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Deborah Jean Priest
M. Ed. (QUT), BSc Ed. (Melb.)
Faculty of Education
Queensland University of Technology
Kelvin Grove Campus, Brisbane.
A Thesis submitted in fulfilment of the requirements leading to the award of
the degree of Doctor of Philosophy
May 2009
ii
CERTIFICATE RECOMMENDING ACCEPTANCE
iii
DEFINITION OF ACRONYMS
ACER Australian Council for Educational Research
ANTA Australian National Training Authority
DEST Department of Education, Science and Training
GSA Graduate Skills Assessment Test
IQ Intelligence Quotient
MCEETYA Ministerial Council on Education, Employment, Training and
Youth Affairs
MYAT Middle Years Ability Test
NAPLAN National Assessment Program: Literacy and Numeracy
NCB National Curriculum Board
NCTM National Council of Teachers of Mathematics
(United States of America)
NNR National Numeracy Review
NMAP National Mathematics Advisory Panel
(United States of America)
NRC National Research Council (United States of America)
POPS Profiles of Problem Solving Test
iv
KEYW ORDS
assessment
cognition
developmental learning
education
engagement
intervention
mathematics
middle year
multiple intelligences
pedagogy
problem solving
problem posing
self-regulation
teaching experiment
underachievement
iv
ABSTRACT
This study reported on the issues surrounding the acquisition of problem-
solving competence of middle-year students who had been ascertained as
above average in intelligence, but underachieving in problem-solving
competence. In particular, it looked at the possible links between problem-
posing skills development and improvements in problem-solving
competence.
A cohort of Year 7 students at a private, non-denominational, co-educational
school was chosen as participants for the study, as they undertook a series
of problem-posing sessions each week throughout a school term. The
lessons were facilitated by the researcher in the students’ school setting.
Two criteria were chosen to identify participants for this study. Firstly, each
participant scored above the 60th percentile in the standardized Middle Years
Ability Test (MYAT) (Australian Council for Educational Research, 2005) and
secondly, the participants all scored below the cohort average for Criterion B
(Problem-solving Criterion) in their school mathematics tests during the first
semester of Year 7.
Two mutually exclusive groups of participants were investigated with one
constituting the Comparison Group and the other constituting the Intervention
Group. The Comparison Group was chosen from a Year 7 cohort for whom
no problem-posing intervention had occurred, while the Intervention Group
was chosen from the Year 7 cohort of the following year. This second group
received the problem-posing intervention in the form of a teaching
experiment. That is, the Comparison Group were only pre-tested and post-
tested, while the Intervention Group was involved in the teaching experiment
and received the pre-testing and post-testing at the same time of the year,
but in the following year, when the Comparison Group have moved on to the
secondary part of the school. The groups were chosen from consecutive
Year 7 cohorts to avoid cross-contamination of the data.
v
A constructionist framework was adopted for this study that allowed the
researcher to gain an “authentic understanding” of the changes that occurred
in the development of problem-solving competence of the participants in the
context of a classroom setting (Richardson, 1999). Qualitative and
quantitative data were collected through a combination of methods including
researcher observation and journal writing, video taping, student workbooks,
informal student interviews, student surveys, and pre-testing and post-
testing. This combination of methods was required to increase the validity of
the study’s findings through triangulation of the data.
The study findings showed that participation in problem-posing activities can
facilitate the re-engagement of disengaged, middle-year mathematics
students. In addition, participation in these activities can result in improved
problem-solving competence and associated developmental learning
changes. Some of the changes that were evident as a result of this study
included improvements in self-regulation, increased integration of prior
knowledge with new knowledge and increased and contextualised
socialisation.
vi
TABLE OF CONTENTS
Certificate ____________________________________________________ ii
Definition of Acronyms __________________________________________ iii
Keywords ____________________________________________________ iv
Abstract ______________________________________________________ v
List of Tables__________________________________________________ x
List of Figures ________________________________________________ xii
List of Appendices ____________________________________________ xiii
Statement of Authenticity _______________________________________ xiv
Acknowledgments _____________________________________________ xv
Chapter 1 - Introduction to the Research Study
1.1 INTRODUCTION ___________________________________________ 16
1.2 DEFINITION OF TERMS _____________________________________ 16
1.3 RATIONALE FOR THE STUDY _________________________________ 18
1.3.1 Summary ____________________________________________ 25
1.4 BACKGROUND TO THE STUDY ________________________________ 26
1.4.1 The Value of Problem Solving in Today’s Society _____________ 27
1.4.2 The Place of Problem Posing in a Responsive Curriculum ______ 28
1.4.3 Disparity in Student Mathematical Performance ______________ 31
1.5 PURPOSE OF THIS STUDY ___________________________________ 32
1.6 SIGNIFICANCE OF THE RESEARCH _____________________________ 32
1.7 THESIS OVERVIEW ________________________________________ 34
Chapter 2 - Theoretical Perspectives
2.1 CHAPTER OVERVIEW ______________________________________ 36
2.2 UNDERSTANDING DEVELOPMENTAL LEARNING ____________________ 36
2.2.1 Information Processing Theory ___________________________ 39
2.2.2 Psychometric Theory ___________________________________ 42
2.2.3 Multiple Intelligences Theory _____________________________ 50
2.2.4 Summary ____________________________________________ 51
2.3 PROBLEM-SOLVING PERSPECTIVES ____________________________ 53
2.3.1 Introduction ___________________________________________ 53
2.3.2 The Power of Teaching through Problem Solving _____________ 56
2.3.3 Can Problem Solving Drive Mathematical Reform? ___________ 57
vii
2.3.4 Issues Related to the Assessment of PSC __________________ 58
2.3.5 Should Specific Problem-solving Strategies be Taught? ________ 59
2.3.6 Student's Understandings of Problem Structures _____________ 60
2.3.7 Summary ____________________________________________ 63
2.4 PROBLEM-POSING PERSPECTIVES ____________________________ 65
2.4.1 Introduction ___________________________________________ 66
2.4.2 Problem Posing as a Tool for Mathematical Reform ___________ 66
2.4.3 Problem-posing Skills for Lifelong Learning _________________ 68
2.4.4 Fostering a Problem-posing Environment ___________________ 70
2.4.5 Connections between Problem Solving and Problem Posing ____ 71
2.4.6 Summary ____________________________________________ 75
2.5 STUDENT UNDERACHIEVEMENT PERSPECTIVES ___________________ 76
2.6 CONSTRUCTIONIST PERSPECTIVES ____________________________ 80
2.7 CONCLUSION ____________________________________________ 83
Chapter 3 - Research Design
3.1 CHAPTER OVERVIEW ______________________________________ 87
3.2 INTRODUCTION ___________________________________________ 87
3.3 RESEARCH QUESTIONS ____________________________________ 92
3.4 RESEARCH DESIGN _______________________________________ 93
3.4.1 Research Design Rationale and Structure __________________ 93
3.4.2 Participants ___________________________________________ 96
3.5 METHODS _____________________________________________ 102
3.5.1 Data Collection _______________________________________ 105
3.5.2 Instruments __________________________________________ 111
3.5.2.1 The Middle Years Ability Test (MYAT) __________________ 111
3.5.2.2 The Profiles of Problem Solving (POPS) Test ____________ 113
3.5.2.3 The Student Survey ________________________________ 115
3.5.2.4 The Problem Criteria Sheet __________________________ 116
3.5.3 Data Analysis ________________________________________ 117
3.5.3.1 Researcher Journal ________________________________ 118
3.5.3.2 Student Surveys ___________________________________ 118
3.5.3.3 Student Workbooks ________________________________ 120
3.5.3.4 Researcher Observations ___________________________ 120
3.5.3.5 Informal Interviews _________________________________ 121
3.5.3.6 The Profiles of Problem Solving (POPS) Test ____________ 122
3.5.4 Reliability and Validity Issues ____________________________ 123
viii
3.5.5 Ethical Issues ________________________________________ 125
3.6 CONCLUSION ___________________________________________ 127
Chapter 4 - The Teaching Experiment
4.1 CHAPTER OVERVIEW _____________________________________ 129
4.2 THE PHILOSOPHICAL UNDERPINNINGS AND STRUCTURE OF THE
TEACHING EXPERIMENT____________________________________ 130
4.2.1 The Philosophical Underpinnings of the Teaching Experiment 130
4.2.2 The Structure of the Teaching Experiment _________________ 133
4.3 THE PRE-TEST AND POST-TEST LESSONS ______________________ 134
4.3.1 Introduction __________________________________________ 134
4.3.2 First Lesson - Pre-test and Initial Survey ___________________ 136
4.3.3 Last Lesson - Post-test and Final Survey __________________ 137
4.4 THE SEVEN TEACHING EPISODES (LESSONS 2-8) ________________ 137
4.4.1 The First Teaching Episode - Lesson 2 ____________________ 138
4.4.2 The Second Teaching Episode - Lesson 3 _________________ 139
4.4.3 The Third Teaching Episode - Lesson 4 ___________________ 143
4.4.4 The Fourth Teaching Episode - Lesson 5 __________________ 143
4.4.5 The Fifth Teaching Episode - Lesson 6 ____________________ 144
4.4.6 The Sixth Teaching Episode - Lesson 7 ___________________ 146
4.4.7 The Seventh Teaching Episode - Lesson 8 _________________ 148
4.5 CONCLUSION ___________________________________________ 148
Chapter 5 - Reporting and Analysis of the Data
5.1 CHAPTER OVERVIEW _____________________________________ 150
5.2 OBSERVATIONS AND INTERVIEWS WITH THREE CASE STUDY STUDENTS 150
5.2.1 Paul _______________________________________________ 152
5.2.2 Andrew _____________________________________________ 161
5.2.3 Nicole ______________________________________________ 170
5.3 STUDENT SURVEYS ______________________________________ 176
5.3.1 Question One ________________________________________ 176
5.3.2 Question Two ________________________________________ 179
5.3.3 Question Three_______________________________________ 182
5.3.4 Question Four ________________________________________ 184
5.4 PROFILES OF PROBLEM SOLVING TEST - THE PRE-TEST AND THE
POST-TEST _____________________________________________ 186
5.4.1 Descriptive Analysis of the POPS Test Results ______________ 190
5.4.2 Paired Samples T-Test Results __________________________ 195
ix
5.4.3 Analysis of Improvement of Scores from the Pre-test and the
Post- test ___________________________________________ 198
5.5 CONCLUSION ___________________________________________ 201
Chapter 6 - Responses to the Research Questions
6.1 CHAPTER OVERVIEW _____________________________________ 204
6.2 RESEARCH QUESTION 1 ___________________________________ 204
6.3 RESEARCH QUESTION 2 ___________________________________ 207
6.4 RESEARCH QUESTION 3 ___________________________________ 209
6.5 CONCLUSION ___________________________________________ 213
Chapter 7 - Limitations and Implications for Future Research
7.1 CHAPTER OVERVIEW _____________________________________ 215
7.2 LIMITATIONS OF THE STUDY ________________________________ 215
7.2.1 Limitations in the Selection of Students ____________________ 216
7.2.2 Limitations in the Timing of the Research __________________ 217
7.2.3 Limitations of the Size of the Control and Intervention Groups __ 218
7.2.4 Limitations of the Withdrawal of Students from their Usual
Classroom Environment ________________________________ 219
7.2.5 Limitations in the Length of the Problem-posing Intervention ___ 220
7.2.6 Limitations of Question Three of the Student Survey _________ 222
7.3 IMPLICATIONS OF THE RESEACH _____________________________ 222
7.4 CONCLUDING COMMENTS __________________________________ 223
REFERENCES _____________________________________________ 225
APPENDICES ______________________________________________ 252
x
L IST OF TABLES
Table 1.1 Percentage Comparison of How Time is Allocated in
Year Eight mathematics Classrooms in Germany, the
United States and Japan
23
Table 1.2 Comparative Problem-solving Scale Scores from the
2003 PISA Test
24
Table 1.3 Overall Combined mathematical Literacy Scores from
the 2003 PISA Test
25
Table 1.4 The Eight Skill Groupings of the Employability Skills
Framework
29
Table 2.1 Comparison of Stage Development in Cognitive
Development Theories
39
Table 2.2 Spearman’s Correlations of Student Scores Between
Subjects
45
Table 3.1 Data Used to Respond to the Three Research
Questions of the Study
107
Table 4.1 Variations to Pre-arranged Lesson Times in 2007 135
Table 5.1 Paul’s Profiles of Problem Solving Pre-test and Post-
test Results
153
Table 5.2 Andrew’s Profiles of Problem Solving Pre-test and
Post-test Results
163
Table 5.3 Nicole’s Profiles of Problem Solving Pre-test and Post-
test Results
171
Table 5.4 Do you enjoy solving problems? 177
Table 5.5 What type of problems do you prefer to solve? 180
Table 5.6 Do you think learning to solve problems is a useful
thing to do?
183
Table 5.7 What things could teachers do to assist you to become
better at solving problems?
185
Table 5.8 Comparison Group Pre-test and Post-test results 188
Table 5.9 Intervention Group Pre-test and Post-test results 189
xi
Table 5.10 Mean Score and Standard Deviation Statistics for each
Aspect of the Profiles of Problem Solving Test for
Students in the Comparison and Intervention Groups
191
Table 5.11 Paired Samples Test for each Subscale of the Profiles
of Problem Solving Test for Students in the
Comparison and Intervention Groups
197
Table 5.12 Numbers of Improvements in Individual Aspect Scores
of Comparison and Intervention Group Students, from
the Pre-test to the Post-test
198
xii
List of Figures
Figure 2.1 A Schematic Diagram of Sternberg’s Triarchic Theory
of Intelligence
42
Figure 3.1 Research Study Framework 95
xiii
L IST OF APPENDICES
Appendix A Project Information Sheet and Parent Consent Form
for Comparison Group
253
Appendix B Project Information Sheet and Parent Consent Form
for Intervention Group
258
Appendix C Student Survey Sheet 263
Appendix D Teaching Experiment Lesson One 266
Appendix E Teaching Experiment Lesson Two 270
Appendix F Teaching Experiment Lesson Three 275
Appendix G Teaching Experiment Lesson Four 280
Appendix H Teaching Experiment Lesson Five 288
Appendix I Teaching Experiment Lesson Six 295
Appendix J Teaching Experiment Lesson Seven 302
Appendix K Teaching Experiment Lesson Eight 309
Appendix L Teaching Experiment Lesson Nine 314
Appendix M Profiles of Problem Solving Assessment Instrument
(Stacey, Groves, Bourke, & Doig, 1993)
317
Appendix N Problem Criteria Sheet 327
Appendix O Participant Pseudonym Code to Psuedonym Name
Conversion for Comparison Group
329
Appendix P Participant Pseudonym Code to Psuedonym Name
Conversion for Intervention Group
331
Appendix Q Marking Scheme for Profiles of Problem Solving Test 333
xiv
STATEMENT OF AUTHENTICITY
The work contained in this document has not previously been submitted for a
degree or diploma at any other higher education institution. To the best of my
knowledge and belief, the document contains no material previously
published or written by another person except where due reference is made
in the document itself.
Deborah Jean Priest
May 2009
15
ACKNOW LEDGEMENTS
I wish to acknowledge the valuable and ongoing support I have received from
Professor Lyn English in the first instance, and also Dr Mal Shield and Associate
Professor Rod Nason. In particular, I wish to thank Professor Lyn English and
Dr Mal Shield for their substantial guidance and encouragement that has been
instrumental in the completion of my PhD journey. In addition, I would like to
thank Dr Mark Bahr for his assistance in becoming familiar with the Statistical
Package for Social Sciences software (SPSS Inc., 2007).
I would also like to thank the Year 7 teachers, the Deputy Principal in charge of
the Year 7 students at the research school, and the Principal for allowing me to
work with their students. I would like to acknowledge you all by name but am
unable to do so as the participants in this study may be more readily identified as
a result. Please accept my deepest appreciation for your cooperation and
assistance.
As those who have previously completed their PhD journeys will fully understand,
there are many activities that must be set aside in order to find the necessary
time to undertake and complete detailed research such as that reported in this
document. My journey to completion would not have been possible without the
understanding of my husband, John and my two daughters Megan and Ashley.
Their patience has been greatly appreciated, cannot be understated and will be
rewarded in the future.
16
Chapter 1
Introduction to the
Research Study
1.1 Chapter Overview
Seven main sections comprise this chapter. The first section is a definition of
terms used frequently throughout this report (see Section 1.2), while the second
section introduces the rationale that led to the overarching question for this study
(see Section 1.3). The third section provides some preliminary background to
the research study including discussion about the value of problem solving and
problem posing in a contemporary mathematics curriculum and introduces the
concept of disparity between a student’s actual mathematical performance and
their potential performance (see Section 1.4). Section four of this chapter
introduces the three research questions investigated in this study (see Section
1.5) while the fifth section considers the significance of this research (see Section
1.6). The final section presents an overview of the chapters in this report (see
Section 1.7).
1.2 Definition of Terms
The following terms, with their associated meaning, are used frequently
throughout this report:
Cognition “refers to the processes or faculties by which knowledge is acquired
and manipulated” (Bjorklund, 2000, p. 3) and “includes conscious, effortful
processes such as those involved in making important decisions and
17
unconscious, automatic processes, such as those involved in recognizing a
familiar face, word or object” (pp.19, 20).
Developmental learning changes refer to cognitive (e.g., Goswami, 2002) and
behavioural (e.g., Lesh & Doerr, 2003) changes that can be attributed to an
intervention or experiences that occurred over a period of time.
Engagement refers to the willing participation of students in activities (Ryan &
Patrick, 2001).
Middle years refer to Years 5 - 9 in Australian schools. Students enrolled in
these year levels are most commonly aged between 10 and 14 years.
Problem posing is the act of creating a new problem for oneself or for peers to
solve. The problem may be presented in an oral, written or other visual format
(English, Fox, & Watters, 2005; Lowrie, 2002).
Problem solving occurs when a specific goal exists that cannot be solved
immediately due to the presence of one or more obstacles (DeLoache, Miller, &
Pierroutsakos, 1998). Problem solving is “getting from givens to goals when a
solution path is not readily apparent” and requires the problem solver to recall
information, draw upon previously learned skills, choose appropriate solution
strategies, and express information in a meaningful way. It involves the
acquisition and utilisation of knowledge, metacognition and socio-cultural
contexts (Lesh & Zawojewski, 2007).
Self-regulation refers to a student’s ability to be actively and productively
involved in an activity that does not intentionally distract or interfere with the
learning of other students (Schunk, 2001).
Underachievement occurs when there is a “distance between the actual
developmental level [of a child] as determined by independent problem solving
and the level of potential development as determined through problem solving
18
under adult guidance or in collaboration with more capable peers” (Vygotsky,
1978, p. 86). In this study, underachieving students will be defined to be those
students who achieve above average results in the MYAT test (Australian
Council for Educational Research, 2005) while also achieving lower than the
average results in the problem-solving criterion of their mathematics tests,
compared to their cohort.
1.3 Rationale for the Study
It could be said that today, children resemble their times more than they
resemble their parents. This is not surprising when we consider that our current
times are typified by dynamic advances in technology and a resultant, ever-
changing job market that requires the workforce to embrace flexibility and
creativity. The responsibility to prepare our students to be effective and
productive citizens in such a world is mandated, in part, to the education system
of the day. In response to this mandate, rigorous reviews of the State-based
education systems in Australia have lead the Australian Federal Government to
move towards a national, futures-focussed curriculum that recognises “that
society will be complex, with workers competing in a global market, needing to
know how to learn, adapt, create, communicate, and interpret and use
information critically” (National Curriculum Board, 2008, p.5).
Two reports the Australian National Numeracy Review Report (National
Numeracy Review, 2008) and Foundations for Success: The final report of the
National Mathematics Advisory Panel (National Mathematics Advisory Panel,
2008) from the United States of America, have provided a foundation for
discussion papers leading to the development of an Australian, national
mathematics curriculum. The establishment of this national mathematics
curriculum is a unique opportunity to redefine not only the appropriate curriculum
content, but also to reconsider and redefine the most appropriate pedagogy to
achieve the desired student outcomes.
19
With mathematics education having a long history of marginalising and
disengaging students through traditional teaching practices, one could argue that
a review of teaching practices is timely (English, 2002; Lesh & Zawojewski, 2007;
Skovsmose & Valero, 2002). Currently not all students are being presented with
mathematics curriculums that allow them to draw on their knowledge to solve
meaningful problems that are relevant to them and to society. Indeed, “an
unintended effect of current classroom practice is to exclude some students from
future mathematics study” therefore creating a need to engage more students in
mathematical activities that are connected meaningfully to real-life contexts
(National Curriculum Board, 2008).
Education departments and national curriculum organisations across Australia
have continued to develop policies to promote contemporary teaching practices
to address this concern, with mixed success. The New Basics Framework is an
example of a recent four-year project in Queensland schools that created new
opportunities to connect the curriculum to real-life contexts (Department of
Education Training and the Arts, 2007). The Framework provided an alternative
organisational and conceptual framework for the curriculum and was intended to
reflect the new demands placed on students, and hence on curriculums,
assessment and pedagogy, by the “new times”.
The New Basic’s trial curriculum was organised around four “clusters”; Life
Pathways and Social Futures; Active Citizenship, Multiliteracies and
Communication Media, and Environments and Technologies. Assessment was
adapted from assessing and reporting against students’ learning outcomes,
through traditional pen and paper tests, to student demonstrations of learning
throughout the transdisciplinary “Rich Tasks”. While some Queensland State
schools have continued, in part, to pursue and support this new direction in
curriculum ideology, broad-scale implementation of the Rich Tasks has not
subsequently occurred across all Queensland schools. Reasons given for the
lack of broad-scale implementation included insufficient professional
20
development for teachers and reduced class time available for the development
of basic student literacy and numeracy skills that will now be measured and
compared between the States of Australia (Department of Education Training
and the Arts, 2007). The first national comparison of literacy and numeracy skills
between students in different States took place in May 2008 as part of the
National Assessment Program Literacy and Numeracy (NAPLAN) (Ministerial
Council on Education, Employment, Training, & Affairs, 2008a).
Despite the Queensland Government’s decision not to proceed with the full
implementation of the New Basics Framework, curriculum organisations are still
calling for meaningful connections to be made between school-based
curriculums and real-life contexts (Department of Education Training and the
Arts, 2007). In 2008, the National Curriculum Board of Australia opened public
discussion about what is important in the teaching of mathematics, by publishing
for public comment, papers about a nationally administered mathematics
curriculum. One such paper, The National Mathematics Curriculum: Framing
Paper argued that “mathematics is important for all citizens” and that “some
students are currently excluded from effective mathematics study” (National
Curriculum Board, 2008, p. 1). The paper stated that equity of opportunity is a
central goal in the construction of a national mathematics curriculum and
included discussion about how specific groups have been excluded and how to
re-engage more students in the study of mathematics. According to the paper,
the students at most risk of disengagement are students in their middle years of
schooling. The paper suggested the alienation and disengagement of these
students is largely attributed to “irrelevant curriculums”, unconnected to real-life
contexts, and “ineffectual learning and teaching processes” (National Curriculum
Board, 2008, p. 5). The report went on to state that it is “imperative that we
reverse this trend” (p. 5).
The concept of irrelevant curriculums is not new (Secada & Berman, 1999) with
Hollingsworth, Lokan and McCrae (2003) reporting that, in Year 8 mathematics
21
lessons, more than seventy-five percent of the problems provided to students
were low in complexity, emphasised procedural fluency, rather than higher-order
critical thinking, and only twenty-five percent of the problems were connected to
real-life contexts. When looking a little further into the senior years of schooling,
Barrington (2006) reported a drop in student participation rates in Year 12
mathematics classes and the National Numeracy Review (NNR) reported a
decline in tertiary students undertaking substantial studies in mathematics which,
in part, has lead to a national shortage of secondary mathematics teachers
(National Numeracy Review, 2008). These reports have major implications for
educators and in particular, teachers of middle-year mathematics; for it is in the
middle years of schooling that students appear to be forming enduring
dispositions and perceptions about the personal relevance of the study of
mathematics that can lead to underachievement, disengagement or both
(National Curriculum Board, 2008).
According to the National Declaration on Education Goals for Young Australians
draft report, Australia has no “inherent advantage – except through the quality of
education” to prepare students for the “radically evolving and uncertain context”
of future life in a global society (Ministerial Council on Education, Employment,
Training, & Affairs, 2008b, p. 4). This is supported in the National Mathematics
Curriculum: Framing paper where the authors stated that “a fundamental goal of
the mathematics curriculum is to educate students to be active, thinking citizens,
interpreting the world mathematically, and using mathematics to help form their
predictions and decisions about personal and financial priorities” (National
Curriculum Board, 2008, p. 3).
The paper defined, as goals of a national mathematics curriculum, four
proficiency strands;
1. understanding (conceptual understanding);
2. fluency (procedural fluency);
22
3. problem solving (strategic competence) and,
4. reasoning (adaptive reasoning).
It stated that problem-solving competence, including “the ability to make choices,
interpret, formulate, model and investigate problem situations, and communicate
solutions effectively”, is central to ensuring a futures orientation to a national
curriculum (National Curriculum Board, 2008, p. 3). The importance of
developing problem-solving competence was previously discussed by Cai (2003)
during his investigation of Singaporean students’ mathematical thinking in
problem solving and problem posing. Cai stated, following his exploratory study,
that problem solving was the most purposeful activity in the study of
mathematics. Later researchers such as Brown and Walter (2005) suggested
that it was the formulation or posing of problems, more so than the solving of
problems, that was fundamental in the development of mathematical skills.
Previous researchers such as Lowrie (2002) had already undertaken some
research into the usefulness of problem posing and had discovered that, when
used as a regular strategy in the study of mathematics, problem posing had the
potential to increase the engagement of underachieving students.
Shimizu (2002) also considered the engagement of students when he
investigated how the structured problem-solving approach to teaching
mathematics in Japanese schools and its associated impact on how Japanese
students perceive their lessons, compared with the pedagogy used by German
and American mathematics teachers and the perceptions of their students. One
of the differences he noted was that fostering mathematical thinking was the
main goal of mathematics lessons for the majority of Japanese teachers whereas
61 percent of American teachers and 55 percent of German teachers had the
development of mathematical skills as their main goal. A second difference he
discussed was the time spent by Japanese, German and American students on
the practice of routine procedures compared to time spent thinking about
23
mathematical problems and inventing new solutions. His data were taken from a
Third International Mathematics and Science Study (TIMSS) video classroom
study (Stigler, Gonzales, Kawanaka, Knoll, & Serrano, 1999) and can be found in
the Table 1.1.
Table 1.1
Percentage Comparison of How Time is Allocated in Year Eight Mathematics
Classrooms in Germany, the United States and Japan
OECD Country Practising routine
procedures
Thinking about
mathematical problems and
inventing new solutions
Japan 40.8 44.1
Germany 89.4 4.3
United States 95.8 0.7
Note. Adapted from " The TIMSS videotape classroom study: Methods and findings from an exploratory research project on eighth-grade mathematics instruction in Germany, Japan, and the United States” by J.W. Stigler, P. Gonzales, T. Kawanaka, S. Knoll & A. Serrano, 1999, Washington, D.C..
According to the study (Stigler et al., 1999), students in Japan spend less than
half the amount of time practising routine procedures and more than ten times
the amount of time working with problems than do their German and American
counterparts. These statistics become more notable when we consider the
performance of Japanese students compared to American and German students
in The Program for International Student Assessment (PISA) test undertaken by
students in twenty-nine member countries of the Organization for Economic
24
Cooperation and Development (OECD) in 2003 (Lemke et al., 2004). The
average country scores for fifteen-year-old students from Japan, Germany,
Australia and the United States on the problem-solving scale are reported in
Table 1.2 while the overall combined mathematical literacy scores can be found
in Table 1.3. Statistics about Australian students have been included for
comparative purposes.
Table 1.2
Comparative Problem-solving Scale Scores from the 2003 PISA Test
OECD Country Average student score
(average = 500, S.D.=100)
OECD Ranking
N=29
Japan 547 3rd
Australia 530 5th
Germany 513 13th
United States 477 24th
Note. Adapted from "International outcomes of learning in mathematics, literacy and problem solving: PISA 2003 results from the U.S. perspective - highlights” by M. Lemke, A. Sen, E. Pahlke, L. Partelow, D. Miller & T. Williams, 2004, Washington, D.C.: National Center for Educational Statistics.
It could be deduced from the results in Tables 1.1, 1.2 and 1.3 that a
mathematics classroom rich in problem-solving opportunities can not only lead to
enhanced performance on international problem-solving testing instruments, it
can also support the development of mathematical literacy and is therefore
worthy of further research in an Australian school context.
25
Table 1.3
Overall Combined Mathematical Literacy Scores from the 2003 PISA Test
OECD Country Average student score
(average = 500, S.D.=100)
OECD Ranking
N=29
Japan 534 4th
Australia 524 8th
Germany 503 16th
United States 483 24th
Note. Adapted from "International outcomes of learning in mathematics, literacy and problem solving: PISA 2003 results from the U.S. perspective - highlights” by M. Lemke, A. Sen, E. Pahlke, L. Partelow, D. Miller & T. Williams, 2004, Washington, D.C.: National Center for Educational Statistics.
1.3.1 Summary
Four foci arose from the preliminary review of the literature surrounding
mathematics education in Australia and internationally:
1. problem solving
2. problem posing
3. middle years and,
4. underachievement
It has been suggested that posing problems can re-engage underachieving
students (Lowrie, 2002) and that middle-year students are at most risk of being
26
disengaged and underachieving in the study of mathematics (National
Curriculum Board, 2008). Problem posing has been attributed as being an
important skill in the development of problem-solving competence (e.g., Cai,
2003; English, 2003; Silver & Cai, 1993a), which is one of the four proficiency
strands that make up the structure of the new national mathematics curriculum
(National Curriculum Board, 2008). International research has reinforced the
value of a mathematics curriculum, rich in problem-solving, to the student
development of mathematical literacy skills and problem-solving competence
(Lemke et al., 2004; Stigler et al., 1999). In light of these observations and to
progress the reform of mathematical curriculums, the following overarching
question was investigated in this present research study:
How might a problem-posing intervention impact upon the development of
problem-solving competence of underachieving, middle-year students?
The decision to investigate this overarching research question was consistent
with international curriculum documents such as those written by the American
National Research Council (e.g., NRC, 2004), and the National Council of
Teachers of Mathematics (NCTM, 2000) that recommended that teachers
provide regular opportunities for students to pose and solve problems within
meaningful contexts. The results of this present study provide education policy
makers, syllabus writers, and teachers with insights into how underachieving,
middle-year, mathematics students may be assisted to develop problem-solving
competence through a problem-posing intervention (e.g., Bjorklund, 2000; Jones
& Myhill, 2004; Kanevsky & Keighley, 2003).
1.4 Background to the Study
This section considers further background information on problem solving,
problem posing and disparity between student results and their potential that was
used to develop the three Research Questions for this study.
27
1.4.1 The Value of Problem Solving in Today’s Society
Problem solving is widely argued as the most purposeful activity in a
mathematics curriculum (Cai, 2003; Cai & Hwang, 2002; Costa, 2005; NCTM,
2000). It is not surprising then to find the States of Australia have been collecting
data about students’ problem-solving performance from all students in Years 3, 5
and 7 for almost ten years (e.g., Queensland Studies Authority, 2005). Despite
these and similar efforts at collecting data, it seems that little of the data have
been converted into reform of the teaching and learning of mathematics (e.g.,
Lowrie, 2002). As international researchers (e.g., Brown & Walter, 2005; Lester,
2003) have indicated, a review of current practices was needed, as was a “fresh
perspective of problem solving … that goes beyond current school curricula and
state standards” (Lesh & Zawojewski, 2007, p. 52). Of equal concern is Lesh
and Zawojewski’s recent review of literature that reported there is a “growing
recognition that a serious mismatch exists (and is growing) between the low-level
skills emphasized in test-driven curriculum materials and the kind of
understanding and abilities that are needed for success beyond school”
(Gainsburg 2003a in Lesh & Zawojewski, 2007 pp. 5-6). In fact, they went so far
as to say that the challenging and novel problems encountered outside of the
school environment, requiring extensive use of mathematics, are frequently
inconsistent with the underlying assumptions of conventional approaches to
solving mathematical problems in schools (Lesh & Zawojewski, 2007).
Indeed, the extent to which our education system is successful in developing
these skills has broad implications for students as they leave the school system.
Universities and employers throughout Australia and overseas are looking to
organisations like the Australian Council for Educational Research (ACER) to
screen prospective students and employees for their problem-solving
intelligence. Testing instruments such as the Commonwealth Government
funded Graduate Skills Assessment (GSA) (ACER, 2003), can now be used by
28
employers and universities to assist in the determination of university placements
and employment suitability.
Further evidence for the value of problem solving in Australian society can be
found in a more recent government initiative, which saw the Department of
Education, Science and Training (DEST) and the Australian National Training
Authority (ANTA) contract a project to establish the Employability Skills
Framework (DEST & ANTA, 2004). The purpose of this project was to inform
educators about employer perspectives on the personal attributes and skills of
desirable employees. The framework specified eight skill groupings that defined
and described employability skills (see Table 1.4).
There is a need to “continue building Australia’s capacity to effectively operate in
the global knowledge-based economy” and “education and training providers will
have a key role in equipping the community for this challenge” (Australian
Chamber of Commerce and Industry, 2002, p. 1). Reports, such as the
Employability Skills Framework (DEST & ANTA, 2004) attempt to address this
need and provide implications for researchers of educational pedagogy. Not only
is the acquisition of problem-solving competence fundamental in acquiring
important mathematical concepts (e.g., Adams, Brower, Hill, & Marshall, 2000;
Bobis, Mulligan, & Lowrie, 2004), it can also impact on the employability of
graduates entering the work place.
1.4.2 The Place of Problem Posing in a Responsive Curriculum
Problem-posing skills are a fundamental building block in the development of
mathematical skills (Brown & Walter, 2005; Lowrie, 2002; NCTM, 2000).
Problem-posing activities are a means to demystify problems and to empower
students to connect with mathematics in a more personal and meaningful way.
However, despite the clear benefits of problem-posing activities, students are not
often given the opportunity to pose their own mathematics problems publicly
(Silver, 1997).
29
Table 1.4
The Eight Skill Groupings of the Employability Skills Framework
Skill Description
Communication Skills that contribute to productive and harmonious relationships between employees and customers
Team Work Skills that contribute to productive working relationships and outcomes
Problem-solving Skills that contribute to productive outcomes
Initiative and enterprise Skills that contribute to innovative outcomes
Planning and organisation Skills that contribute to long-term and short-term strategic planning
Self-management Skills that contribute to employee satisfaction and growth
Learning Skills that contribute to ongoing improvement and expansion in employee and company operations and outcomes
Technology Skills that contribute to effective execution of tasks
Note. From “Employability skills final report: Development of a strategy to support universal recognition and recording of employability skills - A skills portfolio approach.” by Department of Education, Science and Technology and Australian National Training Authority. 2004. Canberra, ACT.
The virtues and benefits to students of posing problems have been known for
some time. Hart (1981) marvelled at how the activity of allowing students to pose
30
their own problems afforded her the opportunity to “open a window” through
which to view students’ thinking. Van Den Brink (1987) expressed a similar view
when he said problem posing provided him with a “mirror” that reflected the
content and character of a student’s mathematical experience. However, Silver
and Cai (1993b) suggested more profound reasons for including problem posing
as a learning activity, as it presents the opportunity to consider students’ views
on issues of morality, justice and human relationships. These virtues and
benefits are as valid today as they were twenty years ago.
Research has been undertaken in recent years that also espouses the benefits of
mathematical problem posing and solving in a balanced mathematics curriculum
(Bjorklund, 2000; Bobis et al., 2004; Brown & Walter, 2005; Cai, 2003; Daniel,
2003; English et al., 2005; Knuth & Peterson, 2002; Stoyanova, 2003). While
problem posing and problem solving feature highly in most Australian and
American policy documents on Mathematics education, in some American
mathematics classrooms the learning of knowledge and processes received over
one hundred times the attention afforded to the development of problem solving
(Stigler et al., 1999).
To address the research that suggests traditional practices in the teaching of
mathematics can contribute to the disengagement of students (e.g., English,
2002; Lesh & Zawojewski, 2007), research into problem posing has continued
(e.g., Brown & Walter, 2005; English et al., 2005). Of particular interest are the
reports by researchers of increased engagement of underachieving students in
the study of mathematics when problem posing was used as a regular teaching
strategy (English, 1997a, 1997b; Lowrie, 2002). However, despite these
findings, connections between a problem-posing intervention and increased
problem-solving competence of students, who achieve above average results in
standardised intelligence tests and who underachieve on problem-solving tests,
are yet to be made. This study has attempted to fill this void in the research.
31
1.4.3 Disparity in Student Mathematical Performance
It is widely accepted that the mathematical abilities of students of different ages
vary enormously; but so do the intellectual abilities of same-aged students (Case,
1998). These differences have been the study of many research projects
investigating intelligence and the means to measure intelligence (e.g., Sternberg,
2002; Vernon, Wickett, Bazana, & Stelmack, 2000). A number of standardised
intelligence tests have been devised over the past one hundred years and have
been used to benchmark cognitive development (e.g., Spearman, 1904;
Wechsler, 1991). These tests distinguish between the mental age of a child and
the chronological age of a child. The power of the message sent to students
when their performance on such tests is alluded to, or even articulated to the
child, cannot be underestimated. What students believe about their intelligence
and mathematical performance has been shown to be a powerful indicator of
achievement outcomes (Stipeck & Gralinski, 1996).
While we can readily accept that mathematical abilities of students vary from
student to student, it is perplexing when intelligence tests suggest a strong
potential for mathematical ability, yet results from classroom tests do not support
this prediction. In particular, the scenario becomes more perplexing when a
student achieves a high predictive score in an intelligence test, scores highly in
routine procedural questions in class tests, yet continues to perform below
average in questions that require significant problem-solving capabilities. This is
an area of research that has received little attention in the corpus of knowledge
connecting students and their problem-solving capabilities, and precipitated one
of the foci of this study.
1.5 The Purpose of this Present Study
The purpose of this present study was to investigate how a problem-posing
intervention might impact on the development of students’ problem-solving
competence, with a particular focus on the engagement of under-achieving,
middle-year students. This present study provided opportunities for selected
32
students, from four different Year 7 classes in the one school, to pose and
explore their own problems over a seven-lesson teaching experiment. Eighteen
participants met the selection process (see Section 3.3.2) and were withdrawn
from their customary Monday morning assembly each week. They met together
as a group in a multi-purpose classroom in their School library. Data from three
of these students was disregarded, due to the multiple absences of these
students from the teaching episodes, leaving data from fifteen students to be
analysed. From the remaining students, three case-study students were chosen
for a detailed investigation of the changes that occurred for them as a result of
the problem-posing intervention (see Section 5.2 for the selection process of the
three case-study students).
To address the purpose of this present study, three research questions were
investigated during the teaching experiment.
Research Question 1
Can, and if so, how can participation in problem-posing activities facilitate the re-
engagement of middle-year mathematics students?
Research Question 2
Can, and if so, how can participation in problem-posing activities facilitate
improved problem-solving competence of middle-year, mathematics students?
Research Question 3
In terms of problem-solving competence, what developmental learning changes
occur during the course of a problem-posing intervention?
1.6 Significance of the Research
Ceci (1996) argued that it is not possible to deduce the intelligence of a person
from their performance on a set of standardised questions such as those found
33
on commonly used Intelligence Quotient (IQ) tests. Indeed, he argued that
cognition occurs within the framework defined by parents, teachers, peers, and
the culture of the time. It follows then that it may not be possible to accurately
deduce students’ mathematical potential from a set of questions presented to
them in a standardised test or examination, as is the current status quo in many
schools across Australia. It has been mooted by several authors that alternative
activities, such as problem posing, may provide teachers with more authentic and
accurate insights into their students’ understandings of mathematical processes
and concepts. Performance at problem-posing tasks may therefore be a more
accurate indicator of student’s mathematical potential (Anderson, 1997; Bobis et
al., 2004; Brown & Walter, 2005).
Siegler (1996) maintained that teachers can influence their students’ cognitive
development in three significant ways. Firstly, they can influence what their
students think about. Secondly, they can influence how their students will
acquire and construct their information and, thirdly, they can influence why their
students engage in the learning process. This view is supported by Tate and
Rousseau (2002) who found that mathematics was the favourite subject of most
Year 1 and 2 students, yet was one of the least favourite by the time they
reached the middle years of schooling. They attributed this phenomenon to
either the students removing themselves from the challenging programs in
mathematics or the teachers removing the challenging programs from them. In
either situation, mathematics teachers clearly have an important role to play in
constructing effective learning opportunities for their students.
The use of a problem-posing intervention has been investigated by many
researchers. For example, Bandura (1997) discussed the impact of problem-
posing opportunities on students’ self-efficacy, while Knuth (2002) considered its
impact on the development of students’ intrinsic motivation to engage in the
learning of mathematics. Graham, Harris and Larsen (2001) looked at how
problem posing could be used in the prevention of writing problems for students
34
with learning difficulties, while Lowrie (2002) focussed on the influence of the
teacher on the types of problems students pose. Contreras (2003) and Lavy and
Bershadsky (2003) investigated a problem-posing approach to solving geometry
problems, while Stoyanova (2003) considered the impact of problem posing on
gifted and talented students. Despite this apparent breadth of problem-posing
research, there appears to be little research into the role of a problem-posing
intervention in assisting underachieving mathematics students who have above-
average performance in standardised intelligence tests. This study has
addressed this shortcoming.
1.7 Thesis Overview
This thesis comprises seven chapters. The first chapter provides an introduction
to the research study while the second chapter provides a report on the relevant
literature pertaining to problem-solving, problem-posing and underachievement
of students in their middle years of schooling. This review highlights where the
shortcomings in the research exist. A discussion about the design and
theoretical foundations of the research study and a detailed description of the
instruments used to collect data, can be found in Chapter Three. This chapter
also includes a section outlining the selection process for participants of this
study and a more detailed description of how three case-study students came to
be chosen from the participant group.
Issues pertaining to reliability and validity of the data collected and the
associated ethical considerations arising from this study are discussed towards
the end of Chapter Three. Chapter Four introduces the structure of the teaching
experiment and discusses each teaching episode in detail. These discussions
are particularly useful in highlighting the situational challenges, and associated
implications for data collection and analysis, that arose throughout the
experiment. The fifth chapter reports on the data collected during the teaching
experiment and contains an in-depth review of the impact of the problem-posing
intervention on three case-study students; Paul, Andrew and Nicole. Chapter Six
35
provides an analysis and synthesis of the data collected throughout the teaching
experiment that enabled the three Research Questions to be answered. The
limitations of this study and the implications of the study’s findings for future
research are discussed in Chapter Seven.
36
Chapter 2
Theoretical Perspectives
2.1 Chapter Overview
This chapter contains a critical review of current literature pertaining to this
present study. The review begins in Section 2.2 with the literature pertaining to
the developmental learning of students. It starts with a brief introduction to the
main theories discussed by education researchers and then focuses on the three
theories that are particularly relevant to educational research related to the
learning of mathematics. The literature surrounding the development of problem-
solving competence and its relevance and role in developing mathematical skills
is reviewed in Section 2.3, while literature about the use of problem-posing as an
intervention to promote student learning is reviewed in Section 2.4. This latter
section concludes with a review of the literature surrounding the relationship
between the development of problem-solving competence and student
opportunities to pose their own problems. Literature related to the possible
causes of underachievement of middle-year students is reviewed in Section 2.5.
The literature surrounding the theoretical framework that underpins this present
study and the investigation of the Research Questions is presented in Section
2.6, while a conclusion for the chapter can be found in Section 2.7.
2.2 Understanding Developmental Learning
“Developing an understanding of the developmental status of students’ thinking
and learning is fundamental to improving that learning” (Cai & Hwang, 2002, p.
401). As student development of problem-solving competence was a goal of this
present study, this section presents an overview of the literature surrounding
developmental learning of students. Links between developmental learning and
37
problem-solving competence are established, as are the areas in the research
where disagreement between authors exists and uncertainty occurs.
Researchers have provided many methods and concepts that increase our ability
to observe, explain and describe the process of student’s developmental
learning. For example, Siegler (1991) said,
all types of thinking involve both products and processes. The products
of thinking are the observable end states – what children know at
different points in development. The processes of thinking are the
initial and intermediate steps, often accomplished entirely inside
people’s heads that produce the products. (p. 3)
He compared children to scientists because they both ask innumerable
elementary questions about the nature of the universe, which seem entirely trivial
to everyone else, and are both given the time by society to pursue their
ruminations. This inquisitive nature of children is the very attribute that lends
itself to the development of problem-solving competence and problem-posing
expertise from a very early age. Siegler (1991) exemplified this view when he
talked about it not being uncommon to see a toddler in a high chair deliberately
drop food from their tray onto the floor to see what happened to the food.
Together with investigations on intelligence and developmental learning,
researchers are gaining a clearer picture of how to assist students to narrow their
“zone of proximal development” (Vygotsky, 1978) in problem-solving
competence. However, it is not clear, from the current research, whether
problem posing is an appropriate teaching strategy for the particular group of
middle-year students who underachieve in problem solving, yet who appear to
have above average intelligence compared to their peers. Whether intelligence
and developmental learning are a function of nature or nurture has been actively
38
debated for many years. In fact, many researchers have published a plethora of
theories, to understand differences in children’s cognition and developmental
learning, that are worthy of review (e.g., Bjorklund, 2000). Despite some
researchers supporting conceptual frameworks of more than one theory, for
example, Sternberg (1999a; 2002) supporting the multiple intelligences and
information processing theories, and Case (1998) supporting the stage and
information processing theories, in general, most researchers’ work aligns with
one of five theories, which are highlighted in Table 2.1. This present study draws
most heavily from the Information Processing Theory, with some reference made
to the Multiple Intelligences Theory, and the Psychometric Testing Theory, where
relevant.
Regardless of which theory a researcher supports, it is helpful to acknowledge
three basic characteristics of developmental learning. Firstly, we can
acknowledge that the brain is capable of finite information storage and
information processing capacity. Secondly, the human brain is constantly
adapting to a changing environment and thirdly, Goswami (2002) would have us
believe that “cognitive skills almost always can be increased, at least to some
degree” (p. 619). These three characteristics will be discussed further, within the
context of the Information Processing Theory, the Multiple Intelligences Theory
and the Psychometric Testing Theory in the next three sections.
39
Table 2.1
Comparison of Stage Development in Cognitive Development Theories
Theory Underpinning Beliefs Leading Researchers
Stage Learning occurs in stages and a child needs to pass through one stage completely before entering the next stage.
(Piaget & Inhelder, 1969); (Case, 1998)
Information Processing
Mental representations, processes, strategies, and knowledge develop over time.
(Sternberg, 2002); (Halford, 2002); (Klahr, 1992); (Deary, 2000); (Lohman, 2000); (Siegler, 1991, 1996)
Psychometric testing
Intelligence can be described in terms of mental factors and psychometric testing instruments can be constructed to reveal such factors.
(Spearman, 1904); (Brand, 1996); (Hernstein & Murray, 1994) ; (Jensen, 1998); (Wechsler, 1991)
Multiple Intelligences
Intelligence is not a unitary concept, but more a multiple one, where intelligence may be domain specific or domain general.
(Gardner, 1999a); (Sternberg, 1997a); (Thelan & Smith, 1998);
Biological, Environmental and Social Factors
Intelligence characteristics are acquired partly through heredity. Cognitive development occurs through the internalisation of concepts experienced through environmental and social contact.
(Vygotsky, 1981); (Feuerstein, 1979); (Rogoff, 1998); (Ceci, 1990); (Grigorenko, 2000); (Vernon et al., 2000)
2.2.1 Information Processing Theory
Information processing theorists argue that thinking is like processing
information. The quality of the thinking is dependent on the processing capability
and memory limitations of the child. In other words, what information the child
40
chooses to use in a particular situation, how the child processes the information
to achieve their desired outcome, and how much of the information they can
retain in memory at anyone time, will be decisive factors in their overall success
at solving problems. Siegler (1996) spoke about an “essential tension” (p. 58)
that exists for children between their limitations to retain and process information
and their automatic striving to find ways to overcome these limitations. He
discussed a variety of strategies commonly used by children in this pursuit which
included:
1. practice and rehearsal to overcome limited memory capacity,
2. increased use of resources such as books or the internet to overcome
limited knowledge, and
3. the use of problem-solving strategies, such as breaking a problem into
smaller sub-problems, to overcome an inability to deal with long
sequences of tasks.
.
According to Siegler (1991) “it is no accident … that the two main theoretical
approaches to cognitive development – the Piagetian and the information
processing approaches – both place great emphasis on problem solving” (p.
252). He said that when children regularly solve problems they are in fact
contributing to their own cognitive development as problem solving requires them
to create solutions for themselves, rather than relying on procedures and
practised routines they have learnt. This active involvement by a child in their
own developmental learning, by engaging in continuous self-modification
(Siegler, 1996), was also supported by Bjorklund (2000) who said “cognitive
development is a constructive process, with children playing an active role in the
construction of their own minds” (p. 481).
Researchers who support an information-processing theory, discuss four change
mechanisms that they believe play a significant role in childhood cognitive
41
development: automatisation (the increasingly efficient execution of mental
processes), encoding (the selection and prioritising of important aspects of
situations), generalisation (the use of prior knowledge of numerous familiar
situations), and strategy construction (the synthesis of change processes to
produce cognitive growth) (see Sternberg, 2000). In previous research
Sternberg (1985) referred to only three information processing components of
general intelligence in his Triarchic Theory of Intelligence (see Figure 2.2), these
being knowledge acquisition components (discrimination between relevant
and irrelevant data), metacomponents (selection and planning of appropriate
strategies) and performance components (combination of the selected data
and appropriate strategy to solve the problem). However, none of these
components explicitly acknowledge the efficiency with which a student solves a
problem as a significant factor of intelligence. The efficiency of execution in the
solution of a problem warrants further investigations where the time allowed for
an assessment of skills is a controlled factor.
While Sternberg’s earlier work is over twenty years old, and has been
superseded by the four change mechanisms to a large extent, researchers (e.g.,
Goswami, 2002; Thomas & Karmiloff-Smith, 2002) still refer to Sternberg’s
Triarchic Theory of Intelligence when discussing childhood cognitive
development. According to Goswami (2002), “individual differences in cognition
derive largely from individual differences in the execution of these three kinds of
components. The components are highly interdependent.” (p. 608)
42
Selective Selective Strategy Strategy Encoding Application Encoding Combination Construction Selection Selective Strategy Inference Comparison Coordination
Figure 2.1. A Schematic Diagram of Sternberg’s Triarchic Theory of Intelligence
(in Siegler, 1991 p. 69).
2.2.2 Psychometric Theory
If we assume that infants come into the world poorly endowed, the
question becomes how they are able to develop as rapidly as they do. But
if we assume that infants come into the world richly endowed, the question
becomes why development takes so long. (Siegler, 1991, p. 3)
This section explores the issues surrounding this nature versus nurture debate
that begun in the late 1800s by researchers such as Sir Francis Galton (1883),
Charles Darwin’s cousin, who popularised the now famous Bell Curve and its
associated normal (Gaussian) distribution. A review of the history of
psychometric theory is relevant to current research as views held by
contemporary proponents of the psychometric theories have not changed
Intelligence
Metacomponents Knowledge Acquisition Components
Performance Components
43
significantly to those of the founding researchers in this field. The Bell Curve,
first introduced by Galton, is still used as a standard tool for comparing students
and prospective employees as well as being used by researchers for interpreting
data in social science research projects.
Galton’s (1883) interest in comparing individuals stemmed from his advocacy for
eugenics, the inter-breeding of intelligent people in order to strengthen the gene
pool of the human species. While Galton initially investigated the distribution of
physical measurements such as weight and height, he later theorised that since
psychological characteristics were based on physiological characteristics, then
human intelligence could also be represented by the Bell Curve. While Galton
had begun founding research into human intelligence, he did not construct broad-
scale instruments to measure intelligence levels of children or adults. This work
was taken up a few years later in France when universal education was
introduced in the late 1890s, as a result of the Industrial Revolution, with
psychologists Alfred Binet, Director of the Sorbonne in France, and Theophile
Simon being engaged to develop a testing instrument to determine which
children needed “special education” (Binet & Simon, 1905).
The first Binet-Simon test (Binet & Simon, 1905) was used in 1905 and included
thirty questions on reasoning, memory, language and problem solving, ordered
by difficulty, and was used to identify children who may experience difficulty with
a common curriculum. The test was based on data from 50 subjects, therefore
lacking validity, and was criticised because it relied heavily on the reading and
language ability of the children. Almost one hundred years later, this same
criticism is leveled at authors of psychometric tests in current use (Bjorklund,
2000; Gardner, 1999b). The Binet-Simon test was revised in 1908 following
further research with 203 subjects and had test items grouped according to age
level rather than increasing difficulty. It was at this stage that Binet and Simon
introduced the concept of mental age (MA), as compared to chronological age
44
(CA), which later resulted in the establishment of the ‘intelligence quotient’ (IQ)
by German psychologist William Stern (1912) and Terman (1916) that is still
used today to define, label and categorise students.
At around the same time as Binet and Simon (1905) established their first test,
research into human intelligence and developmental learning took a different
direction in England with Spearman (1904) investigating the existence of a
general intelligence factor that he abbreviated to a more commonly used
expression, a g factor. According to Spearman, all individual differences in
cognitive ability were due to a general factor that is present at birth and that he
believed was as a result of differences in mental energy. This g factor impacted
upon performance in all cognitive tests, whereas a specific factor, (commonly
called an s factor) could impact upon an individual’s performance in a specific
type of test. To support his proposition, he examined correlations between
student scores on different school subjects (see Table 2.2) and offered the high
positive correlations as evidence of the existence of a single common general
intelligence factor. This suggestion of individuals having specific s factors
maintained momentum and, 85 years later, was paralleled by Ceci’s (1990) view
that the context in which a test occurs is a decisive and determining agent in an
individual’s performance on the test. This position has important implications for
current research where researchers are interested in the participant’s
developmental learning changes as opposed to their connectedness to the
context of the questions used in the assessment instrument or the style of the
questions.
If the position of specific and general factors of human intelligence was to be
accepted, a new testing instrument was needed to measure and compare the
intelligence of individuals. Spearman (1904) in association with Cyril Burt,
another British psychologist, were some of the earliest researchers to develop a
range of intelligence tests, to measure the mental abilities of British school
45
children, that took general and specific intelligence factors into consideration.
They pioneered the concept of factor analysis that other researchers, such as
Thurstone (1938) and Wechsler (1991), further developed many years later.
These tests allowed gender differences to be considered. For example, Halpern
(1997) reported that, on average, boys score higher on tasks that involve visual
and spatial awareness than do girls, while girls perform better than boys at tasks
that require access to long-term memory, fine motor skills, perceptual speed, and
writing and comprehension of complex prose. These findings require current
researchers to consider whether assessment instruments favour the natural
differences of either gender. Without these consderations, the validity of data
could be challenged.
Table 2.2.
Spearman’s Correlations of Student Scores between Subjects
Subject Classics French English Math Pitch Music
Classics - .83 .78 .70 .66 .63
French .83 - .67 .67 .65 .57
English .78 .67 - .64 .54 .51
Math .70 .67 .64 - .45 .51
Pitch .66 .65 .54 .45 - .40
Music .63 .57 .51 .51 .40 -
Note. From "General intelligence, objectively determined and measured” by C. Spearman, 1904, American Journal of Psychology, 15(2), pp. 201-293.
46
By the early 1920s, the use of psychometric testing had expanded to the United
States and was being used as a means to determine which immigrants were
suitable for residency and which should be deported, and later in the 1930s to
determine intelligence levels of American school children. To achieve this goal,
Lewis Terman (1916), a Stanford Professor, revised the French Binet-Simon
(1905) test calling it the Stanford-Binet test. Results from this test were
compared to a standardised sample of 3184 mainly white, urban children from
eleven states in America, chosen by father’s occupation. This revised test was
administered under the assumption that not all children of a particular age think
and reason in the same way or at the same level. Terman’s results were more
reliable for older children aged between twelve and sixteen years than for
younger children, or children in the lower IQ ranges, but he found standard
deviations for children in different age groupings made the interpretation of data
difficult.
The use of intelligence tests continued to grow throughout American schools
over the next eighty years with the most common uses being for the identification
of children with special needs and children with special gifts and talents (Piirto,
2007). The use of IQ tests to investigate differences in intelligence levels of
different ethnic groups became widely provocative with the publishing of
Hernstein and Murray’s (1994) book entitled The Bell Curve: Intelligence and
Class Structure in American Life. The researchers stated in their book that they
had proven that people from minority ethnic backgrounds had lower IQs than
white Americans. Researchers in education, such as Kincheloe, Steinberg and
Gresson (1996), were quick to refute the allegations in their book Measured Lies:
The Bell Curve Examined. They wrote “The Bell Curve … emerges from a
crumbling paradigm often deemed inadequate for the study of human
intelligence” (Kincheloe et al., 1996, p. 28). While this latter book sold widely, it
did not impact on the growth of intelligence testing in American schools. In fact,
47
according to Piirto (2007, p. 14) “the increase in the use of aptitude,
achievement, and personality tests has been marked.”
Psychometric testing began to emerge in Australian schools in the early 1920s
and is now a well-established and accepted part of testing for Australian school
children (Hughes, 2002). The establishment of the Australian Council for
Educational Research in 1930 provided standardised resources for psychometric
testing to be undertaken in schools. By 1936, all Year 6 students in New South
Wales (NSW) were administered intelligence tests to determine which form of
secondary education best suited them. However twelve years later, a United
Nations Educational, Scientific and Cultural Organization (UNESCO) report on
educational psychology services across 41 countries in 1948, estimated that only
20 psychologists were employed across all Australian school systems and were
mostly based in NSW (Korniszewski & Mallet, 1948). Therefore, while a wealth
of data was being collected, the analysis of the data was generally limited to
superficial interpretation by school administrators.
The dominance of psychometric testing in Australian and international schools
has been driven by a widespread need of modern society to quantify individuals’
intellectual capacity. This is evident when one considers that most students in
Australia, and particularly those attending private schools, will not leave formal
schooling without having undertaken at least one Intelligence Quotient (IQ) test
and a dozen more specific intelligence tests (Bjorklund, 2000). The testing is
sometimes undertaken internally by educators using standardised instruments
such as the Middle Years Ability Test (Australian Council for Educational
Research, 2005) or administered privately by organizations such as The Sydney
Child Assessment and Testing Service (SCATS) which provides a private testing
environment for children aged between 3 and 16 years using predominantly the
Wechsler (1991) testing instruments.
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Despite the widespread use and acceptance of psychometric testing in Australia
and internationally, researchers such as Naglieri and Kaufman (2001) raised
concerns about weaknesses in traditional IQ tests when they are used as a tool
to determine giftedness. They said these tests were theoretically old, they were
weak in theory and they were achievement driven. In addition, Jensen (as
reported in Jensen & Miele, 2004) raised concerns about the blatant lack of
understanding among users of IQ tests that has lead to the common misuse of
data generated from such testing instruments. Richhart (2002, p. 16) alerted us
to the impact of test practice on test scores when he said that some critics
“contend that test scores are highly influenced by one’s test-taking competence
and familiarity”. The existence of Core Skills Test (Queensland Studies
Authority, 2009) preparation courses in many Queensland schools, where
students in Year 12 practise tests from previous years to ensure they are familiar
with the test format and timing, could be considered as evidence of the widely
held acceptance of this viewpoint.
While IQ results correlate positively with academic success and employability
(Brody, 1997) and have been strongly supported by researchers such as Jensen
(1998), other researchers argued that IQ tests are limited in what they can
measure and that it is misleading to use an IQ score as a sole indicator of a
child’s overall intelligence. Gardner (1999b) was one of these researchers and
suggested that IQ tests provided at best a distorted view of an individual’s
potential, as they clearly advantaged individuals with strengths in the linguistic
and mathematical intelligences. Individuals with strengths in other intelligence
areas, such as the bodily-kinaesthetic intelligence, are often neglected and
hence do not receive an education sympathetic towards their unique form of
intelligence. Surprisingly, Gardner is not opposed to the use of intelligence tests
for determining intelligence of individuals. He would however, prefer that testing
instruments were constructed to measure and evaluate all of the multiple
intelligences. In additional to these concerns, other critics of IQ tests say they
49
are culturally biased, that is, they are based on knowledge and skills of middle-
class individuals from majority cultures rather than being inclusive of the
traditions, values, predominant language or experiences of minority cultures
(Bjorklund, 2000). The concerns mentioned here are relevant to new research
when IQ testing is used as an instrument to collect data or determine participants
for research studies. The literature would seem to suggest that, at best, IQ test
data can be used as an indication, rather than a definitive measure, of an
individual’s intelligence.
For as long as the existence of general and specific factors has been mooted,
there have been researchers who support the existence of only a single general
factor, or only specific factors, or both. A number of researchers supported the
existence of specific factors but challenged the existence of a general factor.
Debate continues about the existence of a higher-order, general intelligence
factor that oversees and orchestrates these other cognitive factors. For example,
Jensen (1998) is still seeking to demonstrate the factor’s existence, while others
like Ceci (1996) proclaiming the search is “fruitless”. On the other hand, other
researchers, such as Guilford (1988) and Sternberg (2002), have repeatedly
attempted to disprove, without success, the pivotal influence of a g factor in
determining intelligence of individuals however, according to Piirto (2007, p. 15),
“general intelligence is pervasive, even in tests that purport not to measure g-
factor intelligence.”
Brody (2003, p. 319) adds his support to the existence of a g factor when he says
that the “g theory is required to understand the relationships obtained by
Sternberg and his colleagues” who were proponents of information processing
theories of intelligence. That being said, it is now generally accepted that there is
more to intelligence than the general intelligence factor alone (Gottfredson,
2003). The challenge for future research is to develop a theoretical framework
and appropriate testing tools that incorporate the notion of a g factor in
50
combination with the widely accepted multiplicity in intellectual functioning that is
reported by researchers such as Gardner (1999b).
Despite disagreement between researchers about the accuracy of psychometric
testing as an accurate measure of an individual’s intelligence, or whether a
psychometric test is a reliable instrument to measure intelligence for all
individuals, the widespread use of IQ testing remains a feature of our present
education systems both internationally and in Australia. Gottfredson (2003) and
Piirto (2007) supported the use of IQ testing for the purposes of indentifying
individuals for suitable interventions to address their particular developmental
learning needs, however, to measure an individual’s ability within a specific
context and in a specific area of learning, more specific testing instruments are
required (Gardner, 1999b; Sternberg, 2000).
2.2.3 Multiple Intelligences Theory
In contrast to how Sternberg (1999a) emphasised the connectedness of the three
aspects of his Triarchic Theory of Intelligence, Gardner (1999b) emphasised the
separateness of his multiple intelligences. For him, there were up to ten unique
intelligences that represented a modular, brain-based capacity, some of which
were linguistic intelligence, logical-mathematical intelligence, and intrapersonal
intelligence. His was the first theory to account for the diverse range of important
capacities of individuals by considering a diverse range of competences and
based his theory on a diverse range of evidence. His evidence included the
selective damage of specific cognitive abilities following brain trauma and
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