HIROSHIMA JOURNAL OF MATHEMATICS EDUCATION 16: 77-94, 2023

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NETWORKING PRAXEOLOGIES AND THEORETICAL GRAIN SIZESIN MATHEMATICS EDUCATION: CULTURAL ISSUES ILLUSTRATEDBY THREE EXAMPLES FROM THE JAPANESE RESEARCH CONTEXTYusuke Shinno1 and Tatsuya Mizoguchi2
1

Hiroshima University, Japan, 2Tottori University, Japan

Abstract
This study aims to identify the theoretical underpinnings of researchers’ work on networking
theories in terms of the Anthropological Theory of the Didactic. The study is based on the conceptof research praxeology, which allows us to characterise researchers’ practice and knowledge of
networking. Three research examples, conducted in the Japanese educational context, are used toillustrate a means to characterise the four praxeological elements (type of tasks, technique,technology, and theory). The results imply that the notion of theoretical grain sizes (grand,intermediate, and local levels) can be used to deepen our understanding of researchers’ work on
networking theories. Based on our results, different characteristics of networking theories are
discussed according to the cultural issues and specificities of the illustrative examples.
Keywords: 
Networking theories, networking strategies, research praxeology, logos block,theoretical grain sizes

INTRODUCTION
The growth and success of mathematics education as a scientific research field are marked by the existenceof diverse theories. However, such growth also presents a challenge. As Prediger et al. (2008) stated:[T]he more the number of theories grows, the more difficult it will be to get an overview of all of themand of small theoretical bricks using different languages with slightly different meanings. (p. 169)
This motivates networking between different theories wherein the researchers work towards creating a
dialogue between different theoretical approaches while respecting the identity of the different approaches inmathematics education (Prediger & Bikner-Ahsbahs, 2014). Several studies have worked within the context
of networking theories for over a decade, and some have reviewed and reflected researchers’ networking
endeavours (Bikner-Ahsbahs & Prediger, 2014; Kidron et al., 2018). It appears that the latter is a metatheoretical study of the former. For instance, Artigue et al. (2011) use the notion of research praxeology
based on the Anthropological Theory of the Didactic (ATD); and Radford (2008) conceives a dynamic
cultural semiotic space as a semiosphere introducing a triplet comprising a system of basic principles,
methodology, and a set of research questions. In this space, networking of theories takes place and hence
may also inform the dynamic development of the semiosphere. More recently, Tabach et al. (2020) elaborated
Corresponding author: Yusuke Shinno

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Y. Shinno and T. Mizoguchi

on an argumentative grammar for networking theories by adopting Toulmin’s model of argument.
While such meta-level studies clarify the nature of networking theories, attempting them can be difficult
because characterising researchers’ practices and knowledge in networking requires a meta-theoreticalframework; however, this is still under development. Research praxeology has been adopted as a metatheoretical framework in this paper, along with some research examples of networking theories that havetaken place in the Japanese educational context, to contribute to the development of a framework forresearchers’ activities. Many researchers in mathematics education in Japan have worked on different
theoretical approaches developed in Western countries for their studies in the local context. However, only asmall number has explicitly described their networking endeavours. In this study, we attempt a retrospective
analysis, which aims to make the implicit explicit in terms of praxeologies. Thus, the research question inthis study is as follows: How can we characterise researchers’ theoretical work on networking endeavours
in terms of praxeology? Through this research question we expect to formulate a method to characterise thepraxeologies of networking theories in different types of studies. Additionally, this paper discusses some
cultural issues related to the Japanese educational context.

RESEARCH PRAXEOLOGY AND NETWORKING THEORIES
Elements of research praxeology
The concept of praxeology is one of the main constructs of the ATD (Chevallard, 2019). Using this concept,
any human activity can be understood as consisting of two blocks: praxis and logos. Each of these blocks has
two elements: type of tasks (T) and technique (τ) for the praxis block, and technology (θ) and theory (Θ) forthe logos block. A task (T) indicates a problem of a given type; a technique (τ) is a way of performing tasksof this type; a technology (θ) is a way of explaining and justifying the technique; and theory (Θ) explains or
justifies the technology. Artigue et al. (2011) extended the concept of praxeology to researchers’ practices
and knowledge (see also Artigue & Bosch, 2014; Artigue & Mariotti, 2014; Artigue, 2019). Researchpraxeologies of researchers’ activities also comprise four elements (T/τ/θ/Θ) (Artigue & Bosch, 2014). The
type of tasks (T) of a research praxeology often refers to research questions and problems to be studied; thetechnique (τ) comprises the research method that addresses the research questions; the technology (θ)
corresponds to methodological discourse that justifies the choice of the method and explains the results; andthe theory (Θ) includes the main principle and notions of a given theoretical framework.
Networking praxeologies
Networking praxeologies are used to understand the research practices of networking theories (Artigue et al.,2011). The praxis and logos blocks of networking praxeologies can be identified through retrospective
analysis of studies (Artigue & Bosch, 2014; Artigue & Mariotti, 2014; Artigue, 2019). However, suchanalysis is sometimes difficult to practice, because:[T]he theoretical block of networking praxeologies is also in the process of emerging, perhaps at the
moment it is in a less-developed state. (Artigue & Mariotti, 2014, p. 351)
Our study attempts to make the logos block of networking praxeologies more visible, based on ourNetworking Praxeologies and Theoretical Grain Sizes in Mathematics Education

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retrospective analysis of networking strategies and theoretical grain sizes. For this analysis, let us explain
here what we can locate as the four elements (T/τ/θ/Θ) of networking praxeologies.
The praxis block of a networking praxeology also corresponds to the networking questions or problems (T),and method (τ). Unlike research practice in general, the networking endeavour refers to two or moretheoretical approaches. The choice of theoretical frameworks and their usage are part of the type of tasks andthe techniques. The logos block of networking praxeologies is quite different from that of researchpraxeologies in general. The methodological discourse (θ) of networking theories describes, explains, andjustifies networking strategies. Prediger et al. (2008) propose a formulation in terms of: understanding andmaking understandable, comparing and contrasting, combining and coordinating, and integrating locallyand synthesising. However, the proposal of Artigue and Bosch (2014) includes other ingredients based on the
praxeological structure of research strategies. Finally, the theoretical discourse (Θ) explains and justifies the
networking strategies. Identifying the elements that qualify as a theory (Θ) of networking praxeologies isdifficult as “the technological and theoretical discourses are not fully articulated” (Artigue & Bosch, 2014,
p. 260) in the current state of networking praxeologies. This is probably because of the scientific challengeof sharing a common meaning of theory and the role of theory in mathematics education and why it isimportant – or even necessary for researchers – to network theories. Therefore, we attempted to identify
technological and theoretical discourses behind the networking endeavour. The theoretical grain sizes can beused to describe the praxis block of networking praxeologies as well as to interpret the implicit logos blockfor characterizing researchers’ discourses.
Theoretical grain sizes
The notion of grain sizes1 has been used to describe, review, and categorize different levels and magnitudesof theories in mathematics education (e.g., Kieran, 2019; Shinno & Mizoguchi, 2021; Silver & Herbst, 2007;
Watson & Ohtani, 2015). For examples, Silver and Herbst (2007) distinguished three theoretical levels:
grand, middle-range, and local theories. Kieran (2019) also distinguished between three theoretical levels in
the context of task design research: grand, intermediate, and domain-specific theories (or theoretical frames).
These categories allow us to understand and describe the nature and spectrum of a given theory in terms ofdifferent theoretical levels and magnitudes. However, it is not easy to determine the grain sizes of theories ina general way, because one could consider different categorizations from different perspectives.
For Silver and Herbst (2007), grand theories respond “to a need for broad schemes of thought that can help
us organise the field and relate our field to other fields” (p. 60); and middle-range theories grow “from theneed to inform a discrete variety of practices, including individual mathematical thinking, teaching, andlearning in classrooms, or mathematics teacher education” (p. 61). Local theories help mediate connections
between problems, research, and practices, in a particular study’s context. However, for Kieran (2019),
grand theories include the cognitive-psychological, the constructivist, the socio-constructivist, thesociocultural, and other general educational theories. Intermediate theories have a more specialised focus
1 Someone might interpret the meaning of the word grain sizes differently. In this paper, we used this term as a metonymy fordescribing and understanding different levels and magnitudes of theories in mathematics education. The use of this term issimilar to what Watson and Ohtani (2015) mentioned in the reviews of task design research: “Grain size descriptions areintended to be descriptive tools [emphasis is added] for thinking in a structural way about task design, rather than beingprescriptive” (p. 5)

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than grand theories and, as such, can contribute in a more refined way to the design of curricular areas (pp.
271–272). Domain-specific theories “deal with distinct mathematical concepts, procedures, or processes ofmathematical reasoning” (p. 272). The list below summarises Kieran’s (2019) categorisations with examplesof the theoretical grain sizes2.
●

Grand theoretical frames shape the background understanding of research in mathematics education(e.g., the cognitive-psychological, the constructivist, the socio-constructivist, the sociocultural)●

Intermediate level frames3 are located between the grand theories and the domain-specific frames (e.g.,
realistic mathematics education theory (RME), theory of didactical situations (TDS), ATD, lesson
study, variation theory, conceptual change theory)●

Domain-specific frames specify distinct mathematical concepts, reasoning processes, or tools (e.g., aframe for fostering mathematical argumentation within problem-solving, frame for proof problems withdiagrams, frame for learning algebra using technological tools)

Silver and Herbst’s (2007) categories are more general than Kieran’s (2019), despite both considering threelevels of grain size. This is because Kieran aimed to review and discuss different theoretical frameworks
which have been used in task design research, but Silver and Herbst argued for the role of theories inmathematics education research in general. More recently, Shinno and Mizoguchi (2021) distinguished threetheoretical grain sizes in the context of mathematics teacher education as follows:●

Grand theories are well-established frames for research inside and outside mathematics education thathave been developed in a broader context of human activity (e.g., ATD, CHAT)●Intermediate theories are frames that do not specify a feature, but rather a general aspect of teachers’
knowledge or activities (e.g., knowledge for teaching, professional growth, documentational approach)●

Local theories are frames that specify a particular feature of teachers’ knowledge or activities (e.g.,teacher noticing, teacher design).

In this way, the method of distinguishing the grain sizes of theories cannot be absolute but is rather relative
to the research area of the study being undertaken. For the analysis in this paper, we re-categorize Kieran’s(2019) distinctions of grain sizes into three levels (grand, intermediate, and local) by generalising the above
descriptions of the three levels of Shinno and Mizoguchi (2021). In Table 1, there are a few critical points
when re-categorizing the earlier version by Kieran (2019). Despite the local-level being almost identical todomain-specific frames in Kieran (2019), we reconceptualised the other two levels to distinguish the degreeof generality of a given theory in mathematics education. For instance, while both ATD and TDS werecategorised as intermediate-level frames by Kieran (2019), we acknowledge the differences between the twotheories in terms of grain size. Nonetheless, our categorisations cannot distinguish theories for research
inside and outside mathematics education (e.g., ATD and variation theory) as we do not consider theories
developed in other disciplines in this study. In addition, we excluded “the background understanding of2 In Kieran (2019), what she called the grand theoretical frame is based on Cobb (2007). Examples for the intermediate andthe domain-specific frames includes several frames used in the task design research. While Kieran (2019) includes references
to each frame or study, we omitted the references in this table.
3 Kieran (2019) provided an additional explanation of the intermediate-level frames referring to their roots as follows: “Inaddition, intermediate level frames can also be characterized according to whether their roots are primarily theoretical orwhether they are based to a large extent on deep craft knowledge. An example of the former is the Theory of Didactical
Situations and the latter, Lesson Study” (Kieran, 2019, p. 272).

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research in mathematics education”, which might refer to a general philosophical standpoint, from thedescription of grand-level theories. It is because we think background theories (such as constructivism)
providing general principles that shape the universe of what can be a researchable object are relevant to alllevels of grain sizes 4.
Table 1. (Re)categorizations of different theoretical grain sizes

Grain sizes of theories
Descriptions
Grand-level
well-established frames for conducting
research inside and outside mathematics

ExamplesATD, commognitive framework,
variation theory, etc.

education, that have been developed in the
broader context of human activity
Intermediate-level

frames that do not specify any particular

TDS5, theory of realisticknowledge or activities but specify amathematics education (RME), etc.

context rather than a general aspect of theresearch object to be studied
Local-level

frames that specify a particular

A model of problem solving, vanmathematical or didactic knowledge orHiele levels, etc.

activity

METHOD
Seeking an interface between the logos and praxis blocks is an appropriate way to develop the researchers’
discourse of networking praxeologies (Artigue & Bosch, 2014). Three research examples, utilizing multiple
theoretical frameworks for different purposes, are used to illustrate the praxis and logos blocks of networking
praxeologies. The three case studies involved different types of research (comparative study, empirical study,
and curriculum development). While these studies can be considered as cases of networking theory research,
the networking strategies used are not explicitly mentioned in these papers. The theoretical grain sizes werealso implicit. Therefore, we must interpret these implicit aspects. Thus, we selected research examples fromstudies that have been conducted in the Japanese educational context (such as lesson study, classroom
teaching, and curriculum development). One study (Example 1) was selected because it involved a type ofstudy (lesson study) that is culturally situated in Japanese educational research. The other two studies(Examples 2 and 3) were conducted by the author(s), for which we can analyse their theoretical aspects4 For example, Shvarts and Bakker (2021) considered the grain sizes of levels in terms of the philosophical and historical rootsof theories under considerations. They distinguished six different levels by means of vertical analysis for networking
theories: such as epistemic presumption, ontological presumption, grand theory, local theory in mathematics education,
application for teaching, and application for educational design. From this vertical perspective, background theories refer tohigher levels of a certain level of theories.
5 Some researchers would also call TDS a grand-level theory as it can be applied to any situation wherein teaching andlearning take place. Therefore, it should be noted that the distinctions or categorisations of different theoretical grain sizes
are not reserved for the ones mentioned in Table 1.

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retrospectively. While the three research studies are analysed as illustrative examples, they cannot be
conceived as typical or representative, and the results of the analysis cannot be generalised. Our attention isdrawn to the different praxeological models of the research on networking theories. Thus, the primary aim of
this analysis is to make implicit discourses in the logos block explicit by focusing on the theoretical grain
sizes of each theoretical framework to be networked. It allows us to understand the interface between thelogos and praxis blocks of networking praxeologies.
Regarding the praxis block of networking praxeologies, we identify two elements (T/τ) by analysing theaims, questions, and methods of networking in the papers. For the types of networking tasks (T), we
especially focus on the theoretical frameworks and/or models being used for networking, rather than practicalmethods of procedures or analyses in implemented studies (such as data collection or data analysis). Then,the networking strategies used in each study are identified as techniques (τ). Concerning the logos block of
networking praxeologies, the technological discourses (θ) to explain and justify the networking strategies areinterpreted by referring to some relevant quotations from each paper. To better understand the theoretical
discourses (Θ) underlying the technologies in each study, we discuss the role of cultural issues in networking
praxeologies; for example, what is a theory in different educational research traditions? What is the role oftheory regarding educational practice? Why is it necessary to connect or reduce the number of theories? For
such praxeological analyses of research on networking theories, the theoretical grain sizes categorized inTable 1 may play a determinant role for both the praxis and logos blocks.

ILLUSTRATIVE RESEARCH EXAMPLES
Three research examples
Example 1 – Comparing and contrasting
Miyakawa and Winsløw (2009) compared two didactical designs to introduce primary school students toproportional reasoning. The designs were based on two different approaches: didactical engineering inFrance, and lesson study in Japan. The comparison is informative as both approaches are used for planninga lesson, but the method of designing is very different. In the former, the lesson was designed according to
the theoretical principles proposed in a specific theory; however, in the latter, the lesson was created according
to the teachers’ experiences in the classroom. Didactical engineering is a methodology for TDS introduced
by Brousseau (1997) to gain theoretical insight into the functioning of a didactic system. In contrast, lesson
study does not necessarily involve an explicit didactic theory, but often refers to a certain practical orpedagogical approach as a theoretical basis for teachers’ practice. Miyakawa and Winsløw (2009) referred toa Japanese teaching approach called open approach method by Nohda (2000), which is relatively known asa teaching method to enhance multiple ways of thinking in the process of problem-solving, and contrasted itwith the notion of fundamental situation6 in TDS through their analysis and observations. Below, someelements in the process of designing a lesson to introduce students to proportional reasoning are summarisedand contrasted according to theoretical basis, design formats, and realised lessons.
6 A fundamental situation for a concept is a mathematical situation for which the concept constitutes a priori an optimal
solution (Artigue et al., 2014, p. 49).

Networking Praxeologies and Theoretical Grain Sizes in Mathematics Education

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●

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A Japanese case○

Theoretical basis: Open approach (Nohda, 2000)○

Design formats: Lesson study○

Realised lesson: Hatsumon (questioning)7 to enquire about an open problem (Nohda, 2000)

A French case○

Theoretical basis: Fundamental situation (TDS) (Brousseau, 1997)○

Design formats: Didactical engineering○

Realised lesson: adidactical milieu8 in the puzzle situation (Brousseau, 1997)

According to Miyakawa and Winsløw (2009), a comparison of the two didactical designs shows crucial
similarities and differences in a lesson in the following ways.
Both of the designs emphasize the social interaction and independent thinking of students. Both formats
for design require quite similar kinds of analysis, including anticipating student strategies and revising
the design in an experimental cycle. […] In a fundamental situation, they should lead to thepersonalization and institutionalization of a target mathematical knowledge (savoir), consistent withthe ‘official’ mathematical knowledge. […] In the open approach, the aim is for students to apply andtest their mathematical knowledge, through two main processes: the process in which some conditions
and hypotheses from a ‘real world’ problem are formulated mathematically and the process ofgeneralization and systematization after solving a problem. (Miyakawa & Winsløw, 2009, p. 216)
Miyakawa and Winsløw (2009) also discussed how didactical engineering and lesson study differ at the levelof their objectives. Lesson study provides an opportunity for teachers to develop their teaching professions;the main objective is to improve lessons. In contrast, didactical engineering aims to establish scientificknowledge wherein a lesson confirms or rejects/questions the conditions for learning.
Example 2 – Combining and coordinating
Shinno (2018) aimed to characterise the development of mathematical discourses in a series of lessons interms of the semiotic chaining model (triadic nested model) proposed by Presmeg (2006) and the commognitive
framework introduced by Sfard (2008). There were two reasons for using two different theoretical frameworks
in this study. The first is related to the content-specific aspect of students’ difficulties of reification in thelearning of square roots. Shinno (2018) argued that the cognitive account of reification can be used to explain
this difficulty; however, conceptualising it differently from semiotic and discursive points of view may allow
us to arrive at a deeper understanding of the reification phenomenon. The second reason is related to thesetting of the mathematics classroom used in this study. Based on the observations of earlier studies on
Japanese classroom culture (e.g., Emori & Winsløw, 2006), Shinno (2018) mentioned that “the social
interaction between the teacher and the students in a Japanese mathematics classroom can be assumed toconstitute a culturally unique discursive community” (p. 279), and that the semiotic and discursive approaches
can be suitable for analysing the process through which students become familiar with the usage and meaningof square roots through classroom interactions. For example, Shinno (2018) analysed how a new expression
7 Hatsumon (in Japanese) is a general pedagogical term which refers to a teacher’s key questioning in a lesson.
 Within the TDS, milieu is a component of a didactical triangle (teacher, students, and milieu), which constitutes a didacticalsituation. In an adidactical situation, “students accept to take the mathematical responsibility of solving a given problem, andthe teacher refrains from interfering and suggesting the target mathematical knowledge” (Artigue et al., 2014, p. 51). The
adidactical is referring to “the situation has been temporally freed from its didactical intentionality” (ibid.)

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Y. Shinno and T. Mizoguchi√2 ＋√3 becomes a mathematical object rather than a computational process through classroom interactions
to justify or reject a teacher’s question ‘is √2 ＋√3 ＝√2＋3 true?’As shown in Table 1, Shinno (2018) attempts to coordinate the commognitive terms9 with semiotic terms10
in order to gain a multidimensional insight of the reification phenomenon through the networking of the twotheoretical approaches. Shinno (2018) proposed two theoretical benefits of using two theoretical lenses in thestrategy of combining and coordinating:From the viewpoint of the commognitive framework, the implicit meta-discursive rule may become
explicitly identified as the third component of ‘interpretant’ by means of the triadic nested model. Fromthe viewpoint of the triadic nested model, the first and second components, the signifier and signified,can be characterised as distinct features, such as keywords, visual mediators, and endorsed narratives.
(Shinno, 2018, p. 302)
Table 2. Coordinating theoretical terms from the two different frameworks

Commognitive framework (Sfard, 2008)

Triadic nested model (Presmeg, 2006)

Word use (keywords)

Signifier or signified

Visual mediators

Signifier or signified

Endorsed narratives

Signifier or signified

Routines (meta-rules)

InterpretantNote. Adapted from “Reification in the learning of square roots in a ninth-grade classroom: Combining
semiotic and discursive approaches” by Y. Shinno, 2018, International Journal of Science and Mathematics
Education, 16(2), p. 302.

In this empirical study, the semiotic chaining model was used to identify the three components (signifier,signified, and interpretant) in the classroom episodes and then characterised by their discursive features froma commoginive point of view. For example, when a signifier √2＋√3 was reified (objectified), an interpretant‘treating √2 ＋√3 as one irrational number’ was also understood as a meta-discursive rule. In this way,
Shinno (2018) sought to use theoretical terms from the two different frameworks (Table 2) interchangeably
for the analysis. Thus, combining and coordinating the two theoretical frameworks allowed him to gain amulti-faceted insight into the reification phenomenon, meaning a change from process-oriented use to object-

9 The meanings of the commognitive terms are as follows: word use refers to mathematical vocabulary, syntax, and ordinary
words associated with mathematics; visual mediators include physical, diagrammatic, and symbolic mediators ofmathematical objects; endorsed narratives mean a set of mathematical statements, proofs, rules of calculation, which areaccepted within a given community; and routines refer to regularly employed and patterned repetitive activities (e.g.,
calculating, proving, generalising) as well as a set of meta-rules (e.g., how to calculate, how to prove, how to generalise).
10 The model of semiotic chaining by Presmeg (2006) based on Lacan’s inversion of Saussure’s dyadic model (signifier andsignified) as well as Peirce’s triadic model (object/signified, representation/signifier, and interpretant). For this model, therelationship between signifier and signified is considered to have a nested structure, and meaning-making in the twocomponents is understood by the third component interpretant.

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oriented use of a signifier11.
Example 3 – Locally integrating
Shinno et al. (2015, 2018) proposed a theoretical framework for curriculum development of proof insecondary schools in Japan. Unlike the other two research examples, the theoretical framework undertakenin their paper was aimed at being used for developing, designing, or improving a curriculum related tomathematical proof. For this aim, the theoretical framework was constructed by adapting and integrating
different theoretical models and concepts as follows:●

The Mathematical Theorem as a system, consisting of statement, proof, and theory (Mariotti et al.,1997)12●A model of mathematical proofs, comprising knowledge, formulation, and validation (Balacheff,1987)13●

The concept of local organisation and global organisation (Freudenthal, 1971, 1973)14●

The concepts of small theory and large theory (Hanna & Jahnke, 2002)15

On the one hand, the theoretical framework proposed by Shinno et al. (2015, 2018) was a framework for
curriculum development, describing and prescribing mathematical contents and sequences in the Japanese
educational context. On the other hand, it seems that the theoretical models and concepts to be integrated
have different theoretical natures and are not always considered the ones for educational development. For
example, the notion of Mathematical Theorem (Mariotti et al., 1997) is a model to understand mathematicalpractice wherein a proof is carried out, but originally this model does not prescribe curricular contents andactivities to be taught in schools. Therefore, when constructing the framework, Shinno et al. (2015, 2018)reconceptualized some concepts, adapting (or detaching from) their original senses, and integrated them into
one framework so that it is relevant to the Japanese curriculum for proof. Let us briefly explain how they
adapted and integrated different theoretical concepts.
In their framework, the three elements – statement, proof, and theory – were used as the foreground inshaping the framework. Balacheff’s (1987) categorisation of formulation and validation was used tounderstand levels of statement and proof. While the original notion of theory in Mathematical Theorem
11 Regarding the implications of using the two frameworks, Shinno (2018) concluded: “[This study] enables to characterizereification as the change in a set of meta-rules, which means the replacement of the process-oriented use with the objectoriented use of a new signifier. Although the model of semiotic chaining itself conceptualizes reification as the act of making
signifier-signified couple in each node, as far as the reification in the discursive process is concerned, it may be important todetect a set of interpretants (meta-rules)” (Shinno, 2018, p. 311).
12 A Mathematical Theorem consists of a system of relations between a statement, and its proof, and the theory in which theproofs make sense. “[The] existence of a reference theory as a system of shared principles and deduction rules is needed if
we are to speak of proof in a mathematical sense” (Mariotti et al., 1997, p. 182).
13 Balacheff (1987) proposed a theoretical framework composed of knowledge, formulation, and validation, which allow us
to analyse the complex nature of proof and proving. The four levels (naïve empiricism, crucial experiment, generic examples,
and thought experiment) are regarded as the levels of validation.
14 The local organisation by Freudenthal (1973) is a concept to explain short deduction chains consisting of a statement and its
proof in geometry, wherein some properties can be accepted as taken for granted, while the global organisation is a concept
to explain an axiomatized system (such as a system of Euclidean geometry).
15 The distinction between small theory and large theory is similar to that between local and global organization. Hanna andJahnke (2002) mention that “instead of building a large theory (namely, Euclidean geometry) in the course of the curriculum,
it seems to be more appropriate to work in several small theories” (p. 3). Thus, small theory and large theory are considered
concepts rather than part of an operational framework.

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Y. Shinno and T. Mizoguchi(Mariotti et al., 1997) mainly refers to mathematical theory (such as arithmetic, algebraic, or geometricaltheories), Shinno et al. (2015, 2018) reconceptualised it as a component which includes different layers of
the system (such as local and axiomatic theory) to consider the wide range of contents and levels in the
curriculum. They adapted the notions of Freudenthal (1971, 1973) and Hanna and Jahnke (2002).
Their reconceptualization of the element (i.e., theory) of Mathematical Theorem was needed to account for
the axiomatic aspect of geometry curriculum in secondary schools in Japan, as the Japanese geometry
curriculum emphasizes Euclidean geometry (Miyakawa, 2017; Shinno et al., 2018). Miyakawa (2017)
characterised this nature of the system in secondary school mathematics as quasi-axiomatized geometry withrespect to some characteristics such as “the term axiom or postulate is not used; some properties are
introduced after observation; some are admitted implicitly and there is no long list of axioms as in Euclid’s
Elements” (p. 49). This is also related to what Hanna and Jahnke (2002) distinguished as small theory andlarge theory.
Shinno et al. (2018) called a theoretical framework for curriculum development, the reference epistemologicalmodel (REM) and leaned on the elaboration of RME by Bosch and Gascón (2006), which constitutes thebasic theoretical lens for researchers to analyse different types of mathematical knowledge among different
institutions. Thus, Shinno et al. (2018) attempted to integrate different theoretical constructs regarding proof
into one framework (REM), wherein an internal consistency within the framework is created to understandcontents and levels in a curriculum, summarised as follows:
Regarding the proposed REM, we conceptualized that the statements and proofs are interrelated
according to the nature of theory and illustrated how this model helps to describe the evolution of eachelement. This model may help us understand the gaps in the evolution of the formulation of a statement
and how different meanings of proof relate to the distinction between local and (quasi-) axiomaticlevels of theory. (Shinno et al., 2018, p. 30)
For example, there are three levels of the formulation of a statement, i) drawing, diagrams, manipulation, andgesture; ii) ordinary language and words; iii) mathematical words and symbols. Considering a universal
proposition (statement), a universal quantification can be formulated by ordinary words (e.g., all, any) inschools and by mathematical symbols (e.g., ∀) at advanced levels. Based on the proposed framework,
Shinno et al. (2018) implied that there is a gap in the levels of formulation in the Japanese curriculum(especially between the ordinary and mathematical language of a universal proposition), and there is a crucialtransition between the levels for development16.
From the perspective of networking praxeologies
Our retrospective analysis of the three case studies in terms of networking praxeologies and theoretical grain
sizes can shed light on the principal aspects of networking in each study (as summarised in Table 3), and thus
enable the comparison and discussion of the praxis (T/τ) and the logos (θ/Θ). The types of tasks (T) were
found in the choice of the theoretical frameworks in the papers. The techniques (τ) were identified byfocusing on the networking strategies that were most relevant to the studies (based on our interpretations andretrospections). In Example 1, the strategy of comparing and contrasting was chosen and the similarities and16 Regarding linguistic issues, we also discussed how Japanese language may affect the difficulty and gap in the levels of theformulation of quantifications (Shinno et al., 2019).

Networking Praxeologies and Theoretical Grain Sizes in Mathematics Education87

differences between the French and Japanese approaches (TDS and open approach) to didactical designs are
examined. In Example 2, the strategy of combining and coordinating is used to analyse classroom teachingwith a commognitive framework, and with the semiotic chaining model. In Example 3, local integration isused to construct a theoretical framework for the curriculum development of proof in secondary schools inJapan. The strategy in this case also offers a coordination of theoretical concepts (such as the relationshipbetween Freudenthal’s local/global organizations and Hanna & Jahnke’s small/large theories). We think thisis natural since coordinating is a strategy that “can be a starting point for a process of theorizing” (Prediger
et al., 2008, p. 173).
Table 3. Networking praxeologies and theoretical grain sizes in the three research examples

Example 1
To compare TDS and open

Example 2
To combine the model of

Example 3
To construct a theoretical

approach method

semiotic chaining and theframework for curriculum

commognitive framework

development, using theT

Mathematical Theoremτ

A strategy of comparing andA strategy of combining andA strategy of ‘coordinating

contrasting in terms of thecoordinating in terms of theand integrating locally’ intheoretical basis, design

theoretical terminologies

terms of statement, proof,

format, realized lesson

between two frameworks

and theory(see Table 2 (Shinno, 2018,p. 302))

θDiscourses on similarities

Discourses on the

Discourses on how to

and differences on thecompatibilities andintegrate different theoretical

didactic designs based on theinterchangeabilities between

elements, levels, andtwo approaches (see thethe semiotic and theconcepts into a common

quotation above (Miyakawa

commognitive terms

framework

Intermediate-level (TDS) andGrand-level (commognition)

Local-level (all theoretical

local-level (open approach)and intermediate-level

constructs)& Winsløw, 2009, p. 216))

GSa(semiotic chaining)
GS; grain sizesa

To determine the elements of the logos block, we needed to interpret the descriptions explicitly or implicitly
written in the papers. We considered the technological discourses (θ) by focusing on the descriptions to
explain and justify how two or more theories were networked by a certain strategy and why the strategy wasmost relevant to the study undertaken. In the previous subsection, we tried to show such descriptions by thequotations or elaborations from the papers. Table 3 included summarised technological discourses. However,
it is more difficult to determine the theoretical discourses (Θ) because they are mostly implicit in the papers.

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Y. Shinno and T. Mizoguchi

We considered such theoretical underpinnings by focusing on cultural elements included in each study; forexample, what is regarded as a theory in the Japanese educational and research context? What is the role oftheories regarding educational practice? What is the influence of the educational system in attempting toanswer these questions? We develop the discussion related to the cultural issues in the section titled‘Discussion and conclusion’.
From the perspective of theoretical grain sizes
To better understand the networking praxeologies in each study (Table 3), different theoretical grain sizes
were considered. Based on our categorizations of grain sizes mentioned earlier, TDS from Example 1 isconsidered an intermediate-level theory, since it is often used to understand and analyse the epistemologicalprocesses in mathematical classrooms from a systemic perspective. TDS is not a domain-specific theory
which specifies any mathematical knowledge or activity. Rather, it is primarily developed to model thefunctions and mechanisms between teachers, students, and the didactical milieu in a didactical/adidacticalsituation (in the classroom). In contrast, the open approach is seen as a local-level framework or instructional
principle which does not specify any mathematical content but does specify teachers’ teaching methods. For
instance, using an open problem is a characteristic aspect of this approach which is comparable with thenotion of fundamental situation from TDS (Miyakawa & Winsløw, 2009). However, TDS and the open
approach are not fully comparable because the latter does not provide a tool to analyse teaching and learningof mathematics at a general level (e.g., the approach cannot be applied to a lesson with a closed problem).
Therefore, the two frameworks are comparable only if one can focus on a specific notion of one approachthat has a counterpart in the other (such as a fundamental situation and an open problem). Thus, the discourses
on two didactical designs, as cited above from Miyakawa and Winsløw (2009, p. 216), have focused on somecomparable constructs from each framework.
In Example 2, the commognitive framework is viewed as a grand-level theory which is a comprehensive
theory of learning from a discursive point of view. The model of semiotic chaining originates from Peirce’s
semiotics. However, it was introduced by Presmeg (2006) into mathematics education for analysing therelations between signifier, signified, and interpretant in the teaching and learning of mathematics. Thismodel was used as an intermediate-level framework in Example 2, although one could also consider it as acollection of theoretical concepts. Shinno (2018) combined commognitive and semiotic approaches toprovide different accounts of reification in mathematical learning. Since the commognitive frameworkcomprises a much broader range of theoretical constructs, coordination with semiotic chaining is related toonly some semiotic parts of this grand theory. This implies that the strategy of coordinating between different
grain sizes requires an explicit discourse on complementarity; for example, which part of one theory can becoordinated with another theory.
Networking in Example 3 offers local integration, but the connectivity of different theoretical concepts israther complex. All theoretical models or concepts used in the study are regarded as local-level frames whichhave been developed in the context of research on proof and proving. However, this does not mean that all
the theoretical constructs have the same theoretical grain size. For example, the triplet (statement, proof,
theory) has been used as a foreground frame to integrate other theoretical concepts. There have been many
discourses on argumentative connectivity about how to integrate different models or concepts since Shinno

Networking Praxeologies and Theoretical Grain Sizes in Mathematics Education89

et al.’s (2015, 2018) attempt to propose a theoretical framework. A characteristic aspect of such a discussionis that it changes or extends the original meaning of the theoretical terms (statement, proof, theory) to thosewith a broader sense; this allows the inclusion of different contents and levels in the curriculum in secondary
schools in Japan. Additionally, they created new terminologies (for the three different layers of the system;
the logic of the real world, local theory, and quasi-axiomatic theory) through the adaptation of existing
notions (such as small/large theory)17.

DISCUSSION AND CONCLUSION
A deeper understanding of researchers’ work on networking theories
The praxis block corresponds to how researchers work when they choose different theories and network
them. The researchers may consider the grain sizes of theories according to the types of networking tasks (T)for the study undertaken. We then considered networking strategies as a technique (τ) and explored thetechnological discourse (θ) to explain and justify the strategy through a retrospective analysis of threeillustrative research studies as examples. Artigue and Bosch (2014) argue that the characterisation of thelogos block (θ/Θ) of a networking praxeology has not yet been established. In our study, we offered thenotion of theoretical grain sizes to deepen our understanding of the networking praxeologies. As mentioned
in the previous section, the networking techniques are affected by the grain sizes of the theories involved.
While we have only described the grain sizes (e.g., grand, intermediate, or local levels) of the theoreticalframeworks used in the three research examples shown in Table 3, attention was paid to the discourses on
networking different theoretical frameworks, which may have different sizes or are of the same grain sizes.
The characteristics of such networking praxeologies may differ according to the study type, such as empiricalstudies, design studies, or theory development. For example, Bikner-Ahsbahs and Prediger (2014) discussed
the notion of perspective triangulation (Denzin, 1970), meaning a research practice to increase the validityof empirical analysis. Although this notion does not refer to theories, increasing the validity of empiricalstudies can bring forth a benefit from such networking. Combining theories in/for empirical studies often
provides deeper insights into complex phenomena (Prediger et al., 2008), wherein the results of the empiricalanalysis can respond to the original theories. Bikner-Ahsbahs and Prediger (2010, 2014) also mentioned
other perspectives (explicitness, empirical scope, stability, and connectivity) to discuss the empirical andtheoretical benefits of networking theories for different types of studies. Some of these perspectives are
related to what we identified as technological discourse (a justification of the strategies involved in each casestudy). For instance, perspective triangulation or connectivity can support the strategy of combining andcoordinating in Example 2; connectivity can explicate the strategy for locally integrating in Example 3; andexplicitness can be the strategy for comparing and contrasting in Example 1, as well as the strategy for17 For example, a geometric statement “the sum of the interior angles of a triangle is 180°” can be explained by a physicalapproach (such as a measurement or experiment) which is acceptable within the logic of the real world. The same statement
can be proved by previously accepted properties of parallel lines, based on local theory which is often regarding one
particular proof. The quasi-axiomatic theory can be relevant when considering a system of statements and proofs in schoolgeometry (even though the term axiom or postulate is not introduced). More descriptions of the three layers are mentionedin Shinno et al. (2018).

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locally integrating in Example 3. In addition, it is also worth considering how theoretical terminologies maychange or preserve their meaning before and after networking attempts when examining linguistic
connectivity (Shinno, 2017). As shown in Example 2, by coordinating, the theoretical terms in one framework
are interchangeable with those in another framework. In Example 3, some original concepts are locallyelaborated and integrated into new terminologies in the new framework for curriculum development;
however, such new terms do not preserve their meanings in the original sense.
Cultural issues to be considered
Networking praxeologies can provide a model for understanding researchers’ practices and knowledge of
networking theories. This implies that it is important to consider the diversity of research activities related totheories of mathematics education. One of the reasons for this diversity is cultural issues, which may affectthe researchers’ work. This is also the case in the three research examples involving Japanese educational
contexts. If the theory (Θ) in the logos block of networking praxeologies is implicit, its description can only
be based on the researchers’ conception of a theory in research and practice.
For instance, Example 1 indicated the differences between TDS and the open approach; where the open
approach was a practice-oriented theory or a prescriptive theory, which is often used to improve teachers’classroom teaching and facilitate students’ problem-solving activities. In fact, both the descriptive and theprescriptive nature of a theory in mathematics education are often discussed in Japan (e.g., Koyama, 2006;
Yamada, 2011). This is related to what has been discussed in the previous Topic Study Group in ICME-13(Dreyfus et al., 2017); for example, “[i]n Japan, ‘theory’ is always a theory of some practice. A practicedevelops, somebody notices it, reflects upon it, and constructs a theory of practice” (p. 617). This implies two
distinct aspects. One is that a theory emerges and develops from/for some practice. The other is that a theory
is very close to practice and the theory often seems to be a pragmatic one18. However, in the international
context, some researchers might not or cannot accept such a pragmatic theory as a theory in mathematics
education (cf. Miyakawa & Shinno, 2022). Thus, there is miscommunication among international researchers
regarding what is called theory in Japanese educational research (e.g., Zazkis & Zazkis, 2013). This may
open a discussion that reveals the cultural specificity of Japanese (or East Asian) educational research, which
contrasts with the French (or European) tradition of didactics (Blum et al., 2019). Here, we do not discuss
the nature of theory in different research cultures (although such discussion is promising, it lies beyond thescope of the paper), but merely wish to point out that there is a cultural issue that may shape researchers’
knowledge, even in the context of networking theories.
In addition, in Example 2, Shinno (2018) considered the cultural specificity of the Japanese mathematics
classroom to be related to semiotic activities (Emori & Winsløw, 2006); however, this specificity of classroom
is generally attributed to the Japanese teaching pattern (Stigler & Hiebert, 1999). Such a cultural factor may
influence the researcher’s choice of theory or his/her perspective of using a theory for classroom study.
Recently, Funahashi and Hino (2014) have also investigated the interactive classroom pattern, called a
guided focusing pattern, which is often observed in Japanese classroom lessons, in terms of a discursive orcommognitive perspective for their theoretical and methodological framework. Although other theoretical18 Here we use the term pragmatic as an ordinary word without any theoretical underpinning.

Networking Praxeologies and Theoretical Grain Sizes in Mathematics Education

91

approaches can be used to analyse a classroom culture, what we would imply is that the researchers’
perspective of using a theory (or multiple ones) can be affected by the local educational context and by the
researchers’ conceptions of how the theory can be useful for investigating such a cultural specificity.
Furthermore, in Example 3, the specificity of the Japanese mathematics (geometry) curriculum in secondary
schools was considered when adapting some theoretical concepts that were integrated into a framework for
curriculum development. This also implies that the educational contexts (particularly the curriculum) in acountry may affect researchers’ work on theories. For example, given that the Japanese geometry curriculum
is characterised as quasi-axiomatized geometry (Miyakawa, 2017), we could adopt or adapt the related
concepts (such as, local/global organization, small/large theory) in their framework. In addition, developmental
research, which is often aiming at improving educational practices (including task design, curriculum
development, or lesson study) is a dominant type of research and/or study in Japan. Although a theoreticalframework is often constructed for such developmental work, what is important for the framework is to be
useful for development, rather than for scientific research (Miyakawa & Shinno, 2022). This is one reason
why the proposed framework in Example 3 referred to different constructs (models, concepts, frameworks)
to be integrated. For this reason, the researchers see all constructs as theoretical underpinnings of theframework for development, even though some constructs are not usually brought forth as theory for
research. This may have the implication that one can distinguish between theory for research and theory forpractice.
Finally, we attempted to translocate the cultural issues (such as research culture, classroom culture, andcurriculum culture) into the theoretical discourses (Θ), as shown in Table 4, which are implicit in the
networking praxeologies (Table 3). Furthermore, the logos block of networking praxeologies is also related
to the researchers’ conception of the field of mathematics education and the nature of mathematics education
as a scientific discipline. This raises at least two questions: How can we make such implicit discourses more
explicit in networking theories? Why is such a meta-level and a self-referential study important for
researchers? While we believe that the cultural issues in different educational research traditions play a
determinant role, further research is needed to address these questions and issues.
Table 4. Theoretical discourses (Θ) regarding cultural issues in the three research examples

Θ

Example 1
Discourses on the nature of

Example 2
Discourses on a culturally

Example 3
Discourses on the necessity of atheories in different research

unique discursive community in theoretical framework for

traditions (research culture)a Japanese mathematics

curriculum development inclassroom

Japanese mathematics education

ACKNOWLEDGEMENTS
This work was supported by JSPS KAKENHI, Grant Numbers JP20K02913 and JP18H01015. An earlier
version of this paper was presented at the Topic Study Group 57 related to the diversity of theories in ICME14 in 2021.The authors would like to thank the reviewers for their helpful comments to improve the paper.

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Yusuke Shinno

Tatsuya Mizoguchi

School of Education, Hiroshima University, Japan

Faculty of Regional Sciences, Tottori University, Japan

Email: shinno@hiroshima-u.ac.jp

Email: mizoguci@tottori-u.ac.jp

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

https://orcid.org/0000-0002-4399-8988