Young Children's Spatial Structuring Ability and Emerging Number Sense

360 Pages·2009·5.61 MB·English

Young Children’s
Spatial Structuring Ability
and Emerging Number Sense
Nes, Fenna van
Young Children’s Spatial Structuring Ability and Emerging Number Sense / F.T. van Nes-
Utrecht: Freudenthal Institute for Science and Mathematics Education
Dissertation Utrecht University - with references - Met een samenvatting in het Neder-
lands
ISBN 978-90-73346-65-9
Subject headings: spatial sense, number sense, spatial structure, kindergarten, counting,
subitizing, patterning, realistic mathematics education, design research, learning ecology
Cover: Communicatie & Vormgeving, Faculteit Bètawetenschappen, Universiteit Utrecht
Press: All Print, Utrecht
© 2009 FIsme, Universiteit Utrecht
Niets uit deze uitgave mag worden verveelvoudigd en/of openbaar gemaakt door middel van druk,
microfilm of op welke andere wijze ook zonder voorafgaande schriftelijke toestemming van de hou-
der van het copyright.
Young Children’s
Spatial Structuring Ability
and Emerging Number Sense
Ruimtelijk Structureervermogen
en Ontluikend Getalbegrip van Jonge Kinderen
(met een samenvatting in het Nederlands)
Proefschrift
ter verkrijging van de graad van doctor
aan de Universtiteit Utrecht,
op gezag van de rector magnificus, prof. dr. J. C. Stoof,
ingevolge het besluit van het college voor promoties
in het openbaar te verdedigen
op dinsdag, 1 september 2009 des middags te 2:30 uur
door
Fenna Titia van Nes
geboren 31 december 1982 te Jette, België
Promotoren: Prof. dr. J. de Lange
Prof. dr. E.H.F. de Haan
Co-promotoren: Dr. L.M. Doorman
Dr. H.A.A. van Eerde
Dr. J. Nelissen
This dissertation was supported by funding from the Netherlands Organization for Science
Research (NWO) under grant number 051.04.050.
For Koen

Preface
In the final stages of completing this thesis, I began to realize that the four-year period of
working on the MENS research project was very quickly coming to an end. That realization
spurred me on to reflect on this period and its “developmental trajectory”, particularly re-
garding all the people who supported me along the way.
After graduating from university and just before I traveled to Canada, a vacancy at the
Freudenthal Institute caught my attention. It described a Ph.D. position in an interdiscipli-
nary project that combines mathematics education and neuroscientific research. I became
excited about the prospect of contributing to such a unique project; it would relate to my
background in Cognitive Psychology, my interests in mathematics, and my experiences
with working with young children. I looked forward to exploring ways of helping others
learn through better (mathematics) education. I was also attracted to the opportunity of
supplementing my background in traditional, quantitative research with insights into qual-
itative research. As such, the project would offer me learning opportunities for becoming
a competent and a more all-round researcher. It all came together when I heard that I could
start the project as soon as I returned from Canada, in September 2005. Exactly four years
and a few hours later, I completed my work on the project with this thesis.
In light of this background, I would first like to express my gratitude to Prof. Jan de Lange
and Prof. Edward de Haan who were head-supervisors of the project. I greatly appreciate
how, from the very first contacts, Jan de Lange seemed to have an unconditional trust in
my competencies. He warned me that I had to be able to work independently, and in retro-
spect I am very grateful for his way of throwing me in at the deep end: it resulted in a thesis
that in its entirety is the product of my development as a researcher.
This development would not have come as far, however, without the dedicated support that
I felt I could always count on from Michiel Doorman, Dolly van Eerde, and Jo Nelissen.
Their patience, expertise, and committed role as supervisors guided me along the ups and
downs of the research trajectory. Their encouragement and moral support was an enor-
mous driving force. I thank each of my supervisors for all their help and hope they can
catch up on their free time.
All along the data collection phase of the research, I realized how fortunate I was to be
working with the staff, children and parents of the Jenaplanschool Cleophas in Utrecht.
From the very beginning, the staff and parents acknowledged the importance of the re-
search and went out of their way to help me schedule and collect the data. I greatly admire
the staff’s energy and passion for teaching. The repeated warm welcomes at school made
me even more motivated to contribute to ways of helping the children learn and supporting
the teachers in improving their teaching practices.
Mathematics education research at the Freudenthal Institute was a new field of research to
me when I started working on this project. I thank my colleagues at the institute who, each
in their own way, taught me about Realistic Mathematics Education and procedures for
performing design research. Many thanks also to all those people who made it possible for
me to present and discuss my work both at and outside of the institute. Coming into contact
with people from different backgrounds in different settings gave an extra dimension to the
issues that I encountered in mathematics education research.
I had a lot of help finishing the manuscript. In particular, I thank Han Hermsen for his help
in learning to work with Adobe FrameMaker, Nathalie Kuijpers for editing the book, Bas
Holleman for his advice on the screenshots and pictures, and Reyndert Guiljam for his il-
lustrations. Betty Heijman was a great support in finalizing the lay-out and organizing the
publication of the book.
I would like to thank the other “young” researchers (Klasje008) at the institute for their
companionship in working on our projects. During countless lunchtime walks and tea
talks, they encouraged me and brightened up the more demanding times. I wish them all
good luck and enjoyment in completing their research. Thank you, Sylvia van den Boo-
gaard and Iris Verbruggen, for also reminding me that there can be more to a day than
working on a Ph.D. thesis.
This brings me to the people whom I hold most dear to me, my family and friends. I put
them in the spotlight for their endless “behind the scenes” encouragement and enthusiasm.
Koen, I dedicate this book to you for your unconditional support, your enlightening per-
spectives, and, above all, your patience.
Table of Contents
1 Introduction
1.1 Young children, young scientists 11
1.2 Young children’s spatial structuring ability 12
1.3 Structure of the book 14
2 Theoretical Background and Research Questions
2.1 Number sense 17
2.2 Spatial sense 19
2.3 Spatial structures 22
2.4 Motivation for and purpose of the research 27
2.5 Context of the research 29
2.6 The research questions 35
3 Methodology
3.1 Design research 37
3.2 Setting 42
3.3 Procedure 45
3.4 Data analysis 51
3.5 Research validity and reliability 56
4 Developing a Conceptual Schema, the Interview Tasks,
and the Strategy Inventory
4.1 Relating spatial sense and number sense through spatial structure 61
4.2 The interview tasks 63
4.3 The strategy inventory 73
4.4 The final conceptual schema 84
5 The Strategy Inventory as a Foundation for the
Hypothetical Learning Trajectory
5.1 Condensing the strategy scores 90
5.2 Four phases in spatial structuring ability 93
5.3 Relating four phases in a developmental trajectory 97
5.4 Quantitative outcomes of the pre-interviews 100
6 Refining the HLT and Developing Classroom Instruction Activities
6.1 The global HLT and learning goals 103
6.2 Identifying learning moments and defining corresponding
classroom instruction activities 105
6.3 Optimizing the HLT and instructional sequence for
Round 1 of the instruction experiment 118
6.4 The final instructional sequence 122
7 Analysis of Round 1 of the Instruction Experiment
7.1 Reinterpreting micro and macro cycles in design research 131
7.2 The classroom social norms 132
7.3 Key observations from Round 1 133
7.4 RME in a Kindergarten setting 157
7.5 An overarching context 160
7.6 Improvements to the instruction activities for Round 2 162
8 Analysis of Round 2 of the Instruction Experiment
8.1 Retrospective analyses of the HLT 172
8.2 Unexpected issues in learning 189
8.3 The pro-active role of the teacher 200
8.4 General retrospective analysis of the instruction experiment 204
9 The Post-interview and Teacher Evaluations
9.1 Quantitative comparison of the post-interview to
the pre-interview and the LVS scores 207
9.2 Qualitative analysis of the post-interviews 212
9.3 Conclusions from the post-interviews 217
9.4 Teacher evaluations 218
9.5 Summary 225
10 Conclusions, Discussion, and Future Research
10.1 Research question 1: Early spatial structuring ability 227
10.2 Research question 2: Developing a local instruction theory 231
10.3 Discussion 237
10.4 Suggestions for future research 248
References 251
Summary 263
Samenvatting 277
Appendices 293
Curriculum Vitae 353
Index 355