From beliefs to dynamic affect systems in mathematics education: Exploring a mosaic of relationships and interactions

415 Pages·2015·6.57 MB·English

by  Birgit Pepin

Advances in Mathematics Education
Birgit Pepin
Bettina Roesken-Winter E ditors
From beliefs
to dynamic
aff ect systems
in mathematics
education
Exploring a mosaic of relationships and
interactions
Advances in Mathematics Education
Series Editors:
Gabriele Kaiser, University of Hamburg, Hamburg, Germany
Bharath Sriraman, The University of Montana, Missoula, MT, USA
International Editorial Board:
Ubiratan D’Ambrosio (São Paulo, Brazil)
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Christine Knipping (Bremen, Germany)
Oh Nam Kwon (Seoul, Korea)
More information about this series at www.springer.com/series/8392
Birgit Pepin (cid:129) Bettina Roesken-Winter
Editors
From beliefs to dynamic
affect systems in
mathematics education
Exploring a mosaic of relationships
and interactions
Editors
Birgit Pepin Bettina Roesken-Winter
Sør-Trøndelag University College Professional School of Education
Trondheim , Norway Department of Mathematics
Humboldt-Universität zu Berlin
Berlin , Germany
ISSN 1869-4918 ISSN 1869-4926 (electronic)
ISBN 978-3-319-06807-7 ISBN 978-3-319-06808-4 (eBook)
DOI 10.1007/978-3-319-06808-4
Springer Cham Heidelberg New York Dordrecht London
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Forewo rd
From Hidden Dimensions to Dynamic Systems
in Affect Research
Abstract I n this preface, I turn briefl y to the genesis of an infl uential forerunner

to the present volume on affect and mathematics education, namely, B eliefs:
A Hidden Variable in Mathematics Education? , published over a decade ago.
Some brief snapshots of the contents of the earlier manuscripts are presented and
areas then identifi ed in need of further, or more nuanced, exploration are high-
lighted. Foregrounding these issues serves as an expedient link between the
present volume and the earlier work.
Introduction
The M athematisches Forschungsinstitut Oberwolfach (The Mathematical Research
Institute of Oberwolfach) is idyllically situated in Germany’s Black Forest. Over the
years, the Institute has hosted many week-long workshops in which specialist math-
ematics topics are explored by experts in the fi eld. “The Institute brings people
together for a short but intense period, providing them with ideal conditions under
which to pursue research activities which will i nfl uence and stimulate the future
development of the fi eld ” (Mathematisches Forschungsinstitut Oberwolfach n.d.,
emphasis added).
The week of November 21 to November 27 in 1999 marked a particularly impor-
tant time in the mathematics education research calendar. That week, uniquely, the
issues for debate and exploration at the Institute were not confi ned to the fi eld of
mathematics per se. Instead the focus was fi rmly on a topic of great relevance to the
fi eld of mathematics education: M athematical beliefs and their impact on teaching
and learning of mathematics.
v
vi Foreword
Consideration of issues raised during the rich and diverse fl ow of presentations
and discussions extended well beyond the life of the workshop. The tradition of
many of the previous, and indeed subsequent, workshops held at Oberwolfach was
maintained. So, seemingly inevitably, follow up activities were pursued,
c ulminating in the preparation of a volume designed to add to existing knowledge
and literature in the fi eld explored so intensively during the time spent at the
Institute.
The gestation period between conception and birth of the book about the infl u-
ence and impact of beliefs and aspects of the teaching and learning of mathematics
was substantial. Core ideas initially presented and discussed in Oberwolfach were
added to. Finally, in 2002, the manuscript B eliefs: A Hidden Variable in Mathematics
Education? (Leder et al. 2002) was ready for publication.
The Foundations: Inspection
What, more than a decade ago, caught the attention of researchers and practitioners
concerned with beliefs and their impact on the teaching and learning of mathemat-
ics? The “then” state of the art, as captured in Leder et al. (2002), was covered in
three distinct but overlapping sections: Beliefs: Conceptualization and measure-
ment; Teachers’ beliefs; and S tudents’ beliefs. Inevitably, as in the current volume,
affective components beyond beliefs also attracted much attention.
Beliefs: Conceptualization and Measurement
“The contributions in the fi rst section,” Leder et al. (2002, p. 4) clarifi ed,
are particularly concerned with examining what is meant by mathematical beliefs and how
they differ from other, related concepts. The authors draw extensively on existing literature,
highlight consensus and confusion in the ways various terms have been used in earlier
work, and indicate directions for further research without – collectively – offering a unifi ed
view on the main theoretical concepts explored: belief, conception, and knowledge. The
overall thrust is on the diversity of different starting points which typically correspond to
different emphases.
Reports of empirical research studies were largely absent in this section.
Capturing the essence of beliefs was a common theme – a task made more diffi cult
because of the multiple and often loose usage of the term “belief”. This practice, it
was frequently argued, should be challenged. That those working in the fi eld are
unlikely to embrace a single defi nition of the term belief was also generally agreed.
So, it was asked, both explicitly and implicitly, should the fi eld settle for another
option: to aim for a commonly accepted set of defi nitions of beliefs, depending on
the setting, the context, and the audience being addressed? How realistic, and
achievable, is a different, related, and persistently elusive goal: to use more refi ned
Foreword vii
and more comprehensive instruments to tap beliefs, use more careful terminology to
describe and characterize beliefs, and distinguish beliefs from other terms often
used virtually interchangeably?
T he magnitude of this goal was sketched evocatively by one of the reviewers of
Beliefs: A Hidden Variable in Mathematics Education?
“The fi rst stumbling block,” wrote Mason (2004, p. 347) “is to work out what
beliefs actually are, and where they fi t into an entire alphabet of associated inter-
linked terms.” These he listed as:
A is for attitudes, affect, aptitude, and aims; B is for beliefs; C is for constructs, concep-
tions, and concerns; D is for demeanor and dispositions; E is for emotions, empathies, and
expectations; F is for feelings; G is for goals and gatherings; H is for habits and habitus; I
is for intentions, interests, and intuitions; J is for justifi cations and judgements; K is for
knowing; L is for leanings; M is for meaning-to; N is for norms; O is for orientations and
objectives; P is for propensities, perspectives, and predispositions; Q is for quirks and quid-
dity; R is for recognitions and resonances; S is for sympathies and sensations; T is for ten-
dencies and truths; U is for understandings and undertakings; V is for values and views; W
is for wishes, warrants, words, and weltanschauung; X is for xenophilia (perhaps); Y is for
yearnings and yens; and Z is for zeitgeist and zeal. (Mason 2004, p. 347)
Whether written partly in jest or not, Mason’s creative beliefs-alphabet captured
an important challenge faced by researchers and followers of the fi eld: the need to
combat the loose and colloquial usage of the term belief and to adopt more con-
strained terminology.
W hat other directions were nominated in the book’s introductory chapters for
further or more nuanced explorations? Beliefs are not observable but are typically
inferred from observations, from responses to interview probes, or answers to ques-
tionnaires. How can we be sure that the inferences we draw are accurate – a question
that is posed frequently in the book and in the wider research community? Might
other measures, physiological or neurological and/or laboratory based, provide new
insights? How easy is it to make individuals aware of their beliefs about themselves
and about mathematics, and how might these beliefs be changed if that is required
for optimum mathematics learning?
As noted by D. McLeod and S. McLeod (2002), the chapters in this section “are
not the last word on attempts to defi ne beliefs…. Writers in the future will shape the
research using terminology that meets their needs” (p. 119). That many have already
done so is evident from the contributions in this and the other two parts of Leder
et al. (2002). The extent to which later researchers have also adopted this pragmatic
approach can be judged from inspection of the present, new volume, on research on
beliefs and affect more broadly.
Teachers’ Beliefs
The synthesis of the second section, T eachers’ beliefs, read in part:
A ll the authors see a cyclical relationship between changing beliefs and changing practices;
wherever one starts they affect each other. … My reading of these chapters suggests that
viii Foreword
Chapman,1 Llinares, and Philippou and Christou come down on the side of changing beliefs
leading to change in practices, Lloyd and Hart on the side of changing practices leading to
changes in beliefs, and the review chapter by Wilson and Cooney remains with the dialec-
tic…. The chapters provide rich evidence of changing practices in teaching mathematics as
seen through a lens of discourse of beliefs as mental objects that are both cognitive and
affective. They are constituted in teachers’ prior experiences and they need to become the
subject of refl ection and analysis. A whole range of activities can bring about change
towards reform and that change will come about as beliefs change… The picture is rightly
a complex one. (Lerman 2002, p. 235)
Exploring teachers’ beliefs and their development are important topics in
themselves. How these beliefs affect their instructional strategies and in class
behaviors were dominant themes addressed, in different ways, by the contribu-
tors in this section. Typically, however, the proposed implications and applica-
tions for constructive practice have been tentative and nebulous. To quote Wilson
and Cooney (2002, p. 145): “understanding context and developing alternative
explanations for phenomena require researchers to dig deeply. … The human
condition is always beset with a strange mixture of rationality and irrationality
that defy sharp lines of demarcation.” That interviews and observations provide
insights into the ways in which teachers behave and make sense of their world is
clearly acknowledged. But, and under what circumstances, is the information
thus gained suffi cient to predict future actions? Much in this complex fi eld
remains untilled. Welcome additional insights are provided in the present
volume.
E xperimental data, generally based on work with small samples, were reported
in several chapters in this section. While instructive, such fi ndings are normally of
questionable generalizability and unlikely to generate new theories that stretch the
boundaries of the fi eld. What other avenues could be fruitful?
An alternate, or perhaps more appropriately thought of as a complementary
approach to the examination of changing classroom practices and the role of teach-
ers’ beliefs in this process, was advocated by Lerman (2002). Instead of regarding
beliefs and belief systems as psychological constructs, as, he argued, is done by the
authors of the chapters in this section, and indeed often in research about beliefs and
belief systems more widely, why not look “at the issue of changing practices in
mathematics teaching and learning (by) applying a s ociological gaze rather than a
psychological one” (p. 235, emphasis in the original). In parenthesis, it is worth not-
ing that most of the authors of the work contained in the next section in the book
“clearly acknowledge the tremendous infl uence socio-cultural context has on the
formation of beliefs” (Lester 2002, p. 353). Readers of the current volume can
themselves determine whether, and if so how, Lerman’s challenge is taken up in the
current publication.
1 D etails of the work of the authors mentioned in this quotation can be found in Beliefs: A hidden
variable in mathematics education? They are not detailed in the reference list.
Foreword ix
Students’ Beliefs
R esearch on students’ beliefs is relatively new. “It is only during the past 15–20 years
that beliefs research has come to be viewed as an essential ingredient in mathemat-
ics education research programs” according to Lester (2002, p. 346). Core issues
investigated by the contributors to the fi nal section of Leder et al. (2002) were sum-
marized by Lester as follows:
(cid:129) Do students’ beliefs infl uence their interest in and motivation to learn
mathematics?
(cid:129) Why don’t students worry about whether their solutions to word problems make
sense?
(cid:129) Do students’ beliefs infl uence their ability to connect real-world and school
mathematics?
(cid:129) How do classrooms infl uence the development of beliefs?
(cid:129) How do students’ intuitive beliefs about mathematical operations affect their
thinking processes?
O nce again a multiplicity of methods and settings were adopted to explore
students’ beliefs about mathematics and the learning of mathematics. When
included, experimental data were generally again confi ned to small samples.
Inevitably, there were shortcomings in the various studies reported: the scope of
instruments used to gather students’ beliefs was necessarily constrained, and the
need to infer beliefs from students’ answers or actions remained a hurdle.
At the time, and also with hindsight, the fi ndings reported – singly and collec-
tively – were, and are, more appropriately considered as markers of progress in the
fi eld than as end points of research. Lester’s (2002) concluding comments were
indeed apt.
I am skeptical (he wrote) about the credibility and reliability of the data presented by these
fi ve reports because I doubt that they accurately indicate what the students really believe….
I do not think most students really think much about what they believe about mathematics
and as a result are not very aware of their beliefs. So, although I think these researchers are
leading the way in efforts to develop good methodological tools for studying students’
beliefs, a considerable amount of work remains to be done. (p. 353)
Building on the Foundations
S ome years ago, Freud (1940/1986, p. 286) wrote: “the concept of the unconscious
has long been knocking at the gates of psychology and asking to be let in. Philosophy
and literature have often toyed with it, but science could fi nd no use for it.” This
contention is now heavily challenged, and indeed refuted, by the active and sus-
tained research agenda on the interaction between affect and the teaching and learn-
ing of mathematics, spawned in the later part of the twentieth century and continuing
unabated in the twenty-fi rst century. Despite this activity, there continues to be room