Mathematical Literacy on Statistical Measures

232 Pages·2018·13.067 MB·English

by  Christian Büscher

Dortmunder Beiträge zur Entwicklung und
Erforschung des Mathematikunterrichts
Christian Büscher
Mathematical Literacy
on Statistical Measures
A Design Research Study
Dortmunder Beiträge zur Entwicklung
und Erforschung des Mathematik­
unterrichts
Band 37
Reihe herausgegeben von
S. Hußmann
M. Nührenbörger
S. Prediger
C. Selter
Dortmund, Deutschland
Eines der zentralen Anliegen der Entwicklung und Erforschung des Mathematik­
unterrichts stellt die Verbindung von konstruktiven Entwicklungsarbeiten und
rekonstruktiven empirischen Analysen der Besonderheiten, Voraussetzungen und
Strukturen von Lehr­ und Lernprozessen dar. Dieses Wechselspiel findet Ausdruck
in der sorgsamen Konzeption von mathematischen Aufgabenformaten und Unter­
richtsszenarien und der genauen Analyse dadurch initiierter Lernprozesse.
Die Reihe „Dortmunder Beiträge zur Entwicklung und Erforschung des Mathe ­
matikunterrichts“ trägt dazu bei, ausgewählte Themen und Charakteristika des
Lehrens und Lernens von Mathematik – von der Kita bis zur Hochschule – unter
theoretisch vielfältigen Perspektiven besser zu verstehen.
Reihe herausgegeben von
Prof. Dr. Stephan Hußmann
Prof. Dr. Marcus Nührenbörger
Prof. Dr. Susanne Prediger
Prof. Dr. Christoph Selter
Technische Universität Dortmund, Deutschland
Weitere Bände in der Reihe http://www.springer.com/series/12458
Christian Büscher
Mathematical Literacy
on Statistical Measures
A Design Research Study
With a foreword by Prof. Dr. Susanne Prediger
Christian Büscher
Fakultät für Mathematik, IEEM
Technische Universität Dortmund
Dortmund, Germany
Dissertation Technische Universität Dortmund, Fakultät für Mathematik, 2018
Erstgutachterin: Prof. Dr. Susanne Prediger
Zweitgutachter: Prof. Dr. David Pratt
Tag der Disputation: 13.06.2018
Dortmunder Beiträge zur Entwicklung und Erforschung des Mathematikunterrichts
ISBN 978­3­658­23068­5 ISBN 978­3­658­23069­2 (eBook)
https://doi.org/10.1007/978­3­658­23069­2
Library of Congress Control Number: 2018950192
Springer Spektrum
© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2018
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
or information storage and retrieval, electronic adaptation, computer software, or by similar or
dissimilar methodology now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publication does not imply, even in the absence of a specific statement, that such names are exempt
from the relevant protective laws and regulations and therefore free for general use.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authors or the editors give a warranty, express or implied, with respect to the material contained
herein or for any errors or omissions that may have been made. The publisher remains neutral with
regard to jurisdictional claims in published maps and institutional affiliations.
This Springer Spektrum imprint is published by the registered company Springer Fachmedien
Wiesbaden GmbH part of Springer Nature
The registered company address is: Abraham­Lincoln­Str. 46, 65189 Wiesbaden, Germany
Foreword
Statistics education counts as a very important topic for achieving mathematical
literacy, as statistics is applied in many everyday and academic situations, inside
and outside mathematics classrooms. Although this statement seems obvious, it
leaves a lot of open questions. What exactly is the contribution of statistics to
mathematical literacy? How can it be seriously promoted, given the fact that in
many schools a maximum of two weeks per curriculum year are dedicated to
statistics, because arithmetic and algebra take most of the time? How can stu-
dents’ reflections be fostered beyond superficial criticism on manipulations of
graphs?
The PhD thesis submitted by Christian Büscher provides theoretically pro-
found and empirically grounded answers to these questions by developing a
normative conceptualization of mathematical literacy on statistical measures and
by conducting a Design Research study with seventh graders in several iterative
design experiment cycles. As a main theoretical contribution, the study develops
a normative conceptualization of mathematical literacy with two kinds of goals,
mathematizing goals (addressing the ability to use mathematical concepts for
structuring phenomena in the social and natural world) as well as reflective
goals (addressing the ability to reflect and evaluate the role mathematics plays
in society). But whereas for the mathematizing goals, research in mathematics
education has produced a plurality of perspectives and well-elaborated, empiri-
cally grounded theories on teaching and learning, the reflective side of mathe-
matical literacy so far remains underdeveloped. This PhD thesis aims to provide
a contribution towards closing this specification gap by elaborating the notion
of reflective concepts, which – in analogy to the mathematizing concepts of the
mathematizing side of mathematical literacy – need to be developed by learners
to be able to critically reflect on mathematics.
His work on the reflective side of mathematical literacy is interwoven with
another elaboration, this time on the mathematizing side of mathematical litera-
cy. Statistical measures such as the mean and the median are commonly utilized
throughout statistics education research. Christian Büscher substantiates such
work by providing a clear conceptualization of the notion of statistical measure.
This allows him to describe formal, general measures as well as informal, situa-
tive ones used by learners and to formulate a hypothetical learning trajectory for
learners to develop the concept of statistical measure.
One of the main themes running through his work is the trust in learners to
develop complex ideas through their own activities, if supported by careful
design of a teaching-learning arrangement. This is true for the mathematizing
side of mathematical literacy as well as for the reflective side. This, however,
does not remain a purely theoretical standpoint, but is actualized through his
empirical work.
vi Foreword
The PhD thesis follows a Design Research program to develop and empiri-
cally ground reflective learning opportunities along with the mathematizing
learning opportunities which are optimized to let the students develop statistical
knowledge. In the empirical core of the thesis, Christian Büscher presents case
studies for students’ learning pathways towards statistical measures and their
mathematizing concepts and the early germs of reflective concepts. These quali-
tative reconstructions show the complexities of individual processes of guided
re-invention and specifically show how mathematizing and reflective activities
can and should be intertwined as they stabilize each other in many subtle ways.
Overall, Christian Büscher not only provides a theoretical, normative
framework for mathematical literacy and theoretical contributions towards clos-
ing the specification gap of reflection. He rather provides empirical proof that
learners indeed are able, through their own mathematizing and reflective activi-
ties and concepts, to reflect on mathematics while engaging in mathematizing –
while providing design principles and concrete design elements that allow edu-
cators and researchers alike to initiate such activities.
As a whole, the PhD thesis documents a rich and theoretically deep Design
Research project which can substantially contribute to narrowing the gap be-
tween the general goals of statistics education and the classroom reality. The
thesis accounts for the complexity of reflective statistical learning processes
which can be initiated even in a restricted time frame. I hope it will find many
readers and future research that builds upon it.

Dortmund, June 2018

Susanne Prediger
Preface
Although in the following pages I argue that meaningful reflection can be car-
ried out while getting work done, there still remains something to be said for
taking a step back and looking at a work finished. This is the opportunity to
identify all the people that helped along the way.
First, I want to thank Susanne Prediger for her guidance and carefully ad-
ministered nudges into the right directions, while giving me the room to find my
little peculiar niche in mathematics education research. I want to thank Dave
Pratt for taking the time for his deep and always constructive commentary on
the emergent thesis. Susanne Schnell, who accompanied me on the early steps
into statistics education research. The FUNKEN research group, represented
through its leadership team of Bernd Ralle, Stephan Hußmann, Jörg Thiele, and
Claudia Gärtner for the richness of perspectives and for providing a space for
discussing and sharing the woes of Design Research. All members of the work-
ing group Prediger/Hußmann and the whole IEEM for providing an invaluable
pool of expertise and the willingness to share knowledge; in particular Kristina
Penava for being my stochastical companion. The teachers for supporting my
search for students, and the students for the willingness to share their thoughts.
Martin and Heather Buscher for checking the language of the whole manuscript.
And finally, Carina Zindel for all the shared moments of grief and triumph. This
wouldn’t have been possible without you.

Thank you all.
Table of Contents

1 Introduction ................................................................................................... 1
2 Mathematical literacy in statistics ............................................................... 7
2.1 The mathematizing and reflective sides of mathematical literacy ............ 8
2.2 Mathematical literacy in statistics .......................................................... 10
2.2.1 Mathematizing as a statistical activity ............................................ 10
2.2.2 Mathematical literacy as an integrating framework for statistical
literacy and reasoning ..................................................................... 12
2.3 Aims and goals of the mathematizing side of mathematical literacy ..... 13
2.3.1 Two mathematizing aims ................................................................ 13
2.3.2 The mathematizing goal of mastering mathematizing activities ..... 16
2.3.3 The mathematizing goal of developing mathematizing concepts ... 18
2.4 Aims and goals of the reflective side of mathematical literacy .............. 21
2.4.1 Two reflective aims ......................................................................... 21
2.4.2 The reflective goal of mastering reflective activities ...................... 24
2.4.3 The specification gap of reflection.................................................. 26
2.5 The concept of measure as the central learning content ...................... 29
3 The mathematizing concept of measure .................................................... 31
3.1 A conceptualization of measure .............................................................. 31
3.1.1 Measures and the mathematizing concept of average ..................... 31
3.1.2 Conceptualizing measure ................................................................ 32
3.2 Learners’ intuitive situative measures .................................................... 37
3.3 Measures and Informal Statistical Inference .......................................... 39
4 Developing concepts and mastering activities ........................................... 43
4.1 Learning-theoretical background ............................................................ 43
4.1.1 The relationship between concepts and activities ........................... 43
4.1.2 The importance of context .............................................................. 46
4.1.3 Defining this thesis’ central theoretical constructs .......................... 48
4.2 The development of the mathematizing concept of measure.................. 49
4.2.1 A general learning trajectory towards the mathematizing goals ..... 50
4.2.2 The realization gap of reflection ..................................................... 52
5 Summary and research questions .............................................................. 55
6 Methodology ................................................................................................ 59
6.1 Research framework ............................................................................... 59
x Table of Contents
6.1.1 Topic-specific Didactical Design Research .................................... 59
6.1.2 Educational Reconstruction of reflective concepts ......................... 61
6.2 Methods of Data collection .................................................................... 62
6.3 Methods of data analysis ........................................................................ 65
7 Developing Typical ...................................................................................... 75
7.1 Design of a teaching-learning arrangement for Cycle III ....................... 75
7.1.1 Design principles for initiating mathematizing activity .................. 75
7.1.2 The Typical Antarctic Temperatures Problem ................................. 79
7.1.3 The Arctic Sea Ice Problem ............................................................ 91
7.2 Empirical reconstruction of students developing Typical ....................... 98
7.2.1 Maria and Natalie: Developing with a focus on structuring ........... 99
7.2.2 Overarching effects for Maria and Natalie’s learning processes ... 130
7.2.3 Quanna and Rebecca: Developing with a focus on formalizing ... 133
7.2.4 Overarching effects for Quanna and Rebecca’s
learning processes ......................................................................... 150
7.3 Students’ development of measures in Cycle III .................................. 152
7.3.1 Connections to general measures .................................................. 152
7.3.2 The broadening of the contextual neighborhood .......................... 153
7.3.3 Differences in richness of activity ................................................ 153
7.3.4 Unforeseen effects of design principles ........................................ 154
8 Students’ reflective concepts ..................................................................... 155
8.1 Design of a teaching-learning arrangement for Cycle IV ..................... 155
8.1.1 Refined design principles for initiating reflective activities ......... 155
8.1.2 The Antarctic Temperatures problem ............................................ 156
8.1.3 The Arctic Sea Ice Problem .......................................................... 159
8.2 Empirical reconstruction of students’ reflective concepts .................... 160
8.2.1 Kaan and Nesrin: Discovering the perspectivity of measures ....... 160
8.2.2 Jana and Mara: Judging the contextual relevance of measures ..... 174
8.2.3 The effects of design for Cycle IV ................................................ 183
8.3 Identification of situative and general reflective concepts ................... 185
8.3.1 The ambiguity of situative reflective concepts ............................. 187
8.3.2 Three reflective core concepts ...................................................... 187
8.3.3 The possibility and need for development .................................... 188
8.3.4 Connections between reflective concepts and activities ............... 189
8.3.5 An existence proof of reflection in learning processes ................. 190
9 Conclusions ................................................................................................ 193
9.1 Central theoretical results of the thesis ................................................. 194