The aim of the Lexicon Project is to inquire into the pedagogical naming systems describing what is happening in mathematics lesson in nine different countries (eight languages): Australia, Chile, China, Czech Republic, Finland, France, Germany, Japan and the USA. As Janík and Slavík (2007: 267) stress grasping and description of didactical facts (i.e. what is going on) in a lesson meaningfully requires intellectual work based on observer’s/actor’s prior experience. Appropriate terms from pedagogy facilitate interpretation of the facts in a way that they are meaningful to other people. Only then are we able to communicate these facts ad about them. The used terminology forms a common interpretative context and thus is likely to influence not only the perception of pedagogical reality but also contributes to or limits its construction. These issues were at the birth of the international project Lexicon. (Clarke, 2009) More information about the project can be found in (Clarke 2017).

The impact of culture on classroom reality and how it is perceived attracted a lot of attention in the field of mathematics education in the 1980s’ (see e.g. Bruner, 1996). Rezat and Straesser (2012) relate different perception to different actors in education visualized in the form of didactical tetrahedron (Rezat and Straesser, 2012: 648) and discuss conventions about what it means to be doing mathematics, to be a mathematics teacher or a pupil studying mathematics, about school as an institution, about textbooks and their structure and form, public image of mathematics in a society etc. All these have impact not only on mathematics lessons but also on the discourse about these lessons. This may result in a situation when researchers and teachers from different countries may fail to understand each other since the terminology they use refers to very different situations. National Lexicons developed within the project should help teachers, educators and researchers understand each other. The main concern was how to manage to combine the two planned ways of using the Czech (one of several national) lexicon – as a tool describing the structure of Czech lessons and highlighting important parts of lessons and as a tool facilitating discussions between different groups. It resulted in the following questions: What will be the impact of different types of validation on Czech lexicon organization? Are the used terms comprehensible? Do they reflect pedagogical reality, didactical facts? Are they able to describe what is actually happening in a classroom?

It became obvious that it was essential to validate the lexicons to ensure that they would be useful both for use within the Czech Republic (and other involved countries) but also in communication with mathematics educators and teachers from abroad. This validation had to be unique, in order to validate both usability and comprehensibility at home and on international level.

The three levels of validation are national validation, validation by another lexicon team and bilateral comparison of two lexicons. The aim of the national validation was to analyse main differences in the language of teachers and researchers, clarity of the used terms and descriptions for teachers and completeness of the list of terms and descriptions. The main aim of the validation by another lexicon tam was to improve the clarity of descriptions of terms so that national lexicons could be used for further comparative activities done by the whole lexicon community. The aim of the bilateral comparison of two lexicons was to compare different naming systems to describe the events, actions and interactions of the mathematics classroom.

The focus of this contribution will be on three types used for validation of the Czech lexicon.

In the process of national validation, three methods were used: 1. a questionnaire survey among primary and lower secondary school teachers whose aim was to verify whether teachers are familiar with the terms and use them actively, 2. workshops – use of video episodes, 3. interviews conducted with experts during informal meetings as well as after presentation on conferences. Questionnaires: Four different groups of respondents (practicing teachers participating at the conference Two days with didactics of mathematics, practicing lower secondary mathematics teachers from the whole country contacted by e-mail, part-time student teachers at the Faculty of Education of Charles University in Prague, primary teachers in life-long education at the Faculty of Education of University of South Bohemia in České Budějovice. Workshops were organized as follows: watching video (excerpts from lessons), group work with its transcription (possible re-watching of parts if required by the participants), discussion of outputs. Three different groups participated (A. teachers participating at the conference Two days with didactics of mathematics, B. part-time student teachers at the Faculty of Education of Charles University in Prague, C. doctoral students at the Faculty of Education of Charles University in Prague). Another source of data were interviews with mathematics educators and other experts in the area of education. The validation by another lexicon team, the critical review of the lexicon by the Chilean team was conducted. The reviewing team was provided with the lexicon without any further notes and explanations. There was no procedure or template prescribed to the reviewers. The bilateral comparison of two lexicons was conducted in the form of comparison of Czech and French lexicons (Artigue et al., 2017). The analysis came out of ATD, the Anthropological Theory of the Didactic (Chevallard & Sensevy 2014). ATD pays specific attention to the conditions and constraints shaping what is actually taught (mathematical praxeologies) and how it is taught (didactical praxeologies), and to the fact that these depend on the institutional and cultural contexts. The methodology used for the comparison of the Czech lexicon (CL) and the French lexicon (FL) had two parts: a formal comparison and an operational comparison of their use.

National validation: Comparison of results shows that there are considerable differences between terminology researchers and in-service teachers use for description of classroom reality, especially in recently developing fields (e.g. scaffolding, institutionalisation, linking). Majority of in-service teachers have not come across these new innovative theories during their university studies. National validation clearly confirms the important role of the Czech lexicon for teacher education. (Novotná et al., 2016) Validation by another lexicon team: This comparison proves to have been important for both Czech and Chilean teams. The questions and comments of Chilean colleagues were most often (20 out of 23) related to unclarity of some English expressions. The 3 remaining comments were related to a different culture of teaching at Chilean and Czech schools, especially to different roles the teacher plays in the classroom. This critical review resulted in changes in wording of some descriptions in Czech lexicon and adding additional examples and non-examples. Bilateral comparison of two lexicons: The comparison indicates that Czech lexicon (CL) contains fewer terms than French lexicon (FL) (47 vs. 115); CL is not a subset of FL. Both lexicons are structured in similar categories but the distribution of the same terms into categories is not the same differing usually in their interpretation. Two categories in FL do not have their counterpart in CL. Terms in CL are usually significantly shorter than in FL; they are in the active form expressing participants’ actions. FL descriptions are more in nominal form, and quite often they make explicit the purpose of described actions. Differences were identified also between examples and non/examples in CL and FL, mainly in their number. The identified differences have their origin in the different educational culture in the two countries.

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 Kaur, B., Ho, W.K., Toh, T.L., & Choy, B.H. (Eds.), Proceedings of the 41st Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 113-120). Singapore: PME. Bruner, J. (1996). The culture of education, Cambridge, Mass.: Harvard University Press. Chevallard, Y., & Sensevy, G. (2014). Anthropological approaches in mathematics education, French perspectives. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 38–43). New York: Springer. Clarke, D.J. (2009). Mind your language: Speaking in and about the mathematics classroom. In Mathematics of Prime Importance (pp. 34-49). Brunswick: MAV. Clarke, D.J. (2017). Using cross-cultural comparison to interrogate the logic of classroom research in mathematics education. In Kaur, B., Ho, W.K., Toh, T.L., & Choy, B.H. (Eds.), Proceedings of the 41st Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 13-28). Singapore: PME. Janík, T., & Slavík, J. (2007). Fakty a fenomény v průniku didaktické teorie, výzkumu a praxe vzdělávání. Pedagogika, 57(3), 263-274. Novotná, J., Moraová, H., Hošpesová, A., Žlábková, I., & Bureš, J. (2016). How do we understand each other when we describe classroom activities – Lexicon. In Krejčí, I., Flégl, M., & Houška, M. (Eds.), Proceedings of ERIE 2016 (pp. 440-447). Praha: CULS. Rezat, S., & Sträßer, R. (2012). From the didactical triangle to the socio-didactical tetrahedron: artifacts as fundamental constituents of the didactical situation. ZDM, 44(5), 641-651. Acknowledgement: The research was partly supported by the project ARC-DP140101361 The Lexicon Project: Analysing pedagogical naming systems from different cultures to reconceptualise classroom practice and advance educational theory supported by the Australian Research Council.