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

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.


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 1 3 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 (1915Barthes ( -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.

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.

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 crossprofessional study of clergy, teaching and clinical psychology noted: "among our trio of professions, this language of practice seems particularly well-developed 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 professional 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 traditions for developing professional languages in the context of teaching exist.
The lack of a professional language for describing mathematics 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 language 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 preparation 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 teachers 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 learning have evolved differently so far, in different cultural contexts. On the other hand, professional communities in education 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 medical 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 practitioners 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. Professional 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 sharing 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 specific type of classroom lesson (Cramer et al., 2021). These examples already indicate how complex the study of 'professional 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, language 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 Documentational Approach to Didactics, also raises questions: In how far is a professional language an 'artefact' or can become an 'instrument' for describing mathematics classrooms teaching and learning?
"The instrumental approach distinguishes between an artefact, product of the human activity, designed for a goaldirected 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.

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 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).

Capturing language as a resource
To address these lexical limitations of the Lexicon Project, 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 interactions (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 mathematics 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 comparing 'our distinct stories' and 'our professional languages' about teaching and learning in mathematics classes. The narrative approach allows us to study and compare language as a resource across different cultural and linguistic contexts. Accordingly, we pose the following research question: How does a cultural comparative approach of narratives 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 linguistic traditions. This is the focus and the contribution of this article.

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.

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 practical, 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 situations, is embedded in methodological and theoretical traditions 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 learning. Fifth, narratives are 'theoretical' in the sense that their thick descriptions actually capture significant dynamics of teaching and learning, which allows us to theoretically conceptualize 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 narratives 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 narratives in education research (Clandinin & Connelli, 2000), our intention was to make it visible again that, for each team, each video tells a particular story generated through the selection, arrangement and interpretation 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 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 pedagogicaldidactical and mathematical language, where pedagogicaldidactical 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.

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 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 context and structure of the lesson, followed by a chronological 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 narratives 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 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 language 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.

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 lexicon 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 themselves, 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-references 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 comparative analyses of the narratives also indicated varying resource systems for describing classrooms. This motivated us to conduct a more detailed analysis, an analysis of selected episodes, whose results are presented in Sect. 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 showing captivating features of the lesson and a priori appropriate 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 narrative 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, interpretative analysis of these episodes was to understand better 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.

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 underlined portions are terms of the lexicon) and a commentary, summarizing the interpretation of the episode in the different narratives (the Czech, French and German narrative of the episode).

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.

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. 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 (1988Goldschmidt ( , 1999 points out, the French language lacks such physical connotations and uses terms of Latin or Greek origin instead.

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.

Discussion
Our cultural comparative approach of narratives, even if limited 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 describing 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 language influences our descriptions and perceptions of classroom 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 complexity 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 narratives 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 teaching methods and teacher-student interactions, staying close to observable actions, without going into detail about the mathematical activities carried out. The French narratives 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 technical terms in the lexicons helps us to express more deeply what is at stake in these episodes. For example, the German lexicon includes two terms for describing the handling of errors in class: Fehleranalyse (literally "error-analysis", in English: analysis of mistakes) and Fehlerkultur (literally "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 homework) 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 consideration 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 processes supporting pupils learning of this content".

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 classroom 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 didactic intentions and techniques in situ. Specifically, narratives 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 particularly to this phenomenon, and offer culturally complementary, 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 comparative 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 systematically 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 communities 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 developing cross-cultural comparisons. The potential of using teachers' professional language is studied in a crosscultural comparative approach, and provides methodologically 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 comparisons, 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 initial education. Cultural comparative approaches can inspire teachers to discover other professional perspectives on classroom 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.