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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.
48
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 the
existence of individuals who present as low-achieving in IQ testing, yet who
display extraordinary abilities within one intelligence domain, such as Mozart who
was a musical genius but particularly ordinary in his performance at school tests
(Gardner, 1999b).
51
Like Gardner, Bjorklund (2000), when discussing the multiplicity of intelligence,
acknowledged the existence of both domain specific and domain general
intelligences. He said,
Intelligence, I believe, is multifaceted, with intellectual functioning varying
considerably as a function of a person’s knowledge and the context in
which the cognitive operations were acquired and are assessed…..The
most tenable position from my point of view is that some aspects of human
intelligence are domain general in nature, whereas others are domain
specific in nature. (Bjorklund, 2000, p.435)
Like Bjorklund (2000), Gardner (1999b) stated that although each of his multiple
intelligences had its roots in biology, each intelligence was flexible and could be
improved with education given an appropriate facilitation of learning experiences.
The implications of this statement are noteworthy for research that involves
teaching experiments with student participants in classrooms as facilitating
developmental learning is a key goal for many educators. Ritchhart (2002, p. 13)
extended the notion of appropriate learning experiences to beyond the classroom
and encouraged educators to consider intelligence in the context of the real world
where it will ultimately be applied rather than “the artificial world of school and
testing”. While such variables are difficult to control or accommodate in an
intervention with children, mention should be made in the limitations of such
research reports to ensure appropriate interpretation of the findings can be
made.
2.2.4 Summary
In section 2.2 of this chapter, the five major cognitive development theories that
researchers refer to today were identified. The three theories that are particularly
relevant to educational research, and that relate to problem solving and the
learning of mathematics, were then discussed in more detail. While the theories
52
were presented and discussed separately, there are some common threads that
lend support for the design of this present study. For example, Sternberg’s
(1988) Triarchic Theory sits well within the information processing theory due to
his definition of its three distinct information processing components; knowledge
acquisition components, metacomponents, and performance components.
However, the Triachic Model had its foundations in biological and social factors,
thus making an alignment with the biological, social and environmental theories,
and could be said to have links with the Psychometric Theory due to Sternberg’s
discussion of an overarching, general intelligence factor. Similarly, there is
agreement between the Psychometric theorists (e.g., Hutton, Wilding, & Hudson,
1997) and the Information Processing theorists (e.g., Siegler, 1996) that
intelligence is dependent on cognitive factors related to memory capacity and
function, and cognitive processing speed. A discussion of these shared ideas
and beliefs between the theories serves to inform what underpins our
understanding of childhood developmental learning with respect to the
development of problem-solving competence in a mathematical context.
Information processing theorists (e.g., Halford, 2002; Sternberg, 2002) suggested
that it is a student’s ability to select the correct information with which to work,
their ability to process that information, and how much of the information they can
retain at any one time, that will be the significant factors in developing problem-
solving competence. In addition, the amount of time a student is given to
undertake a task can determine the level of success they are able to
demonstrate in specific skills. These are important factors to consider when
designing a study that measures problem-solving competence and
developmental learning changes. Further, (Siegler, 1991 p. 59) suggested that
cognitive change occurred as a result of “self-modification” and (Bjorklund, 2000)
said it is a process where students best construct their own learning by taking an
active role. It follows therefore, that a traditional approach to teaching, where the
teacher instructs and the student’s feedback what has been instructed to them,
53
may not be the most beneficial if long term cognitive change is a goal of the
learning experience. The National Council of Teachers of Mathematics (2000)
and Ritchhart (2002) supported the need for learning opportunities for students
that involve experiential learning of real world, novel problems where they have
opportunities to be risk takers in the advancement of their own developmental
learning. More research in this area is being called for with problem posing and
problem modelling being two forms of intervention that have been identified as
worthy of further investigation as a means to promote the active engagement of
students in the development of mathematical skills (e.g., Lesh & Zawojewski,
2007).
A review of developmental learning theories highlighted the contribution of
problem-solving activities to the development of general, cognitive growth, and
as such, a review of the research surrounding the development of problem-
solving competence can be found in the following section.
2.3 Problem-solving Perspectives
This section reports on the current literature connecting problem solving to the
developmental learning of students, with particular emphasis on how it can be
used as a tool to develop mathematical learning and why the ability to be a
competent problem solver is so important. The benefits to students of teaching
mathematics from a problem-solving perspective are considered as are the ways
that problem-solving can drive mathematical reform. Issues surrounding the
development and assessment of problem-solving competence and student’s
understandings of problem structures are also reviewed.
2.3.1 Introduction
There is an increasing need for students to be able to solve real world, novel
problems (e.g., NCTM, 2000) and while substantive literature can be found which
expounds the virtues of providing students with opportunities to solve problems,
54
there is inadequate research into practical teaching strategies for teachers in
classrooms to use. Despite this, it is heartening to see significant research has
taken place in recent years surrounding the use of problem posing and problem
modelling as a means to facilitate increased problem-solving competence (e.g.,
Brown & Walter, 2005; English, 2003; English et al., 2005; Stoyanova, 2003).
However, little research has focussed on the specific group of students that
teachers would be expecting to succeed in problem solving, that is, the students
who perform above average, compared to their peers, in academic intelligence
tests. The reasons for underachievement of these students are still an
unexplained phenomenon (Baker, Bridger, & Evans, 1998; Lesh & Zawojewski,
2007).
As socio-cultural contexts change Lesh and Zawojewski (2007) suggested that it
follows then that teaching and learning strategies need to change to
accommodate them. Students are now looking towards an ever-changing job
market characterised by dynamic advances in technology, and one in which they
will need higher-order thinking skills and increased problem-solving ability to
cater for the inevitable ambiguities they will encounter. Researchers, who
question the ability of our curricula to prepare students for such a world, have
asked, “What if education were less about acquiring skills and more about
cultivating the dispositions and habits of mind that students will need for a lifetime
of learning, problem solving, and decision making?” (Ritchhart, 2002 p. xxii) This
question appears to have gone unanswered in the literature.
In 2003, Lester and Kehle undertook a review of the literature relating to problem
solving over the previous ten years. They found for example, that some
researchers had reported on how students use specific strategies for different
types of problems (e.g., Bjorklund, 2000; Ceci, 1996), while others had reported
on children’s use of talking with peers to solve problems (e.g., Teasley, 1995)
and the effect of metacognitiion in the development of problem-solving
55
competence (e.g., Kramarski, Mevarech, & Arami, 2002). As a result of their
meta-review, Lester and Kehle reported that minimal progress had been made,
and the literature undertaken in that time offered little to the pedagogical reform
of mathematics teaching (Lester & Kehle, 2003). Similar findings were reported
by Lesh and Zawojewski (2007) who reported there was still a pressing need to
improve the way students view problem solving and mathematics, while others
reported the need to increase students’ abilities to solve unfamiliar, novel
problems (English & Larson, 2005; English et al., 2005; Accreditation Board for
Engineering and Technology 2004 in Lesh & Zawojewski, 2007).
According to Lesh and Zawojewski (2007), this lack of significant reform in the
teaching of mathematical problem solving is not surprising as many developed
countries, such as the United States of America, have experienced a strong
swing back to curricula that emphasise basic skills. They reported this swing as
a result of a world-wide emphasis on high-stakes, standardised testing (Lipman,
2004; Stecher, 2002) which, according to the American National Research
Council (as cited in Kilpatrick, Swafford, & Findell, 2001, p. 4), has resulted in
mathematics curricula that are “shallow, undemanding, and diffuse in content
coverage”, with students becoming “notably deficient in their ability to apply
mathematical skills to solve even simple problems”. It is also worth noting that a
compromising feature of these standardised tests is that the results of the tests
are sometimes used to determine teacher pay rates, ongoing employment and
funding for schools in the United States of America and in some States of
Australia. According to Stecher (2002) this focus on basic computational skills,
leading to high-stakes test results, has resulted in the systematic marginalisation
of innovative and experimental teaching practices and less focus on conceptual
understanding. In addition, it has lead to a decline in the amount of problem-
solving research being undertaken (Lester & Kehle, 2003; Stein, Boaler, & Silver,
2003).
56
When reviewing the historical literature on the development of mathematical
understanding and knowledge, it becomes apparent that the bulk of this literature
presupposes that the development of mathematical understanding occurs along
a one-dimensional line (e.g., Piaget & Beth, 1966; Vygotsky, 1978). More recent
mathematical research however, particularly research surrounding models and
modelling perspectives of learning (e.g., Hamilton, Lesh, Lester, & Yoon, 2006;
Lesh & Zawojewski, 2007; Lester & Kehle, 2003), assumes that mathematical
understanding develops from a multi-dimensional perspective and that students’
final understandings of mathematics are a direct result of many influences within
the learning process. Van de Walle (2004) refers to this deeper understanding
as “relational” knowledge and attributes it with higher levels of intrinsic
motivation, improved attitudes and improved problem-solving competence of
students. According to Lesh and Zawojewski (2007), this more integrated and
multi-dimensional view of mathematical learning requires new methodologies of
teaching and warrants further research.
2.3.2 The Power of Teaching through Problem Solving
From the literature it can be seen that many researchers (e.g., Lesh &
Zawojewski, 2007) believe that problem solving is a powerful way to learn
mathematics, as opposed to simply a means to challenge talented mathematics
students or apply mathematical knowledge and procedures that have already
been taught in a traditional drill and practice fashion. English and her colleagues
(2005) supported this position and extended it to encompass problem posing and
solving using mathematical modelling when they said:
57
As children work interactively in solving the modelling problems they share
ideas, question one another’s claims, ask ‘what-if’ questions, contemplate
numerous decisions, and consider alternate courses of action. In so
doing, children generate important mathematical ideas and processes and
use a variety of representations to display their findings. The multifaceted
nature of these problems makes them ideal vehicles for advancing
children’s learning in numerous directions. (p. 11)
According to the National Council of Teachers of Mathematics (2000) the majority
of mathematics teachers remain supporters and practitioners of traditional
teaching methodologies. However, researchers have repeatedly called for
innovation and reform in mathematics education for over four decades (e.g., Ben-
Chaim, Fey, Fitzgerald, Benedetto, & Miller, April, 1997; Brown & Walter, 1983;
Bruner, 1965; Cattell, 1971; Collins, 1986; Ritchhart, 2002). Nearly forty years
ago, Freire (1970, p. 45) argued that education was suffering from “narration
sickness” whereby the teacher “narrates” the subject content to passive students
who are expected to memorise and regurgitate it. Little had changed by 1996
with Roberts (1996) referring to this pedagogy as the “banking” model where
teachers “deposit” ideas into willing receptacles (students). He lamented that it is
still widespread in modern education where “students are treated as acquiescent
automatons to be controlled in both thought and action” (Roberts, 1996, p. 2).
2.3.3 Can Problem Solving Drive Mathematical Reform?
Despite the limited success of reformists, optimists like Costa (2005) continued to
promote problem solving as a valid teaching strategy and encouraged teachers to
work toward producing “effective problem solvers [who] know how to ask
questions to fill in the gaps between what they know and what they don’t know”
(p. 5). One of the most ardent supporters of a “liberating” education, Costa, is still
“marketing” this form of education today:
58
We hope that students will learn how to take intellectual as well as
physical risks. Students who are capable of being different, going
against the grain of the common, thinking of new ideas and testing
them with their peers as well as teachers are more likely to be
successful in an era of innovation and uncertainty. (p. 8)
One of the apparent obstacles in the broad scale use of problem solving as a
teaching strategy appears to be the view by teachers that problem solving is not
a pedagogically sound or motivating approach for the introduction, development,
and application of mathematical concepts and skills. Rather, it is more commonly
seen as a tool to consolidate and reinforce mathematical knowledge and
procedures (Adams et al., 2000; Bobis et al., 2004). Earlier researchers such as
Pejouhy (1990), recognised that the destiny of students’ problem-solving
successes, and the success of any mathematics curriculum reforms, ultimately
lay in the hands of classroom teachers. To effect reform teachers must be aware
that understanding mathematics and doing mathematics are different (Canobi,
Reeve, & Pattison, 2003; Cassel & Reid, 1996) and researchers need to
construct new tools to measure the constructs required for understanding
mathematics (Lesh & Zawojewski, 2007).
2.3.4 Issues Related to the Assessment of Problem-Solving Competence
While the main emphasis in developing problem-solving competence has
surrounded the acquisition of problem-solving strategies (Bjorklund, 2000), there
are many students who are very good at numeration and computation but who do
not demonstrate a commensurate ability in problem-solving skills (English, 2003;
Sternberg, 2002; Thomas & Karmiloff-Smith, 2002). Students with these
particular characteristics are worthy of further research studies to determine the
nature of this phenomenon.
59
Recent research has shown that using the wrong strategy to solve a problem can
either slow the solution process or inhibit a student’s opportunity to solve
problems (e.g., Bjorklund, 2000; Cai, 2000; Ceci, 1996). An awareness and
understanding of strategies and when to use them can make the difference
between a student being a successful problem solver or not (e.g., Goswami,
2002; Thomas & Karmiloff-Smith, 2002). A number of researchers (e.g.,
Kramarski et al., 2002) have tried different interventions to assist students to
improve their problem-solving competence. Kramarski and his colleagues
investigated the notion of teaching students metacognitive strategies to assist
them to self-regulate and monitor their problem-solving processes. They
believed this form of instruction might have the additional benefit of improving
students’ self-efficacy with regard to mathematical problem solving. Bentley
(1996) had previously examined the benefits of teaching metacognitive practices,
and while this research focussed on middle-year students in Years 6 and 7, he
did not consider students of differing intelligence levels within his study. This is a
shortcoming in the research on problem solving and worthy of further
investigations.
2.3.5 Should Specific Problem-solving Strategies be Taught?
While the specific instruction of problem-solving strategies might therefore seem
prudent, we are reminded by Bobis, Mulligan, and Lowrie (2004) that:
the teaching of specific strategies may increase the likelihood of success
in problem solving but unfortunately teachers tend to model specific
strategies in whole-class situations. This can lead to the perception that
you need to solve a problem the way the teacher has completed it. (p.
47)
This is consistent with Lesh and Zawojewski’s (2007) findings. They reported that
there is insufficient evidence to suggest conclusively that prescribing the problem-
60
solving strategies of “experts” to “novices” is effective in improving the novices’
problem-solving performances. This was also reported earlier by Schoenfeld
(1992) and Lester (1994) in their literature reviews when they summarised their
findings by saying that teaching students to use general problem-solving
strategies was found to be unsuccessful and has done very little to improve the
ability of students to solve problems. In fact, Lester and Kehle (2003) suggested
that researchers have made much rumination, but few practical suggestions, to
improve school practice in this area and there is a need for further research to
address the unanswered questions on this issue. Two primary concerns reported
by several researchers were the “relative ineffectiveness of instruction to improve
students’ ability to solve problems” (Lester & Kehle, 2003, p. 510) and the decline
of research focussing on problem solving (Lester & Kehle, 2003; Stein et al.,
2003). Lesh and Zawojewski (2007) added their additional concern about the
complete separation of problem-solving activities from the learning of “substantive
mathematical concepts”. They alternatively emphasised a “synergistic
relationship” between the developmental learning of mathematics and problem
solving (Lesh & Zawojewski, 2007).
2.3.6 Students’ Understandings of Problem Structures
The research into the development of students’ awareness and understanding of
how problems are constructed began almost thirty years ago. Rumelhart (1980)
wrote about how students encoded problems using a rich, embedded schema.
He said that once the problem is encoded, understanding it and solving it, are
nearly the same thing. Van Essen and Hamaker (1990) also followed this line of
research and reported that representing the problem in a meaningful format is the
most important step in the problem-solving process, since it is during this time
that the student determines the actual structure of the problem. English and
Halford (1995) referred to the important role of “reasoning by analogy” in
identifying these structures. Such reasoning occurs when a student discovers the
“source” or “base” problem that has all the “base” elements that corresponds to
61
the stated problem, but without any of the superfluous, distracting information.
Some years later, Lester and Kehle (2003) summarised similar findings and said
that “good problem solvers know more than poor problem solvers and what they
know, they know differently – their knowledge is well connected and composed of
rich schemas” (p. 507).
Whether this form of analogical reasoning is available to students has been
debated over the years (e.g., Bernardo, 2001; DeLoache et al., 1998; Goswami,
1996). Followers of the Piagetian model would suggest that it is not available to
adolescents (Inhelder & Piaget, 1958) but more recent research suggests it may
be present from birth (Goswami, 1996) and is indeed the basis for problem
solving and other reasoning tasks (Bernardo, 2001). The conflict in findings from
this present review of literature suggests that more research is needed to
establish whether analogical reasoning is available to middle-year students as a
tool for cognitive development. This position is supported by Cai and Hwang
(2002) who said that, “the more information teachers can obtain about what their
students know and how they think, the more opportunities they can create for
student success” (p. 401). They believe that research into the teaching of
mathematics through problem solving can fill some of the gaps in the current body
of research.
There is “general agreement that new perspectives are needed regarding the
nature of problem solving and its role in school mathematics” (Lester & Kehle,
2003, p. 509). Lesh and Zawojewski (2007) and English, Fox and Watters
(2005) would like to see researchers focus their problem-solving investigations
on the bridge that links the problem-solving instruction in schools with the real-life
problem solving that occurs beyond the classroom. The ability of students to be
able to transfer their mathematical and problem-solving knowledge to real-life
situations is a goal of many researchers (e.g., English et al., 2005) and was a
particular focus of a report by the President of the NCTM, which proposed
62
“building bridges of mathematical understanding” (as cited in Price, 1996, p. 3).
Lesh and Zawojewski (2007) discussed this goal in their recent research and
wrote:
The notion of transfer is important when considering situated knowledge,
because one might argue that the knowledge and routines are developed
for only the situation at hand. However, in everyday contexts, people
quickly move from creating mathematical procedures to dealing with not
only the original context, but also isomorphic situations routinely ... (p. 36)
The findings of The Connected Mathematics Project, conducted at Michigan State
University and focussing on middle-year students, also valued the methods of
instruction that assist students to construct meaning and build effective networks
of skills and understanding (Ben-Chaim et al., April, 1997). The findings from the
two-year study indicated that the more time students spend working in the project,
the more their results in the state-run standardised mathematics tests improved,
and indeed the gap between their results and those students who were not
participating in the project widened. These findings provide evidence to refute the
claims of those who suggest that for students to succeed in broad-scale
standardised testing, teachers are required to devote the majority of class time to
the development of basic skills and less time to activities such as problem solving
(see Lipman, 2004; McNeil & Valenzuela, 2000).
The context in which students explored and solved problems was investigated by
Gravemeijer (1994) in the Netherlands during his longitudinal study entitled The
Realistic Mathematics Education Study. This study used everyday contexts to
pose problems to students that caused “internal conflict” which in turn lead to
students creating “abstract conceptualisations” of the problems. The findings
from the study recommended that familiar contexts when setting problems for
students to solve as it was more likely to result in higher intrinsic interest levels
63
and engagement by the participants. Following similar lines, but taking a more
conservative approach to the research undertaken in the Netherlands, the major
research projects in this area in the United States of America have focussed on
the more traditional methods for developing problem-solving competence that
included using concrete materials to conceptualise the problems (e.g., Ball &
Cohen, 1996; Cobb, 1997). One such study in Texas was developed following
the publication of the Third International Mathematics and Science Study (TIMSS)
results (Adams et al., 2000). The researchers in the study worked with 350
teachers from over 100 schools to determine which practices should be used to
promote mathematics and science reform in middle-school education. Not
surprisingly, the researchers reported that reform would require increased
emphasis on developing student’s problem solving and reasoning abilities, but fell
short of providing a detailed report of practical suggestions for teachers. These
findings suggest there is still a need for investigations into intervention strategies,
such as a problem-posing teaching experiment, to fill this void in the research of
the development of problem-solving competence.
In Australia, research projects such as the Literacy and Numeracy Development
in the Middle Years of Schooling Project (Luke et al., 2003) have investigated a
variety of issues surrounding the learning of mathematics. The aim of this project
was on the development of literacy and numeracy skills amongst middle-year
students. However, the report suggested in its findings that students in
classrooms where higher-order thinking tasks, such as problem-solving tasks,
were commonplace, and where the content was linked to real-life situations with
which the students were familiar, demonstrated superior numeracy skills to those
students in classrooms where lower order, more abstract tasks were prevalent.
2.3.7 Summary
There are repeated calls in the literature for the need for mathematics curriculum
reform (e.g., Adams et al., 2000; Groves, Mousley, & Forgasz, 2006; NCTM,
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2000). The findings of these studies suggested that problem solving should
receive more intensive focus in the mathematics curriculum since it facilitates in
students a deeper understanding and connectedness of mathematical concepts
(e.g., Ben-Chaim et al., April, 1997). Problem solving is, after all, an everyday
activity, at some level, and a necessary skill for individuals to be able to adapt to
our dynamic society where change and innovation are considered to be routine.
The ability to reason by analogy, and recognise and transfer similar structures
between classes of problems, has been demonstrated as particularly effective in
improving students’ performances at solving problems (English & Halford, 1995).
Teaching students which strategies to use for problems of a particular structure,
has however, produced conflicting reports about its effectiveness and long-term
benefits to improve students’ problem-solving competence (e.g., Bobis et al.,
2004).
Problem solving is a skill that can be learnt by students of all achievement levels
in mathematics (e.g., Groves et al., 2006). Underachieving students have been
shown to develop quite sophisticated and complex strategies for solving problems
following intervention programs. The review of literature has highlighted multiple
directions for future studies. However, there is significant evidence to suggest
that further study in the use of intervention strategies with underachieving middle-
year students is particularly warranted. More importantly, there is significant
indication in the current research findings to suggest that not only can
underachieving students improve their problem-solving competence but they can
also “construct important mathematical ideas through solving novel problems”
(Groves et al., 2006, p. 200). To fill the void in the research, a future study with
underachieving, middle-year students who have been ascertained with above
average intelligence would provide evidence to move this area of research
forward.
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Research has been conducted into student attitudes to solving novel problems
based on real-life and abstract scenarios (e.g., Bjorklund, 2000; English & Larson,
2005; English et al., 2005), and the findings of these reports suggest that making
connections between classroom mathematics and real-life mathematics has a
positive impact on student outcomes. In addition, discussion surrounding the
facilitation of such teaching practices indicates that some changes to classroom
teacher goals and teaching methodologies may be required in order to integrate
meaningful problem-solving activities into the current curriculum (e.g., Costa,
2005; Lester, 2003; Ritchhart, 2002). It became apparent when reviewing the
literature for this section, that there were a number of reports into the merits of
problem posing by researchers who were investigating ways to develop students’
problem-solving competence. A review of the literature pertaining to problem-
posing perspectives can therefore be found in the following section.
2.4 Problem-posing Perspectives
The formulation of a problem is often more essential than its solution,
which may be merely a matter of mathematical or experimental skill. To
raise new questions, a new possibility, to regard old problems from a new
angle, requires creative imagination and marks real advances. (Albert
Einstein, n.d.)
These words from an eminent and past mathematician encapsulate the contents
of this following section. Einstein’s words have held meaning for recent
researchers who also believe that providing students with opportunities to pose
problems can provide them with meaningful learning opportunities that can result
in improved problem-solving competence (e.g., Brown & Walter, 2005; Lesh &
Doerr, 2003; Lesh & Zawojewski, 2007). This section reports on the literature
surrounding problem posing with particular emphasis on problem posing as a life-
long learning tool and for mathematical reform. Literature reporting on how to
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foster a problem-posing environment in the classroom is then reviewed followed
by a review of the connections that have been reported in the literature between
problem solving and problem posing.
2.4.1 Introduction
According to Silver (1994), problem posing refers to one of three distinct
mathematical activities:
1. Pre-solution posing, whereby a student poses a question from given or
perceived stimuli;
2. Within-solution posing, whereby a student reformulates a given problem
while it is being solved; and
3. Post-solution posing, whereby a student changes the conditions or goal
of a problem at the end of the solving process in order to generate more
problems.
In other words, problem posing can happen before, during or after problem
solving (Silver, Mamona-Downs, Leung, & Kenney, 1996). However, problem
posing involves far more than simply creating new problems. For example, it
provides students with opportunities to develop creativity (Silver, 1997), improve
self-efficacy, (Bandura, 1997; Marat, 2005) and to share problems with peers,
have them critiqued and hence refine problems using peer feedback (English,
Cudmore, & Tilley, 1998).
2.4.2 Problem Posing as a Tool for Mathematical Reform
“Problem posing … has the potential to create a totally new orientation toward the
issue of who is in charge [of the learning process] and what has to be learned”
(Brown & Walter, 2005, p. 5). Brown and Walter (2005) believed that problem
posing can change the orientation of mathematics lessons from a goal of finding
the correct answer and recording an appropriate method, to finding the correct
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question from a infinite array of imaginative possibilities. In fact, Lesh and
Zawojewski (2007) reported that the development of systems for interpreting
problems is equally as important, if not more so, than simply developing problem-
solving processes, while Lesh and Doerr (2003) reported the ability to “see” a
problem is as important as the ability to “do” a problem. With such strong support
from researchers, one would imagine that a revolution in mathematics education
might have already taken place. However, Ritchhart (2002, p. xxi) lamented that:
The fact is that most schools today do not try to teach for intelligence.
Rather than working to change who students are as thinkers and
learners, schools for the most part work merely to fill them up with
knowledge ….This form of schooling serves to bind rather than free the
mind.
Brown and Walter (2005) say that rather than students looking inward at
problems, through a lens with which they have become accustomed to use, and
which narrows their focus, students should be concentrating on developing
divergent forms of thinking which may present unexpected discoveries and new
and deeper opportunities for learning. This would create a major shift for the
students from a state of being to a state of becoming. As an example, Brown and
Walter (2005) cite the following problem:
Given two equilateral triangles, find a third one whose area is equal to the
sum of the areas of the other two. (p. 112)
This problem stimulates a plethora of problem-posing opportunities in order to find
a solution. Students need to ask themselves questions about the dimensions of
the triangle’s sides, the ratio between the side lengths of each of the triangles and
what variables they may use to reference the triangles and their dimensions.
Along the way, attributes of equilateral triangles will need to be revisited by the
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students, as will algebraic theory. This is a perfect example of the type of
learning that encourages reflective practice whereby students continually cycle
through iterative steps of thinking when they detect weaknesses in their previous
solution strategies and thinking (Hamilton, Lesh, Lester, & Yoon as cited in Lesh
& Zawojewski, 2007).
It can be seen from this example that much discussion and questioning would
need to occur within and between students before a satisfactory method of
solution and justification is determined. These forms of learning experiences
have been attributed with combating mathematics anxiety since posing and
solving your own problems is far less threatening than simply solving someone
else’s problem (e.g., Brown & Walter, 2005).
2.4.3 Problem-posing Skills for Lifelong Learning
Patton (2002) believed that problem posing was not merely an activity to be
undertaken in a classroom by students. He saw far reaching applications of
problem posing for adults and indeed for the advancement of our society. He
referred to problem posers as “problem pioneers” and defined them as
“individuals with a problem that is not only new, but that is destined to be
important to an entire field or the community at large some time in the future” (p.
111). He saw problem posing as an essential aspect of planning for the future
and he maintained that “those who concern themselves ahead of time with the
effects of a potential calamity such as war, famine, earthquakes, landslides, or
hurricanes are in effect problem pioneers” (p. 123).
From less dramatic settings, Patton (2002) cited several examples of problem
pioneers who have made substantial contributions to society. He talked about the
world-renowned paediatrician and author, Benjamin Spock, who posed many
questions, and solutions to questions, about how parents cope with breakdowns
in family ties as a result of raising a family in today’s society. Patton also
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discussed Freeman Dyson who posed significant questions about the research
being undertaken by two independent radiation scientists. The questions Dyson
posed allowed each radiation scientist to see the merit in the other researcher’s
work and inspired them both to move forward together to eventually discover a
new theory of radiation, for which the two scientists then received a Nobel Prize.
These examples, and others in Patton’s paper, serve as significant motivation to
educators and the broader community to value the activity of problem posing and
to ensure that students are provided with ample opportunities to practice and
hone these skills while they are still at school.
Problem posing is a natural inclination for all people; however, some people are
more naturally inquisitive and inclined to pose more problems than others (Costa,
2005; Gonzales, 1998). Inquisitive people might pose questions like; “Why can’t
we see wind?” or “When will humans first inhabit a planet other than the Earth?”
or “What makes the waves roll up onto the beach each day?”. In fact, Costa
(2005, p. 7) believed that, “all human beings have the capacity to generate novel,
original, clever or ingenious products, solutions, and techniques – if the capacity
is developed”.
There are many examples in history of famous, inquisitive people who posed
very important questions that were later investigated by many researchers, and
for many of which we now have the answers. One such example is that of Albert
Einstein who in 1911 posed the question; What if light does not travel in straight
lines but is indeed deflected around objects of huge mass such as the sun? (as
cited, in Kline, 1996) It was not until eleven years later, in 1922, that his original
question received its first independent, experimental verification and Einstein was
finally heralded worldwide as the brilliant mathematician and physicist that he
was. Had he not posed the question all those years earlier, it is unclear where
our current theories on radiation might be today. Einstein and Infeld (1938)
believed that posing problems was one of the key components of exploring
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mathematics. They said that in scientific enquiry, posing a problem well was
often a more important task than solving the problem.
2.4.4 Fostering a Problem-posing Environment
Stoyanova (2003) suggested that it was not only Einstein that supported the
notion of problem posing from as far back as the early twentieth century. She
said that “prominent scientists like Einstein, Darwin, Wertheimer, and many others
have placed far greater emphasis on the importance of posing significant
questions than on attempts to solve them” (p.32). Despite the historic discussion
of the significance of problem posing, it has only been recently that problem
posing has begun to receive broad based and renewed attention by researchers
(e.g., Brown & Walter, 2005; Cai, 2003; Contreras, 2003; Crespo, 2003; English
et al., 2005).
Although these reports, and others such as the those published by the NCTM
(2000), have championed the cause for an increased focus on problem-posing
activities for mathematics students, sadly it seems that problem-posing activities
still receive limited if any regular attention (Bobis et al., 2004). Despite this,
researchers like English (1997b) and Lowrie (2002) continue to propose that
greater emphasis be given to problem posing, particularly within the framework of
realistic contexts. On discussing the promotion of a problem-posing classroom,
English (1997b, p. 173) said “one of the main strengths of such an environment is
that it can empower all students to explore problem situations and to pursue lines
of inquiry that are personally satisfying”. Knuth and Peterson (2002, p. 579)
added their support to these comments when they said that “providing students
with ample opportunities to engage in problem posing strengthens their problem-
solving abilities and, perhaps more important, promotes the development of their
mathematical thinking”.
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2.4.5 Connections between Problem Solving and Problem Posing
It is widely accepted that problem solving is a worthwhile activity in which
students should be actively engaged (e.g., English & Larson, 2005; Lester &
Kehle, 2003; NCTM, 2000). The level of success at solving problems varies
dramatically between students (e.g., Lesh & Zawojewski, 2007), and a variety of
strategies have been trialled to address this issue (e.g., Brown & Walter, 2005;
English et al., 2005). While problem posing has been investigated in recent
years by a number of authors, those investigations have focussed more on
students from a specific year level (e.g., Lowrie, 2002), a particular strand of
mathematics (e.g., Contreras, 2003; Lavy & Bershadsky, 2003), or on the
process of problem posing in general, without consideration of the particular
cognitive attributes of the students within the study groups (e.g., Brown & Walter,
2005; Cai, 2003). In addition, the majority of the research conducted on middle-
year students was conducted in the 1990s (e.g., Gonzales, 1994; Silver & Cai,
1996), thus creating a void of recent research into the development of problem-
solving competence of middle-year students.
Over the past thirty years, significant research progress has been made in many
aspects of problem solving. However, according to Stoyanova (2003) “in
mathematics education research, problem posing has long been under the
shadow of problem solving” (p. 32). Silver and Cai (1996) concurred with
Stoyanova and added that “far less is known about the cognitive processes
involved when solvers generate their own problems or about the instructional
strategies that can effectively promote productive problem posing” (p. 522). In
their research study, Silver and Cai worked with 509 middle-year students, who
represented the range of student intelligences found within a mainstream school
setting, and investigated the correlation between the students’ problem-solving
and problem-posing performances. Their findings were convincing as they
reported a high correlation between the students’ problem-solving and problem-
posing performances throughout the eight open-ended tasks that were used to
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determine the students’ mathematical problem-solving abilities. This is consistent
with an earlier report by Silver and Jinfa (1996) who reported that middle-year
students, who were regarded as good problem solvers, were the ones who
generated the more complex and challenging problems, compared to their weak
problem-solving peers.
Rather than looking at differences in problem-solving ability, other authors such
as Cai and Hwang (2002) and Becker, Sawada and Shimizu (1999) have
considered cultural differences that impact upon a student’s ability to pose and
solve problems, with particular emphasis on differences between Asian and
American students. Interestingly, these studies showed that Asian students are
far more likely to use symbolic representations in their methods of solutions, while
American students are far more likely to seek out concrete representations to
assist them with their problem posing and solving. In addition, Cai and Hwang
found a high correlation between problem-solving and problem-posing ability of
Chinese students, but did not find a similarly strong correlation amongst the
American students in the study. These findings are in direct contrast to those
found by Silver and Cai (1996).
Researchers, such as Mestre (2000), have suggested that problem-posing
activities provide far more opportunity for students to demonstrate their
mathematical understandings than do solving problems. When solving problems
he reported that:
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the subject is provided with a problem that not only has been formulated in
advance by an expert, but also is well defined and solvable, [whereas]
problem-posing requires that the subject perform the job of the expert in
constructing a suitable problem, a job that entails combining a viable story
line with appropriate surface features in ways that embody specific
concepts. Hence, to do well at posing problems, the subject must be well
versed in how concepts apply across a wide range of problem contexts,
whereas it is possible for the solver to avoid the meaning of concepts
altogether by solving problems by means-ends analysis. (p. 160)
He goes on to recommend that teachers should not only provide problems for
students to solve, they should also provide situations or contexts within which to
construct a variety of problems to solve.
Problem-posing research has also included a focus on the pre-service teachers’
perspectives, with the premise that if pre-service, mathematics teachers have
been introduced to the underlying theory and supporting literature of problem
posing during their internships, then they may facilitate this style of teaching in
their classrooms when they become fulltime teachers (Crespo, 2003; Gozales,
1994). Crespo (2003) reported success in this endeavour when she found that
the pre-service teachers who had adopted the new teaching practices
incorporating problem-posing strategies, viewed mathematical problems as far
more useful in providing challenges for students and for probing their
mathematical thinking.
Fosnot and Dolk (2001) were also interested in what mathematics teachers did in
the classroom. They undertook a meta-study of recently completed research to
investigate what had been discovered about effective practice in the modern
mathematics classroom. As a result of their study, they developed the notion of
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‘situations for mathematizing’ as being a useful focus for teachers. In a middle-
school environment where teachers are able to teach across syllabuses, they
recommended teachers be on the constant lookout for these situations as they
encourage students to ask mathematical questions, look for and see patterns,
and become inquisitive. In particular, they said that by students posing their own
smaller problems, in the process of solving bigger problems, the bigger problem
was “owned” and it became alive and meaningful to the students.
This transdisciplinary approach to problem posing may serve to build the
connections between “school maths”, intuitive knowledge and everyday, familiar
contexts. These were noted by English (1996) as lacking from students
participating in her study. One of the recommendations from her report
suggested “getting children into the habit of recognising mathematical situations
wherever they might be” (p. 238). Bernardo (2001) followed a similar research
path and reported that “an important objective of education is to increase a
student’s capability to competently address varied problems in a changing
environment” (p.137). This was also discussed in Section 2.3. He said that one
of the challenges for students in meeting this goal is the ability to transfer
knowledge and skills from one problem in one particular setting, to another
analogous problem in another type of setting, thus creating a mapping between a
familiar problem and a problem that is being constructed (Bernardo, 2001).
Bernardo (2001), and other authors (Cai, 1997; Silver & Cai, 1996), have cited
students’ lack of understanding of problem structures as the inhibiting factor in
developing these transfer skills. He investigated a problem-posing intervention as
a means to assist students to engage more readily with problem structures. This
experience, although limited to word problems associated with probability,
allowed the high school students in his study to recognise common characteristics
in the structures of various analogous problems and hence assisted them to
become more competent at solving problems based on probability.
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2.4.6 Summary
Within this review of problem-posing literature, it has become apparent that
problem-posing activities should be seen an essential part of a mathematics
curriculum and as an important activity in the community at large. In this way,
teachers can assist their students to become “mathematicians in a mathematics
community” (Fosnot & Dolk, 2001) and make links between theory and the
practice (Mestre, 2000). Mestre (2000, p. 167) supported his position by saying
that “in traditional physics instruction, we train students to solve many problems,
believing that in solving problems correctly, the student also understands the
underlying concepts. … Instructors often delude themselves that problem-solving
proficiency implies conceptual understanding.”
From a student’s perspective, reports from the research suggested that a
problem-posing intervention has not only the ability to impact positively on
students’ problem-solving outcomes, but it can also have a positive effect on
students’ self-efficacy, creativity, sense of ownership of the learning process, and
divergent thinking. Despite the persuasive nature of this research, Silver’s (1994)
findings drew our attention to the fact that there was insufficient research on how
students respond to problem-posing intervention programs. This present study
addresses this issue.
Problem-posing research has been conducted with students, pre-service
teachers, and teachers. Authors such as Gonzales (1998) have told us that
problem-posing skills develop naturally for students when they are investigating
mathematical scenarios. Research foci on problem-posing studies have included
cultural difference, context, strategy preference, mathematics anxiety, teaching
style and student enjoyment of the learning process. While some smaller
research projects have been undertaken looking at differences between
participant students of higher and lower problem-solving ability, there appears to
be little research focussed on students who have been ascertained as high-
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achieving in standardised intelligence tests, yet underachieving in the problem-
solving criterion of school-based mathematics tests. In addition, there is some
disagreement throughout the research about whether a consistent, positive
correlation exists between a student’s problem-solving and problem-posing
abilities in middle-year students (e.g., Silver & Cai, 1993b). This study contributes
to the shortcomings in the body of knowledge regarding both of these issues.
2.5 Student Underachievement Perspectives
While problem-posing intervention, as a suitable strategy to assist in the
development of problem-solving competence, was a primary focus of this
research, a secondary focus was the group of middle-year students who
underachieve in problem solving yet who demonstrate superior scores to their
peers in mathematical intelligence tests. Therefore, the themes emerging from
the literature surrounding underachievement in middle-year students are reported
in this section.
Underachievement of students is all about potential and little, if anything, to do
with a lack of ability (Jones & Myhill, 2004). The view adopted throughout this
present study was that underachieving students have latent, unrealised talents
that may be impacted upon by appropriate intervention strategies. Delisle (1992)
made a salient distinction between “gifted non-producers” and “gifted
underachievers” that is worthy of mention before continuing with this review. He
noted that “gifted non-producers” are academically, but not psychologically, at
risk. They are typically self-assured and have independently chosen to withdraw
from the set work due to its irrelevance to them or due to boredom (see also
Gentry, Gable, & Springer, 2000). They are frustrated individuals who have
choices to make concerning how they react (Schultz, 2000). Schultz (2000, p.
42) asked “Do they change their habits to fit in with the crowd, or flex their
independence and stand up for their specific needs?”. He says teachers rarely
cater adequately for the “under-challenged” student and that teachers generally
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respond negatively to a student who says they are not learning anything new in
their class. This results in “a power struggle that leads ultimately to the
emotional and social downfall of the under-challenged student” (p. 43). By
contrast, “gifted underachievers” often do not complete work because of low self-
esteem or because they are dependent learners requiring ongoing instruction
and reinforcement from a teacher. In a group of students who have been
ascertained as above average in intelligence, yet who appear to have
underachieved in aspects of their mathematical testing, it is worth noting that
both types of underachievers may exist within the group.
Gender differences in students have also been reported upon in recent studies
on underachievement. Jones and Myhill (2004, p. 531) reported that:
the identity of the underachiever has become synonymous with the
stereotypical identity of boys. Teachers know what underachievement
looks like: it looks like a boy who is bright, but bored. … [whereas]
underachievement in girls is often overlooked or rendered invisible. … It
becomes a matter of concern if teachers perceive boys as the vessel of
potential and of latent ability, while the high achievement of girls is seen to
be about performance, not ability.
Reis and Siegle (2006) also undertook a study that focussed on gender
differences and made a contrasting comparison. They reported that girls who
underachieved almost always had serious boyfriends who came first, before
school work, whereas boys who underachieved often got into trouble and spent
hours playing videogames or watching television. These findings were followed
a year later by an international study conducted on 1,700 German, Canadian and
Israeli middle-year students. Boehnke (2007), investigated whether
underachievement was caused as a consequence of high peer pressure. With
the assumption that students underachieve to avoid social exclusion, Boehnke
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reported that the primary victims of peer pressure that resulted in
underachievement in mathematics were girls, while the perpetrators of the peer
pressure were more likely to be boys. These gender differences are useful to
consider when undertaking future studies about underachievement of students in
a co-educational setting.
Whether we are considering a male or a female student, according to Reis and
Siegle (2006), there are three degrees of underachievement:
1. Pervasive and devastating – which results in student drop-outs and “life
failures”;
2. Moderate – which results in failing grades in some areas; and
3. Minimal – which results in lower grades than expected across some or all
subjects.
The literature suggests that it is members of these last two categories that are
most able to be positively influenced by the adaptation and modification of
situational variables in and out of the school setting (e.g., Cummins & Sayers,
1995; Gootman, 2001; Simons-Morton, Crump, Haynie, & Saylor, 1999).
Dispositional factors such as gender (e.g., Reis & Siegle, 2006; Younger &
Warrington, 1996), boredom (e.g., Kanevsky & Keighley, 2003), and
computational proficiency (Johnson, 2000) require individual intervention
strategies for maximum improvement of student performance and as such, are
not part of this present study. However, since attention to the common
situational factors, such as pedagogy and classroom structure, have been shown
to have a positive influence on students affected by dispositional factors (e.g.,
Gootman, 2001; Kanevsky & Keighley, 2003; Simons-Morton et al., 1999), a
group-intervention process may be an appropriate strategy for use in further
research.
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Family, school, and the unique attributes of preadolescents impact upon the
development of underachievement (Baker et al., 1998). Researchers such as
Cummins and Sayers (1995) suggested that underachievement in mathematics
is commonly the result of traditional teaching pedagogy whereby the subject is
“taught sequentially through drill and memorisation” (p. 145) with a classroom
structure adopting an emphasis on straight rows of desks and chairs, equanimity,
and acquiescent behaviour. Cummins and Sayers considered this form of
learning to “encourage passivity, and to be ineffective and xenophobic” (p. 145),
and is particularly unhelpful to primary and middle-year boys (Jones & Myhill,
2004; Younger & Warrington, 1996). Kanevsky and Keighley (2003, p. 1)
supported this position and indicated that some students develop a “growing
sense of moral indignation” toward this style of pedagogy and respond by
disengaging from the learning process.
This review on recent literature has uncovered a varying range of views
regarding the underlying causes of underachievement in middle-year students.
Some researchers (e.g., Boehnke, 2007; Jones & Myhill, 2004; Reis & Siegle,
2006) have focussed on the gender differences while others (e.g., Cummins &
Sayers, 1995; Gootman, 2001; Kanevsky & Keighley, 2003) have categorised the
differences into situational and dispositional factor categories. With general
agreement from researchers that middle-year students are particularly vulnerable
to underachievement in mathematics, it could be expected that a large body of
research may have already reported specific teaching strategies and
interventions to assist these students to reach their potential. Some guidance for
future research may be found from researchers of developmental learning, such
as Lincoln and Denzin (2000) who have reported that empowering students to be
in control of their own learning may address issues of underachievement and
disengagement of learners, or from Bjorklund (2000) who emphasised the
importance of students playing an active role in their own education. If this line of
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advice is to be followed a constructionist (as opposed to constructivist)
theoretical perspective would be helpful as a foundation for future research.
2.6 Constructionist Perspectives
According to Crotty (1998), the predominant paradigm of research in the western
world has it roots in ancient Greek philosophy and has previously focussed
around a belief that “objective truth” combined with suitable methods of inquiry,
can lead us to “accurate and certain knowledge of that truth” (p. 42). The
assumption then is that truth and meaning are contained within objects and
artefacts and are independent of human consciousness, interaction and thought.
This stance is rejected by followers of constructionism (e.g., Schwandt, 2001)
who prefer to view truth and reality as being constructed through the interactions
between human beings, and conveyed within a generally social context, as is
clearly the case with students in a classroom. This is particularly relevant in the
learning of mathematical problem solving as the development of mathematical
concepts and problem-solving competence are more interdependent,
contextually situated and socially constructed than traditional theorists have lead
us to believe (Lesh & Zawojewski, 2007). Indeed, recent research has reverted
to a focus on the social perspectives of group learning of mathematics as a
means to investigate how students develop cognitive models and ways of
thinking (e.g., Greeno, 2003; Zawojewski, Lesh, & English, 2003). However,
rather than the main learning of mathematical concepts arising from student-
student interactions, with the teacher as a facilitator to ensure the intended
learning occurs, it is still the case that most learning in mathematics classrooms
is dependent solely on teacher-student interactions (Lesh & Zawojewski, 2007).
Constructionism began to emerge as a theoretical perspective in the 1980s as a
result of a series of computing technology projects undertaken by Seymour
Papert of the Epistemology and Learning Group at the MIT Lab (see Papert,
1980). He used the foundation of constructivism that supports learning as
“building knowledge structures” and added the concept that learning occurs most
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felicitously when the learner is consciously involved in constructing a “public
entity” (Papert, 1991). Constructivism, as Piaget and Inhelder (1969) saw it,
required knowledge to be passed from one person to the next as a whole,
whereas Papert’s view of constructionism held that knowledge was constructed
by individuals in the process of their practice and was not contained solely within
the mind of an individual (Papert, 1991). A fundamental premise of
constructionist theory is therefore that best learning occurs when students are
actively engaged in creating objects that are meaningful to them and that they
can share with their peers (Crotty, 1998). According to the NRC (as cited in
Kilpatrick et al., 2001, p. 135) “to become proficient, they [students] need to
spend sustained periods of time doing mathematics – solving problems,
reasoning, developing understanding, practicing skills – and building connections
between their previous knowledge and their new knowledge”. However, in
mathematics classrooms, it is clear that students do not often explore and create
novel problems of their own, about topics of interest to them, or share them with
their peers (Lesh & Zawojewski, 2007).
As mentioned earlier, the embedded theory of knowledge in constructionism is
founded on the principle that all knowledge “is contingent on human practices,
being constructed in and out of interaction with human beings” (Crotty, 1998, p.
42) and brings subjectivity and objectivity together in a way that no other single
theory does. Therefore, constructionists believe that understanding and
interpretation come from construction rather than discovery alone. This
construction centres on the “collective generation of meaning” facilitated by the
experiences of the learner (Schwandt, 2001 p. 42). For a constructionist study to
investigate developmental and affective changes in middle-year students, two
theoretical frameworks can provide a clear structure around which to develop
further research projects; the post-modern theoretical framework and the critical
theoretical framework. Critical postmodernists hold the view that as researchers
we do not simply seek to investigate our world; we seek to create change as a
result of our investigations and findings (Tierney, 1997). Indeed, as Tierney
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wrote, it is the role of the critical postmodernist researcher “to assemble new
practices, languages and ways of seeing and hence acting in the world so that
individuals and groups will not of necessity need to subsume their identities into a
homogenous mass” (p. 24). A teaching experiment, incorporating a problem-
posing intervention, could provide such an opportunity for students to create their
own learning, through posing and publicly sharing their own problems, in a forum
that supported, encouraged and celebrated a student’s individuality.
Throughout a teaching experiment, a constant awareness remains that
conducting research with child participants is inevitably based on a perceived
power differential between the participants and the adult researcher and that the
identities of both the researcher and the participants frame the social situation
created by the intervention (Broido, 2002). This awareness is important when
conducting an experiment and analysing the data, as the research is not
independent of the researcher’s and participant’s assumptions about the
research process and anticipated findings of the research study. Consistent with
the framework of critical theory, the researcher must attempt to empower the
participants to take control of their learning experiences and hence have a sense
of “emancipation” from the normal constraints of the teacher/student relationship
(Lincoln & Denzin, 2000).
Issues of social justice, power and identity are at the base of critical theory. As
Lincoln and Denzin (2000) write, “critical theorists claim that society, in its current
form, is oppressive” (p. 1056). While the word oppressive may seem overly
dramatic when discussing a middle-year mathematics classroom, it is clear that
the traditional teaching methodologies, employed by practising mathematics
teachers, have the “power” in the classroom held predominantly by the teacher
(e.g., Roberts, 1996; Stein et al., 2003). From a management and safety
perspective, it is important that the adult in charge is in clear control of what is
occurring in the classroom. However, this does not necessarily preclude the
students from sharing in the decision making process of how learning can occur
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and indeed in playing a significant role in customising the learning experience for
themselves. A problem-posing intervention, where students choose how they
want to pose and share problems, could contribute to addressing this issue and
could be used as a means to partly restructure a power base in the classroom. A
teaching experiment could therefore promote a liberating and transformative
learning environment (Broido, 2002) that dismantles traditional oppressions, thus
paving the way for “jointly constructed knowledge” and “respectful participation”
(Alvesson & Skolberg, 2000).
Together, the underlying premises and truths, held in the critical and postmodern
theories, define the relationship between the researcher and the participants as
being interactive, subjective, and interdependent. They acknowledge the reality
of the mathematics classroom as being pluralistic, complex, and difficult to
quantify. The theories acknowledge that research is context specific and that
any findings and interpretations of the data are most appropriately viewed in such
a context.
2.7 Conclusion
This review of the literature has considered five fields of research; cognitive
developmental theories, problem solving, problem posing, student
underachievement, and constructionist perspectives. There are shortcomings in
all five fields of research that leave open a possible focus on underachieving,
high-ability, middle-year students, and the correlation between a problem-posing
intervention and the development of the students’ problem-solving competence.
The review of research into cognitive developmental theories demonstrated that
students learn best when they are actively engaged in their learning (e.g.,
Bjorklund, 2000). For the learning of mathematics, Kelly and Lesh (2000)
argued that:
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Doing mathematics involves (more than anything else) interpreting
situations mathematically, that is, it involves mathematizing. When this
mathematization takes place, it is done using constructs [conceptual
models]. …These constructs must be developed by the students
themselves; they cannot be delivered to them through their teachers’
presentations. (p. 215)
The need for curriculum reform in mathematics teaching has been heralded for
many years, with recent advocates such as Lester (2003) and Groves and her
colleagues (2006) supporting the position stated in the reports by the NCTM
(2000) and the NRC (2005) that both call for problem solving to receive far more
attention than it currently does in mathematics curriculums. The review provides
evidence to suggest that problem-solving activities are a rich form of instructional
strategy for the development of skills in mathematics (e.g., English, 2002). While
there is little difference of opinion by researchers to say that problem solving
should be an integral part of a mathematics curriculum, the most appropriate way
to develop problem-solving competence has created conflict amongst
researchers. Some suggest that the most traditional method of teaching problem
solving, through teaching strategies alone, does not necessarily result in
improved problem-solving competence (e.g., Bobis et al., 2004), and the NRC
(2004) warns us that significant classroom time can be lost if student-focussed
activities are allowed to become unfocussed. Despite this warning, other
researchers suggest that teaching problem solving through problem-posing and
modelling activities is worthwhile (e.g., English et al., 2005), and the NRC (2005)
acknowledges that issues, such as losing valuable class time, are not
insurmountable.
Silver (1997) was one of the researchers who investigated possible correlations
between problem-posing activities and the development of problem-solving
competence, calling for further research into a possible connection. There is
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disagreement among the researchers about the strength of the correlation
between problem posing and problem solving. Earlier researchers such as Silver
and Cai (1993a) were not convinced about the link whereas more recent
researchers reported that the correlation was evident through their research (e.g.,
Lavy & Bershadsky, 2003; Lowrie, 2002). This review of the literature has
demonstrated that further research into this possible correlation is warranted
(e.g., Gonzales, 1998).
Consideration has also been given to the research surrounding the possible
reasons for underachievement of middle-year students. Groves and her
colleagues (2006) looked at the attributes of the students they were investigating
and reported that underachieving students are capable of developing
sophisticated strategies for solving problems. Cummins and Sayers (1995)
reported that underachievement is commonly the result of a traditional teaching
pedagogy, while Jones and Myhill (2004) found the physical arrangement of the
classroom played a significant role in the underachievement of middle-year boys.
Reis and Siegle (2006) investigated the varying degrees of underachievement
and how they may individually be addressed, while Gootman (2001) and others
(e.g., Simons-Morton et al., 1999) investigated the impact of situational and
dispositional variables in underachievement. Kanevsky and Keighley (2003)
continued the research into situational and dispositional variables, finding that
improvements in underachievement for students demonstrating dispositional
variables, were less likely to occur through group intervention than individual
attention. They did however, support group intervention as a beneficial strategy
for addressing situational variables. Despite this consideration of research into
why students underachieve, there was little research to be found on practical
strategies to assist underachieving students with high ability in mathematics. This
is an area in need of further research.
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The review of constructionist perspectives established a need for research into
teaching practices that allow students to become more active and responsible for
their own learning. A teaching experiment, involving problem posing with
students as authors, could provide an opportunity for them to become more
engaged in the learning process. In addition it could provide students with an
opportunity to be acknowledged as individuals in terms of how they prefer to
construct their own learning (Gardner, 1999b). The traditional power base in the
classroom would become shared between the facilitator and the participants.
This sharing of power is a strong tenant of critical theorists (e.g., Broido, 2002)
who suggest it is a requirement for improved performance of students who may
currently be underachieving due to their disenfranchisement from the learning
process.
In summary, the need for students to be competent at solving problems is a
recurring message in the literature, suggesting that teaching practices that
facilitate problem-solving opportunities are to be encouraged. There is a need
for further research to be conducted into ways students can improve their
problem-solving competence and into the links between a problem-posing
intervention and the development of problem-solving competence, which are
currently unclear. In addition, little consideration has been given to students who
underachieve but are of high ability, in the studies of problem solving. This
particular group of students could therefore become a focus group of future
studies.
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Chapter 3
Research Design
3.1 Chapter Overview
The purpose of this present study was to investigate and explain the links
between a problem-posing intervention and the development of problem-solving
competence of middle-year students who have been ascertained as above
average in standardised intelligence tests, yet below average in the problem-
solving criterion of their school mathematics tests. This chapter outlines how this
study was structured to achieve this aim.
Five main sections comprise this chapter. The first section provides an
introduction to the design of the present study (see Section 3.2) while the second
section provides the reasoning that lead to the establishment of the three specific
research questions (see Section 3.3). The third section contains the rationale
and structure of the research design and the selection process of participants
(see Section 3.4), while the fourth section of this chapter describes the methods
for the collection and analysis of the data and the associated considerations that
were given to reliability, validity and ethical issues (see Section 3.5). The final
section presents a conclusion of this chapter (see Section 3.6).
3.2 Introduction
A compelling theme arising from the literature was that students appear to
experience better outcomes when they are actively engaged in the learning
process (e.g., Bjorklund, 2000). The philosophical foundations of
constructionism are centred on this premise and extend to a student’s
construction of objects, such as problems, that can be shared with others
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(Papert, 1991). This philosophy of learning provided an ideal framework within
which to construct the design of this present study.
The review of the literature in Chapter 2 considered five fields of research;
developmental learning theories, problem solving, problem posing, student
underachievement, and constructionist perspectives. The review uncovered a
number of shortcomings in these fields of research. It was discussed in Chapter
1 that traditional practices of teaching mathematics have been held responsible
for the marginalisation of some students and, as a result, there have been
repeated calls for a review of mathematics pedagogy (e.g., English, 2002; Lesh &
Zawojewski, 2007; NRC, 2004). Research that can inform new teaching practices
is an area needing further attention by researchers, while teaching practices, that
actively engage students in directing their own learning, have been suggested as
a useful focus for future research studies (e.g., Bjorklund, 2000). A student-
centred approach to learning, underpinned by a ‘Critical’ theoretical perspective,
was therefore incorporated into the design of this present study and will be
discussed in more detail later in this chapter.
The literature review provided evidence to suggest that problem-solving activities
are a rich form of instructional strategy for the development of skills in
mathematics (e.g., English, 2002), yet there is a decline in research focussing on
problem solving (Lester & Kehle, 2003; Stein et al., 2003). Lesh and Zawojewski
(2007) have however, pursued their investigations on this topic and expressed
concerns about “substantive mathematical concepts” being taught in isolation of
problem-solving activities. They emphasised a synergistic relationship between
problem solving and learning. While the focus of this present study is on
developing problem-solving competence in apparent isolation of the student’s
mathematics curriculum, it is through the focus of developing problem-solving
competence that ultimately students may become more successful at learning the
“substantive mathematical concepts” that Lesh and Zawojewski discussed in their
report. In this way, it can be justified to withdraw students from their normal
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mathematics classes to become involved in the problem-posing intervention
incorporated into this present study.
While there is little difference of opinion by researchers to say that problem
solving should be an integral part of a mathematics curriculum, the most
appropriate way to develop problem-solving competence has created conflict
amongst researchers and warrants further investigation (see Chapter 2). Despite
some researchers suggesting that significant classroom time can be lost if a less-
traditional approach to teaching mathematics is pursued and student focus on the
task at hand is lost (e.g., Bobis et al., 2004; NRC, 2004), other researchers
suggested that teaching problem solving through problem-posing activities is
worthwhile (e.g., English et al., 2005; Lavy & Bershadsky, 2003; Lowrie, 2002). A
consistent viewpoint of whether or not a problem-posing intervention can assist all
students to develop their problem-solving competence, and as a result their
mathematical skills, cannot be found in the literature. This study has therefore
investigated the link between a problem-posing intervention and the development
of problem-solving competence.
To determine a focus group for the present study, a review of the literature into
underachievement was conducted. The desire to provide a learning environment
that allows students to be actively engaged in learning and to “assemble new
practices” (Tierney, 1997) that have measureable and positive outcomes, is
consistent with a post-modernist theoretical perspective and is a goal of this
present study. Some researchers reported that underachievement is commonly
the result of a traditional teaching pedagogy (e.g., Cummins & Sayers, 1995),
while others discussed that middle-year students were particularly at risk of
underachievement as a result of disengagement, for a number of situational and
dispositional reasons (e.g., Gootman, 2001; Jones & Myhill, 2004; Kanevsky &
Keighley, 2003). While these views are generally accepted, there was little
research found that provided practical ways for educators to address
underachievement in middle-year students who have demonstrated a high level of
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mathematical ability in routine knowledge and procedural aspects of a
mathematics curriculum, or indeed those students who have demonstrated
above-average intelligence in intelligence tests. This group of students therefore
formed the focus group for this present study, that is, middle-year students who
have performed below the average for their cohort in problem-solving
competence but who have also demonstrated above average results in
intelligence testing.
A research study that empowered students to become more active and
responsible for their own learning (critical theory), that expected positive learning
outcomes to be achieved for the participants (post-modern theory), and that
incorporated a problem-posing intervention with middle-year students as a means
to develop problem-solving competence and re-engagement of these students in
the learning process, could be facilitated through a teaching experiment. This
present study incorporated these features into a seven-episode teaching
experiment with Year 7 students that took place in the final term of their school
year.
The constructionist approach to social science research supports opportunities to
collect both qualitative and quantitative data. Gergen (1996) supported this
triangulation approach to data collection when he raised concerns about the sole
use of quantitative approaches to test changes in learning development. He
wrote, “there is nothing about a social constructionist psychology that rules out
empirical research” (p. 5) however, “from a constructionist perspective, the
traditional attempt [using only quantitative approaches] to test hypotheses about
universal processes of the mind …. seems at a minimum misguided, and more
tragically, an enormous waste of resources” (p. 5). A number of researchers
have supported Gergen in this position. For example, Lesh and Zawojewski
(2007 p. 51) said that:
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general theories of learning are useful for guiding the design of worthwhile
learning activities and theoretical models, but useful educational models
for learning and teaching in specific contexts, often need to draw on and
build from more than a single practical or theoretical perspective.
Further, Bell (2004, p. 3) reported that “learning is too complex a phenomena to
be the sole province of any one discipline, theoretical perspective, or research
method”. Support for multiple approaches to data collection in an educational
setting with students was also offered by a number of other researchers (e.g.,
Brown & Campione, 1998; Bruner, 1996; Goldman-Segall, 1998; Love, 2002).
The repeated recommendation to collect both qualitative and quantitative data
precipitated the use of two theoretical frameworks for this present study that are
discussed in the following section.
The review of the literature pertaining to the development of problem-solving
competence highlighted that the bulk of research undertaken in this area
presupposes that problem solving is a skill acquired after a particular cycle of
learning has been facilitated. The cycle usually consists of:
1. Learning mathematical knowledge
2. Learning problem solving strategies
3. Acquiring metacognitive strategies
4. Applying correct strategy for solution, and
5. Unlearning ineffective beliefs and dispositions while learning effective
ones (Lesh & Zawojewski, 2007).
The research questions that have arisen from the literature will challenge the
validity of this learning cycle and address the shortcomings of the assumptions
underpinning the cycle. As Lesh and Zawojewski reported, “research based on
these preceding assumptions have been unimpressive” (2007, p. 88).
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3.3 Research Questions
The following research questions related to the focus group of students for this
present study, who scored highly on a widely used intelligence test but who were
considered by their teachers to be underachieving in problem-solving
competence, as evidenced by the results of their Year 7 mathematics tests.
The overarching question that was responded to in this study was:
How might a problem-posing intervention impact upon the development of
problem-solving competence of underachieving, middle-year students?
The review of literature related to this overarching question generated the
following more specific research questions.
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?
Research Question 1 is at the heart of this study. The response to this question
provides further evidence to determine whether a problem-posing intervention
can have a beneficial impact on the engagement of middle-year students in the
learning process. As the link between engagement and underachieving has
been established in the literature (e.g., English, 2002; Lesh & Zawojewski, 2007),
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if the engagement of these students can be increased through a problem-posing
intervention, then an opportunity exists to address their underachievement.
Directions for curriculum reform are provided by the response to Research
Question 2. The study findings associated with this response also contributed
information to the debate in the literature between those who believe there is a
strong correlation between a problem-posing intervention and problem-solving
competence (e.g., Lavy & Bershadsky, 2003; Lowrie, 2002) and those who are
yet to be convinced (e.g., Silver, 1997). It was a goal of this study to investigate
a teaching experiment that may result in increases in problem-solving
competence. However, other developmental learning changes that resulted from
the teaching experiment may also contribute to the effectiveness of mathematical
teaching programs. The investigation of Research Question 3 allows data related
to developmental learning changes that were collected throughout this study, to
be analysed.
3.4 Research Design
3.4.1 Research Design Rationale and Structure
As writers like Crotty (1998) point out, there are four main elements to consider
when designing a research proposal. The study should be informed by a
philosophy of beliefs (epistemology) that are underpinned by theoretical
perspectives leading to a suitable methodology for appropriate methods of
data collection.
In order to determine these four elements, Crotty suggests researchers consider
the following four questions:
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1. What methods do we propose to use?
2. What methodology governs our choice and use of methods?
3. What theoretical framework lies behind the methodology in question?
4. What epistemology informs the theoretical perspective?
Crotty (1998, p. 2)
These four questions are answered in the following section. The design of this
present study incorporated two mutually exclusive groups of students to enable
comparisons between the cohorts, identifying developmental learning changes
and other changes that resulted from the problem-posing intervention. The 31
participants, 16 in the Comparison Group and 15 in the Intervention Group, were
drawn from consecutive year cohorts at the one school. While Comparison and
Intervention students from the same year cohort may have been considered
more ideal in terms of the control of experimental variables, the possible cross-
contamination of students who were in the Intervention Group, sharing
information with the Comparison Group students, would have been counter
productive to this research. This compromise was supported by Lesh and his
colleagues who said that, “planning research often involves trade-offs similar to
those that occur when an automobile is designed to meet conflicting goals (such
as optimising speed, safety, and economy)” (2000, p. 19). Students in the
Comparison Group were pre-tested and post-tested in 2006, while students in
the Intervention Group were involved in the teaching experiment and tested in
the following year (see Figure 3.1). Three case study students were selected
from the Intervention Group (see Section 3.4.2 for further information on how
these students were chosen).
The NCTM-2000 Standards (2000, p. 18) highlighted the multifarious nature of
teaching mathematics and reported that the teaching of mathematics “must
balance purposeful, planned classroom lessons with the ongoing decision-
making that inevitably occurs as teacher and students encounter unanticipated
95
discoveries or difficulties that lead them to unchartered territory”. This being the
case, it was clear that the methodology chosen for this study needed the
flexibility contained within design-based research (Brown, 1992) and also the
rich, contextualised and flexible structure afforded by a teaching experiment
(Malara, 2002). Both methods rely on the cyclical nature of classroom-based
research that incorporates the refinement of the instruction process throughout
the intervention.
Comparison Group
Term 4 2006
Intervention Group
Term 4 2007
Pre-test
Post-test
Teaching
Experiment
Researcher observation
Researcher journal
Student workbooks
Informal interviews
Pre-test & Survey
Post-test & Survey
Figure 3.1. Research Study Framework
Teaching is a fluid and dynamic undertaking that occurs in response not only to
work programs and lesson plans, but also to the characteristics of the students in
the class, the weather conditions and the activities that are occurring within the
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school environment on any one day. For example, a teacher may become aware
at the start of the year that there is a student in the class who has performed
particularly poorly in testing for mathematics in the previous year. This may
influence how the teacher talks to that particular student (to be mindful of the
self-esteem of the student), where the teacher sits the student in the classroom,
(to assist the teacher to closely monitor and assist the student) and to the
modification of tasks for the student to undertake (individualising the work
program). From a whole-class perspective, on any given day, the “mood” of the
class may be impacted by the weather, resulting in students being particularly
unsettled and unable to self-regulate their behaviour. In response, a teacher
may provide opportunities for students to do more structured tasks. These
examples are typical factors that need to be accommodated by a teacher through
the refinement of lesson plans both within and between lessons. This flexibility
was built into the design of this present study and is further discussed in Chapter
4.
As the purpose of this present study was to gain an “authentic understanding”
(Richardson, 1999) of the changes that occurred as a result of a problem-posing
intervention, without a loss of scientific rigour, the design of this study blended
elements of both qualitative and quantitative methodologies. The significant
strengths of combining both qualitative and quantitative methodologies have
been reported upon by a number of researchers including Bruner (1996) and
Brown and Campione (1998). Discussion related to the instruments and
methods used for data collection can be found later in this chapter.
3.4.2 Participants
As mentioned previously, the participants for this present study, typically aged
between 11 and 12 years of age, were drawn from two consecutive Year 7
cohorts. Year 7 is the first year of secondary studies in New South Wales,
Victoria, Tasmania and the Australian Capital Territory but is the final year of
primary school studies for students in Queensland, South Australia, Western
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Australia and the Northern Territory. While the middle-years of schooling
generally refer to students in Years 5 to 9, students in Year 7 were chosen as
this was the final year of primary schooling at the research school. After Year 7,
students move into the senior part of the school with different lesson times,
separate classrooms and recreation areas. The new Year 8 students do not
commonly come into regular contact with students in the current Year 7 cohort
and are less likely to have communication with them. Had students from the
same Year 7 cohort be chosen to be participants in both the Comparison and
Intervention Groups, there would be a chance of contamination of the data due to
conversations held between Intervention Group students returning from a
teaching episode with the researcher and Comparison Group students who
remained in their usual classes. In addition, if the number of students who
satisfied the selection criteria in any one year were to be divided equally to create
the Comparison and Intervention Groups, then the numbers available to each
group would be substantially reduced.
Two specific criteria were used to select the Year 7 participants for this study.
The criteria were chosen in consultation with the Head of Department –
Curriculum and the Coordinator of Year 7 Mathematics at the research school,
who were both wanting to seek answers to the apparent underachievement of
students who seemed, from intelligence testing, to have the potential to have a
greater competence in problem solving than that which they had demonstrated in
their mathematics tests. The Head of Department – Curriculum, the Coordinator
of Year 7 Mathematics and the researcher felt that this particular group of
students could gain most immediate and significant benefit from the findings of
this present research study (see also Lesh & Zawojewski, 2007).
The first selection criterion required the participants to have scored above the
60th percentile in the standardized Middle Years Ability Test (MYAT) (Australian
Council for Educational Research, 2005), which is routinely administered to all
students at the research school, in their first term of Year 7. The MYAT is a
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general ability test that provides an overall, norm-referenced, ability score
combined with a profile of the student’s performance. Further details on this test
can be found in Section 3.5.1.1.
The second criterion required the students to have scored below the Year 7
cohort average for Criterion B (Problem Solving Criterion) in their school
mathematics tests during the first semester of Year 7. Meeting with the students
who satisfied the two criteria in 2006 and 2007 and the researcher were arranged
by the Coordinator of Year 7 Mathematics in July of both years. That is, students
who were eligible to participate in the Comparison Group had their meeting in
July of 2006 and students who were eligible to participate in the Intervention
Group had their meeting in July of 2007. It was at this stage of the year that the
Semester 1 mathematics results for these consecutive cohorts had been
finalised. The students who attended the meetings were given a brief overview
of the purpose of the study and the voluntary nature of their participation should
they wish to become involved in the study.
At the meetings, students were given an information letter and permission slip
(see Appendices A and B) to take home to their parents. Students of each group
were asked to return the signed permission slips within two weeks of the
meetings for them to be included in the study. This time frame was chosen to
provide the students with time to remember to hand the sheets to their parents
and to provide parents with time to contact the researcher should they have any
questions about their son’s or daughter’s involvement. No attempt was made to
choose equal numbers of male and female participants or participants that were
representative of all ethnic groups at the school. These considerations were
outside the boundaries of this present study. However, students who met the
two criteria for the Intervention Group but who had a sibling or cohabitant that
that had been a participant in the Comparison Group, were not chosen for
participation in this present study. Their inclusion would have opened up
opportunities for cross-contamination of the data.
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As mentioned previously, an aim of this study was to investigate students who
had been ascertained as above average in the MYAT test (Australian Council for
Educational Research, 2005) but who had performed below the average of their
cohort in the problem-solving aspect of their mathematics test. From a
mathematician’s point of view this would suggest that any student scoring in the
51st percentile or higher would be considered above the average and therefore a
potential participant. However, testing instruments such as the MYAT test are
useful in providing a strong indication of intelligence only, rather than an inflexible
and unquestionable representation of a student’s ability (e.g., Bjorklund, 2000;
Gardner, 1999b).
Reliability and validity factors, beyond the control of a test administrator such as
the pre-disposition of a candidate at the time of taking the test can influence test
results (see Section 3.5.3). Therefore, rather than choosing students who were
in the 51st percentile or higher, the researcher allowed for a 10% margin in
testing results and chose students scoring in the 61st percentile and above. The
size of the margin was chosen to be as large as possible without reducing the
potential number of candidates for each of the Comparison and Intervention
Groups to below 24 students, the size of a usual class of students in the research
school. As it happened, the size of the two groups was further reduced by the
number of students who did not return their permission forms, those who were
absent on one or both of the pre-testing and post-testing days, or by Intervention
Group students who were absent for more than two of the teaching episodes. As
a result of the attrition, the final Comparison Group was comprised of 16 students
while the final Intervention Group was comprised of 15 students.
To undertake a deeper investigation of the research questions associated with
this study, three students were identified early in the teaching experiment as
case study students. Initially, all participants were potential case study students.
However, the three case study students were selected for their potential to
provide data to assist in responding to the research questions of this present
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study. For example, students who began the teaching experiment as particularly
disengaged were considered potential case study students to assist with the
response to Research Question 1. In addition to being initially disengaged, other
qualities such as their score in the MYAT test (Australian Council for Educational
Research, 2005) or their behaviour during their disengagement, would provide
some useful comparisons between case study students. The first two students
were identified within the first teaching episode due to their specific
characteristics.
The first case study student, Andrew, had the lowest MYAT (Australian Council
for Educational Research, 2005) score of the Intervention Group students
(scoring in the 76th percentile) and was noticeably disengaged from the class
activities in the first teaching episode. In fact, he was far more interested in
talking to his neighbours than posing or solving problems. The second case
study student, Nicole, had the highest MYAT score of the Intervention Group
students (scoring in the 94th percentile) and was noticeably less inclined than her
peers to participate in posing problems. She was however, very quiet and did
not disturb other students, preferring to watch what other students were doing.
The characteristics displayed by these two students were consistent with the
findings of Jones and Myhill (2004) who reported that underachieving boys are
often typified by rowdy behaviour and disengagement while underachieving girls
are often “rendered invisible” because they quietly disengage from activities.
The study of these two students produced useful data to respond to Research
Questions 1 and 3.
The third case study student, Paul, was identified in the first teaching episode but
only confirmed as a case study student by the fourth teaching episode. His
initial enthusiasm, yet inability to organise his thoughts, drew attention at the start
of the teaching experiment. While his levels of enthusiasm did not waver or
diminish throughout the teaching experiment, his ability to become more
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organised in his thoughts and coherent in his explanations became apparent
within a few teaching episodes. These observations were consistent with
Sternberg’s (2000) proposition that four change mechanisms; automatisation,
encoding, generalisation and strategy construction play a substantial role in
developmental learning and discussion about how Paul demonstrated each of
these mechanisms can be found in Chapter 4. Paul had the median MYAT
(Australian Council for Educational Research, 2005) score of the Intervention
Group students (scoring in the 85th percentile) and was the most willing and
enthusiastic student to participate in the problem-posing activities and to share
his problems with his peers. Data collected about Paul were particularly useful in
responding to Research Questions 2 and 3.
Throughout the entire teaching experiment, observational data primarily about
the three case study students were collected using three audio-video tape
recorders. The cameras were labelled Camera One, Camera Two and Camera
Three. This enabled accurate transcripts to be made of conversations between
students and the researchers, and between the students themselves. All
extracts of the transcripts used in this document are described using the number
of the camera, the teaching session number from which the recording was made
and the time into the recording. The time is displayed as mm:ss which refer to
the number of minutes (mm) and seconds (ss) into the recording of a particular
session. As soon as the case study students were identified, the two fixed-
position cameras were positioned to ensure that footage of these students was
captured as well as the activities of surrounding students. Paul and Andrew sat
at adjacent tables which enabled a single fixed camera to make recordings of
both students simultaneously. The second camera was focussed on the table
where Nicole sat. The third camera was mobile and was used to capture
informal interviews with the three case study students as they occurred as well
as recording details of general class activity. This arrangement of recorders
ensured that accurate transcripts could be made of the conversations the three
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case study students had with their peers and with the researcher. In addition, it
also allowed the researcher to repeatedly review each teaching episode from
different perspectives in order to respond to the research questions.
3.5 Methods
An introduction to the teaching experiment, the methods of data collection and
the testing instruments used in this present study are discussed in this next
section. This is followed by a detailed description of how the data were analysed
and the reliability and validity issues were taken into account throughout the data
collection process. A more detailed description of each teaching episode within
the teaching experiment can be found in Chapter 4.
The teaching experiment took place in the library classroom of the research
school. The experiment was not able to be conducted in the students’ usual
classrooms as participants were drawn from four different Year 7 classes. The
sourcing of participants from across the four classes was necessary to obtain
enough students to proceed with the study. The withdrawal of students from
their usual classrooms, classmates and teachers was therefore, a necessary
construct of the research design. The undertaking of research with the students
out of their everyday classroom setting and environment does have implications
for the validity of the data if it is to be used to make statements about student
learning in their usual classroom environment. However, the library classroom in
their school library was a room with which the students were very familiar and in
which they often undertook activities such as language or research lessons. The
students were familiar with working in groups other than their ‘homeroom’
groups, as alternative Year 7 groupings occurred for language classes and sport.
Therefore, the use of the library classroom with students from four different Year
7 classes was deemed to be an acceptable compromise in attempts to create an
authentic and familiar classroom setting for the students in both groups.
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While the withdrawal of students from their usual class groupings could be seen
as a limitation of this present study, it is also a feature supported by the
philosophical underpinnings of this study. The participating students in the
Comparison and Intervention Groups have all been ascertained as
underachieving in their usual classroom environments. This present study gave
the students in the Intervention Group an opportunity to work in a different
environment where they could feel safer to explore the concept of mathematical
problem solving from a unique and different perspective. Problem posing offers
students an opportunity to incorporate “ingenuity” and “playfulness” into learning
that is not available in a traditional mathematics classroom (Brown & Walter,
2005). Brown and Walter said that it was through posing problems that students
can discover that “a slight turn of phrase, or recontextualising the situation, … will
transform it [a problem] from one that appears dull to one that ‘glitters’” (Brown &
Walter, 2005, p. 5). They add that a problem-posing environment is “the
beginnings of a mechanism for confronting the rather widespread feelings of
mathematical anxiety” that can lead to underachievement (Brown & Walter, 2005,
p. 5).
The research methods chosen for this study were designed to be consistent with
both the theoretical frameworks of postmodernism and critical theories, whereby
postmodernists are looking to effect positive change as a result of research and
intervention, and critical theorists are concerned with issues related to equity,
identity and power sharing (e.g., Alvesson & Skolberg, 2000; Broido, 2002). In
the design of the teaching experiment the freedom of the students to choose how
to engage with the learning activities was a focus of the study. However, it was
framed by the resources available in the library classroom, the customs of the
School and the ‘duty of care’ expected of all registered teachers in the State of
Queensland. For example, students were required to remain quiet at the start of
each teaching episode while a roll was taken and they were expected to respect
each other and listen while students were sharing their problems with the group.
For reasons of student safety, the researcher was required to remain in the
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classroom and to supervise the students at all times. It was inevitable then that
the researcher would need to direct the students in some activities such as the
whole-class sharing of problems, when to pack up, when to hand in workbooks
and when the students should return to their next class. Student direction by the
researcher was however kept to a minimum.
While for practical purposes, such as those mentioned in the previous paragraph,
the researcher facilitated each teaching episode; the students were given many
options on how to engage with the learning in order to encourage a student-
focussed approach to the teaching experiment (see Chapter 4). As the
mathematical teaching environment with which these students were familiar was
traditional and one where the teachers provided limited, if any, choices about
how to work within the classroom, the level of freedom provided in this study was
deemed to be significant and consistent with a critical theorist approach.
Students were able to choose where to sit, how to pose their problems, whether
to pose problems individually, in pairs or in small groups, whether to pose their
problems to the students sitting near to them or to the whole class or not at all, or
whether to become involved in providing feedback to the problems posed by their
peers. This was a genuine attempt to emancipate the students from their usual
structured way of learning where the teacher held all of the power in the
classroom (Lincoln & Denzin, 2000).
The teaching experiment consisted of a preliminary week for pre-testing and
conducting the first student survey (see Appendix C for the Survey Sheet and
Appendix D for the lesson plan related to the preliminary week lesson), followed
by seven, weekly teaching episodes lasting one hour each (see Appendices E to
K for the lesson plans of each consecutive teaching episode), finalised by a
session for post-testing and the repeated administration of the student survey
(see Appendix L for the final lesson plan). A lesson plan was prepared in
advance for each teaching episode. Each subsequent lesson plan was refined
both prior to the lesson occurring, as a result of student feedback and researcher
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observations of the previous teaching episode, and during the lesson depending
on the activities that were happening on the day in the research school that could
not have been predicted. For example, during a teaching episode, some
students may have arrived late due to heavy rain and road closures. This would
necessitate the researcher speaking to the individual students as they arrived to
provide them with an explanation of what the other students were doing (The
rationale for the construction of each lesson plan can be found in Chapter 4 and
in the lesson plans found in Appendices D to L.)
3.5.1 Data Collection
Both qualitative and quantitative data, in a variety of forms, were collected
throughout this study (see Table 3.1). The different forms of data were collected
for all students in the Intervention Group and were used in various ways to
respond to the three research questions. That is, all students in the Intervention
Group undertook the surveys (see Appendix C), the pre-test and the post-test
(see Appendix M), had their work books reviewed weekly, were videotaped and
were equally encouraged to pose their problems and share them with their peers.
However, for practical purposes, transcripts of the videotaped conversations and
informal interviews were only completed for the three case study students. Since
the researcher was also the facilitator of the teaching episodes, the recording of
observational data could not be undertaken in great detail during each teaching
episode. Therefore, detailed observations of the students in the Intervention
Group were undertaken primarily through a weekly review of videotapes from the
three audio-video recorders. Lesh and Clarke (2000, p. 137) supported the
collection of observational data when they said that “teacher’s observations can
lead to rich portrayals (or models) of their students”. According to Burns (1995,
p. 260) “observation serves to elicit from people their definitions of reality and the
organizing constructs of their world”. The data collected from observations
throughout the teaching experiment were invaluable in determining any
developmental learning changes that occurred in students.
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The qualitative data collected from the observations, student surveys, workbooks
and the informal interviews provided evidence of transformation in student
engagement and changes to students’ attitudes towards solving problems. For
example, the length of time taken by a student to begin posing problems, their
ability to remain focussed or the quality of the problems posed in their
workbooks, could all indicate whether a student was engaged in learning or not.
In addition, a student’s response to question in an informal interview could
indicate if a student had been actively engaged in the posing of problems.
The review of the tapes and transcripts provided a rich source of data on
individual students that in turn, provided the opportunity to make detailed
investigations of Paul’s, Andrew’s and Nicole’s learning development throughout
the teaching experiment. In addition, it provided supporting data to be used in
the refinement of subsequent teaching episodes. The data collected from the
student workbooks, and the observations of students, provided ongoing insights
into the students’ development learning and engagement during the teaching
episodes in the experiment (Confrey & Lachance, 2000). The level of
mathematical content and level of sophistication of the problems posed by the
students, provided evidence of how successfully the students’ could
“mathematize” situations (Fosnot & Dolk, 2001). This ability to recognise
mathematical situations was noted by English (1996) as being important and may
result in greater connections being made by students between their “school
maths”, intuitive knowledge and everyday, familiar contexts.
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Table 3.1
Data Used to Respond to the Three Research Questions of the Study
Research Question Qualitative/Quantitative Data Instruments
1. Can, and if so, how can
participation in problem-posing
activities facilitate the re-
engagement of middle-year
mathematics students?
Qualitative Informal interviews
Observations
Workbook analyses
Student survey
2. Can, and if so, how can
participation in problem-posing
activities facilitate improved
problem-solving competence of
middle-year mathematics
students?
Quantitative and
Qualitative
*Profiles of Problem Solving test results
Informal interviews
Observations
Workbook analyses
3. In terms of problem-solving
competence, what developmental
learning changes occur during the
course of a problem-posing
intervention?
Qualitative Informal interviews
Observations
Workbook analyses
Note. * From “Profiles of Problem Solving” by K. Stacey, S. Groves, S. Bourke, and B. Doig, 1993, Australian Council of Educational Research, Hawthorne, Vic.
The review of student workbooks on a weekly basis provided opportunities to
identify emergent themes in the problems posed by the students. An emergent
theme was considered to occur when a substantial number of students were
demonstrating a similar characteristic in their problems. For example, the first
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problems posed by the students were typically simplistic and minimalist. That is,
they required at most two steps to solve and were posed at a level that a student
of average ability in Year 4 may have been able to solve. It was at this point that
a refinement to the subsequent teaching episodes was planned as it seemed
apparent that students had little concept about what constituted a “good” problem
(see definition below). The workbooks from the 15 students were collected at
the end of each teaching episode and provided data about the development of
student learning and the levels of engagement that students demonstrated
throughout the teaching experiment. They also provided invaluable evidence to
respond to Research Questions 1 and 3. In addition, the lesson by lesson review
of these workbooks allowed the subsequent activities in the teaching experiment
to be adjusted where necessary, as “even researchers experienced in teaching
may not know well enough what progress students will make or know well
enough their mathematical thinking and power of abstraction to formulate
learning environments prior to teaching” (Steffe, Thompson, & von Glasersfeld,
2000, p. 279).
For the students, a “good” problem was defined to be one that was high in
interest factor, was appropriately challenging for the intended audience and had
sufficient data embedded, within it, to be solved. This concept of a “good”
problem was revisited in two ways during each teaching episode to provide a
scaffold around which students could pose problems that their peers would want
to solve. In the first instance, students were given opportunities each week to
self-rate their own problem by completing a self-rating sheet that described the
three qualities (see Appendix N for a copy of the self-rating sheet). In the second
instance, students were given opportunities to rate their peers’ problems using a
simple card system. Each student was provided with three cards upon which
were written the numbers 1 on one card, 2 on another card and 3 on the third
card. After a student had posed a problem to the class, other students were
asked to provide feedback to that student by holding up the card that, in their
opinion, had the number of qualities of a good problem that were demonstrated
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by the problem they had just heard. The use of these cards was further refined
in subsequent teaching episodes as the students began asking for more specific
feedback from their peers on each individual quality of a good problem, rather
than the problem as a whole.
Informal interviews with students were conducted during the teaching episodes,
throughout the teaching experiment. It was important to conduct the interviews in
an informal manner, while the students were working, to ensure they were
immersed in the problem-posing tasks and therefore responding about their
immediate thought processes, rather than from what they thought they might
have been thinking at the time. Students were asked to discuss what they were
thinking as they worked through a particular problem-posing task. An example of
some of the questions asked of students are: “What has been the biggest
challenge for you in constructing this problem today?” or “How did you go about
constructing your problem today?”. The responses were recorded for later
analysis.
The data from these interviews with case study students supplemented the
researcher’s observations and provided data to demonstrate change processes
occurring in the learning development and attitudes towards problem solving of
the students throughout the teaching experiment. Mestre (2000, p. 167)
commented on this form of data collection saying that “although these
methodologies are extremely time consuming to administer and the data are
difficult to analyse, they can provide detailed information of a subject’s
conceptual knowledge and its links both to contexts in which it can be applied
and to conditions of applicability”. He further said that multiple methods, such as
those used in this present study, provide researchers with different data, and it is
common that more than one method is required to satisfactorily respond to all of
the research questions in a particular study (Mestre, 2000).
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At the completion of each lesson, general observations and notes about the
lesson were recorded in a researcher journal. The journal observations
contextualised and recorded any events that impacted upon the behaviour and
concentration of students during individual teaching episodes. These events
included any incidents that took place before a teaching episode, or distractions
that occurred nearby the library classroom on a particular day, during a particular
episode. They were recorded to monitor the “mood” of the classroom throughout
the experiment and thereby contextualise student behaviours. The data
collected from the observations were taken into consideration when
determinations were being drawn about students’ behaviours during each
teaching episode.
The quantitative data collected in the pre-testing and post-testing were necessary
to provide empirical evidence of transformation in problem-solving competence of
the students. The triangulation of data obtained from both the qualitative and
quantitative methods used in this experiment is consistent with Gardner’s
(1999b) call for a variety of assessment instruments to provide multiple and
different opportunities for individuals to demonstrate their understanding of
concepts. For example, one student may be able to successfully demonstrate
the understanding of a concept by undertaking an oral presentation to peers
whereas another student may prefer to quietly respond to some questions in an
essay or short response format. In the context of this study, accommodation of
different learning styles and intelligence types is not possible with pre-testing and
post-testing alone. Kelly and Lesh (2000, p. 229) agreed and said that “pretest-
posttest designs … tend to presuppose that the best way to get complex systems
to evolve is to get them to conform to a single one-dimensional conception of
excellence”, which in this study would have prohibited discussion about
observations of developmental learning demonstrated throughout the teaching
experiment.
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3.5.2 Instruments
The instruments used to collect the data in this present study are discussed in
this section. The first instrument, the Middle Years Ability Test (Australian
Council for Educational Research, 2005), was used to assist in the participant
selection process. The second instrument, the Profiles of Problem Solving
assessment instrument (Stacey et al., 1993), was used as both the pre-test and
the post-test for this study. The third instrument, the Student Survey Sheet, was
created by the researcher to collect attitudinal data from the students that may
assist in responding to Research Question 1. These instruments are described
in greater detail I the following sub-sections.
3.5.2.1 The Middle Years Ability Test (MYAT)
The Middle Years Ability Test (2005) is an intelligence testing instrument
developed by the Australian Council for Education Research (2005) and was
used to identify participants for this study. The normative data, provided to
administrators of the MYAT test by the Australian Council for Education
Research, to be used for the assessment of students came from the
achievement scores of over 2000 Australian middle-year students tested in July
and August of 2004. This test was chosen as it was specially designed for
Australian middle-year students and was routinely administered to students in the
research school at the beginning of Year 7, thus negating the need for the Year 7
cohort to undertake additional testing as a whole group. In addition, unlike other
IQ tests, the MYAT test contains 25 out of the 75 questions contained in the test
that assess non-verbal reasoning skills. As mentioned earlier, Gardner (1999b)
has suggested that intelligence tests can distort the view about an individual’s
potential but, he was not against them if they made attempts to address
intelligences other than linguistic and mathematical intelligences. The MYAT test
goes some way to addressing this issue.
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The MYAT test (Australian Council for Educational Research, 2005) contains a
pair of parallel general ability tests that were designed to assist teachers to
assess the general abilities and skills of their middle-year students. Each of the
two tests contains 75 items based on literacy, numeracy and non-verbal
reasoning skills that are arranged in order of difficulty from simplest to most
difficult. Students respond to each question by marking their choice of five
possible answers for each question on an answer sheet. Only one answer of
each set of five possible answers is correct. Consideration was given to the
debate in the literature about the validity of administering intelligence tests that
have time as a component of the testing conditions, such as is the case with the
MYAT test (Australian Council for Educational Research, 2005). Opponents,
such as Goswami (2002), suggest the reliability of results from such timed
instruments is questionable, whereas supporters, such as Hutton and his
colleagues (1997), believe that speed in processing information, does indeed
correlate highly with intelligence. To address this reliability issue, a 10% margin
of consideration in the student results was made before considering potential
participants for this present study.
The Head of Department – Curriculum and the Coordinator of Year 7
Mathematics at the research school reported that the MYAT test had provided a
sound indication of student intelligence based on comparisons with the
classroom performances of students generally across all subjects. However,
they noted that there seemed to be some inconsistencies with some students
who had high predictive scores on the MYAT test but whose performance on the
problem-solving criterion of their mathematics tests was below the average score
of their peers. The specific nature of why this situation occurs is beyond the
scope of this present study, however, strategies to overcome this potential
underachievement of students have been addressed in the findings of this
present study (see Section 3.4.2).
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3.5.2.2 The Profiles of Problem Solving (POPS) Test
A number of criteria were considered before choosing the instrument to be used
for the pre-testing and post-test of participants in this study. Firstly, it was
important that the test was specifically designed and written for Australia middle-
year students. While Australia is not typified by a mono-culture, it is reasonable
to at least ensure that the language and contexts used in the test are the
language and contexts that the students would commonly experience in their
day-to-day activities. It is worth noting that students who had English as a
second language were not considered as potential participants for this study.
Their inclusion could have created additional variables and lead to results that
were more about a student’s ability to understand and read the English language
than they were about their ability to pose or solve problems.
The second consideration, when choosing a testing instrument, was that the
instrument should measure five different aspects of problem-solving competence:
1. Correctness of answer – a fundamental criterion for developing a high
level of problem-solving competence
2. Method used – the use of a systematic plan that realistically can lead to
the solution of a problem is integral in developing problem-solving
competence
3. Accuracy – without the ability to complete accurate calculations a
problem cannot be solved
4. Extracting information – whether a student can identify and filter useful
information from extraneous information provided in a problem, and
5. Quality of explanation – a student’s ability to effectively communicate
their mathematical ideas, processes and calculations as well as the final
answer
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These five aspects of problem solving are consistent with Polya’s (1957) four
stages of solving a problem that are still commonly referred to in schools as the
‘See, Plan, Do, Check’ scaffold for solving problems. (The ‘Accuracy’ and
‘Extracting Information’ are combined in the ‘Do’ aspect.) While the majority of
test instruments written in Australia focussed on a student’s ability to solve
problems across the five strands of mathematics; number, algebra, space,
measurement, and chance and data, the Profiles of Problem Solving (POPS) test
(Stacey et al., 1993) was the only test that specifically addressed the five aspects
of problem solving.
The third consideration when choosing a testing instrument was that it needed to
be a reliable and valid instrument to use in the context of this study. The POPS
(Stacey et al., 1993) test incorporated everyday contexts that were within the
typical experiences of an Australian middle-year student, for example, money
and birthday candles. This instrument had pre-established validity and reliability
(Impara & Plake, 1998; McLellan, 1998; Medina-Diaz, 1998) and was a useful
tool in determining changes in problem-solving competence of the students
between the start and end of the teaching experiment. The only testing
instrument that satisfied all the three considerations of assessing the five
problem-solving criteria, being designed in Australian for middle-year students
and having verified validity and reliability, was the POPS test and hence it was
used as the pre-testing and post-testing instrument for this study.
The POPS test took approximately 40 minutes to administer, with 32 minutes
being specifically set aside for the students to answer the questions. While
authors, such as Ridgeway and colleagues (2000), suggested that
understandings related to problem-solving competence are difficult to measure,
and that standardised tests can fall short of capturing the true nature of students’
problem-solving competence, the use of multiple data collection methods,
including both qualitative and quantitative methods used in this present study,
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provided for triangulation of the data and hence, increased faith in the results of
this study (Mestre, 2000).
3.5.2.3 The Student Survey
The purpose of the student survey was to determine if there had been a shift in
student attitudes towards solving novel problems as a result of the problem-
posing intervention. The four questions in the survey were designed by the
researcher to be simple to understand and answer by the students. The four
questions in the survey are listed below.
Question One: Do you enjoy solving problems?
This question was included in the survey as it has direct links with the
dispositional state of students and the possible reasons for their
underachievement in this criterion of their routine mathematics tests (e.g.,
Gootman, 2001; Kanevsky & Keighley, 2003).
Question Two: What type of problems do you prefer to solve?
This question was included in the survey to determine if students were discerning
about the types of problem they like to solve. The student responses to this
question had the potential to provide some data to explain possible
disengagement of students in the Intervention Group. In addition, possible
correlations were made possible between the similarity or difference of
responses to this question in the initial survey compared to the final survey, and
the observations of student engagement in the classroom throughout the
teaching experiment.
Question Three: Do you think learning to solve problems is a useful thing to do?
Students’ attitudes to problem solving have been demonstrated by Kanevsky and
Keighley (2003) and Gootman (2001) as pivotal is their willingness to solve them.
The responses to this question in the initial survey, when compared to responses
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to this question in the final survey, had the potential to provide an insight into
whether involvement in a problem-posing intervention could facilitate a change in
student’s attitude and possible their levels of engagement in classroom activities.
Question Four: What things could teachers do to assist you to become better at
solving problems?
This question was included in the survey out of respect for, and interests of, the
opinions of the participants in this study. It is commonplace for teachers to make
decisions for students on the premise that they are more wise and
knowledgeable about what and how students should learn (e.g., Roberts, 1996;
Stein et al., 2003). However, the philosophical framework, upon which this study
was founded, supported a liberating and transformative learning environment
(Broido, 2002) that respected the views of the students (Alvesson & Skolberg,
2000). Student responses to this question had the potential to direct refinement
in the lesson plans for each teaching episode and to provide direction for future
research into problem solving and problem posing.
3.5.2.4 The Problem Criteria Sheet
The problem criteria sheet (see Appendix N) was designed to allow students to
become reflective and mindful about the qualities of the problems they were
posing for their peers and to provide the researcher with a regular opportunity to
provide feedback to the students on their problems. The sheet was made up of
two sections; one for the student to self-rate their problems and a second
identical section below for the ‘teacher’ to provide feedback. The word ‘teacher’
was used on the sheet as opposed to ‘researcher’ as a genuine attempt to make
the students feel that this experience was as similar to a normal teaching
environment as possible.
Each week, the researcher would glue a new criteria sheet into the students’
workbooks so that it would be in place when the students completed posing their
new problem during the next teaching episode. Each lesson the students would
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be given time to pose problems and then to share them. They received
immediate peer feedback during each lesson from the 1, 2, 3 card system
described in Section 3.5.1. Students then self-rated their posed problems by
ticking one of three boxes in each of the three criteria, interest factor, challenge
level and do-ability. In addition, students were given an opportunity to make any
comments they wanted to about their problem in a section alongside the boxes
before handing their workbooks to the researcher. This data contained in the
‘Comment’ section, provided clarification for why a student rated their problem as
they did and provided data that could be correlated with other data sources to
respond to Research Question 1.
The researcher reviewed the posed problems and the self-ratings recorded in the
problem criteria sheets of the student workbooks each week. The researcher
then completed the ‘teacher’ section of the criteria sheets, rating each student’s
posed problem against the three criteria, and provided each student with an
individualised comment that included affirmations like “Your problem is very
interesting this week, well done!” and constructive feedback like “When you are
posing your next problem, remember to include all of the information that a
person would need to solve the problem. By making the time to do the problem
yourself, before you pose it to your peers, you will be able to ensure it is do-able.”
3.5.3 Data Analysis
Data analysis is a “dynamic and creative process” through which researchers
attempt to make sense of what they are investigating (Taylor & Bogdan, 1998, p.
141). The qualitative data collected from this teaching experiment were analysed
from a number of perspectives to identify emergent themes. The quantitative
data collected from students in the Comparison Group and the Intervention
Group were analysed according to accepted statistical methods both within and
between groups. This section will look at how the data from all sources were
analysed singularly and in combination and, where applicable, how emergent
themes were identified.
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3.5.3.1 Researcher Journal
The purpose of the researcher journal was to record details about the “mood” of
the class and to identify external factors that impacted upon the teaching
experiment. Trends across the class were recorded in the researcher’s journal at
the conclusion of each teaching episode and were used to determine if the
students were displaying group behaviours that may be as a result of external
influences on the day, such as an upcoming test or an extremely hot day.
Possible correlations between atypical behaviours of students and situational
factors were reviewed on a lesson by lesson basis. For example, if the
engagement of some students was noted in the tapes and video recordings as
lower than in previous weeks, the researcher could consider possible correlations
with entries for that teaching episode in the researcher journal. A possible
scenario may have been that this observation correlated to an entry in the
researcher journal mentioning these same students arrived late to the teaching
episode due to a vaccination program being conducted at the research school on
that morning. It would therefore be reasonable to suggest that some of the
disengagement of these students may have been due to the factors external to
the teaching experiment. This triangulation of data was of particular importance
in identifying the apparent trends of engagement, attributable to the problem-
posing intervention, throughout the teaching experiment and provided information
to respond to Research Question 1.
3.5.3.2 Student Surveys
There were four questions in the student survey (see Appendix C and Section
3.5.2.3). The students’ responses to Question One were categorised as a
positive, a negative or a neutral response (see Table 5.4). The ‘closed’ nature of
the wording of this question made these categories obvious and appropriate.
The administration of the survey sheets did not pre-empt possible student
responses to Questions Two, Three and Four and students were not prompted in
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any way while they were responding to the questions. The categories shown in
Tables 5.5, 5.6 and 5.7 were determined after the students completed both the
initial and final surveys, as natural groupings became evident.
However, as is often the case when categorising data, a conflict can arise
between the number of categories finally chosen and the ‘fit’ of the data into the
categories. The categories were recorded in the tables as they arose in the
responses. For this present study it was decided that unless a student response
naturally belonged to a category that had previously been recorded (as a result of
another student making similar comments), then a new category would be
created in the table. This process ensured that all students’ views were
represented and not ‘lost’ in the compromises that are sometimes made when
categories are made known to students in advance of a survey, and is consistent
with the critical theorist framework that underpins this present study.
Students were asked to explain their answers to Questions One, Two and Three.
These responses were used to interpret if the students may have changed their
view from the start to the end of the teaching experiment as a result of the
problem-posing intervention. These responses could be correlated with data
collected from student workbooks, observations and the pre-test and post-test.
For example, a student may have written in their first survey that they disliked
solving all problems because they were too hard. This comment may have
contrasted to the comment they wrote in their final survey which could have been
“I now like solving problems because I understand how they are made and I am
better at knowing where to start to solve them”. It would be reasonable to
deduce from this change in attitude that something that occurred in the problem-
posing intervention had affected the change of view. The opportunity then
existed to correlate this attitudinal change with any changes in the quality of
problems progressively written in this student’s workbook, their self-ratings,
researcher observations of student engagement and results from the pre-test and
post-test.
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3.5.3.3 Student Workbooks
The student workbooks were collected each week and the problems posed by
the students and their self-ratings on the problem criteria sheets were reviewed
by the researcher. Weekly feedback was provided to the students through the
researcher entries in the problem criteria sheets. At the conclusion of the
teaching experiment, the student workbooks of the three case study students
were analysed from a number of perspectives. In response to Research
Question One, the comments that students wrote, and any changes in the quality
of the problems posed by the students, could be correlated with other data
sources. For example, if a student had increased their score in the post-test as
compared to their score in the pre-test, the workbook may or may not show a
corresponding improvement in the quality of problems posed by the student
throughout the teaching experiment. This information would assist in responding
to Research Question Two. Analysis of workbook responses was also very
helpful when comparing the students’ answers in the initial and final surveys with
their comments and self-ratings in their problem criteria sheets. It was, for
example, useful to determine if a change in answer to Question One on the
student survey correlated or contrasted with any changes in students’ comments
in the problem criteria sheets of their workbook.
The themes that students choose to write about when they posed their problems
was not a focus of this present study, however, the reading of the problems on a
weekly basis inevitably provided guidance to the researcher about what themes
interested students and therefore, informed refinements in subsequent teaching
episodes. An awareness of which themes engage students emerged from this
study as a useful focus for further research.
3.5.3.4 Researcher Observations
The audio-video tape recordings of each teaching episode allowed detailed and
unobtrusive observations to be made of the three case study students as well as
providing secondary data about the “mood” of the class. Following each teaching
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episode, the three video cassettes from the audio-video recorders were
electronically transferred to digital video disks (DVDs) and the cassettes were re-
used in the following teaching episode.
Each DVD was viewed and any conversations between a case study student and
peers, or informal interviews between the researcher and a case study student,
were transcribed. These data were then correlated with data from other sources,
such as the workbooks and the student surveys, to respond to Research
Questions 1 and 3. For example, the length of time a student took to begin
writing a problem in each teaching episode could be determined by reviewing the
tapes. This data, in combination with other data, such as the quality of the
problems posed in the student’s workbook or the comments they wrote in their
problem criteria sheet, could provide evidence about the student’s levels of
engagement throughout the teaching experiment.
The DVDs were also used to collect data on the case study students’ body
language and behaviour throughout the teaching episodes. Developmental
learning changes involve changes in internal processes that essentially are only
identifiable by the ‘products’ produced by an individual (e.g., Bjorklund, 2000;
Siegler, 1991). These products can be physical artefacts such as posed
problems written in workbooks and test results, or they can manifest as changes
in social behaviour such as a student’s body language or their ability to
communicate effectively or willingly with other individuals or groups of individuals.
As the DVDs were viewed, notes were made about each case study student’s
behaviours and types of interaction with peers and the researcher. These data
were used to respond to Research Question 3.
3.5.3.5 Informal Interviews
As mentioned in the previous section, informal interviews were conducted with
the case study students throughout the teaching experiment and were recorded
by one of three audio-video recorders positioned around the room. The
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questions were designed to uncover some of the conscious and unconscious
thought processes that had been experienced by the students throughout the
teaching experiment. The questions asked in the informal interviews were not
scripted to ensure they were in context with each individual’s circumstances at
the time of the interviews. Analysis of this type of data is often difficult (Mestre,
2000) but it does provide insights into an individual’s developmental learning
(Confrey & Lachance, 2000). As mentioned in the previous section, the
interviews were transcribed and extracts from these transcripts were used in
conjunction with the pre-test and post-test data, to respond to Research Question
3.
3.5.3.6 The Profiles of Problem Solving (POPS) Test
The quantitative data collected from the five problem-solving aspects of the pre-
test and the post-test were analysed using the Statistical Package for Social
Sciences (SPSS) Version 15 (SPSS Inc., 2007). The data for both the
Comparison and Intervention Groups were entered using the SPSS software and
a paired t-test was performed to establish the means and standard deviations of
the paired sets of data. The results of the paired t-test provided opportunities to
compare the means and standard deviations of each problem-solving aspect
between the two groups. To assist with the analysis of the paired t-test results,
the mean scores for the Comparison and Intervention Groups, within each
problem-solving aspect, were graphed on single sets of axes.
The data from the POPs test (Stacey et al., 1993) were further analysed by
considering the number of students, in each of the Comparison and Intervention
Groups, who improved their score in the individual problem-solving aspects from
the pre-test to the post-test. This provided a further opportunity to compare and
contrast the data sets from the two groups and to explore the impact that the
problem-posing intervention may have had on students in the Intervention Group.
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3.5.4 Reliability and Validity Issues
Issues of validity and reliability were considered in order to “trust” the results of
this educational research (Merriam, 1998) and to determine the quality of the
research study design. That is, trustworthiness was a major consideration
throughout the design and implementation of the study, including the collection,
analysis, interpretation and presentation of data and results. The qualitative
analysis required the establishment of a “strong chain of evidence” to be
collected in the survey sheets, audio-video tape recordings, student workbooks,
and the researcher’s journal, and by the laying of an “audit trail” that provided a
complete documentation of the research process for potential replication by other
researchers (Borg, Gall, & Gall, 1999).
The quantitative analysis was undertaken using accurate mathematical
techniques and commonly used statistical tests for the purpose of analysing data
in social science studies. Issues of instrument validity were addressed in
multiple ways; by ensuring the testing instrument was designed for Australian
middle-year students; the pre-test and post-test questions were of a format with
which the students were familiar, yet contained questions that were still novel in
nature; and by ensuring the testing instrument assessed the subject of this study,
that is, aspects of problem-solving competence. This was possible because the
testing instrument, the Profiles of Problem Solving (POPS) test (Stacey et al.,
1993), provided information on all five aspects of problem solving rather than
providing information about a student’s strength within a particular strand of the
mathematics curriculum in Queensland.
Validity of the design of the study was addressed predominantly through the
choice of a teaching experiment for the study’s intervention. The teaching
experiment provided the opportunity for students to become participants within a
familiar environment and with familiar peers. The students were not required to
attend school outside of their usual school hours thus minimising any disruption
to the students’ usual weekday routines. In addition, the teaching experiment
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provided an opportunity to collect multiple sources of data and to observe the
students over an extended period of time.
Reliability of the study’s findings was maximised by appropriate and consistent
participant selection, whereby the 16 Comparison Group students and the 15
Intervention Group students came from cohorts who attended the same school,
in the same year level (albeit consecutive years), with the same mathematics
teacher and the same mathematics curriculum. In addition, the same selection
process was used for both groups and the Profiles of Problem Solving (POPS)
testing instrument (Stacey et al., 1993) was administered to the students in the
two groups at the same time of the school year to ensure similar cognitive
development and maturation of the students in the two groups. Students from
both groups undertook their pre-testing and post-testing during the morning of
the testing days to ensure that students were not tired from other testing or
activities that may have naturally occurred on the school day. The effect of
students guessing correct solutions to the test did not have a major impact on the
reliability of the data as it was the process of solution that was analysed more so
than the solutions themselves.
Practising of test questions was not allowed with either the Comparison or
Intervention Group students, with the exception of students having undertaken
the pre-test eight weeks prior to undertaking the post-test. McLellan highlighted
her concerns about the “practice effect” (Burns, 1995) of individuals undertaking
the same pre-test and post-test. She wrote, “a limitation of the POPS test is the
consumable nature of the product” (1998, p. 2). In this study, the administration
of the POPS test ensured that neither group received feedback on their pre-test
results or was made aware that the post-test contained the same questions as
the pre-test.
The reliability of the POPS test (Stacey et al., 1993) has been demonstrated by
the authors and widely accepted by researchers (e.g., Booker & Bond, 2001;
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Impara & Plake, 1998; McLellan, 1998; Medina-Diaz, 1998). The test has been
widely used in Australia and internationally as a pre-testing and post-testing
instrument for measuring the problem-solving competence of middle-year
students. It was originally trialled with over 200 students from a selection of
schools that represented students from a diverse range of socio-economic
backgrounds in Australia and is accepted as a reliable testing instrument.
Subsequent to the original trial, a number of authors had considered the
instrument for its internal validity and reliability and determined the instrument to
be sound in both aspects (e.g., Booker & Bond, 2001; Impara & Plake, 1998;
McLellan, 1998; Medina-Diaz, 1998). Indeed, McLellan commended the
development methodology of the instrument when she wrote, “the number of
children used in the development of this assessment tool is impressive,
considering the individual involvement required from each child and the depth of
analysis applied to each child’s performance” (1998, p. 2).
The reliability of the testing instrument is further strengthened by the prescriptive
method of marking provided by the authors. To demonstrate the reliability of
their instrument, Stacey, Groves, Bourke and Doig, authors of The Profiles of
Problem Solving (1993), provided a random sample of fifty test responses to five
independent markers consisting of three of the authors, one primary teacher and
one secondary teacher, three of whom marked the tests twice. Using a Pearson
Product Moment correlation (Burns, 1995, pp. 185,193) between the various
pairs of markers, the calculated correlation coefficients were all above 0.95
(Stacey et al., 1993). According to Stacey et al. (1993, p. 57)., “these high
correlations indicate that the marking scheme is sufficiently explicit to enable
markers to make consistent judgements, even in relatively subjective areas”.
3.5.5 Ethical Issues
This study received ethics approval number 4193H from the QUT University
Human Research Ethics Committee.
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Conducting ethical research requires a researcher to be aware of ethical issues
such as; researcher/participant relationships, confidentiality, informed consent,
storage, analysis and reporting of data, and privacy. As Glesne and Peshkin
said, “by their nature, ethical dilemmas defy easy solutions” (1992, p. 124). This
being said, every attempt was made to minimise the impact of these issues on
the participants and the research outcomes of this study. Ethical considerations,
that were specific to the research site or subjects, were carefully considered. While
this study involved participants who were unable to give their own consent to the
research, consent was sought from the parents or guardians of the participants.
All testing and permission sheets, and study results, were stored in a secure
office. Thus, this research was of Low Risk in terms of Ethics Clearance.
The participants in this present study were provided with full anonymity through
the use of pseudonyms in the reporting process and at no time were the students
required to participate involuntarily. The choice of a pseudonym for each
participant was influenced by a pseudonym code allocated in a particular way by
the researcher. This process allowed the researcher to back-track to a data
source, while at the same time protecting the participants from identification by
readers of the research report.
Students in the Comparison Group received their pseudonym code in the order
of receipt of their signed permission notes. The first student received the code of
a06 with the next being b06, c06, d06, and so on, with the first character of the
code following the alphabetic order and the “06” representing 2006, the year of
testing. Similarly, participants in the Intervention Group also received their
pseudonym code in the order of receipt of their permission notes. The first
participant received the code A07, with the next being B07, C07, D07, and so
on, following the alphabetic order and the “07” representing 2007, the year of the
teaching experiment with the Intervention Group.
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In recognition of the participants of this study being people and not objects, the
pseudonym codes were then converted to given names using the letter of the
pseudonym code to be the starting letter of the names that were chosen at
random (see Appendices E and F). For example, the Comparison Group male
student with the code a06 became known as Adam, while the Intervention
Group, female student with the code G07 became known as Georgia. This use
of names, to refer to the students, in the reporting of this study was more
appropriate than the use of an alphanumeric code that could be perceived as de-
humanising or disrespectful to the students. This process is in keeping with the
theoretical framework underpinning this study and is more user-friendly to the
reader. While gender was not a factor being controlled or investigated in this
present study, the use of pseudonym names was kept gender consistent.
3.6 Conclusion
Three specific research questions arose from the literature review. Research
questions 2 and 3 provided an opportunity to investigate developmental learning
issues that relate to the development of problem-solving competence. The first
question however, provided an opportunity to investigate affective factors that,
through research, have been linked to underachievement in problem-solving
competence. The importance of considering affective factors was reported by
Leder and Forgasz (2003, p. 95) who said that “ it is now widely accepted that
cognitive as well as affective factors – such as attitudes, beliefs, feelings, and
moods – must be explored if our understanding of the nature of mathematics
learning is to be enhanced.”
A single method of data collection would have provided insufficient evidence to
respond to the three research questions and hence, a triangulation approach
(Gergen, 1996) to data collection was adopted to collect both qualitative and
quantitative data. This approach was consistent with Gardner’s (1999b) view of
the existence of multiple intelligences whereby he suggested that multiple forms
of data should be collected before making determinations about changes in the
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developmental learning status of students. The collection of multiple forms of
data was also consistent with a constructionist approach to educational research
(Schwandt, 2001) whereby learning occurs progressively, as opposed to
summatively, through active student participation in the construction of
knowledge and artefacts for sharing.
The data in this study was collected during an eight-week, teaching experiment
that occurred in the participants’ school, in a familiar classroom, albeit not their
‘homeroom’. With much of the previous research on problem posing and
problem solving occurring in non-school settings, findings were open to criticism
of sustainability in everyday classroom situations (Lester & Kehle, 2003). The
design of this study addressed the need “for classroom research that is more
speculative and in which some of the constraints of typical classrooms are
relaxed while others remain in force” (Confrey & Lachance, 2000, p. 231).
The design of this study was informed by two theoretical perspectives that work
closely together; postmodern theory and critical theory (Lincoln & Denzin, 2000).
These two perspectives underpinned the teaching experiment that aimed to
effect positive change for its participants as a result of the intervention (post-
modernist theory) (see Tierney, 1997), and at the same time allowed individuals
to experience liberation and recognition throughout the learning process (critical
theory) (see Broido, 2002; Stein et al., 2003).
Issues of reliability and validity of data were addressed by the collection of
multiple sources of data, and the choice of testing instrument for the pre-test and
post-test. The selection of methods provided opportunities to correlate findings
and thus created a higher level of confidence in the data obtained from the study.
Ethical considerations about the choice of participants and their anonymity, and
the collection and storage of data have been discussed and the study was
acknowledged by the QUT University Human Research Ethics Committee as
being of low risk and therefore received ethics clearance.
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Chapter 4
The Teaching Experiment
4.1 Chapter Overview
This chapter describes in detail the structure and evolution of the teaching
experiment, and the unforeseen events and circumstances that impacted upon
the individual teaching episodes and which resulted in refinements of subsequent
teaching episodes. While this chapter was designed to complement the
information provided in Chapter 3, about the methodology used in this present
study, it was inevitable that some references to what occurred during the
teaching experiment, and what ultimately influenced some refinements to
subsequent teaching episodes, were discussed in this chapter. However,
detailed discussion about results is reported in Chapter 5.
Chapter 4 comprises four main sections. The first section introduces the general
overall structure of the experiment and briefly re-visits the philosophical
underpinnings that were considered in its design (see Section 4.2). The second
section looks specifically at the first and last lessons in which the students of the
Intervention Group undertook their pre-test and initial survey, and post-test and
subsequent survey respectively (see Section 4.3), while the third section looks at
the development of the seven individual teaching episodes and how experiences
from each episode influenced the refinement of subsequent teaching episodes
(see Section 4.4). The final section presents a conclusion of this chapter (see
Section 4.5).
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4.2 The Philosophical Underpinnings and Structure of the Teaching
Experiment
4.2.1 The Philosophical Underpinnings of the Teaching Experiment
Chapter 3 provided a rationale for the choice of a teaching experiment as the
appropriate intervention for investigating the three research questions presented
in Section 3.2. The teaching experiment methodology provided the ideal
framework to support constructionist beliefs that students learn best when they
are creating artefacts to share with peers, and when they are actively engaged in
their own learning (e.g., Crotty, 1998; Papert, 1991). Students were able to be
observed and interviewed over the eight-week timeframe of the intervention
process to investigate changes in their problem-solving competence,
engagement and developmental learning. During the design of the teaching
experiment, the following five student learning goals were established to enable
the research questions to be responded to and to direct the development of each
teaching episode.
1. Pose problems
2. Become increasingly engaged and interested in posing problems
3. Increase problem-solving competence
4. Choose to pose problems individually, in pairs, or in small groups, and
5. Choose to share problems independently, via a chosen peer, or via the
researcher
The two major foci of this study were a problem-posing intervention and problem-
solving competence, hence the development of Goals 1 and 3. The group of
students being investigated in this present study were ascertained as
underachieving, therefore, Goal 2 was developed to respond to the connections
established in the literature review between underachievement and engagement
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(e.g., Jones & Myhill, 2004; Reis & Siegle, 2006). The constructionist
perspective underpinning this present study supported the fundamental premise
that best learning occurs through human interaction and the creation of
meaningful artefacts to share within a human-interaction framework (Crotty,
1998). Goals 4 and 5 were developed to ensure this focus was present
throughout the teaching experiment. These five goals provided the impetus for
the development of each of the teaching episodes (see lesson plans for each
teaching episode in Appendices D to L) and will now be further discussed in turn.
Goal 1: Pose problems
This goal was at the heart of the teaching experiment, as a problem-posing
intervention was being investigated as an appropriate strategy to improve the
problem-solving competence of the participants. There has been considerable
debate over the use of problem-posing instruction to improve students’ problem-
solving competence (e.g., Brown & Walter, 2005; Silver, 1997) and this present
study has contributed to the body of knowledge surrounding this debate.
Students were given opportunities in each of the seven teaching episodes to
pose problems from a variety of stimuli sympathetic to their interests,
experiences, and maturation. The choices of each stimulus are discussed in
Section 4.3. While students worked individually or collaboratively to pose their
own problems they created an opportunity for all three of the research questions
to be investigated.
Goal 2: Become increasingly engaged and interested in posing problems.
This study was underpinned by two theoretical frameworks; the post-modern and
the critical frameworks. A significant premise of post-modernist research, such
as this present study, is the desire to create change as a result of research
investigations (e.g., Tierney, 1997) as without this perceived opportunity of
change, there seems little point in pursuing the line of research. To fully
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investigate Research Question 1, the increased engagement and interest of
students in posing problems was seen to be an important goal of the teaching
experiment.
Goal 3: Increase problem-solving competence.
Similarly, as for Goal 2, the problem-posing intervention in this teaching
experiment was designed to provide opportunities for developmental learning
changes in the students, as well as affective changes. The establishment of this
goal ensured the focus of the teaching experiment extended to developmental
learning changes and provided data that informed the response to Research
Question 3.
Goal 4: Choose to pose problems individually, in pairs, or in small groups.
The second theoretical framework that underpinned this study was the critical
framework (e.g., Broido, 2002). Critical theorists support research that
celebrates, encourages, and supports individuality. In this way, students can feel
valued and safe and a transformative learning environment is created (Lincoln &
Denzin, 2000). In this teaching experiment, individuality was extended to both
how the students preferred to work with their peers (Goal 4) and how they
wanted to express themselves in front of their peers (Goal 5). By focussing on
both of these goals all students were given an opportunity to work within their
preferred learning style using their dominant intelligence(s) (Gardner, 1999b).
Goal 5: Choose to share problems independently, through a chosen peer or
through the teacher.
Constructionism, as a belief system, underpins this teaching experiment by
students being actively engaged in posing problems for peers to solve. For
students to be able to share their problems (artefacts) with their peers,
consideration had to be given to ways in which the students felt most comfortable
in doing so. Therefore, students were given a variety of suggestions on how
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they might share their problems and were empowered to suggest alternative
methods of sharing. For example, if the problem they had posed was a written
one, they could choose to read the problem themselves, or have a friend or the
researcher read it for them. Both Goal 4 and Goal 5 ensured that the
epistemology and underpinning theoretical perspectives of this study were
supported throughout the teaching experiment.
In summary, five goals were developed for this teaching experiment to ensure
that:
1) the data collected would provide evidence to respond to the research
questions, and
2) the theoretical perspectives that informed the design of this study, were
supported throughout the teaching experiment.
4.2.2 The Structure of the Teaching Experiment
In 2006, a one-hour session in week one, and a one hour session in week eight
of Term 4 were used to pre-test and post-test the Year 7 students at the research
school who met the participant criteria for the Comparison Group. Students were
excused from their weekly assembly in both weeks to undertake the testing. Pre-
testing and post-testing was undertaken in the same way for students in the
Intervention Group in the first and last lesson of the teaching experiment in 2007,
with the additional aspect of the students in the Intervention Group completing
both initial and final survey sheets.
In Term 4 of 2007, the researcher was initially provided with one hour per week
access to the Year 7 students at the research school who met the participant
criteria for the Intervention Group (same criteria as for the Comparison Group).
Term 4 consisted of eight teaching weeks. With one hour required for each of
the pre-testing and post-testing sessions, only six weeks would have remained
between the first and last week of Term 4, in which to conduct the teaching
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experiment with the Intervention Group. Following negotiation with the Head of
Department – Curriculum and the Coordinator of Year 7 Mathematics at the
research school, permission was given to have an additional hour with the
Intervention Group students in the final week of the term since all school-based
assessment would be completed and students would be engaged in end-of-year
activities. Thus, the researcher had nine hours access to the Intervention Group
over the eight weeks of Term 4 in 2007. Lesson plans for all nine lessons can be
found in Appendices D to L.
The one-hour teaching episodes began at 8.30 a.m., fifteen minutes before
school started, and continued through the forty-five minute assembly. This timing
was agreed to by the senior staff at the research school to ensure that no
students were absent from any of their regular teaching classes during the
teaching experiment. Situating this study in a working school environment, as
required with the experimental approach, was an important aspect of this present
study but required some accommodation of atypical circumstances due to
unforeseen events (see Section 4.4). In addition, some variations to the timing of
lessons occurred for a variety of normal, school-based reasons. A summary of
the variations appears in Table 4.1.
4.3 The Pre-test and Post-test Lessons
4.3.1 Introduction
The design of this study included both qualitative and quantitative data collection
and analysis. The same testing instrument and survey sheet were used in the
first and last lessons, however, the students of both the Comparison and
Intervention Groups were unaware, until the final lesson, that the pre-test and
post-test were identical, or that the two surveys contained identical questions.
Discussion about the validity of using the same tests and surveys, and the
corresponding reliability issues of data collected from these methods, has
previously been discussed in Section 3.4.3.
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Table 4.1
Variations to Pre-arranged Lesson Times in 2007
Lesson Number Proposed date
Resultant date
Reason for change
1.Pre-test and
first survey
8th
October
8th
October
No change
2. First teaching
episode
15th
October
15th
October
No change
3. Second
teaching episode
22nd
October
29th
October
Pupil-free day at the research
school on 22nd October
4. Third teaching
episode
29th
October
30th
October
Catch-up day for 29th October
5. Fourth
teaching episode
5th
November
5th
November
No change
6. Fifth teaching
episode
12th
November
12th
November
No change
7. Sixth teaching
episode
19th
November
19th
November
No change to date/ room change
due to a library stock take
8. Seventh
teaching episode
26th
November
26th
November
No change
9. Post-test and 26th 26th No change to date / lesson time
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final survey November November was from 9.30 a.m. to 10.30 a.m.
4.3.2 First Lesson – Pre-test and Initial Survey (see Appendix D for lesson
plan)
The research room was set up with video cameras in place and the desks
arranged separately to ensure that students were unable to easily see a
neighbour’s work. Each desk had a survey sheet and a pen placed upon it.
Students were asked not to bring their pencil cases to the research room as they
would likely contain erasers, pencils and correction fluid that would provide a
student with an opportunity to erase working when he or she felt it was not
leading to a correct solution. Marking of the standardised Profiles of Problem
Solving test (Stacey et al., 1993) required the marker to consider all written work
to investigate the method used by the student, the accuracy of calculations, the
ability of the student to extract information, and their quality of explanation, as
well as the correctness of their final answer (see Appendix Q for the Profiles of
Problem Solving marking scheme).
All students were present for the pre-test and the initial survey. The first ten
minutes of the session were spent introducing the research to the students and
their role in the research, together with the pseudonym system that was used to
protect their identity. The provision of pseudonyms to the students was
deliberately held over until the following week to avoid distractions during this
testing session. Students were given ten minutes to complete the survey sheet.
This time appeared to allow all students to write a response to each of the four
questions. No students asked for additional time. The pre-tests were then
handed out to the students face down until all students had a copy. The test was
then completed in the required thirty-two minutes.
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4.3.3 Last Lesson – Post-test and Final Survey (see Appendix L for lesson
plan)
This lesson followed on directly from the final teaching episode and therefore
occurred an hour later in the day than the pre-test and initial survey had been
conducted. The same process of administration was followed for this lesson as
occurred with the first lesson. Students made comment about the survey sheet
questions and the Profiles of Problem Solving test (Stacey et al., 1993) questions
being the same as they had seen eight weeks earlier, but no students seemed to
be adversely concerned.
4.4 The Seven Teaching Episodes (Lessons 2-8)
The teaching episodes began with an opportunity for the students to adapt a
given novel problem to create a new problem. This experience served to
introduce the concept of students being authors of problems and gave them their
first opportunity to seek feedback from their peers by reading their problems to
the class.
Each subsequent teaching episode gave students the opportunity to pose a
variety of novel problems to their peers and to receive feedback from both their
peers and the researcher. In some of the teaching episodes unforeseen issues
arose that necessitated the refinement of subsequent teaching episodes. As
seen in Table 4.1 some adjustments were made to the location or timing of a
teaching episode, however, other changes were made to subsequent lesson
plans as a result of students misinterpreting tasks. For example, the first
problems posed by students were superficial and involved only simple
computations to be solved. In response, the stimuli provided in the subsequent
teaching episode was more abstract thus encouraging students to be more
divergent in their thinking and creative in their problem posing. In addition,
factors that made a problem interesting, such as increased detail and a challenge
factor, were discussed. Further examples and explanations of lesson plan
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refinements can be found in Sections 4.4.1 to 4.4.7. Details of how particular
students engaged with the learning activities in the teaching episodes can be
found in Chapter 5.
4.4.1 The First Teaching Episode - Lesson 2
The detailed lesson plan for this teaching episode can be found in Appendix E.
On this morning, rather than the students being dressed in their usual school
uniform, they were dressed in their sports uniform. It quickly became apparent
that this day was allocated by the school for the taking of sporting group
photographs; for example, the girls’ hockey team photograph, or the boys’ rugby
team photograph. The usual practice was for students in the different teams to
leave their classes, as they were called for by the photographer. Therefore,
throughout this first teaching session, different students left the research room for
up to five minutes to have their photographs taken in the adjacent hall. The
students appeared to be familiar with this practice and, on returning to the
research room, they immediately resumed their work. During this teaching
session, only three students were called for photographs and hence, the process
had a minimal impact on the Intervention Group as a whole.
The purpose of this first teaching session was twofold:
1) to encourage the students to consider more than one solution to a given
novel problem, and
2) to begin the process of posing their own problems by modifying a given
novel problem.
Students were provided with a novel problem and then given opportunities to
consider each other’s answers, as well as an opportunity to challenge and/or
support them. This part of the lesson laid the ground work to validate and ‘give
permission’ to the students to think independently from both their peers and the
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teacher, and prepared them for their first attempts at posing problems for their
peers to solve.
Three fixed video cameras were used in this lesson as a means to capture
discussions between peers, informal interviews with students, and to monitor
student engagement. While the cameras were focussed on the case study
students, vision and conversations of some neighbouring students were also
captured by the cameras. Following the lesson, the video tapes were reviewed
and it became apparent that the sound quality from the fixed locations was poor
due to the level of background noise in the room. It was therefore decided to use
a research assistant, for each subsequent teaching episode, to enable one of the
video cameras to be mobile and hence capture significant conversations and
informal interviews from a closer proximity. This proved to be a successful
solution to the issue.
4.4.2 The Second Teaching Episode - Lesson 3
The detailed lesson plan for this teaching episode can be found in Appendix F.
This teaching episode occurred two weeks after the first one, due to the
scheduled session in the previous week falling on a “pupil-free day” when
students were not required to attend school. To add to the challenge of the two-
week break between sessions, the heavy rain throughout the previous evening
and the morning had caused localised flooding and hence many school buses
were running late. This resulted in three students being absent, and five
students being late for the session. The students who arrived more than five
minutes late were clearly unsettled however, following individual briefing on what
they were required to do, all late students settled into posing their problems. It is
worth noting that two of the students who arrived particularly late were either
unable to complete their problems and solutions in the remaining time, or wrote
very superficial problems during the remainder of the lesson.
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Written below, are two examples of superficial problems written by students who
arrived late to this session.
Ethan, who was 30 minutes late to this session, wrote (without corrections to
Ethan’s spelling or grammatical mistakes):
If there are 40 skittles inside one pack and Liam ate 6, Kip ate 20, Elliot ate 4,
how many are left for us to eat?
Andrew, who was also 30 minutes late to this session, wrote (without corrections
to Andrew’s spelling or grammatical mistakes):
If there are 50 skittles and there are 10 different colours and 5 of each what is
the percentage of green?
These two problems contrasted with one written by Felicia and transcribed
below, who was on time for the start of this session. She wrote (without
corrections to her spelling or grammatical mistakes):
Maddie has two piles of scittles. In one pile, for every red scittle, there are 3
green scittles. In the other pile, for every green scittle, there are 3 red skittles.
Both piles have 40 scittles. How many green scittles are there?
Both Andrew’s and Ethan’s problems required only one or two steps to solve and
involved basic arithmetic, although Ethan did incorporate the concept of
percentage. The problems written by the two boys, contrasted with the question
written by Felicia. Her question incorporated the concepts of ratio and or
fractions, and involved several multiplications and additions to solve the problem,
once it had been determined how to solve the problem.
The main purpose of this teaching episode was to introduce the concept that a
“good” problem satisfies the following three criteria:
1) the problem contains an interest factor,
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2) the problem is appropriately challenging for its intended audience, and
3) the problem has enough detail to be “do-able”.
In addition, students were introduced to reflective practice through a weekly
rating system of their problems and through immediate peer feedback (see
Appendix N for the Problem Criteria Sheet).
As students each shared their posed problems with the group, peers were
encouraged to ‘rate’ each problem by holding up one of three cards provided to
them at the start of the lesson. One card had the number one printed on it, while
the other two cards had the numbers two and three respectively printed on them.
It was explained to the students that a problem that satisfied all three of the
accepted qualities of a ‘good’ problem should receive a ‘three-rating’, whereas a
problem that satisfied only two or one of the criteria should receive a ‘two-rating’
or ‘one-rating’ respectively.
After each problem had been read to the group, and when students had rated the
problem by holding their chosen card up, selected students who held differing
numbers were asked to provide a brief justification of their choice of rating.
Students were constructive with their feedback and did not use the opportunity to
diminish a peer’s self-esteem.
For example, Paul’s problem, which was confusing to other students and not do-
able in its present form, is recorded below, followed some of the feedback from
his peers. Paul’s problem is written as it appears in his workbook without
corrections made to his spelling or grammar.
The skittles company wanted to create a new couler (pink) although the
maximum amount of skittles is 40 and there are already 37. the minimum
amount of skittles in one couler is 4.
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Courtney rated Paul’s problem with a “2” card and when asked to justify her
answer she replied, “it was interesting <pause> and confusing and kind of do-
able, I think”. Ethan rated Paul’s problem with a “1” card and said “it was like,
<pause> hard <pause> and I didn’t get what the question was”. Paul smiled at
this latter comment, despite it being associated with a “1” rating. He appeared,
from his body language, to be proud of being acknowledged for creating a
challenging problem.
This mutual respect, such as was developed between the students during the
teaching experiment, was a phenomenon reported by Crotty (1998, p.45) who
said about humans engaging in learning with other humans, “it is in and out of
this interplay that meaning is born”. The method of immediate peer feedback,
incorporated into the teaching experiment, was consistent with the constructionist
beliefs that underpin this form of intervention. The students’ general willingness
and involvement in the problem posing and the peer rating was testament to how
“safe” and valued they felt during the session.
Towards the end of this and subsequent teaching episodes, students were asked
to self-rate their posed problems against the three criteria of a ‘good’ problem by
completing a problem criteria sheet (see Appendix N). At the conclusion of each
teaching episode, the student workbooks were collected and the researcher then
rated the students’ problems on the same problem criteria sheet. Workbooks
were returned to students at the start of each subsequent teaching episode and
time was allocated for students to read the teacher ratings and comments before
beginning the new teaching episode.
This was the first teaching episode that students were provided with a variety of
objects as stimuli about which to pose their problems. These objects were
chosen to facilitate problem posing across some or all of the strands of the
mathematics syllabus including; number, algebra, measurement, chance and
data, and space and were chosen as objects with which they would be familiar.
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In subsequent teaching episodes the stimuli were chosen to respond to issues
that arose in a previous teaching episode. For example, students may not have
been inspired to pose problems about everyday objects found in the classroom,
such as bulldog clips, as they held little interest for them. However, if a student
was asked to pose a problem about an object that was meaningful to them, such
as an iPod, they were more likely to be engaged in posing a problem.
4.4.3 The Third Teaching Episode - Lesson 4
The detailed lesson plan for this teaching episode can be found in Appendix G.
This lesson occurred one day after the previous lesson, since the scheduled
Monday session had been used to catch-up the session missed on the ‘pupil-free
day’. Problems posed in the previous lesson had been superficial and focussed
on very basic counting strategies, so a goal of this lesson was to challenge the
students to pose more complex problems using a seemingly, unquantifiable
stimulus, thereby creating an opportunity for the students to think more deeply.
Therefore, a close-up, colour photograph of a patch of grass was provided to
each student as the stimuli for their problem posing for this teaching episode.
Counting the blades of grass was an unlikely option for students; hence they
were encouraged to think differently about the type of problem they would be
able to pose.
4.4.4 The Fourth Teaching Episode - Lesson 5
The detailed lesson plan for this teaching episode can be found in Appendix H.
After the posed problems from the previous week were read, it was noted that
the setting out of student’s solutions lacked detail. Therefore, as a prelude to the
problem posing of this lesson, the researcher posed a problem to the students
about the patch of grass photograph (stimuli from the previous lesson) and asked
for two volunteers to solve it on a double-sided whiteboard (one student either
side). This created the opportunity to focus on the quality of a solution, in
addition to the quality of a problem, and thereby assisted the students to refine
their posing ability. By considering the steps involved in a quality solution, the
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students could then focus on posing problems that required a more detailed
mathematical solution, rather than a one or two-step superficial solution that
lacked challenge or interest.
The stimuli for this lesson were three familiar shapes; a triangle, a circle and a
square, all of the same proportion, but with no recorded dimensions. These were
chosen because they were familiar to the students, yet an immediate challenge
was present because of the lack of dimensions. Once again, the aim of this
choice of stimuli was to encourage the students to think broadly and deeply,
rather than superficially, about the type of problem they may be able to pose.
To encourage the students to focus on the three individual attributes of a good
problem, students were asked to hold up their 1, 2, 3 rating cards three times for
each problem; once for interest factor, once for challenge level and once for ‘do-
ability’, thus providing more immediate and specific feedback to the problem
posers. For example, students could now determine if their peers thought the
problem they had posed was very interesting, moderately interesting or not
interesting at all. This refinement was consistent with the problem criteria sheet
(see Appendix N) on which the students self-rated, and the researcher rated,
each problem posed by a student on a 3-point scale for each of the qualities of a
‘good’ problem.
4.4.5 The Fifth Teaching Episode - Lesson 6
The detailed lesson plan for this teaching episode can be found in Appendix I.
This teaching episode occurred on another rain-soaked morning which meant
school buses ran late again. Five students arrived between five and thirty
minutes late to the classroom, in varying states of concentration.
In the previous session, as students read their problems to their peers, it was
apparent that the length of the problems had increased substantially on
previously shared problems. This may have been as a result of students
becoming more engaged with writing their problems, or as a result of the
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students wanting their peers to be more interested in their problems. It may also
have been as a result of a student perception that a long problem was a “good”
problem.
An example of one of these longer problems was written by Georgia and is
recorded below. Georgia wrote (without corrections made to spelling or
grammatical mistakes):
Bob usually travels to school on his skateboard. There is a new shopping
centre that is known to be a shortcut to his school. It just so happens that
today he is late for school. He gets ready quickly and decides to go through
the shopping centre. However, as he arrives to the shopping centre he can
not skateboard through it. If he were to walk it would take him twice as long if
he skateboarded. This is equivilent to his normal trip (not going through the
shop). If it takes him twenty minutes on his normal trip what would it be if he
could travel through the shopping centre on his skateboard, it would take him
twice as long if he did that. What would be the ratio on a skateboard through
the shopping centre, to his normal trip, to the WALKING through the shopping
centre.
After Georgia read her problem there seemed to be a slight pause before anyone
held up a rating card. This may have been because they were still trying to
determine what the problem was about or because they were trying to assimilate
the information provided within the problem. This is what Sternberg (2000)
referred to as the “encoding” phase of solving a problem. When students were
asked to provide feedback after the problem was shared with the group, students
commented on how hard it was to remember all the details of the problem when
they could only ‘hear’ it rather than ‘read’ it for themselves.
The feedback, provided by the students, suggested that they needed some re-
direction on how to make a problem interesting, challenging and do-able, without
it being a very long problem. Without this re-direction it would have been difficult
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for all students to share their problems with their peers in the time provided for
each session. In addition, the refinement of the lesson plan for this session
allowed students to think differently by focussing on more concise problems.
They were now challenged to make a problem interesting, using as few words as
possible.
To accommodate this refinement, students were introduced to an example
problem consisting of one short sentence (see Appendix I) and asked to solve it
and then rate it against the three criteria for a ‘good’ problem established earlier
in the teaching experiment. The purpose of this activity was to show the students
that a problem could be interesting, challenging and do-able, despite it only
requiring one sentence to write.
The stimuli chosen for this lesson, a skateboard and an iPod, were both visual
and very meaningful to the students and were objects with which all of them were
familiar. Students were asked to try to pose a problem that could be written in
one or two short sentences.
At the conclusion of this teaching episode, students were reminded that the
library was having its stock-take during the following week so the venue for the
next session would be changed to an alternative classroom with which the
students were familiar.
4.4.6 The Sixth Teaching Episode - Lesson 7
The detailed lesson plan for this teaching episode can be found in Appendix J.
When the students were collected from their classrooms for this teaching
episode, it became apparent that a number of them were missing. Georgia,
Joanne and Leah, who all held school leadership positions, had been asked to
guide parents of prospective new students around the school campus. Quick
negotiation with the Year 7 teachers enabled other students, not participating in
this present study, to be sent to takeover from the leaders and Georgia, Joanne
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and Leah were able to re-join the Intervention Group within ten minutes of the
start of the session.
In previous teaching episodes, despite students being free to move around the
room, all students chose a seating position at the start of each lesson and
remained seated for the duration of the lesson. In an effort to maintain student
interest, the methodology for this teaching episode incorporated visual stimuli
and required the students to move around the room and interact differently than
they had in previous sessions. (Modifying the structure and design of
consecutive lessons is a strategy commonly used by experienced teachers to
ensure students remain engaged throughout a ‘unit’ of work.) The furniture in the
alternative classroom was re-arranged to allow students to initially sit in a semi-
circle of chairs at the front of the room, with all of the desks in two long rows
behind them. This allowed some visual problems to be laid out on the tables
without the students initially being able to see them directly.
The initial consideration of short problems in the previous session had not
successfully resulted in the students posing more concise problems that were
any easier to rate than long ‘wordy’ problems, when presented orally. The
methodology adopted for this session addressed the issue by taking advantage
of student preferences to work with visual aids (e.g., Rose, 2007). Students
began by attempting to solve a selection of visual problems presented to them by
the researcher on the board (see Appendix J for examples). They were then
asked to stand up and move to the desks at the back of the room where twelve
visual problems were placed in a row. (These three-dimensional problems had
been constructed by Year 7 students in previous years during a problem-posing
unit of work.) After attempting to solve, and ultimately discussing, all of the three-
dimensional, visual problems, students were asked to construct their own visual
problems at home and share them with their peers in the following session.
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4.4.7 The Seventh Teaching Episode - Lesson 8
The detailed lesson plan for this teaching episode can be found in Appendix K.
This session was the first of two consecutive lessons with the Intervention Group
on this day and it occurred on the final Monday of the school year. The purpose
of this session was to provide the students with an opportunity to showcase and
share their visually-posed problems. The method of peer feedback was varied
this week to add a level of competition and culmination to the teaching
experiment. Students were consulted about, and gave approval for, the changed
method of feedback. After all students had assembled their problems along the
row of tables in the classroom, they vacated the room and stood quietly in the
corridor. Only one student at a time was permitted to re-enter the room to place
a 1, 2 or 3 card alongside the problems they felt were the best; three points being
allocated for the best, two points for the second best and one point for the third
best. The word ‘best’ was defined to be a problem that most addressed the three
criteria of a ‘good’ problem (see Section 4.4.2). When all students had
completed this process, all students re-entered the room and the scores were
tallied and shared.
4.5 Conclusion
Research undertaken with students in a school setting is often beset by
unexpected dispositional factors (see for example Reis & Siegle, 2006) and
situational factors (see for example Gootman, 2001) that cannot be pre-empted
during the design of the research methodology. Factors such as rain, late school
buses, school photographs and final week activities all impacted to varying
degrees on the implementation of the teaching experiment. The design of the
teaching experiment, and the flexibility of the staff at the teaching site, ensured
that any negative impact caused by these factors was minimised. As is the
normal case for school teachers and researchers in school settings (Steffe et al.,
2000), refinements were made to subsequent teaching episodes as factors
arose. For example, when the library chose a Monday to have their stock-take,
an alternative room was chosen as the research room.
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During the problem-posing intervention, students were encouraged to choose
how they wished to work; individually, in pairs or in small groups. The research
intervention was contingent on human interaction between the students and with
the researcher and occurred in a way consistent with constructionist beliefs (e.g.,
Broido, 2002). The teaching experiment provided the students with varied
opportunities to assemble new practices and ways of seeing problems (Tierney,
1997) and provided them with choice and control over the direction of their
learning.
Time to accommodate pre-testing and post-testing is acknowledged as
problematic, as was finding a regular intervention time to ensure that students
would not miss any of their normal school classes. A student arriving late to a
session was a regular occurrence for a variety of valid reasons, but the reasons
were always beyond the control of the students and not as a result of their
reluctance to participate in the teaching experiment. Given the context of the
teaching experiment, and the time of the year the research was conducted, the
setting for this intervention was suitable to investigate the impact of a problem-
posing intervention on the development of problem-solving competence.
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Chapter 5
Reporting and Analysis of
the Data
5.1 Chapter Overview
This chapter reports on the qualitative and quantitative data collected during the
teaching experiment and comprises four sections that follow the Chapter
Overview. Further discussion and interpretation of the data, reported in this
chapter, can be found in Chapter 6 where responses to the three research
questions are presented. The first section of this chapter reports the data
collected from observations of, and informal interviews with, three case-study
students during the teaching experiment (see Section 5.2), while the second
section reports on student surveys data collected from all of the students in the
Intervention Group (see Section 5.3). This is followed in Section 5.4 by a report
on the quantitative data, collected from the Profiles of Problem Solving
assessment instrument (Stacey et al., 1993), collected from students in the
Comparison and the Intervention Groups. The final section presents a
conclusion of this chapter (see Section 5.5).
5.2 Observations and Interviews with the Three Case Study Students
This section will begin with a brief re-introduction to the three case study
students, Paul, Andrew and Nicole, followed by an in-depth review of their
individual progress throughout the teaching experiment. As did their peers, these
three students completed the Profiles of Problem Solving (POPS) (Stacey et al.,
1993) pre-test and post-test that was scored out of a possible 53 marks. Paul
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scored 31 and 40 respectively, while Andrew scored 42 and 38 respectively and
Nicole scored 37 on both the pre-test and the post-test.
Paul had a median result in the Middle Years Ability Test (MYAT) (Australian
Council for Educational Research, 2005), compared to the other students in
the Intervention Group, yet he demonstrated one of the highest increases in
problem-solving competence as measured by the POPS (Stacey et al., 1993)
test at the end of the teaching experiment. The POPS test allowed data to be
collected on five aspects of problem-solving competence; correctness of
answer, method used to obtain an answer, accuracy of calculations, ability to
extract useful information and the quality of the student explanation of their
answer (see Section 3.4.3 and Section 5.4). Paul was of particular interest in
the investigation of Research Questions 2 and 3.
Andrew was chosen as a case study student for his increased engagement
in learning activities throughout the teaching experiment. He was of particular
interest in the investigation of Research Questions 1 and 3. His engagement
behaviours changed significantly from being off-task, to a student who was
interested to listen to his peer’s problems and one who became focussed
while writing increasingly mathematical problems of his own. These latter
problems, posed by Andrew, required more than two mathematical
calculations to be completed, in order to find a solution.
Nicole began the teaching experiment as the participant with the highest
MYAT (Australian Council for Educational Research, 2005) score of all
participants in the Intervention Group. While her POPS (Stacey et al., 1993)
test results did not indicate improvement in problem-solving competence, the
problems Nicole wrote in her workbook and posed to the group, consistently
became more mathematically sophisticated throughout the teaching
experiment. That is, the problems Nicole progressively posed, required an
increasing number of mathematical calculations to be undertaken in order to
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solve her problems. In addition, Nicole did not actively participate in posing
problems in her workbook or sharing them with her peers in the first few
teaching episodes. These behaviours steadily changed throughout the
teaching experiment. Nicole was of particular interest in the investigation of
Research Questions 1 and 3.
5.2.1 Paul
As mentioned in Section 3.4.2, one of the selection criteria for inclusion in this
study was for the participants to have scored above the 60th percentile in the
MYAT (Australian Council for Educational Research, 2005). Paul had a MYAT
score equal to the median result for all students in the Intervention Group. He
scored in the 85th percentile compared to his peers with the following group
statistics: N=15; range: 76 to 94; mean = 85 and median = 85. His pre-test and
post-test results from the POPS test (Stacey et al., 1993) recorded in Table 5.1.,
show that Paul improved his overall score from 31 to 40 marks out of a possible
53 marks, improving in four of the five criteria.
In the first criterion, ‘Correctness of answer’, Paul’s high score remained
unchanged with 11 marks out of a possible 13 marks in both the pre-test and the
post-test. However, his score for the ‘Method used’ aspect increased from 6
marks to 10 marks, out of a possible 14 marks (see Table 5.1). Information
Processing theorists, such as Sternberg (2002) and Halford (2002), discussed an
individual’s chosen methods of solution to a problem in terms of change
mechanisms that they said were largely responsible for improvements in
developmental learning. In Paul’s case, improvements in the ‘Method used’
criterion would align with an increasingly efficient execution of mental processes
(automatisation) and an improved ability to select and prioritise important aspects
of a problem (encoding) (see Section 2.2.1). Sternberg (2000) said it was the
synthesis of these change processes that resulted in cognitive growth and
developmental learning. The improved results in the Paul’s ‘Method used’ score,
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suggested that Paul may have experienced some developmental learning as a
result of the problem-posing intervention.
Table 5.1
Paul’s Profiles of Problem Solving Pre-test and Post-test Results
Pre-test result Post-test result
Correctness of answer (maximum 13 marks)
11 11
Method used (maximum 14 marks)
6 10
Accuracy (maximum 10 marks)
6 8
Information extraction (maximum 8 marks)
6 8
Quality of explanation (maximum 8 marks)
2 3
Total test score (maximum 53 marks)
31 40
Paul increased in his score for ‘Accuracy’ and ‘Information extraction’ by 2 marks.
In this latter criterion, Paul achieved the maximum marks possible by the end of
the teaching experiment. In the fifth criterion, he demonstrated a modest
improvement in his ‘Quality of explanation’. While Paul could determine the
correct answer for most questions in the POPS test (Stacey et al., 1993), his
increased ability, to demonstrate skills in these latter three criterion, following the
problem-posing intervention, needed further analysis. During Paul’s schooling,
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he may have only been encouraged or expected to achieve the correct answer.
That is, the product, rather than the process, may have always been the primary
focus of his problem solving. Lokan and McCrae (2003) reported that only 25%
of problems provided to students in Year 8 classes in Australia were rich in
complexity emphasising procedural fluency. It is reasonable to suggest then,
that this may also be the experience for Paul. This more traditional approach to
teaching mathematics has been identified and criticised by the Australian
National Curriculum Board (2008) which referred to such teaching as “ineffectual”
and “irrelevant” in the pursuit of mathematical learning that is connected and
meaningful to a futures-focussed society.
An alternative viewpoint, to explain Paul’s improved scores in the individual
criterion of the POPS (Stacey et al., 1993) test, may come from consideration of
the amount of time dedicated to solving problems in Pauls’ normal mathematics
classes. In particular, within the amount of time spent solving problems, it would
be interesting to compare the amount of time that was focussed on finding the
correct answer as compared to ‘thinking’ about problems and on how they are
constructed and solved. This fundamental distinction was discussed in Shimizu’s
(2002) findings and in the report of data collected in the Third International
Mathematics and Science Study (TIMSS) (Stigler et al., 1999). Shimizu reported
that, for Japanese students, it was the time spent thinking about mathematical
problems and finding alternative solutions that most influenced a student’s ability
to become a more competent problem solver.
The TIMSS study presented some very pertinent data for comparing
mathematical pedagogy that supports the mathematical teaching practices in
Japanese schools. It reported that Japanese students were ranked 3rd in the
comparative problem-solving scale scores of the test, compared to American
students who ranked 24th. In addition, it reported that Japanese students spent
45 times more class time thinking about mathematical problems and inventing
new solutions than did their American peers (see Tables 1.1 and 1.2). While
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these findings are reported about Japanese and American students, and not
Australian students, it could be argued that it may have been the opportunity to
focus on the process, rather than the product in the teaching experiment, that
allowed Paul to improve his ability to articulate his methods of problem solution.
Paul was an enthusiastic participant from the very first teaching episode. He was
full of self-confidence and very willing to share his posed problems with his
peers. He enjoyed working in pairs or by himself and once he began posing his
problems he focussed on the task at hand and ignored any distractions around
him. An early example of this was captured on Camera Three (time 06:11) in the
third teaching episode when a neighbouring student made several attempts to
tease him for his industriousness. It was clear that Paul found problem posing to
be a liberating activity and he felt empowered to demonstrate his individuality.
This may have been as a result of the critical, post-modernist framework
underpinning this study (Broido, 2002; Lincoln & Denzin, 2000), or it may have
been as a result of Paul looking for public notoriety or reinforcement from his
peers and the researcher, as discussed by Schultz (2000).
In Paul’s case there is more evidence from this present study to support the latter
explanation. When students were asked for volunteers to pose their problems to
the class, Paul would always volunteer with enthusiasm despite not always
knowing exactly what he would be required to do. This often resulted in him not
being able to communicate his thoughts coherently. For example, during the
fourth teaching episode, the lesson began with students being given a problem
posed by the researcher. Students were asked to deconstruct it and then to try
to solve it. Camera Three captured the following scene:
05:17 Researcher: “Now, I’m going to ask for two brave people.” (Paul
immediately waved his hand in the air before he knew what he was going to be
asked to do.)
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06:00 Paul and another student were chosen to record their full solution on
either side of a mobile whiteboard in front of the class.
<Both students recorded separate solutions on individual sides of the
whiteboard. The other student was first to discuss his solution with the class.>
16:13 Paul attempted to describe his thinking to the class as he went through
the steps of his solution. He stumbled over his words and scratched his head
twice while pausing.
Paul’s solution was discussed by the group and subsequently all students were
provided a researcher solution for comparison (see Appendix H for researcher
solution). The researcher solution provided some guidance to the students about
the level of mathematical detail required to effectively communicate a solution to
a reader. Paul immediately appeared to understand the difference between how
he was accustomed to writing a solution and the researcher solution. This was
evident by Paul’s ensuing discussions with his peers and his regular reference to
the researcher’s solution in subsequent teaching episodes, and was
demonstrated in his improved scores for the ‘Method Used’ and ‘Quality of
Explanation’ criterion of his post-test.
From the fourth teaching episode, Paul began to seek preliminary feedback from
the researcher before he posed his problems to the class. An example of such a
conversation was captured by Camera Three in the second teaching episode
when students were provided with a small packet of colourful lollies (Skittles), as
well as several other objects to use as stimuli about which to pose a problem:
13:57 Paul to Researcher: “Mine will be hard if it’s just er <pause> like spoken,
but, if like, <pause> ‘cos, no-one will understand it ‘cos it’s got heaps of
colours.”
14:03 Researcher read the following problem, as written in Paul’s workbook
(without corrections to Paul’s spelling or grammatical mistakes):
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If a new couler was created (pink) and the maximum amount of skittles is 40.
The creators wanted pink to have the same amount as the red even if they had
to take some skittles from the yellow and change them to pink. How many
skittles were in green, red, pink, orange, purple, yellow?
14:32 Researcher to Paul: “Have another read of it Paul because I’m thinking
about the third criteria. Is it do-able? In your mind you can say ‘I know what I
mean?’ but other people have to. If we publish that in a Maths book other
people have got to read it and understand it.”
14:48 Paul nodded and began to refine his problem immediately.
Paul did not complete his refined problem due to lack of time but he had begun to
consider the feedback he had received from the researcher. His partly
completed, refined problem, as written in his workbook was as follows (without
corrections to Paul’s spelling or grammatical mistakes):
The skittles company wanted to create a new couler (pink) although the
maximum amount of skittles is 40 and there are already 37. the minimum
amount of skittles in one couler is 4.
From early in the teaching experiment, students were encouraged to ‘rate’ each
other’s problems and their own. This process was made simple by including only
three criteria for deliberation, all of which were fully explained to the students to
ensure their understanding. Students were expected to consider how interesting
the problems were, whether they were appropriately challenging to their peers,
not too hard and not too easy, and whether there was sufficient information
contained within the problem for a solution to be determined (see Section 3.5.1
for discussion about peer and self-rating and see Appendix N for the Problem
Criteria Sheet template). Paul self-rated this problem as two out of three in the
three criteria of a ‘good’ problem; interest factor, challenge level and do-ability.
He wrote in the General Comments section of the Criteria Sheet, “Make it much
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less confusing”. This demonstrated that he had been reflective about the
feedback he received from the other students and it supported Crotty’s (1998)
findings which suggested that best learning occurs when students are actively
engaged in creating meaningful artefacts to share with their peers.
Paul’s problems were always high in ‘interest factor’. In the first teaching
episode, students were presented a problem about some people going on a
holiday and they had asked their neighbour’s children to care for their dog and
cat. Students were asked to re-write the problem to pose a new problem (see
Appendix E). Most students simply changed the amount of money being offered
to care for the pets, or changed the frequency of the care needed by each pet.
Paul posed the following problem, as written in his workbook (without corrections
to Paul’s spelling or grammatical mistakes):
My neighbours are going to Chad, they have a pet lion and a pet tiger, the tiger
needs to eat 5 females every fortnight and the lion needs to eat 2 females
every second day. If we had to look after them for 5 weeks …….
The gruesome theme drew smiles and laughter from the students sitting at Paul’s
desk when he read the first line to them. This problem, and later problems
written by Paul and other male students, often contained more gruesome themes
compared to those written by the female students in the group. This is consistent
with the findings of Jones and Myhill (2004) who said that boys typically like to
explore more aggressive games and activities while female students are more
interested in collaborative and nurturing themes and activities.
Despite Paul’s problem not being completed in the time provided, he self-rated
the problem as two out of three for interest factor and challenge, and three out of
three for do-ability which was interesting considering it was incomplete. He did
not write any general comments in the space provided on the criteria sheet. It
was during this teaching episode that it became clear that Paul had some
difficulty in getting his stream of ideas into a logical sequence and order to create
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a complete, meaningful problem in the time provided. In the first few teaching
sessions he often used more time to pose an imaginative problem than he did to
consider if it were actually do-able.
Paul’s subsequent problems were no less entertaining. In the fourth teaching
episode students were provided with a triangle, a square and a circle as stimuli
for posing a problem. Providing stimuli each session assisted the students to
focus on the posing of problems rather than the thinking about a topic about
which to pose a problem. Most students wrote about blocks of land or animal
enclosures that referred directly to the shapes provided. However, on his second
attempt, Paul wrote the following problem (transcribed without corrections to
Paul’s spelling or grammatical mistakes):
Every day the cadbury factory in Tasmania sells 50032 boxes of chocolate.
They get all there cocoa beans from a factory called cocoa bean Hevean. CbH
has had some problems with the bank lately and havn’t been able to sell their
cocoa beans. For every 100 boxes they sell Cadbury factory earns $200.
Because they havn’t been getting cocoa beans from CbH they havn’t been able
to sell any chocolates. How much money would the cadbury factory have lost if
CBh didn’t sell them coca beans for a fortnight?
When asked how the stimuli took his thoughts to chocolates, Paul said the
square reminded him of a box of chocolates and the circle and the triangle
reminded him of chocolates in the box. This was the first problem where Paul
clearly demonstrated development in his mathematical thinking and analogical
reasoning. Sternberg (2002) discussed developmental learning, such as that
typified by Paul’s newly demonstrated skills, as an increasingly efficient
execution of mental processes (automatisation), an increased ability to select
and prioritise relevant data from a situation being mathematised (encoding), and
an increased ability to draw on prior knowledge to inform current thinking
(generalisation). Paul had incorporated mathematical information into an
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elaborate storyline in a meaningful way. After he read his problem to the class,
Paul was able to clearly describe the steps that would be needed to solve the
problem. He had successfully transferred his knowledge gained from peer and
researcher feedback of his previous problems to create a new and meaningful
novel problem. This finding supports Mestre’s (2000) research that suggested
posed problems provide evidence of whether the author has become ‘well-
versed’ in how mathematical concepts can apply across a range of problem
contexts.
Paul continued to demonstrate the development of his mathematical thinking
throughout the teaching experiment. Each successive problem posed by Paul,
albeit containing grammatical and spelling errors, was increasingly mathematical
in content while remaining entertaining to his peers. Brown and Walter (2005)
reported that problem posing has the potential to encourage students to think
divergently, which in turn has the potential to promote developmental learning
opportunities. Their finding is supported by Paul’s progress in posing meaningful
and detailed problems as he moved further into the teaching experiment.
In the fifth teaching episode students were provided with a cartoon about
skateboarding and an iPod as stimuli for this session (see Appendix I). Paul
posed the following highly-entertaining problem (transcribed without corrections
to his spelling or grammatical mistakes):
At school kids were getting really angry and went on a strike, because the
teachers didn’t want any kids leaving the school they decided to let the kids run
the school for 1 Day. The kids created a poster to show how mean the
teachers had been the teachers didn’t care. The kids were soo angry about
how rude the teachers were being they decided to destroy the school. In the
school there was 100 bricks 30 lights and 30 desks. To Destroy 1 brick it takes
1 minute and a half to destroy 7 desks it takes 9 minutes and to destroy 13
lights it takes 17 minutes. How long does it take to Destroy the whole school?
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This problem showed that Paul was now able to incorporate some increasingly
sophisticated mathematical challenges into his problems. Not only did he
challenge his peers to extract information and do some multiplications to solve
his problem, he also challenged them to do additions and conversions of units.
Paul had begun to demonstrate what Sternberg (2002) referred to as the
“synthesis of change processes” that results in cognitive growth. Previously Paul
had not been able to coherently explain his thoughts, whether they were in
describing how he would solve a peer’s problem or how he was going to pose
one of his own problems. However, Paul was now able to demonstrate
sequencing of logical thoughts to achieve a meaningful product, that is, an
interesting, challenging and ‘do-able’ problem. Despite the problem being
implausible, Paul recorded an accurate solution to the problem in his workbook.
According to studies undertaken by Bernardo (2001), analogical reasoning is
available to all students to different degrees, however, it can be further
developed through appropriate intervention strategies. Paul’s increasing
understanding of problem structures had enabled him to ‘map’ between his
previous problems and his new problems, thus suggesting that his experiences in
the problem-posing intervention has facilitated some developmental learning
changes for him. His problems had become ‘do-able’ as well as highly
interesting to his peers.
5.2.2 Andrew
In the selection process, Andrew had the lowest MYAT score of all participants in
the Intervention Group, scoring in the 76th percentile. He was by far the most
lacking in confidence and most easily distracted student at the start of the
teaching experiment, thus making him a suitable candidate for a case study
student. During the first few teaching episodes, Andrew did not pose complete
problems or work earnestly at the task. Shultz (2000), when talking about “gifted
underachievers”, said that students, such as Andrew, typically do not fully
engage with work for a number of reasons including possible low self-esteem,
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while Boehnke (2007) suggested disengagement may be due to the individual’s
fear of social exclusion. While Andrew seemed quite confident in his interactions
and was liked by his peers, he initially seemed self-doubting and reticent about
his ability to pose or solve problems.
In the first few teaching episodes, when volunteers were asked to share their
problems, Andrew would not raise his hand. When he was asked about what he
thought of his problems, he replied that they were not very good and not worth
sharing with his peers. However, as the teaching experiment unfolded, Andrew’s
self-esteem and confidence developed and, in the third teaching episode, he first
offered to share with his peers a problem he had posed. When his peers were
asked to ‘rate’ his problem using the 1, 2, 3 card system, Andrew initially looked
anxious. As the cards were raised and Andrew saw that there were some 2’s
and 3’s amongst them, he smiled then immediately dropped his head as if he did
not want to show his peers that he was so pleased.
As the teaching experiment continued to unfold, Andrew not only engaged with
the problem-posing activities, he also gained in self-confidence and actively
ignored students who tried to distract him. As the teaching episodes progressed,
his problems became far more interesting, including detailed story lines about
topics that were familiar to himself and his peers. In addition, he began to enjoy
the opportunity to pose and share his problems and joined Paul as one of the first
students to offer to share his problems with the group when given an opportunity.
At the completion of the fourth teaching episode, students were asked if they
were enjoying the sessions. Andrew responded by saying that it was good to
learn how to solve problems by learning how to “write” them. This was an
interesting response because Andrew had himself made the connection between
posing and solving problems despite no direct mention being made of this
connection in the question. The ability for problem-posing activities to impact on
a student’s self-efficacy was reported by Bandura (1997) as significant, while
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Knuth (2002) reported its positive impact on the development of students’
intrinsic motivation to engage in learning activities. Both authors associate
problem-posing activities with these two positive changes that were increasingly
demonstrated by Andrew throughout the teaching experiment. It would seem
reasonable to suggest then that we may be able to attribute Andrew’s increased
engagement to the problem-posing activities undertaken as part of this current
study.
Andrew’s pre-test and post-test results are shown in Table 5.2. While the test
results would appear to indicate that he made no gains in problem-solving
competence as a result of the teaching experiment, the observations made of
Andrew by the researcher and the developing quality of problems posed in his
workbook, present a different picture.
The session in which the pre-test occurred was the first session the students
came together as a group and met the researcher. All students, including
Andrew, quietly and conscientiously completed the student survey and the pre-
test. As a group, they were far more reserved in this session than they were
throughout the teaching experiment or in the final session (the post-test session)
by which time they had become familiar with the researcher, the teaching
experiment, posing problems and being withdrawn from their usual class
activities. On the day of the post-test, Andrew appeared particularly distracted.
His distraction may have been due to the final session taking place in the final
school week of the year, or to the lack of problem-posing opportunities in this
session. Jones and Myhill (2004) and Baker (1998) reported that perceived
underachievement, such as Andrew’s, has little to do with a lack of ability and is
more likely to do with situational factors such as the timing of a test. While a
number of the students were clearly excited and more animated at the start of the
post-test lesson than they were at the start of the pre-test lesson, most students
were able to re-focus and apply themselves to complete the post-test to the best
of their ability. Andrew did not have the same ability to moderate his physical
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and emotional responses to the situational factors of the day. He seemed
distracted and fidgeted throughout the session.
Table 5.2
Andrew’s Profiles of Problem Solving Pre-test and Post-test Results
Pre-test result Post-test result
Correctness of answer (maximum 13 marks)
11 10
Method used (maximum 14 marks)
11 8
Accuracy (maximum 10 marks)
8 8
Information extraction (maximum 8 marks)
8 8
Quality of explanation (maximum 8 marks)
4 4
Total test score (maximum 53 marks)
42 38
From the earliest teaching sessions Andrew had demonstrated how easily he
was impacted by situational factors. His engagement in the first teaching
episode was characterised by fiddling with pens, being easily distracted by any
form of movement or noise around him, and with minimal writing. He required re-
direction from the researcher on three occasions during this session. An
example of a three minute time segment, captured on Camera One in the first
session, can be seen below:
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22:30 laughing with neighbours – off task
23:11 “I have an idea” – finally started writing
23:48 put hand up for assistance but researcher was with another student
24:06 put hand down and started to write
24:29 looked up and chatted to neighbour
24:32 began writing again
24:38 neighbour wanted to swap pens but Andrew playfully put his pen
behind his back
24:53 pen situation resolved – started writing again
25:46 Andrew finished modifying the researcher-provided problem and looked
up (see Appendix I for original problem provided to students)
Andrew’s problem, written below without grammatical corrections, was
superficial, unclear and lacked detail. He did not provide a worked solution,
choosing instead to simply record how he would divide the money.
Tom and Sue have asked to look after the neighbours dogs they have left for a
holiday 5 days and are giving the $100 to share between them Tom has to look
after fifi for 50 mins Jane has to look after Pluto for 25 mins everyday.
Tom = $75
Jane = $25
As students completed writing their problems, volunteers were sought to read
their problems to the class. Andrew did not offer to read his problem. At the end
of the session he chose not to complete his first self-rating criteria sheet despite
other students around him taking the time to fill in their sheets.
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During the second teaching episode Andrew’s engagement in the sessions
began to change noticeably. He started to respond to questions from the
researcher without shrugging his shoulders or looking away. An example of such
a response was recorded on Camera One:
07:40 Researcher: “What was the first thing you thought to do Andrew?”
07:43 Andrew: “Um, well, condense five squared to find out how much the
perimeter <pause> ‘cos the whole thing is 72 you have to find out how much
more than five squared it is.”
This was followed later in the lesson by another example of engagement
captured on Camera Two:
13:03 Researcher: “Andrew, tell me what were you thinking?” (Students had
been asked to determine a strategy to calculate how many dots were in a piece
of aboriginal art as a warm-up exercise for their subsequent problem posing.
See Appendix H for details.)
13:05 Andrew: “If you really want to, you can count up the outside ones of the
circle and times it by how many rings are in the circle.”
Later in the lesson, when students were given an opportunity to pose their first
novel problem, using a photograph of some grass as a stimulus (see Appendix
H), Andrew began to work immediately. He continued to work uninterrupted until
the researcher asked for volunteers to read their problems aloud. The following
dialogue was captured on Camera Two:
48:18 Researcher: “Andrew, have a go at yours.” (Andrew smiles)
48:21 Andrew: “In a grass patch there is a quarter green grass and the
remainder is dead. If the grass patch is a square and one side is 18cm, what is
the area of dead grass?”
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48:36 Researcher: “How many steps do you think there are to mathematically
solve that?”
48:41 Andrew: “Three” (Andrew smiles with confidence)
This problem was quite different from Andrew’s previous problem because it
made sense and the grammar and spelling were correct. Andrew also took the
time to record a fully worked solution in his workbook. When asked to self-rate
his problem Andrew gave himself a two out of three for each of the three criteria;
interest factor, challenge level and do-ability. In addition, he wrote the following
sentence in the General Comments section of the criteria sheet, “I think my
problem is better than last because it does actually have steps to take to work it
out.” This comment and Andrew’s increased engagement demonstrated that he
had responded to feedback and had become engaged with the process of posing
meaningful novel problems. Andrew had demonstrated what Brown and Walter
(2005) reported when they said that posing problems puts the poser in charge of
the learning process which in turn has the potential to develop more divergent
forms of thinking. From the problems Andrew had written in his workbook, it
could be concluded that Andrew had either chosen to apply himself to the task at
hand or that he had begun to think about problems differently to the ways he had
done in the past.
Andrew continued to be fully engaged and on-task over the remaining teaching
episodes. From the third teaching episode he began to seek feedback from his
neighbours about his posed problems, before the class had been asked to share
their problems. By the fifth teaching episode he was also seeking preliminary
feedback from the researcher before posing his problem to his peers. An
example of this was captured on Camera One at 35:16 and resulted in Andrew
writing the following problem, with a full solution, in his workbook:
If Andrew is riding his skateboard at 17 km/hr and 50-50 grinds down the
handrail and increases his speed by 5% how fast is he going? If Andrew then
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lands and his speed drops by 2% what is his speed after the trick and while he
is doing it?
This problem, posed by Andrew, could be considered to be the “observable end
state” in the process of thinking as discussed by Siegler (1991). Seigler reported
about processes of thinking as being the initial and intermediate steps to
developmental learning accomplished inside an individual’s mind and that are
unseen by an observer except through the end product, which in this case was
Andrew’s problems. If we accept Siegler’s position, then the change in the
‘quality’ of Andrew’s problems could be considered as evidence of developmental
learning. Andrew had consciously and unconsciously become involved in the
cognitive processes that resulted in him posing increasingly sophisticated
problems for his peers to solve. He became increasingly efficient in the
execution of his mental processes (automisation), he was clearly able to select
appropriate data and prioritise how to present it in the form of a coherent problem
(encoding), he responded to prior feedback from his peers and the researcher in
the posing of subsequent problems (generalisation) and was able to coherently
discuss solutions for his problems and his strategies for constructing them
(strategy construction).
The increased demonstration of these four skills, automisation, encoding,
generalisation and strategy construction, was identified by Sternberg (2000) as
playing a significant role in developmental learning. It is therefore reasonable to
suggest that we may be able to attribute some developmental learning changes
demonstrated by Andrew to the problem-posing intervention in this study.
Andrew’s ability to remain on-task was repeatedly challenged by distractions
around the room during the teaching episodes. As time progressed he moved
from a position of ignoring these distractions to actively discouraging them. For
example, in the fifth teaching episode, Camera Three captured the following
incident:
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41:25 Blair, who had finished posing his problem, started to move irritably from
side to side on his seat in an attempt to distract Andrew who was still writing.
41:28 Andrew said to Blair, without looking up from his writing, “Don’t pay me
out!”
41:30 Blair smiled and replied, “Sorry, I’m not paying you out, I’m just rocking.”
In the sixth teaching episode students were investigating and posing visual
problems (see Appendix J). Andrew was particularly engaged during this
teaching episode and was clearly not prepared to be interrupted by other
students. Towards the end of the lesson the researcher asked all students to sit
quietly for an evaluation of the lesson, but some students continued to chat to
each other about their problems. Camera One captured the following scene:
45:15 Andrew puts his finger to his lips and said “Ssh!” to the students around
him. Students responded and the researcher continued to talk without any
further student interruption.
While involved in the problem-posing intervention, Andrew had become a fully-
engaged student with an ability to self-moderate his behaviour and remain on
task without re-direction. This outcome would not be unexpected by English
(1997b) who reported that a problem-posing classroom can empower and
encourage students to pursue “lines of inquiry” that are personally satisfying to
them, as was clearly the case with Andrew. He had even begun to ‘take some
risk’ with regard to how he was perceived by his peers transforming from a
‘follower’ in the first teaching episode to a ‘leader’ in the final teaching episodes
by encouraging others to engage in the problem posing when they were off-task.
The problem-posing intervention had served to re-engage Andrew to work in a
focussed way with novel problems. His increased focus and change of
behaviour was noticeable and increasingly consistent while he was engaged in
problem posing. In the absence of an opportunity to pose problems, Andrew’s
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behaviour returned to being affected by situational factors and he disengaged
with set tasks. This was evident in the final session of the teaching experiment
when Andrew was required to solve problems in the POPS test (Stacey et al.,
1993). For Andrew, it seemed that it was the opportunity to pose problems and
receive feedback on those problems from his peers that allowed him to engage in
learning throughout the teaching experiment.
5.2.3 Nicole
In the selection process, Nicole had the highest MYAT score of all participants in
the Intervention Group. She scored in the 94th percentile compared to her peers.
During the teaching episodes, students had four opportunities for the researcher
to ‘rate’ their problems against the three qualities of a ‘good’ problem; interest
factor, appropriate level of difficulty and do-ability (see section 3.5.2.4). Nicole’s
researcher-rating range was 5.5 to 7.5 out of a possible 9 marks, compared to
the group average researcher-rating range of 4.6 to 6.7. Interestingly, while
Nicole posed consistently sophisticated problems during the intervention, her
pre-test and post-test results from the POPS test did not vary significantly. Her
pre-test and post-test results are shown in Table 5.3.
Nicole’s ‘Quality of explanation’ mark decreased from 4 to 2 marks out of a
possible 8 marks. As she had previously demonstrated an increased capacity in
this criterion and the pre-test and post-test were identical test papers, it can be
assumed that either dispositional or situational factors had influenced Nicole
during the post-test. The ability of such factors to have a substantial influence on
an individual’s preparedness or ability to demonstrate skills, was reported by
Gootman (2001, p. 5) who said that “many children haul the baggage of
dysfunction straight into the classroom” where it can manifest in
underachievement or disengagement.
The students in this present study were not immune from day-to-day challenges
that can arise at any given time on any given day, and the exact nature of
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situational factors impacting Nicole on this particular day were not known.
However, dispositional factors such as Nicole being bored or simply disinterested
in completing a test that she had already seen and completed seven weeks
earlier, cannot be dismissed as contributing to Nicole’s decreased score on the
post-test in this criterion. Gentry, Gable and Springer (2000) reported that
boredom is a major contributor for the withdrawal from activities by middle-year
students and can therefore not be discounted as contributing to Nicole’s
disengagement from the activities in the final lesson of the teaching experiment.
Table 5.3
Nicole’s Profiles of Problem Solving Pre-test and Post-test Results
Pre-test result Post-test result
Correctness of answer (maximum 13 marks)
9 9
Method used (maximum 14 marks)
10 10
Accuracy (maximum 10 marks)
6 8
Information extraction (maximum 8 marks)
8 8
Quality of explanation (maximum 8 marks)
4 2
Total test score (maximum 53 marks)
37 37
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At the start of the teaching experiment, Nicole was an acquiescent student who
preferred to pose problems by herself. While she was willing to pose problems,
when students were first given the opportunity to rate each other’s problems
using the 1, 2, 3 cards, she did not participate in the rating process. By the
second teaching episode she was scanning the students sitting at her table to
see what ratings they were giving. Only then would she lift her hand from the
table with her chosen rating card. As she lifted a card she kept her wrist on the
table which meant the cards were never raised high enough for them to be
clearly seen. By the third teaching episode, Nicole had begun to lift both her
hand and her wrist from the table to raise a rating card and she did not always
wait to see which cards her peers were raising. It appeared that she had begun
to feel ‘safe’ in the teaching environment.
Nicole continued to become more engaged and enthusiastic about rating other
student’s problems as the teaching experiment progressed. The behaviours,
demonstrated by Nicole, are consistent with those reported by Alvesson and
Skolberg (2000) who found that students would participate more readily in
activities if they felt ‘respected’ and if the learning environment was ‘supportive’
and involved ‘jointly constructed knowledge’, as was the case in these problem-
posing sessions. In the first two teaching episodes Nicole did not volunteer to
share her problems with her peers and once she had posed her problems, she
would not refer back to them. That is, she would quietly look around the room, or
would ask to read a neighbour’s problem. By the third teaching episode, Nicole
had begun to seek ideas from her neighbour before posing a problem and had
willingly offered to read her problem to her neighbour. She had still not
volunteered to read her problems to the class. The following scene was captured
during the third teaching episode by Camera One. The students had just been
given a photograph of some grass as a stimulus for their problem posing (see
Appendix G).
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20:53 Nicole to her neighbour while she looked at the photograph of the grass:
“I don’t know what to do.”
22:43 started to pose a problem in her workbook.
23:04 looked up and mouthed silently to herself, “I don’t know what I am doing.”
24:09 stopped writing and doodled on her page.
24:29 started writing again
24:29 Nicole lifted her head with a form of non-focussed, rapid-eye movement.
25:12 started writing again
25:33 looked up and scratched her head with a quizzical look on her face
25:35 started writing again
26:23 stopped writing, looked up and began to chew the top of her pen
26:26 started writing again
26:36 stopped writing, looked up and flicked her pen through her hair
27:03 started writing again
28:03 read her problem to herself
28:48 read her problem to her neighbour and laughed happily when she has
finished
29:10 made some changes to her problem
Nicole appeared to gain confidence from her neighbour’s positive feedback. This
spurred her to seek further feedback from the researcher that was captured
again by Camera One.
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32:27 Nicole put her hand up to attract the researcher’s attention
32:39 Nicole to researcher while pointing to her problem: “Can you see what
you think of that?”
<researcher read problem and offered some feedback>
34:45 Nicole nodded and smiled and, when the researcher had moved away,
tore the page out of her workbook. She then re-wrote the problem on a fresh
page.
Nicole continued to work on her problem and its solution despite the class being
asked to stop work so volunteers could pose their problems to their peers. She
did not look up to listen to her peers or to rate their problems with her rating
cards. Her completed problem is recorded below:
My friend Oscar pulls out grass for a job. Yesterday, he had a patch of grass
32m x 21m and he was pulling the grass out at a rate of half a square metre per
20 mins. If he got paid $1.25 for each hour he did, how much money did he
earn?
On her self-rating criteria sheet, Nicole gave herself a three out of three rating for
challenge level and do-ability, and a two out of three for interest factor. In the
General Comments section she wrote, “I liked it. I think it’s my best one so far.”
Gardiner (1999b), who emphasised the ‘separateness’ of intelligences, reported
on students such as Nicole. If the multiple intelligences viewpoint is accepted, it
could be suggested that Nicole’s perceived disposition towards problem posing
may have been more related to her naturally dominant intelligence of being
‘intrapersonal’ rather than her response to the activity itself. Despite Nicole
having a more introverted personality than that of her peers, she continued to
become more engaged with posing problems over the remaining teaching
episodes. She took less time to start posing her problems and the problems
were always interesting and challenging. Rather than sitting quietly and looking
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around the room, Nicole became proactive in seeking help from the researcher to
start a problem. The following scene was captured by Camera Three in the fifth
teaching episode after the students had been provided with a skateboard cartoon
and an iPod as stimuli (see Appendix I):
35:35 Nicole to researcher: “I kind of know what I want to do. I just don’t know
how to do it.”
<researcher read what Nicole had written so far>
35:49 Researcher: “Okay, tell me what it is you want the reader to find out?”
35:52 Nicole: I was going to say she has this CD collection and she has to
download all of them but she doesn’t know if she’s got enough room <on her
iPod>.”
After considering the advice provided, Nicole posed the following problem in her
workbook:
Leanne has just bought a new iPod that can hold 8MB. She has already
downloaded 204 songs, and she still has 5.7MB left. She has 27 albums that
she wants to download to the iPod. If the average number of songs on each
album is 18 – 22, will she have enough room?
Nicole rated her problem on her self-rating Criteria Sheet as two and a half out of
three for interest, one and a half out of three for challenge and three out of three
for do-ability. In the General Comments section she wrote, “I liked this one
because it involved music.” It was during this teaching episode that Nicole first
volunteered to pose her problem to the class. In addition, she became more
actively engaged in listening and rating her peer’s problems as recorded on
Camera 2 starting at 38:49. Problem posing had created an opportunity for
Nicole to develop not only her mathematical skills, but also to become more
personally involved with the process of her learning and to improve her self-
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efficacy. This ability of problem-posing activities to positively influence an
individual’s self-efficacy has previously been reported by Bandura (1997) and
Marat (2005). While Nicole was initially an introverted student, her involvement
in the problem-posing intervention in this present study, had encouraged her to
become a more outwardly focussed and socially interactive student.
5.3 Student Surveys
There were four questions in the survey that the students in the Intervention
Group responded to prior to the pre-test and then again prior to the post-test.
(see Appendix C). While the student responses were narrative in nature, it was
possible to identify common emergent themes for which a tally was recorded to
indicate the level of student support for each similar type of response (see
Section 3.5.2.3 for the methodology of identifying these emergent themes). All
student responses were accommodated in the categories chosen for each
question as seen in Tables 5.4 to 5.7. Most students responded to each
question with two or three sentences, hence their responses were simple to
interpret and to categorise. The data from each question are reported in turn,
beginning with Question One and responses from the case study students as
well as others students in the Intervention Group are reported here.
5.3.1 Question One
The students’ responses to Question One, as summarised in Table 5.4, were
enlightening and supported the existence of a dispositional shift towards
engagement with problem solving as a result of the problem-posing intervention.
This position is supported by the dramatic increase in affirmative responses to
‘Do you enjoy solving problems?’ from 2 before the intervention, to 11 at the
conclusion of the teaching experiment.
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Table 5.4
Do you enjoy solving problems?
Pre-intervention Post-intervention
Indicated affirmative response 2 11
Indicated negative response 8 2
Neutral or undecided 5 2
Note. The number of students in the Intervention Group was 15.
A typical set of responses to Question One included those from Blair and Felicia
who wrote:
Blair prior to the teaching experiment:
It really depends on the difficulty. If a problem is easy and interactive most
people will enjoy it and accept they can do it. But if it is hard, and requires
special practice, then I usually end up simply becoming frustrated and lose
patience for the problem…
Blair at the completion of the teaching experiment:
Depends on their do-ability, interest factor and challenge. Problems that are
too challenging and are not very interesting quickly lose their charm, while
fun, interesting and challenging problems are great.
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Felicia prior to the teaching experiment:
I hate solving problems. I hate it because it takes so long to do and I hate
having to set out all of my working…because it always makes me confused
with my answer.
Felicia at the completion of the teaching experiment:
Now doing these sessions I feel more confident solving problems and I enjoy
them more.
Blair’s and Felicia’s responses indicated that, as a result of the problem-posing
intervention, they had adopted new attitudes to solving problems. Blair’s view
had changed from, a good problem being an “easy” problem, to a good problem
being a “challenging” problem, while Felicia’s view had changed from hating
solving problems, to enjoying solving them more. Brown and Walter (2005, p. 5)
reported similar findings on the potential of a problem-posing intervention to
“create a totally new orientation” of students’ thinking towards mathematical
problem solving. This position was further supported by many of the other
students whose responses in the post-intervention survey included (presented
without spelling or grammatical corrections):
Paul at the completion of the teaching experiment:
I enjoy problem solving much more now that I know how to figure them out.
Hayley at the completion of the teaching experiment:
The problems were fun to solve as we got to make our own.
Rodney at the completion of the teaching experiment:
I do enjoy solving problems now because they are fun and if you do more of
them it is funner and you become smarter.
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Georgia at the completion of the teaching experiment:
I found after these sessions that I really, really like working out problems and
creating them.
The affirmative responses to Question One on the survey had increased from
13% of the Group before the problem-posing intervention, to 73% of the Group at
the completion of the teaching experiment. This result, combined with the
observations of students throughout the teaching experiment, could suggest that
problem-posing activities have the potential to re-engage middle-year
mathematics students.
5.3.2 Question Two
The responses to this question in the initial survey, compared to those in the final
survey, indicated that the way students viewed mathematical problems had
changed (see Table 5.5). For example, in the first survey, a problem with few
words and requiring only a few calculations to solve was the most highly-rated
category by students. However, by the end of the teaching experiment, the most
highly-rated category was the visual problems category. This change in student
preferences cannot easily be interpreted without looking further into the
supporting comments that students wrote in their responses to this question on
the survey sheets. However, it would seem that, as a result of the problem-
posing intervention, the students may now have been looking at the structure of
the problems from an informed, discriminating position as problem posers rather
than simply as problem solvers. They were now able to see the problems more
clearly for how they were constructed rather than for simply what the problems
were asking them to find. The responses suggested that the students were now
looking at problems through a different ‘lens’. Lesh and Doerr (2003) reported
this ability to “see” a problem as being equally important to being able to “do” a
problem.
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Table 5.5
What type of problems do you prefer to solve?
Pre-intervention Post-intervention
Number problems – problems with few words
that require several mathematical calculations 5 1
Word problems – problems that require
substantial reading to extract data for solving 2 2
Visual problems – problems that can be
solved without the use of calculations or reading more than a few words
2 6
Short problems – problems that contain no
more than two short sentences and that require only one or two calculations
3 3
Real-life problems – problems that are
written about familiar real-life contexts 2 1
No preference 1 2
Note. The number of students in the Intervention Group was 15.
In the first survey, conducted before the students had begun the intervention, the
students’ responses to Question Two, focussed on the level of difficulty and
length of problems. Some examples of responses written by the students are
recorded below:
Joanne prior to the teaching experiment:
Easy problems.
Nicole prior to the teaching experiment:
I like solving problems that are kind of hard, but not impossible.
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Rodney prior to the teaching experiment:
I like to solve easy but not too easy [problems] and if I finish the easy ones I
like doing hard ones…
Kelly prior to the teaching experiment:
I prefer to solve problems that are shorter because I figure it out faster….
Andrew prior to the teaching experiment:
I prefer to answer just questions that get straight to the point because they
are simple.
Following the problem-posing intervention, the responses to this question had
changed notably. They now focussed on the structures and content of the
problems. Some examples of responses written by the students are recorded
below:
Blair at the completion of the teaching experiment: (without spelling
corrections)
Problems in my field of interest are the best for me. Problems with soccer,
snowboarding, archetecture and others with pictures and diagrams are
fantastic.
Paul at the completion of the teaching experiment: (without spelling
corrections)
I like to solve problems that have twists in them because it’s a challenge and
once I figure out what it is I feel forfilled.
182
Georgia at the completion of the teaching experiment:
We have learnt different types of problems…. I really like looking at a
problem and thinking outside the box.
Brown and Walter (2005, p. 165) discuss the potential of a problem-posing
intervention to reorientate students’ thinking from ‘narrow’ and ‘inward’ to
‘divergent’ thus creating new learning opportunities. They said that problem
posing “...is also a handmaiden of other aspects of mathematical activity – from
problem solving to greater personal understanding”. The responses to Question
Two in the final survey supported this finding and suggested that students had
consciously and unconsciously made new personal discoveries about their own
problem-solving identities. Not only were these discoveries about problem
structures, they were also about their own feelings towards different types of
problems and their solutions. Responses to this survey question suggested that
involvement in a problem-posing intervention has the potential to change the way
that students feel about problems, which can then impact upon the engagement
of these students in solving problems.
5.3.3 Question Three
The responses to this question, as summarised in Table 5.6 above, did not vary
substantially between the initial and final surveys. In the initial survey, 13 out of
the 15 students wrote that problem solving is a skill used in ‘real-life’ situations
that is needed for life outside of school. The students’ responses were largely
unchanged in the final survey. The one student who seemingly changed her
view gave the following responses to this question:
Hayley prior to the teaching experiment: (without grammatical corrections)
Yes, because you learn a new skill and you are faced with problems every
single day of our live. They help you with your estimation and trial and error.
183
Hayley at the completion of the teaching experiment: (without
grammatical corrections)
No, not really, because you wouldn’t use it unless you wanted to be a
crossword person or something. You would use it but it’s not really a must
like maths and english.
Further, more specific questioning may have allowed an informed interpretation
of the apparent change in Hayley’s position on this question. Her first response
appeared considered while her second response seemed flippant and conceived
without due attention to the question or the importance of her response.
Table 5.6
Do you think learning to solve problems is a useful thing to do?
Pre-intervention Post-intervention
Indicated affirmative response 15 14
Indicated negative response 0 1
Neutral or undecided 0 0
Note. The number of students in the Intervention Group was 15.
Like Nicole and Andrew, Hayley may simply have been distracted by the
situational factors occurring in the final week of the year, thus influencing her
readiness to be meaningfully engaged in this final survey. Schultz (2000) offered
an alternative viewpoint to inconsistent responses, reporting that gifted
underachieving students make conscious and sub-conscious choices about how
184
they react in situations and are far more likely to choose to “withdraw” from
engagement in an activity if they feel “under-challenged”. In Hayley’s case, it
may be that she simply felt disinterested in a task that she had seen before and
that was not, in her view, as important as other activities in which she may
otherwise be engaged. Indeed, it may have been a combination of both factors
that resulted in her quite different responses to the same question.
5.3.4 Question Four
The responses to this question were varied within each survey and between
surveys. (A summary of the results is presented in Table 5.7.) However, in the
first survey it seemed that students were of the view that there was a lock-step
process to learning how to solve problems. Two typical responses are recorded
over page:
Rodney prior to the teaching experiment: (without grammatical and
spelling corrections)
They [teachers] could have a couple of problems then do the first one with us
then the students could do them by them selfs that way the children know
how to do it and it would be easier.
Oliver prior to the teaching experiment:
They [teachers] teach us the different ways to solve problems. They also
teach us about the different types of problems.
Oliver’s view had changed by the time he responded to the same question at the
end of the problem-posing intervention. His second response is recorded over
the page:
185
Oliver at the conclusion of the teaching experiment: (without spelling
corrections)
…They [teachers] make us write our own problems so if we no how to write
the problems we know how to solve them.
This new response indicated that Oliver had adopted a positive view on the
potential of problem-posing to assist him to become a more competent problem
solver.
Table 5.7
What things could teachers do to assist you to become better at solving problems?
Pre-intervention Post-intervention
Make problems about real life situations 1 1
Teach us the steps to solve problems 4 3
Start with simple problems first 2 0
Help students understand the problems 3 4
Provide a positive class environment 1 0
Do problem solving regularly in class 3 5
Let students pose problems 0 1
Allow students to use visual aids 0 1
No help required 1 0
Note. The number of students in the Intervention Group was 15.
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Paul, who made one of the largest improvements in the POPS test (Stacey et al.,
1993) following his participation in the problem-posing intervention, wrote a
similar response to Oliver when he answered Question Four in the final survey.
His response was:
Paul at the conclusion of the teaching experiment: (without grammatical
corrections)
Well they [teachers] could … let us figure out how to create a problem so that
you know how to fix it
Silver (1997) had reported the potential that problem-posing activities have to
demystify problems, over a decade ago. The responses to this question, from
both Oliver and Paul, add support to Silver’s findings and remind us of the need
to consider problem-posing activities as worthwhile for inclusion in mathematics
teaching practice. In addition, as mentioned earlier in Paul’s case, we can go
further to add that significant developmental learning changes can occur during
the course of a problem-posing intervention.
5.4 Profiles of Problem Solving Test – The Pre-test and the Post-test
The Profiles of Problem Solving test (POPS) (Stacey et al., 1993) is designed to
assess mathematical problem solving skills and to monitor the effectiveness of
intervention that has been specifically designed to improve student’s problem-
solving competence. In the test, students were presented, at the start and the
end of the seven-week teaching experiment, with a variety of novel problems that
simulated real-life situations. The design of the test allowed data to be collected
on five aspects of problem-solving competence:
1. Correctness of answer
2. Method used to obtain an answer
3. Accuracy of calculations
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4. Ability to extract useful information, and
5. Quality of the student explanation of their answer.
The data collected in the pre-testing and post-testing of students in the
Comparison and Intervention Groups are recorded in Table 5.8 and Table 5.9
respectively with the post-test data being ‘bolded’ for easy discrimination from the
pre-test data. The data contained in the tables provides an opportunity to
consider individuals’ subscale score changes that occurred between the pre-test
and the post-test. Each table shows both pre-test and post-test results, in each
of the five problem-solving aspects, for each student in the respective groups. In
addition, the total score possible for each individual aspect is recorded at the top
of each column. According to the authors of the POPS test (Stacey et al., 1993),
the final two sub-columns, under the heading ‘Total’, can be used to determine if
a student has developed an increased overall competence in solving problems.
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Table 5.8
Comparison Group Pre-test and Post-test results
Correctness /13
Method /14
Accuracy /10
Extraction /8
Explanation /8
Total /53
Pre-test
Post-test
Pre-test
Post-test
Pre-test
Post-test
Pre-test
Post-test
Pre-test
Post-test
Pre-test
Post-test
Laura 10 9 8 6 9 6 8 6 4 2 39 29
Jack 5 6 6 4 4 3 7 3 2 1 24 17
Fay 9 10 9 10 8 8 8 8 6 5 40 41
Gayle 8 9 6 9 8 6 8 7 4 3 34 34
Penny 6 11 7 11 6 8 8 8 4 2 31 40
Diane 7 10 5 5 4 8 5 7 3 2 24 32
Matt 11 10 12 8 8 6 8 8 5 3 44 35
Nola 11 10 11 10 8 7 7 8 4 2 41 37
Ben 9 10 6 4 6 6 8 8 3 2 32 30
Adam 8 8 6 6 6 6 7 7 2 2 29 29
Kyle 7 10 7 8 6 6 8 8 5 4 33 36
Imogen 10 9 8 8 7 6 8 8 1 1 34 32
Clare 8 7 7 5 5 6 8 7 2 1 30 26
Oliver 9 8 8 7 8 6 8 8 3 1 36 30
Ellen 11 8 8 8 8 8 8 6 4 2 39 32
Helen 6 9 3 4 7 8 5 3 0 0 21 24
Note. The number of students in the Comparison Group was 16.
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Table 5.9
Intervention Group Pre-test and Post-test results
Correctness /13
Method /14
Accuracy /10
Extraction /8
Explanation /8
Total /53
Pre-test
Post-test
Pre-test
Post-test
Pre-test
Post-test
Pre-test
Post-test
Pre-test
Post-test
Pre-test
Post-test
Leah 13 11 11 10 9 8 8 8 6 6 47 43
Blair 8 9 6 8 7 9 7 8 2 5 30 39
Joanne 8 10 10 8 8 6 8 6 4 4 38 34
Paul 11 11 6 10 6 8 6 8 2 3 31 40
Courtney 10 11 10 11 7 8 8 8 6 6 41 44
Hayley 11 9 11 8 8 8 7 8 5 4 42 37
Nicole 9 9 10 10 6 8 8 8 4 2 37 37
Rodney 9 8 8 9 7 8 8 8 4 5 36 38
Oliver 9 9 8 9 5 8 8 8 5 6 35 40
Georgia 6 7 6 9 6 6 7 8 4 5 29 35
Ethan 11 13 9 11 8 8 7 8 2 5 37 45
Kelly 8 10 6 7 8 6 8 6 4 3 34 32
Andrew 11 10 11 8 8 8 8 8 4 4 42 38
Felicia 11 11 11 11 8 10 8 8 7 8 45 48
Danielle 10 10 10 10 8 8 8 8 5 3 41 39
Note. The number of students in the Intervention Group was 15.
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5.4.1 Descriptive Analysis of the POPS Test Results
The mean and standard deviation scores of students for each aspect in the pre-
test and post-test were calculated using the student version of the Statistical
Package for Social Sciences (SPSS Inc., 2007) and compared between the
Comparison Group and the Intervention Group. These scores can be found in
Table 5.10. The mean score data in Table 5.10 indicate a trend of improvements
in each of the five subscales comprising the POPS (Stacey et al., 1993) test for
Intervention Group students that was not apparent consistently for students in the
Comparison Group. This trend suggested that the problem-posing intervention
may be responsible for some aspects of the development of problem-solving
competence amongst the Intervention Group students.
The ‘Quality of Explanation’ aspect of the test, the fifth subscale, produced some
of the most interesting comparisons between students in the Comparison Group
and students in the Intervention Group. While the increase in scores from the
pre-test to the post-test is only modest for students in the Intervention Group, the
decline in the scores from the pre-test to the post-test, for students in the
Comparison Group, is noteworthy. In their test administration book, the authors
of the POPS (Stacey et al., 1993) test stated that, from their trials, the ‘Quality of
Explanation’ aspect had, by far, the lowest score of the five problem-solving
aspects. They attributed this to its perceived “non-essential nature that results in
it receiving little attention in classroom mathematics” (Stacey et al., 1993 p. 62).
That is, students generally perceive the goal of problem solving to be achieving
the correct answer as opposed to focussing on the method or quality of the
explanation that leads to the correct answer. This focus on the ‘destination’, as
opposed to the ‘journey’, limits opportunities for students to explore the nature of
problems and generate important mathematical concepts and ideas (English et
al., 2005).
191
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
Comparison Intervention Comparison Intervention
Problem-solving aspects Pre- test
M
Post-test M
Pre- test M
Post-test M
Pre- test
SD
Post-test SD
Pre- test
SD
Post-test SD
Correctness of answer 8.60 9.00 9.67 9.87 1.84 1.36 1.76 1.46
Method used 7.60 7.27 8.87 9.27 1.92 2.22 2.03 1.28
Accuracy of calculations 6.73 6.40 7.27 7.80 1.58 1.30 1.10 1.08
Extraction of information 7.60 7.13 7.60 7.73 0.83 1.36 0.63 0.70
Quality of explanation 3.47 2.20 4.27 4.60 1.36 1.15 1.49 1.55
Note. The number of students in the Intervention Group was 15 and the number of students in the
Comparison Group was 16. All results in this table are correct to two decimal places and were calculated using the Statistical Package for Social Sciences (Student Version, Version 15) by SPSS Inc., 2007, Prentice Hall, Melbourne, Victoria. * From “Profiles of Problem Solving” by K. Stacey, S. Groves, S. Bourke, and B. Doig, 1993, Australian Council of Educational Research, Hawthorne, Victoria.
The reflective practice of focussing on the ‘journey’ with peers, as experienced by
the Intervention Group students throughout the teaching experiment required
them to think critically, to detect their weaknesses and to self-regulate improved
practices. It may be reasonable to suggest that this opportunity resulted in these
students having an enhanced ability to explain their solutions. Students in the
Comparison Group did not have this same opportunity. According to Hamilton
and his colleagues (2006), opportunities for reflective practice, such as those
found in the problem-posing intervention, can lead to other significant gains such
as reduced mathematics anxiety and improved performance in solving problems.
The responses to the Question One of the Student Survey supported this
192
statement showing a substantial improvement in student attitudes towards
problem solving following the teaching experiment (see Section 5.3.1).
The difference in improvements between the Comparison Group and the
Intervention Group, in the ‘Quality of Explanation’ aspect of problem solving, is
not surprising. Throughout the teaching experiment, students in the Intervention
Group were repeatedly encouraged and given opportunities to discuss their
reasoning and to explain their methods of solutions with their peers. Brown and
Walter (2005) discussed the benefits of changing the “orientation” of learning
mathematics from finding a correct solution to finding a good question as being
very useful in developing divergent ways of thinking and deeper opportunities for
learning.
On their own, the standard deviations of the subscale scores for both the
Comparison and Intervention Groups portray a limited interpretation of the data.
However, when combined with the mean score data, they are able to contribute
to a deeper understanding of the differences between the Comparison and
Intervention Groups that may be attributed to the problem-posing intervention.
Earlier in this section, it was noted that the mean score of students in the
Comparison Group, for the ‘Method used’ aspect of the POPS (Stacey et al.,
1993) test, had decreased from 7.60 to 7.27, while the mean score of students in
the Intervention Group had increased from 8.87 to 9.27. In themselves, these
results are only mildly persuasive in suggesting that the problem-posing
intervention may have contributed to the improvement in this aspect for the
students in the Intervention Group. However, these results become more
convincing when we consider that, not only had the students in the Intervention
Group improved their mean score in this aspect, they had also produced more
consistent results following the problem-posing intervention than they had in the
pre-test. This can be seen from the reduction in the standard deviation of their
scores from 2.03 to 1.28. This contrasts to the reduced mean score of students
193
in the Comparison Group combined with an increase in standard deviation of
their scores, between the pre-test and the post-test, from 1.92 to 2.22.
The more consistent and improved scores of the Intervention Group may have
been as a result of the students’ improved ability to understand problem
structures. Bernardo (2001) mooted this potential of problem-posing activities, to
influence problem-solving competence, when he reported on students’ abilities to
transfer knowledge and skills from one problem in one setting, to another
problem in another setting. He and other researchers, such as Cai (1997),
suggested that the experience students gained from posing problems allowed
students to recognise common characteristics in the structures of analogous
problems.
During the teaching experiment, students were required to focus on the
structures of problems as they posed them. This was facilitated through the
posing of the problems and the recording of solutions in their workbooks, as well
as by the ‘rating’ of their own and their peers’ problems. In addition, students
were regularly challenged to determine how many steps would be required to
solve their problems. This focus on the steps required to solve their problems, in
combination with the peer rating of ‘do-ability’ and ‘level of challenge’, required
the students to look back at the structure of their problems. Stoyanova (2003, p.
39) reported on this opportunity when she said “problem-posing activities provide
environments that seem to engage students in a natural way in reflective
mathematical abstraction”. In the absence of similar findings with the
Comparison Group, it would seem reasonable to suggest that the problem-
posing intervention may have contributed to a focus on problem structures that
had in some way influenced the Intervention Group’s improved and more
consistent scores in the ‘Method used’ aspect of the POPS (Stacey et al., 1993)
test.
194
It has been noted earlier in this section that it may have been the focus on the
process (method of construction and method of solution) of problem solving as
opposed to the product (the solution) that allowed the students in the Intervention
Group to improve their scores in the ‘Quality of Explanation’ aspect of the POPS
(Stacey et al., 1993) test. In comparison to the students in the Intervention
Group, the mean score for the students in the Comparison Group fell from 3.47
to 2.20. Not only did their mean score fall, the standard deviation of the
individual scores for this problem-solving aspect also fell which indicates a
reduced spread of the scores. The trend of reduced scores is therefore more
consistent between the students in the Comparison Group and the reduced
mean score cannot be attributed to a few ‘outlier’ scores that may have reduced
the mean score for the entire group. It is worth noting, that by contrast, the
standard deviation for students in the Intervention Group varied only slightly
between the pre-test and the post-test.
In summary, both the measure of central tendency, the mean scores, and the
measure of spread, the standard deviation of scores, suggested that the students
in the Intervention Group had undergone some form of change, perhaps
developmental learning and perhaps affective, that had influenced their scores in
the post-test. This same pattern of change was not apparent for students in the
Comparison Group. It is reasonable to suggest then that it was the involvement
of the Intervention Group students in the problem-posing intervention that was
responsible for the apparent trend of improvement in problem-solving
competence.
195
5.4.2 Paired Samples T-Test Results
To further investigate the impact of the intervention in this study on students’
scores of the POPS (Stacey et al., 1993) test a paired samples t-test was
conducted using the student version of the Statistical Package for Social
Sciences (SPSS Inc., 2007). This test provided an opportunity to consider
another parameter of interest, the difference between the means within the
Comparison and Intervention Groups and adds further weight to the discussions
in Section 5.4.1. Results of the paired samples t-test can be seen in Table 5.11.
While no significant differences were found in students’ scores on the POPS test
for the Intervention Group, a paired samples t-test did demonstrate however, that
for the Comparison Group there was a statistically significant decrease in scores
on the Quality of Explanation subscale from the pre-testing (M=3.47, S.D.=1.36)
to the post-testing (M=2.20, S.D.=1.15, t=6.97, p<.05). The Eta Squared Statistic
(.78) indicated a large effect size.
As mentioned in Section 5.4.1 the decrease in scores on the Quality of
Explanation subscale for the Comparison Group may be attributed to a lack of
involvement in the teaching intervention and the study by the members of this
group. That is, the only involvement of the participants in this group in the study
was to complete the pre-test and the post-test which were separated by eight
weeks in which time there was no discussion or further involvement with the
study. In comparison, participants in the Intervention Group spent time together
as a unique group during the eight-week teaching experiment leading to
increased opportunity to develop an interest in and a sense of commitment to the
study. This interest and commitment may in turn have resulted in participants in
the Intervention Group striving to achieve their potential in the post-test whereas
students in the Comparison Group may have had limited interest in completing
the post-test as it may have appeared irrelevant or meaningless to them.
196
The transformative nature of teaching experiments was reported by Confrey and
Lachance (2000) who commented on the benefits of facilitating student-centred
learning environments that were free of the usual restraints in traditional
classroom settings. Unlike students in the Comparison Group, students in the
Intervention Group were provided with the student-centred learning opportunities
throughout the teaching experiment in which they could share, with their peers,
explanations of how they solved problems, thus raising their awareness of this
important aspect of problem-solving competence. The responses to the Student
Surveys from students in the Intervention Group supported the increased interest
and commitment to solving problems students (see Section 5.3).
Apart from the statistically significant decrease in Comparison Group scores on
the Quality of Explanation subscale of the POPS (Stacey et al., 1993) test from
the pre-test to the post-test no other significant differences were found.
197
Table 5.11
Paired Samples Test for each Aspect of the *Profiles of Problem Solving Test for
Students in the Comparison and Intervention Groups
Paired Differences
95% Confidence Interval of the Difference
M S.D. Std. Error
Mean
Lower Upper t df Sig.
(2-tailed)
Comparison pre to post
Correctness of answer
-.40 2.06 .53 -1.54 .74 -.75 14 .47
Method used
.33 2.06 .53 -.81 1.47 .63 14 .54
Accuracy of calculations
.33 1.76 .45 -.64 1.31 .73 14 .48
Extraction of information
.47 1.41 .36 -.31 1.25 1.28 14 .22
Quality of explanation
1.27 0.70 .18 .88 1.66 6.97 14 .00
Intervention pre to post
Correctness of answer
-.20 1.32 .34 -.93 .53 -.59 14 .57
Method used
-.40 2.03 .52 -1.52 .72 -.76 14 .46
Accuracy of calculations
-.53 1.51 .39 -1.37 .30 -1.37 14 .19
Extraction of information
-.13 1.06 .27 -.72 .45 -.49 14 .63
Quality of explanation
-.33 1.50 .39 -1.16 .50 -.86 14 .40
Note. The number of students in the Intervention Group was 15 and the number of students in the Comparison Group was 16 although subjects with unchanged scores were excluded from these statistics.
All results in this table are correct to two decimal places and were calculated using the Statistical Package for Social Sciences (Student Version, Version 15) by SPSS Inc., 2007, Prentice Hall, Melbourne, Victoria. * From “Profiles of Problem Solving” by K. Stacey, S. Groves, S. Bourke, and B. Doig, 1993, Australian Council of Educational Research, Hawthorne, Victoria.
198
5.4.3 Analysis of Improvement Scores from the Pre-test to the Post-test
When we compare the number of improvements in individual problem-solving
aspect scores of students from the pre-test to the post-test (see Tables 5.8 and
5.9), we can detect changes that may be consistent with developmental learning
occurring as a result of participation in the problem-posing intervention. A
summary of the number of improvements can be found in Table 5.12.
Table 5.12
Numbers of Improvements in Individual Sub-scale Scores of Comparison and
Intervention Group Students, from the Pre-test to the Post-test
Problem-solving aspects Comparison Group Intervention Group
Correctness of answer 8 6
Method used 5 8
Accuracy of calculations 4 7
Extraction of information 2 5
Quality of explanation 0 7
Note. The number of students in the Intervention Group was 15 and the number of students in the
Comparison Group was 16.
From considering only the number of students with improved scores in the
‘Correctness’ aspect of the test, it would seem that the Comparison Group’s
results demonstrated they had achieved greater improvements than had the
students in the Intervention Group. However, the pre-test mean score in this
aspect for the Comparison Group was 8.60 compared to a higher pre-test mean
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score of 9.67 achieved by students in the Intervention Group (see Table 5.10). In
addition, twice as many students in the Intervention Group, compared to students
in the Comparison Group, had pre-test scores of 11 or more out of a possible 13
marks (see Tables 5.8 and 5.9) thus reducing the range of values to which they
could improve.
In the ‘Method Used’ aspect of the test, a similar situation to that described for
the ‘Correctness’ aspect of the test can be seen. The pre-test mean score in this
aspect for the Comparison Group was 7.60 compared to a pre-test mean score
of 8.87 achieved by students in the Intervention Group (see Table 5.10). As with
the ‘Correctness’ aspect of the test, twice as many students in the Intervention
Group, compared to students in the Comparison Group, had pre-test scores of
11 or more (see Tables 5.8 and 5.9). It is reasonable to suggest that both of
these factors would have made it more difficult for students in the Intervention
Group to achieve high scores than students in the Comparison Group for both of
these aspects. Despite this possible impedance to improved scores, 8 students
in the Intervention Group improved their score in the ‘Method used’ aspect
compared to only 5 students in the Comparison Group.
Improvements between the groups in the ‘Accuracy of Calculations’ aspect of the
test are consistent with those for the ‘Method Used’ aspect. The Intervention
Group began with a higher mean score of 7.27 compared to the Comparison
Group’s mean score of 6.73, but still managed to have more students improve
their score in this aspect of the test. Seven students made improvements in the
Intervention Group compared to only four students making improvements in the
Comparison Group. This trend was continued when we looked at the increased
ability of students to extract useful and relevant information from data presented
in the problems contained in the test. Interestingly, the mean score of 7.60 out of
a possible 8 marks for the ‘Extraction’ aspect of the test was the same for both
groups in the pre-test. However the Intervention Group were able to improve
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their mean score to 7.73 in the post-test while the Comparison Group’s mean
score fell to 7.13.
Initially it may seem unusual that a mean score in an identical pre-test and post-
test, undertaken by the same students in a group, could fall. However, the
students in the Comparison Group may not have developed a sense of
ownership or respect for the teaching experiment as they had not committed the
same amount of time as had students in the Intervention Group, thus possibly
impacting the students’ willingness to wholeheartedly complete the post-test to
the best of their abilities. On the other hand, the students in the Intervention
Group had experienced a routine of coming together as a group over an eight-
week period, thus having a sustained opportunity to investigate the nature of
problems and ultimately to pose their own problems to their peers. This coming
together to share the learning experience is discussed by Crotty (1998) who
attributes it with the development of interpretation and understanding of
knowledge in a way that traditional forms of teaching are unable to compete.
Crotty’s (1998) constructionist (as opposed to constructivist) proposition, that
best learning occurs when students are actively engaged in activities that allow
them to create artefacts [problems] to share with peers in a group situation, is
consistent with findings from the NRC (2001) that stated that students need to
spend “sustained periods of time” involved with problem solving to become more
competent at solving problems and to make connections between their prior
knowledge and their new knowledge.
The final problem-solving aspect of the POPS (Stacey et al., 1993) test
measured the students’ ability to explain their solutions. As with the first three
problem-solving aspects, students in the Intervention Group had a higher mean
score in the pre-test compared to the students in the Comparison Group. While
students in the Intervention Group improved their mean score from 4.27 to 4.60
out of a possible 8 marks, the mean score for the Comparison Group fell from
3.47 to 2.20. As was discussed for the ‘Extraction of Information’ aspect, it is not
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easy to explain why a mean score can fall substantially from an identical pre-test
to a post-test when both tests were taken under the same conditions by two
groups of students selected with the same selection criteria. On the basis that
the only difference between the Comparison and Intervention Groups was the
problem-posing intervention, it is reasonable to suggest that the intervention may
have contributed to the improvement in mean scores for the Intervention Group
and therefore to the overall problem-solving competence of the students in that
group.
5.5 Conclusion
A report on both the qualitative and quantitative data, collected throughout the
teaching experiment, was presented in this chapter. Specific responses to the
three research questions are addressed in the following chapter. The data
collected from the video-tapes, student interviews and the student workbooks,
combined with the data from the surveys, pre-tests and post-tests has provided
evidence to determine the extent to which a problem-posing intervention can
assist in the development of problem-solving competence in underachieving
middle-year students.
This chapter began with an introduction to the three case-study students; Paul,
Andrew and Nicole, who were chosen for the unique developmental learning and
behaviour changes that occurred for them as a result of the problem-posing
intervention. Paul was a student who began the teaching episode as
disorganised in his thinking, as it related to problem solving. He demonstrated
the most improvement in his problem-solving competence from the start to the
end of the teaching experiment. Andrew, the student with the lowest Middle
Years Ability Test (MYAT) score (Australian Council for Educational Research,
2005) and Nicole, the student with the highest MYAT score, both began the
teaching experiment as disengaged in the problem-posing activities. Despite
Andrew being easily distracted by his peers and Nicole being a passive non-
participant at the start of the teaching experiment, they both responded well to
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the problem-posing intervention and as a result, became engaged with learning
and enthusiastic about posing problems to their peers.
A report on the data obtained from the student surveys, undertaken at the start of
the teaching experiment and then again at the completion of the teaching
experiment, is found in Section 5.3. The analysis of data from Question One
provided some interesting findings including a substantial dispositional shift of
students from negative to positive feelings about problem solving as a result of
the problem-posing intervention. Student responses to Question Two suggested
that, as a result of their involvement in the teaching experiment, students had
begun to look at problem structures more knowledgeably. It could be suggested
that the students’ foci had shifted from considerations of the difficulty of problems
to the characteristics of the problem, as the discerning factor in enjoyment. Data
from Question Three of the initial and final surveys confirmed that students
remained constant in their view that solving problems was a worthwhile activity.
An investigation of the data from Question Four of the surveys showed students
were more divergent in their views about how a teacher could assist them to
become a better problem solver as a result of the problem-posing intervention.
Section 5.4 began with a brief reintroduction to the problem-solving testing
instrument used to determine changes in problem-solving competence from the
start to the end of the teaching experiment. The descriptive statistics of the data
were discussed in Section 5.4.1 and then further explored using a paired
samples t-test in Section 5.4.2 to determine if the problem-posing intervention
could be attributed with improved problem-solving competence of the students in
the Intervention Group. The changes in mean scores of the five problem-solving
aspects tested in the POPS test (Stacey et al., 1993) were compared within and
between the Comparison and Intervention Groups.
Students in the Intervention Group had improved mean scores in each of the five
problem-solving aspects between the pre-test and the post-test although these
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improvements were not statistically significant. Therefore, analysis of the
quantitative data revealed that the problem-posing intervention did not
significantly improve students’ test scores on the POPS (Stacey et al., 1993) test
for the Intervention Group. However, the qualitative data suggested that the
intervention did generally enhance students’ attitudes towards problem solving
thus paving the way for students to become more engaged in problem-solving
activities.
For the Comparison Group, the quantitative data analysis revealed that there
were no significant differences in students’ scores on the POPS (Stacey et al.,
1993) test except for the subscale Quality of Explanation in which scores
significantly decreased from the pre-test (M=3.47) to the post-test (M=2.20).
Possible reasons for this finding might be attributed to teachers’ lack of attention
given to this aspect when teaching mathematics. For the Intervention Group, the
problem-posing intervention focused on students’ quality of explanations when
undertaking problem-solving activities and this may be reflected in the students
maintaining consistent scores on this subscale in both the pre-test (M=4.27) and
the post-test (M=4.60).
The analysis of both the qualitative and quantitative data, presented in this
chapter, uncovered interesting findings and valuable evidence that lead to the
research questions being responded to with confidence in Chapter 6.
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Chapter 6
Responses to the
Research Questions
6.1 Chapter Overview
In this chapter the research questions are responded to using the data presented
and discussed in Chapter 5 and reflect findings about students who have English
as their first language. Section 6.2 provides some discussion and a response to
Research Question 1, while Section 6.3 and Section 6.4 provide discussion and
responses to Research Questions 2 and 3 respectively. The final section,
Section 6.5, addresses the overarching question to be responded to as a result
of this study; How might a problem-posing intervention impact upon the
development of problem-solving competence of underachieving, middle-year
students?.
6.2 Research Question 1
The first research question was:
Can, and if so, how can participation in problem-posing activities facilitate the
engagement of middle-year mathematics students?
An investigation of the data collected in this study suggests an affirmative
response to this question. As mentioned in Chapter 3, a number of data sources
provided an opportunity for triangulation of the data which added an increased
level of certainty when responding to this research question. Accurate details of
informal interviews between the three case study students and the researcher,
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and between three case study students and their peers, were captured by three
audio-visual recorders used throughout the teaching experiment. In addition, the
problems posed by these students in their workbooks, and their associated self-
ratings and comments, added a more personal depth to the data collected from
the students. This qualitative data, in conjunction with the student responses
recorded in the student surveys, provided convincing evidence of the potential for
problem-posing activities to re-engage previously disengaged students.
In Chapter 5, Paul, Andrew and Nicole were presented as three case-study
students (see Sections 5.2.1, 5.2.2 and 5.2.3). While Paul was consistently
engaged in learning throughout the teaching experiment, Andrew and Nicole
began the teaching experiment as disinterested in learning. In Andrew’s case,
he was easily distracted from learning and lacking in self-efficacy. His increased
engagement and increased levels of public participation in sharing his posed
problems with his peers, supported Bandura’s (1997) findings that problem-
posing activities have the potential to improve self-efficacy. Similarly, Van de
Walle (2004) investigated middle-year students, like Andrew, and attributed
improved attitudes and higher levels of intrinsic motivation to multi-dimensional
pedagogies found in classrooms where students are given options about how
they want to learn, such as those incorporated into the teaching experiment in
this present study.
The transcripts from the videotapes of the three cameras used in the experiment,
showed significant changes in the sentiment of the conversations held by Andrew
and Nicole with the researcher and with their peers. Both of these students
seemed to increasingly enjoy the opportunity to pose and share their problems
throughout the problem-posing intervention. From the data collected in this
study, it could be concluded that the non-traditional methodologies, underpinned
by constructionist beliefs and built into the teaching experiment, provided an
environment in which Andrew and Nicole felt increasingly ‘safe’ and willing to
‘expose’ their thinking to their peers. This conclusion was also supported
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through the research reported in Chapter 2. Broido (2002) reported on the
potential of “liberating” and “transformative” learning environments, such as was
offered to the Intervention Group in this teaching experiment, to assist students
to re-engage with learning. Similarly, Alvesson and Skolberg (2000) suggested
that a learning environment that supported “respectful participation” and “jointly
constructed knowledge” had the potential to re-engage students. In Andrew’s
case, he not only became engaged in learning, he also became active in
encouraging others to be more engaged so that he could focus more fully on the
problem-posing activities provided in the teaching experiment (see Section
5.2.2). In Nicole’s case, she was able to show her re-engagement in a physical
way through the use of the ‘1,2,3 rating cards’ as well as through the improved
quality of the problems she posed in her workbook and her willingness to share
her problems with her peers (see Section 5.2.3).
An analysis of the answer to the first question in the initial and final student
surveys also provided supporting evidence for a response to Research Question
1. In the first survey, only two out of fifteen students, compared to eleven out of
the fifteen students at the end of the teaching experiment, said they enjoyed
solving problems (see Table 5.4). While Question One in the survey asked
students about their enjoyment of solving problems and therefore was not a
direct measure of student engagement, it did however provide an opportunity to
consider changes in how students viewed their experiences of solving problems,
and it has been argued that there is a direct relationship between enjoyment of
activities and student engagement in activities (e.g., Adams et al., 2000;
Bjorklund, 2000).
The data collected from the first survey question that demonstrated a positive
dispositional shift to enjoying problem solving, was consistent with findings by
Brown and Walter (2005), Lesh and Doerr (2003) and Lesh and Zawojewski
(2007) who all reported that opportunities to pose problems can provide students
with meaningful learning environments in which they are more likely to engage.
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In summary, the survey responses add confidence to the response that
participation in problem-posing activities can facilitate the engagement of
disengaged, middle-year mathematics students.
6.3 Research Question 2
The second research question was:
Can, and if so, how can participation in problem-posing activities facilitate
improved problem-solving competence of middle-year, mathematics
students?
The qualitative and quantitative data collected in this present study, presented
contrasting evidence to respond to this research question. There were two
primary data sources, the pre-test and post-test results, as well as three
secondary sources, informal interviews, observations and student workbooks that
were used to support this statement.
Paul, one of the three case-study students in this teaching experiment,
demonstrated one of the largest improvements in problem-solving competence,
as measured by the difference between his pre-test and post-test scores. Paul
improved in four of the five problem-solving aspects being tested in the Profiles
of Problem Solving (POPS) test (Stacey et al., 1993) and scored the same result
in the fifth aspect in both the pre-test and the post-test. His overall score
increased from 31 marks at the start of the problem-posing intervention, to 40
marks out of a possible 53 marks at the end (see Table 5.1). Knuth and
Peterson (2002) reported similar improvements for students in their study when
they said that, as a result of providing students with many opportunities to pose
problems, students improved their problem-solving competence as well as
developed their mathematical thinking.
An improvement by one student in this present study is insufficient to suggest
that participation in this problem-posing intervention resulted in improved
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problem-solving competence for all participants. Indeed, Nicole, another case
study student, made no overall improvement in her total score (see Table 5.3),
while Andrew’s score fell from 42 marks to 38 marks out of a possible 53 marks
(see table 5.2). Further investigations of the data, collected from the Comparison
and Intervention Groups, did however suggest that a trend of improvement was
evident for the Intervention Group that was not apparent for students in the
Comparison Group (see Tables 5.10 and 5.11). The graphical presentations of
the mean scores in each of the five problem-solving aspects of the POPS test,
for both the Comparison and Intervention Groups, showed a trend of improved
scores for students in the Intervention Group that were not apparent for the
students in the Comparison Group. The visual representation of the mean-score
data were persuasive and suggested that the problem-posing intervention could
be responsible for the increases in problem-solving competence of students in
the Intervention Group. In addition, when a comparison of the number of
students who improved their scores in each problem-solving aspect is made
between the two groups, evidence can be found to suggest that the problem-
posing intervention has contributed to the improvements of students in the
Intervention Group (see Table 5.10).
Throughout the teaching experiment, students were given a number of
opportunities to solve researcher and peer problems. Some students, like Paul,
took every opportunity to tell his peers how he would solve a particular problem.
Costa (2005, p. 8) acknowledged students, like Paul, as risk takers who are
prepared to go “against the grain, … thinking of new ideas and testing them with
their peers” as being more successful at solving problems than those who don’t.
Early in the teaching experiment, Paul attempted to solve a researcher-provided
problem on a whiteboard at the front of the classroom (see Section 5.2.1). He
was unable to fully explain either his thinking or his incorrect solution. His
perceived confusion of his ideas typified his early replies to similar researcher
questions about how he would solve other problems. However, towards the end
of the teaching experiment, Paul demonstrated that some developmental
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learning changes had taken place (discussed further in Section 6.4).
Gravemeijer (1994) discussed opportunities to pose, explore and solve problems,
as opportunities to create “internal conflict” that ultimately can lead to a student
developing more abstract and detailed conceptualisations of problems. In Paul’s
case, it appeared that as a result of his experiences in the problem-posing
intervention, he had learnt how to order his thoughts in a coherent and
appropriate order to pose meaningful, challenging and interesting problems and
generate correct solutions. This was demonstrated in the problems he
progressively posed in his workbook and in his responses to questions in the
POPS (Stacey et al., 1993) test that resulted in his improved overall score.
In summary, the qualitative and quantitative data collected to determine if, and if
so, how can a problem-posing intervention have an impact on student’s problem-
solving competence, provided contrasting findings. The numbers of students in
the Intervention Group who improved their overall POPS (Stacey et al., 1993)
test scores was, in itself, insufficient to demonstrate with certainty that the
problem-posing intervention was solely responsible for the Intervention Group’s
improvements in problem-solving competence. However, when the individual
problem-solving aspect scores were compared between students in the
Comparison Group and the Intervention Group, the suggestion that the
intervention may have been responsible for improved problem-solving
competence began to emerge. When considering all the evidence presented in
this study, support can be given to Silver and Cai’s (1996) findings which state
that participation in problem-posing activities can result in the improved problem-
solving competence of some middle-year, mathematics students.
6.4 Research Question 3
The third research question was:
In terms of problem-solving competence, what developmental learning
changes occur during the course of PPI?
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A number of student developmental learning changes were identified throughout
the teaching experiment. These included changes in self-regulation for some
students. For example, Andrew increasingly demonstrated an improved ability to
self-regulate as the experiment proceeded (see Section 5.2.2). Self-regulated
learning is a complex interactive process involving both cognitive and
motivational functions (Boekaerts, 1997) and can be influenced by instructional
conditions and task difficulty (Paris & Newman, 1990), such as those presented
throughout the teaching experiment. It is noteworthy that while Andrew was able
to self-regulate throughout the teaching sessions that involved problem-posing
activities, he was unable to demonstrate the same degree of self-regulation in the
post-test session. Possible explanations for this apparent anomaly can be found
in Section 5.2.2.
When students are actively engaged in activities that provide opportunities to
integrate prior knowledge with new knowledge, they are then able to construct
new meaning that results in cognitive change (e.g., Goswami, 2002; Siegler,
1996). At the start of the teaching experiment, Paul appeared to be full of
enthusiasm but lacking in specific cognitive tools that allowed him to order his
thoughts logically (see Section 5.2.1). Sternberg (2000) referred to these tools
as the metacomponents and performance components of general intelligence
that, when lacking in development, can impede an individual’s progress in solving
problems.
Evidence of Paul’s development in these components can be seen by comparing
his solutions in the pre-test questions to those in the post-test, and by comparing
his early entries to his later entries in his workbook. New and deeper
opportunities for developmental learning, as a result of involvement with problem-
posing activities, were reported by Brown and Walter (2005) who attributed
problem posing with presenting unexpected discoveries about an individual’s
learning potential. Throughout the problem-posing intervention, Paul had
discovered unexpected abilities that neither he nor his teachers had previously
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identified. Since Paul was always a willing participant in all of the teaching
episodes, we can reasonably suggest that the problem-posing intervention was
responsible for the development of learning skills that resulted in his increased
problem-solving competence.
The opportunity to pose problems provided Paul with the knowledge of problem
structures that allowed him to make the necessary connections with prior
knowledge, resulting in cognitive change. This opportunity to pose problems,
that results in cognitive change, is discussed at great length by Siegler (1991)
and is generally supported by both Piagetian and Information Processing
theorists (e.g., Piaget & Inhelder, 1969; Sternberg, 2002). However,
Psychometric theorists would say that if performance in tests, such as the POPS
test (Stacey et al., 1993), is to be used as a measure of intelligence, then Paul’s
improved overall score from 31 to 40 out of a possible 53 marks is more to do
with improved memory capacity and cognitive speed than it is to do with
developmental learning changes (Hutton et al., 1997). This is an unlikely
explanation for Paul’s improved score and it is more likely that the problem-
posing intervention activities have provided the new knowledge that Paul needed
to create new meaning in the process of solving problems.
Lesh and Doerr (2003) reminded us that conceptual and developmental learning
changes include social perspectives as well as psychological perspectives. This
study was underpinned by contructionism theory (Schwandt, 2001) that stressed
the importance of the social interaction in the production of shared artefacts,
which in this case were the problems posed by the students. This opportunity
for social interaction, throughout the teaching experiment, had a notable impact
on a number of students and their levels of involvement in the activities. Nicole
was a student who became increasingly engaged in the learning process as a
result of the problem-posing intervention. Her Middle Years Ability Test (MYAT)
(Australian Council for Educational Research, 2005) results that lead to her
selection as a participant in this study, put her in the 94th percentile of students.
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However, her perceived talent did not manifest itself in a correspondingly high
score in the pre-test, nor was she more enthusiastic or engaged than other
students at the start of teaching experiment.
Evidence from researcher interviews and observations recorded by the cameras
in the research room, provided evidence that Nicole had progressively developed
socially-oriented learning skills as a result of her participation in the problem-
posing activities (see Section 5.2.3). This potential for problem-posing activities
to encourage students “to explore problem situations and to pursue lines of
enquiry that are personally satisfying” was reported by English (1997b). At the
beginning of the teaching experiment, Nicole was disinterested in being involved
with the problem-posing, perhaps because she felt it was of no interest to her.
However, as she observed her peers, she became drawn into the problem-
posing activities and began to actively participate by posing and sharing her
problems and by providing feedback to her peers about their problems. Her
increased and contextualised socialisation within the research room, created
opportunities for Nicole to engage with the problem-posing activities and resulted
in her writing increasingly sophisticated problems in her workbook. It would
seem that the problem-posing intervention was the contributing factor that
provided this opportunity to Nicole, resulting in socially-oriented learning
development.
Evidence of a number of developmental learning changes was found during the
analysis of the data collected in this study. Some students, like Andrew, became
more self-regulated while others, like Nicole, demonstrated more socially-
oriented developmental learning changes. Paul, through his practice at posing
problems, learnt how to combine existing knowledge with a new knowledge of
problem structures that resulted in substantial increases in his problem-solving
competence. The findings of this study have demonstrated that, in terms of
problem-solving competence, a problem-posing intervention has the potential to
facilitate developmental learning changes amongst middle-year students.
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6.5 Conclusion
The purpose of this study was to investigate and explain the links between a
problem-posing intervention and the development of problem-solving
competence in middle-year students who had been ascertained as above-
average in standardised intelligence tests, yet underachieving in the problem-
solving criterion of school mathematics tests. In Sections 6.2 and 6.3, evidence
was presented to suggest that a problem-posing intervention could facilitate the
re-engagement of students who were previously disengaged in learning and
could facilitate increased problem-solving competence of some underachieving
middle-year students. The opportunity to be withdrawn from their traditional
classrooms, provided the students in the Intervention Group with a learning
environment that encouraged individuality and one in which they could feel safe
to explore solving problems in a unique way. A focus on posing problems, such
as occurred in this present study, was supported by Brown and Walter (2005, p.
168) who said that “problem generation has the potential to redefine in a radical
way who it is that is in charge of one’s education”.
A number of developmental learning changes were identified in Section 6.4, such
as self-regulation and the creation of new meaning through the combination of
prior and new knowledge that occurred as a result of student involvement in the
problem-posing intervention. From the analysis of the data in Chapter 5 and a
review of the evidence used to respond to the research questions in Chapter 6, a
response can be presented for the overarching question of this study:
How might a problem-posing intervention impact upon the development of
problem-solving competence of underachieving, middle-year students?
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Involvement in a problem-posing intervention can:
1. facilitate the re-engagement of middle-year mathematics students
2. improve the problem-solving competence of middle-year mathematics
students, and
3. can facilitate developmental learning of middle-year mathematics
students.
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Chapter 7
Limitations and
Implications for Future
Research
7.1 Chapter Overview
In this final chapter the limitations and implications of this study are discussed.
Consideration is given to limitations regarding the selection of students (see
Section 7.2.1), the timing of the research (see Section 7.2.2), the size of the and
Intervention Groups (see Section 7.2.3), the withdrawal of students from their
usual classroom environment (see Section 7.2.4), the length of the problem-
posing intervention (see Section 7.2.5) and the usefulness of Question Three in
the Student Survey (se Section 7.2.6). In Section 7.3 the implications of this
research are discussed to inform other researchers of potential avenues for
future research. Concluding comments can be found in Section 7.4.
7.2 Limitations of the Study
While every care was taken to ensure the reliability and validity of data obtained
from this study, inevitably as often happens with educational research involving
children, unexpected events occurred that may have had an impact on the
research outcomes of this study. Chapter 4 contains a detailed discussion of the
types of events, such as flooding rains that could not be predicted or planned for
in the design of the teaching experiment. In addition, the number of students
made available for the experiment by the research school, and the dispositional
and situational factors that can influence middle-year students on any given day
for any number of reasons (e.g., Gootman, 2001; Reis & Siegle, 2006), cannot
216
be disregarded as having had an influence on the ‘trustworthiness’ of the data
(Merriam, 1998). While it is less than ideal to work with small sample sizes,
useful analysis of the data was possible through the triangulation of the
qualitative and quantitative data (see Chapters 4, 5 and 6).
7.2.1 Limitations in the Selection of Students
Two specific criteria were used to select participants for this study. Each student
in the Comparison and Intervention Groups was required to have scored above
the 60th percentile in the routinely administered Middle Years Ability Test (MYAT)
(Australian Council for Educational Research, 2005) and they were required to
have scored below the mean score of their cohort in the problem-solving criterion
of their school mathematics tests. While these criteria determined the initial
number of students available for this research, we are reminded that students
who had English as a second language were not considered as potential
participants for this study due to their potential language barrier in undertaking
the tasks contained within the teaching experiment.
When potential participants were identified, they were given a participant
information and permission notice to take home to their parents (see Appendices
A and B). If parents approved of their child’s participation the students were
required to return their signed forms to the school by a nominated date.
Unfortunately, some students did not return their forms by the due date and
hence were unable to become participants in the study. This factor reduced the
sample sizes of both Comparison and Intervention Groups by two students.
Once the groups were confirmed, further reductions in sample sizes occurred
due to some students being at school for the pre-test and not for the post-test.
In addition, throughout the teaching experiment, a number of students in the
Intervention Group were unable to attend all of the teaching sessions due to
illness. Students who had missed more than two of the eight sessions were
deemed to be ineligible for data analysis to maintain a high level of reliability and
217
validity of the data. These factors resulted in the Comparison Group being
reduced to 16 students and the Intervention Group being reduced to 15 students.
7.2.2 Limitations in the Timing of the Research
There were two limitations to the research findings arising as a result of the
timing of the study. In the first instance, senior staff at the School, while being
very supportive of this research project, preferred the students not to miss any
timetabled lessons. In addition, it was deemed inappropriate to ask students to
use their break times or after school to participate in the study. The weekly
assembly, lasting 45 minutes, occurred in the first session of each Monday. This
session was considered to be the most suitable time for the students to be
withdrawn however; the research required a 60 minute block of contact each
week with the students. The compromise chosen was to have participants start
the day 15 minutes earlier than other Year 7 students at the School. All
participating students in the Intervention Group confirmed at the start of the
research they were able and willing to arrive at school every week for the start of
the sessions.
As the teaching experiment progressed, there were a number of unexpected
factors that impacted upon student arrival times at the research room. These
included road works, parents and students who slept in, specialist medical
appointments, school photographs, pupil-free days and students being asked to
accept awards at assembly. Daily notices on the preceding Fridays were used to
remind students of the sessions on each forthcoming Monday and the students
were collected from the playground and classrooms before each teaching
episode began. These initiatives were successful in minimising some of the
disruptions to the start of the sessions however, other disruptions were
unavoidable. On the occasion when students were not at school on the
scheduled Monday due to a pupil-free day, the ‘lost’ session occurred on the
218
following Tuesday. This was a satisfactory exchange of times and resulted in
minimal impact to the flow of the teaching sessions.
The research was conducted in the final term of the year. This time of the year
was chosen so that the students would have completed at least three
mathematics tests, one in each of the preceding three terms. Data from these
mathematics tests were used as a selection criterion for participation in this
study. As detailed in Section 4.3.3, the final session of the teaching experiment
was used for administering the post-test. As this week was the final week of the
year for both the Comparison and Intervention Groups, the atypical excitement
and distraction of the students may have influenced their willingness and ability
to focus on completing the post-test to the best of their abilities. To address this
set of situational distractions the students were encouraged to consider the
significance of their involvement in the study. They were reminded that the
research may be used to help other students to develop their problem-solving
competence. Some students, like Paul, became settled almost immediately on
entering the research room while others, like Andrew and Nicole, found it
extremely difficult to focus on completing the post-test.
7.2.3 Limitations of the Size of the Comparison and Intervention Groups
As a result of the smaller than expected sample sizes, the number of suitable
statistical tests to investigate the data was limited. However, a quantitative
analysis was achieved by comparing the mean scores and standard deviations of
the five aspects of the Profiles of Problem Solving (POPS) test (Stacey et al.,
1993) between the Comparison and Intervention Groups, and by analysing the
number of ‘improved‘ problem-solving aspect scores from the pre-test to the
post-test of individuals in both groups. Additional confidence in findings came
from the evidence contained in the qualitative data collected throughout the
experiment (see Chapter 5). This triangulation of data sought to overcome the
limitations of a smaller than expected sample size in each of the groups.
219
The smaller than anticipated number of students participating in this study was
unexpected and may have impacted upon the trustworthiness of the findings of
this study. Findings from further studies, or findings from a replication of this
study, may benefit from being undertaken in a research site where a larger
number of initial suitable participants were available to allow for attrition of
participants throughout the study.
7.2.4 Limitations of the Withdrawal of Students from their Usual
Classroom Environment
In an ideal research study environment, as many variables as possible would be
kept constant to ensure that changes detected at the conclusion of a study could
be attributed directly to the intervention. However, practically speaking this is not
always possible, particularly when investigations of changes in students are
being conducted. In this present study, there were two factors that necessitated
the withdrawal of students from their usual class groups and rooms.
Firstly, students satisfying the participant requirements were found in all four of
the year 7 classes and all of these students were required to collectively
constitute the Comparison and Intervention Groups. However, students were
familiar with being withdrawn in different groupings, from their usual ‘home’
groups, for language and sport lessons so their withdrawal was deemed to have
minimal impact on the results of the study.
Secondly, withdrawal of the students from their usual ‘home’ rooms to a different
teaching room was required to house the participant group and to facilitate the
teaching experiment. As mentioned in Section 3.5, the classroom in which the
study was conducted, the library classroom, while not being a Year 7 ‘home
room’ was a room with which the students were familiar and to which some of the
students were regularly withdrawn for their language classes. This factor was
also deemed to have minimal impact on the results of the study.
220
The intervention was not directly associated with any part of the students’ usual
mathematics curriculum. That is, there was no intention in this present study for
students to develop specific mathematical skills or knowledge contained in the
Year 7 Mathematics work program. The Semester 1 assessment of Mathematics
at the research school in both 2006 and 2007 provided specific data on the
previous problem-solving competence of the students in the consecutive Year 7
cohorts. These data were used in part as selection criteria to determine students
who met the participant requirements of this study.
It was the intention of this study to determine if, and if so, how can a problem-
posing intervention develop problem-solving competence. The problem-solving
competence was not limited to that used to solve mathematical problems found
within the Year 7 Mathematics Syllabus. However, a number of researchers
have reported that the development of problem-solving competence can be a
powerful way to learn mathematics (e.g., English et al., 2005; Lesh &
Zawojewski, 2007) so it is reasonable to suggest that students may have
developed their understanding of some mathematical concepts as a result of
their participation in the teaching experiment. By demonstrating that a problem-
posing intervention may develop problem-solving competence and by accepting
the proposition that developing problem-solving competence can develop
students’ understandings of some mathematical concepts, it follows that this
present study may also provide useful evidence for researchers investigating
ways to reconsider and redefine appropriate pedagogy for the teaching of
mathematics to middle-year students.
7.2.5 Limitations in the Length of the Problem-posing Intervention
The intervention lasted a total of eight weeks with the post-testing occurring
immediately at the end of the last teaching episode. One could argue that this
immediate post-testing may not have accurately detected significant or
longitudinal changes in engagement, developmental learning or enhanced
problem-solving competence of the participants. In addition, some of the
221
changes recorded between the pre-test and post-test may have been as a result
of familiarity with the testing instrument. This possibility was reported by
Ritchhart (2002) who said we cannot underestimate the impact of test practice on
test scores. However, if we were to accept this position we may not be able to
explain why some student scores fell between the pre-test and the post-test. In
addition, if we were to have used a ‘parallel’ test as the post-test, questions could
be raised about the validity and reliability of the comparative data between the
pre-test results and the post-test scores. Lesh and his colleagues (2000, p. 19)
discussed this type of “trade-off”, to meet conflicting goals, and suggested that it
was a legitimate part of planning the design of an educational research study.
Ideally, this present study would have been conducted over a longer timeframe
than eight weeks. This would have enabled the students to deeper explore
problems, to have additional practice at mathematising different situations and to
become increasingly aware of changes in their own developmental learning, thus
perhaps influencing the engagement of more students. A study conducted over
a longer timeframe may also have produced more consistent and reliable data for
analysis.
Access to student participants in educational research projects is governed by
many factors including the willingness of the research school to participate in the
study and the enthusiasm and willingness of the staff at the school to release the
students for involvement in the intervention. The number of students who meet
the selection criteria and the possible impact of their withdrawal from regular
classes, can also impact upon the time made available for an educational study.
This study was conducted over the longest period made available by the
research school and it could be suggested that had it been a longer timeframe, it
may have added an additional level of ‘trust’ to the research question responses.
222
7.2.6 Limitations of Question Three of the Student Survey
The data collected from this question did not generate detailed information about
why the students thought solving problems was a useful activity. The student
responses clearly showed that students agreed that solving problems was useful
but did not provide detailed information to explain their response. A more helpful
re-wording of the question may have been: If you believe solving problems is a
useful activity write down three reasons why it is useful. The comparison of the
reasons given between the Comparison and Intervention Groups may have been
supportive in interpreting the other data collected in this study.
7.3 Implications of the Research
The findings of this research show that, notwithstanding the limitations mentioned
in Section 7.2, a problem-posing intervention can impact upon the development
of problem-solving competence in some middle-year students who are
ascertained as being of above average intelligence yet underachieving in
problem solving. A group of students with these aforementioned attributes had
not been investigated previously and hence this study serves to add to the body
of research related to problem-posing activities, mathematical pedagogy and
solving novel problems.
While these results would not surprise researchers of the field who confidently
applaud the benefits of problem-posing activities in the pursuit of improved
problem-solving competence (e.g., English, 2003; English et al., 2005), others
such as Silver and Cai (1993b) have been previously unconvinced about a
positive correlation between a student’s problem-solving competence and their
involvement with problem-posing activities. The findings of this study add weight
to the argument that problem-posing activities are worthwhile. However, a
longitudinal study of a larger group of students would be very useful in tracking
the sustainability and consistency of positive impacts that occur as a result of a
problem-posing intervention. Practically speaking, it is difficult to access and
223
track a large group of students for a number of years. If this were undertaken,
the quantitative data may be able to ‘stand alone’ in defending the premise that
incorporating problem-posing activities into a mathematics work program for
middle-year students can result in measurable improvements in their problem-
solving competence. This is a challenge for future research.
During the sixth teaching session of the teaching experiment, students seemed
less engaged by the opportunity to pose worded problems than they had done in
previous weeks (see Section 4.4.5). As a result, a move to a focus on creating
and posing visual problems was planned for the seventh teaching session. This
strategy was employed to increase student motivation by providing diversity to
the teaching methodology while remaining within the parameters of the research
design. The immediate re-engagement of students was clearly evident and was
sustained to the end of the teaching experiment (see Sections 4.4.6 and 4.4.7).
Evidence of the student interest in visual problems was supported by the
substantial increase in the number of students recording visual problems as their
most preferred problems to solve in Question Two of the student surveys (see
Table 5.5 and Section 5.3.2). This was an unexpected finding and one that is
worthy of further research. It may be that it was simply the change in the type of
activity that re-engaged the students or it may be that providing problem-posing
activities, rich in visual stimuli, may be a useful teaching strategy in sustaining
student engagement in problem-solving activities. Further research would clarify
which of these alternatives is more likely.
7.4 Concluding Comments
The education of students in mathematics has been studied over many years by
many researchers (e.g., Anderson, 2007; Brown & Walter, 2005; English &
Larson, 2005). A large body of this research focussed on the teaching practices
used by teachers of mathematics (e.g., English, 2002; Lesh & Zawojewski,
2007), the training and professional development of teachers of mathematics
(e.g., Crespo, 2003; Fosnot & Dolk, 2001) and the importance of problem solving
224
in mathematics work programs (e.g., Cai, 2003; Costa, 2005; NRC, 2004).
Despite a number of studies being undertaken by various researchers in different
countries (e.g., Cai, 2003; Hollingsworth et al., 2003; Shimizu, 2002) and
recommendations for changes to curricula being made, Lester (2003) called for a
fresh view of problem solving as little had changed in teaching practices in
mathematics classrooms. Lesh and Zawojewski (2007) went further and stated
that a serious mismatch still existed between how problem-solving was
experienced by students in schools and the type of skills they required when they
entered the work force.
Over the past decade a number of researchers have begun to address this
anomaly through their research of mathematical modelling as a tool to
contextualise problem-solving experiences (e.g., English et al., 2005; Lesh &
Zawojewski, 2007; Lester & Kehle, 2003). Throughout this research, significant
discussion focussed on the importance of interpreting problems (Lesh &
Zawojewski, 2007) and being able to “see” what a problem is before being able
to solve it (Lesh & Doerr, 2003). English (2003) and Brown and Walter (2005)
were amongst leading researchers who researched the potential of problem-
posing activities to assist students to understand the underlying structure of
problems and to therefore transfer their knowledge between analogous
problems. Their work informed the design and foci of this study.
This study set out to investigate problem-posing activities for a group of students
who often confound their teachers, that is, students who appeared to be
particularly intelligent, but who were not able to perform well at solving problems
in a test situation. The findings of this research have provided some persuasive
evidence to suggest that involvement in problem-posing activities can have a
positive impact for this group of students. It would seem then that this line of
research is fruitful and should be pursued in the endeavour of preparing our
brightest students to be our future problem solvers.
225
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252
Appendices
253
APPENDIX A
PROJECT INFORMATION SHEET AND PARENT CONSENT FORM FOR
COMPARISON GROUP
254
PARTICIPANT INFORMATION for QUT RESEARCH PROJECT
“A problem-posing intervention in the development of problem-solving
competence of underachieving, middle-year students.”
Researcher Contacts
Assistant Professor Deborah Priest Chief Investigator
Mobile Number xxxxxxxx Email xxxxxxx
Description This project is being undertaken as part of a Doctor of Philosophy (PhD) research study by Deborah Jean Priest. The purpose of this project is to investigate the issues surrounding the acquisition of problem-solving skills of middle-year students. In particular, it will look at the links between problem-posing skills development and improvements in problem-solving competence.
The data collection for this research project will occur in Term 4 of 2006, with selected students from the current Year 7 cohort, and again in Term 4 of 2007, with selected students from next years Year 7 cohort.
The students from the current Year 7 cohort will simply be asked to complete a 40 minute problem solving test, containing 6 questions, in week 1 of Term 4 and then the same test again at the end of week 8 of Term 4. The students from the current Year 7 cohort will represent the “comparison” group.
Next year, the selected students (current Year 6 students) will also undertake the same problem solving test in weeks 1 and 8 of term 4, but those students will receive a weekly problem-posing lesson, from the researcher, throughout Term 4 of 2007. The selected students from 2007 will represent the “intervention” group. The changes, if any, in problem-solving competence between the two groups participating in the two eight week periods, will then be compared to determine if the a problem-posing intervention has resulted in improved problem-solving competence.
255
Participation Your child has been selected to participate in the study as part of the “comparison” group.
The school has given the researcher permission to conduct this research and to approach you to request permission for your child to take part. Your child’s participation in this project is voluntary. If you do agree for your child to participate, your child can withdraw from participation at any time during the project without comment or penalty. Your decision for your child to participate will in no way impact upon their current or future relationship with QUT or with your child’s grades for school assessment.
Each student participating in this project will complete the test in class time, in a classroom at the school, and will be supervised by a teacher. The test has proven to be a valid and reliable measure of students’ problem-solving ability and looks at five aspects of problem solving: correctness of answer, method used, accuracy, extraction of information and quality of explanation. Please contact the researcher directly if you wish to ask any questions about the test. Expected benefits Results from this study will provide the researcher with crucial insights into how teachers and parents can assist their children to become more confident and competent problem solvers. Staff at the school have taken a keen interest in the project to date and will be reviewing the findings of the researcher.
Problem-solving competence is at the centre of contemporary curricula, including the International Baccalaureate Middle Years Program, which is currently being implemented at the school. Through this research project, the researcher hopes to demonstrate that particular forms of teaching intervention can be used by teachers to assist students to become more confident and competent at solving non-routine problems.
Risks There are no risks beyond normal day-to-day living associated with your child’s participation in this project. Confidentiality All comments and responses will be treated in strictest confidence. The names of individual students and their personal results in the testing will only be available to the researcher. The information obtained from the data will be discussed in the researcher’s thesis, but pseudonyms (replacement names) will be used to reference students and their results. This will ensure the anonymity of the participating students. All tests will be stored in a locked cabinet at the office of the researcher at the University and no other person, other than the researcher, will have access to the individual tests. Consent to Participate I would like to ask you and your child to sign the written consent form (enclosed) to confirm your agreement for your child to participate. Please ask your child to return the complete consent form to their class teacher as soon as possible before the end of Term 3.
256
Questions / further information about the project Please contact the researcher, Deborah Priest, to have any questions answered or if you require further information about the project. Please retain this sheet for future reference. Concerns / complaints regarding the conduct of the project QUT is committed to researcher integrity and the ethical conduct of research projects. However, if you do have any concerns or complaints about the ethical conduct of the project you may contact the QUT Research Ethics Officer on 3864 2340 or [email protected]. The Researcher Ethics Officer is not connected with the research project and can facilitate a resolution to your concern in an impartial manner.
257
CONSENT FORM for QUT RESEARCH PROJECT
“A problem-posing intervention in the development of problem-solving
competence of underachieving, middle-year students.”
Statement of consent By signing below, you are indicating that you:
• have read and understood the information document regarding this project;
• have had any questions answered to your satisfaction;
• understand that if you have any additional questions you can contact the researcher;
• understand that your child is free to withdraw at any time, without comment or penalty;
• understand that you can contact the Research Ethics Officer on 3864 2340 or [email protected] if you have concerns about the ethical conduct of the project;
• have discussed the project with your child;
• agree for your child to participate in the project. Parent/Guardian Name
Signature
Date / /
Statement of Child consent Your parent or guardian has given their permission for you to be involved in this research project. This form is to seek your agreement to be involved.
By signing below, you are indicating that the project has been discussed with you and you agree to participate in the project. Student’s Name
Signature
Date / /
258
APPENDIX B
PROJECT INFORMATION SHEET AND PARENT CONSENT FORM FOR 2007
INTERVENTION GROUP
259
2007 PARTICIPANT INFORMATION for QUT RESEARCH PROJECT
“A problem-posing intervention in the development of problem-solving
competence of underachieving, middle-year students.”
Researcher Contacts
Assistant Professor Deborah Priest Chief Investigator
Mobile Number xxxxxxxx Email xxxxxxx
Description This project is being undertaken as part of a Doctor of Philosophy (PhD) research study by Deborah Jean Priest. The purpose of this project is to investigate the issues surrounding the acquisition of problem solving skills of middle-year students. In particular, it will look at the links between problem-posing skills development and improvements in problem-solving competence.
The data collection for this research project began in Term 4 of 2006, with selected students from last years Year 7 cohort, and it will occur again in Term 4 of 2007, with selected students from this years Year 7 cohort.
The students from last year were simply asked to complete a 40 minute problem-solving test, containing 6 questions, in week 1 of Term 4 and then the same test again at the end of week 8 of Term 4. The students from the 2006 Year 7 cohort represent the “comparison” group.
This year, the selected Year 7 students will undertake the same problem-solving test in weeks 1 and 8 of term 4, but, in addition, will receive a weekly problem-posing lesson, from the researcher, throughout Term 4 of 2007. The selected students from 2007 will represent the “intervention” group. The changes, if any, in problem-solving competence between the two groups participating in the two eight week periods, will then be compared to determine if the a problem-posing intervention has resulted in improved problem-solving competence.
260
Participation Your child has been selected to participate in the study as part of the “intervention” group.
The school has given the researcher permission to conduct this research and to approach you to request permission for your child to take part. Your child’s participation in this project is voluntary. If you do agree for your child to participate, your child can withdraw from participation at any time during the project without comment or penalty. Your decision for your child to participate will in no way impact upon their current or future relationship with QUT or with your child’s grades for school assessment.
Each student participating in this project will complete the test in class time, in a classroom at the school, and will be supervised by a teacher. The test has proven to be a valid and reliable measure of students’ problem-solving ability and looks at five aspects of problem solving: correctness of answer, method used, accuracy, extraction of information and quality of explanation. Please contact the researcher directly if you wish to ask any questions about the test. Expected benefits Results from this study will provide the researcher with crucial insights into how teachers and parents can assist their children to become more confident and competent problem solvers. Staff at the school have taken a keen interest in the project to date and will be reviewing the findings of the researcher.
Problem-solving competence is at the centre of contemporary curricula, including the International Baccalaureate Middle Years Program, which is currently being implemented at the school. Through this research project, the researcher hopes to demonstrate that particular forms of teaching intervention can be used by teachers to assist students to become more confident and competent at solving non-routine problems.
Risks There are no risks beyond normal day-to-day living associated with your child’s participation in this project. Confidentiality All comments and responses will be treated in strictest confidence. The names of individual students and their personal results in the testing will only be available to the researcher. The information obtained from the data will be discussed in the researcher’s thesis, but pseudonyms (replacement names) will be used to reference students and their results. This will ensure the anonymity of the participating students. All tests will be stored in a locked cabinet at the office of the researcher at the University and no other person, other than the researcher, will have access to the individual tests. Consent to Participate I would like to ask you and your child to sign the written consent form (enclosed) to confirm your agreement for your child to participate. Please ask your child to return the complete consent form to their class teacher as soon as possible before the end of Term 3.
261
Questions / further information about the project Please contact the researcher, Deborah Priest, to have any questions answered or if you require further information about the project. Please retain this sheet for future reference. Concerns / complaints regarding the conduct of the project QUT is committed to researcher integrity and the ethical conduct of research projects. However, if you do have any concerns or complaints about the ethical conduct of the project you may contact the QUT Research Ethics Officer on 3864 2340 or [email protected]. The Researcher Ethics Officer is not connected with the research project and can facilitate a resolution to your concern in an impartial manner.
262
2007 CONSENT FORM for QUT RESEARCH PROJECT
“A problem-posing intervention in the development of problem-solving
competence of underachieving, middle-year students.”
Statement of consent By signing below, you are indicating that you:
• have read and understood the information document regarding this project;
• have had any questions answered to your satisfaction;
• understand that if you have any additional questions you can contact the researcher;
• understand that your child is free to withdraw at any time, without comment or penalty;
• understand that you can contact the Research Ethics Officer on 3864 2340 or [email protected] if you have concerns about the ethical conduct of the project;
• have discussed the project with your child;
• agree for your child to participate in the project. Parent/Guardian Name
Signature
Date / /
Statement of Child consent Your parent or guardian has given their permission for you to be involved in this research project. This form is to seek your agreement to be involved.
By signing below, you are indicating that the project has been discussed with you and you agree to participate in the project. Student’s Name
Signature
Date / /
263
APPENDIX C
STUDENT SURVEY SHEET
264
Student Survey Sheet:
Name: ____________________________ Date: ____________________
Question One:
Do you enjoy solving problems? Please explain your answer in a few sentences.
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
Question Two:
What type of problems do you prefer to solve? Please explain your answer in a
few sentences.
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
265
Question Three:
Do you think learning to solve problems is a useful thing to do? Please explain
your answer in a few sentences.
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
Question Four:
What things could teachers do to assist you to become better at solving
problems?
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
266
APPENDIX D
TEACHING EXPERIMENT LESSON ONE
267
TEACHING EXPERIMENT LESSON 1 – YEAR 7
Date: October 8, 2007 Topic: Student Survey and Pre-testing
Year Level: 7 Duration: 60 minutes
Rationale:
The purpose of this lesson is to allow the students to complete a survey containing 4 attitudinal questions and then a problem-solving pre-test, before they are involved in a teaching experiment during Term 4. Both the survey and the test are identical to the survey and test that the students will complete in their final session at the end of the teaching experiment. Students WILL NOT be made aware that they will be completing the same survey sheets or tests at the start and end of the teaching experiment. This is to ensure that as many variables as possible are controlled throughout this experiment.
The results from both the surveys and the tests will allow the researcher to determine if there are noteworthy changes and/or emergent themes in student attitudes to problem solving or their problem-solving competence, as a result of the students’ involvement in the teaching experiment.
Key Investigations:
Establish student attitudes and/or problem-solving competence before the students are involved in the teaching experiment.
268
THE MULTIPLE INTELLIGENCES CHECKLIST
Visual/Spatial Verbal/Linguistic Bodily/Kinaesthetic
� Charts � graphs � diary √ visual metaphors
√ 3D
√ sketching
√ mind maps
� stories � retelling � book making � research � speeches √ reading
� drama
� activities � hands on � body language � crafts � drama � mime � PE
Logical/Math Intrapersonal Interpersonal
√ timeline
√ coding
√ geometry
√ measuring
√ classifying
√ money
√ time
√ individual study
� journals √ reflection
� self-esteem activities √ individual reading
√ individual projects
� cooperative learning � sharing � group work � discussion � peer editing � brainstorming
269
Time Lesson Plan - Learning Activities Resources
5
mins
Introduce myself, the research I am undertaking, the students’ participation and the 8 week teaching experiment.
13
mins
Ensure students are well spread out around the room and that they each have a pen with which to write.
Hand out a survey sheet to each student and ask them to write their names on the top of the sheet and then complete the attitudinal surveys individually and in silence.
Collect the survey sheets.
One survey sheet
for each
participant. Spare
pens.
40
mins
Pre-tests to be handed out face down while students remain in silence. Inform students about the administration process of the test. Ask the students to write their names on the top of the front page of the
test and turn the first page over. Administer the test according to the requirements specified in the POPS
Administrator’s Handbook. Collect the test booklets from the students.
Pre-tests for each
student.
2
mins
Ask students to assist in returning the furniture in the room to the way it usually is.
Thank the students, remind the students to arrive on time next lesson, and allow students to return to their classes.
270
APPENDIX E
TEACHING EXPERIMENT LESSON TW O
271
TEACHING EXPERIMENT LESSON 2 – YEAR 7
Date: October 15, 2007 Topic: Introduction to Problem Posing
Year Level: 7 Duration: 60 minutes
Rationale:
The students begin the teaching experiment by considering a problem that is novel in nature but one to which they can relate. This problem does not have one “correct” answer, although it is likely that a number of students will choose to follow a similar path. The student’s justifications of their own answers will be a focus of the lesson. This problem has been chosen to begin the process of thinking “outside” the box and empowering and validating students to do so. It is also designed to dispel the notion that every problem has only one answer.
The lesson will continue with the students being asked to modify the problem to generate a different problem for their peers to solve. They will be reminded that it must be a problem that they can solve. This will begin the process of giving the students some autonomy in their learning as supported by the Critical Theory (Tierney, 1997).
Key Investigations:
• Is there always one correct answer to a problem? • Is one answer “better” than another? • How many problems can be generated from one base problem?
272
THE MULTIPLE INTELLIGENCES: CHECKLIST
Visual/Spatial Verbal/Linguistic Bodily/Kinaesthetic
� charts � graphs � diary � visual metaphors � 3D √ sketching
√ mind maps
√ stories
√ retelling
� book making � research � speeches √ reading
√ drama
� activities √ hands on
√ body language
� crafts √ drama
� mime � PE
Logical/Math Intrapersonal Interpersonal
√ timeline
� coding � geometry � measuring √ classifying
√ money
√ time
√ individual study
� journals √ reflection
√ self-esteem activities
√ individual reading
√ individual projects
√ cooperative learning
√ sharing
√ group work
√ discussion
√ peer editing
√ brainstorming
273
Time Lesson Plan - Learning Activities Resources
7 mins
Introduction.
Welcome students back and once again show appreciation for their involvement.
Hand out a new exercise book for each student and ask them to write their names on the front cover.
Remember to emphasise no summative assessment contribution for their school grades.
Remember to emphasise we are going to have loads of fun! Ask students if there are any questions.
One new exercise book for each student.
3 mins
Tell students they can work individually, in pairs or a small group for the first activity.
Ask students to move places if they need to, but remind them that they must all write their own solutions in their own exercise book.
Show students a variety of “concrete” resources at the front table if they want to use them at any time.
Bags of “matchsticks”, plastic discs, connectable blocks, small timber blocks, and fake coins
7 mins
Hand out the problem about Jane and Tom (see Problem for Lesson One on next page ).
Ask students to open their exercise books and write today’s date on the first page.
Ask students to read the problem, determine a solution, and write down their solution, including some notes to justify it, in their exercise books. I will go to each student and stick the problem into their exercise books (to save time).
as above +
glue stick, photocopies of Problem for Lesson One for each student
8 mins
Ask students to stop work. Ask students to write their final answer on the yellow post-it-notes, that I
handed out while they were working, and ask a member of each group, or individual, to stick the note on the whiteboard.
I will quickly sort the solutions to put those with the same answer together in a visual way on the board.
Ask students why there are different answers. (expect students to say incorrect process used, but hope for students to say different interpretations were made)
Ask students if there is a problem with having different answers. Ask for volunteers to justify their answers for the class (validate different
solutions that have appropriate, but alternate interpretations).
Pad of yellow post-it-notes
10 mins
Ask students to look again at the problem and ask them if they could modify it in some way to create a new problem that they could solve.
Ask students to write their new problem down, as well as the solution to it, stating any assumptions they made along the way.
All resources that were available at the start of the lesson
20 mins
Ask students to stop work. Ask students for volunteers to share their problems. (either they read, or a
friend or I read – their choice) Ask each volunteer to share their thinking that resulted in the new problem
being posed.
274
Repeat previous two steps for as long as time allows.
5
mins
Thank all students for their contributions of posed problems and collect exercise books.
Ask students what they think they have learned during the lesson and record them on the board (visual reinforcement).
Add any outcomes that they have missed. Tell students to look forward to some more creative fun next lesson and
then allow students to leave.
Problem for Lesson Two
Tom and Jane are brother and sister. They live next door to Mr and Mrs Lee who
are about to leave on a two week trip to China. Mr Lee has asked Tom and Jane
to look after Fifi (their cat) and Pluto (their dog) while they are away. Mrs Lee
has just given Tom and Jane’s parents $200 to share between the children
based on how much work they each contribute to the care of the pets.
Fifi needs combing every second day for 15 minutes and needs fresh food and
water daily.
Pluto needs walking every day for 30 minutes and also needs fresh food and
water daily.
If Jane looks after Pluto and Tom looks after Fifi, and they both do an equally
good job, how should Tom and Jane’s parents share the $200 between the
children? Remember to fully justify your solution stating clearly any assumptions
you have made.
275
APPENDIX F
TEACHING EXPERIMENT LESSON THREE
276
TEACHING EXPERIMENT LESSON 3 – YEAR 7
Date: October 29, 2007 Topic: Problem Posing
Year Level: 7 Duration: 60 minutes
Rationale:
The lesson will begin with a review of the previous lesson. In particular, we will discuss the following attributes of a “good problem”: interest factor, challenge level and “do-ability”. We will look at some problems and discuss whether they meet the criteria we have discussed. In particular, I want to draw the students’ attention to the different types of problems; such as long, wordy problems compared to pictorial or design problems and what makes each type challenging or interesting. This process will allow the students to decide on what types of problems they would like to create, and should liberate them from institutionalised thinking patterns.
I will then draw the students’ attention to the problems that they wrote last lesson and I will ask them to self-rate their problem. This is designed to develop some reflective and metacognitive skills which will inform their future problem posing. I will then tell the students that, in addition to the students’ self-ratings, I will also rate their problems from this week. I have chosen to do this because I want to maximise the motivation of the students during their problem posing. In the final week I will announce some awards; such as Most Entertaining Problem Poser, Most Improved Problem Poser; Best Overall Problem Poser.
Key Investigations:
• What constitutes a “good problem”? • Identify different types of problems.
277
THE MULTIPLE INTELLIGENCES: CHECKLIST
Visual/Spatial Verbal/Linguistic Bodily/Kinaesthetic
� Charts � graphs � diary √ visual metaphors
√ 3D
√ sketching
√ mind maps
√ stories
√ retelling
� book making � research � speeches √ reading
√ drama
√ activities
√ hands on
√ body language
√ crafts
√ drama
� mime � PE
Logical/Math Intrapersonal Interpersonal
√ timeline
√ coding
� geometry √ measuring
√ classifying
� money √ time
√ individual study
� journals √ reflection
√ self-esteem activities
√ individual reading
√ individual projects
√ cooperative learning
√ sharing
√ group work
√ discussion
√ peer editing
√ brainstorming
278
Time Lesson Plan - Learning Activities Resources
5 mins
Introduction.
Welcome students to class. Ask students what they liked and disliked about the last lesson. Introduce the 1,2,3 card system of peer feedback.
Video cameras in place and all resources laid out.
5 mins
Hand out exercise books to students. Discuss what attributes were displayed by the students’ problems from
last week. Reminding students about interest factor, challenge level and “do-ability”. Write these up on the board.
Ask students to open their books and look at the criteria rating sheet that I have stuck into each book. Tell students about the self-rating and teacher-rating that will occur for the remainder of the term.
Ask students to self-rate their problem from last week. Discuss briefly with the students and be positive about improved ratings
as improved problem-posing competence is achieved.
Student exercise books with self-rating criteria sheets already attached to the first weeks problem.
15 mins
Ask students if they would like to work individually, in pairs or a small group for the activity.
Ask students to move places if they need to, but remind them that they must all write their own solutions in their own exercise book.
Ask students to open their exercise books and write today’s date on the next new page.
Ask students to write a problem based on the stimuli provided, determine a solution, and write down their solution, and some notes to justify their solution, in their exercise books. Emphasise the attributes of a good problem that were discussed earlier.
Sets of stimuli on each table:
Mini packets of Skittles; set of three different sized bulldog clips; one new Post-it-note pad (enough for each student)
25 mins
Ask students to stop work. Ask students for volunteers to share their problems. (either they read, or a
friend or I read – their choice) Ask other students to “rate” the problem by holding either a 1, 2 or 3 card
in the air. Repeat previous two steps for as long as time allows. Ask students to self-rate their problems on the criteria sheets that have
been attached to their books before the lesson began.
Cards with 1, 2 and 3 written on them.
8 mins
Summarise the main points that arose from the lesson. Points to summarise may include imagination of the students, length of problems, similarity or difference of problems, or student engagement. Use language that will ensure the students leave the room feeling valued, empowered and heard.
Ask students what they think they have learned during the lesson.
2 mins
Thank all students for their contributions of posed problems and collect exercise books.
Remind students that I will rate their problems before next lesson. Tell students to look forward to some more creative fun next lesson and
then allow students to leave.
279
PHOTOGRAPH OF LESSON STIMULI PROVIDED TO EACH STUDENT
280
APPENDIX G
TEACHING EXPERIMENT LESSON FOUR
281
TEACHING EXPERIMENT LESSON 4 – YEAR 7
Date: October 30, 2007 Topic: Problem Posing
Year Level: 7 Duration: 60 minutes
Rationale:
The lesson will begin with a review of the previous lesson and a reminder of the self-rating and teacher-rating for the students’ problems. In particular, I will summarise my thoughts on their problems in general. This may draw the student’s attention to the attributes of a “good” problem that have been discussed over the last two lessons.
We will re-visit the notion of different types of problems as most students constructed similar problems during the last lesson. For this lesson I will provide an unexpected object for stimuli. I hope that students will think beyond a simple addition or multiplication type of problem to write. The students still seem to be thinking superficially and without imagination.
The lesson will continue with the students being asked to write a problem for their peers to solve that is based around the stimuli provided. I expect some students to be confounded by the stimuli, but I will assist them during the lessons with prompts and starting point ideas for them to develop.
Key Investigations:
• How can I write a problem about a seemingly unquantifiable object?
282
THE MULTIPLE INTELLIGENCES: CHECKLIST
Visual/Spatial Verbal/Linguistic Bodily/Kinaesthetic
� Charts � graphs � diary √ visual metaphors
� 3D √ sketching
√ mind maps
√ stories
√ retelling
� book making √ research
� speeches √ reading
� drama
� activities � hands on √ body language
� crafts √ drama
� mime � PE
Logical/Math Intrapersonal Interpersonal
√ timeline
√ coding
√ geometry
√ measuring
√ classifying
√ money
√ time
√ individual study
� journals √ reflection
√ self-esteem activities
√ individual reading
√ individual projects
√ cooperative learning
√ sharing
√ group work
√ discussion
√ peer editing
√ brainstorming
283
Time Lesson Plan - Learning Activities Resources
3 mins
Introduction.
Welcome students to class. (Exercise books, scoring cards and photographs to be on desks before students enter the room)
Briefly discuss the problems written from the last lesson. Give students a few minutes to read the comments I have written about
their problems and ask any questions they may have.
Room to be prepared with video cameras in place and all resources laid out.
7 mins
Ask students to look at the sheet of three questions. Discuss with students about the attributes of each problem and what
makes them interesting, challenging and do-able, or not. Draw students’ attentions to the fact that the final problem is do-able if the
original painting were available, however, discuss ways of getting an answer that may be close to the real answer.
A sheet with 3 questions on and an answer sheet for each student.
10 mins
Discuss today’s stimuli asking students what their initial thoughts are. Ask students to open their exercise books and write today’s date on the
next new page. Ask students to write a problem, determine a solution, and write down
their solution, and some notes to justify their solution, in their exercise books.
Emphasise the attributes of a good problem that were discussed earlier.
Student exercise books with rating criteria sheets already attached to the first weeks problem. Photograph of some grass.
30 mins
Ask students to stop work. Ask for volunteers to share their problems. (either they read, or a friend, or
I read – their choice) Ask other students to rate the problems with the 1, 2 or 3 card system. Repeat previous two steps for as long as time allows. Ask students to self-rate their problems on the criteria sheets that have
been attached to their books before the lesson began.
Cards with 1, 2 and 3 written on them.
8 mins
Summarise the main points that arose from the lesson. Points to summarise may include imagination of the students, length of problems, similarity or difference of problems, or student engagement. Use language that will ensure the students leave the room feeling valued, empowered and heard.
Ask students what they think they have learned during the lesson.
2 mins
Thank all students for their contributions of posed problems and collect exercise books.
Remind students that I will rate their problems before next lesson. Tell students to look forward to some more creative fun next lesson and
then allow students to leave.
284
PROBLEM 7.1
CONSIDER THE INFORMATION GIVEN BELOW TO WORK OUT THE NUMBER OF JELLY BEANS
IN A FULL JAR.
PROBLEM 7.2
THE FIGURE BELOW IS MADE UP OF 5 CONGRUENT SQUARES. THE PERIMETER OF THE
FIGURE IS 72 CM. FIND THE NUMBER OF SQUARE CM IN THE AREA OF THE FIGURE.
285
PROBLEM 7.3
TO THE BEST OF YOUR ABILITY DETERMINE HOW MANY DOTS MAKE UP THIS PICTURE.
286
PROBLEM 7.1 SOLUTION
LET THE NUMBER OF JELLY BEANS IN A FULL JAR BE N. SINCE THERE ARE 7 JARS ON
EACH SIDE OF THE SCALES WE CAN CONSIDER ONLY THE CONTENTS OF THE JARS OR
LEFTOVER JELLY BEANS.
6N + ¼ N = 375
6.25N = 375
N = 375/6.25
N= 60
THERE ARE 60 JELLY BEANS IN A FULL JAR.
PROBLEM 7.2 SOLUTION
SINCE THIS SHAPE IS MADE UP OF 5 CONGRUENT (SAME) SQUARES, THEN WE CAN
CONSIDER EACH EDGE TO BE OF EQUAL LENGTH. LET THE LENGTH OF ONE EDGE BE L.
TOTAL NUMBER OF OUTSIDE EDGES IS 12.
TOTAL PERIMETER = 72CM.
12L = 72
=> L= 6CM
FIVE SQUARES MAKE UP THE SHAPE. EACH SQUARE HAS AN AREA OF LXL= 6X6 CM2
= 36 CM2
THE AREA OF THE ENTIRE SHAPE = 5X36 CM2
= 180 CM2
PROBLEM 7.3 SOLUTION
? - Discuss
287
LESSON 3 STIMULI
288
APPENDIX H
TEACHING EXPERIMENT LESSON F IVE
289
TEACHING EXPERIMENT LESSON 5 – YEAR 7
Date: November 5, 2007 Topic: Problem Posing
Year Level: 7 Duration: 60 minutes
Rationale:
The lesson will begin by providing students with the opportunity to read the comments that I wrote in their books from last lessons problem. I will summarise my thoughts on their problems in general and make mention of some students who have improved or whose problem had addressed the attributes of a “good” problem. This is important to make a link between last weeks lesson and where we are headed today.
I have written a problem based on the photo of grass from last lesson. I want the students to read it and deconstruct it in order to get some additional ideas about how to create an interesting story line and embed the information for solving the problem. I also want them to experience what looks like a formidable problem but in actual fact is very straight forward and “do-able”.
The lesson will continue with the students being asked to write a problem, for their peers to solve, that is based around the three shapes I have glued in the students’ books. Once again, some students may be unsure of how to start, but I will assist them during the lessons with prompts and starting point ideas for them to develop.
This week I will ask students to use the 1, 2, 3 card system three times to rate their peer’s (volunteers) problems in each of the three separate attributes of do-ability, challenge and interest factor. I hope this will provide students with some immediate, yet more specific, feedback from their peers. Last week I felt the students where getting a little restless, so each week I will ask for volunteers and try to share the opportunities across all students in the remainder of the teaching experiment and encourage those who may be less willing to volunteer.
Key Investigations:
• What makes a problem look difficult? • What makes a problem interesting to read and do? • How can students make their problems “do-able” yet challenging?
290
THE MULTIPLE INTELLIGENCES: CHECKLIST
Visual/Spatial Verbal/Linguistic Bodily/Kinaesthetic
� charts � graphs � diary √ visual metaphors
� 3D √ sketching
√ mind maps
√ stories
√ retelling
� book making � research � speeches √ reading
� drama
� activities √ hands on
√ body language
� crafts √ drama
� mime � PE
Logical/Math Intrapersonal Interpersonal
√ timeline
√ coding
√ geometry
√ measuring
√ classifying
� money √ time
√ individual study
� journals √ reflection
√ self-esteem activities
√ individual reading
√ individual projects
√ cooperative learning
√ sharing
√ group work
√ discussion
√ peer editing
√ brainstorming
291
Time Lesson Plan - Learning Activities Resources
3 mins
Introduction.
Welcome students to class. (Exercise books, scoring cards and my problem to be on desks before students enter the room)
Briefly discuss the problems written from the last lesson and mention notable problems and posers.
Give students a few minutes to read the comments I have written about their problems.
Room to be prepared with video cameras in place and all resources laid out.
10 mins
Give students a problem that I wrote about last weeks stimuli and have students deconstruct it and then try to solve it.
Ask for two volunteers to use the whiteboard to share their solutions. Write up other students’ final answers and then discuss.
Consider Mrs Priest’s solution and discuss the setting out example and the importance of good quality written communication.
Ask students to place both their completed worksheet and my printed solution in the back of their books so that I may glue them in before next lesson.
Researcher problem on grass and the associated worksheet. (see below)
20 mins
Direct students’ attention to the three shapes I have glued in their books (triangle, square and circle).
Ask students to write a problem, determine a solution, and write down their solution, in their exercise books. Remind students to think of the three attributes of a good problem and the need to communicate their answers more effectively.
Student exercise books with rating criteria sheets and printout of 3 shapes triangle, square and circle already attached.
20 mins
Ask students to stop work. Ask students for volunteers to share their problems. This week we will focus on each attribute of a quality problem individually.
I will ask other students to rate the problems with the 1, 2 or 3 card system for interest factor, then do-ability, and then challenge.
Repeat previous two steps for as long as time allows.
Cards with 1, 2 and 3 written on them.
5 mins
Summarise the main points that arose from the lesson. Ask students to self-rate their problems on the criteria sheets that have
been attached to their books before the lesson began.
2 mins
Thank all students for their contributions of posed problems and collect exercise books.
Remind students that I will rate their problems before next lesson.
292
Lesson 4 Stimuli
293
Researcher Problem (stimuli – photograph of grass)
Mrs Priest noticed brown patches appearing in her lawn. According to Mr White, her neighbour, the rain has brought lawn grubs to a lot of lawns in the area and the lawn will need to be sprayed. Mrs Priest rang a man at a lawn maintenance company and asked him how much it would cost to remove the grubs. The man offered to give Mrs Priest a quote over the phone and asked her what size lawn she had. She replied, “it is twice as long as it is wide and it takes 24 of my paces to walk the full length”. The man on the phone then asked Mrs Priest how long each of her paces was. She excitedly replied “I have just got a new pedometer and had to calculate that! My pace is about ¾ of a metre”. If the cost of lawn grub removal is $5.20/m2 what will the total cost of removing the lawn grubs be if the man does his maths correctly?
What makes this problem challenging? (Challenge)
Write down the facts that will help you solve this problem. (Do-ability)
What makes the problem interesting? (Interest Factor)
Write your solution below:
294
POSSIBLE SOLUTION:
Length in paces: 24 paces
Length in metres: 24 x ¾ = 18m
Width: ½ of 18m = 9m
Area of lawns = LxW
= 18 x 9 m2
= 162 m2
Cost to remove grubs = Area x cost/ m2
= 162 x 5.20
= $842.40
The total cost to remove the lawn grubs is $842.40.
295
APPENDIX I
TEACHING EXPERIMENT LESSON S IX
296
TEACHING EXPERIMENT LESSON 6 – YEAR 7
Date: November 12, 2007 Topic: Problem Posing
Year Level: 7 Duration: 60 minutes
Rationale:
To start this lesson I have written a very short problem based on the three shapes from last lesson. I want the students to read it and consider it in terms of the three attributes of a “good” problem. Last week I asked the students to look at a long, wordy problem and this problem is designed to provide a contrast so that students may become open to cleverly designed, short problems. It will be worth discussing that the interest factor may be minimised by posing a short problem.
The lesson will continue with the students being asked to write a problem for their peers to solve that is based around the pictures that I have put on the desks. The pictures are of a skateboard cartoon and an iPod. I chose these objects as they are meaningful artefacts for the students and most of them will have had first-hand experience with either or both objects.
This week I will again ask students to use the 1, 2, 3 card system to rate their peer’s problems in each of the three separate attributes of do-ability, challenge and interest factor. The immediate, yet more specific, feedback from their peers appeared to be appreciated last week. I will again ask for volunteers to read their problems (different to those who read their problem out last week).
Key Investigations:
• Can a short problem, even a one-line problem, be challenging, interesting and do-able all at the same time?
297
THE MULTIPLE INTELLIGENCES: CHECKLIST
Visual/Spatial Verbal/Linguistic Bodily/Kinaesthetic
� charts � graphs � diary √ visual metaphors
� 3D √ sketching
√ mind maps
√ stories
√ retelling
� book making � research � speeches √ reading
� drama
� activities √ hands on
√ body language
� crafts √ drama
� mime � PE
Logical/Math Intrapersonal Interpersonal
√ timeline
� coding � geometry � measuring √ classifying
√ money
√ time
√ individual study
� journals √ reflection
√ self-esteem activities
√ individual reading
√ individual projects
√ cooperative learning
√ sharing
√ group work
√ discussion
√ peer editing
√ brainstorming
298
Time Lesson Plan - Learning Activities Resources
3 mins
Introduction.
Welcome students to class. (Exercise books, scoring cards and my problem to be on desks before students enter the room)
Briefly discuss the problems written from the last lesson and mention notable problems and posers.
Give students a few minutes to read the comments researcher has written about their problems.
Room to be prepared with video cameras in place and all resources laid out.
10 mins
Give students a problem that I wrote about last weeks stimuli and have students deconstruct it and then try to solve it.
Ask for two volunteers to use the whiteboard to share their solutions. Write up other students’ final answers and then discuss.
Consider researcher’s solution and discuss the setting out example and the importance of communication.
Ask students to place both their completed worksheet and my printed solution in the back of their books so that I may glue them in before next lesson.
Researcher’s problem on the triangle, square and circle and the associated worksheet. (see below)
20 mins
Direct students’ attention to the two pictures I have placed on their desks of a skateboard cartoon and an iPod.
Ask students to write a problem, determine a solution, and write down their solution, in their exercise books.
Remind students to think of the three attributes of a good problem. Emphasise the need to communicate their answers more effectively in a
written format when they have completed posing their problem.
Student exercise books with rating criteria sheets already attached. Pictures of a skateboard cartoon and an iPod. (5 sets)
20 mins
Ask students to stop work. Ask students for volunteers to share their problems (different students
from last week). Again this week we will focus on each attribute of a quality problem
individually. I will ask other students to rate the problems with the 1, 2 or 3 card system for interest factor, do-ability and challenge.
Repeat previous two steps for as long as time allows.
Cards with 1, 2 and 3 written on them.
5 mins
Summarise the main points that arose from the lesson. Ask students to self-rate their problems on the criteria sheets that have
been attached to their books before the lesson began.
2 mins
Thank all students for their contributions of posed problems and collect exercise books.
Remind students that I will rate their problems before next lesson and that I look forward to providing them with some feedback.
REMIND STUDENTS ABOUT THE ROOM CHANGE NEXT LESON DUE TO THE STOCKTAKE IN THE LIBRARY 19/11.
Room booking for Ray 2.2
299
Researcher’s Problem (stimuli – THREE SHAPES)
If I cut the largest circle out of a square with area 36 m2, what percentage of the square remains?
What makes this problem challenging? (Challenge)
Write down the facts that will help you solve this problem. (Do-ability)
What makes the problem interesting? (Interest Factor)
Write your solution below:
300
SOLUTION (with example of setting out)
Let side length of square = S
Area of square = SxS
SxS = 36 m2
S=6m
Radius of square = 3m � 6m �
Area of circle = ∏ r2
= 28.27 m2
Area remaining = 36 – 28.27
= 7.72 m2
% area remaining = 7.72 x 100
36
=21.46%
301
Lesson 6 Stimuli
OR
302
APPENDIX J
TEACHING EXPERIMENT LESSON SEVEN
303
TEACHING EXPERIMENT LESSON 7 – YEAR 7
Date: November 19, 2007 Topic: Problem Posing
Year Level: 7 Duration: 60 minutes
Rationale:
I noticed last week that some students were starting to become a little restless after either finishing their problem posing early or simply with the sameness of each lessons format. This week I am changing the lesson and will need the students to move around the room. As with all teaching, lesson delivery needs to respond to students and their needs. I sense a need for change and will provide it with an opportunity to work quite differently. This week we have also been moved to a different classroom, so it is an ideal time for this weeks activity.
The lesson will begin with the students being asked to consider some quizzles on the board (see examples at the end of the lesson plan). In my experience, students are highly motivated to solve this type of visual problem. To further encourage students to make their own visual puzzles and to allow the students to move around the room, I will show them 16 visual picture puzzles that have been created by some Year 7 students I taught a number of years ago. The puzzles will be placed around the room so students can move freely and come back to a problem at any time. We will review the intended answers and consider the validity of any other answers that students may have come up with.
In the second half of the lesson I will ask the students to begin thinking of a visual, 3D puzzle of their own. They can brainstorm with me, their neighbour or in small groups. Each student or group of students will need to create a problem of their own for the class to do next week. This public, visual sharing will allow students to use their own initiative, creativity and interest area which is consistent with Critical Theory. I hope the students will find this activity to be an exciting way to round-off the teaching experiment.
Key Investigations:
• Can visual problems have the three attributes of a good problem? If so, what do they “look” like?
304
THE MULTIPLE INTELLIGENCES: CHECKLIST
Visual/Spatial Verbal/Linguistic Bodily/Kinaesthetic
√ charts
� graphs � diary √ visual metaphors
� 3D √ sketching
√ mind maps
√ stories
√ retelling
� book making √ research
� speeches √ reading
� drama
√ activities
√ hands on
√ body language
√ crafts
√ drama
� mime � PE
Logical/Math Intrapersonal Interpersonal
√ timeline
√ coding
� geometry � measuring √ classifying
� money √ time
√ individual study
� journals √ reflection
√ self-esteem activities
√ individual reading
√ individual projects
√ cooperative learning
√ sharing
√ group work
√ discussion
√ peer editing
√ brainstorming
305
Time Lesson Plan - Learning Activities Resources
3 mins
Introduction.
Welcome students to class. Briefly discuss the problems written from the last lesson and mention
notable problems and problem posers. Give students a few minutes to read the comments I have written about
their problems.
Video cameras in place, all resources laid out. 3D puzzles laid out in order on tables behind the students’ desks.
10 mins
Talk to the students about visual problems such as quizzles. Ask students how these problems might rate with the three attributes of a “good” problem. Discuss which attribute they feel is particularly served well by the visual presentation.
Give students an opportunity to do 8 quizzles from the board. Discuss the answers and the strategies the students used to solve them.
Selection of quizzles.
20 mins
Direct students’ attention to the 3D, visual puzzles that are laid out on the desks behind them and the answer sheet stuck in their exercise books.
Ask students to move along the line of puzzles and record answers to them in their books. They may go back to any, do them in any order, or change their answer if they need to.
Student exercise books, answer sheets attached. Set of 16 x 3D visual puzzles.
20 mins
Ask students to stop work and return to their seats. Ask students to share their answers to each puzzle as I hold them up one
by one. Discuss what made the problem obvious or difficult, or what caught students’ attention and focus.
Ask students to write a final comment about their enjoyment, or otherwise, of this activity on the bottom of their answer sheets.
5 mins
Ask students to begin thinking about their own visual problem. Tell students they will each need to bring a constructed problem to next weeks lesson for everyone to share and attempt to solve.
Encourage students to talk to other students, their parents or friends in order to get some really interesting ideas that will satisfy the three attributes of a “good” problem.
2 mins
Thank all students for their contributions of posed problems and collect exercise books.
Remind students that they should all try to be on time for the session.
306
Some examples of quizzles:
Solution: ‘Once upon a time’
Solution: ‘A balanced meal’
Solution: ‘Makeup’
Once
____
Time
P
U
E
K
A
M
Meal
307
3-D V ISUAL PROBLEMS STUDENT ANSW ERS
1. ______________________________________________________________
2. ______________________________________________________________
3. ______________________________________________________________
4. ______________________________________________________________
5. ______________________________________________________________
6. ______________________________________________________________
7. ______________________________________________________________
8. ______________________________________________________________
9. ______________________________________________________________
10. _____________________________________________________
11. _____________________________________________________
12. _____________________________________________________
13. _____________________________________________________
14. _____________________________________________________
15. _____________________________________________________
16. _____________________________________________________
308
3-D Visual Problems Correct Answers
1. Tulip
2. Chain Letter
3. Apricot
4. Middle Man
5. Phone Number
6. Zebra Crossing
7. Address
8. Monkey
9. Cross-country
10. Six Feet Under
11. Centigrade
12. Sticks and stones can break my bones
13. Accountant
14. Time flies when you are having fun
15. Bibliography
16. Abandon Ship
309
APPENDIX K
TEACHING EXPERIMENT LESSON E IGHT
310
TEACHING EXPERIMENT LESSON 8 – YEAR 7
Date: November 26, 2007 Topic: Pictorial Problem Posing
Year Level: 7 Duration: 60 minutes
Rationale:
This lesson is the final lesson of the teaching experiment before the students undertake their post-test and final attitudinal survey. It will close with the presentation of awards for the Most Improved, Most Entertaining and Best Overall problem posers, as well as awards for the best 3D puzzles as judged by the students themselves. The students should be highly motivated during this lesson as a result of these culminating activities.
In previous weeks, students have only been able to share their problems by reading them out. Written problems are often best read and re-read rather than just heard. Some students have told me that they felt other students would have liked their problems more if they had more time to read them. With time being limited for each teaching episode, I needed to find an alternative format for problem posing that would allow all of the students to share each others problems within a teaching-episode (60 minutes) timeframe. Hence, last week I introduced visual problems to the students and gave them an opportunity to investigate what makes them interesting and challenging, but still do-able. The students enjoyed the session enormously (as can be seen from the comments in their work books). For example,
Oliver wrote, “It was fun and very challenging. It was a lot better than normal work.”
Hayley wrote, “I thought it was a great idea and it was so much fun.”
Felicia wrote, “I thought this lesson was so much fun! I think it was the best lesson so far! It was so cool!”
The students will have a chance to pose and present their own problems for each other during this lesson.
Key Investigations:
• Students will investigate if they can create problems for their peers that are interesting, challenging, and do-able.
• They will investigate other student’s 3D problems.
311
THE MULTIPLE INTELLIGENCES: CHECKLIST
Visual/Spatial Verbal/Linguistic Bodily/Kinaesthetic
� charts � graphs � diary √ visual metaphors
� 3D √ sketching
√ mind maps
√ stories
√ retelling
� book making � research � speeches √ reading
√ drama
√ activities
√ hands on
√ body language
√ crafts
√ drama
� mime � PE
Logical/Math Intrapersonal Interpersonal
√ timeline
√ coding
� geometry √ measuring
√ classifying
� money √ time
√ individual study
� journals √ reflection
√ self-esteem activities
√ individual reading
√ individual projects
√ cooperative learning
√ sharing
√ group work
√ discussion
√ peer editing
√ brainstorming
312
Time Lesson Plan - Learning Activities Resources
5 mins
Introduction.
Welcome students to class. (Exercise books to be laid out on front desk before students enter the room)
Ask students to place their 3D problem on top of one of the sequentially numbered pieces of paper that have been set out on the row of desks.
Settle students onto the chairs at the front of the room. Briefly remind the students of what is going to happen in this lesson
including the viewing of their 3D problems.
Video cameras in place and all resources laid out. Sheets with 1, 2, 3,…15 to be laid across 15 desks in the room ready for student puzzles.
15 mins
Remind the students about the three attributes of a “good” problem. Ask the students to consider this for each problem as they attempt to work out the answers of each others problems.
Hand out student’s exercise books and ask students to open to the answer sheet that has already been glued in and then to move to the puzzles and attempt to “solve” them. They may go back to any, do them in any order, or change their answer if they need to.
Student puzzles and student exercise books with answer sheets already attached.
15 mins
Ask students to stop work and return to their seats at the front of the room. Ask the students to turn their chairs around so they are facing the desks on which the puzzles are placed.
Ask students to share their answers to each puzzle as I hold the puzzles up one by one. Discuss what made the problem obvious or difficult or what caught students’ attention and focus. If the puzzle was not able to be solved, ask the author of the problem to share the answer with the group.
Ask students to comment on their thoughts about the problems; in particular, what made them interesting, challenging and do-able.
18 mins
Tell students that their puzzles are now going to be peer assessed. All students will leave the room and collect a set of 1, 2, and 3 cards from the desk outside of the classroom. Students will be asked to enter the classroom one by one to place their voting cards on the problems they thought were the best. 3 for the best problem, 2 for the next best problem and 1 for the third best problem.
Students remaining outside of the classroom will not be able to see on which puzzles the students place their cards as the blinds will be drawn in the classroom. A research assistant will remain in the room to ensure that no students remove/change other students’ cards. I will remain outside of the classroom with the students to ensure they remain orderly and quiet while the voting takes place inside the classroom.
When the voting is completed, all students will re-enter the room to witness the tallying. A student volunteer to write scores on the board.
Sets of three cards with 1, 2 and 3 written on them. One set for each student.
5 mins
Announce the “best”, second “best” and third “best” problems as reviewed by the student peers.
Announce the three major awards; Most Improved Problem Poser, Most Entertaining Problem Poser and Best Overall Problem Poser as reviewed by me and hand out certificates.
Certificates for student winners.
2 mins
Collect exercise books, thank students for their puzzles and input today. Have students help to quickly rearrange the room.
313
3-D Visual Problems Student Answers
1. ______________________________________________________
2. ______________________________________________________
3. ______________________________________________________
4. ______________________________________________________
5. ______________________________________________________
6. ______________________________________________________
7. ______________________________________________________
8. ______________________________________________________
9. ______________________________________________________
10. -______________________________________________________
11. _____________________________________________________
12. _____________________________________________________
13. _____________________________________________________
14. _____________________________________________________
15. _____________________________________________________
314
APPENDIX L
TEACHING EXPERIMENT LESSON N INE
315
Date: November 26, 2007 Topic: Student Survey and Post-testing
Year Level: 7 Duration: 60 minutes
Rationale:
The purpose of this lesson is to allow the students to complete a survey containing 4 attitudinal questions and then a problem-solving post-test, following the completed teaching experiment. Both the survey and the test are identical to the survey and test that the students completed in their first session at the start of the teaching experiment. The results from both of these items will allow the researcher to determine if there are noteworthy changes and/or emergent themes in student attitudes to problem solving or their problem-solving competence, as a result of the PPI.
Key Investigations:
• Have student attitudes and/or problem-solving competence changed as a result of the teaching experiment in which the students participated?
THE MULTIPLE INTELLIGENCES: CHECKLIST
Visual/Spatial Verbal/Linguistic Bodily/Kinaesthetic
� Charts � graphs � diary √ visual metaphors
√ 3D
√ sketching
√ mind maps
� stories � retelling � book making � research � speeches √ reading
� drama
� activities � hands on � body language � crafts � drama � mime � PE
Logical/Math Intrapersonal Interpersonal
√ timeline
√ coding
√ geometry
√ measuring
√ individual study
� journals √ reflection
� self-esteem activities √ individual reading
� cooperative learning � sharing � group work � discussion � peer editing � brainstorming
316
√ classifying
√ money
√ time
√ individual projects
Time Lesson Plan - Learning Activities Resources
15 mins
Welcome students back and tell them about the significance of this lesson.
Ensure students are well spread out around the room and that they each have a pen with which to write.
Hand out a survey sheet to each student and ask them to write their names on the top of the sheet and then complete the attitudinal surveys individually and in silence.
Collect the survey sheets.
One survey sheet for each participant. Spare pens.
40
Mins
Post-tests to be handed out face down while students remain in silence. Remind students about the administration process of the test. Ask the students to write their names on the top of the front page of the
test and turn the first page over. Administer the test according to the requirements specified in the POPS
Administrator’s Handbook. Collect the test booklets from the students.
A set of post-tests for the group and a POPS Administrator’s Handbook.
5 mins
Ask students to assist in returning the furniture in the room to the way it usually is.
Thank the students for their valued input, participation and enthusiasm throughout the teaching experiment and allow students to return to their classes.
317
APPENDIX M
PROFILES OF PROBLEM SOLVING ASSESSMENT INSTRUMENT
318
319
320
321
322
323
324
325
326
327
APPENDIX N
PROBLEM CRITERIA SHEET
328
PROBLEM CRITERIA SHEET
DATE :
Interest Factor Challenge Level “Do-ability” Comments
Self -
rating
□ Very entertaining,
original and everyone wanted to solve it (3 points)
□ At least half the
class seemed interested and wanted to solve it (2 points)
□ Not many people
seemed interested in solving my problem or it was not original (1 point)
□ Not too hard and not
too easy. More than 3 steps had to be taken in order to solve the problem. (3 points)
□ Some felt it was too
hard or too easy. At least 2 steps had to be taken in order to solve the problem. (2 points)
□ Very little challenge
for most Year 7 students. (1 point)
□ Problem was very
well presented and clear. All information required to solve the problem was provided and there was no ambiguity. (3 points)
□ Most students felt the
problem was clear and felt they could solve it without extra information or clarification (2 points)
□ The problem was
confusing and not many students felt confident they could do it without more help or information. (1 point)
Self-rating: /9 points
Teacher
-rating
□ Very entertaining,
original and everyone wanted to solve it (3 points)
□ At least half the
class seemed interested and wanted to solve it (2 points)
□ Not many people
seemed interested in solving my problem or it was not original (1 point)
□ Not too hard and not
too easy. More than 3 steps had to be taken in order to solve the problem. (3 points)
□ Some felt it was too
hard or too easy. At least 2 steps had to be taken in order to solve the problem. (2 points)
□ Very little challenge
for most Year 7 students. (1 point)
□ Problem was very
well presented and clear. All information required to solve the problem was provided and there was no ambiguity. (3 points)
□ Most students felt the
problem was clear and felt they could solve it without extra information or clarification (2 points)
□ The problem was
confusing and not many students felt confident they could do it without more help or information. (1 point)
Teacher-rating: /9 points
329
APPENDIX O
PARTICIPANT PSEUDONYM CODE TO PSEUDONYM NAME CONVERSION FOR
COMPARISON GROUP
330
COMPARISON GROUP
Pseudonym Code Pseudonym Name
ao6 Adam
b06 Ben
c06 Clare
d06 Diane
e06 Ellen
f06 Fay
g06 Gayle
h06 Helen
i06 Imogen
j06 Jack
k06 Kyle
l06 Laura
m06 Matt
n06 Nola
o06 Olive
p06 Penny
331
APPENDIX P
PARTICIPANT PSEUDONYM CODE TO PSEUDONYM NAME CONVERSION FOR
2007 INTERVENTION GROUP
332
2007 INTERVENTION GROUP
* Please note that data from students I07, M07, and Q07 were excluded from this study as these students
were absent for more than two of the seven teaching episodes.
Pseudonym Code Pseudonym Name
A07 Andrew
B07 Blair
C07 Courtney
D07 Danielle
E07 Ethan
F07 Felicia
G07 Georgia
H07 Hayley
J07 Joanne
K07 Kelly
L07 Leah
N07 Nicole
O07 Oliver
P07 Paul
R07 Rodney
333
APPENDIX Q
MARKING SCHEME FOR THE PROFILES OF PROBLEM SOLVING TEST
334
335
336
337
338
339