Gérard Sensevy

Ivy Kidron

John Monaghan

Marianna Bosch

Susanne Prediger

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INTRODUCTION 
DIFFERENT THEORETICAL PERSPECTIVES AND APPROACHES 
IN MATHEMATICS EDUCATION RESEARCH - STRATEGIES AND 
DIFFICULTIES WHEN CONNECTING THEORIES 
Susanne Prediger1, Marianna Bosch2, Ivy Kidron3, John Monaghan4, Gérard Sensevy5 
1Dortmund University (Germany), 2 Universitat Ramon Llull (Spain), 3 Jerusalem College of Technology (Israel), 
4University of Leeds (United Kingdom), 5 INRP Lyon (France) 
 
 
A large diversity of different theoretical perspectives and research paradigms charac-
terize the European mathematics education research community. Since CERME 4, 
the ‘Theory Working Group’ has explored differences between these theories, their 
expression in different research practices and possible ways to deal with this diversity 
(see Artigue et al. 2006, Bosch et al. 2008 and Prediger et al. 2008).  
Exploiting diversity as a rich resource for grasping complex realities (Bikner-
Ahsbahs & Prediger 2006) requires developing strategies for connecting theories or 
research results obtained using different theoretical approaches. In 2007, the Theory 
Working Group continued its efforts in this direction and reflected on opportunities 
and difficulties of what we call ‘networking theories’. We noted different intentions 
behind researchers efforts to network theories. In some cases, the goal is to investi-
gate the complementary insights that are offered when we analyze given data with 
different theories (Kidron, 2008). In other cases, the intention is to explore the in-
sights offered by each theory to the other theories and, at the same time, to highlight 
the limits of such an endeavour (Kidron et al., 2008; Radford, 2008).  
The call for papers for the Theory Working Group at CERME 6 was guided by the 
idea of avoiding an overly abstract discussion without a concrete basis. That is why 
we called for papers with concrete case studies in which two or more theoretical ap-
proaches were connected. After an intensive peer review process, 15 substantial pa-
pers were chosen for discussion in the working group and for publication in these 
proceedings. The most important issues arising in the discussion of these case studies 
can be sketched under some key words structured according to the landscape of net-
working strategies as proposed by Prediger, Bikner-Ahsbahs & Arzarello, 2008).  
 
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Main issues arising in comparing and contrasting: Dimension of comparison 
Comparing theories requires categories for comparison. A variety of categories have 
been suggested by Prediger, Arzarello & Bikner-Ahsbahs (2008). The discussion this 
year was influence by the following: 
• the delimitation of empirical data and the kind of questions that arise, as well as 
the concrete formulation of results (see Ligozat & Schubauer-Leoni in this vol-
ume); 
• the distinction between theoretical approaches and perspectives (discussed by 
Wedege in this volume); 
• an ontological characterization of theories such as that proposed by Winsløw (in 
this volume) called the GOA-Model, which distinguishes theories according to na-
ture of their objects of research, namely groups (G) structured by certain relation-
ships, the organisation (O) of knowledge and practice, and artefacts (A) used to 
access and communicate in and about O. 
• an epistemological characterization of theories such as that proposed by Radford 
(2008), distinguishing between their basic principles, their methodology, and the 
paradigmatic questions that are approached. 
Main issues arising in combining and coordinating: Compatibility 
In order to combine or coordinate different theories, it appears to us that the theories 
must, in some sense, be compatible; but what exactly does this mean? In working 
group discussions of the case studies presented in the papers, different levels were 
posited as possible locations for potential incompatibilities:  
• the level of general principles, e.g. epistemological principles about how to inter-
pret mathematical knowledge;  
• the level of basic ‘paradigms’, the potential danger of hastily combining stability-
oriented with transformation–oriented perspectives; 
• the level of central constructs: although the sense or denotation of constructs may 
not be identical over different theories, they should not be contradictory (Gellert in 
this volume shows an interesting example of networking around the construct 
“rules”); 
• the level of practical consequences: if coordinating theories in empirical work 
leads to contradictory practical consequences with regard to learning, then there is 
a need to continue reflection (see Bergsten & Jablonka in this volume); 
• the level of ontology: this does not seem to present as many difficulties as some of 
the above since different grain sizes of analyses and focuses might help in com-
bining theories (see, for example, Jungwirth in this volume).  
In the working group discussion it was suggested that when paradigmatic research 
questions and/or objects diverge in different perspectives, the combination of these 
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perspectives in the course of analysing an empirical phenomenon might produce inc-
ommensurable, but not contradictory, results, as shown by the paper of Bergsten & 
Jablonka (in this volume). This raises the question of whether it is acceptable that dif-
ferent results can, without contradiction, lead to radically opposed interpretations. 
On the other hand, we found some aspects that facilitate the connection of theories. 
Theories might be linked more easily when they are not too strong with respect to 
their grammar or their methodologies (i.e. when they are at an early level of elabora-
tion) or when they are complementary with respect to their hypothetical scope or em-
pirical load (see Jungwirth in this volume). 
Main issue arising in integrating and synthesizing: Substrategies 
The working group discussion regarding strategies for integrating and synthesizing 
theories led to the tentative proposal to identify substrategies which included: ‘brico-
laging’ (that is adapting non-conflicting principles, notions or local analysis methods 
of different grand theories); ‘subordinating’ (see Gellert); ‘zooming in and out’ (see 
Jungwirth); and ‘metaphorical structuring’, the use of single concepts based on meta-
phors from one theory that converge into another (see Gellert with regard to rules).  
As Radford (2008) stated, although connections between theories are possible, there 
is a limit to what can be connected and this limit is determined by the goal of the con-
nection and the specificities of the theories that are being connected. In the following, 
we differentiate between different goals in the networking process. 
Networking with different aims 
In order to link theories beyond comparing and contrasting, we discussed the aims of 
the papers.:  
• Some of the papers propose networking strategies with the aim of understanding an 
empirical phenomenon that seems difficult to entirely grasp within one single the-
ory. These can be described as having an initial combining strategy that ends up 
with the construction of local coherence between the notions or principles used. In 
this sense, Arzarello, Bikner and Sabena (in this volume) combine theories for ana-
lysing data about a failed teaching strategy and integrating them (very) locally for 
the purpose of making sense of the situation described. The paper of Schäfer (in 
this volume) combines theories for constructing a local theory that improved his 
potential to approach a ‘practical’ question about low achieving students. Wedege 
(in this volume) presents a study in which some aspects of two theoretical perspec-
tives are coordinated. Stadler (in this volume) coordinates different perspectives 
within one empirical study, describing how a research interest in the transition be-
tween mathematics studies at secondary and tertiary levels generates the need for 
different theoretical approaches.  
• A different goal presented by some papers is to network with the aim of dealing 
with new problems. For example Ligozat & Schubauer-Leoni’s and Sensevy’s pa-
pers are hybrids which borrow constructs from distinct theories for local integra-
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tion with conversions in order to address specific research problem, the issue of 
joint action of the teacher and the students. 
• Networking is also an important tool to elaborate existing theories with the aim of 
increasing their scope by questioning them from the outside. Artigue, Bosch, 
Gascón & Lenfant (in this volume) show how a theory can evolve locally when an 
effort is made to approach a question formulated by another theory. The strategy 
here is to work within one theoretical framework and develop it in interaction with 
others, for instance by enlarging the set of paradigmatic research questions or its 
empirical unit of analysis. The work of Jungwirth (in this volume) presents a 
method of synthesizing local theories for ‘zooming in and out’ of the data.  
• Other papers consider networking with the aim of satisfying the need for an 
enlarged framework in relation to some new domain of research, assuming the ex-
isting frames are insufficient. For instance, Lagrange & Monaghan (in this volume) 
incorporated Saxe’s four parameters model in order to understand the situation of 
teachers using technology. To these authors, the existing frameworks they consid-
ered for viewing teachers’ activities in technology-based lessons are insufficient 
because they focus on teachers’ established routines but technology interferes with 
these routines.  
Different kinds of dialogues 
Within these aims we may distinguish different kinds of dialogues between theories. 
We use the word ‘dialogue’ not only to describe that which enables mutual under-
standing in the way we communicate our theories but also to emphasize differences 
in the use of language. Different kinds of dialogues were offered in the papers by Li-
gozat & Schubauer-Leoni, by Sensevy and by Artigue et al. One important charac-
terization is that the dialogues in these papers are between neighbouring approaches - 
theoretical approaches which were born in the same educational and didactic culture, 
which may be considered as belonging to the same ‘paradigm’. Even so, when we 
explore the dialogues in depth important differences between the theories can be seen 
and some interesting questions arise: 
• Do these “neighbouring approaches” use the same words with the same meanings? 
For instance, is the word milieu in the Anthropological Theory of the Didactic 
(ATD) equivalent to the a didactic milieu in the Theory of Didactic Situations 
(TDS)? The same question could be asked in relation to other terms, e.g. institu-
tion or contract. The question could arise also for theories which are not necessar-
ily neighbouring approaches. 
• Do the different theories deal with different ways of addressing similar issues? For 
instance, comparing the Joint Action Theory in Didactics (JATD), as described in 
both Ligozat & Schubauer-Leoni and Sensevy’s papers, with ATD and TDS, we 
may ask what is the difference between ATD media milieu dialectic, TDS a didac-
tic and didactic situations, and JATD dialectic between contract and milieu. Sen-
sevy states that in order to situate JATD in relation to TDS and ATD it can be ar-
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gued that whereas these two theories initially focus, from a logical point of view, 
on the nature of knowledge (what is the knowledge which is taught?), JATD ini-
tially focuses on the diffusion process (what is going on when a specific piece of 
knowledge is taught?). The aim of the networking is to construct a new theory 
JATD which makes use of existing theories, ATD and TDS. Therefore we may 
ask what supplementary insights and/or what new questions/problems are offered 
to ATD and TDS by JATD’s analysis of the diffusion process? For example, the 
JATD may raise the following question: within the contract-milieu dialectic how 
may the teacher link the topogenesis and the chronogenesis processes with respect 
to the piece of knowledge at stake, and how might these processes lead the 
teacher, in specific cases, to enact a new learning game? In this question there are 
some notions from ATD and TDS which are reconceptualized in that they are used 
in a new way, and there is a new notion (learning game). From an abstract view-
point, this kind of question is not impossible in ATD and TDS, and it is clearly 
understandable in these two theories. But the probability that this question is 
raised in these two theories is not high because their fundamental concerns are not 
focused on the problems of didactic joint action even though they are interested in 
didactic action. 
In Artigue et al. (in this volume) the notion of ‘minimal unit of analysis’ appears as a 
basic aspect of the modelling of educational phenomena proposed by each theory. 
Starting from the way each perspective reformulates a given research question, we 
could specify what units of analysis are considered in each case and how they can be 
connected. The authors add that this could be a good way to improve our capacity for 
describing and comparing not only the concrete research or practical problem formu-
lated by each theory but also the types of problems that can be proposed, the kind of 
empirical data needed and the set of ‘acceptable answers’ that can be provided. When 
we choose a specific unit of analysis, we make decisions not only about the empirical 
data we consider but also about our different priorities with regard to the focus of the 
analysis (Bosch & Gascón, 2005).  
 
Final remarks 
The discussions that took place in our working group about affordances and con-
straints of different networking strategies made us aware that the theoretical frame-
works used in our research are ‘living entities’ that evolve through our studies. Some 
have been around and have developed for many decades, others are less mature. They 
are our working tools, providing us with new ways of looking at reality, new descrip-
tions of empirical phenomena, new methods of analysis and new possible answers to 
the difficulties of teaching and learning mathematics. They are imbedded in research-
ers’ social, cultural and institutional inheritances and their development is also im-
pregnated with the personal interactions between researchers and the cooperative 
work done in our community. When we embody ‘theories’ into research practices 
that, at the same time, use theories and produce them, it becomes clear that our reflec-
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tions about ‘networking theories’ are methodological reflections, referring to the kind 
of tools we can or cannot use, the basis and the aim of our research, as well as the 
kind of rules we follow.  
Considering the networking of theories as the networking of research practices may 
lead us further not only in our capacity to collaborate between different groups of re-
searchers (and thus accumulate efforts and results) but also to gain insight about the 
very nature – and the rationale – of our own research in mathematics education. 
 
References  
Artigue, M.; Bartolini-Bussi, M., Dreyfus, T., Gray, E., and Prediger, S.: 2006, ‘Dif-
ferent theoretical perspectives and approaches in research in mathematics educa-
tion’, in M. Bosch. (ed.), Proceedings of CERME 4, Sant Feliu de Guíxols 2005, 
Fundemi IQS – Universitat 2006, 1239-1244.  
Arzarello, F.; Bosch, M.; Lenfant, A.; and Prediger, S.: 2008, ‘Different theoretical 
perspectives in research from teaching problems to research problems’, in D. Pitta-
Pantazi and G. Phillipou (eds.), Proceedings of CERME 5, Cyprus 2007, ERME, 
1618-1627. (+ other contributions of the working group in the same proceedings).  
Bikner-Ahsbahs, A. and Prediger, S.: 2006, ‘Diversity of Theories in Mathematics 
Education - How can we deal with it?’ ZDM - The International Journal on 
Mathematics Education 38(1), 52-57. 
Bosch, M., Gascón, J. (2005). La praxéologie comme unité d’analyse des processus 
    didactiques. In: Mercier, A., Margolinas, C. (coord.), Balises en Didactique des 
    Mathématiques, Grenoble: La Pensée Sauvage (pp. 107-122). 
Kidron, I. (2008). Abstraction and consolidation of the limit procept by means of in-
strumented schemes: the complementary role of three different frameworks. Edu-
cational Studies in Mathematics 69(3), 197-216. Springer. 
Kidron, I., Lenfant, A., Bikner-Ahsbahs, A., Artigue, M. and Dreyfus, T. (2008). 
Toward networking three theoretical approaches: the case of social interactions.  
Zentralblatt fur Didaktik der Mathematik, 40(2), 247-264. Springer.  
Prediger, S,; Bikner-Ahsbahs, A.¸ Arzarello, F.: 2008, ‘Networking strategies and 
methods for connecting theoretical approaches – First steps towards a conceptual 
framework’, ZDM - The International Journal on Mathematics Education, 40 (2), 
165-178. 
Prediger, S.; Arzarello, F.; Bosch, M.; Lenfant, A.: 2008, eds., ‘Comparing, Combin-
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ZDM - The International Journal on Mathematics Education, 40 (2). 
Radford, L.:2008, 'Connecting theories in mathematics education: challenges and 
possibilities', ZDM - The International Journal on Mathematics Education, 40 (2). 
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INTRODUCTION DIFFERENT THEORETICAL PERSPECTIVES AND APPROACHES IN MATHEMATICS EDUCATION RESEARCH -STRATEGIES AND DIFFICULTIES WHEN CONNECTING THEORIES

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INTRODUCTION DIFFERENT THEORETICAL PERSPECTIVES AND APPROACHES IN MATHEMATICS EDUCATION RESEARCH -STRATEGIES AND DIFFICULTIES WHEN CONNECTING THEORIES

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