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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)
i
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
ii
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.
iii
Keywords
Mathematics education, maps, map reading, graphics tasks, gestural behaviours,
assessment, spatial reasoning
iv
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
v
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
vi
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
vii
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
viii
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
ix
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
x
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
xi
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
xii
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.
xiv
xv
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
1
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.
3
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
4
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.
42
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
58
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.
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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)
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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
120
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
127
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
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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.