Table Of ContentThe Mathematics in Our Hands
Christina M. Krause
The Mathematics
in Our Hands
How Gestures Contribute to
Constructing Mathematical
Knowledge
With a foreword by Angelika Bikner-Ahsbahs
and Ferdinando Arzarello
Christina M. Krause
University of Duisburg-Essen
Germany
Dissertation University of Bremen, 2015
Date of thesis defense: April 21st, 2015
Assessors:
Prof. Dr. Angelika Bikner-Ahsbahs (Doctoral supervisor) (University of Bremen, Ger-
many)
Prof. Dr. Ferdinando Arzarello (University of Turin, Italy)
ISBN 978-3-658-11947-8 ISBN 978-3-658-11948-5 (eBook)
DOI 10.1007/978-3-658-11948-5
Library of Congress Control Number: 2015956041
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Foreword
“anyway the exponential function would ,wouldn't go up again' because one but it would
flatten further that would than haven u-h ,as its asymptote” is Tim’s commentary as he
watches the computer screen in front of him. On the computer screen we find a coordi-
nate system. Starting from the first quadrant, a parabola is swept out, crossing the y-axis
in the curve’s lowest point. However, Tim does not directly refer to the parabola but to
the graph of an exponential function, that is not visible on the screen. In his commentary,
Tim wants to say that the visual piece of the curve on the screen cannot represent an
exponential function, “… because one but it would flatten further that would than haven
u-h ,as its asymptote”. The last part of his utterance is accompanied by a gesture: directly
in front of the computer screen, Tim’s finger draws the graph of an exponential function
in the air, passing from the right top downwards, a bit up again before going to the left
and approximating the x-axis.
Such situations often happen in mathematical learning situations. Even when unob-
served, students may use gestures to approach mathematical ideas or clarify aspects,
specifically when solving problems. This does not only happen with graphical or geo-
metrical topics. Learners may represent all kinds of things by gestures, for instance if
they indicate the substitution of an algebraic expression by a “grasping hand” as is de-
scribed in this book. Such representational gestures are relevant tools for constructing
mathematical knowledge, as has been known for quite some time. However, what has
not been known is how gestures enable students to advance their epistemic processes. In
her thesis, Christina Krause investigates this research problem. Her main approach is to
merge two models, an epistemic action model, representing the three epistemic actions
of gathering and connecting mathematical meaning as a basis for structure seeing, and
the semiotic bundle model, connecting semiotic means, such as gestures, language and
representations. Empirically based, she identifies ‘specifying gestures’, specifying the
Where, the What and the How of mathematical ideas and also Relationships between
mathematical aspects. In the example above, Tim not only specifies what he refers to
verbally, the exponential function, but also its graphical style, hence the How and, in
terms of direction, also the Where. Through comparison, Tim also constitutes a relation-
ship between two graphical representations: one represented on the screen, the other by
tracing the style of its shape with a gesture in front of the screen in the air. In the inves-
tigated data more than 90% of the gestures referring to mathematical objects were spec-
ifying gestures, and on average 40% of them specified several things at the same time.
For her empirical investigation, Christina Krause uses given data about pairs of students
solving mathematical problems. These data, already transcribed, were re-worked in de-
tail and enriched by pictures of gestures and inscriptions. In her fine-grained analysis,
v
vi Foreword
she focused on so called “epistemic-dense episodes”, the most relevant parts for structure
seeing. For that she clarified the epistemic role of gestures: two planes of analyses are
distinguished, one in which gestures represent mathematical aspects related to speech
and inscriptions, the second in which gestures relate to epistemic actions. For the latter,
Christina Krause is able to reconstruct two different types of epistemic functions affect-
ing the epistemic processes: performing functions and forming function. This frame of
concepts linked to results from gesture studies allows her to reconstruct processes of
coming to know mathematics as they take place in epistemic-dense situations. The re-
sults show that gestures may prepare, shape, substitute or directly advance epistemic
actions. In the example above, Tim’s gesture sources parts of the argumentation out from
speech to visualize them by gesturing. By this, gesturing simplifies verbal argumenta-
tion.
Gestures do not only advance epistemic insights, they may even assist to create them
and to elaborate them. For example, as the abovementioned “grasping hand” was re-
peated, the idea of substitution afforded by a task on continued fractions evolved towards
new insights. Hence, gestures may indicate on which idea a student is working, thus
providing diagnostic potential for teachers’ noticing. For this fresh area of research,
Christina Krause has made a significant contribution by showing how gestures are used
spontaneously to complement speech and advance mathematical insights.
She succeeds not only in presenting her research results in an exciting manner, her book
is also shaped in an attractive way. This shaping is not just an end in itself: it makes her
fine-grained gesture analyses accessible and transparent to the reader. Her analyses do
not only affect the reader by consistent argumentation but they often happen to be much
more directly convincing, e.g. when one realizes that while reading the meaning of the
“grasping hand” is grasped by gestural imitation.
At first sight the data base seems rather small, but the added value of this research lies
in the theoretical and methodological deepness having led to a framework for studying
a new topic in a fresh field of research. Only now it will be possible to explore the epis-
temic function of gestures systematically with qualitative as well as quantitative meth-
ods. Therefore, we wish that the national and international community of researchers
interested in gesture research and beyond will adopt the results of this dissertation and
will be inspired by this book and its exciting contribution to our understanding of math-
ematics education practices and research.
July 2015, Angelika Bikner-Ahsbahs (University of Bremen),
Ferdinando Arzarello (University of Torino)
Acknowledgments
There are several people and institutions that made it possible for me to carry out this
study and to write this book by supporting me in many different ways:
First, I want to thank my supervisor, Angelika Bikner-Ahsbahs, for the great opportunity
she offered when accepting me as her PhD-student. I appreciate all the trust she set in
me and thank her for all the creative ideas, the support, and the encouraging words right
when I needed them. But especially, I thank her for the contagious curiosity and
enthusiasm she herself showed in my research topic, and in the results that finally arose.
Second, I want to thank Ferdinando Arzarello as my second advisor and cooperation
partner of the project that made my studies possible in the first place. Despite his
numerous engagements and the spatial distance between Bremen and Turin, he found
the time to sit down with me, providing helpful advices as an expert in the field of
semiotics and for the study of gestures in mathematics education.
I thank the Central Research Development Fund of the University of Bremen for making
this study possible by funding the project “Epistemic processes from a semiotic point of
view – On the role of signs in the construction of mathematical knowledge” as
collaboration project with the University of Turin. Additional thanks are provided to the
principle investigators of the project “Effective knowledge construction in interest-dense
situations”, Angelika Bikner-Ahsbahs, Tommy Dreyfus, and Ivy Kidron, for their consent
to use data of the project. Their project was funded by the German-Israeli-Foundation
for Scientific Research and Development (GIF); therefore this research was also
supported by the GIF. Also, I thank the German team of the GIF-project, Angelika
Bikner-Ahsbahs, Julia Cramer, Thomas Janßen, and Jakob Priwitzer for data collection
and preparation, and the entire project group (Angelika Bikner-Ahsbahs, Tommy
Dreyfus, Ivy Kidron, Miriam Adi, Julia Cramer, Nava Gilboa, and Raz Harel) for their
task designs.
I also thank the entire mathematics education working group at the University of Turin
for their collaboration in the project and the hospitality during my three stays at their
department. Thank you for providing many helpful advices and for introducing me into
a multimodal analysis in general and the study of gestures in particular. Especially, I
need to send out a very warm ‘grazie’ to Cristina Sabena for being a great professional
support and such a kind and friendly person I appreciate to have gotten to know through
this project.
vii
viii Acknowledgements
The mathematics education working groups of the University of Bremen always
provided a very supportive and collaborative environment for me. Special thanks are
given to my PhD-student colleagues Julia Cramer, Thomas Janßen, Jonathan von
Ostrowski, Jenny Cramer, Stephanie Lachky, and Daniela Behrens (in order of
appearance) for all the talks we had and for their support. Furthermore, I want to
acknowledge my current working group and collaborators at the University of Essen for
backing me in the very last stage.
Benjamin Rott, Johannes Kautzsch, and Daniela Behrens, thank you for reading (more
or less mature) parts of the writing during the last couple of years. Raja Herold, thanks
a lot for finding my file after my laptop’s ‘hiccup’. Thanks to my friends for being patient
in those periods I hided somehow from social live. Especially to mention Johannes once
more: thank you for cake and cookies, coffee, and words in long office hours.
I also thank my family for their steady support and for always encouraging me. I found
a lot of words in this book, but I do not find proper words to thank my parents as much
as they deserve it. Therefore, just a simple “Danke für alles!“. To my brother thanks also
for the technical advice during the last couple of years, and double thanks for two
wonderful nephews that always make me smile.
Last, but quite the opposite of least: Thank you, Jan. Thanks for your patience, your
hugs, your words, for simply being there and supporting me with the best timing
possible.
July 2015
Abstract
When learning mathematics, students do not only use language to communicate, but also
gestural and written modes of expression. Gestures seem to play an important role in a
pre-explicit stadium of an idea, when it is not yet explicitly accessible to the students.
This is especially the case when learners construct new mathematical knowledge in so-
cial interaction(cid:86). While previous studies mostly examined these processes of social
con-struction of knowledge on the basis of verbal utterances, the present work
addresses the question: How do gestures contribute to processes of constructing
mathematical knowledge in social interaction(cid:86)?
To answer this question, I aimed to enrich a pre-existing theory of epistemic processes
with a semiotic perspective based on empirical findings. Thereby, a more extensive com-
prehension of social processes of constructing mathematical knowledge shall be at-
tained. Three pairs of high performing students were filmed while solving three tasks
that differ both in mathematical topic, as well as in the variety of representations pro-
vided for solving the task. Following an interpretive approach, the traditional, speech-
based reconstruction of the epistemic processes by means of an epistemic action model
was extended by an analysis of gestures in their synergy with language and inscriptions.
The contribution of gestures to the epistemic processes is captured by two functions of
gestures: The representational function of gestures concerns the ways in which gestures
can refer to a mathematical object in processes of knowledge construction. I systemati-
cally expose how gestures can enrich the verbal utterance and how speech, gesture, and
inscription act together to shape the mathematical object. The epistemic function of ges-
tures concerns the ways in which gestures can contribute to the accomplishment of the
three epistemic actions ‘gathering’, ‘connecting’, and ‘structure-seeing’ (Bikner-
Ahsbahs, 2005). I found that gestures not only support, but can also prepare, and even
realize these epistemic actions. More specifically, I differentiate between four forming
and six performing epistemic functions of gestures: The forming functions are related to
the ways in which gestures can provide visual access to the mathematical object of an
epistemic action; the performing functions of gestures concern how gestures can actively
affect the accomplishment of epistemic actions. A comparison of the different data sets
reveals that gestures influence the epistemic process more significantly than previously
thought, and gives an idea about what this influence may depend on.
ix
Zusammenfassung
Beim Lernen von Mathematik nutzen Schüler nicht nur Sprache, um sich mitzuteilen,
sondern auch gestische und schriftliche Ausdrucksformen. Gesten scheinen eine
besondere Rolle zu spielen, wenn Ideen für die Lernenden noch nicht ‚greifbar‘, noch
nicht explizit zugänglich sind. Dies ist vor allem dann der Fall, wenn Lernende neues
mathematisches Wissen in sozialer Interaktion konstruieren. Während bisherige Studien
diese Prozesse sozialer Wissenskonstruktion zumeist auf Basis der verbalen Äußerungen
untersuchten, stellt die vorliegenden Arbeit die Frage: Wie tragen Gesten zu Prozessen
der Konstruktion mathematischen Wissens in sozialer Interaktion bei?
Zur Beantwortung dieser Frage wurde an eine bestehende Theorie epistemischer
Prozesse angeknüpft, um diese auf empirischer Grundlage durch eine semiotische
Perspektive anzureichern und so ein umfassenderes Verständnis sozialer Prozesse der
Konstruktion mathematischen Wissens zu erlangen. Hierzu wurden leistungsstarke
Schülerpaare bei der Bearbeitung dreier Aufgaben gefilmt, die sich sowohl in der
mathematischen Thematik, wie auch in der Vielfalt und Art der zur Verfügung stehenden
Repräsentationen unterscheiden. Einem interpretativen Ansatz folgend wurde die
traditionelle, sprachbasierte Rekonstruktion der sozialen epistemischen Prozesse durch
ein epistemisches Handlungsmodell um eine Analyse der Gesten im Zusammenspiel mit
Sprache und Inskriptionen erweitert. Der Beitrag von Gesten zu den
Erkenntnisprozessen wird anhand zweier Funktionen der Gesten gefasst: Die
Repräsentationsfunktion von Gesten betrifft die Arten und Weisen, in denen Gesten in
Prozessen der Wissenskonstruktion dazu beitragen können, auf ein mathematisches
Objekt zu verweisen. So wird systematisch herausgestellt, wie Gesten die sprachliche
Äußerung anreichern können und wie Sprache, Geste und Inskription das mathematische
Objekt hierdurch gemeinsam formen. Die Erkenntnisfunktion von Gesten bezieht sich
darauf, wie Gesten zu der Ausführung der drei epistemischen Handlungen Sammeln,
Verknüpfen und Struktursehen (Bikner-Ahsbahs, 2005) beitragen können. Es hat sich
gezeigt, dass Gesten diese epistemischen Handlungen nicht nur unterstützen, sondern sie
auch vorbereiten und sogar selbst realisieren können. Hierbei wurden vier ausformende
und sechs ausführende epistemische Funktionen von Gesten unterschieden: Die
ausformenden Funktionen beziehen sich auf die Arten und Weisen, in denen Gesten dazu
beitragen mathematische Objekte epistemischer Funktionen visuell zu gestalten; die
ausführenden Funktionen darauf, wie Gesten aktiv auf die Ausführung epistemischer
Handlungen wirken können. Ein Vergleich der Analysen verschiedener Datensätze führt
zu der Vermutung, dass Gesten mehr Einfluss auf den epistemischen Prozess haben als
bisher angenommen und gibt Hinweise darauf, wovon dieser Einfluss abhängen kann.
x
