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In the setting of the transformation of our societies to become multicultural ones, we intend in this paper to examine situations in a mathematics classroom where respect for, and exclusion of, the ‘different’ co-exist. Conducting an instrumental case study, we follow two Grade 4 groups while they are solving a mathematics generalization task. We examine how the interaction that took place between the members of the teams mirrors the two faces of the same coin (i.e. the culture of the classroom) and at the same time reflects the change in the primary classroom that necessitates openness and genuine respect for the ‘different’. On the one hand, we will demonstrate the dynamic of the harmony that was apparent between the members of one of the groups since the ‘different’ was not a barrier but a source of mathematical ideas that contributed to the solution. On the other hand, the lack of openness in the other group resulted in good opportunities for ‘doing mathematics’ being missed and put the self-confidence of one of its members at risk. Keywords: classroom diversity and conflict, multicultural mathematics classroom, collaborative problem-solving

Vzdělání není možné bez obsahu. Vyučování a učení ve školách se věnuje „něčemu“, ne „ničemu“ nebo „čemukoliv“. Čím více heterogenní, multikulturní a jazykově různorodá je výuka, tím více je během vyučování a učení nutná sémantizace vzdělávacího obsahu. V tomto článku rozvíjíme teorii transformace obsahu a představujeme koncept operačního isomorfismu a jeho potenciálu v oblasti didaktiky, výzkumu výuky a vzdělávání a vzdělávání učitelů. Pomocí případové studie výuky matematiky ilustrujeme, jak mohou studenti těžit ze sémantizačních a instrumentalizačních postupů ve třídě a jak tyto procesy nabývají podoby extenzionální dekonstrukce a intenzionálního rozšiřování nebo zhušťování. There is no education without content. Teaching and learning in schools is devoted to “something,” not to “nothing” or “anything.” The more heterogeneous, multicultural, and multilingual the classroom, the more semantization of educational content within teaching and learning is needed. Therefore, the role of various languages in the classroom is revisited. In this article, we elaborate on the theory of language-sensitive content transformation and introduce the concept of operational isomorphism and its potential in the field of didactics, research on teaching, and learning and teacher education. Using a case study of mathematics teaching, we illustrate, how students can benefit from semantization and instrumentalization practices in the classroom and how these processes take the form of extensional deconstruction and intensional expansion or intensional condensation.

In his article, M. Hejný (this issue) provided the theoretical background for the development of a diagnostic tool which can be used for the characterisation of a mathematics teacher’s educational style. The goal of this study is to describe the tool and show how it can be applied. The tool consists of 20 parameters divided into four areas: (A) beliefs, (B) experience, (C) personality and (D) abilities / competences. The tool has been applied to two short case studies and one longer one. This tool can also inspire a teacher for the improvement of his/her teaching style.

Through visualisation, geometry can mediate understanding of some demanding arithmetic and algebraic concepts, relationships, processes and situations for pupils. This thesis is explained by the method of genetic parallel and of a didactic analysis of two educationally interesting problem situations. Theoretical considerations are illustrated by several real experiences. Suggestions for the application of theoretical results are given in conclusion.

As student diversity increases in many schools across the globe, teacher diversity may not be keeping pace with the ever changing student population. Along with this comes an increasing number of issues and queries around race and education. The ideas surrounding how race impacts the education of students have been shrouded in controversy given the sensitive nature of the topic and its troubled past in some countries. Countries across the world struggle to recruit a teacher workforce that matches that of the student population especially in pockets where there are a larger number of minorities students. Many education programs, policies, and interventions more recently have been reasserting the ways that addressing issues with race in schools can be a positive change, specifically seeking to diversify teacher recruitment and match teachers or mentors with groups of students based on race to serve as built- in role models. Indeed, our increased awareness of teacher recruitment practices and the need for a more diverse teaching force is based on some of those assumptions that student/teacher race matching is and can be a positive thing. This pilot study seeks to examine these assumptions by looking at how race may actually affect teacher student relationships. This study focuses on students and teachers in a low-income area of the United States to assess student perceptions of their teachers on several key attributes of quality teaching. The aim of the study is to see if classrooms of students with similar races to that of their teacher perceive their teachers differently. More directly, do students share more favorable perceptions of their teachers if they are of the same racial background? This study finds that students perceive that teachers of the same race are more effective and have more positive relationships with them. This finding is consistent for both white and black teachers. The paper concludes with a call for more research and a continued push to diversity the teacher workforce worldwide.

International comparison studies (TIMSS and PISA) and teachers’ expressed views indicate a decreasing trend in Czech pupils’ competence in geometry. The present study describes an innovative educational strategy for teaching 3D geometry - the didactical environment Cube Buildings and how it fosters the development of spatial sense in the earliest school age group. The results of one experiment in this environment are included as a partial result of a longitudinal action research project of one of the authors. The findings illustrate how a pupil can build problem-solving strategies and how knowledge is spread among pupils.

In the past five years our research team has studied and elaborated a teaching method called the schema-oriented education. Pupils do not receive mathematical knowledge from the teacher. Instead, they build own mathematical schemata through problem-solving and class discussion. Problems are nested in various semantic and structural teaching environments. This article describes in detail the building of a schema in relation to the solution of a system of two linear equations x + y = a, x – y = b in three of such environments.

The mathematical learning environment Bus is based on pupils’ real-life experience and has been elaborated by M. Hejný to develop pupils’ understanding of number as a state, an address, an operator of change and of comparison when it is grasped by the specific language of this environment. The main result of our study is a description of the process in which individual 7-8 years old pupils lose their tendency to use irrelevant aspects of language and effectively select and structure those that are key in a given situation.