The Nature and Development of Processes in Mathematical Investigation Yeo Boon Wooi Joseph ...

567 Pages·2014·3.87 MB·English

The Nature and Development of Processes
in Mathematical Investigation

Yeo Boon Wooi Joseph

A thesis submitted to the
National Institute of Education,
Nanyang Technological University
in fulfilment of the requirement for the degree of
Doctor of Philosophy

2013
“Investigation [is] just a vehicle for other learning… This other
learning might be seen as learning to be mathematical.”
Barbara Jaworski (1994, p. 4)

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TABLE OF CONTENTS

ACKNOWLEDGEMENTS .......................................................................................xxv 
ABSTRACT .......................................................................................................... xxvi

PART ONE: CURRENT STATE OF RESEARCH ON MATHEMATICAL
INVESTIGATION

CHAPTER 1: INTRODUCTION TO THE PRESENT RESEARCH STUDY ............2 
1.1  Background of the Research ..................................................................2 
1.2  Rationale for Carrying Out the Present Research ..................................9 
1.3  Statement of the Research Problem .......................................................9 
1.4  Research Questions ..............................................................................10 
1.5  Significance of the Research Study .....................................................11 
1.6  Concluding Remarks for This Chapter ................................................12

CHAPTER 2: REVIEW OF RELEVANT LITERATURE ........................................13 
2.1  Literature Review on Mathematical Investigation ...............................14 
2.1.1  Mathematical Problems and Investigative Tasks .....................14 
2.1.2  Problem Solving and Mathematical Investigation as Processes
..................................................................................................21 
2.1.3  Problem Solving and Mathematical Investigation as Activities
..................................................................................................31 
2.1.4  Two Types of Investigative Tasks: Type A and Type B .........36 
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2.1.5  Empirical Studies on Mathematical Investigation ...................41 
2.1.6  Summary of Literature Review on Mathematical Investigation
..................................................................................................48 
2.2  Literature Review on Cognitive Processes ..........................................48 
2.2.1  Thinking Processes ..................................................................49 
2.2.2  Models of Cognitive Processes in Mathematical Investigation
..................................................................................................52 
2.2.3  Processes and Outcomes in Mathematical Investigation .........61 
(a)  Stage 1: Understanding the Task .............................. 62 
(b)  Stage 2: Problem Posing ........................................... 63 
(c)  Stage 3: Specialising and Using Other Heuristics .... 68 
(d)  Stage 4: Conjecturing ................................................ 71 
(e)  Stage 5: Justifying ..................................................... 72 
(f)  Stage 6: Generalising ................................................ 75 
(g)  Stage 7: Checking ..................................................... 77 
(h)  Stage 8: Extension..................................................... 78 
2.2.4  Empirical Studies on Cognitive Processes ...............................81 
2.2.5  Summary of Literature Review on Cognitive Processes .........88 
2.3  Literature Review on Metacognitive Processes ...................................89 
2.3.1  Nature of Metacognition ..........................................................90 
2.3.2  Differences between Metacognitive and Cognitive Processes
..................................................................................................94 
2.3.3  Models of Metacognitive Processes .........................................95 
2.3.4  Empirical Studies on Metacognitive Processes .......................99 
2.3.5  Summary of Literature Review on Metacognition ................104 
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2.4  Literature Review on Research Methodologies .................................105 
2.4.1  Methods of Data Collection for Processes .............................106 
2.4.2  Thinking-Aloud Method ........................................................109 
2.4.3  Methods of Data Analysis for Processes ...............................112 
2.4.4  Method for Development of Processes: Teaching Experiment
................................................................................................114 
2.4.5  Summary of Literature Review on Research Methodologies 118 
2.5  Concluding Remarks for This Chapter ..............................................119

PART TWO: RESEARCH METHODOLOGY AND DEVELOPMENT OF
DATA ANALYSIS TOOLS

CHAPTER 3: RESEARCH METHODOLOGY .......................................................121 
3.1  Theoretical Background .....................................................................121 
3.2  Theoretical Mathematical Investigation Models ...............................123 
3.2.1  Theoretical Investigation Model for Cognitive Processes .....123 
3.2.2  Theoretical Investigation Model for Metacognitive Processes
................................................................................................130 
3.3  Initial Exploratory Study to Set Direction of Research for Present
Study ..................................................................................................133 
3.4  Research Questions ............................................................................137 
3.5  Definitions of Terms ..........................................................................138 
3.6  Research Design .................................................................................140 
3.6.1  Research Framework .............................................................140 
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3.6.2  Sample for Main Study ..........................................................142 
3.6.3  Teaching Experiment .............................................................145 
3.6.4  Selection of Investigative Tasks for Lessons, Pretest and
Posttest ...................................................................................150 
3.6.5  Data Collection ......................................................................154 
3.7  Pilot Study ..........................................................................................156 
3.8  Main Study .........................................................................................157 
3.9  Quality of Present Study ....................................................................158 
3.10  Concluding Remarks for This Chapter ..............................................160

CHAPTER 4: DEVELOPMENT OF CODING SCHEME ......................................161 
4.1  Transcribing of Thinking-Aloud Protocols ........................................161 
4.2  Purpose of Coding Scheme ................................................................165 
4.3  Phase 1: Development of Initial Coding Scheme Based on Theory
............................................................................................................166 
4.4  Phase 2: Development of Revised Coding Scheme Based on Empirical
Data ....................................................................................................169 
4.4.1  Coding of Transcripts of Thinking-Aloud Protocols .............169 
4.4.2  Some Issues with Initial Coding Scheme ...............................172 
4.4.3  Revised Coding Scheme ........................................................178 
4.4.4  Parsing of Protocols into Stages and Episodes ......................182 
4.5  Phase 3: Refinement of Revised Coding Scheme Based on More
Empirical Data ...................................................................................188 
4.6  Phase 4: Development of Final Coding Scheme Based on New Data
from Posttests in Main Study .............................................................188 
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4.6.1  Fine-tuning the Revised Coding Scheme ...............................189 
4.6.2  Final Coding Scheme .............................................................190 
4.7  Phase 5: Inter-Coder Reliability Test for Final Coding Scheme .......200 
4.8  Concluding Remarks for This Chapter ..............................................205

CHAPTER 5: DEVELOPMENT OF DATA ANALYSIS TOOLS .........................206 
5.1  Protocol Analysis Method..................................................................206 
5.2  Investigation Pathway Diagram and Investigation Timeline
Representation....................................................................................207 
5.3  Summary Tables of Processes and Outcomes ...................................213 
5.4  Inter-Coder Reliability Tests for Quality of Problems Posed and
Conjectures Formulated .....................................................................215 
5.5  Concluding Remarks for This Chapter ..............................................218 

PART THREE: DATA ANALYSIS AND FINDINGS

CHAPTER 6: DATA ANALYSIS OF MATHEMATICAL INVESTIGATION
PATHWAYS ......................................................................................220 
6.1  The First Research Question ..............................................................220 
6.2  Investigation Pathways for Type A Tasks .........................................221 
6.3  Investigation Pathways for Type B Tasks .........................................246 
6.4  Investigation Pathways Across Both Types of Tasks ........................271 
6.5  Summary of Answer to Research Question 1 ....................................273 
6.6  Concluding Remarks for This Chapter ..............................................274 
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CHAPTER 7: DATA ANALYSIS OF MATHEMATICAL INVESTIGATION
PROCESSES AND OUTCOMES .....................................................275 
7.1  The Second Research Question .........................................................275 
7.2  Mathematical Investigation Processes and Outcomes for Type A Tasks
............................................................................................................276 
7.2.1  Understanding the Task (Stage 1) ..........................................278 
7.2.2  Problem Posing (Stage 2).......................................................281 
7.2.3  Specialising (Stage 3) ............................................................284 
7.2.4  Conjecturing (Stage 4) ...........................................................291 
7.2.5  Justifying and Generalising (Stages 5 and 6) .........................308 
7.2.6  Checking (Stage 7) .................................................................312 
7.3  Mathematical Investigation Processes and Outcomes for Type B Tasks
............................................................................................................320 
7.3.1  Understanding the Task (Stage 1) ..........................................322 
7.3.2  Problem Posing (Stage 2).......................................................325 
7.3.3  Using Other Heuristics (Stage 3) ...........................................333 
7.3.4  Conjecturing (Stage 4) ...........................................................341 
7.3.5  Justifying and Generalising (Stages 5 and 6) .........................343 
7.3.6  Extension (Stage 8) ................................................................345 
7.3.7  Specialising and Using Other Heuristics (Stage 3) for Extension
of Task ...................................................................................352 
7.3.8  Conjecturing (Stage 4) for Extension of Task .......................363 
7.3.9  Justifying and Generalising (Stages 5 and 6) for Extension of
Task ........................................................................................367 
7.3.10  Checking (Stage 7) .................................................................368 
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7.4  Summary of Answer to Research Question 2 ....................................373 
7.4.1  Processes and Outcomes of Mathematical Investigation .......373 
7.4.2  Validation and Refinement of Mathematical Investigation
Models....................................................................................381 
7.5  Concluding Remarks for This Chapter ..............................................388

CHAPTER 8: DATA ANALYSIS OF DEVELOPMENT OF MATHEMATICAL
INVESTIGATION PROCESSES ......................................................389 
8.1  The Third Research Question ............................................................389 
8.2  Proficiency in Mathematical Investigation ........................................390 
8.2.1  Investigation Scoring Rubric (ISR) .......................................390 
8.2.2  Actual Scoring for Sample Students ......................................395 
8.2.3  Analysis of the Pretest and Posttest Scores ............................399 
8.3  Development of Investigation Processes ...........................................406 
8.3.1  Understanding the Task (Stage 1) ..........................................407 
8.3.2  Problem Posing and Extension (Stages 2 and 8) ...................412 
8.3.3  Specialising and Using Other Heuristics (Stage 3) ................419 
8.3.4  Conjecturing (Stage 4) ...........................................................429 
8.3.5  Justifying and Generalising (Stages 5 and 6) .........................435 
8.3.6  Checking (Stage 7) .................................................................443 
8.4  Summary of Answer to Research Question 3 ....................................449 
8.5  Concluding Remarks for This Chapter ..............................................453

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CHAPTER 9: CONCLUSION AND IMPLICATIONS ...........................................454 
9.1  Main Contributions of Present Research ...........................................454 
9.1.1  Relationship between Investigation and Problem Solving ....454 
9.1.2  Mathematical Investigation Models, Coding Scheme and
Scoring Rubric .......................................................................455 
9.1.3  Effect of Processes on Outcomes in Mathematical Investigation
................................................................................................457 
9.1.4  Development of Processes in Mathematical Investigation ....460 
9.2  Limitations of Present Study ..............................................................462 
9.3  Implications and Recommendations for Teaching and Research ......463 
9.3.1  Implications and Recommendations for Teaching ................463 
9.3.2  Implications and Recommendations for Research .................468 
9.4  Epilogue .............................................................................................470

REFERENCES .........................................................................................................471

APPENDIX A:  Initial Small Study Test Instrument ...............................................479 
APPENDIX B:  Exploratory Study Test Instrument ...............................................480 
APPENDIX C:  Outlines of Familiarisation Lesson and Developing Lessons .......481 
APPENDIX D:  Investigative Tasks for Lessons, Pretest and Posttest ...................483 
APPENDIX E:  Task Analyses for the Pretest and Posttest Items ..........................486 
APPENDIX F:  Instructions for Thinking Aloud ....................................................511 
APPENDIX G:  Instructions for Invigilator during Pretest and Posttest .................512 
APPENDIX H:  Checklist on Investigation Processes .............................................514 
APPENDIX I:  Sample Coded Transcript for Posttest Task 1 (Kaprekar) .............516 
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