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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. [ABSTRACT FROM AUTHOR]

A significant distinguishing characteristic of the International Classroom Lexicon Project was the documentation of classroom pedagogical practices of mathematics teachers in the original languages of ten communities. These lexicons provide us with an opportunity to compare identified teacher practices within these communities. In this paper, we explore the challenges of conducting a cross-lexicon comparison to explore the possibility of moving towards an international lexicon by focusing on the Australian, Chilean, Czech and French lexicons. We focus on two clusters of terms, namely, those related to assessment and those related to mathematics. Inspired by theories and studies related to networking and boundary crossing, strategies were conceived and tested in order to accomplish the following objectives: to support a meticulous and comprehensive comparison of the lexical items; to classify the complexity of similarity of lexical items; and to reveal connections amongst the four lexicons in a bid towards developing an international lexicon. This analysis has the potential to contribute significantly to the study and promotion of mathematics teachers' reflective practice and highlights the cultural underpinning of the way classroom practices are perceived. [ABSTRACT FROM AUTHOR]

The Learner's Perspective Study (LPS) offers an extremely rich source of data for analyses of classroom situations. In this paper we focus on events in mathematics classrooms through the lens of the Theory of Didactical Situations in Mathematics, namely, on the situation of institutionalization. During the phase of institutionalization, the teacher tends to help pupils to transform a piece of their knowledge from the means of resolving a problem to a reference for further uses in new situations. A detailed analysis of transcripts of communication between a Czech teacher and her 14- to 15-year-old pupils resulted in the classification of institutionalization into four types, as follows: asking additional questions referring to statements from the stage of validation with the aim of adding new knowledge to the repertoire; presenting a mathematical proof of a statement from validation; asking pupils to solve a similar problem; and setting problems for pupils, whose solution will allow them to apply the acquired knowledge. These are presented and illustrated in the paper. The combination of video-recordings from three cameras (one recording the teacher, one a pair of pupils and one the whole class), together with interviews with the teacher and the pair of pupils, materials prepared by the teacher, and pupils' productions, allows us to gain a deeper understanding of the role of institutionalization in mathematics education. [ABSTRACT FROM AUTHOR]

With the myriad theories generated through research over the years, a continuing challenge for researchers is to navigate the multitude of theories in order to communicate their research, integrate empirical results, and make progress as a field by building upon empirical research. The Social Unit of Learning project was purposefully designed so that researchers from multiple disciplines with different theoretical perspectives could work together to examine the complexity of the mathematics classroom. In this paper, we reflect on the multiple analytical accounts generated from the project, drawing from the notions of complementarity and commensurability. Two parallel analyses, applying the commognitive framework and the theory of representations respectively, are used as illustrative examples for discussion regarding complementarity and commensurability. The paper addresses two focal questions, as follows: in what ways do divergence or contradiction in incommensurable analytical accounts reflect methodological discrepancies or fundamental differences in the underpinning theories? Furthermore, in what ways do the accounts generated by the parallel analyses predicated on different theories lead to differences in instructional advocacy? The answers to these questions provide empirically-grounded insights into the consideration of incommensurability in educational research, and suggest ways in which researchers and practitioners might apply the notion of complementarity to reconcile or exploit incommensurable analytical accounts that have resulted in different instructional advocacies. [ABSTRACT FROM AUTHOR]

The aim of the paper is to show whether the systems of mathematics teacher education in two countries whose historical development may seem quite similar (the Czech Republic and Hungary were parts of the Habsburg Monarchy, and they were part of the Eastern European communist block) are also very similar, or whether each country follows its own, unique path. In this paper we build on the general need for more international comparative studies in the area of teacher education. We present a historical review of development of mathematics teacher education in both countries and analyse and compare current teacher education programs, their structure and organization, the demands on mathematics teachers and their qualifications for teaching at various levels. In the paper we pinpoint a number of aspects resulting in differences in the two systems, including historical development and traditions, (de)-centralization, globalization and international cooperation, etc. The comparison based on these aspects shows that the systems in both countries are very different. The analysis described in this paper could become the starting point for further research using the data from international surveys (e.g. PISA). Such research would enable us to make conjectures about the effects of various ways of organizing teacher education. [ABSTRACT FROM AUTHOR]

The goal of this study is to show a novel way of using large-scale data (N = 6203) to identify pupils' strategies when solving missing value number equations. It is based on the assumption that wrong numerical results appearing more frequently than would be the case if they were consequences of random guessing can be expected to be underlain by a pupil's rational, theoretically justifiable, solving procedure. We determined all of the numerical results suggested by the pupils in the study for three problems with the same format and classified them based on their occurrence. Possible solving procedures leading to them were inferred using a priori analysis. Next, we identified pupils' strategies based on the anticipated solving procedures in two problems of the same format and validated them by comparing them to the pupils' results in the third problem and to the pupils' whole test result. We identified six strategies, some of them underlain by clearly identified procedures. Our analysis showed a high predictive power of the identified pupils' strategies for their achievement in the mathematics test written 3 years later and in a more difficult problem of the same format. Our study has implications for test makers and teacher practice. [ABSTRACT FROM AUTHOR]

Research has shown that word problem difficulty is influenced by various linguistic, numerical and general variables, but results in this area are inconsistent. This paper focuses on the question of how the order of numerical data, the context, the position of the unknown transformation, and the length of the statement in an additive word problem, influence the achievement and reasoning of primary pupils. These variables were varied in two word problems requiring additive reasoning and these variants were solved by pupils in Grades 4 and 5 from four Prague primary schools. Item Response Theory was used both for the division of the pupils into four equally able groups each solving a different variant of the word problem, and for the quantitative interpretation of data. The pupils' solutions were also analysed in a qualitative way. For a complex additive problem with several states and transformations, we noticed a significant influence of the order of data and the position of the unknown transformation on the pupils' achievement and types of mistakes. No such influence of context (with which the pupils do not have immediate experience) or the length of the text (prolonged by superfluous information) was found. Some implications for teaching mathematics and preparing tests are given. [ABSTRACT FROM AUTHOR]

The article reports the results of a longitudinal research study conducted in three mathematics classes in Czech schools with 62 pupils aged 12-18 years. The pupils were exposed to the use of selected heuristic strategies in mathematical problem solving for a period of 16 months. This was done through solving problems where the solution was the most efficient if heuristic strategies were used. The authors conducted a two-dimensional classification of the use of heuristic strategies based on the work of Pólya () and Schoenfeld (). We developed a tool that allows for the description of a pupil's ability to solve problems. Named, the Culture of Problem Solving (CPS), this tool consists of four components: intelligence, text comprehension, creativity and the ability to use existing knowledge. The pupils' success rate in problem solving and the changes in some of the CPS factors pre- and post-experiment were monitored. The pupils appeared to considerably improve in the creativity component. In addition, the results indicate a positive change in the students' attitude to problem solving. As far as the teachers participating in the experiment are concerned, a significant change was in their teaching style to a more constructivist, inquiry-based approach, as well as their willingness to accept a student's non-standard approach to solving a problem. Another important outcome of the research was the identification of the heuristic strategies that can be taught via long-term guided solutions of suitable problems and those that cannot. Those that can be taught include systematic experimentation, guess-check-revise and introduction of an auxiliary element. Those that cannot be taught (or can only be taught with difficulty) include the strategies of specification and generalization and analogy. [ABSTRACT FROM AUTHOR]

One of the manifestations of learning is the student's ability to come up with original solutions to new problems. This ability is one of the criteria by which the teacher may assess whether the student has grasped the taught mathematics. Obviously, a teacher can never teach the ability to invent new solutions (at least not directly): he/she can ask for it, expect it, encourage it, but cannot require it. This is one of the fundamental paradoxes of the whole didactical relationship, which Guy Brousseau modelled in one of the best-known concepts in didactics of mathematics: the didactical contract. The teacher cannot be confident that the student will learn exactly what the teacher intended to teach and hence the student must re-create it on the basis of what he/she already knows. In this paper, the importance and the role of situations affording mathematical creativity (in the sense of production of original solutions to unusual situations) are demonstrated. The authors present an experiment (with 9-10-year-old children) that makes it possible to show how certain situations are more favourable (for all children) to express some characteristics of mathematical creativity. [ABSTRACT FROM AUTHOR]

Many students have difficulties with basic algebraic concepts at high school and at university. In this paper two levels of algebraic structure sense are defined: for high school algebra and for university algebra. We suggest that high school algebra structure sense components are sub-components of some university algebra structure sense components, and that several components of university algebra structure sense are analogies of high school algebra structure sense components. We present a theoretical argument for these hypotheses, with some examples. We recommend emphasizing structure sense in high school algebra in the hope of easing students' paths in university algebra. [ABSTRACT FROM AUTHOR]