# Language as a resource for teachers describing mathematics classroom teaching and learning: a comparative approach

Michèle Artigue, Christine Knipping, Jarmila Novotná, Birte Specht

<figure><img src="image_1.png" /></figure>

## To cite this version:

Michèle Artigue, Christine Knipping, Jarmila Novotná, Birte Specht. Language as a resource for teachers describing mathematics classroom teaching and learning: a comparative approach. ZDM, 2023, 55 (3), pp.597-610. 10.1007/s11858-023-01476-5. hal-04234731

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# Language as a resource for teachers describing mathematics classroom teaching and learning: a comparative approach

Michèle Artigue<fnref n="1" /> · Christine Knipping<fnref n="2" /> ID · Jarmila Novotná<fnref n="3" />,<fnref n="4" /> · Birte Specht<fnref n="5" />

Accepted: 21 February 2023 / Published online: 28 March 2023

© The Author(s) 2023

## Abstract

Describing mathematics classroom teaching and learning is part of a professional expertise. Comparing ‘our distinct stories’ and ‘our professional languages’ about teaching and learning in mathematics classes around the world, reveals cultural and linguistic underpinnings of our ‘stories’ and ‘languages’. Analysing and understanding these ‘stories’ and ‘languages’ in the form of narratives of classroom lessons, is a methodological approach that allows us to study and compare language as a resource across different cultural and linguistic contexts. We based our approach on results of the Lexicon Project, which set out to document the terms and the professional vocabulary that teachers use for describing the phenomena of middle school mathematics classrooms around the world, but enlarged this approach by narratives and a narrative methodology. Our cultural comparative approach based on these narratives revealed not only technical terms that define and make explicit didactical intentions, techniques and approaches, but also offered narrative descriptions of the enactment of the didactic intentions and techniques in situ. This approach provides a deeper understanding of the potential of language and turned out to be valuable for understanding how language orients teachers’ visions and analyses of mathematics classrooms.

**Keywords** Resources · Teachers · Classrooms · Teaching · Learning

## 1 Introduction

In line with the aim of this special issue, this paper focuses on the potential of a cross-cultural comparative approach when studying language as a resource for teachers (and researchers) describing mathematics classrooms, teaching and learning (Planas, 2018; see also Planas et al., 2022).

Our capacity and our inclination to act in classroom settings are significantly shaped by what our discourse establishes conceivable, and therefore as possible. In short, teachers’ classroom activity is influenced by

those practices they are able to name. (Mesiti et al., 2021, p. 17)

Identifying, reconstructing and comparing naming systems of teaching practices, as done in the International Classroom Lexicon Project (abbreviated as Lexicon Project or LP in the following) and conceptualized as different lexicons (Mesiti et al., 2021), enables us to come to an understanding of how a common professional language for describing classroom situations might constitute a resource for teachers. A ‘professional language’ allows decomposing of practice and it mirrors at the same time how this decomposing depends on the existence of a specific language and its structure for describing practice (Grossman et al., 2009; see also Mesiti et al., 2021). Professional development depends on language (Adler, 1999; González-Forte et al., 2022; Schicke, 2011). But, so far, a common ‘professional language’ has not been a general focus in mathematics education and therefore is rather underdeveloped in most cultural contexts of our field (Dobie et al., 2021; Grossman et al., 2009).

At the same time, ‘professional languages’ reduced to naming systems turn out to be too limited to understand how these orient teachers’ visions and analyses of mathematics

<footnote>Christine Knipping
knipping@math.uni-bremen.de</footnote>

<footnote n="1">¹ Université Paris Cité, Paris, France</footnote>

<footnote n="2">² Universität Bremen, Bremen, Germany</footnote>

<footnote n="3">³ Charles University, Prague, Czech Republic</footnote>

<footnote n="4">⁴ Université de Bordeaux, Bordeaux, France</footnote>

<footnote n="5">⁵ Carl-Von-Ossietzky Universität Oldenburg, Oldenburg, Germany</footnote>
# Language as a resource for teachers describing mathematics classroom teaching and learning: a comparative approach

Michèle Artigue<fnref n="1" /> · Christine Knipping<fnref n="2" /> ID · Jarmila Novotná<fnref n="3" />,<fnref n="4" /> · Birte Specht<fnref n="5" />

Accepted: 21 February 2023 / Published online: 28 March 2023

© The Author(s) 2023

## Abstract

Describing mathematics classroom teaching and learning is part of a professional expertise. Comparing ‘our distinct stories’ and ‘our professional languages’ about teaching and learning in mathematics classes around the world, reveals cultural and linguistic underpinnings of our ‘stories’ and ‘languages’. Analysing and understanding these ‘stories’ and ‘languages’ in the form of narratives of classroom lessons, is a methodological approach that allows us to study and compare language as a resource across different cultural and linguistic contexts. We based our approach on results of the Lexicon Project, which set out to document the terms and the professional vocabulary that teachers use for describing the phenomena of middle school mathematics classrooms around the world, but enlarged this approach by narratives and a narrative methodology. Our cultural comparative approach based on these narratives revealed not only technical terms that define and make explicit didactical intentions, techniques and approaches, but also offered narrative descriptions of the enactment of the didactic intentions and techniques in situ. This approach provides a deeper understanding of the potential of language and turned out to be valuable for understanding how language orients teachers’ visions and analyses of mathematics classrooms.

**Keywords** Resources · Teachers · Classrooms · Teaching · Learning

## 1 Introduction

In line with the aim of this special issue, this paper focuses on the potential of a cross-cultural comparative approach when studying language as a resource for teachers (and researchers) describing mathematics classrooms, teaching and learning (Planas, 2018; see also Planas et al., 2022).

Our capacity and our inclination to act in classroom settings are significantly shaped by what our discourse establishes conceivable, and therefore as possible. In short, teachers’ classroom activity is influenced by

those practices they are able to name. (Mesiti et al., 2021, p. 17)

Identifying, reconstructing and comparing naming systems of teaching practices, as done in the International Classroom Lexicon Project (abbreviated as Lexicon Project or LP in the following) and conceptualized as different lexicons (Mesiti et al., 2021), enables us to come to an understanding of how a common professional language for describing classroom situations might constitute a resource for teachers. A ‘professional language’ allows decomposing of practice and it mirrors at the same time how this decomposing depends on the existence of a specific language and its structure for describing practice (Grossman et al., 2009; see also Mesiti et al., 2021). Professional development depends on language (Adler, 1999; González-Forte et al., 2022; Schicke, 2011). But, so far, a common ‘professional language’ has not been a general focus in mathematics education and therefore is rather underdeveloped in most cultural contexts of our field (Dobie et al., 2021; Grossman et al., 2009).

At the same time, ‘professional languages’ reduced to naming systems turn out to be too limited to understand how these orient teachers’ visions and analyses of mathematics

<footnote>Christine Knipping
knipping@math.uni-bremen.de</footnote>

<footnote n="1">¹ Université Paris Cité, Paris, France</footnote>

<footnote n="2">² Universität Bremen, Bremen, Germany</footnote>

<footnote n="3">³ Charles University, Prague, Czech Republic</footnote>

<footnote n="4">⁴ Université de Bordeaux, Bordeaux, France</footnote>

<footnote n="5">⁵ Carl-Von-Ossietzky Universität Oldenburg, Oldenburg, Germany</footnote>
classrooms. For understanding and analyzing these visions and the intricate embedding of ‘stories’ in ‘language’ we rely on the concept of narrative, which is a prominent approach in the social sciences and also is applied in professional and vocational education (Schicke, 2011). Connelly and Clandinin argue that “knowing a teaching and learning situation is a matter of the recollections from one’s narratives that are called forth by the situation” (Clandinin, 2019, p. 7, see also Connelly & Clandinin, 1990).

This view of narratives and the ‘Narrative Turn’ in social science research in the 1970s, is associated with the work of Roland Barthes (1915–1980), the French semiologist (Czarniawska, 2004). We are committed to this turn in our approach to portraying teachers’ resources.

... narrative is present in every age, in every place, in every society; it begins with the very history of mankind and there nowhere is nor has been a people without narrative. All classes, all human groups, have their narratives ... it is simply there, like life itself. (Barthes, 1977, p. 79)

In our comparative work, we rely on narratives as cultural objects, as these are written in and about a specific cultural context. We claim that these narratives play an important role in the development and enacting of teachers’ professional language for describing classroom situations. The outcomes of the Lexicon Project (Artigue et al., 2017; Clarke et al., 2017; Mesiti et al., 2021) are, as mentioned above, another important source of our attempts to understand language as a resource for teachers describing mathematics classrooms, teaching and learning. This project set out to document the professional vocabulary that teachers use for describing the phenomena of middle school mathematics classrooms around the world. An important commitment in the LP was giving teachers a voice and building on their experiences when identifying teachers’ naming systems for classroom phenomena (Mesiti et al., 2021). It was also important to look at teaching practices and language from a cross-professional perspective (Grossman et al., 2009).

The generated lexicons showed significant differences, which showed that they are cultural artefacts, products of the particular pedagogical history of each community that indicate forms of conventionalized pedagogical practice in these communities (Mesiti & Clarke, 2018). To better understand and capture language as a resource we decided to go beyond these naming systems and also look into the narratives that professionals tell, when asked to describe instances of teaching and learning in mathematics classrooms. So, mutual narratives, agreed on by both researchers and teachers in our lexicon teams, became our focus. The first comparisons turned out to be promising for understanding how the narratives, seen as cultural objects, exposed visions of the classroom and of associated phenomena (Artigue et al., 2017).

Approaching narratives as windows into teachers’ resource systems (Trouche et al., 2019) is an enlargement of our former perspective. In total we investigated nine narratives, three narratives for a mathematics lesson in three national contexts, the Czech Republic, France and Germany. Based on these exemplary narratives, this paper explores the potential of such a cross-cultural comparative approach. We illustrate this potential by discussing three episodes (one from each cultural context of the nine narratives) that provide insights into teachers’ naming systems and narratives as a resource when talking about classrooms and different perspectives on teaching and learning.

## 2 Describing mathematics classrooms teaching and learning in a professional language

As indicated above, a professional language for describing classroom situations constitutes a resource for teachers and researchers, together. But, so far, a common ‘professional language’ has not been a general focus in mathematics education. This section draws attention to existing literature that addresses the ‘language need’ for describing mathematics classroom teaching and learning, as a resource and common enterprise in mathematics educational research and in teachers’ professional practice. We attempt to portray how far and in which directions this discussion has gone actually in mathematics education.

### 2.1 The ‘language need’ for describing mathematics classroom teaching and learning

As a professional community, teachers and researchers together need a language for describing mathematics classrooms teaching and learning. Dobie et al. (2021) “inquire as to whether having a name for an action, idea, or other phenomenon helps teachers to see it happening in their classrooms” (Dobie et al., 2021, p. 331). Rousseau and Morvan (2000) comment furthermore that the procedure of naming of concepts is crucial for the development of any scientific domain.

Studying and theorizing professional languages in our field of mathematics education seems to be at an early stage.

While some professions are able to boast of a professional language that is well-documented and comprehensively understood by members of its profession, Grossman and her colleagues (2009), in their cross-professional study of clergy, teaching and clinical psychology noted: “among our trio of professions, this language of practice seems particularly well-developed

<figure><img src="image_1.png" /></figure>
classrooms. For understanding and analyzing these visions and the intricate embedding of ‘stories’ in ‘language’ we rely on the concept of narrative, which is a prominent approach in the social sciences and also is applied in professional and vocational education (Schicke, 2011). Connelly and Clandinin argue that “knowing a teaching and learning situation is a matter of the recollections from one’s narratives that are called forth by the situation” (Clandinin, 2019, p. 7, see also Connelly & Clandinin, 1990).

This view of narratives and the ‘Narrative Turn’ in social science research in the 1970s, is associated with the work of Roland Barthes (1915–1980), the French semiologist (Czarniawska, 2004). We are committed to this turn in our approach to portraying teachers’ resources.

... narrative is present in every age, in every place, in every society; it begins with the very history of mankind and there nowhere is nor has been a people without narrative. All classes, all human groups, have their narratives ... it is simply there, like life itself. (Barthes, 1977, p. 79)

In our comparative work, we rely on narratives as cultural objects, as these are written in and about a specific cultural context. We claim that these narratives play an important role in the development and enacting of teachers’ professional language for describing classroom situations. The outcomes of the Lexicon Project (Artigue et al., 2017; Clarke et al., 2017; Mesiti et al., 2021) are, as mentioned above, another important source of our attempts to understand language as a resource for teachers describing mathematics classrooms, teaching and learning. This project set out to document the professional vocabulary that teachers use for describing the phenomena of middle school mathematics classrooms around the world. An important commitment in the LP was giving teachers a voice and building on their experiences when identifying teachers’ naming systems for classroom phenomena (Mesiti et al., 2021). It was also important to look at teaching practices and language from a cross-professional perspective (Grossman et al., 2009).

The generated lexicons showed significant differences, which showed that they are cultural artefacts, products of the particular pedagogical history of each community that indicate forms of conventionalized pedagogical practice in these communities (Mesiti & Clarke, 2018). To better understand and capture language as a resource we decided to go beyond these naming systems and also look into the narratives that professionals tell, when asked to describe instances of teaching and learning in mathematics classrooms. So, mutual narratives, agreed on by both researchers and teachers in our lexicon teams, became our focus. The first comparisons turned out to be promising for understanding how the narratives, seen as cultural objects, exposed visions of the classroom and of associated phenomena (Artigue et al., 2017).

Approaching narratives as windows into teachers’ resource systems (Trouche et al., 2019) is an enlargement of our former perspective. In total we investigated nine narratives, three narratives for a mathematics lesson in three national contexts, the Czech Republic, France and Germany. Based on these exemplary narratives, this paper explores the potential of such a cross-cultural comparative approach. We illustrate this potential by discussing three episodes (one from each cultural context of the nine narratives) that provide insights into teachers’ naming systems and narratives as a resource when talking about classrooms and different perspectives on teaching and learning.

## 2 Describing mathematics classrooms teaching and learning in a professional language

As indicated above, a professional language for describing classroom situations constitutes a resource for teachers and researchers, together. But, so far, a common ‘professional language’ has not been a general focus in mathematics education. This section draws attention to existing literature that addresses the ‘language need’ for describing mathematics classroom teaching and learning, as a resource and common enterprise in mathematics educational research and in teachers’ professional practice. We attempt to portray how far and in which directions this discussion has gone actually in mathematics education.

### 2.1 The ‘language need’ for describing mathematics classroom teaching and learning

As a professional community, teachers and researchers together need a language for describing mathematics classrooms teaching and learning. Dobie et al. (2021) “inquire as to whether having a name for an action, idea, or other phenomenon helps teachers to see it happening in their classrooms” (Dobie et al., 2021, p. 331). Rousseau and Morvan (2000) comment furthermore that the procedure of naming of concepts is crucial for the development of any scientific domain.

Studying and theorizing professional languages in our field of mathematics education seems to be at an early stage.

While some professions are able to boast of a professional language that is well-documented and comprehensively understood by members of its profession, Grossman and her colleagues (2009), in their cross-professional study of clergy, teaching and clinical psychology noted: “among our trio of professions, this language of practice seems particularly well-developed

<figure><img src="image_3.png" /></figure>
in clinical psychology but less developed in teaching”
(p. 2075). (Mesiti et al. 2021, p. 18)

Much earlier Lortie (1975) also outlines a lack of pro-
fessional language in teaching, as does Lampert (2000) for
the United States. Mesiti et al. (2021) further comment that
this is not only true for the US context, but can also be seen
in Australia, unlike China and Japan, where stronger tradi-
tions for developing professional languages in the context
of teaching exist.

The lack of a professional language for describing math-
ematics classroom teaching and learning has consequences
for our profession. For example, mathematics teachers'
noticing of incidents in the classroom depends on such a
language (Sherin et al., 2011, see also Adler, 1999). Also,
professional development (González-Forte et al., 2022), in
general and specifically of novice teachers, depends on lan-
guage and has been a goal in further developed professions
as law or medicine (Schicke, 2011). As Mesiti et al. (2021)
point out:

This has implications for each education community’s
conception of accomplished practice and for the prepa-
ration and ongoing professional development of their
teachers. Our articulation of teaching standards and
good teaching models is limited to those practices we
can adequately name and describe, and professional
learning opportunities are likely to be missed by teach-
ers with only limited language to reflect on teaching
practice and the phenomena of the classroom. (Mesiti
et al. 2021, p. 18)

The wording ‘community’ in this reference draws our
attention to two important dimensions, we need to consider
in this context: (1) the cultural dimension of language in
general and (2) professional languages specifically. As the
research literature indicates, professional languages related
to describing mathematics classroom teaching and learn-
ing have evolved differently so far, in different cultural con-
texts. On the other hand, professional communities in edu-
cation are not only constituted by teachers, but also include
researchers in the field. This is clear in medicine; medical
practitioners, such as physicians, surgeons or dentists, are
the bulk of the profession. But an essential part of the medi-
cal profession are doctors who not only practice but also
undertake research in the field, as well as other scientists
who do medical research but do not practice medicine. The
professional language in medicine is shared by practition-
ers and researchers. Similarly, looking at the professional
language(s) for describing mathematics classroom teaching
and learning requires to not only studying teachers’ use of
language, but also including the technical language used in
research for describing mathematics classroom teaching and
learning.

These considerations shed light on another issue. Profes-
sional language consists of terms from everyday language as
well as technical terms. For example, ‘didactical contract’,
a common notion in French Didactics, is a technical term,
describing the implicit or explicit rules that govern the shar-
ing of responsibilities between the teacher and the student, in
relation to the knowledge at stake (Brousseau, 1997). ‘Roter
Faden’, in contrast, is a term of everyday German, used for
describing the existence of a coherent theme, which is also
used in German professional contexts for describing a spe-
cific type of classroom lesson (Cramer et al., 2021). These
examples already indicate how complex the study of ‘profes-
sional languages’ can be and what it might include.

This observation, and a general lack of language emphasis in the Resource Approach to Mathematics Education, motivated this special issue. The starting point for it was to take language as “critical for analyzing, formulating, thinking and theorizing (Vygotsky, 1934) in mathematics, as in mathematics education” (call for this special issue). As Adler pointed out regarding her own research on language issues in a multilingual setting: “Firstly it was important to expand a conceptualization of resources beyond the physical and material to include social and cultural resources” (Adler, 2019, p. vii). Understanding language as a resource in the context of mathematics classrooms (Planas, 2018; Ruthven, 2019) may also be recognized by Gueudet as going beyond a focus on material resources: “The move towards a focus on resources should not be limited to material resources, easier to observe” (Gueudet, 2019, p. 38). It seems that professional languages in this context are a different kind of resource, not like resources such as designed teaching activities, used in a classroom.

Reflecting on the chapters in Trouche et al. (2019) Adler
emphasises:

Coming from a multilingual context, I was struck by
an apparent absence of reflection on language as a
resource or concern across many chapters. For those
working in multilingual contexts in particular, lan-
guage is a critical resource, and a substantial focus
of attention in research on language and mathematics
education—a large research domain in itself. (Adler,
2019, p. vii)

The recent focus on language in the Resource Approach
to Mathematics Education, which is grounded in the Docu-
mentational Approach to Didactics, also raises questions:
In how far is a professional language an ‘artefact’ or can
become an ‘instrument’ for describing mathematics class-
rooms teaching and learning?

"The instrumental approach distinguishes between an
artefact, product of the human activity, designed for a goal-
directed human activity, and an instrument developed by a
subject using the artefact. The instrument is a mixed entity,
in clinical psychology but less developed in teaching”
(p. 2075). (Mesiti et al. 2021, p. 18)

Much earlier Lortie (1975) also outlines a lack of pro-
fessional language in teaching, as does Lampert (2000) for
the United States. Mesiti et al. (2021) further comment that
this is not only true for the US context, but can also be seen
in Australia, unlike China and Japan, where stronger tradi-
tions for developing professional languages in the context
of teaching exist.

The lack of a professional language for describing math-
ematics classroom teaching and learning has consequences
for our profession. For example, mathematics teachers'
noticing of incidents in the classroom depends on such a
language (Sherin et al., 2011, see also Adler, 1999). Also,
professional development (González-Forte et al., 2022), in
general and specifically of novice teachers, depends on lan-
guage and has been a goal in further developed professions
as law or medicine (Schicke, 2011). As Mesiti et al. (2021)
point out:

This has implications for each education community’s
conception of accomplished practice and for the prepa-
ration and ongoing professional development of their
teachers. Our articulation of teaching standards and
good teaching models is limited to those practices we
can adequately name and describe, and professional
learning opportunities are likely to be missed by teach-
ers with only limited language to reflect on teaching
practice and the phenomena of the classroom. (Mesiti
et al. 2021, p. 18)

The wording ‘community’ in this reference draws our
attention to two important dimensions, we need to consider
in this context: (1) the cultural dimension of language in
general and (2) professional languages specifically. As the
research literature indicates, professional languages related
to describing mathematics classroom teaching and learn-
ing have evolved differently so far, in different cultural con-
texts. On the other hand, professional communities in edu-
cation are not only constituted by teachers, but also include
researchers in the field. This is clear in medicine; medical
practitioners, such as physicians, surgeons or dentists, are
the bulk of the profession. But an essential part of the medi-
cal profession are doctors who not only practice but also
undertake research in the field, as well as other scientists
who do medical research but do not practice medicine. The
professional language in medicine is shared by practition-
ers and researchers. Similarly, looking at the professional
language(s) for describing mathematics classroom teaching
and learning requires to not only studying teachers’ use of
language, but also including the technical language used in
research for describing mathematics classroom teaching and
learning.

These considerations shed light on another issue. Profes-
sional language consists of terms from everyday language as
well as technical terms. For example, ‘didactical contract’,
a common notion in French Didactics, is a technical term,
describing the implicit or explicit rules that govern the shar-
ing of responsibilities between the teacher and the student, in
relation to the knowledge at stake (Brousseau, 1997). ‘Roter
Faden’, in contrast, is a term of everyday German, used for
describing the existence of a coherent theme, which is also
used in German professional contexts for describing a spe-
cific type of classroom lesson (Cramer et al., 2021). These
examples already indicate how complex the study of ‘profes-
sional languages’ can be and what it might include.

This observation, and a general lack of language empha-
sis in the Resource Approach to Mathematics Education,
motivated this special issue. The starting point for it was
to take language as “critical for analyzing, formulating,
thinking and theorizing (Vygotsky, 1934) in mathematics,
as in mathematics education” (call for this special issue). As
Adler pointed out regarding her own research on language
issues in a multilingual setting: “Firstly it was important to
expand a conceptualization of resources beyond the physical
and material to include social and cultural resources” (Adler,
2019, p. vii). Understanding language as a resource in the
context of mathematics classrooms (Planas, 2018; Ruthven,
2019) may also be recognized by Gueudet as going beyond
a focus on material resources: “The move towards a focus
on resources should not be limited to material resources,
easier to observe” (Gueudet, 2019, p. 38). It seems that pro-
fessional languages in this context are a different kind of
resource, not like resources such as designed teaching activi-
ties, used in a classroom.

Reflecting on the chapters in Trouche et al. (2019) Adler
emphasises:

Coming from a multilingual context, I was struck by
an apparent absence of reflection on language as a
resource or concern across many chapters. For those
working in multilingual contexts in particular, lan-
guage is a critical resource, and a substantial focus
of attention in research on language and mathematics
education—a large research domain in itself. (Adler,
2019, p. vii)

The recent focus on language in the Resource Approach
to Mathematics Education, which is grounded in the Docu-
mentational Approach to Didactics, also raises questions:
In how far is a professional language an ‘artefact’ or can
become an ‘instrument’ for describing mathematics class-
rooms teaching and learning?

"The instrumental approach distinguishes between an
artefact, product of the human activity, designed for a goal-
directed human activity, and an instrument developed by a
subject using the artefact. The instrument is a mixed entity,
comprising the artefact and a scheme of use (Vergnaud, 1998) of this artefact" (Gueudet, 2019, p. 20).

Furthermore: How can we theoretically understand and empirically study our professional languages for describing mathematics classrooms teaching and learning?

A first stance to answer these questions has been taken by the Lexicon Project, which we will present next.

## 2.2 Learning from the professional lexicons of mathematics teachers around the world

In this section, we present the Lexicon Project, which was established in 2014 to identify the key pedagogical terms from various educational communities around the world. The LP set out to document "the professional vocabulary available to middle-school mathematics teachers in Australia, Chile, China, the Czech Republic, Finland, France, Germany, Japan, Korea, and the United States" (Mesiti et al., 2021, p. i; see also Clarke, 2015). When identifying naming systems for classroom phenomena, mixed teams of (senior and junior) researchers, and experienced teachers were formed, and further teachers, including novice teachers and teacher educators were consulted in the process of the study and involved in the validation of the lexicons.

The main outcome of the LP certainly is the collection of ten lexicons, which represent the pedagogical history and values of their community. As such, the lexicons provide an insight into language as a resource for teachers. These resources have evolved in different cultural contexts and have become locally institutionalised as a professional language of mathematics teachers. They make clear that teachers in one language have access to terms and constructs that may not be available to others, and that the actions, interactions and events of mathematics classrooms are described and categorised in different ways by different communities (Mesiti et al., 2019).

In the following paragraphs we briefly summarize some results of the LP, focusing on three cultural contexts—Czech Republic, France and Germany—to illustrate the potential and the insights provided by the comparative lexicon approach. Despite the differences observed between the Czech, French and German lexicons in number of terms and structure, all three undeniably offer linguistic resources for teachers to describe the practice of mathematics classrooms. All of them include terms that make it possible to identify phases in a lesson, to distinguish between different types of tasks proposed to students as learning progresses and between different pedagogical methods, to describe the pedagogical and didactic management of the class by the teacher, and to evaluate learning.

The three lexicons are, however, very different and, as is usually the case in comparative studies, comparison allows for a better understanding of the resources offered by each

of them as well as some of their limitations. Comparison shows that the French lexicon differs from the other two in its mathematical and didactic focus. This difference is expressed by the existence of a category specifically dedicated to mathematical activities, and also by the vocabulary available to differentiate types of tasks. In addition, the French lexicon differs from the other two in the number of terms with origin in didactic research or innovative practices that have emerged in the network of the Institutes for Research on Mathematics Teaching (IREM) where mathematicians, didacticians and teachers normally collaborate. These technical terms encapsulate, in one word or short expression, a body of knowledge and experience. In this respect, the French lexicon contrasts with the Czech lexicon, which is clearly more pedagogical and whose vocabulary is much less technical, much more descriptive and closer to everyday language. The German lexicon occupies an intermediate position. Also, cross-cultural comparison has particularly highlighted the way in which the German lexicon exploits the possibilities offered by its language. The German language allows the creation of compound words that are richer in meaning than their constituents, and may combine the concrete and the abstract, which results in technical terms that are both concise and accessible. For example, "The word Anschauungsmaterial is formed by combining two nouns Anschauung and Material. This new word derives its meanings from its components, but at the same time becomes a new entity" (Cramer et al., 2021, p. 228). The linguist Georges-Arthur Goldschmidt (1988) describes this phenomenon and points out another aspect that comes with it. Not only are words commonly composed by two or more familiar words in German, but each component is generally understandable and has a very concrete, material or even physical meaning. Thus, "It seems that the scientific research was never tempted to distance itself from a common understanding" (Goldschmidt, 1988, p. 23, our translation).

Zusammenfassung, for example is compound by the words zusammen (together) and fassen (grasp together), carries the concrete meaning of physically catching something. Zusammenfassung is characterized in the German lexicon as: "Written or oral summary of fundamental content and results, by the teacher" (Cramer et al., 2021, p. 253). The English translation 'summary' of the German term Zusammenfassung obviously loses this physical connotation.

The differences we have identified in our comparison of lexicons result from a combination of factors. In particular, they are marked by the didactic traditions in which these lexicons are embedded, with very different histories, as is shown in the chapters devoted to the three traditions in Blum et al. (2019). That said, comparing lexicons also raises many questions, such as: Does the absence of analogous terms in different lexicons mean that there is no analogue in the teachers' linguistic resources? Does this

<figure><img src="image_2.png" /></figure>
comprising the artefact and a scheme of use (Vergnaud, 1998) of this artefact" (Gueudet, 2019, p. 20).

Furthermore: How can we theoretically understand and empirically study our professional languages for describing mathematics classrooms teaching and learning?

A first stance to answer these questions has been taken by the Lexicon Project, which we will present next.

## 2.2 Learning from the professional lexicons of mathematics teachers around the world

In this section, we present the Lexicon Project, which was established in 2014 to identify the key pedagogical terms from various educational communities around the world. The LP set out to document "the professional vocabulary available to middle-school mathematics teachers in Australia, Chile, China, the Czech Republic, Finland, France, Germany, Japan, Korea, and the United States" (Mesiti et al., 2021, p. i; see also Clarke, 2015). When identifying naming systems for classroom phenomena, mixed teams of (senior and junior) researchers, and experienced teachers were formed, and further teachers, including novice teachers and teacher educators were consulted in the process of the study and involved in the validation of the lexicons.

The main outcome of the LP certainly is the collection of ten lexicons, which represent the pedagogical history and values of their community. As such, the lexicons provide an insight into language as a resource for teachers. These resources have evolved in different cultural contexts and have become locally institutionalised as a professional language of mathematics teachers. They make clear that teachers in one language have access to terms and constructs that may not be available to others, and that the actions, interactions and events of mathematics classrooms are described and categorised in different ways by different communities (Mesiti et al., 2019).

In the following paragraphs we briefly summarize some results of the LP, focusing on three cultural contexts—Czech Republic, France and Germany—to illustrate the potential and the insights provided by the comparative lexicon approach. Despite the differences observed between the Czech, French and German lexicons in number of terms and structure, all three undeniably offer linguistic resources for teachers to describe the practice of mathematics classrooms. All of them include terms that make it possible to identify phases in a lesson, to distinguish between different types of tasks proposed to students as learning progresses and between different pedagogical methods, to describe the pedagogical and didactic management of the class by the teacher, and to evaluate learning.

The three lexicons are, however, very different and, as is usually the case in comparative studies, comparison allows for a better understanding of the resources offered by each

of them as well as some of their limitations. Comparison shows that the French lexicon differs from the other two in its mathematical and didactic focus. This difference is expressed by the existence of a category specifically dedicated to mathematical activities, and also by the vocabulary available to differentiate types of tasks. In addition, the French lexicon differs from the other two in the number of terms with origin in didactic research or innovative practices that have emerged in the network of the Institutes for Research on Mathematics Teaching (IREM) where mathematicians, didacticians and teachers normally collaborate. These technical terms encapsulate, in one word or short expression, a body of knowledge and experience. In this respect, the French lexicon contrasts with the Czech lexicon, which is clearly more pedagogical and whose vocabulary is much less technical, much more descriptive and closer to everyday language. The German lexicon occupies an intermediate position. Also, cross-cultural comparison has particularly highlighted the way in which the German lexicon exploits the possibilities offered by its language. The German language allows the creation of compound words that are richer in meaning than their constituents, and may combine the concrete and the abstract, which results in technical terms that are both concise and accessible. For example, "The word Anschauungsmaterial is formed by combining two nouns Anschauung and Material. This new word derives its meanings from its components, but at the same time becomes a new entity" (Cramer et al., 2021, p. 228). The linguist Georges-Arthur Goldschmidt (1988) describes this phenomenon and points out another aspect that comes with it. Not only are words commonly composed by two or more familiar words in German, but each component is generally understandable and has a very concrete, material or even physical meaning. Thus, "It seems that the scientific research was never tempted to distance itself from a common understanding" (Goldschmidt, 1988, p. 23, our translation).

Zusammenfassung, for example is compound by the words *zusammen (together)* and *fassen (grasp together)*, carries the concrete meaning of physically catching something. Zusammenfassung is characterized in the German lexicon as: "Written or oral summary of fundamental content and results, by the teacher" (Cramer et al., 2021, p. 253). The English translation 'summary' of the German term Zusammenfassung obviously loses this physical connotation.

The differences we have identified in our comparison of lexicons result from a combination of factors. In particular, they are marked by the didactic traditions in which these lexicons are embedded, with very different histories, as is shown in the chapters devoted to the three traditions in Blum et al. (2019). That said, comparing lexicons also raises many questions, such as: Does the absence of analogous terms in different lexicons mean that there is no analogue in the teachers' linguistic resources? Does this

<figure><img src="image_4.png" /></figure>
absence mean that what these terms refer to is excluded
from their field of perception? The results of recent
research (Mesiti et al., 2022) about mathematically-related
terms in the Australian, Czech, Chilean and French lexicons,
encourage us not to draw hasty conclusions. Lexicon
terms are like bricks. They are constituents of teachers'
linguistic resources, but linguistic resources extend any
terminology (Goldschmidt, 1988).

## 2.3 Capturing language as a resource

To address these lexical limitations of the Lexicon Pro-
ject, we have taken up the idea of narratives introduced in
Artigue et al. (2017) and used the videos of mathematics
classroom lessons made for the LP as stimuli to provoke
narratives. As did the lexicons, the narratives reveal how
language regulates individuals’ experiences and interac-
tions (Cole & Engeström, 1993). Documenting language
use in narratives allows us to mirror what teachers (and
researchers) are ‘noticing’ in classroom contexts (Sherin
et al., 2011). This is also reflected in research in math-
ematics education on how learners and teachers use the
resource of language (Adler, 1999; González-Forte et al.,
2022; see also Planas et al., 2022).

These considerations bring us to the focus of this article
and the methodology we used for capturing and compar-
ing ‘our distinct stories’ and ‘our professional languages’
about teaching and learning in mathematics classes. The
narrative approach allows us to study and compare lan-
guage as a resource across different cultural and linguis-
tic contexts. Accordingly, we pose the following research
question:

How does a cultural comparative approach of narra-
tives help us to understand language as a resource for
teachers describing mathematics classrooms teaching
and learning?

Language as a resource for teachers is discussed in this
article primarily in respect to teachers’ and researchers’
common conceptualizing of classroom practice, instruction
and learning, across different cultural contexts and linguis-
tic traditions. This is the focus and the contribution of this
article.

## 3 Methodology and methods

In this section, we present the methodology and methods of
comparing narratives, which allow us to reveal cultural and
linguistic underpinnings of our ‘stories’ about mathematics
classrooms and ‘languages’ we use for describing these.

## 3.1 Narratives

Connelly and Clandinin argue that knowledge of a teaching
and learning setting is a matter of the individual’s narratives
elicited by the setting.

These recollections are personal, in that they are
derived from a person’s narrative, and they are practi-
cal, in that they are aimed at meeting the demands
of a particular situation. These recollections are, of
course, also theoretical, both in the sense of containing
conceptual content, and they are cultural, in the sense
that individual narratives are embedded in cultural and
historical narratives. (Connelly & Clandinin, 1985, pp.
184–185)

This understanding of narratives and their significant role
for describing and understanding teaching and learning situ-
ations, is embedded in methodological and theoretical tradi-
tions of the social sciences, as mentioned above. Clandinin
and Connelly highlight several advantageous points of the
narrative approach. First, narratives allow us to grasp our
understanding of classroom teaching and learning situations.
Second, narratives are ‘called forth by the situation’, making
the teaching and learning understandable. Third, they allow
us to grasp ‘personal’ (i.e. different) views on a teaching and
learning context, within and across cultural comparisons.
Fourth, as narratives are ‘aimed at meeting the demands of
a particular situation’ they give meaning to our professional
language, when describing classroom teaching and learn-
ing. Fifth, narratives are ‘theoretical’ in the sense that their
thick descriptions actually capture significant dynamics of
teaching and learning, which allows us to theoretically con-
ceptualize them. Sixth, narratives are ‘cultural’ according
to Clandinin and Connelly. Narratives are more than just
personal, individual stories, but imbricated in cultural and
historical underpinnings. This means that comparing narra-
tives allows us to reveal and understand this footing.

Like Trgalova et al. (2019) we follow “... the narrative
inquiry (Clandinin & Connelly, 1990), which involves the
reconstruction of the experience of a person in relation to
both the other and a social background” (Trgalova et al.,
2019, p. 216). We adopt an approach that has been suggested
by Artigue et al. (2017).

With this new artefact, in line with the use of nar-
ratives in education research (Clandinin & Connelli,
2000), our intention was to make it visible again that,
for each team, each video tells a particular story gener-
ated through the selection, arrangement and interpre-
tation of events which contain meaning for the teller,
without forgetting that the stories we produce are
socio-cultural products. Introducing these narratives
also allowed us to include the additional information
absence mean that what these terms refer to is excluded
from their field of perception? The results of recent
research (Mesiti et al., 2022) about mathematically-related
terms in the Australian, Czech, Chilean and French lexicons,
encourage us not to draw hasty conclusions. Lexicon
terms are like bricks. They are constituents of teachers'
linguistic resources, but linguistic resources extend any
terminology (Goldschmidt, 1988).

## 2.3 Capturing language as a resource

To address these lexical limitations of the Lexicon Pro-
ject, we have taken up the idea of narratives introduced in
Artigue et al. (2017) and used the videos of mathematics
classroom lessons made for the LP as stimuli to provoke
narratives. As did the lexicons, the narratives reveal how
language regulates individuals’ experiences and interac-
tions (Cole & Engeström, 1993). Documenting language
use in narratives allows us to mirror what teachers (and
researchers) are ‘noticing’ in classroom contexts (Sherin
et al., 2011). This is also reflected in research in math-
ematics education on how learners and teachers use the
resource of language (Adler, 1999; González-Forte et al.,
2022; see also Planas et al., 2022).

These considerations bring us to the focus of this article
and the methodology we used for capturing and compar-
ing ‘our distinct stories’ and ‘our professional languages’
about teaching and learning in mathematics classes. The
narrative approach allows us to study and compare lan-
guage as a resource across different cultural and linguis-
tic contexts. Accordingly, we pose the following research
question:

How does a cultural comparative approach of narra-
tives help us to understand language as a resource for
teachers describing mathematics classrooms teaching
and learning?

Language as a resource for teachers is discussed in this
article primarily in respect to teachers’ and researchers’
common conceptualizing of classroom practice, instruction
and learning, across different cultural contexts and linguis-
tic traditions. This is the focus and the contribution of this
article.

# 3 Methodology and methods

In this section, we present the methodology and methods of
comparing narratives, which allow us to reveal cultural and
linguistic underpinnings of our ‘stories’ about mathematics
classrooms and ‘languages’ we use for describing these.

## 3.1 Narratives

Connelly and Clandinin argue that knowledge of a teaching
and learning setting is a matter of the individual’s narratives
elicited by the setting.

These recollections are personal, in that they are
derived from a person’s narrative, and they are practi-
cal, in that they are aimed at meeting the demands
of a particular situation. These recollections are, of
course, also theoretical, both in the sense of containing
conceptual content, and they are cultural, in the sense
that individual narratives are embedded in cultural and
historical narratives. (Connelly & Clandinin, 1985, pp.
184–185)

This understanding of narratives and their significant role
for describing and understanding teaching and learning situ-
ations, is embedded in methodological and theoretical tradi-
tions of the social sciences, as mentioned above. Clandinin
and Connelly highlight several advantageous points of the
narrative approach. First, narratives allow us to grasp our
understanding of classroom teaching and learning situations.
Second, narratives are ‘called forth by the situation’, making
the teaching and learning understandable. Third, they allow
us to grasp ‘personal’ (i.e. different) views on a teaching and
learning context, within and across cultural comparisons.
Fourth, as narratives are ‘aimed at meeting the demands of
a particular situation’ they give meaning to our professional
language, when describing classroom teaching and learn-
ing. Fifth, narratives are ‘theoretical’ in the sense that their
thick descriptions actually capture significant dynamics of
teaching and learning, which allows us to theoretically con-
ceptualize them. Sixth, narratives are ‘cultural’ according
to Clandinin and Connelly. Narratives are more than just
personal, individual stories, but imbricated in cultural and
historical underpinnings. This means that comparing narra-
tives allows us to reveal and understand this footing.

Like Trgalova et al. (2019) we follow “... the narrative
inquiry (Clandinin & Connelly, 1990), which involves the
reconstruction of the experience of a person in relation to
both the other and a social background” (Trgalova et al.,
2019, p. 216). We adopt an approach that has been suggested
by Artigue et al. (2017).

With this new artefact, in line with the use of nar-
ratives in education research (Clandinin & Connelli,
2000), our intention was to make it visible again that,
for each team, each video tells a particular story gener-
ated through the selection, arrangement and interpre-
tation of events which contain meaning for the teller,
without forgetting that the stories we produce are
socio-cultural products. Introducing these narratives
also allowed us to include the additional information
<figure><img src="image_3.png" /><figcaption>Fig. 1 Methodological approach</figcaption></figure>

about the sessions and their context in the description, that each team considered necessary to generate the story. (Artigue et al., 2017, p. 115)

In the narrative approach used and presented here, narratives are understood as methodological artefacts. Narratives in our understanding go beyond the methodological tools generated in the Lexicon Project, which was limited to the terms used by teachers. When delving into a narrative methodology our goal was to capture teachers' and researchers' professional views on mathematics classrooms teaching and learning. We also found, when first experimenting with this approach, that our discourses as researchers and teachers generally contain mathematical and pedagogical-didactical language when we speak about mathematics lessons. But, as it had been decided at the beginning of the Lexicon Project that the national lexicons should not contain specifically mathematical terms, this language was missing. Through narratives we intended to capture this mix of pedagogical-didactical and mathematical language, where pedagogical-didactical refers to teaching and learning processes in general. Didactic language is more specific to each discipline.

We also hypothesized that this mixing might be culture dependent. Our examples in this article illustrate concretely how this mixing of pedagogical and mathematical language looks.

Last, but not least, we decided to write descriptive narratives. We wanted to limit our narratives to thick descriptions, not drawing inferences about the 'why'. Our exchanges with colleagues from other countries in the Lexicon Project had shown how easily we misinterpreted classroom situations when making such inferences. Another reason was that, due to the spirit of the LP, we wanted to make a clear distinction between these narratives and the didactic analyses of the sessions guided by our theoretical frameworks, which as researchers we could produce and would be, of course, another kind of story.

## 3.2 The creation of narratives and their structure

The descriptive narratives that are at the core of the approach were stimulated by videos of mathematics classroom lessons made for the Lexicon Project and inspired by the lexicons produced around these videos (see Fig. 1 with a focus on the

<figure><img src="image_4.png" /></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 1 Methodological approach</figcaption></figure>

about the sessions and their context in the description, that each team considered necessary to generate the story. (Artigue et al., 2017, p. 115)

In the narrative approach used and presented here, narratives are understood as methodological artefacts. Narratives in our understanding go beyond the methodological tools generated in the Lexicon Project, which was limited to the terms used by teachers. When delving into a narrative methodology our goal was to capture teachers' and researchers' professional views on mathematics classrooms teaching and learning. We also found, when first experimenting with this approach, that our discourses as researchers and teachers generally contain mathematical and pedagogical-didactical language when we speak about mathematics lessons. But, as it had been decided at the beginning of the Lexicon Project that the national lexicons should not contain specifically mathematical terms, this language was missing. Through narratives we intended to capture this mix of pedagogical-didactical and mathematical language, where pedagogical-didactical refers to teaching and learning processes in general. Didactic language is more specific to each discipline.

We also hypothesized that this mixing might be culture dependent. Our examples in this article illustrate concretely how this mixing of pedagogical and mathematical language looks.

Last, but not least, we decided to write descriptive narratives. We wanted to limit our narratives to thick descriptions, not drawing inferences about the 'why'. Our exchanges with colleagues from other countries in the Lexicon Project had shown how easily we misinterpreted classroom situations when making such inferences. Another reason was that, due to the spirit of the LP, we wanted to make a clear distinction between these narratives and the didactic analyses of the sessions guided by our theoretical frameworks, which as researchers we could produce and would be, of course, another kind of story.

## 3.2 The creation of narratives and their structure

The descriptive narratives that are at the core of the approach were stimulated by videos of mathematics classroom lessons made for the Lexicon Project and inspired by the lexicons produced around these videos (see Fig. 1 with a focus on the

<figure><img src="image_6.png" /></figure>
French team). As with the lexicons, these narratives were
team productions and the teachers in the teams were given
a predominant voice in case of disagreement.

The narratives include a general description of the con-
text and structure of the lesson, followed by a chronologi-
cal description of the lesson. As explained in Artigue et al.
(2017), when introducing this methodological artefact the
intention of the authors was to make visible the different
stories that each team notices and tells when watching the
same video in line with the use of narratives in education
research (Clandinin & Connelli, 2000). These stories are
socio-cultural products (Artigue et al., 2017). The narra-
tives aim at capturing the stories by informal descriptions
and by using terms from the lexicons. For the research
reported here, three narratives have been produced for each
of the Czech, French and German videos, by the co-authors,
together with teachers<fnref n="1" /> (see Fig. 1). The instruction given
was: “Describe the lesson for a colleague teacher giving her/
him the feeling that she/he is there. Use the lexicon terms.”
The narratives were written first in the native language of
each team and afterwards translated into English, our lan-
guage of international communication both in the Lexicon
Project and in this particular study. These translated versions
are used in this paper. However, we regularly came back to
the original versions of the narratives and lexicon terms in
the development of the comparative analysis.

## 3.3 Comparing narratives and connecting them to the lexicons

Looking at these narratives a descriptive and eclectic style
was evident. The narratives were not analytic, but vague
and linguistically multifaceted. They illustrated some lexи-
con terms, but mainly employed linguistic alternatives not
contained in the lexicon. Given this result, we decided to
identify in a first global comparative analysis the terms used
in the narratives or close variations of these. The lexicon
terms found in these narratives, as in the lexicons them-
selves, denoted different elements “from lesson phases, types
of tasks, and of mathematical activities to the forms that
interactions take in the classroom” (Artigue et al., 2021,
p. 193), which had led seven Lexicon teams, including the
Czech and the French team, to organize them into categories.
The French and the Czech team found it useful to use these
categories (see Appendix 1) in their narratives, which we
included in our comparative analyses.

The German lexicon does not provide such categories
(Cramer et al., 2021, p. 232), but a system of cross-refer-
ences was adopted instead (Cramer et al., 2021, p. 223).

So, for the German narratives and our global comparative
analyses we used the categories that had been suggested by
the Czech and French teams. Through comparison of the
three different narratives for each country’s lesson it became
evident that at times similar terms and categories were used
to describe the lesson, but sometimes other categories and
terms were used. The narratives reflected different cultural
discourses about the same classroom situations. These com-
parative analyses of the narratives also indicated varying
resource systems for describing classrooms. This moti-
vated us to conduct a more detailed analysis, an analysis of
selected episodes, whose results are presented in Sect. 4.

## 3.4 Analysing selected episodes

For a comparative analysis of narratives, we identified as
a first step two episodes for each narrative as shown in
the third row of Fig. 1. These were selected by each team
according to the following criteria: short episodes show-
ing captivating features of the lesson and a priori appro-
priate to illustrate and discuss the potential of narratives;
episodes offering complementary perspectives on the lesson
(see more details in Sect. 4). For each episode, we listed
which lexicon terms or close variants were used in the nar-
rative of the episode. After that we listed again all lexicon
terms that could be associated with the episode, but were
not used in the narratives. We then investigated what can be
learned from the terms in addition to the descriptive form
of the narratives. Our intention in doing this refined, inter-
pretative analysis of these episodes was to understand bet-
ter the resource potential offered by the different narratives.
Comparative analyses of these refined interpretations finally
allowed us to characterize this resource potential. We will
illustrate the value of our comparative narrative approach
and its resource potential in the next section.

# 4 Illustrating comparisons of narratives via episodes—first results

In this section, we illustrate via episodes similarities and
differences of our Czech, French and German narratives
and the potential of our comparative narrative analyses. We
reveal not only diverse perspectives on each episode, but
also show how language and linguistic specifics shape our
views and descriptions of classroom situations.

In the following, after a brief introduction, we present for
each episode the corresponding part of one narrative (e.g.
the Czech narrative of the Czech episode, Fig. 2, the under-
lined portions are terms of the lexicon) and a commentary,
summarizing the interpretation of the episode in the differ-
ent narratives (the Czech, French and German narrative of
the episode).

<footnote n="1">¹ The narratives already produced in the Czech-French comparison have only been slightly revised.</footnote>
French team). As with the lexicons, these narratives were
team productions and the teachers in the teams were given
a predominant voice in case of disagreement.

The narratives include a general description of the con-
text and structure of the lesson, followed by a chronologi-
cal description of the lesson. As explained in Artigue et al.
(2017), when introducing this methodological artefact the
intention of the authors was to make visible the different
stories that each team notices and tells when watching the
same video in line with the use of narratives in education
research (Clandinin & Connelli, 2000). These stories are
socio-cultural products (Artigue et al., 2017). The narra-
tives aim at capturing the stories by informal descriptions
and by using terms from the lexicons. For the research
reported here, three narratives have been produced for each
of the Czech, French and German videos, by the co-authors,
together with teachers<fnref n="1" /> (see Fig. 1). The instruction given
was: “Describe the lesson for a colleague teacher giving her/
him the feeling that she/he is there. Use the lexicon terms.”
The narratives were written first in the native language of
each team and afterwards translated into English, our lan-
guage of international communication both in the Lexicon
Project and in this particular study. These translated versions
are used in this paper. However, we regularly came back to
the original versions of the narratives and lexicon terms in
the development of the comparative analysis.

3.3 Comparing narratives and connecting them to the lexicons

Looking at these narratives a descriptive and eclectic style
was evident. The narratives were not analytic, but vague
and linguistically multifaceted. They illustrated some lexи-
con terms, but mainly employed linguistic alternatives not
contained in the lexicon. Given this result, we decided to
identify in a first global comparative analysis the terms used
in the narratives or close variations of these. The lexicon
terms found in these narratives, as in the lexicons them-
selves, denoted different elements “from lesson phases, types
of tasks, and of mathematical activities to the forms that
interactions take in the classroom” (Artigue et al., 2021,
p. 193), which had led seven Lexicon teams, including the
Czech and the French team, to organize them into categories.
The French and the Czech team found it useful to use these
categories (see Appendix 1) in their narratives, which we
included in our comparative analyses.

The German lexicon does not provide such categories
(Cramer et al., 2021, p. 232), but a system of cross-refer-
ences was adopted instead (Cramer et al., 2021, p. 223).

So, for the German narratives and our global comparative
analyses we used the categories that had been suggested by
the Czech and French teams. Through comparison of the
three different narratives for each country’s lesson it became
evident that at times similar terms and categories were used
to describe the lesson, but sometimes other categories and
terms were used. The narratives reflected different cultural
discourses about the same classroom situations. These com-
parative analyses of the narratives also indicated varying
resource systems for describing classrooms. This moti-
vated us to conduct a more detailed analysis, an analysis of
selected episodes, whose results are presented in Sect. 4.

**3.4 Analysing selected episodes**

For a comparative analysis of narratives, we identified as
a first step two episodes for each narrative as shown in
the third row of Fig. 1. These were selected by each team
according to the following criteria: short episodes show-
ing captivating features of the lesson and a priori appro-
priate to illustrate and discuss the potential of narratives;
episodes offering complementary perspectives on the lesson
(see more details in Sect. 4). For each episode, we listed
which lexicon terms or close variants were used in the nar-
rative of the episode. After that we listed again all lexicon
terms that could be associated with the episode, but were
not used in the narratives. We then investigated what can be
learned from the terms in addition to the descriptive form
of the narratives. Our intention in doing this refined, inter-
pretative analysis of these episodes was to understand bet-
ter the resource potential offered by the different narratives.
Comparative analyses of these refined interpretations finally
allowed us to characterize this resource potential. We will
illustrate the value of our comparative narrative approach
and its resource potential in the next section.

4 Illustrating comparisons of narratives
via episodes—first results

In this section, we illustrate via episodes similarities and
differences of our Czech, French and German narratives
and the potential of our comparative narrative analyses. We
reveal not only diverse perspectives on each episode, but
also show how language and linguistic specifics shape our
views and descriptions of classroom situations.

In the following, after a brief introduction, we present for
each episode the corresponding part of one narrative (e.g.
the Czech narrative of the Czech episode, Fig. 2, the under-
lined portions are terms of the lexicon) and a commentary,
summarizing the interpretation of the episode in the differ-
ent narratives (the Czech, French and German narrative of
the episode).

<footnote n="1">¹ The narratives already produced in the Czech-French comparison have only been slightly revised.</footnote>
## 4.1 Czech episode

The Czech episode illustrates how a teacher introduces a new method and builds on her students' prior knowledge and experience. The Czech narrative and the Czech terminology used both emphasize an interactive organization of the lesson, where the teacher and the students both actively engage in investigating a new method. The Czech episode is represented in similar, but also distinct ways in the three narratives.

The Czech description of the episode is comparatively short, as are the Czech narratives in general. It is written in active form, expressing the teacher's or the students' actions (e.g. *Teaching Methods*), without entering into a discussion of possible functions of actions. *Explanation*, a key term in the Czech lexicon, is used in the description of this episode (here *explanation in the form of a teacher's monologue*), but its specific function is not indicated. It is also noteworthy that it first seems that in the Czech narrative of this episode, and in general in the Czech narratives, the focus is on interactions. As highlighted in Artigue et al. (2017), the Czech lexicon is mostly a lexicon in which the mathematics discipline is hardly visible. We see this also in the language used in the narrative of this episode. The absence of a technical language more specific to the mathematics discipline is related to a long-lasting influence of general didactics on the Czech educational culture, which can be traced back to the Didactica Magna by Comenius in the twelfth century (Blum et al., 2019).

But, with a second, closer look, we see that the actions described in the Czech Episode (see Fig. 2) are closely bonded to the mathematical content at stake and the *Processes Supporting Pupils Learning* of this content. But, no specific linguistic terms are used for these descriptions, except the term *linking*. *Linking* is connotated in the Czech lexicon in a mathematical sense, so it has a technical meaning more specific than general.

As in the Czech description *Teaching Methods* play a central role in the German narrative of the Czech episode,

at the same time the structure of the lesson is another focus in the German narrative. Even though *Roter Faden*, a term in the German lexicon, is not used in the German narrative of this episode, the focus on the structure could have well been described by this metaphorical term. “... the term *Roter Faden* translates literally as *red thread* but the sense of the term is *main theme* or perhaps better *leitmotif*” (Cramer et al., 2021, p. 223). In addition, the mathematical content is emphasised in the German narrative, which is evident in the use of the terms *abstraction* and *reasoning*. But terms related to mathematical content are not as often explicitly employed as in the French narrative, where five terms are used to characterize the *Mathematical Activities*, an important category of the French Lexicon. *Pupils’ Individual Work* (1) and *Stages of a Lesson* (3) are also described in the French narrative, but not as much emphasized as the mathematical content and the *Teaching Methods* (9). The numbers in brackets indicate the occurrence of terms belonging to this category. The didactical and mathematical description of the episode is overt in the French narrative of the episode.

## 4.2 French episode

The French episode “Correction of homework” has been selected, because it shows some relevant aspects of this mathematics lesson: one can see the expertise of the teacher, not just asking for answers, but also for *justifications* like a proof of the rules. She enriches the phase of correction in order to deepen students’ knowledge (*validating*). An ordinary task (*correction of homework*) is central in this episode, but the way the teacher handles it is noteworthy (Fig. 3).

The French episode illustrates how a teacher handles an ordinary task, the correction of homework, in a promising way, asking for justifications of general rules and validating in class whether solutions are pertinent or not. Common in all three narratives of this episode is that the *TM* and the *Pedagogical and Didactical Management of the Classroom* are a focus. Differences can be seen in regard of the insights provided by the analyses of the *Mathematical Activities*.

<caption>**Fig. 2** Czech episode</caption>

**Description of the Czech episode in the Czech narrative**

The next stage is a *teacher's communication of information*. The teacher introduces a new method: the comparison method for solving a set of two equations with two unknowns. She gives an *explanation in the form of a teacher's monologue* on how the presentation will be organized and why she chose this procedure. She continues by solving the system of equations and gives *explanations* for every step in the solution process. For some steps the teacher opens the space for students and asks them to solve the parts that they are able to master with their previous knowledge, for example expressing one variable in terms of the other. For finding the principle of the comparison method of solving a set of two equations for two unknowns, the teacher tries to guide the students by posing questions for *clarification*. She also does *linking* by recalling a previous approach for solving the same task, where they used a table.

<figure><img src="image_5.png" /></figure>
## 4.1 Czech episode

The Czech episode illustrates how a teacher introduces a new method and builds on her students' prior knowledge and experience. The Czech narrative and the Czech terminology used both emphasize an interactive organization of the lesson, where the teacher and the students both actively engage in investigating a new method. The Czech episode is represented in similar, but also distinct ways in the three narratives.

The Czech description of the episode is comparatively short, as are the Czech narratives in general. It is written in active form, expressing the teacher's or the students' actions (e.g. *Teaching Methods*), without entering into a discussion of possible functions of actions. *Explanation*, a key term in the Czech lexicon, is used in the description of this episode (here *explanation in the form of a teacher's monologue*), but its specific function is not indicated. It is also noteworthy that it first seems that in the Czech narrative of this episode, and in general in the Czech narratives, the focus is on interactions. As highlighted in Artigue et al. (2017), the Czech lexicon is mostly a lexicon in which the mathematics discipline is hardly visible. We see this also in the language used in the narrative of this episode. The absence of a technical language more specific to the mathematics discipline is related to a long-lasting influence of general didactics on the Czech educational culture, which can be traced back to the Didactica Magna by Comenius in the twelfth century (Blum et al., 2019).

But, with a second, closer look, we see that the actions described in the Czech Episode (see Fig. 2) are closely bonded to the mathematical content at stake and the *Processes Supporting Pupils Learning* of this content. But, no specific linguistic terms are used for these descriptions, except the term *linking*. *Linking* is connotated in the Czech lexicon in a mathematical sense, so it has a technical meaning more specific than general.

As in the Czech description *Teaching Methods* play a central role in the German narrative of the Czech episode,

at the same time the structure of the lesson is another focus in the German narrative. Even though *Roter Faden*, a term in the German lexicon, is not used in the German narrative of this episode, the focus on the structure could have well been described by this metaphorical term. “... the term *Roter Faden* translates literally as *red thread* but the sense of the term is *main theme* or perhaps better *leitmotif*” (Cramer et al., 2021, p. 223). In addition, the mathematical content is emphasised in the German narrative, which is evident in the use of the terms *abstraction* and *reasoning*. But terms related to mathematical content are not as often explicitly employed as in the French narrative, where five terms are used to characterize the *Mathematical Activities*, an important category of the French Lexicon. *Pupils’ Individual Work* (1) and *Stages of a Lesson* (3) are also described in the French narrative, but not as much emphasized as the mathematical content and the *Teaching Methods* (9). The numbers in brackets indicate the occurrence of terms belonging to this category. The didactical and mathematical description of the episode is overt in the French narrative of the episode.

## 4.2 French episode

The French episode “Correction of homework” has been selected, because it shows some relevant aspects of this mathematics lesson: one can see the expertise of the teacher, not just asking for answers, but also for *justifications* like a proof of the rules. She enriches the phase of correction in order to deepen students’ knowledge (*validating*). An ordinary task (*correction of homework*) is central in this episode, but the way the teacher handles it is noteworthy (Fig. 3).

The French episode illustrates how a teacher handles an ordinary task, the correction of homework, in a promising way, asking for justifications of general rules and validating in class whether solutions are pertinent or not. Common in all three narratives of this episode is that the *TM* and the *Pedagogical and Didactical Management of the Classroom* are a focus. Differences can be seen in regard of the insights provided by the analyses of the *Mathematical Activities*.

<caption>**Fig. 2** Czech episode</caption>

**Description of the Czech episode in the Czech narrative**

The next stage is a *teacher's communication of information*. The teacher introduces a new method: the comparison method for solving a set of two equations with two unknowns. She gives an *explanation in the form of a teacher's monologue* on how the presentation will be organized and why she chose this procedure. She continues by solving the system of equations and gives *explanations* for every step in the solution process. For some steps the teacher opens the space for students and asks them to solve the parts that they are able to master with their previous knowledge, for example expressing one variable in terms of the other. For finding the principle of the comparison method of solving a set of two equations for two unknowns, the teacher tries to guide the students by posing questions for *clarification*. She also does *linking* by recalling a previous approach for solving the same task, where they used a table.

<figure><img src="image_7.png" /></figure>
<caption>Fig. 3 French episode</caption>

### Description of the French episode in the French narrative

The *correction* of the homework has a *dialogic form*, one student being asked for each exercise. After cleaning the board, the teacher *launches this phase*, making clear its stake and *explicit* the meaning of the notations $3^2$ et $3^3$.

<figure><img src="image_6.png" /></figure>

When the teacher *asks for* a result, Jason begins to *calculate* mentally. Ultimately the teacher clarifies the task pointing out that he simply wants the final result. Jason now provides the *expected answer*: "3 exponent 5". The teacher *validates* this answer and *asks the class to formulate a general rule*, in form of a sentence (*generalization*). He then *helps* the class to produce a precise *formulation*. For the next calculation (b) another student is called on. After also *validating* his answer, the teacher asks the student to *make explicit* the *implicit* exponent 1, *validates* the answer and comments. Calculation d) mobilizes a negative exponent and the teacher takes this opportunity to *recall* that the operations can be performed both with positive and negative numbers. For calculation e), the teacher *asks for a justification* of the result. The student called on for this task refers back to the *definition* of the power notation and *explains* the performed simplification. Again, the teacher makes the student *formulate* a corresponding general rule (*generalizing*). For reasons of *time management*, the *correction* of the next exercises is quickly done. For the two first ones using the same rule, the teacher *corrects an error* without insisting on it. He then tightly guides the correction of the last exercise while writing it on the board. He finally *asks* for the decomposition of the two powers, then suggests to change the order of the terms to arrive at $2 \times 3 \times 2 \times 3$, then grouped in $6 \times 6$, which is $6^2$. The teacher now *formulates* himself the corresponding rule, then re-*institutionalizes* the three rules, *recalling* that they have already been copied in the notebook.

More than the other two descriptions, the French narrative makes visible the didactic expertise of the teacher, e.g. by describing the way she carefully manages the progressive transformation of symbolic expressions. *Defining* and *generalizing* illustrate the acknowledgement of definitions and their role in proofs and the didactic management of generalization processes in the French narrative. The term *making explicit* (*expliciter*) highlights another important point: the sensitivity of the teacher for students’ difficulties with the algebraic symbolism, resulting from differences with arithmetic symbolism.

In contrast, the Czech narrative of the episode focuses more on the visible actions of the teacher without pointing out potential goals, intentions or the progressive transformation. The German narrative comments on some of these intentions, e.g. using the terms *reviewing*, *structuring* or *reasoning* and *showing*. But, in the French narrative the didactic and mathematical expertise is teased out in detail. This is embedded in a French tradition, which is marked by a historical investment of leading mathematicians in educational issues, and the development of didactics of mathematics as a genuine research field closely tied to the discipline of mathematics (Blum et al., 2019).

Both the French and the German narratives comment on the *dialogic form* of teaching. Furthermore, in both narratives the *validation* of students’ productions is emphasized. Note that the episode is labelled as *comparing homework* in the German narrative and as *correction of homework* in the French description. Last, but not least it should be emphasized that the term *institutionalisation* of the French lexicon is used, as in other French narratives. The term *institutionalisation* illustrates that technical terms coming from research are often distant from everyday language and used in a nominal form. These technical terms explicitly point to the purpose and function of the described teacher actions and to teachers’ beliefs related to these functions. Often, these technical terms are not immediately clear and understandable for a common audience, unlike the German technical terms, which are often formulated in everyday language with a physical connotation. As Goldschmidt (1988, 1999) points out, the French language lacks such physical connotations and uses terms of Latin or Greek origin instead.
<caption>Fig. 3 French episode</caption>

### Description of the French episode in the French narrative

The *correction* of the homework has a *dialogic form*, one student being asked for each exercise. After cleaning the board, the teacher *launches this phase*, making clear its stake and *explicit* the meaning of the notations $3^2$ et $3^3$.

<figure><img src="image_8.png" /></figure>

When the teacher *asks for* a result, Jason begins to *calculate* mentally. Ultimately the teacher clarifies the task pointing out that he simply wants the final result. Jason now provides the *expected answer*: "3 exponent 5". The teacher *validates* this answer and *asks the class to formulate a general rule*, in form of a sentence (*generalization*). He then *helps* the class to produce a precise *formulation*. For the next calculation (b) another student is called on. After also *validating* his answer, the teacher asks the student to *make explicit* the *implicit* exponent 1, *validates* the answer and comments. Calculation d) mobilizes a negative exponent and the teacher takes this opportunity to *recall* that the operations can be performed both with positive and negative numbers. For calculation e), the teacher *asks for a justification* of the result. The student called on for this task refers back to the *definition* of the power notation and *explains* the performed simplification. Again, the teacher makes the student *formulate* a corresponding general rule (*generalizing*). For reasons of *time management*, the *correction* of the next exercises is quickly done. For the two first ones using the same rule, the teacher *corrects an error* without insisting on it. He then tightly guides the correction of the last exercise while writing it on the board. He finally *asks* for the decomposition of the two powers, then suggests to change the order of the terms to arrive at $2 \times 3 \times 2 \times 3$, then grouped in $6 \times 6$, which is $6^2$. The teacher now *formulates* himself the corresponding rule, then re-*institutionalizes* the three rules, *recalling* that they have already been copied in the notebook.

More than the other two descriptions, the French narrative makes visible the didactic expertise of the teacher, e.g. by describing the way she carefully manages the progressive transformation of symbolic expressions. *Defining* and *generalizing* illustrate the acknowledgement of definitions and their role in proofs and the didactic management of generalization processes in the French narrative. The term *making explicit* (*expliciter*) highlights another important point: the sensitivity of the teacher for students’ difficulties with the algebraic symbolism, resulting from differences with arithmetic symbolism.

In contrast, the Czech narrative of the episode focuses more on the visible actions of the teacher without pointing out potential goals, intentions or the progressive transformation. The German narrative comments on some of these intentions, e.g. using the terms *reviewing*, *structuring* or *reasoning* and *showing*. But, in the French narrative the didactic and mathematical expertise is teased out in detail. This is embedded in a French tradition, which is marked by a historical investment of leading mathematicians in educational issues, and the development of didactics of mathematics as a genuine research field closely tied to the discipline of mathematics (Blum et al., 2019).

Both the French and the German narratives comment on the *dialogic form* of teaching. Furthermore, in both narratives the *validation* of students’ productions is emphasized. Note that the episode is labelled as *comparing homework* in the German narrative and as *correction of homework* in the French description. Last, but not least it should be emphasized that the term *institutionalisation* of the French lexicon is used, as in other French narratives. The term *institutionalisation* illustrates that technical terms coming from research are often distant from everyday language and used in a nominal form. These technical terms explicitly point to the purpose and function of the described teacher actions and to teachers’ beliefs related to these functions. Often, these technical terms are not immediately clear and understandable for a common audience, unlike the German technical terms, which are often formulated in everyday language with a physical connotation. As Goldschmidt (1988, 1999) points out, the French language lacks such physical connotations and uses terms of Latin or Greek origin instead.
<caption>Fig. 4 German episode</caption>

### Description of the German episode in the German narrative

#### Interruption 1 – Return to homework / correcting mistakes

Now there is an interruption – a teachable moment: A student raises his hand and mentions that in the homework there is only 3m of an angled aluminium bar and that in the solution it is 4.5m. The teacher returns to the homework and decides to go through it again in detail according to the “cooking recipe” (step by step progression), they have discussed before. A student dictates the equation:

$3 = 2x + 2y$. In a form of *dialogical teaching*, the teacher now wants to *analyse* the *mistake* and correct it (*embracing mistakes*). He emphasises that no one – even not himself – noticed the mistake. A student *demonstrates* her approach: She transforms the equation, subtracts $2y$ and notes the equation: $3 - 2y = 2x$ on the board. She then divides the equation by 2 and notes: $1,5 - y = x$.

The teacher now changes the numbers on the board, following comments from the students. Once again, he makes a mistake (not changing a 3 to a 1,5), which a student sees quickly. The teacher values the comment of the student and corrects the mistake on the board, where is now noted:

$$ 
\begin{array}{l}
\text{long side: } y \qquad \text{short side: } x \\
\\
\begin{array}{|c|c|c|c|c|c|c|c|c|c|}
\hline
y & = & -x + 1,5 & & & & & & \\
y & = & 2x & & & & & & \\
\hline
-x + 1,5 & = & 2x & +x & & & & & \\
1,5 & = & 3x & : & 3 & & & & \\
0,5 & = & x & & & & & & \\
y & = & 2 \cdot 0,5 & & & & & & \\
y & = & 1 & & & & & & \\
\hline
\end{array}
$$

$$ L = \{(0, 5/1)\}. $$

The teacher then asks, whether the equation and solution are now correct. A short control takes place and the solution is confirmed. Finally, the teacher acknowledges again that a student found this mistake, which he did not see. He explicitly asks for the student, who found the mistake and approves that the mistake was pointed out (*embraces mistakes*).

## 4.3 German episode

The German episode has been selected for its teachable moment. This means a specific moment, where intense mathematical discussions that lead to mathematical learning occur. In the episode, a profound teaching situation emerges, where the mathematical knowledge at stake is explicitly a topic (Fig. 4).

In the German narrative of this episode the focus is on the systematic discussion of the error and its source. This *analysis of mistakes* and *embracing mistakes*, which are initiated by a student and valued by the teacher, indicates an “error-culture”. Errors are portrayed in the German narrative as a learning opportunity in the classroom. The *embracing of mistakes* is considered as a valuable part of the *Pedagogical and Didactical Management of the Classroom*. These terms, in German, are compound words: *Fehleranalyse* would be *error-analysis* and *Fehlerkultur* would be *error-culture* (*embracing mistakes*). For each term, both parts are used in everyday language, however, compounded they make up a technical term. In this way

concrete terms become terms for technical concepts that are concise and accessible.

The explanation of specific steps of a procedure, here *demonstrated* by a student, and the class discussion guided by the teacher, labelled as *dialogical teaching*, are also described as an essential part of the *Pedagogical and Didactical Management of the Classroom* in this narrative. In the German narrative this episode is described as a teachable moment.

The Czech narrative of this German episode is impressively short. In two sentences the actions of the teacher and the students are summarized and labelled as *teacher-controlled solving of a task* and characterized as a *Teaching Method*. The fact that the description in the Czech narrative is rather short can be seen as evidence that this episode is not considered as significant for the entire lesson.

The French narrative of the episode, in contrast to the Czech narrative, describes this situation as *dealing with a didactical incident*, which also emphasizes the teacher's role in it. Both the German and the French narratives interpret the episode as an instance where the learning of mathematics
<caption>Fig. 4 German episode</caption>

## Description of the German episode in the German narrative

### Interruption 1 – Return to homework / correcting mistakes

Now there is an interruption – a teachable moment: A student raises his hand and mentions that in the homework there is only 3m of an angled aluminium bar and that in the solution it is 4.5m. The teacher returns to the homework and decides to go through it again in detail according to the “cooking recipe” (step by step progression), they have discussed before. A student dictates the equation:

$3 = 2x + 2y$. In a form of *dialogical teaching*, the teacher now wants to *analyse* the *mistake* and correct it (*embracing mistakes*). He emphasises that no one – even not himself – noticed the mistake. A student *demonstrates* her approach: She transforms the equation, subtracts $2y$ and notes the equation: $3 - 2y = 2x$ on the board. She then divides the equation by 2 and notes: $1,5 - y = x$.

The teacher now changes the numbers on the board, following comments from the students. Once again, he makes a mistake (not changing a 3 to a 1,5), which a student sees quickly. The teacher values the comment of the student and corrects the mistake on the board, where is now noted:

$$ 
\begin{array}{l}
\text{long side: } y \qquad \text{short side: } x \\
\\
\left| \begin{array}{l} y = -x + 1,5 \\ y = 2x \end{array} \right| \\
\\
-x + 1,5 = 2x \mid +x \\
1,5 = 3x \mid : 3 \\
0,5 = x \\
y = 2 \cdot 0,5 \\
y = 1
\end{array}
$$

$$ L = \{(0, 5/1)\}. $$

The teacher then asks, whether the equation and solution are now correct. A short control takes place and the solution is confirmed. Finally, the teacher acknowledges again that a student found this mistake, which he did not see. He explicitly asks for the student, who found the mistake and approves that the mistake was pointed out (*embraces mistakes*).

## 4.3 German episode

The German episode has been selected for its teachable moment. This means a specific moment, where intense mathematical discussions that lead to mathematical learning occur. In the episode, a profound teaching situation emerges, where the mathematical knowledge at stake is explicitly a topic (Fig. 4).

In the German narrative of this episode the focus is on the systematic discussion of the error and its source. This *analysis of mistakes* and *embracing mistakes*, which are initiated by a student and valued by the teacher, indicates an “error-culture”. Errors are portrayed in the German narrative as a learning opportunity in the classroom. The *embracing of mistakes* is considered as a valuable part of the *Pedagogical and Didactical Management of the Classroom*. These terms, in German, are compound words: *Fehleranalyse* would be *error-analysis* and *Fehlerkultur* would be *error-culture* (*embracing mistakes*). For each term, both parts are used in everyday language, however, compounded they make up a technical term. In this way

concrete terms become terms for technical concepts that are concise and accessible.

The explanation of specific steps of a procedure, here *demonstrated* by a student, and the class discussion guided by the teacher, labelled as *dialogical teaching*, are also described as an essential part of the *Pedagogical and Didactical Management of the Classroom* in this narrative. In the German narrative this episode is described as a teachable moment.

The Czech narrative of this German episode is impressively short. In two sentences the actions of the teacher and the students are summarized and labelled as *teacher-controlled solving of a task* and characterized as a *Teaching Method*. The fact that the description in the Czech narrative is rather short can be seen as evidence that this episode is not considered as significant for the entire lesson.

The French narrative of the episode, in contrast to the Czech narrative, describes this situation as *dealing with a didactical incident*, which also emphasizes the teacher's role in it. Both the German and the French narratives interpret the episode as an instance where the learning of mathematics
is actually possible for students. In contrast to the Czech
narrative, the French description focuses on the *PDMC* in
this situation, pointing out how the teacher is *dealing with
errors* and *valuing* students' contributions. Unlike many
other parts of the French narratives, in the description of this
episode no lexicon terms are used related to the *Mathematical Activities*. But still, the mathematical content is a focus in
the French narrative of this episode, in line with the French
didactic tradition. In the German narrative this is also the
case, the *Mathematical Activities* are described in detail, but
not with specific terms. As in the French narrative the possible function of teacher actions and teachers' beliefs related
to these functions are intertwined in the German narrative.

# 5 Discussion

Our cultural comparative approach of narratives, even if lim-
ited to three lexicons and three cultural contexts, as in this
study, shows that understanding language, used as a resource
for teachers describing mathematics classrooms teaching and
learning, is complex. Regularities and differences do not
only exist in technical vocabulary, when it comes to describ-
ing and naming what one sees in a classroom. Also, our
comparative approach demonstrates that technical terms
only constitute a part of our means to describe what is going
on in classrooms in a professional way.

In part 4 we illustrated, via different narratives of a Czech,
a French and a German classroom episode, the potential of a
comparative narrative analysis. Beyond revealing different
perspectives on the episodes, we also indicated how lan-
guage influences our descriptions and perceptions of class-
room situations. Comparing these descriptions enables us to
come to an understanding of how complex it is to develop
a common professional language. It becomes evident that a
conceptualization, in the form of different lexicons, might
constitute a start, but is too limited to represent the complex-
ity of language as a resource for teachers. We conclude our
article by coming back to our research question:

How does a cultural comparative approach of narra-
tives help us to understand language as a resource for
teachers describing mathematics classrooms teaching
and learning?

As expected, the three narratives of each episode are
different. The Czech narratives primarily describe teach-
ing methods and teacher-student interactions, staying close
to observable actions, without going into detail about the
mathematical activities carried out. The French narra-
tives describe the mathematical-didactic management of
the classroom by the teacher, going into great detail about
the mathematical activity. The mathematical content at
stake is also strongly present in the German narratives,

combined with a marked attention to the structure of the
lessons, the teaching methods and the way in which the
teachers exploit learning opportunities, even unanticipated
ones.

Although different in their description of the teacher’s
pedagogic and didactic management of the classroom, these
narratives do not propose conflicting views of the episodes
at stake, as could be observed for example in research on
the networking of theories (Bikner-Ahsbahs et al., 2014).
The visions they propose are complementary and mutually
enriching, as shown in the syntheses that conclude their
analyses.

These analyses also show how the existence of techni-
cal terms in the lexicons helps us to express more deeply
what is at stake in these episodes. For example, the Ger-
man lexicon includes two terms for describing the handling
of errors in class: Fehleranalyse (literally “error-analysis”,
in English: analysis of mistakes) and Fehlerkultur (liter-
ally “error-culture”, in English: embracing mistakes). Both
terms are connected in the German lexicon, making explicit
that the management of errors is associated with a didactic
approach, “which includes the valuing of errors as learning
opportunities” (see Sect. 4).

Also, the term *Hausaufgabenvergleich* (*comparing home-work*) and the term *Auswertung* (*comparing results*) in the German lexicon, make explicit the existence of a named didactic technique, valuing students’ results and work. This contrasts the French lexicon entry *correction*, which names the phase of a lesson, “aimed at presenting expected solutions and possibly rectifying errors”. In the French lexicon the emphasis is on correcting errors rather than on valuing students’ approaches. Also, the term *gérer des erreurs* (*dealing with errors*) seems to emphasize this.

At the same time, the narrative of the French episode
demonstrates that even though these French terms have a
specific connotation, it is valued in the narrative how the
teacher exploits a correction activity to consolidate and
deepen students’ learning. This is an example of how the
use of technical language might be adapted in narratives, in
order to better capture the didactic intentions and techniques
in situ. Narratives seem to be closer to what is at stake in
a situation, compared to technical vocabulary defined in a
lexicon.

As pointed out in Sect. 4, the comparative analyses of the
Czech narratives have drawn our attention to the potential
of the Czech lexicon, which differentiates several forms of
explanation: *výklad formou monologue* (*explanation in the
form of a teacher’s monologue*), *výklad spojený s demonstrací* (*explanation with demonstration*), *interaktivní výklad* (*interactive explanation*), while the other lexicons only have
one general term *explaining*. Again, the absence of a term in
some lexicon does not necessarily imply the absence of con-
sideration in general, which the narratives show quite well.
is actually possible for students. In contrast to the Czech
narrative, the French description focuses on the *PDMC* in
this situation, pointing out how the teacher is *dealing with
errors* and *valuing* students' contributions. Unlike many
other parts of the French narratives, in the description of this
episode no lexicon terms are used related to the *Mathematical Activities*. But still, the mathematical content is a focus in
the French narrative of this episode, in line with the French
didactic tradition. In the German narrative this is also the
case, the *Mathematical Activities* are described in detail, but
not with specific terms. As in the French narrative the possible function of teacher actions and teachers' beliefs related
to these functions are intertwined in the German narrative.

# 5 Discussion

Our cultural comparative approach of narratives, even if lim-
ited to three lexicons and three cultural contexts, as in this
study, shows that understanding language, used as a resource
for teachers describing mathematics classrooms teaching and
learning, is complex. Regularities and differences do not
only exist in technical vocabulary, when it comes to describ-
ing and naming what one sees in a classroom. Also, our
comparative approach demonstrates that technical terms
only constitute a part of our means to describe what is going
on in classrooms in a professional way.

In part 4 we illustrated, via different narratives of a Czech,
a French and a German classroom episode, the potential of a
comparative narrative analysis. Beyond revealing different
perspectives on the episodes, we also indicated how lan-
guage influences our descriptions and perceptions of class-
room situations. Comparing these descriptions enables us to
come to an understanding of how complex it is to develop
a common professional language. It becomes evident that a
conceptualization, in the form of different lexicons, might
constitute a start, but is too limited to represent the complex-
ity of language as a resource for teachers. We conclude our
article by coming back to our research question:

How does a cultural comparative approach of narra-
tives help us to understand language as a resource for
teachers describing mathematics classrooms teaching
and learning?

As expected, the three narratives of each episode are
different. The Czech narratives primarily describe teach-
ing methods and teacher-student interactions, staying close
to observable actions, without going into detail about the
mathematical activities carried out. The French narra-
tives describe the mathematical-didactic management of
the classroom by the teacher, going into great detail about
the mathematical activity. The mathematical content at
stake is also strongly present in the German narratives,

combined with a marked attention to the structure of the
lessons, the teaching methods and the way in which the
teachers exploit learning opportunities, even unanticipated
ones.

Although different in their description of the teacher’s
pedagogic and didactic management of the classroom, these
narratives do not propose conflicting views of the episodes
at stake, as could be observed for example in research on
the networking of theories (Bikner-Ahsbahs et al., 2014).
The visions they propose are complementary and mutually
enriching, as shown in the syntheses that conclude their
analyses.

These analyses also show how the existence of techni-
cal terms in the lexicons helps us to express more deeply
what is at stake in these episodes. For example, the Ger-
man lexicon includes two terms for describing the handling
of errors in class: Fehleranalyse (literally “error-analysis”,
in English: analysis of mistakes) and Fehlerkultur (liter-
ally “error-culture”, in English: embracing mistakes). Both
terms are connected in the German lexicon, making explicit
that the management of errors is associated with a didactic
approach, “which includes the valuing of errors as learning
opportunities” (see Sect. 4).

Also, the term *Hausaufgabenvergleich* (*comparing home-work*) and the term *Auswertung* (*comparing results*) in the German lexicon, make explicit the existence of a named didactic technique, valuing students’ results and work. This contrasts the French lexicon entry *correction*, which names the phase of a lesson, “aimed at presenting expected solutions and possibly rectifying errors”. In the French lexicon the emphasis is on correcting errors rather than on valuing students’ approaches. Also, the term *gérer des erreurs* (*dealing with errors*) seems to emphasize this.

At the same time, the narrative of the French episode
demonstrates that even though these French terms have a
specific connotation, it is valued in the narrative how the
teacher exploits a correction activity to consolidate and
deepen students’ learning. This is an example of how the
use of technical language might be adapted in narratives, in
order to better capture the didactic intentions and techniques
in situ. Narratives seem to be closer to what is at stake in
a situation, compared to technical vocabulary defined in a
lexicon.

As pointed out in Sect. 4, the comparative analyses of the
Czech narratives have drawn our attention to the potential
of the Czech lexicon, which differentiates several forms of
explanation: *výklad formou monologue* (*explanation in the
form of a teacher’s monologue*), *výklad spojený s demonstrací* (*explanation with demonstration*), *interaktivní výklad* (*interactive explanation*), while the other lexicons only have
one general term *explaining*. Again, the absence of a term in
some lexicon does not necessarily imply the absence of con-
sideration in general, which the narratives show quite well.
Another example of this is the German lexicon, which is
poor in terms describing the nature of mathematical tasks.
But, this lack of vocabulary does not prevent mathematical
content from being very present in the German narratives,
for example via mathematical activities. This phenomenon
can also be observed in the Czech narrative (as pointed out
in Sect. 4). The Czech narrative of the Czech episode shows
that “the pedagogical interactions described are closely
bonded to the mathematical content at stake and the pro-
cesses supporting pupils learning of this content”.

# 6 Summary of results and further perspectives

Summarizing our results, we can state that narratives as a
cultural object and their comparisons reveal the complexity
of a common professional language for describing class-
room teaching and learning. Different lexicons and related
categories, can constitute a start, but are alone too limited
to represent the complexity of language as a resource for
teachers. Technical terms in the lexicons do define and make
explicit didactical intentions, techniques and approaches,
but narratives capture more the enactment of the didac-
tic intentions and techniques in situ. Specifically, narra-
tives describe didactical intentions along the rationale of
the teaching–learning situation, even if technical terms are
absent in the given lexicon. The comparative analyses of the
different narratives of the classroom episodes point particu-
larly to this phenomenon, and offer culturally complemen-
tary, rather than conflictive, views on classroom situations
and how teachers use language as a resource for describing
these. Hence, the two approaches, terms in a lexicon and
descriptive narratives, work together to offer a fuller and
more analytical representation of teaching practice within
different cultural contexts.

This suggests going further in this direction of compara-
tive analyses in future studies and research. Comparative
approaches of narratives are promising to understand better
teachers’ resources and specifically language as a resource
for teachers describing classroom teaching and learning.
Such a comparative approach also allows us to study more
precisely the variations of descriptions of the same situation,
with the same or similar terms (e.g., the term *explaining*
above). In respect to resources, studying more systemati-
cally the possible functions and uses of terms, or clusters
of terms, in narratives is valuable. The inclusion of more
cultural contexts, other and different collectives and com-
munities of teachers, will also be fruitful.

Our research study not only offers cultural results from
our comparative narrative approach, but it also provides
more globally a methodological strategy for develop-
ing cross-cultural comparisons. The potential of using

teachers' professional language is studied in a cross-
cultural comparative approach, and provides methodo-
logically a resource for describing and making sense of
classroom phenomena. In the methodology and methods
developed for this study, indeed, some elements have
played an essential role:

* the accessibility of videos of classroom sessions from different cultural contexts and languages at stake;

* the production of descriptive narratives for these videos by collectives from different cultural contexts, which allows us to overcome the idiosyncrasies of narratives produced by a single person;

* the preliminary development and application of a lexicon of professional terms for each context;

* cross-case studies of episodes engaging an interpretative research work building on both: lexicons and narratives.

The comparative narrative approach, suggested here as a
general methodological strategy for cross-cultural compari-
sons, can also be used for cultural comparisons focused on
more specific domains. For instance, focusing on reasoning
and proof, as is currently envisaged in a German-Japanese
project, can reveal insightful and new results in a globally
increasing research field.

This study also suggests an original use of narratives in
mathematics education research. Narratives, so far used in
teacher education and professional development research
to investigate teachers' identities and knowledge, have only
been used to a limited extent in mathematics education
research (see, for instance da Ponte, 2001; Tatsis, 2011).
Narratives have also been used for understanding students'
learning processes, considering the role of the narrative
mode of thinking in mathematics learning (for instance
Healy & Sinclair, 2007; Sinclair et al., 2009). However, to
our knowledge, there are not many examples of the use of
narratives in cross-cultural comparison. Our research tends
to show that narratives can be also effective tools to address
such comparisons.

Last, but not least, the comparison of lexicons and the
production of narratives can be an inspiring means for the
professional development of teachers, also during their ini-
tial education. Cultural comparative approaches can inspire
teachers to discover other professional perspectives on class-
room situations. Or teachers might discover that a term,
which seems apparently familiar, has a different meaning in
another cultural context. This might lead teachers to question
their own professional language, the meaning of familiar
terms and their usage. A better understanding for the value
of a well-developed and shared professional language might
be an outcome, which could broaden teachers' perspectives
on teaching and learning and thus be an essential resource.
Another example of this is the German lexicon, which is
poor in terms describing the nature of mathematical tasks.
But, this lack of vocabulary does not prevent mathematical
content from being very present in the German narratives,
for example via mathematical activities. This phenomenon
can also be observed in the Czech narrative (as pointed out
in Sect. 4). The Czech narrative of the Czech episode shows
that “the pedagogical interactions described are closely
bonded to the mathematical content at stake and the pro-
cesses supporting pupils learning of this content”.

# 6 Summary of results and further perspectives

Summarizing our results, we can state that narratives as a
cultural object and their comparisons reveal the complexity
of a common professional language for describing class-
room teaching and learning. Different lexicons and related
categories, can constitute a start, but are alone too limited
to represent the complexity of language as a resource for
teachers. Technical terms in the lexicons do define and make
explicit didactical intentions, techniques and approaches,
but narratives capture more the enactment of the didac-
tic intentions and techniques in situ. Specifically, narra-
tives describe didactical intentions along the rationale of
the teaching–learning situation, even if technical terms are
absent in the given lexicon. The comparative analyses of the
different narratives of the classroom episodes point particu-
larly to this phenomenon, and offer culturally complemen-
tary, rather than conflictive, views on classroom situations
and how teachers use language as a resource for describing
these. Hence, the two approaches, terms in a lexicon and
descriptive narratives, work together to offer a fuller and
more analytical representation of teaching practice within
different cultural contexts.

This suggests going further in this direction of compara-
tive analyses in future studies and research. Comparative
approaches of narratives are promising to understand better
teachers’ resources and specifically language as a resource
for teachers describing classroom teaching and learning.
Such a comparative approach also allows us to study more
precisely the variations of descriptions of the same situation,
with the same or similar terms (e.g., the term *explaining*
above). In respect to resources, studying more systemati-
cally the possible functions and uses of terms, or clusters
of terms, in narratives is valuable. The inclusion of more
cultural contexts, other and different collectives and com-
munities of teachers, will also be fruitful.

Our research study not only offers cultural results from
our comparative narrative approach, but it also provides
more globally a methodological strategy for develop-
ing cross-cultural comparisons. The potential of using

teachers' professional language is studied in a cross-
cultural comparative approach, and provides methodo-
logically a resource for describing and making sense of
classroom phenomena. In the methodology and methods
developed for this study, indeed, some elements have
played an essential role:

* the accessibility of videos of classroom sessions from different cultural contexts and languages at stake;

* the production of descriptive narratives for these videos by collectives from different cultural contexts, which allows us to overcome the idiosyncrasies of narratives produced by a single person;

* the preliminary development and application of a lexicon of professional terms for each context;

* cross-case studies of episodes engaging an interpretative research work building on both: lexicons and narratives.

The comparative narrative approach, suggested here as a
general methodological strategy for cross-cultural compari-
sons, can also be used for cultural comparisons focused on
more specific domains. For instance, focusing on reasoning
and proof, as is currently envisaged in a German-Japanese
project, can reveal insightful and new results in a globally
increasing research field.

This study also suggests an original use of narratives in
mathematics education research. Narratives, so far used in
teacher education and professional development research
to investigate teachers' identities and knowledge, have only
been used to a limited extent in mathematics education
research (see, for instance da Ponte, 2001; Tatsis, 2011).
Narratives have also been used for understanding students'
learning processes, considering the role of the narrative
mode of thinking in mathematics learning (for instance
Healy & Sinclair, 2007; Sinclair et al., 2009). However, to
our knowledge, there are not many examples of the use of
narratives in cross-cultural comparison. Our research tends
to show that narratives can be also effective tools to address
such comparisons.

Last, but not least, the comparison of lexicons and the
production of narratives can be an inspiring means for the
professional development of teachers, also during their ini-
tial education. Cultural comparative approaches can inspire
teachers to discover other professional perspectives on class-
room situations. Or teachers might discover that a term,
which seems apparently familiar, has a different meaning in
another cultural context. This might lead teachers to question
their own professional language, the meaning of familiar
terms and their usage. A better understanding for the value
of a well-developed and shared professional language might
be an outcome, which could broaden teachers' perspectives
on teaching and learning and thus be an essential resource.
## Appendix 1: Categories in the Czech and French lexicons (adapted from Mesiti et al., 2022, pp. 144 and 217)

<table><thead><tr><th>Lexicon</th><th>Category names and number of associated terms</th></tr></thead><tbody><tr><td>Czech Lexicon (57 terms, 11 categories)</td><td>Didaktický test—didactical test (4)<br/>Domácí úkol—homework (2)<br/>Druhy úloh—type of problems (4)<br/>Fáze vyucovací hodiny—stages of a lesson (2)<br/>Hodnocení—assessment (7)<br/>Organizace vyucování—organisation of a lesson (9)<br/>Organizacní formy vyucování—organization forms of instruction (5)<br/>Procesy podporuící ucení záku—processes supporting pupils' learning (5)<br/>Samostatná práce záku—pupils individual work (3)<br/>Vyucovaci metody—teaching methods (10)<br/>Vyuzití didaktických prostředku—use of didactical means (6)</td></tr><tr><td>French lexicon (116 terms, 6 categories)</td><td>Termes généraux—general terms (11)<br/>Nature des taches, activités—nature of tasks, activities (17)<br/>Phases de séances—lesson phases (12)<br/>Formes d'organisation pédagogique—forms of pedagogical organisation (9)<br/>Activités mathématiques—mathematical activities (19)<br/>Gestion pédagogique et didactique de la classe—pedagogical and didactical management of the classroom (48)<br/>Organisation—organisation (12)<br/>Interaction—interaction (23)<br/>Exploitation et évaluation—exploiting and assessing (13)</td></tr></tbody></table>

**Funding** Open Access funding enabled and organized by Projekt DEAL.

**Data availability** According to the chief editor of this journal data availability is not mandatory for this article.

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

## References

Adler, J. (1999). The dilemma of transparency: Seeing and seeing through talk in the mathematics classroom. *Journal for*

Research in Mathematics Education, 30(1), 47–64. https://doi.org/10.2307/749629

Adler, J. (2019). Foreword. In L. Trouche, G. Gueudet, & B. Pepin (Eds.), *The 'resource' approach to mathematics education* (pp. v–viii). Springer.

Artigue, M., Chevalarias, T., Debertonne-Dassule, F., Grugeon-Allys, B., Horoks, J., & Pilet, J. (2021). French lexicon. In C. Mesiti, M. Artigue, H. Hollingsworth, Y. Cao, & D. J. Clarke (Eds.), *Teachers talking about their classrooms. Learning from the professional lexicons of mathematics teachers around the world* (pp. 193–218). Routlege.

Artigue, M., Novotná, J., Grugeon-Allys, B., Horoks, J., Hošpesová, A., Moraová, H., Pilet, J., & Žlábková, I. (2017). Comparing the professional lexicons of Czech and French mathematics teachers. In *Proceedings of the 41st annual meeting of the international group for the psychology of mathematics education PME* (vol. 2, pp. 113–120). http://www.igpme.org/publications/current-proceedings/

Barthes, R. (1977). Introduction to the structural analysis of narratives. In R. Barthes (Ed.), *Image-music-text (trans. Stephen Heath)* (pp. 79–124). Collins.

Bikner-Ahsbahs, A., Artigue, M., & Haspekian, M. (2014). Topaze effect: a case study on networking of IDS and TDS. In A. Bikner-Ahsbahs & S. Prediger (Eds.), *Networking of theories as a research practice in mathematics education. Advances in mathematics education*. Springer. https://doi.org/10.1007/978-3-319-05389-9_12

Blum, W., Artigue, M., Mariotti, M.-A., Strässer, R., & Van den Heuvel-Panhuizen, M. (Eds.). (2019). *European traditions in didactics of mathematics*. Springer Open. https://doi.org/10.1007/978-3-030-05514-1

Brousseau, G. (1997). *Theory of didactical situations in mathematics*. Kluwer.

Clandinin, J. (2019). *Journeys in narrative inquiry—The selected works of D. Routledge*.

Clandinin, J., & Connelly, F. M. (2000). *Narrative inquiry: Experience and story in qualitative research*. Wiley.

Clarke, D. J. (2015). Comparative Research in mathematics education: Boundary crossing and boundary creation. In *Proceedings of the 39th conference of the international group for the psychology of mathematics education PME* (vol. 2, pp. 169–176). http://www.igpme.org/publications/current-proceedings/

Clarke, D. J., Mesiti, C., Cao, Y., & Novotná, J. (2017). The Lexicon Project: Examining the consequences for international comparative research of pedagogical naming systems from different cultures. *Proceedings of the Tenth Congress of the European Society for Research in Mathematics Education CERME, 10, 1610–1617*.

Cole, M., & Engeström, Y. (1993). A cultural-historical approach to distributed cognition. In G. Salomon (Ed.), *Distributed cognitions: Psychological and educational considerations* (pp. 1–46). Cambridge University Press.

Connelly, F. M., & Clandinin, D. J. (1985). Personal practical knowledge and the modes of knowing: Relevance to teaching and learning. In E. Eisner (Ed.), *Learning and teaching the ways of knowing: Eighty-fourth Yearbook of the National Society for the study of education (Part II, chap. X)* (pp. 174–198). University of Chicago Press.

Connelly, F. M., & Clandinin, D. J. (1990). Stories of experience and narrative inquiry. *Educational Researcher, 19*(5), 2–14.

Cramer, J. Ch., Knipping, Ch., Reid, D. A., & Specht, B. J. (2021). German lexicon. In C. Mesiti, M. Artigue, H. Hollingsworth, Y. Cao, & D. J. Clarke (Eds.), *Teachers talking about their classrooms. Learning from the professional lexicons of mathematics teachers around the world* (pp. 232–254). Routlege.

Czarniawska, B. (2004). *Narratives in social science research*. Sage.
## Appendix 1: Categories in the Czech and French lexicons (adapted from Mesiti et al., 2022, pp. 144 and 217)

<table><thead><tr><th>Lexicon</th><th>Category names and number of associated terms</th></tr></thead><tbody><tr><td>Czech Lexicon (57 terms, 11 categories)</td><td>Didaktický test—didactical test (4)<br>Domácí úkol—homework (2)<br>Druhy úloh—type of problems (4)<br>Fáze vyucovací hodiny—stages of a lesson (2)<br>Hodnocení—assessment (7)<br>Organizace vyucování—organisation of a lesson (9)<br>Organizacní formy vyucování—organization forms of instruction (5)<br>Procesy podporuící uceni záku—processes supporting pupils' learning (5)<br>Samostatná práce záku—pupils individual work (3)<br>Vyucovaci metody—teaching methods (10)<br>Vyuzití didaktických prostredku—use of didactical means (6)</td></tr><tr><td>French lexicon (116 terms, 6 categories)</td><td>Termes généraux—general terms (11)<br>Nature des taches, activités—nature of tasks, activities (17)<br>Phases de séances—lesson phases (12)<br>Formes d'organisation pédagogique—forms of pedagogical organisation (9)<br>Activités mathématiques—mathematical activities (19)<br>Gestion pédagogique et didactique de la classe—pedagogical and didactical management of the classroom (48)<br>Organisation—organisation (12)<br>Interaction—interaction (23)<br>Exploitation et évaluation—exploiting and assessing (13)</td></tr></tbody></table>

**Funding** Open Access funding enabled and organized by Projekt DEAL.

**Data availability** According to the chief editor of this journal data availability is not mandatory for this article.

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

## References

Adler, J. (1999). The dilemma of transparency: Seeing and seeing through talk in the mathematics classroom. *Journal for*

Research in Mathematics Education, 30(1), 47–64. https://doi.org/10.2307/749629

Adler, J. (2019). Foreword. In L. Trouche, G. Gueudet, & B. Pepin (Eds.), *The 'resource' approach to mathematics education* (pp. v–viii). Springer.

Artigue, M., Chevalarias, T., Debertonne-Dassule, F., Grugeon-Allys, B., Horoks, J., & Pilet, J. (2021). French lexicon. In C. Mesiti, M. Artigue, H. Hollingsworth, Y. Cao, & D. J. Clarke (Eds.), *Teachers talking about their classrooms. Learning from the professional lexicons of mathematics teachers around the world* (pp. 193–218). Routlege.

Artigue, M., Novotná, J., Grugeon-Allys, B., Horoks, J., Hošpesová, A., Moraová, H., Pilet, J., & Žlábková, I. (2017). Comparing the professional lexicons of Czech and French mathematics teachers. In *Proceedings of the 41st annual meeting of the international group for the psychology of mathematics education PME* (vol. 2, pp. 113–120). http://www.igpme.org/publications/current-proceedings/

Barthes, R. (1977). Introduction to the structural analysis of narratives. In R. Barthes (Ed.), *Image-music-text (trans. Stephen Heath)* (pp. 79–124). Collins.

Bikner-Ahsbahs, A., Artigue, M., & Haspekian, M. (2014). Topaze effect: a case study on networking of IDS and TDS. In A. Bikner-Ahsbahs & S. Prediger (Eds.), *Networking of theories as a research practice in mathematics education. Advances in mathematics education*. Springer. https://doi.org/10.1007/978-3-319-05389-9_12

Blum, W., Artigue, M., Mariotti, M.-A., Strässer, R., & Van den Heuvel-Panhuizen, M. (Eds.). (2019). *European traditions in didactics of mathematics*. Springer Open. https://doi.org/10.1007/978-3-030-05514-1

Brousseau, G. (1997). *Theory of didactical situations in mathematics*. Kluwer.

Clandinin, J. (2019). *Journeys in narrative inquiry—The selected works of D*. Routledge.

Clandinin, J., & Connelly, F. M. (2000). *Narrative inquiry: Experience and story in qualitative research*. Wiley.

Clarke, D. J. (2015). Comparative Research in mathematics education: Boundary crossing and boundary creation. In *Proceedings of the 39th conference of the international group for the psychology of mathematics education PME* (vol. 2, pp. 169–176). http://www.igpme.org/publications/current-proceedings/

Clarke, D. J., Mesiti, C., Cao, Y., & Novotná, J. (2017). The Lexicon Project: Examining the consequences for international comparative research of pedagogical naming systems from different cultures. *Proceedings of the Tenth Congress of the European Society for Research in Mathematics Education CERME, 10, 1610–1617*.

Cole, M., & Engeström, Y. (1993). A cultural-historical approach to distributed cognition. In G. Salomon (Ed.), *Distributed cognitions: Psychological and educational considerations* (pp. 1–46). Cambridge University Press.

Connelly, F. M., & Clandinin, D. J. (1985). Personal practical knowledge and the modes of knowing: Relevance to teaching and learning. In E. Eisner (Ed.), *Learning and teaching the ways of knowing: Eighty-fourth Yearbook of the National Society for the study of education (Part II, chap. X)* (pp. 174–198). University of Chicago Press.

Connelly, F. M., & Clandinin, D. J. (1990). Stories of experience and narrative inquiry. *Educational Researcher, 19*(5), 2–14.

Cramer, J. Ch., Knipping, Ch., Reid, D. A., & Specht, B. J. (2021). German lexicon. In C. Mesiti, M. Artigue, H. Hollingsworth, Y. Cao, & D. J. Clarke (Eds.), *Teachers talking about their classrooms. Learning from the professional lexicons of mathematics teachers around the world* (pp. 232–254). Routlege.

Czarniawska, B. (2004). *Narratives in social science research*. Sage.
Da Ponte, J. P. (2001). Professional narratives in mathematics teacher education. In *Proceedings of the 2001 annual meeting of the Canadian mathematics education study group* (pp. 61–65). University of Alberta.

Dobie, T., Sherin, M., & White, S. (2021). A Lexical Snapshot—An investigation into the evolving terminology of middle-school mathematics teachers in the United States. In C. Mesiti, M. Artigue, H. Hollingsworth, Y. Cao, & D. J. Clarke (Eds.), *Teachers talking about their classrooms. Learning from the professional lexicons of mathematics teachers around the world* (pp. 323–333). Routlege.

Goldschmidt, G. A. (1988). *Quand Freud voit la mer. Freud et la langue allemande!* Buchet/Chastel.

Goldschmidt, G. A. (1999). *Als Freud das Meer sah—Freud und die deutsche Sprache (Translated by B. Große).* Ammann.

González-Forte, JM., Planas, N., & Fernández, C. (2022). Towards a newer notion: noticing languages for mathematics content teaching. In *Proceedings of the 45th conference of the international group for the psychology of mathematics education PME* (vol. 2, pp. 315–322).

Grossman, P., Compton, C., Igra, D., Ronfeldt, M., Shahan, E., & Williamson, P. (2009). Teaching practice: A cross-professional perspective. *Teachers College Record, 111*(9), 2055–2100.

Gueudet, G. (2019). Studying teachers’ documentation work: Emergence of a theoretical approach. In L. Trouche, G. Gueudet, & B. Pepin (Eds.), *The ‘resource’ approach to mathematics education* (pp. 17–42). Springer Nature.

Healy, L., & Sinclair, N. (2007). If this is our mathematics, what are our stories? *International Journal of Computers for Mathematical Learning, 12*(1), 3–21.

Lampert, M. (2000). Knowing teaching: The intersection of research on teaching and qualitative research. *Harvard Educational Review, 70*(1), 86–99.

Lortie, D. C. (1975). *Schoolteacher*. University of Chicago Press.

Mesiti, C., Artigue, M., Grau, V., & Novotná, J. (2022). Towards an international lexicon. *ZDM Mathematics Education, 54*(2), 239–255. https://doi.org/10.1007/s11858-022-01349-3

Mesiti, C., Artigue, M., Hollingsworth, H., Cao, Y., & Clarke, D. J. (Eds.). (2021). *Teachers talking about their classrooms. Learning from the professional lexicons of mathematics teachers around the world*. Routlege.

Mesiti, C., & Clarke, D. J. (2018). The professional, pedagogical language of mathematics teachers: A cultural artefact of significant value to the mathematics community. In *Proceedings of the 42nd conference of the international group for the psychology of mathematics education PME* (vol. 3, pp. 379–386). http://www.igpme.org/publications/current-proceedings/

Mesiti, C., Novotná, J., Clarke, D. J., Hošpesová, A., & Hollingsworth, H. (2019). Speaking about the mathematics classroom: A

comparison of the professional lexicons of teachers in Australia and the Czech Republic. In *Proceedings of the 43rd conference of the international group for the psychology of mathematics education PME* (vol. 3, pp. 89–96). http://www.igpme.org/publications/current-proceedings/

Planas, N. (2018). Language as resource: A key notion for understanding the complexity of mathematics learning. *Educational Studies in Mathematics, 98*(3), 215–229. https://doi.org/10.1007/s10649-018-9810-y

Planas, N., Adler, J., & Mwadzaangati, L. (2022). What is mathematics teaching talk for? A response based on three sites of practice in mathematics education. *ZDM Mathematics Education*.

Rousseau, D., & Morvan, M. (2000). *La dénomination*. Odile Jacob.

Ruthven, K. (2019). The construct of ‘resource system’ as an analytic tool in understanding the work of teaching. In L. Trouche, G. Gueudet, & B. Pepin (Eds.), *The ‘resource’ approach to mathematics education* (pp. 43–60). Springer Nature.

Schicke, H. (2011). *Organisationsgebundene pädagogische Profesionalität—Initiierter Wandel*. Budrich UniPress Ltd. https://doi.org/10.3224/86388002

Sherin, M., Jacobs, V., & Philipp, R. (Eds.). (2011). *Mathematics teacher noticing: Seeing through teachers’ eyes*. Routledge.

Sinclair, N., Healy, L., & O Reis Sales, C. (2009). Time for telling stories: Narrative thinking with dynamic geometry. *ZDM—the International Journal on Mathematics Education, 41*(4), 441–452.

Tatsis, K. (2011). Language as an identity shaping tool. The case of in-service Greek teachers. *Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education CERME, 7*, 1376–1385.

Trgalova, J., Sokhna, M., Assis, C., Alturkmani, M., Espindola, E., Hammoud, R., & Sayah, K. (2019). Teachers’ resource systems: Their constitution, structure and evolution. In L. Trouche, G. Gueudet, & B. Pepin (Eds.), *The ‘resource’ approach to mathematics education* (pp. 197–256). Springer Nature.

Trouche, L., Gueudet, G., & Pepin, B. (Eds.). (2019). *The ‘resource’ approach to mathematics education*. Springer.

Trouche, L., Gueudet, G., & Pepin, B. (2020). Documentational approach to didactics. In S. Lerman (Ed.), *Encyclopedia of mathematics education* (2nd ed., pp. 237–247). Springer.

Vergnaud, G. (1998). Towards a cognitive theory of practice. In A. Sierpinska & J. Kilpatrick (Eds.), *Mathematics education as a research domain: A search for identity* (pp. 227–241). Kluwer Academic Publisher.

Vygotsky, L. S. (1934/1978). *Thought and language*. MIT Press. (Original work published 1934).

**Publisher's Note** Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Da Ponte, J. P. (2001). Professional narratives in mathematics teacher education. In *Proceedings of the 2001 annual meeting of the Canadian mathematics education study group* (pp. 61–65). University of Alberta.

Dobie, T., Sherin, M., & White, S. (2021). A Lexical Snapshot—An investigation into the evolving terminology of middle-school mathematics teachers in the United States. In C. Mesiti, M. Artigue, H. Hollingsworth, Y. Cao, & D. J. Clarke (Eds.), *Teachers talking about their classrooms. Learning from the professional lexicons of mathematics teachers around the world* (pp. 323–333). Routlege.

Goldschmidt, G. A. (1988). *Quand Freud voit la mer. Freud et la langue allemande!* Buchet/Chastel.

Goldschmidt, G. A. (1999). *Als Freud das Meer sah—Freud und die deutsche Sprache (Translated by B. Große).* Ammann.

González-Forte, JM., Planas, N., & Fernández, C. (2022). Towards a newer notion: noticing languages for mathematics content teaching. In *Proceedings of the 45th conference of the international group for the psychology of mathematics education PME* (vol. 2, pp. 315–322).

Grossman, P., Compton, C., Igra, D., Ronfeldt, M., Shahan, E., & Williamson, P. (2009). Teaching practice: A cross-professional perspective. *Teachers College Record, 111*(9), 2055–2100.

Gueudet, G. (2019). Studying teachers’ documentation work: Emergence of a theoretical approach. In L. Trouche, G. Gueudet, & B. Pepin (Eds.), *The ‘resource’ approach to mathematics education* (pp. 17–42). Springer Nature.

Healy, L., & Sinclair, N. (2007). If this is our mathematics, what are our stories? *International Journal of Computers for Mathematical Learning, 12*(1), 3–21.

Lampert, M. (2000). Knowing teaching: The intersection of research on teaching and qualitative research. *Harvard Educational Review, 70*(1), 86–99.

Lortie, D. C. (1975). *Schoolteacher*. University of Chicago Press.

Mesiti, C., Artigue, M., Grau, V., & Novotná, J. (2022). Towards an international lexicon. *ZDM Mathematics Education, 54*(2), 239–255. https://doi.org/10.1007/s11858-022-01349-3

Mesiti, C., Artigue, M., Hollingsworth, H., Cao, Y., & Clarke, D. J. (Eds.). (2021). *Teachers talking about their classrooms. Learning from the professional lexicons of mathematics teachers around the world*. Routlege.

Mesiti, C., & Clarke, D. J. (2018). The professional, pedagogical language of mathematics teachers: A cultural artefact of significant value to the mathematics community. In *Proceedings of the 42nd conference of the international group for the psychology of mathematics education PME* (vol. 3, pp. 379–386). http://www.igpme.org/publications/current-proceedings/

Mesiti, C., Novotná, J., Clarke, D. J., Hošpesová, A., & Hollingsworth, H. (2019). Speaking about the mathematics classroom: A

comparison of the professional lexicons of teachers in Australia and the Czech Republic. In *Proceedings of the 43rd conference of the international group for the psychology of mathematics education PME* (vol. 3, pp. 89–96). http://www.igpme.org/publications/current-proceedings/

Planas, N. (2018). Language as resource: A key notion for understanding the complexity of mathematics learning. *Educational Studies in Mathematics, 98*(3), 215–229. https://doi.org/10.1007/s10649-018-9810-y

Planas, N., Adler, J., & Mwadzaangati, L. (2022). What is mathematics teaching talk for? A response based on three sites of practice in mathematics education. *ZDM Mathematics Education*.

Rousseau, D., & Morvan, M. (2000). *La dénomination*. Odile Jacob.

Ruthven, K. (2019). The construct of ‘resource system’ as an analytic tool in understanding the work of teaching. In L. Trouche, G. Gueudet, & B. Pepin (Eds.), *The ‘resource’ approach to mathematics education* (pp. 43–60). Springer Nature.

Schicke, H. (2011). *Organisationsgebundene pädagogische Profesionalität—Initiierter Wandel*. Budrich UniPress Ltd. https://doi.org/10.3224/86388002

Sherin, M., Jacobs, V., & Philipp, R. (Eds.). (2011). *Mathematics teacher noticing: Seeing through teachers’ eyes*. Routledge.

Sinclair, N., Healy, L., & O Reis Sales, C. (2009). Time for telling stories: Narrative thinking with dynamic geometry. *ZDM—the International Journal on Mathematics Education, 41*(4), 441–452.

Tatsis, K. (2011). Language as an identity shaping tool. The case of in-service Greek teachers. *Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education CERME, 7*, 1376–1385.

Trgalova, J., Sokhna, M., Assis, C., Alturkmani, M., Espindola, E., Hammoud, R., & Sayah, K. (2019). Teachers’ resource systems: Their constitution, structure and evolution. In L. Trouche, G. Gueudet, & B. Pepin (Eds.), *The ‘resource’ approach to mathematics education* (pp. 197–256). Springer Nature.

Trouche, L., Gueudet, G., & Pepin, B. (Eds.). (2019). *The ‘resource’ approach to mathematics education*. Springer.

Trouche, L., Gueudet, G., & Pepin, B. (2020). Documentational approach to didactics. In S. Lerman (Ed.), *Encyclopedia of mathematics education* (2nd ed., pp. 237–247). Springer.

Vergnaud, G. (1998). Towards a cognitive theory of practice. In A. Sierpinska & J. Kilpatrick (Eds.), *Mathematics education as a research domain: A search for identity* (pp. 227–241). Kluwer Academic Publisher.

Vygotsky, L. S. (1934/1978). *Thought and language*. MIT Press. (Original work published 1934).

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