# International transposition of lesson study in mathematics

Koji Otaki

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# International transposition of lesson study in mathematics:
A report of the case involving the theory of didactic situations

Koji Otaki / Hokkaido University of Education, Japan

**Abstract:** Within the framework of the anthropological theory of the didactic, the present study investigates into the institutional transposition from lesson study in mathematics (LSM) in Japan to a European counterpart. For that, in the first part, I focus on several examples from some LSM projects alongside the theory of didactic situations (TDS). The similarity and dissimilarity between the Japanese original LSM and the LSM with the TDS are analyzed by a reference praxeological model, that is also created in this paper. Main findings are: 1) despite the ostensible replacement of theoretical frameworks, the transposed LSM keeps some fundamental assumptions; and 2) the participants' behavior pattern in LSM, too, is essentially preserved even though part of it is inconsistent with the TDS's posture on research. And then, in the second part, we discuss a cultural condition under which LSM steadily lives in the Japanese teaching profession.

**Keywords:** Anthropological theory of the didactic (ATD) · Praxeology · Institutional transposition · The paradidactic · Didactic paradigm · Teaching profession · Sustainability of lesson study

## 1. Introduction: studying the international diffusion of lesson study in mathematics

Since the publication of *the teaching gap* (Stigler & Hiebert, 1999), Japanese “lesson study” has been attracting internationally mathematics educators’ and mathematics teacher educators’ interests as a promising format for allowing schoolteachers to develop professionally. Consequently, *lesson study in mathematics* (LSM) has been conducted in many counties outside Japan (e.g., Huang, Takahashi, & Ponte, 2019; Quaresma et al, 2018). Within the framework of the ATD, i.e. the *anthropological theory of the didactic*, this paper investigates into such international diffusion of LSM. Especially, I will focus on a special kind of such diffusion of LSM together with the TDS, i.e. the *theory of didactic situations*, which is different from a typical theoretical framework of the Japanese original LSM called the *structured problem-solving approach* (Stigler & Hiebert, 1999). There are two reasons why I study LSM with the TDS. Firstly, some previous studies discussed the relationship between LSM and the TDS through the notion
of didactic engineering (e.g., Clivaz, 2015; Miyakawa & Winsløw, 2009). Secondly, the TDS is a relatively understandable framework for me among different non-Japanese theories which are networked with LSM. Here is my first research question: *how is LSM diffused together with the TDS into European countries?*

Moreover, this paper is also interested in the *sustainability of lesson study* outside Japan, which are, implicitly or explicitly, crucial for every project which aims to import lessen study from Japan (e.g., Østergaard & Winsløw, 2021). For approaching this, in the second part of this paper, I will investigate into a cultural condition under which the Japanese teaching profession has sustained lesson study so far. There are several papers which focus on the cultural aspect of LSM (e.g., Capone et al., 2024: Chen, 2017; Isoda & Baldin, 2023), but they seem to be much more interested in cultural conditions for the *structure and functioning* of LSM than cultural conditions for the *sustainability* of LSM. Here is my second research question: *what is a cultural condition for the sustainability of LSM in Japan?*

Before entering the next section, for the sake of readability, I want to list up four acronyms which frequently appear in this paper.

ATD: Anthropological Theory of the Didactic

LSM: Lesson Study in Mathematics

TDS: Theory of Didactic Situations

SPSL: Structured Problem-Solving Lesson

## 2. Theoretical framework

First and foremost, I want to highlight that the TDS is *not* an analytic tool of this paper, but a part of the *object of study*, which is supposed to be analyzed from the perspective of another framework. The theoretical framework of this paper is the ATD. Especially, I mainly draw on the following four notions: *praxeology*, *institutional transposition*, the *paradidactic*, and the *teaching profession*.

### 2.1 Praxis, logos, and a praxeology

Within the framework of the ATD, the reality of knowledge is modeled through describing and articulating the system of *praxeology* without forgetting that every item of knowledge is produced for doing something. In the ATD's vocabulary, the word “praxeology” does not mean the philosophical theory of human action, but is a form of systems to be studied in didactics in which any human activity is involved (cf. Chevallard, 2019). Any praxeology expresses a complex of its
praxis part for actions and its logos part for discourse while following its etymological construction: praxis + logos = praxeology. The praxis part Π of a determined praxeology φ consists of the type of tasks T which includes all tasks tᵢ brought together into T by their own pattern (T = {tᵢ}), and the technique τ which signifies the way for accomplishing tᵢ in (part of) T: Π = [T / τ]. And, the logos part Λ of φ consists of the technology θ as a discourse for explaining, justifying, controlling, and/or evaluating the technique τ (technē + logos = technology), and the theory Θ as a discourse for explaining, justifying, controlling, and/or evaluating the technology θ: Λ = [θ / Θ]. Based on them, any praxeology is defined by the praxeological quadruple: φ = [Π / Λ] = [T / τ / θ / Θ].

The functioning of a praxeology generates and activates a variety of works denoted by σ; their
size, materiality, and stability do not matter. In the cases of mathematical praxeologies, a work is
any mathematical concept, algorithm, proof, and so on. Conversely, a praxeology can also be
generated and activated by works as part of a set of tools for study, i.e. a so-called *didactic milieu*.
Besides, different praxeologies involving a variety of works can be integrated as a bundle of
praxeologies or a *praxeological organization* linked up by its praxeological component, which is
often a technology or a theory. For instance, a mathematical praxeology generated by the type of
tasks “to calculate derivatives of polynomial functions” is organized as the mathematical
(praxeological) organization of differential calculus, along with other mathematical praxeologies
related to derivatives of different functions of other kinds.

## 2.2 *Institutional transposition of praxeologies*

Every praxeological organization is produced by some *institution*, e.g., research communities and different professions. In the ATD's terminology, an institution is any ensemble of persons who are *subjected* to its corresponding *institutional contract* the rules of which its members are supposed to follow (cf. Chevallard, 1992a). At the same time, some parts of a praxeological organization in an institution are often not produced by the institution itself, but copied from other institutions with required adjustment to their original structure, functioning, and/or utility. This changing process of praxeologies from an institution to another is what the ATD calls the *institutional transposition* of praxeologies (cf. Chevallard, 2019). In the cases that arrival institutions are school institutions, such processes are called *didactic transposition*: e.g., transposition of the notion of complex number from university to upper secondary school. In this study, we are interested in the institutional transposition of the *LSM praxeology* from the Japanese teaching profession to the European one.

## 2.3 *Paradidactic praxeologies*
Needless to say, the object of study in didactics was originally the *teaching of a given targeted work* σ (e.g., a mathematical concept), which is, in the ATD's parlance, called a *didactic praxeology* of σ denoted by $D(σ)$ (cf. Bosch & Gascón, 2006). This means that it excessively studies *didactic systems* $S(X, Y, σ)$ where a group *X* of *students* x studies something σ to study, that is, a *didactic stake*, with the help of a team *Y* of *didactic helpers y* (e.g., teachers and supervisors) (cf. Chevallard, 2019). But, after the emergence of the *didactic transposition theory* in the 1980s, didactics started including *study praxeologies of didactic praxeologies* into its research range; e.g., working for reforming the current state of mathematics education in a country (cf. Otaki, 2024). This implies that a new variant of didactic systems appeared as expanded part of the object of study in didactics. It is what is called *paradidactic system* (Winsløw, 2012), which is any study system that has any didactic praxeology $D(σ)$ of any targeted work σ as its study stake. The expression $S(X, Y, D(σ))$ is a representation of the nested structure inherent in paradidactic systems. It signifies a social system where a group $X$ of *paradidactic students* x studies a didactic praxeology $D(σ)$ as a *paradidactic stake* with the help of a team $Y$ of *paradidactic helpers* n. A representative of $X$ is a curriculum development committee who prepares and sets up didactic praxeologies to be realized at school while deciding targeted works to be taught. In this case, $Y$ can be a group of professional scholars of targeted works like mathematicians, and (part of) its paradidactic stake is the set of didactic praxeologies, or simply a *didactic organization* so to speak, actually performed and observed until that time. Any study praxeology of any didactic praxeology activated in any paradidactic system $S(X, Y, D(σ))$ is called a *paradidactic praxeology* (Winsløw, 2012). Here we can observe the emergence of paradidactic phenomena or *the paradidactic* as a full-fledged area of the object of study in didactics.

## 2.4 *Teaching profession and professions in general*

The teaching profession is a representative place where paradidactic systems are activated. In the ATD's terminology, a *teaching profession* in a given *society* is a subinstitution of the society—clearly, a society is an institution—, which consist of all schoolteachers at a given school level in the society. Moreover, according to the ATD's attitude of generalization, we can define an *institutional profession* of a given institution, which is all people who occupy the same *position* in the institution (Otaki, 2024). Let me emphasize here that the expression “institutional profession” means not only professions in usual sense like the institution of medical doctors, but also any non-occupational “community” of people. For example, all math-haters of a society (implicitly) construct the math-hating profession in the society.
# 3. Methodology

## 3.1 Epistemological analysis: Constructing a reference praxeological model under epistemological vigilance

For studying the research questions abovementioned, we should firstly clarify our *reference model* of LSM, which allows us to identify the transposition process of LSM. The expression “reference model” in the ATD’s terminology signifies any theoretical lens for modeling a given *system* at stake (cf. Bosch & Gascón, 2006). In other words, it is any theoretical *hypothesis* about the structure, functioning, and/or utility of the system. For example, we, implicitly or explicitly, need a reference model about what school algebra is, when we analyze current school algebra (cf. Bosch, 2015). Let me underline here that a reference model of a given system is *not* any precise depiction nor idealized one of the system. It is any representation of a *possible form* of the system, which is useful for comprehending the current status of the system, especially thereby allowing us to recognize absent elements in the state of the system. And, such remark on absence *never* intends to reprove the actuality of the system, but just to clarify our statements on the system. The proper use of reference models is based on the epistemological modesty of refraining from value judgements as meticulously as we can.

However, constructing scientifically sound reference models is usually difficult, since they are supposed to meet the requirement of the following *emancipatory principle* (cf. Bosch, 2015): every didactician should make themselves detached from all prevailing models of given systems created by others who are not didacticians, e.g., curriculum developers and teacher educators, as much as possible. In other words, a reference model should not be any pseudo-theoretical “translation” of non-didacticians’ spontaneous models. How do we construct a (relatively) appropriate reference model while following this principle? Clearly, using a solid theoretical framework and collecting data from different institutions are minimal conditions for emancipation, which this paper tries to meet.

About such emancipation, the author also takes on a possibly advantageous position by chance. Firstly, I do not have any professional experience as a certified schoolteacher nor any stable collaboration with schoolteachers, that is to say, I have substantially not been a member, or a *subject*, of the teaching profession. Secondly, I can relatively easily access to the information on LSM in Japan as a teacher educator, a didactician, and a Japanese-speaking person. The combination of these positional conditions may be rare. On the one hand, most Japanese
didacticians are former schoolteachers and/or professionally intimate with schoolteachers in the
position of leading them, thereby having a risk to subject themselves to the teaching profession.
On the other hand, didacticians from other countries usually cannot gather thoroughly data-sets
about the original, Japanese lesson study. This possible, methodological advantage may allow me
to increase the possibility of objectifying and disenchanting LSM. At least, this kind of self-
analysis is crucial as part of so-called *epistemological vigilance*, which is necessary when given
researchers are, intentionally or unintentionally, involved or implicated in their own object of
study—here LSM—, i.e. when their study is devoted to some field of social science (Bourdieu,
Chamboredon, & Passeron, 1991).

3.2 Case report: Establishing existence assertions

Description of the institutional transposition of LSM, in principle, consists of depiction of “original LSM” before transposition and “transposed LSM” after transposition. About transposed LSM, I focus on LSM involving the TDS, and draw on data from papers published in research journals or books which I could find out. At first glance, this “case report” status seems to be a weakness of this paper, because it is “limited” at two levels of the object of study and of the data source. In my view, this feeling is based on the following principle, which could function as an epistemological obstacle in didactics: *any scientific assertion must be a universal statement*. For example, a possible schema of universal, didactic statements is that *every* student is able to understand a mathematical concept under a set of certain didactic conditions. But, we are often interested in, consciously or unconsciously, another kind of statements, which can be called *existence statements*. For instance, such statements can be formed in the following style, implicitly or explicitly: there *exist* some students who have a kind of (mis)understanding about a mathematical concept under a set of certain didactic conditions. Furthermore, reaching existence assertions is, from a methodological perspective, even one of main tasks in research:

> “The design work that accompanies the use of didactical engineering has been primarily put
> in service of understanding the economy and ecology of didactical systems, producing
> didactic phenomena, and establishing *existence theorems*.” (Artigue, 2017, p. 148: the italics
> are by the author.)

In a similar manner, this paper modestly aims to establish some existence theorem about the
international transposition of LSM without rashly and recklessly jumping on universal theorems
about it.

3.3 *Ecological analysis: Exploring generic institutional conditions*
An outstanding but often ignored feature of the ATD's methodology is its "ecological" ambition. In its terminology, the word *ecology* means the set of all conditions (and constraints<fnref n="1" />) under which a given didactic phenomenon appears or disappears. And especially, the ATD focuses on generic, *institutional* conditions for phenomena at stake. For instance, its study of elementary algebra at school identified the western civilization's attitude of preferring spoken language to written language, so called logocentrism or phonocentrism, as a cultural condition for the shortage of the treatment of multiple parameters in school algebra (cf. Bosch, 2015).

Within the framework of the ATD, such ecological conditions are “measured” by the *scale of levels of didactic codeterminacy*: Humankind ⇔ Civilization ⇔ Society ⇔ School ⇔ Pedagogy ⇔ Didactic system (cf. Chevallard, 2019). According to this line, a paradidactic “completion” of the scale was recently proposed (Otaki, 2024). Adjusting it to this paper’s interest in the teaching profession, different levels of conditions for the functioning of paradidactic systems are inserted into the original scale as follows: … ⇔ Society ⇔ *Noosphere* ⇔ *Teaching profession* ⇔ *Paradidactic system* ⇔ School ⇔ …<fnref n="2" />.

About the ecology of the sustainability of LSM, previous studies have concentrated on conditions for it at the level of the teaching profession. For example, Miyakawa and Winsløw (2013) studied the *open lesson* event which is an opportunity for facilitating LSM conducted by the Japanese teaching profession and noosphere, with the help of the notion of *paradidactic infrastructure*. According to this line of ecological analysis, I explore more generic conditions for the sustainability of LSM at the level of Japanese society involved in oriental civilization.

### 3.4 Research questions reconstructed in the framework of the ATD

I want to reformulate here the said research questions in the precise manner of the ATD.

*Q1: How is LSM praxeology transposed together with the TDS into European countries?*

*Q1-1: What praxeological structure does original LSM before transposition have?*

*Q1-2: What praxeological structure does transposed LSM after transposition have?*

*Q2: What is a generic, institutional condition for the sustainability of LSM at the level of Japanese society?*

<footnote n="1">¹ A constraint is any condition which a given person or institution *cannot* change nor control.</footnote>

<footnote n="2">² The *noosphere* of a given school system is the set of all people who move on and/or think about the school system (e.g., Chevallard, 1992b).</footnote>
# 4. A reference paradidactic model of LSM

## 4.1 Metatheoretical remarks: On the rationale of a reference model

Before constructing our reference model, I firstly want to add some comments on the validity of reference models. It is functionally or pragmatically displayed through the process of drawing on the models for analysis and/or design. Apparently, the rationale of any reference model cannot be validated logically or deductively before using it in advance. In other words, the construction of a reference model seems to be inevitably creative process the nature of which cannot be captured in any methodological or systematic way. But, of course it does not come from vacuum but is based on different data. Then, to clarify some sources of ingredients of my reference model of LSM is probably helpful for understanding it and expecting its functioning. There are two categories of my information sources which work as the bases of construction of the reference model of LSM. The first category is my experience attending LSM. As a mathematics teacher educator, I have had the opportunity to participating in LSM by in-service teachers and preservice teachers. This experience is useful for grasping an actual or real picture of LSM as a whole, from the perspective of the *position of insider* of LSM. On the other hand, I also rely on the experience of the didactic engineering of inquiry-based study process framed within the ATD. In such engineering, didacticians with the ATD often design and analyze so called *study and research paths*, that, roughly speaking, are inquiry processes *with* the reference to media like the Internet and *without* a predetermined fragment or piece of knowledge to be taught (cf. Barquero & Bosch, 2015). To invoke the didactic engineering of study and research paths as a source of my reference model of LSM allows me to occupy the *position of outsider* of LSM in Japan. Besides, the combination of LSM and the didactic engineering of study and research paths also has another important reason or merit, but we will come back this issue later (Subsection 6.5).

## 4.2 On the praxis part

Our reference type of tasks of LSM, $T_R$, is to design a didactic organization for teaching a relatively large mathematical organization like a theme, a domain, and so on. This reference type of tasks mainly comes from the didactic engineering of *finalized study and research paths* (cf. Bosch, 2019), which are study and research paths with bodies of knowledge to be taught (e.g., Vásquez, Barquero, & Bosch, 2021). A main reason of this choice is for relativizing the motivation of usual LSM. However, to suppose a large praxeological organization as a didactic stake is not
so unrealistic for LSM. In fact, in usual LSM, schoolteachers write *lesson plans*<fnref n="3" /> (gakushū sidō an in Japanese), full versions of which are supposed to include the design of a sequence of lessons organized by a theme (*tangen* in Japanese).

About our reference technique τ<sub>R</sub>, I draw on the notion of *dialectic* in the ATD—e.g., appearing in the expression of the dialectic of questions and answers (cf. Bosch & Winsløw, 2015)—, while applying and rebuilding the idea of the *dialectics of lesson study* (Otaki, Asami-Johansson, & Hakamata, 2019). Let me highlight here four dialectics of lesson study. The first dialectic is based on the both of LSM and didactic engineering; the second dialectic comes from the didactic engineering of study and research paths; and the third and four dialectics are rooted from LSM. Let me emphasize here again that any reference model is a *hypothetically created possible reality* helpful for analyzing and producing systems or phenomena at stake. In other words, it is *always under construction*. Accordingly, I have no intention that I give you the complete, exhaustive, and necessary-and-sufficient list of dialectics of LSM. The following is a list which is possibly (or hopefully) well-descriptive for the study of our research questions.

The first one is the *dialectic of didactic systems and didactic stakes*. Obviously, the object of study in paradidactic systems is basically didactic systems which bring into being didactic praxeologies D(σ) of their didactic stakes σ in question. However, no one can design proficiently and analyze deeply D(σ) without making sense of σ disciplinarily and epistemologically well.

The second one is the *dialectic of the standard problematic and the primordial problematic*. The standard problematic means the following question schema (Chevallard, 2011): how should we actually teach a *predetermined* mathematical work? This is a very basic way of questioning in the current teaching profession. The standard problematic assumes that someone already resolves problems of another kind, which is related to the construction of knowledge to be taught: what mathematical works do or should we teach and why? This is called the primordial problematic (ibid.; Chevallard, 2019).

The third one is the *dialectic of paradidactic situations and metadidactic situations*. Here, I define the notions of paradidactic situation and metadidactic situation as different states of paradidactic systems S(X, Y, D(σ)). In the paradidactic situations, schoolteachers X study didactic praxeologies D(σ) (and their didactic stakes σ) in question on their own responsibility, whereas, in the metadidactic situations, X are supposed to be “taught” about D(σ) by paradidactic helpers Y. Here I just want to indicate, by the adjective *metadidactic*, the “didactic” functioning of

<footnote n="3">³ You can see an example in the appendix of Miyakawa (2015).</footnote>
paradidactic systems. From this perspective, we can characterize the paradidactic situations as “adidactic” situations of the corresponding paradidactic systems with the help of the TDS’s language.

The fourth one is the *dialectic of the paradidactic student and the didactic helper*. On the one hand, schoolteachers in paradidactic systems act in different paradidactic situations, that is, *predidactic situations* (e.g., lesson planning), *didactic observation situations*, and *postdidactic situations* (e.g., lesson reflection) (Winsløw. 2012). On the other hand, the schoolteachers take on the *position of teacher* in different situations of (purely) didactic systems involving their own students who they are responsible for.

### 4.3 On the technological part

About our reference technology $\theta_R$, I draw on the notion of the *disciplinary paradigm of mathematical modeling* (cf. Gascón & Nicolás, 2021), based on the didactic engineering of finalized study and research paths. This idea for constructing $\theta_R$ is being inspired by recent works about the relationship between LSM and didactic paradigms taken up by Spanish researchers (e.g., Lendínez, García, Lerma, & Abril, 2024). In such works, a didactic theory like the TDS is interpreted as a didactic paradigm *independent by itself*; e.g., the TDS as a didactic paradigm. By contrast, in this paper, I assume that any didactic theory appears as a concrete (and often partial) realization of corresponding didactic paradigms.

Roughly speaking, in the ATD’s context, a didactic paradigm is identified as “a set of rules prescribing, however implicitly, what is to be studied—what the didactic stakes *O* can be—and what the forms of studying them are” (Chevallard, 2015, p. 174). According to Gascón’s theory of didactic paradigms as a subtheory of the ATD, any didactic paradigm *in mathematics* can be described by: 1) its epistemological model of mathematics; 2) its teaching end in mathematics education; and 3) its didactic means for teaching mathematics<fnref n="4" />. Let me illustrate this paradigm schema while describing the structure of the paradigm of mathematical modeling. The epistemological model of the paradigm is a form of *constructivism*, in which mathematics is regarded as a *human activity*, and thereby underlining its *instrumental value* helpful for accomplishing various tasks. Accordingly, the teaching end of the paradigm is to make it possible that the students produce and utilize mathematical works as *tools* for answering their *questions* (via modeling certain systems in question). The didactic means of the paradigm is to prompt students to study questions which inevitably activates a mathematical organization along with

<footnote n="4">⁴ This model of didactic paradigms often also includes the fourth element of *didactic phenomenon* supplementally, which a corresponding paradigm is interest in (e.g., Gascón & Nicolás, 2021).</footnote>
targeted mathematical works. The reason why this paradigm is named “mathematical modeling” is that, in the ATD’s epistemology of mathematics, the utility of mathematics in human activities is based on the fact that mathematics (and especially elementary algebra in there) is a *privileged tool for modeling any system in question*, whether extramathematical or intramathematical (cf. Bosch, 2015).

## 4.4 On the theoretical part

Finally, we discuss the theory in our reference paradidactic praxeology. Our reference theory $\Theta_R$ is the *pedagogical paradigm of questioning the world* (Chevallard, 2015). Let me use, as indicators of the paradigm, the following three students’ attitudes, which are encouraged by the didactic contract under the paradigm: *Herbartian*, *procognitive*, and *exoteric* (ibid.). In the first one, “Herbartian” means the attitude of actively studying questions even if they look difficult, complicated, and/or unfamiliar for them, whenever and wherever time and circumstances permit. The second, procognitive attitude means the tendency of gathering different items of information supposedly helpful for answering questions at stake from everywhere accessible. This deeply pertains to the use of “media” during inquiry into questions. And, the third, exoteric attitude is related to the effort to study questions and to create answers to them, a straightforward measure of which is time spent to the study. Under this attitude, students try to create their own answers to questions and to refine them *as much as they can*, while recognizing that the ideas of the “perfect” answer to a question and the “complete” mastery of a work are fiction.

## 4.4 Summary

Here is my reference praxeological model of LSM for this paper (Table 1). Let me emphasize here again that this is *not* any idealization nor realistic tracing of the actual state of LSM, but just reference for studying LSM.

<table><caption>Table 1. Our reference paradidactic praxeology of LSM</caption><tr><td>$T_R$</td><td>✓ To design and analyze a didactic organization of a mathematical organization</td></tr><tr><td>$\tau_R$</td><td>✓ The dialectic of didactic systems and didactic stakes<br>✓ The dialectic of the standard problematic and the primordial problematic<br>✓ The dialectic of paradidactic situations and metadidactic situations<br>✓ The dialectic of the paradidactic student and the didactic helper</td></tr><tr><td>$\theta_R$</td><td>The didactic paradigm of mathematical modeling<br>✓ Mathematics as a human activity<br>✓ Mathematics as tools</td></tr></table>
<table><tr><td></td><td>✓ Inherent formative utility (mathematics as modeling tools)<br/>✓ The study of questions<br/>✓ Finalized at a mathematical organization</td></tr><tr><td>Θ<sub>R</sub></td><td>The pedagogical paradigm of questioning the world<br/>✓ Herbartian attitude<br/>✓ Procognitive attitude<br/>✓ Exoteric attitude</td></tr></table>

Before closing this section, I want to briefly describe the relationship between the *praxis* part and the *logos* part, especially between the technique and the technology. Simply speaking, a didactic paradigm can explain given dialectics of paradidactic praxeologies as follows: the dialectics are conducted for designing and analyzing didactic praxeologies under the paradigm.

# 5. Japanese original form of LSM

## 5.1 On the praxis part

Japanese original LSM usually treats, as its unit of design and analysis, the teaching of a small praxeological piece, like a mathematical concept and a mathematical method, through a lesson (45-50 min) or a sequence of a few lessons. In short, the type of tasks of Japanese LSM is to design and analyze a didactic praxeology for teaching a relatively small mathematical work. In the case of LSM in Japan, this tendency explicitly appears lesson plans, which mainly focus on single lessons, even though a full version of its format also includes a rough plan of a whole theme. And, open lesson events for lesson study is also compatible with the said type of tasks (cf. Miyakawa & Winsløw, 2013). In there, attending schoolteachers are supposed to observe a single lesson, and then discuss the lesson.

About the first technical dialectic of didactic systems and didactic stakes, a central action in Japanese LSM is writing lesson plans practice which is located at the pole of didactic systems. And then, the schoolteachers in LSM are also supposed to study $\sigma$ by themselves. In Japanese LSM, the schoolteachers' study of $\sigma$ is called *kyōzai kenkyū*, which literally means the study of teaching materials. Needless to say, this study of $\sigma$ is supposed be more “transcendent” than the study by their students mathematically and epistemologically.

About the second dialectic of the standard problematic and the primordial problematic, the standard problematic is a very basic way of questioning in Japanese original LSM, since
knowledge to be taught, which are written in curricular documents and textbooks, is usually
selected and organized at the outer side of the Japanese teaching profession in advance. By
contrast, for the very same reason, the primordial problematic disappears from the schoolteachers'
range of professional questions in Japanese LSM. Indeed, schoolteachers in original LSM are not
responsible for this problematic, thereby having a few freedom which only includes the making
decision of problems and of teaching orders relating to predetermined works as didactic stakes.

About the third dialectic of paradidactic situations and metadidactic situations, many segments of
the running process of original lesson study systems, S($\mathcal{X}$, $\mathcal{Y}$, $\mathcal{D}(\sigma)$), are undoubtedly
autonomous paradidactic situations under $\mathcal{X}$'s control for designing and analyzing lessons, while
$\mathcal{Y}$ is almost absent. However, in some parts of it, $\mathcal{Y}$ are supposed to play important roles as
"knowledgeable others" (cf. Seino & Foster, 2021) for supporting $\mathcal{X}$'s lesson study. In the case of
Japan's LSM, post lesson discussion systemically includes knowledgeable others like teacher
educator, facilitator, or didactician who explicitly activates the metadidactic aspect of paradidactic
systems by giving their comments on the lessons observed.

About the fourth dialectic of the paradidactic student and the didactic helper, the pole of the
paradidactic student is typically represented by the so-called *lessen study cycle*, which is a variant
of the so-called PDCA cycle. And, this cycle implicitly assumes that a schoolteacher in a given
LSM process actually teaches in a corresponding lesson.

## 5.2 On the technological part

Japanese LSM (at least, representative part of it) is a methodology for designing and analyzing
the so-called *structured problem-solving lessons* (hereafter, SPSLs). How much compatible are
SPSLs in Japan with the paradigm of mathematical modeling? In terms of epistemological model,
we can see that SPSLs presuppose at least two principles pertaining to the constructivism, even
though epistemological principles of SPSLs are usually latent in schoolteachers' description about
SPSLs. Firstly, SPSLs are supposed to activate students' mathematical activities in the process of
teaching. About this point, an outstanding moment of the process of SPSLs is the moment of
teachers' problem-posing (*hatsumon* in Japanese) at the initial part of the lessons. In the beginning
of an SPSL process, the teacher is supposed to pose a problem to students, whereby the students
are invited to mathematical activities for solving the initial problem. The assumption of
mathematics as the *activity of solving mathematical problems* indirectly but clearly appears in
different variations of the lesson format or pattern of SPSLs. For instance, in the pattern of SPSLs
manifested by the Japanese mathematics educator Kazuhiko Souma, the three phases out of all
the six phases are devoted to the process relating to the initial problem-posing (cf. Asami-
Johansson, 2022). Secondly, the SPSL includes students' communication and discussion in its process. This epistemological assumption is reified as the notion of *neriage*, which means "to socially construct something". In fact, some variations or patterns of SPSLs consider that the *neriage* moment is most important in their lesson processes (e.g., Takahashi, 2021). Here we can identify the assumption of mathematics as the *activity of socially constructing mathematical works*. By contrast, in comparison with the principle of mathematical activity, the viewpoint of "mathematics as tools" is much weaker, or at least more implicit and peripheral from the perspective of the SPSL. About it, there is only a semi-registered notion of *yosa* (usefulness) of a work. In this sense, the epistemological stance of the SPSL is closer to epistemological *monumentalism* (cf. Chevallard, 2015), where any mathematical work "is approached as a monument that stands on its own, that students are expected to admire and enjoy, even when they know next to nothing about its *raisons d'être*, now or in the past." (p. 175; the italics are original.)

About the teaching end, the SPSL's one is different from the teaching end of the paradigm of mathematical modeling. This difference can be clarified by the ATD's notion of *formative utility* (Chevallard, 2019). There are mainly two kinds of formative utility. The first one is the *inherent formative utility*, where students study works as tools for performing tasks which are principal for the work: e.g., studying the notion of probability as a tool for measuring the degree of uncertainty of events of certain kinds. And the second one is the *transcendent formative utility*, where students study works for getting something "beyond" the functions inherent in the works like ability, competence, sense, and so on. Clearly, the teaching end of the paradigm aims to activate the inherent formative utility of mathematical works by emphasizing the functionality of the works as a modeling tool. By contrast, the teaching end of the SPSL is to *cultivate students' mathematical mind*. Typically, the notion of *sūgakutekina kangaekata* (an expression of "mathematical thinking" in the Japanese teaching profession) has been underlined by Japanese many mathematics schoolteachers, mathematics educators, and didacticians in mathematics education, while deeply relating to the SPSL (cf. Asami-Johansson, 2021). For instance, Takahashi (2021) said that:

> "Japanese educators and researchers have also spent decades focusing on how to nurture students' ability to think mathematically. Years of collaborative effort have produced a specialized pedagogical approach known as Mondai Kaiketsu Gakushuu, or 'Teaching Through Problem-solving (TTP)'." (p. 5: From this paper's point of view, the TTP is a variation of the structured problem-solving.)

About the didactic means, indeed, the structured problem-solving approach makes normal the study of a question by students which begins with the teacher's *hatsumon*, but on the other hand,
the process or path of the study is predetermined or finalized by small mathematical works at stake. This explicitly appears in the format of lesson plans, the focus of which is essentially in single lessons as we have seen. Besides, in the professional vocabulary for describing SPSLs, there are few convenient, technical terms for depicting the teaching process of relatively large mathematical and didactic organizations which the paradigm of mathematical modeling concerns.

### 5.3 On the theoretical part

Within the framework of the SPSL, the first Herbartian attitude is implicitly assumed as important, because it considers that the starting point of students’ learning of any mathematical work is an appropriate question which is supposed to be introduced through the initial *hatsumon* moment. This means that there is no learning without students’ attitude toward questions.

About the second procognitive attitude, in SPSLs, their didactic milieus are determined by teachers, that is to say, there are no students’ freedom of gathering information from different media. In fact, “not to open textbooks without teachers’ permission” is a commonly observed clause of the didactic contracts of SPSLs. This clause is usually contracted to avoid spoiling the process of students’ own problem-solving. Needless to say, using the Internet for collecting information is totally forbidden. Consequently, the attitude cultivated by SPSLs is completely opposite to the procognitive attitude, that is named the *retrocognitive* attitude (Chevallard, 2015), under which students only refer to their own knowledge acquired at past lessons. The *retrocognitive* attitude is a typical sign of the *pedagogical paradigm of visiting works* (ibid.).

About the third exoteric attitude, the opposite, *esoteric* attitude (ibid.) makes students believe that the perfectness of an answer to a question and the completeness of the mastery of a work are achievable (and even have already been achieved in some works to some extent), thereby inducing laziness to achieve them as easily and quickly as possible. SPSLs are more compatible with the esoteric attitude than the exoteric attitude. That is symbolically realized in the relationship between lessons and works in SPSLs. Any lesson basically corresponds to a small work. This implies the following message, whether the SPSL intends it or not: students must “master” a given mathematical work within a short period of didactic time.

### 5.4 Summary

Table 2 is the summary of the praxeological structure of original LSM in Japan. In there, the symbol “■” indicates the items different from the reference model of LSM. This is the form of LSM before transferring from the Japanese teaching profession to other countries’ ones.
<table><caption>Table 2. Original LSM in Japan</caption><tr><td>T<sub>LSM-SPSL</sub></td><td>■ To design and analyze a didactic praxeology of a small mathematical work (“lesson” study)</td></tr><tr><td>τ<sub>LSM-SPSL</sub></td><td>✓ The dialectic of didactic systems and didactic stakes (kyozai kenkyu, lesson plan)<br>■ The repression of the primordial problematic<br>✓ The dialectic of paradidactic situations and metadidactic situations (knowledgeable other)<br>✓ The dialectic of the paradidactic student and the didactic helper (lesson study cycle)</td></tr><tr><td>θ<sub>LSM-SPSL</sub></td><td>The paradigm of mathematical modeling of the SPSL<br>✓ Mathematics as a human activity (neriage, hatsumon)<br>■ Mathematics as the object of study<br>■ Transcendent formative utility (mathematical thinking)<br>✓ The study of questions (hatsumon)<br>■ Finalized at a small mathematical work</td></tr><tr><td>Θ<sub>LSM-SPSL</sub></td><td>The paradigm of questioning the world of the SPSL<br>✓ Herbartian attitude (hatsumon)<br>■ Retrocognitive attitude<br>■ Esoteric attitude</td></tr></table>

# 6. International transposition of LSM

## 6.1 Analyzed data

In this section, we discuss an example of the structure of LSM praxeologies transposed under European contexts. Our data-sources are the following three papers which report LSM realizations, giving relatively concrete descriptions of them, with the TDS:

Bahn & Winslow (2018)—Doing and investigating lesson study with the theory of didactical situations;

Jessen, Bos, Doorman, & Winslow (2023)—Lesson study in mathematics with TDS and RME as theoretical support; two cases from the European TIME project;

García, Lendínez, Lerma, & Abril (2024)—Mechanisms and evidence of prospective teachers’ learning through enquiry-oriented practices: the case of a lesson study intervention.
They are only research papers which I can find in scientific journals or books, and surely report results of seminal LSM projects in the European context.

## 6.2 On the praxis part

In this subsection, I will analyze the *praxis* part of the lesson study praxeologies reported in the above papers in terms of the elements of our reference LSM model. About the types of tasks, according to Japanese manner, all praxeologies of the papers set out the following one: to design and analyze a didactic praxeology of a *small mathematical work*. A crucial indicator of that is that all the papers basically consider one lesson (or several lessons at most) with subtle variations of didactic time. For instance, the second lesson study in Jessen et al. (2023) revolves around one lesson relating to the notion of piecewise defined function as follows: “The target knowledge of *the lesson* is to describe a skiing slope in terms of a piecewise defined function.” (p. 60; the italics are added.).

About the techniques, all the lesson study realizations, more or less, activated the dialectics of didactic systems and didactic stakes; of paradidactic situations and metadidactic situations; and the paradidactic student and the didactic helper. All the reports of the lesson study experiences included participants’ study of lesson process and knowledge to be taught in there, supports by researchers or facilitators, and lesson implementation and observation of it. That is most explicitly reported in Bahn and Winsløw’s (2018) second example of lesson study about the operation of subtraction. By contrast, the dialectic of the standard problematic and the primordial problematic malfunctions. Or, at least, there are no descriptions of participants’ inquiries into the primordial problematic. This quite contrasts to the explicit presence of the standard problematic as follows: “*what situations should we generate in the classroom for children to understand the functionality of written numerals?*” (Garcia et al. 2024, p. 1188; the italics are original.).

## 6.3 On the technological part

Next, let me analyze the *logos* part of the lesson study praxeologies in the papers. The main task of this subsection and the next subsection is to reveal paradigmatic aspects of the TDS at the disciplinary, technological level and the pedagogical, theoretical level, respectively, with the help of our reference paradidactic model of LSM. About the technological dimension, firstly, the TDS assumes a variant of constructivist epistemology of mathematics. That symbolically appears in the notion of (mathematical) *situation*. From the perspective of the TDS, mathematical knowledge is modeled by different types of situations where the corresponding types of *activities* take place, especially the situations of *action, formulation, and validation* (cf. Brousseau, 1997). And then,
the idea of *optimality*, which is unobtrusive but crucial in many papers within the framework of the TDS, highlights the instrumental value of mathematical knowledge.

About the teaching end, let me begin with the difference between the teaching end and the *research end* (Gascón & Nocolás, 2019). In general, the aim of the domain of intervention and that of the domain of science are different. So, in principle, we cannot directly take out teaching ends from scientific theories in didactics. For this reason, I want to interpret that teaching ends of didactic theories are indirectly and implicitly represented by targeted didactic phenomena which the theories attempt to explain the mechanism of and to find out conditions for. According to this view, the teaching end of a didactic theory can be “rationally constructed” in the following form: the teaching end of a determined theory is to produce its targeted phenomenon. This means that the teaching end and the phenomenon to produce of a determined, didactic theory can be identified simultaneously. In the case of the TDS, the phenomenon of *fundamental situation* is a crucial study target, the conditions for which are supposed to be revealed through research (cf. Brousseau, 1997). A fundamental situation of a mathematical work, either a praxeological element or a praxeological organization, is a sequence of mathematical situations where the work implicitly appears as a hopefully useful idea for answering a question problematic for students, and then is gradually constructed and confirmed as a dependable, optimal tool for answering the question. A major project of the TDS is to find out such fundamental situations as for many works and at many school levels as possible. According to this line, the teaching end of the TDS is more compatible with the inherent formative utility than the transcendent formative utility.

About the didactic means, a typical one of the TDS is a particular kind of situations, that is, *adidactic situations* (cf. Brousseau, 1997). An adidactic situation is any “paradoxical” situation where students are surely heteronomous under the “didactic” contract of the existence of a work to be taught by a teacher, whereas the students retain relatively high degrees of autonomy, or “adidacticity”, for studying the work which the students are supposed to encounter and learn. And, about the scale of works at stake, the TDS basically considers larger mathematical organizations. That clearly appears some representatives of fundamental situations. For example, Brousseau, Brousseau, & Warfield (2014) is a synthesis of quite a long sequence of mathematical and didactic situations for teaching rationals and decimals.

## 6.4 On the theoretical part

According to the above didactic means, the TDS assumes that students autonomously study questions for learning predetermined mathematical works. That is a sign illustrating that the TDS presupposes students’ Herbartian attitude. Moreover, in the framework of the TDS, this
presupposition is also marked by the notion of *devolution* (cf. Brousseau, 1997). In the situation of devolution, a teacher devolves the responsibility of controlling the study process to students. Especially, the TDS presumes that the students can take the responsibility of facing a problem.

By contrast, didactic scenarios based on the TDS often probably nurture the retrocognitive and esoteric mindset. That is because it does usually not suppose any active use of media nor any genuinely open-ended inquiry in accordance with students' interests.

## 6.5 Summary and synthesis

Here is the summary of the analysis abovementioned (Table 3).

<table><caption>Table 3. LSM with the TDS</caption><tr><td>T<sub>LSM-TDS</sub></td><td>■ To design and analyze a didactic praxeology of a small mathematical work</td></tr><tr><td>τ<sub>LSM-TDS</sub></td><td>✓ The dialectic of didactic systems and didactic stakes<br>■ The repression of the primordial problematic<br>✓ The dialectic of paradidactic situations and metadidactic situations<br>✓ The dialectic of paradidactic students and didactic helpers</td></tr><tr><td>θ<sub>LSM-TDS</sub></td><td>The paradigm of mathematical modeling of the TDS<br>✓ Mathematics as a human activity (situation)<br>✓ Mathematics as tools (optimality)<br>✓ Inherent formative utility (fundamental situation)<br>✓ The study of questions (adidactic situation)<br>✓ Finalized at a mathematical organization (fundamental situation)</td></tr><tr><td>Θ<sub>LSM-TDS</sub></td><td>The paradigm of questioning the world of the TDS<br>✓ Herbartian attitude (devolution)<br>■ Retrocognitive attitude<br>■ Esoteric attitude</td></tr></table>

We can point out at least three properties of the transposition of LSM that we have seen. Firstly, the *logos* parts of LSM superficially change from the knowledge about SPSLs to the TDS, but essentially share and keep a similar structure. Especially, the similarity between the theoretical elements are noteworthy. Even about the technological elements, the epistemological view of mathematics as a human activity activated by problematic questions is stably preserved. They might be a reason why the conductors of the mentioned European LSM projects, chose or accepted the TDS as the theoretical framework in LSM. They probably thought, explicitly or implicitly, that the TDS does not distort the essence of LSM.
Secondly, they share the scale of didactic stakes that the participants design and analyze, while focusing on the smaller mathematical works rather than larger mathematical organizations. This is quite a surprising fact, because the didactic engineering with the TDS is originally supposed to focus on much larger organizations than the LSM of SPSLs does. This praxeological “contradiction” between the *praxis* part of the LSM with the TDS and the *logos* part of it may be, at least partially, due to a strict copy of the formats of lesson plan and lesson study cycle, which are quite specialized for the teaching of small works.

Thirdly, all the lesson study processes analyzed above did not handle the primordial problematic. This repression, too, seemingly conflicts with the TDS’s research mind. That is because the TDS-didacticians are supposed to problematize mathematical praxeologies to be taught that is dominant in current school system. A reason for this repression is the same at the second property which we just discussed. The original LSM for designing and analyzing SPSLs does not deal with such questions, so that the other LSM praxeologies followed that. Besides, ecologically speaking, each society underlying the LSM praxeologies probably gives the *semiprofessional* status to its teaching profession (cf. Chevallard, 2022). In the societies, that is, Denmark, Japan, and Spain, the primordial problematic does basically not fall within the jurisdiction of the teaching profession.

Let me return here to the issue of the rationale of our reference paradidactic model of LSM. Why did I rely on the didactic engineering within the framework of the ATD? I have already answered this question partially at the subsection 4.1. Here I want to give you another portion of my answer to this question. It is related to the change of pedagogical paradigms from the traditional paradigm of visiting works to the oncoming paradigm of questioning the world. Needless to say, LSM in Japan has been originally conducted for responding to schoolteachers’ professional demands and needs under the paradigm of visiting works, the key notion of which is the *study* of predetermined works. That clearly appears the difference of our reference model and the abovementioned original form of LSM. However, the didactic paradigm in our recent educational environment is slowly but surely moving to the paradigm of questioning the world, the key notion of which is *inquiry* into unfinalized questions. Accordingly, our didactic reality is changing in multiple aspects; e.g., the inclusion of Generative AI. In my view, LSM, too, cannot avoid adjusting itself to this new didactic paradigm. The reference paradidactic model of LSM that I proposed is a possible organization which an upcoming lesson study could take on.

# 7. On the ecological dimension of the sustainability of LSM
Let us focus on here a much more implicit, transparent, and naturalized property of the transposed
LSM than the said ones. It is not pertinent to the structure and functioning of LSM itself, but
related to the *institutional conditions* for its *sustainability*. On the one hand, LSM is quite a
*spontaneous* phenomenon in Japan, but on the other hand, in the said European cases, the
realization of LSM needs some *project* in initial or in-service teacher education which
intentionally “imposes” LSM on (prospective) schoolteachers. This difference is *not* self-evident.
I draw on an ecological analogy for justifying this statement. Let us suppose that a plant or animal
is accidentally or deliberately transported from a continent to another. There are four possibilities
of its life: 1) artificially alive with special care; 2) spontaneously alive without special care; 3)
dead in spite of special care; and 4) spontaneously dead without special care. Obviously, our
European examples of LSM is of the first type. Why did not occur the second type? This is a
paraphrase of the second research question manifested at the subsection 3.4.

For answering this question, it is useful to examine the institutional contract of the Japanese
teaching professions at primary and lower secondary schools, which are the main habitats of LSM
in Japan. There are many clauses of the institutional contract of them, but we will pay attention to
three clauses in the Japanese teaching profession's contract, which could be crucial for LSM, as
follows: 1) *exoteric attitude*; 2) *master-disciple relationship*; and 3) *learning by adaptation*.

About the idea of exoteric attitude, I have already described its meaning at the subsection 4.4—
therefore I also use this notion reflexively for representing a schoolteacher's attitude. It means
some high degree of volition and openness to further study. According to the institutional contract
of the Japanese teaching profession, schoolteachers should keep on improving their own didactic
scholarship and skills as much as possible. This means that, in principle, all schoolteachers should
be exoteric professionals, who indefinitely regard themselves as professionally “immature” while
involving themselves in hard training or self-cultivation (keiko and/or shugyō in Japanese). This
clause explains why lesson study is that popular and sustainable at the organizational level in
Japan; and why many Japanese schoolteachers are, at the individual level, that eager in lesson
study to such an extent that they voluntarily devote to it their after-work hours, days off, even
dinner party with beer! For instance, Miyakawa and Winsløw (2019) report a monthly meeting of
schoolteachers held after working hours. In all likelihood, a well-institutionalized form of the
exoteric attitude is a crucial condition for sustainable lesson study.

About the master-disciple relationship, Japanese schoolteachers should have their “masters” who
mentally work as their role models of teaching professionals. This clause typically becomes
visible in the situations of post lesson discussion together with knowledgeable others who can
function as “one-time masters” there, but that is not everything. Rather, a “genuine master” is supposed to be more stable and long-lasting. For instance, a master of a schoolteacher x can be a teacher educator who is x's ex-supervisor in x's initial teacher education, a charismatic schoolteacher who synthesizes his knowledge and skills as a “theory” which x actually uses, and/or a researcher who knows some teaching method very well which x wants to know. In fact, it is not rare that a schoolteacher said as follows: “I am a disciple (deshi in Japanese) of ō (a person)”, “ō is my master (shishō in Japanese)”, or something like that.

My description of the third clause is based on the notion of *learning by adaptation* (Bosch & Winsløw, 2015). Learning by adaptation is a study format where students are supposed to learn something important through relevant observation and imitation without the help of any explicit didactic stake nor action. This clause typically appears in the situation of open lesson in the cycle of lesson study, where schoolteachers observe lessons, thereby being supposed to learn something about teaching. Besides, the learning process of board work (*bansho* in Japanese) in teacher education gives an almost perfect example of the learning by adaptation. *Bansho* is quite a professional technique or skill (*waza* in Japanese) of writing on black (or recently, white) board for organizing lessons, but surprisingly, there are usually no courses for the training of board work in initial teacher education (see also Asami-Johansson, Otaki, & Akamoto, 2023). Therefore, aspiring schoolteachers have to learn how to write on black board through only observing lessons and imitating board work in there. In Japan, this kind of learning method is traditionally expressed “stealing a skill” (*waza wo nusumu*) with no negative, illegal nuance, or “learning by seeing the (master’s) back” (*senaka wo mite manabu*).

We have seen a minimal set of clauses which could represent the institutional contract relating to lesson study in the Japanese teaching profession, and which can be summarized as the following statement: any teaching professional should *continue to learn* about teaching *through observation and imitation of lessons* while *respecting other professionals* as their great masters. So, where does this institutional contract come from? A plausible answer is that there exists an archetypal or reference profession for instituting the teaching profession, which one can “copy” its institutional contract from. I want to call it an *archeprofession* while referring to the ATD’s use of the prefix “arche-” (cf. Strømskag & Chevallard, to appear). An encompassing and traditional profession of the institution of Japanese society is what we can characterize as the profession of *geidō* (cf. Nishiyama, 1980/2024), the idea of which you might meet through the name of the sport *jūdō*, that is one of the Olympic events and especially popular in France. Actually, the said terms *shugyō* (self-cultivation), *deshi* (disciple), *shishō* (master), and *waza* (art) are representative word usage in this comprehensive profession, and the above contract expresses a typical tendency in its culture.
It is a historical institution which has been established in the 18th century while having grown up ever since the end of the 8th century. The word geidō consists of two Chinese characters of “gei” meaning technique or art and “dō” meaning (never-ending) way or path. Roughly speaking, the part of gei stands for professional skills and the part of dō signifies the process of training the skills metaphorically. There are many officially registered areas of geidō like Japanese tea ceremony (sadō or chadō), Japanese flower arrangement (ikebana or kadō), Japanese fencing (kendō) and so on. But also, we can observe institutional structures which can be regarded as branches of geidō everywhere in Japan. For instance, this notion often includes the art of craftsmen like sculptors of Buddhist statues (būssūri) and the art of popular entertainment like Japanese theater (kabuki). And we can easily find some features of geidō in more ordinary professions; e.g., the chefs of Japanese traditional foods like sushi, tempura, and so on. From a historical perspective, Japanese traditional “mathematics” (wasan) was also carried out under a cultural influence from the mind of geidō before Japan’s introducing western mathematics in the 19th century (e.g., Mikami, 1984). Even there are several papers which point out the relationship between the culture of geidō and the modern school mathematics education (e.g., Hirabayashi, 2006; Winsløw & Emori, 2006). The teaching profession is not an exception. In short, the Japanese teaching profession is an implicit, unregistered branch of the archiprofession of geidō in some way<fnref n="5" />. In fact, we can find out, on the Internet, some neologisms like *kyōshidō* (teacher-dō) and *jugyōdō* (lesson-dō), which reflect the interpretation of “the teaching profession as a branch of geidō”.

## 8. Final remarks: lesson study in danger?

In this paper, we studied the mechanism and ecology of the international transposition of LSM. Especially, we have analyzed the localization of LSM along with the TDS. According to the study, LSM is transposed in accordance with the following regulations like conservation laws, even though its logos part superficially changes from the discourse on SPSLs to the TDS: 1) its technological dimension keeps a variant of constructivism under the paradigm of mathematical modeling which regards school mathematics as finalized problem-solving activities; and 2) its theoretical dimension maintains the retrocognitive and esoteric tendency, and takes over the monumentalistic principle which enhances interest in fragmented mathematical works and represses the primordial problematic<fnref n="6" />. Besides, it was also clarified that the traditional, social

<footnote n="5">⁵ In relation to this, Asami-Johansson, Otaki, & Akamoto (2023) roughly discussed the didactic dimension of geidō in LSM with the idea of the *study paradigm of dō*.</footnote>

<footnote n="6">⁶ We cannot enter into its details due to limitations of space, but we can also observe the same tendency in the</footnote>
system of geidō could underlie the professional passion of Japanese schoolteachers to LSM. These findings contribute to the accumulation of scientific knowledge about LSM.

Moreover, this paper also has implication on the intervention in the professional development of schoolteachers, even if it is very modest or looks roundabout. For justifying that, let me suppose that a facilitating instance ź who is responsible for initial or in-service teacher education in a country wishes to make an LSM-like organization sustainable outside Japan. According to the findings, ź needs, first and foremost, to activate the exoteric attitude of the country's teaching professionals. In the case of Japan, the fuel to LSM has been supplied spontaneously from the institutional contract of the archiprofession of geidō. How can ź allows the country's schoolteachers to be professionally exoteric? According to the ecological way of thinking in the ATD, to transplant the Japanese culture of geidō is apparently a bad move. That is because any teaching profession is involved in the institutional ecosystem of a given society where the teaching profession exists.

When we apply to our issue Chevallard's (2015, p. 183) opinion relating to the ancient Greek notion of *skholē* (leisure), a possible solution is to carve out *time* to (reflexively) invite the schoolteachers themselves to the authentic inquiry into their own professional questions, that is, to the paradigm of questioning the world. Indeed, Japanese schoolteachers have relied on a “trick” of converting their leisure time into *paradidactic time* for lesson study (See the section 7), while following the geidō's logic of self-cultivation. About this, due to different reasons, the time devoted to lesson study by Japanese schoolteachers seems to be decreasing recently. If this is actually true, then, in (near) future, the life of lesson study may be going to weaken even in Japan. Nowadays, the sustainability of lesson study can be problematic in Japan, too.

## Acknowledgements

I would like to thank the reviewers for their critical yet constructive comments. And I would also like to express my gratitude to Yukiko Asami-Johansson and Toshihiko Ogawa who kindly discussed, with me, part of the very initial idea of this paper, respectively. Besides, this work is supported by JSPS KAKENHI (JP18H01015, JP21K18532).

<footnote>first LSM example of Jessen et al. (2023) with the theoretical framework of the so-called *realistic mathematics education*.</footnote>
## References

Artigue, M. (2017). The challenging relationship between fundamental research and action in mathematics education. In G. Kaiser (Ed) *Proceedings of the 13th International Congress on Mathematical Education* (pp. 145–163). Springer. https://doi.org/10.1007/978-3-319-62597-3_10

Asami-Johansson, Y. (2021). The didactic notion of “mathematical activity” in Japanese teachers’ professional scholarship: A case study of an open lesson. *Journal of Research in Mathematics Education, 10*(1), 88–112. https://doi.org/10.17583/redimat.2021.4598

Asami-Johansson, Y. (2022). Conditions and constraints for transferring Japanese structured problem solving to Swedish mathematics classroom. *Recherches en Didactique des Mathématiques. 41*(3), 347–391. La Pensée Sauvage.

Asami-Johansson, Y., Otaki, K., & Akamoto, J. (2023). Board work as a resource for Lesson Study in mathematics. In D. Paul, C. Csaba, P. Hanna, G. Katalin, & K. Eszter (Eds.), *Proceedings of the Thirteenth Congress of the European Society for Research in Mathematics Education* (CERME13: pp. 4874–4881), Budapest, Hungary.

Bahn, J. & Winsløw, C. (2019). Doing and investigating lesson study with the theory of didactical situations. In R. Huang, A. Takahashi, J. P. da Ponte (Eds), *Theory and practice of lesson study in mathematics: An international perspective* (pp. 83–101). Springer. https://doi.org/10.1007/978-3-030-04031-4_5

Barquero, B. & Bosch, M. (2015). Didactic engineering as a research methodology: From fundamental situations to study and research paths. In A. Watson & M. Ohtani (Eds), *Task design in mathematics education* (pp. 249–272). Springer. https://doi.org/10.1007/978-3-319-09629-2_8

Bosch, M. (2015). Doing research within the anthropological theory of the didactic: the case of school algebra. In S. Cho (Ed.), *Selected regular lectures from the 12th international congress on mathematical education* (pp. 51–69). Springer. https://doi.org/10.1007/978-3-319-17187-6_4

Bosch, M. (2019). Study and research paths: A model for inquiry. In B. Sirakov, P. N. de Souza, & M. Viana (Eds.), *Proceedings of the International Congress of Mathematicians: Rio de Janeiro 2018* (Vol. 3, pp. 4001–4022). World Scientific. https://doi.org/10.1142/9789813272880_0210

Bosch, M. & Gascón, J. (2006). Twenty-five years of the didactic transposition, *ICMI Bulletin*, 58, 51–65.

Bosch, M. & Winsløw, C. (2015). Linking problem solving and learning contents: The challenge of self-sustained study and research processes. *Recherches en Didactique des Mathématiques*,
35(3), 357–399.

Bourdieu, P., Chamboredon, J.-C., & Passeron, J.-C. (1991). *The craft of sociology: Epistemological preliminaries* (edited by B. Krais and translated by R. Nice). Berlin: Walter de Gruyter.

Brousseau, G. (1997). *Theory of didactical situations in mathematics: Didactique des mathématiques, 1970–1990* (edited and translated by N. Balacheff, M. Cooper, R. Sutherland, V. Warfield). Kluwer Academic Publishers. https://doi.org/10.1007/0-306-47211-2

Brousseau, G., Brousseau, N., & Warfield, V. (2014). *Teaching fractions through situations: A fundamental experiment*. Springer. https://doi.org/10.1007/978-94-007-2715-1

Capone, R., Adesso, M. G., Manolino, C., Minisola, R., & Robutti, O. (2024). Culturally crafted Lesson Study to improve teachers’ professional development in mathematics: A case study in Italian secondary school. *Journal of Mathematics Teacher Education, 27*(3), 607–636. https://doi.org/10.1007/s10857-023-09578-3

Chen, X. (2017). Theorizing Chinese lesson study from a cultural perspective. *International Journal for Lesson and Learning Studies, 6*(4), 283–292. https://doi.org/10.1108/IJLLS-12-2016-0059

Chevallard, Y. (1992a). Fundamental concepts in didactics: Perspectives provided by an anthropological approach. In R. Douady & A. Mercier (Eds), *Recherches en Didactique des Mathématiques, Selected Papers* (pp. 131–167). La Pensée Sauvage.

Chevallard, Y. (1992b). A theoretical approach to curricula. *Journal für Mathematikdidaktik, 13*(2/3), 215–230.

Chevallard, Y. (2011). La notion d'ingénierie didactique, un concept à refonder. Questionnements et éléments de réponse à partir de la TAD. In C. Margolinas, M. Abboud-Blanchard, L. Bueno-Ravel, N. Douek, A. Fluckiger, P. Gibel, F. Vandebrouck, & F. Wozniak (Eds.), *En amont et en aval des ingénieries didactiques* (pp. 81–108). La Pensée sauvage.

Chevallard, Y. (2015). Teaching mathematics in tomorrow’s society: A case for an oncoming counterparadigm. In S. J. Cho (Ed.) *The proceedings of the 12th international congress on mathematical education: Intellectual and attitudinal challenges* (pp. 173–187). Springer. https://doi.org/10.1007/978-3-319-12688-3_13

Chevallard, Y. (2019). Introducing the anthropological theory of the didactic: An attempt at a principled approach. *Hiroshima Journal of Mathematics Education, 12*, 71–114. https://doi.org/10.24529/hjme.1205

Chevallard, Y. (2022). Challenges in advances in teacher education within the ATD. In Y. Chevallard, B. Barquero, M. Bosch, I. Florensa, J. Gascón, P. Nicolás, & N. Ruiz-Munzón (Eds.), *Advances in the anthropological theory of the didactic* (pp. 81–89). Springer.
https://doi.org/10.1007/978-3-030-76791-4_7

Clivaz, S. (2015). French didactique des mathématiques and lesson study: A profitable dialogue? *International Journal for Lesson and Learning Studies, 4*(3), 245–260.
https://doi.org/10.1108/IJLLS-12-2014-0046.

García, F. J., Lendínez, E. M., Lerma, A. M., & Abril A. M. (2024). Mechanisms and evidence of prospective teachers’ learning through enquiry-oriented practices: The case of a lesson study intervention. *ZDM Mathematics Education, 56*, 1183–1196. https://doi.org/10.1007/s11858-024-01620-9

Gascón, J. & Nicolás, P. (2019). Research ends and teaching ends in the anthropological theory of the didactic. *For the Learning of Mathematics, 39*(2), 42–47.

Gascón, J. & Nicolás, P. (2021). Paradigm crisis in the step from tertiary to secondary mathematics education. *International Journal of Mathematical Education in Science and Technology, 53*(5), 1153–1169. https://doi.org/10.1080/0020739X.2021.2005169

Hirabayashi, I. (2006). A traditional aspect of mathematics education in Japan. In F. K. S. Leung, KD. Graf, F.J. Lopez-Real (Eds), *Mathematics education in different cultural traditions: A comparative study of East Asia and the West* (pp. 51–64). Springer. https://doi.org/10.1007/0-387-29723-5_3

Huang, R., Takahashi, A., & Ponte, J. P. da [Eds.] (2019), *Theory and practice of lesson study in mathematics: An international perspective*. Springer. https://doi.org/10.1007/978-3-030-04031-4

Chen, X. (2017). Theorizing Chinese lesson study from a cultural perspective. *International Journal for Lesson and Learning Studies, 6*(4), 283–292. https://doi.org/10.1108/IJLLS-12-2016-0059

Isoda, M. & Baldin, Y. Y. (2023). Japanese lesson study, its nature and its impact on the teaching and learning mathematics. *Revista Paradigma*, *XLIV*, 5–35.
https://doi.org/10.37618/PARADIGMA.1011-2251.2023.p5-35.id1410

Jessen, B., Bos, R., Doorman, M., & Winsløw, C. (2023). Lesson study in mathematics with TDS and RME as theoretical support: Two cases from the European TIME project. *International Journal for Lesson and Learning Studies, 12*(1), 52–64. https://doi.org/10.1108/IJLLS-01-2022-0009

Lendínez, E. M., García, F. J., Lerma, A. M., & Abril, A. M. (2023). Increase in self-efficacy in prospective teachers through theory- based lesson study. *Journal of Mathematics Teacher Education, 27*, 717–742. https://doi.org/10.1007/s10857-023-09597-0

Mikami, Y. (1984). *Bunkashijyō yori mitaru Nihon no sūgaku* (in Japanese). Kōseisha Kōseikaku.

Miyakawa, T. (2015). What is a good lesson in Japan? An analysis. In M. Inprasitha, M. Isoda, P.
Wang-Iverson, & B.-H. Yeap (Eds.) *Lesson Study: Challenges in Mathematics Education* (pp. 327–349). Singapore: World Scientific. https://doi.org/10.1142/9789812835420_0021

Miyakawa, T. & Winsløw, C. (2009). Didactical designs for students’ proportional reasoning: An “open approach” lesson and a “fundamental situation”. *Educational Studies in Mathematics*, 72(2), 199–218. https://doi.org/10.1007/s10649-009-9188-y

Miyakawa, T. & Winsløw, C. (2013). Developing mathematics teacher knowledge: The paradidactic infrastructure of “open lesson” in Japan. *Journal of Mathematics Teacher Education, 16*(3), 185–209. https://doi.org/10.1007/s10857-013-9236-5

Miyakawa, T. & Winsløw, C. (2019). Paradidactic infrastructure for sharing and documenting mathematics teacher knowledge: a case study of “practice research” in Japan. *Journal of Mathematics Teacher Education, 22*(3), 281–303. https://doi.org/10.1007/s10857-017-9394-y

Nishiyama, M. (1980/2024). *Gei: Hiden denju no sekai* (in Japanese). Kōdansya.

Østergaard, C. H. & Winsløw, C. (2021). Sustainable lesson study and paradidactic infrastructure: The case of Denmark. *Hiroshima Journal of Mathematics Education, 14*, 1-15. https://www.jasme.jp/hjme/

Otaki, K. (2024). The paradidactic: A (would-be) fundamental theory of noospheres. In I. Florensa, N. Ruiz-Munzón, K. Markulin, B. Barquero, M. Bosch, & Y. Chevallard (Eds.), *Extended abstracts 2022: Proceedings of the 7th International Conference on the Anthropological Theory of the Didactic (CITAD7)*, (pp. 45–56). Birkhäuser Cham.

Otaki, K., Asami-Johansson, Y., & Hakamata, R. (2020). Theoretical preparations for studying lesson study: Within the framework of the anthropological theory of the didactic. In H. Borko & D. Potari (Eds.), *Conference Pre-proceedings of the Twenty-Fifth ICMI study: Teachers of Mathematics Working and Learning in Collaborative Groups* (pp. 150–157). Lisbon, Portugal: University of Lisbon.

Quaresma, M., Winsløw, C., Clivaz, S., Ponte, J. P., Shúilleabháun, A. N., & Takahashi, A. [Eds.] (2018). *Mathematics lesson study around the world: Theoretical and methodological issues*. Springer. https://doi.org/10.1007/978-3-319-75696-7

Seino, T. & Foster, C. (2021). Analysis of the final comments provided by a knowledgeable other in lesson study. *Journal of Mathematics Teacher Education, 24*, 507–528. https://doi.org/10.1007/s10857-020-09468-y

Souma, K. (1997). *Sūgakuka mondai kaiketsu no jugyō* (in Japanese). Meigi tosho.

Stigler, J. W. & Hiebert, J. (1999). *The teaching gap: Best ideas from the world's teachers for improving education in the classroom*. New York, NY: Free Press.

Strømskag, H., & Chevallard, Y. (to appear). Didactic transposition and the knowledge to be
taught: towards an archeorganisation for concave/convex functions. *International Journal of Mathematical Education in Science and Technology*. https://doi.org/10.1080/0020739X.2024.2305879

Takahashi, A. (2021). *Teaching mathematics through problem-solving: A pedagogical approach from Japan*. Routledge.

Vásquez, S., Barquero, B., & Bosch, M. (2021). Teaching and learning combinatorics in secondary school: a modelling approach based on the Anthropological Theory of the Didactic. *Quadrante*, 30(2), 200–219. https://doi.org/10.48489/quadrante.23878

Winsløw, C. (2012). A comparative perspective on teacher collaboration: The cases of lesson study in Japan and of multidisciplinary teaching in Denmark. In G. Gueudet, B. Pepin, & L. Trouche (Eds.), *From text to ‘lived’ resources: Mathematics curriculum materials and teacher development* (pp. 291–304). Springer. https://doi.org/10.1007/978-94-007-1966-8_15

Winsløw, C. & Emori, H. (2006). Elements of a semiotic analysis of the secondary level classroom in Japan. In F. K. S. Leung, KD. Graf, F.J. Lopez-Real (Eds), *Mathematics education in different cultural traditions: A comparative study of East Asia and the West* (pp. 553–566). Springer.