in this area considers gestural behaviours in relation to different

types of spatial tasks, beyond that of map tasks, in order to determine whether gestural use is a

support mechanism for activities that do not require spatial navigation and movement. These

tasks could include activities which require the interpretation of other graphics and could also

move beyond the two-dimensional representation of graphic and explore three-dimensional

virtual worlds.

Second, it would be worthwhile to interrogate students’ sense making on mapping tasks from an

internal representation perspective. This research could include more detailed analysis of

142

students’ thinking in terms of visualisation and the extent to which students both visualise

(internal representations) and gesture (external representations) as they solve mapping tasks.

Other researchers (e.g., Lowrie & Hill, 1997; Pirie & Kieran, 1994) have noted that visual

imagery is most effective in situations where students are unable to routinely or symbolically

solve a task. The notions of problem-solving preference (Lowrie & Hill, 1997) or folding back to

more concrete representations (Martin, 2008; Pirie & Kieren, 1994) resonate with the findings of

this study in terms of students’ gesture use on map tasks. Consequently, studies that build on the

current study could determine students’ preference for using gesture on particular types of

graphics tasks, and monitor the way students utilise gesturing in situations where they cannot

effectively solve tasks using non-gestural approaches. These future studies should encourage

participants to verbalise their thinking and include explicit questioning as a way of evoking

internal representations. They should also establish research designs that monitor students’ sense

making across a variety of spatial tasks of varying complexity.

Third, a mixed-method study which utilises a pre- and post-test design for the quantitative

aspects of the study could be explored. This design would allow for an experimental design with

a control group and a treatment group. The treatment would be a training program which

involved explicit instruction regarding the benefit and nature gestural behaviours on mapping

tasks. Another design could involve participants solving map tasks of their own accord in the

first instance and then establishing a set of protocols where they are required to either gesture or

specifically not gesture (visualise) as they solve such tasks. In both designs, student performance

when gesturing or visualising would be measured against non-treatment measures. Related

designs could also encourage students to provide other external representations (e.g., drawing) to

solve tasks rather than gesturing. These studies would provide further evidence for the resilience

of Pirie and Kieran’s (1994) Model of dynamical growth of mathematical understanding.

Finally, from a qualitative perspective, future research could consider student engagement with

different types of map tasks and indeed different types of graphics tasks. It would be beneficial

to ensure that interview techniques allowed for both semi-structured and open-ended questioning

in order to ensure that participants’ internal representations were considered. Students could also

be encouraged to pose their own problems in relation to tasks they are solving as a way of

143

determining students’ level of understanding on such tasks. By using such procedures students’

representational thinking would not be predetermined. The analysis of these data could then be

triangulated with data produced from the alternate quantitative designs to produce a more

detailed understanding of when and why students revert back to more concrete visual processing

on particular types of mapping tasks.

6.6. Chapter Summary

Maps represent space in a two-dimensional form through the use of mathematics conventions

which include coordinate grids, pictorial maps with landmarks, and street map grids. Students

were untroubled by the various representations displayed in the respective tasks and typically

had a sound understanding of the mapping skills required to interpret map tasks. When students

encountered difficulties in decoding graphics information it was predominantly because they

were unable to process the mathematics concepts associated with the tasks. These concepts

included the ability to sequence direction and movement within the map. Gestural behaviours

were particularly useful in situations where students were encountering such challenges, with the

act of gesturing often allowing the students to monitor their approaches. Gesturing allowed the

students to use a concrete tool to assist in processing information. This study builds on previous

work in the field by highlighting the important role(s) multiple forms of communication have in

students’ task solutions and explanations. These verbal and non verbal behaviours provide useful

mechanisms to evaluate students’ interpretation of map tasks.

144

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Appendices

Appendix A. Mass Testing Protocol

Protocol for Mass testing in 2nd year of study

My name is XXX and I work out at XXX University.

At the University, we are finding out about how much children know about diagrams. Diagrams

can be things like different kinds of graphs, maps, flowcharts etc.

The test you are about to complete is called the GLIM (Graphical Languages in Mathematics)

test and you completed this test last year in year 4 as a part of a research project that we are

doing out at the University. You will also be asked to complete the test in Year 6 next year. The

diagrams or pictures that you will see in the booklet have been taken from tests that primary

school children are often asked to do. However, what you do in this test has nothing to do with

school and won’t be used in your school report. The information that we gain from you doing

this test, will hopefully help us to help teachers teach their students about how to use different

types of diagrams.

• Students use their own pencil and eraser • Put an example of how to complete the front of the booklet up on the board • Ask students to have something they can read if they complete the test early • On question 28, bring students attention to the typo and ask them to cross out ‘row’ and

change it to ‘road’

Tell the students the following;

- Read each question very carefully - Only mark one answer on each question - Double check your work when you have completed the test - When you have completed the test, read your book quietly and I will come and check

your work - You have an hour to complete the test.

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Appendix B. Interview Protocol for GLIM

GLIM Interview B1 - 2nd year of interviews Tasks 1 6

Introduction Welcome the student and thank him/her for coming. Introduce yourself, “My name is xxxxx and

I work at Charles Sturt University or CSU.”

Explain the project. “At the university we are finding out about how much children know about

diagrams. Diagrams are special types of pictures.” Show them some examples of diagrams and

ask if they remember doing the interview last year.

“I am going to be recording and video taping whilst you do the interview. I do this to help me

remember what you said and did. We will use the information you tell us to help teachers to

teach their students about how to use different kinds of diagrams.”

Explain that the interview has nothing to do with school or their reports. Highlight the point, “It

doesn’t matter if you get the question right or wrong, what is important is that you are able to

explain to me how you worked the answer out.”

Ask if the student has any questions and if they are happy to participate. If they are, Start

interview, if not, take the child back to class and tell them if they change their mind to let you

know.

The Interview – Part A

Turn on the video and tape recorder. Say “This is (name) from (class) and we are doing interview

B1 on (date). Stop audio tape.

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“Today we are going to be looking at different sorts of tasks. All these tasks have words and

some have words and numbers. They all have a diagram or a picture.”

Show an example of task from GLIM test. “Which part of this task do you think is the diagram

(or picture)? Show another one, if they have no problem with this move on to the interview

booklet.

“I would like you to work out the answers to tasks 1 and 2 and then we will come back and have

a talk about them. You can use the space under the task or on the spare page if you need to do

any working out.”

When the child has completed the two tasks, start the audio tape and say “(name) could you read

out task 1 and tell me how you worked it out.” Use open questions as much as possible, see sheet

for some suggestions, and try to gain as much information as possible without putting the child

under any undue pressure. If the child has done any working out ask them about it and describe

what they have drawn and where it is on the page.

“That’s a great explanation/idea” or something similar. Let’s look at task number 2 now. How

did you work this one out?”

“That is great (name). Is there anything else you would like to tell me?”

“Now, out of the two tasks, which one did you find harder to work out?

“What made it hard for you?” Use open questions to elicit the child’s thoughts. Avoid leading

questions. Continue on with tasks 3&4 following this procedure, then tasks 5&6.

The Interview – Part B

After all six tasks have been completed, refer to tasks 1 6 only, “Have you seen any of these

types of diagrams or pictures before? Can you remember where? Would you be able to draw an

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example for me?” Turn to the spare page at the back of the booklet. Encourage them to draw

something but if they won’t don’t worry too much.

Next section

Ask the child about the chosen task from one of the six languages, e.g. axis, opposed position or

retinal list. “Let’s look at task X again.”

“How do you think this task could be made easier?”

“How do you think this task could be made harder?”

If they are reluctant to answer or are having trouble, suggest “If you were going to give this task

to a friend to answer, what could you do to it to make it easier/harder?”

“Is there anything we’ve done here today that you’d like to tell me more about?”

Thank the child for coming and doing the interview and let them know we will be doing second

part of interview either tomorrow or next week.

End of Interview Stop tape and video and note the counter on the video. Complete details on the cover of the

booklet and any details on video and audio tape covers. Fast forward audio tape so side B is

ready for interview B2.

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GLIM Interview B2 - 2nd year of interviews Tasks 7 12

Before student arrives

Complete part 2 on the front of the booklet including video counter. Select the child’s video and

have ready in camera. Select child’s audio tape and have ready on side B. have booklet open at

task 7.

Introduction

Welcome student and thank them for coming.

“This second interview is very similar to the first one. We are going to be looking at tasks 7

through to 12 this time.”

The Interview – Part A

Turn on the video and tape recorder and say “This is (name) from (class) and we are doing

interview B2 on (date).” Stop audio tape.

The same procedure as first interview, complete tasks 1 & 2, then come back ask them how they

worked it out, use open questions as much as possible.

Ask which of the two tasks was the harder one? What made it hard for the student?

Continue on for tasks 3 & 4 and 5 & 6.

The Interview – Part B

After all six tasks have been completed refer to tasks 7 12 only. “Have you seen any of these

types of diagrams or pictures before? Can you remember where? Would you be able to draw an

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example for me?” Turn to the spare page at the back of the booklet. Encourage them to draw

something but if they won’t don’t worry too much.

Next section

Ask the child about the chosen task from one of the six languages, e.g. map, connection,

miscellaneous “Let’s look at task X again.”

“How do you think this task could be made easier?”

“How do you think this task could be made harder?”

If they are reluctant to answer or are having trouble, suggest “If you were going to give this task

to a friend to answer, what could you do to it to make it easier/harder?”

“Is there anything we’ve done here today that you’d like to tell me more about?”

“Thank you so much for helping with the graphics project this year. We will be doing another

two interviews next year when you are in Year 6. This will be the final lot of interviews for the

project. I look forward to seeing you again then.”

End of Interview Stop tape and video and note the counter on the video. Complete details on the cover of the

booklet and any details on video and audio tape covers.

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Appendix C. Information Package for Parents

INFORMATION PACKAGE FOR PARENTS/ GUARDIANS OR CAREGIVERS

Project title: How primary school students become code-breakers of information graphics in

mathematics.

During 2005 to 2007, Associate Professor Carmel Diezmann, Queensland University of

Technology and Associate Professor Tom Lowrie, Charles Sturt University and staff from QUT

and CSU will be undertaking a mathematics research project that focuses on children’s

understanding of graphics. This research project is federally funded by the Australian Research

Council. The aim of this project is to understand how primary students learn about general

purpose graphical languages that are important in mathematics (eg graphs, diagrams, charts and

maps). During the first year of the project the students will undertake a mathematical test based

on graphical languages (GLIM test) and a spatial abilities test based on Raven’s Standard

Progressive matrices. The following two years will involve a GLIM test per annum. The

estimated time per test will not exceed two hours.

In 2005, Year 5 students (QLD) and Year 4 students (NSW) will participate in the study. These

students will continue to participate in the study during 2006 and 2007. Over the three-year

period the children will:

• solve mathematical problems • be video-taped or audio-taped whilst describing how they solved these problems in an

interview. The interviews will be conducted at their school by research staff from either Queensland

University of Technology or Charles Sturt University. Video-tapes, audio-tapes, photographs or

work samples from the project may be used in reporting the outcomes of this research, in

curriculum materials or in teacher education programs.

The results of this study will provide a comprehensive understanding of the development of

students mathematical graphics skills during the primary years and the influence of outside

experiences that contributes to this knowledge. Knowledge of this graphical development will

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help teachers and educators design ways to improve the teaching of mathematics in the primary

years as well as assist in the selection and design of appropriate print and electronic resources to

support children’s mathematical learning.

Your child’s participation in this research study is voluntary and he or she may withdraw at

any time without comment or penalty. There are no out-of the ordinary risks associated with

this research and there will be no discomfort to your child. In all reporting of the research

and any publications, the identity of the children and the school will be anonymous unless

prior written permission has been granted.

The universities require informed consent for all participants and we are seeking consent for

your child to participate in the study. The tests and interviews will be administered by QUT

(Qld) or CSU (NSW) staff in consultation with the class teacher. Any queries or questions

about this project should be directed to the Chief Investigators:

Professor Carmel Diezmann Professor Tom Lowrie

Faculty of Education Head School of Education

Queensland University of Technology Charles Sturt University

Victoria Park Road PO Box 588

KELVIN GROVE 4059 WAGGA WAGGA 2678

Ph: 07-3864 3803 Ph: 02-6933 2440

[email protected] [email protected]

Both Queensland University of Technology and Charles Sturt University’s Ethics in

Human Research Committees have approved this study. If you have any complaints or

reservations about the ethical conduct of this project, you may contact the Committee through

the Executive Officer:

• Qld: QUT Research Ethics Officer, Office of Research, Queensland University of Technology, GPO Box 2434, Brisbane 4001 (phone 07 3864 2340 or fax: 07 3864 1304) OR

• NSW: CSU Executive Officer, Ethics in Human Research Committee, Academic Secretariat, Charles Sturt University, Private Mail Bag 29, Bathurst NSW 2795 (ph: 02

6338 4628 or fax: 02 6338 4194)

Students will not receive feedback on their results during the project. However results will be

provided to parents, guardians or caregivers at the conclusion of the project, if requested.

Thank you for considering your child’s participation in this study. This project has the support

of your school principal and your child’s class teacher. If you agree to your child’s participation

in this project, please complete the relevant section of the accompanying Parent/Guardian or

Caregiver Consent Form and return to your child’s teacher by the nominated date.

Yours Sincerely,

Associate Professor Tom Lowrie

23/05/05

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Appendix D. The Use of Chi Square Procedures

Chi square analysis was an appropriate procedure to determine potential differences between two

variables since the data were coded dichotomously and categorically. Since the total sample was

less than 50 participants, Hair, Anderson, Tatham and Black (1998) suggested that the Fisher’s

exact test be undertaken in order to ascertain relationships on categorical data that result from

classifying objects in two different ways—that is, correctness and purposeful gesture use. This

procedure allows for an analysis to be undertaken which examines the significance of the

association between task correctness and gesture use. This relationship was measured through a

probability value, with a p = .05 level used for determining statistically significance.