types and levels of data arrangement and representation in statistics as modeled by grade 4

306 Pages·2006·8.01 MB·English

TYPES AND LEVELS OF DATA ARRANGEMENT AND
REPRESENTATION IN STATISTICS AS MODELED BY
GRADE 4 – 7 LEARNERS

by

HELENA MARGARETHA WESSELS

submitted in accordance with the requirements
for the degree of

DOCTOR OF EDUCATION

in the subject

DIDACTICS

at the

UNIVERSITY OF SOUTH AFRICA

PROMOTER: PROF HD NIEUWOUDT

JOINT PROMOTER: PROF DCJ WESSELS

FEBRUARY 2006
ii

Student Number: 449 364 8

I declare that

TYPES AND LEVELS OF DATA ARRANGEMENT AND
REPRESENTATION IN STATISTICS AS MODELED BY GRADE
4 – 7 LEARNERS

is my own work and that all the sources that I have used or quoted have been indicated
and acknowledged by means of complete references.

__________________ _________________
Helena M Wessels Date
iii
SUMMARY

The crucial role of representation in mathematical and statistical modeling and problem
solving as evident in learners’ arrangement and representation of statistical data were
investigated with focus points data arrangement, data representation and statistical
thinking levels. The representation tasks required learners to arrange and represent
data through modeling, focusing on spontaneous representations. Successful
transnumeration determine the ultimate success of a representation and the ability to
organise data is regarded as critical. Arrangement types increased in sophistication with
increased grade level and the hierarchical nature of arrangement types became
apparent when regarded in the context of an adapted SOLO Taxonomy framework. A
higher level arrangement strategy pointed to a higher SOLO level of statistical thinking.
Learners in the two tasks produced a rich variety of representations which included
idiosyncratic, unsophisticated responses as well as standard statistical representations.
The context of the two tasks, the quantitative versus qualitative nature of the data in the
tasks, and the statistical tools or representational skills learners have at their disposal,
played an important role in their representations. Well-planned data handling activities
develop representational and higher order thinking skills. The variety of responses and
different response levels elicited in the two tasks indicate that the nature of the tasks
rather than the size of the data set play a conclusive role in data tasks. Multiple
representations by an individual were an indication of successful modeling, are effective
in problem solving and are associated with good performance. The SOLO model which
incorporates a structural approach as well as a multimodal component proved valuable
in the analysis of responses. Using this model with accompanying acknowledgement of
different problem solving paths and the contribution of ikonic support in the concrete
symbolic mode, promote the in-depth analysis of responses.

This study contributes to the research in the field of data representation and statistical
thinking. The analysis and results led to an integrated picture of Grade 4-7 learners’
iv
representation of statistical data and of the statistical thinking levels evident in their
representations.

Key words: statistics education, mathematics education, modeling, representation,
arrangement types, transnumeration, statistical thinking levels, multiple representations.
v
OPSOMMING

Die deurslaggewende rol van voorstelling in wiskundige en statistiese modellering en
probleemoplossing soos blyk uit leerders se ordening en voorstelling van statistiese
data, is ondersoek. Die fokus was op data ordening, data voorstelling en statistiese
denkvlakke. Die datatake het van leerders vereis om data deur modellering spontaan te
orden en voor te stel. Suksesvolle transnumerasie het die uiteindelike sukses van ‘n
voorstelling bepaal. Die vermoë om data te orden is krities in hierdie proses. Meer
gesofistikeerde ordeningstipes is by leerders in hoër grade aangetref en die hiërargiese
aard van ordeningstipes het duideliker geword in die konteks van ‘n aangepaste SOLO
Taksonomie raamwerk. ‘n Hoër ordeningstrategie het gedui op ‘n hoër SOLO vlak van
statistiese denke. Leerders se wye verskeidenheid voorstellings in die twee take het
idiosinkratiese, ongesofistikeerde response sowel as standaard voorstellings ingesluit.
Die konteks van die take, die kwalitatiewe versus kwantatiewe aard van die data en die
statistiese kennis of voorstellings-vaardigheid van leerders het ‘n belangrike rol in hulle
voorstellings gespeel. Goed beplande datahanteringsaktiwiteite ontwikkel beter
voorstellings- en hoër-orde denkvaardighede. Die verskeidenheid response en
verskillende responsvlakke wat by leerders ontlok is, is ‘n aanduiding dat die aard van
die take ‘n meer beduidende rol speel as die grootte van die datastel. Veelvuldige
voorstellings was ‘n aanduiding van suksesvolle modellering. Veelvuldige voorstellings
deur een individu is effektief in probleemoplossing en word geassosieer met goeie
resultate. Die SOLO model sluit ‘n strukturele sowel as multimodale komponent in en
was daarom ‘n waardevolle instrument in die analise van die response. Die gebruik van
hierdie model met sy gepaardgaande erkenning van verskillende
probleemoplossingsweë en die bydrae van ikoniese ondersteuning in die konkreet-
simboliese modus bevorder die in-diepte analise van response.

Hierdie studie dra by tot die literatuur in die veld van datavoorstelling en statistiese
denke. Die analise en resultate gee ‘n meer geïntegreerde beeld van Graad 4 – 7
vi
leerders se voorstelling van statistiese data en van die statistiese denkvlakke in hul
voorstellings.

Sleutelwoorde: statistiekonderwys, wiskundeonderwys, modellering, voorstelling,
ordeningstipes, transnumerasie, statistiese denkvlakke, veelvuldige voorstellings.
vii
ACKNOWLEDGEMENTS

I wish to thank the following people and institutions for their support:
• Thanks to my heavenly Father for the ability, opportunity and strength to start and
complete this work.
• My promoter, Prof Hercules Nieuwoudt, for his critical, constructive assistance.
• My co-promoter, Prof Dirk Wessels, who gave generously of his time and
knowledge; encouraging and guiding me through crucial times in the development of
my ideas and in the production of this document.
• Prof Peter Hewson, Drr Norman Webb and Mindy Kalchman of the Wisconsin
Center for Education Research at the University of Wisconsin-Madison for valuable
contributions during the semester spent in Madison.
• Prof John Pegg, Drr Chris Reading and Rosemary Callingham for their mentorship
in using the SOLO model for the analysis of the statistical thinking levels during my
visit to the University of New England in Australia.
• The National Research Foundation and National Science Foundation (US) for
financial support.
• Dr Hennie Gerber of the Department of Computer Services at Unisa and Dr John
Barnard of Melbourne, Australia for assistance with the quantitative analysis of the
data.
• My parents, Klasie and Idelette Henning, who gave me a love and respect for
education, and who unfailingly encouraged me to realise my dreams.
• My children Idelette, Helgard and Henning for their love, support and their patience
with my impossible hours.
• My colleagues at Laerskool Lynnwood for their friendship and support.

I dedicate this work to my husband Dirk for believing in me -
thank you for your unselfish support and encouragement.

Soli Deo Gloria
viii

The financial assistance of the National Research Foundation (NRF) of South Africa
(GUN numbers 2050502 and 2053491), the National Science Foundation (NSF) of the
USA, and the role of the Spatial Orientation and Spatial Insight Research Group (SOSI)
at Unisa is hereby acknowledged. Any opinions, findings and conclusions or
recommendations are those of the author and do not necessarily reflect the views of the
supporting organisations.
ix

The study of representation in mathematical learning allows us - at least
potentially - to describe in some detail students’ mathematical development
in interaction with school environments and to create teaching methods
capable of developing mathematical power (English, 2002:198).
x
TYPES AND LEVELS OF DATA ARRANGEMENT AND
REPRESENTATION IN STATISTICS AS MODELED BY
GRADE 4 – 7 LEARNERS

TABLE OF CONTENTS

CHAPTER 1: RESEARCH DESIGN

1.1 INTRODUCTION AND ORIENTATION 1
1.2 MOTIVATION FOR THE STUDY 4
1.3 RESEARCH FOCUS 7
1.4 AIM AND OBJECTIVES OF THE RESEARCH 8
1.5 RESEARCH DESIGN 9
1.5.1 Research methodology 9
1.5.2 Qualitative research 9
1.5.3 Research principles 11
1.5.3.1 Legal principles 12
1.5.3.2 Ethical principles 12
1.5.3.3 Philosophical principles 13
1.5.3.4 Procedural principles 14
1.5.4 Criteria for data generation 16
1.5.5 Phases of the research 18
1.5.5.1 Phase One: Literature study 18
1.5.5.2 Phase Two: Empirical study 19
1.5.5.3 Phase Three: The analysis and interpretation of findings 20
1.6 VALUE OF THE RESEARCH 20
1.7 AN OVERVIEW OF THE RESEARCH 21

CHAPTER 2: MODELING AND PROBLEM SOLVING IN MATHEMATICS AND
STATISTICS EDUCATION

2.1 INTRODUCTION 23
2.2 MODELING IN MATHEMATICS EDUCATION 25
2.2.1 Definitions of modeling 26
2.2.2 The process of modeling 29
2.2.3 Emergent models 34
2.2.4 Modeling in the school curriculum 36