How elementary students learn to mathematically analyze word problems

221 Pages·2014·3.45 MB·English

How elementary students learn to
mathematically analyze word problems:
The case of addition and subtraction
ELENA POLOTSKAIA (ARKHIPOVA), DEPARTMENT OF INTEGRATED STUDIES IN
EDUCATION, FACULTY OF EDUCATION, MCGILL UNIVERSITY, MONTREAL
June, 2014
A thesis submitted to McGill University in partial fulfillment of the requirements of the degree of Doctor
of Philosophy, Mathematics Education

The complexity “is a hallmark of educational settings.”
(Cobb, Confrey, Lehrer, & Schauble, 2003, p. 9)

© Elena Arkhipova, 2014
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TABLE OF CONTENTS
Abstract .......................................................................................................................................... vii
Résumé ........................................................................................................................................... vii
Acknowledgements ......................................................................................................................... ix
Preface .............................................................................................................................................. x
Chapter 1 Introduction and Problem ...............................................................................................1
1.1 Societal and practical need .....................................................................................................1
1.2 Need for research ...................................................................................................................2
1.3 Context of the study ...............................................................................................................4
1.4 Research objectives ................................................................................................................5
1.5 The terminology used in the study .........................................................................................5
Simple additive word problems ........................................................................................5
Mathematical expression .................................................................................................6
Mathematical structure of the problem ..........................................................................7
Mathematical analysis ......................................................................................................8
Mathematizing a problem ................................................................................................8
Modeling ...........................................................................................................................8
Representation .................................................................................................................8
Schema .............................................................................................................................9
Chapter 2 Theoretical exploration................................................................................................. 10
2.1 Two paradigms of additive problem solving ....................................................................... 10
2.2 Epistemology of additive problems ..................................................................................... 11
2.2.1 Classifications of word problems .................................................................................. 12
Classification 1 ................................................................................................................... 13
Classification 2 ................................................................................................................... 13
Classification 3 ................................................................................................................... 13
Classification 4 ................................................................................................................... 14
Classifications of word problems through the lens of the Relational Paradigm ............... 14
2.2.2 Position of the unknown and structure inversion ........................................................ 14
2.2.3 Mathematical structure of a problem and cognitive load ........................................... 16
Simple combine problems ................................................................................................. 17
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Simple Compare problems ................................................................................................ 18
Composition-of-two-transformations ............................................................................... 19
2.2.4 Summary of problems’ classifications .......................................................................... 21
2.3 Word problem as a phenomenon related to natural language .......................................... 29
2.3.1 Construction/integration model................................................................................... 30
2.3.2 Stages of development and knowledge schemas ........................................................ 31
Nesher and colleagues’ model .......................................................................................... 31
Riley and Greeno’s model.................................................................................................. 32
Okamoto and Case’s model ............................................................................................... 32
Fuson and colleagues’ model ............................................................................................ 33
Summary of stages and possible schemas ........................................................................ 33
2.3.3 Text and schema interplay ........................................................................................... 34
Sequential and holistic relational schemas ....................................................................... 34
Consistent and inconsistent problems from the reading point of view ............................ 37
2.4 Problems, their wording and knowledge development ...................................................... 39
2.5 Models of knowledge development in relation to additive problems ................................ 40
2.5.1 Current model .............................................................................................................. 40
2.5.2 Davydov’s model .......................................................................................................... 41
2.5.3 Research in neuro-education ....................................................................................... 43
2.5.4 Equilibrated Development model ................................................................................ 44
2.6 Teaching approaches ........................................................................................................... 45
2.6.1 Existing approaches ...................................................................................................... 45
2.6.2 Contemporary approach in Quebec ............................................................................. 46
2.6.3 Particular teaching strategies ....................................................................................... 47
Modeling as a teaching approach ..................................................................................... 47
Realistic mathematics ........................................................................................................ 48
Manipulatives and mathematical expressions .................................................................. 48
Schematic representations ................................................................................................ 49
Didactic management and meta-cognitive regulation ...................................................... 50
Continuous and discrete reasoning about numbers ......................................................... 50
2.6.4 Summary of teaching approaches ................................................................................ 51
2.6.5 Equilibrated Development Approach ........................................................................... 51
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2.7 Models for interpreting students’ production in problem solving ...................................... 54
2.7.1 Summary of interpretative models .............................................................................. 56
2.8 Restatement of the research questions .............................................................................. 56
Chapter 3 Research methods ........................................................................................................ 59
3.1 Research methodology ........................................................................................................ 59
3.2 Context ................................................................................................................................ 61
3.3 Data collection methods...................................................................................................... 62
3.3.1 Participant selection ..................................................................................................... 62
3.3.2 Instrument: problem-solving tasks ............................................................................... 63
3.3.3 Individual interviews with students ............................................................................. 70
3.3.4 Collecting data about teaching ..................................................................................... 71
3.3.5 Summary of the data collection ................................................................................... 72
3.4 Data analysis methods ......................................................................................................... 73
3.5 Limitations of applied methods ........................................................................................... 78
Chapter 4: Results.......................................................................................................................... 80
4.1 Phase 1: Ways of mathematizing ........................................................................................ 80
4.1.1 First session, November ............................................................................................... 81
4.1.1.1 Mental representation .......................................................................................... 81
Sequential mental representation ................................................................................ 81
Structure substitution or shift of meaning .................................................................... 83
Holistic representation .................................................................................................. 84
Complex structure and its mental representation ........................................................ 84
4.1.1.2 Students’ presumptions about the task ................................................................ 86
Draw-and-count presumption ....................................................................................... 86
Numbers-are-important presumption .......................................................................... 87
Grouping presumption .................................................................................................. 88
4.1.1.3 Mathematizing ...................................................................................................... 88
Mimicking ...................................................................................................................... 88
Tacit mental model ........................................................................................................ 89
Mathematical equation .................................................................................................. 89
Objects model ................................................................................................................ 89
4.1.1.4 Summary of the first session ................................................................................. 90
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Students’ difficulties ...................................................................................................... 91
4.1.2 Second session, January ............................................................................................... 92
4.1.2.1 Mental representation .......................................................................................... 93
Sequential representation ............................................................................................. 93
Volatile mental representation ..................................................................................... 94
Holistic representation .................................................................................................. 95
Emergent mental representation .................................................................................... 96
Misinterpretations (structure substitution) .................................................................. 99
No solution .................................................................................................................... 99
4.1.2.2 Presumptions about the task ................................................................................ 99
4.1.2.3 Mathematizing ..................................................................................................... 100
Incorrect use of keywords ........................................................................................... 100
Tacit holistic model ..................................................................................................... 101
AA diagrams ................................................................................................................ 101
4.1.2.4 Summary of the second session ........................................................................... 102
Students’ difficulties .................................................................................................... 103
4.1.3 Third session, March................................................................................................... 104
4.1.3.1 Mental representation .......................................................................................... 105
Mental holistic representation .................................................................................... 105
Volatile mental representation ................................................................................... 109
Emergent mental representations .............................................................................. 111
4.1.3.2 Students’ presumptions about the task .............................................................. 112
4.1.3.3 Mathematizing ..................................................................................................... 113
Mimicking ................................................................................................................... 113
Using diagrams as templates ....................................................................................... 113
Mental modeling .......................................................................................................... 115
Incorrect keyword use ................................................................................................. 116
AA Diagram modeling ................................................................................................ 116
Use of multiple ways of mathematizing ...................................................................... 116
4.1.3.4 Summary of the third interview session.............................................................. 116
Students’ difficulties .................................................................................................... 118
4.1.4 Fourth session, May ................................................................................................... 119
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4.1.4.1 Mental representation .......................................................................................... 119
Mental holistic representation .................................................................................... 119
Emergent mental representation ................................................................................ 120
Volatile mental representation ................................................................................... 122
Structure substitution.................................................................................................. 122
4.1.4.2 Students’ presumptions about the task .............................................................. 123
4.1.4.3 Mathematizing ..................................................................................................... 124
AA diagram as template .............................................................................................. 124
AA diagram as model .................................................................................................. 126
Mental holistic model .................................................................................................. 128
4.1.4.4 Summary of the fourth interview session ........................................................... 129
Students’ difficulties .................................................................................................... 130
4.1.5 Summary of the interviews ........................................................................................ 131
4.1.5.1 Mental representations and presumptions ........................................................ 131
4.1.5.2 Mathematizing .................................................................................................... 132
4.1.5.3 Overview .............................................................................................................. 134
4.2 Phase 2: Learning .............................................................................................................. 134
4.2.1 Dynamics of knowledge development ....................................................................... 135
4.2.2 Teaching implemented ............................................................................................... 138
4.2.3 Links between learning and teaching ......................................................................... 141
4.2.4 Summary..................................................................................................................... 149
4.3 Summary of Chapter 4 ....................................................................................................... 150
Chapter 5 Discussion ................................................................................................................... 151
5.1 Answering Question 1 ....................................................................................................... 151
5.1.1 Mimicking ................................................................................................................... 151
5.1.2 Use of tacit mental model .......................................................................................... 153
5.1.3 Using diagram as template ......................................................................................... 153
5.1.4 Multiple uncoordinated means .................................................................................. 154
5.1.5 AA diagram as model .................................................................................................. 156
5.1.6 Theoretical outcomes ................................................................................................. 156
5.1.6.1 Factors affecting students’ production in word problem solving ....................... 156
5.1.6.2 Students’ mathematizing processes .................................................................... 157
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5.2 Answering Question 2 ....................................................................................................... 162
5.2.1 Dynamics of students’ reasoning development ......................................................... 162
5.2.2 The dynamic of the “indifferent” group ..................................................................... 164
5.2.3 Causes of the particular dynamics observed in the study .......................................... 164
5.2.4 Theoretical outcome .................................................................................................. 166
Chapter 6 Conclusions ................................................................................................................. 169
6.1 How students mathematize problems .............................................................................. 170
6.2 The Equilibrated Development Approach as the positive cause of knowledge
development in students......................................................................................................... 171
6.3 Various learning challenges and their causes ................................................................... 172
6.4 Theoretical value of the study ........................................................................................... 174
6.5 Value of the study to the larger experiment ..................................................................... 174
6.6 Limitations of the study ..................................................................................................... 175
6.7 Perspectives on teaching and future research .................................................................. 175
Bibliography ................................................................................................................................. 176
Appendix 1 Tables ....................................................................................................................... 190
Appendix 2 Written problem-solving test ................................................................................... 208
List of tables ................................................................................................................................. 209
List of figures ............................................................................................................................... 209
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Abstract
Mathematical problem solving, and more specifically the ability to
mathematically analyze and model a situation, is one of the most important aspects of
teaching and learning mathematics in school. Today, researchers agree that the problem-
solving and mathematizing phenomena are extremely complex and that research is
needed to better understand the cognitive processes involved at a phenomenological
level. The lack of nuanced understanding of the ways of reasoning students might employ
to analyze and model a problem prevents teachers from effectively meeting their needs.
Within the context of a larger study on the development of mathematical
reasoning in early grades of elementary school, I studied how grade two elementary
school students solve additive problems to answer the following questions:
1. What kind of mathematizing do students use to solving additive word
problems?
2. What are the relationships between the instruction implemented and
students’ development of mathematizing processes?
Applying the grounded theory methodology, I analyzed multiple observations of
students solving additive problems throughout one school year. I suggest models for six
strategies of mathematizing, which I describe in detail. I describe the dynamics of change
in the learners’ ways of reasoning and the relationships between this change and the
teaching implemented.
Résumé
La résolution de problèmes mathématiques, et plus particulièrement la capacité
d'analyser et de modéliser mathématiquement une situation, est l'un des aspects les plus
importants de l'enseignement et l'apprentissage des mathématiques à l'école. De nos jours,
les chercheurs s'accordent à dire que les phénomènes de résolution de problèmes et de la
mathématique d’une situation sont extrêmement complexes et que la recherche est
nécessaire pour mieux comprendre au niveau phénoménologique des processus cognitifs
impliqués. Le manque de compréhension nuancée du raisonnement que des apprenants
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pourraient employer pour analyser et modéliser un problème empêche les enseignants de
répondre à leurs besoins de façon efficace.
Dans le contexte d'une plus grande étude concernant le développement de
raisonnement mathématique dans les premières années de l'école primaire, j'ai étudié les
élèves de deuxième année du primaire en train de résoudre des problèmes additifs pour
répondre aux questions suivantes:
1. Quels sont les moyens de mathématisation utilisent les élèves pour
résoudre des problèmes écrits ayant des structures additives?
2. Quel est le rapport entre l'enseignement mis en œuvre et le développement
des processus de mathématisation des élèves?
En adoptant la méthodologie de la théorisation ancrée, j'ai observé et analysé des
élèves à résoudre des problèmes additifs au cours d'une année scolaire. J'ai modélisé six
stratégies de mathématisation, que j'ai décrites en détail. J'ai décrit la dynamique du
changement des modes de raisonnement chez les apprenants, ainsi que les relations entre
ce changement et l'enseignement mis en œuvre.
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Acknowledgements
I would like to express my greatest appreciation for the encouragement and support that
my supervisors, Professor Annie Savard and Professor Viktor Freiman, have given me all
these years. I would like to thank the members of my PhD committee, Maria Mellone and
Kara Jackson, for their care throughout my studies.
I am grateful to my family members for their patience and unwavering support.
I would like to specially thank the school board and each of the 12 students who
participated in my study.
I acknowledge the financial support provided by the Quebec Ministry of Education,
Leisure, and Sport.
My special thanks go to Anna-Maria D’Aviro. Without her help, this journey would have
never started.
Thank you to Alana Duncan for the linguistic revision of this study.