HIROSHIMA JOURNAL OF MATHEMATICS EDUCATION 16: 21-26, 2023

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Guest Editorial
RETHINKING THE DIVERSITY OF THEORIES IN MATHEMATICSEDUCATION. CONTRIBUTIONS RELATED TO THE TOPIC STUDYGROUP 57 OF ICME 14
Angelika Bikner-Ahsbahs
Bremen University (Germany) and Western Norway University of Applied Sciences (Norway)

Ivy Kidron
Jerusalem College of Technology (Israel)

Yusuke Shinno
School of Education, Hiroshima University (Japan)

Takeshi Miyakawa
School of Education, Waseda University (Japan)
This Special Issue emerged from Topic Study Group 57 (TSG57) on the “Diversity of Theories inMathematics Education” conducted at ICME 14 in July 2021 (Bikner-Ahsbahs et al., in print). Based on thepresentations and discussions in the TSG57 at the conference, five papers have been extended and elaborated
for the special issue. To introduce this special issue, we would like to look back briefly at what we havediscussed in the TSG. Main topic was the Networking of Theories in mathematics education research andthis also resonates with this special issue.
In recent years, the theory networking enterprise has advanced (Kidron et al. 2018). Networking ofTheories means to relate different theories, to make use of them in research and scientifically reflect thisprocess. Key assumption of this approach consists of respecting theoretical identities that are often based on
cultural backgrounds and at the same time requiring from researchers to keep a critical stance on theoreticalcompatibility in the theoretical networking practice. To keep this balance, Prediger et al. (2008) havedeveloped pairs of networking strategies that express increasing degrees of integration; these areunderstanding other and making understandable own theories, comparing and contrasting theories,
combining and coordinating theories, and local integration and synthesizing of theories. Many authors inthis issue recount how they used these strategies in their research.
The TSG57 addressed three subthemes in three sessions: Reasons for the diversity of theories;
methodological approaches in design research related to the diversity of theories; and reconsidering specificcommitments in the diversity of theories. In the first session, the TSG articulated a double aim, to respect
theory development coming from various educational cultures and research areas in the field and at the sametime to overcome theoretical fragmentation in the field. This requires a balance of keeping the diversity
needed but simultaneously bridging theories to make communication across the theory cultures possible.
Two themes were discussed, first, networking of theories can be a source of (as well as it may unpack)‘tensions’ between theoretical discourses, depending on basic assumptions and, second, networking oftheories can offer ‘flexibility’ when relating basic assumptions and thus creating a new discourse. From thesecond session we learned that each theory is built on the three commitments, ontology, axiology, and22epistemology, in specific ways that underlie how teaching and learning are understood, in research as well asin practice. Main postulation raised in the discussion was to increase scientific sensitivity for thesecommitments, specifically in terms of theoretical choices in transformative research. From the third session
we have learned to make these commitments visible in research and to look for methodological ways topractice this sensitivity, as for instance by a vertical analysis grounded in historical or philosophicalconsiderations. A crucial part of the discussion resulted in the insight that each theory on teaching andlearning entails a certain understanding of ethics grounded in this theory.
Finally, two questions were raised: To what extent is it necessary or even mandatory to take ethicalissues of the teaching/learning into consideration for theorizing and what are the relevant concepts of ethics
for that? More generally, what are the pitfalls when such commitments go unnoticed?
This Special Issue does not address all the aspects and questions mentioned in the TSG57, but somerelevant aspects are captured by the publications. All the five papers focus on and ‘rethink’ the diversity oftheories from different perspectives. In the first contribution “Facing the Challenge of Theoretical Diversity:
The Digital Case” Michèle Artigue explores the advancement in the field of technology in mathematics
education in terms of research on two concepts conducted by networking of theory approaches. In the second
article “The Role of a Priori Analysis in Theories”, Ivy Kidron explores the notions of a priori analyses
related to two theories showing how the epistemological grounding of theories determines them. In the third
paper “Ethics in the Mathematics Classroom”, Luis Radford elaborates the concept of communitarian ethics
for the Theory of Objectification making explicit how theory and ethics are entwined. The fourth contribution“Networking Praxeologies and Theoretical Grain Sizes in Mathematics Education: Cultural Issues Illustratedby Three Examples from the Japanese Research Context” by Yusuke Shinno and Tatsuya Mizoguchi is acomparison of three Japanese networking examples that shows clearly the relevance of culture when
networking theories. The final contribution by Angelika Bikner-Ahsbahs et al. is a networking case on design
research leading to local integration of two theoretical approaches that allows to explain how learners
transformed their behavior when acting with a digital artifact.
In the following, we are going to summarize each article and thereby extract their main contributions
to the field.

SUMMARIES OF THE ARTICLES IN THE SPECIAL ISSUE
The theoretical diversity offered by different theoretical cultures is a real challenge in mathematics
education research. The analysis in the paper by Artigue (2023) provides an important overview of thetheoretical landscape and its main tendencies in theorizing the use of technology in mathematics education.
Artigue aims to contribute to the reflection on the question: “To what extent, are we now better equipped tomeet the challenge of theoretical diversity in relation to technology-based mathematics teaching andlearning?” To this end, two conceptual tools are introduced in the paper. Both tools have proven theireffectiveness in addressing issues of theoretical diversity: The scale of networking strategies and the conceptof research praxeology (introduced in Artigue et al., 2011). Two case studies are described: The Instrumental
Approach (IA) and the Documentational Approach to Didactics (DAD). The emergence and development of

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these approaches illustrate well the global dynamic of the field towards increasing theoretical diversity, thequestions raised by this dynamic and the insightful efforts made to deal with it. The evolution of teachers’
documentational work was induced by technological advances. One important contribution of Artigue’s
paper is her analysis that demonstrates how knowledge has advanced both, in terms of understanding thediversity challenge and in terms of developing strategies to address it.
Kidron (2023) investigates the role of a priori analysis in different theories in the context of networking
theories. A priori analysis here means an analysis, which is often carried out prior to the experiment or datacollection. While the a priori analysis was initially proposed as a part of the methodology of didactic
engineering developed within the Theory of Didactical Situations (TDS) (Artigue, 2020), it is today sharedin different theories. As the methodology is often considered a part of the theory (Radford, 2008), thecomparison of the methodologies reveal the nature of each theory as well as the relationship between them.
In the previous work of networking theories, several case studies have been carried out by analyzing thecommon empirical data from different theoretical perspectives to shed light on their differences,commonalities, and complementarities (see Prediger & Bikner-Ahsbahs, 2014), that is to say, these studies
consist of the comparative analyses of the processes and results of a posteriori analyses. Kidron’s paper is
unique in that it focuses on a priori analysis instead of a posteriori analysis. It shows by comparing the apriori analyses of TDS and the theory of Abstraction in Context (AiC), each theory has a different focus,
while both theories place importance on the epistemological perspective: on the one hand, AiC focuses on
the learner’s construction of knowledge with a question “What is the epistemic process of the student?”; on
the other hand, TDS focuses on the didactic system with a question “How this [epistemic] process ispossible?” One of the important contributions of Kidron’s paper is that it shows the relevance of analyzing apriori analyses in the context of networking theories. As the focus of the theory affects the data collection,
the study of a priori analyses allows us to better understand the choice of data.
Radford (2023) begins his elaboration of educational ethics by an example, sensitizing readers forseeing teaching and learning as a deeply ethical form of activity. Although always present, ethics seem tohave lost sight in mathematics education. Radford sees this being grounded in the conception of mathematics
education that has been reduced “to a matter of acquiring knowledge, making the question of being andbecoming peripheral aspects of teaching and learning” (p. 60). He traces two fundamentally distinguishable
forms of ethical action back to Hobbes (1841) and Kant (2006). With reference to Hobbes, the notion of
ethics consists of contractual obligations that regulate acting in society. Under the maxim of universal reason,
Kant anchors moral action, hence, ethical acting, in the individual. According to Radford, Hobbes’ conceptionof ethics informs the transmissive idea of teaching and learning, while ethics according to Kant rather inform
constructivist approaches. Thus, ethics “appear framed by the way in which we understand teaching andlearning” (Radford, 2023, p. 60), hence, by theories of teaching and learning. Radford elaborates the concept
of communitarian ethics for the Theory of Objectification as an example. His contribution is unique as hemakes explicit how the Theory of Objectification and communitarian ethics are interrelated by elaborating
the theory’s relational conception of the social and how this hands over to ethics. He defines communitarian
ethics through three dimensions: responsibility, commitment, and care. Making use of Lévinas (1982)’
understanding of responsibility he regards responsibility as “living and acting with and for others” (p. 69).
The key function of communitarian commitment is “to participate in the creation of the classroom common

24work” (p. 69) of joint labor. A core ingredient of educational ethics is care, meaning here that “the importanceof caring for the Other is to go beyond ourselves and to be dragged powerfully into the world, to position
ourselves there, with-the-Other” (p. 70). Communitarian ethics are thus relational and, consistent with the
Theory of Objectification, permanently materializing themselves within the classroom.
Networking of theories is a way to bring different theoretical perspectives into a dialogue often used to
unpack implicit theoretical assumptions. However, there is a lack of meta-theoretical knowledge for
conducting a networking theories study. With their article, Yusuke Shinno and Tatsuya Mizoguchi (2023)
want to contribute to filling this gap by answering the question How can we characterise researchers’
theoretical work on networking endeavours in terms of praxeology? Leaning on the work of Artigue andBosch (2014), they use the concept of praxeology introduced by the Anthropological Theory of the Didactic(Chevallard, 2019) and elaborate the concept of grain size of theories to characterize the researchers’
theoretical work in three Japanese research and development examples that follow networking theories
approaches. Their networking technique, one of the elements of praxeology, consists of networking strategies.
Shinno and Mizoguchi use the strategy of comparing and contrasting theories to explore a design approach(lesson study or didactical engineering), with the strategy of combining and coordinating theories they
investigate an empirical classroom study, and they use local integration for the case of a curriculum
development study where the reference epistemological model (Bosch & Gascón, 2006) plays a key role.
Shinno and Mizoguchi identify and compare the three networking praxeologies of the examples to describe
the theoretical discourses involved. Their contribution is unique as it shows that these discourses are
intrinsically related to culture, i.e., classroom culture, research culture, and curriculum culture. As reflectedby the authors, these forms of culture are specific in the Japanese culture of mathematics education in thattheory for research and theory for practice (p. 7) are distinguished.
Bikner-Ahsbahs et al. (2023) address theoretical networking between Activity Theory (Leontyev, 2009)
and Instrumental Approach (Rabardel, 2002) to investigate the role of digital feedback in supporting theteaching and learning of negative numbers with a multimodal algebra learning (MAL) system. Within
Activity Theory, tools such as the MAL system play an essential role as cultural objects. From the perspectiveof Instrumental Approach, the process of how the MAL system becomes an individual instrument can beconceived as instrumental genesis, shaped by two interrelated processes: instrumentation andinstrumentalization. Bikner-Ahsbahs et al. (2023) examine the teaching and learning of negative numberswith the MAL system using Activity Theory and a model about feedback loops. They characterize the roleof digital feedback as mediating between different types of action expressed by a four-layered model. One of
the important contributions of Bikner-Ahsbahs et al.’s paper is their theorizing of an empirical phenomenon
at the boundary of the two theories leading to ‘local integration’, a relatively challenging strategy of theory
networking that allows to go beyond the mere understanding of a special empirical phenomenon (Prediger etal., 2008). In their networking work, after coordinating the two theories with the help of the layered model,
they clearly demonstrate how local integration in terms of Radford’s (2008) theory concept is achieved.
Four of the five articles are about the networking of theories, and two papers address philosophicalcommitments underlying educational theories. While Artigue elaborates the state of the art of research andits advancement in technology in mathematics education realized by theory networking approaches, theother three networking papers contribute directly to advance its meta-theoretical sphere. Shinno andHIROSHIMA JOURNAL OF MATHEMATICS EDUCATION 16: 25-26, 2023

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Mizoguchi elaborate that cultural backgrounds affect notions of theory and thus also notions of theory
networking. Bikner-Ahsbahs et al. explore the role of digital feedback at the boundary of two theories as a
new phenomenon and show the advancement made by the take-up of the networking strategy of local
integration. Kidron, through theory networking, contributes to our understanding of epistemologicalcommitments of theories and their methodological and conceptual consequences. Grounded in philosophicalconsiderations, Radford improves our knowledge on ethical commitments, underlying theories on teaching/learning in general. All the five papers advance the field by meta-theoretical considerations included into
research resulting in deepening our understanding of what can make a difference in the diversity of theories
in mathematics education.
Acknowledgement
We would like to thank all the reviewers who contributed to this special issue. As part of the editorial
process for this special issue, we have invited at least one external reviewer for each paper who was not
involved as an author of a paper or as a guest editor of the issue, but who had expertise in the research areaof theories in mathematics education. The contributions of all the reviewers were crucial to the elaborationof each paper and to the publication of this special issue.
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Angelika Bikner-Ahsbahs,
Bremen University, Germany & Western Norway University of Applied Sciences, Norway
bikner@math.uni-bremen.dehttps://orcid.org/0000-0002-0407-4897
Ivy Kidron

Yusuke Shinno

Jerusalem College of Technology, Israel

Hiroshima University, Japanivy@jct.ac.il

shinno@hiroshima-u.ac.jp

https://orcid.org/0000-0002-8986-0167
Takeshi Miyakawa
Waseda University, Japantmiyakawa@waseda.jp
https://orcid.org/0000-0002-1076-3592

https://orcid.org/0000-0001-6999-5075