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The paper is a sequel to the article (Novotná et al., 2014), where the authors present the results of a 4-month experiment whose main aim was to change pupils’ culture of problem solving by using heuristic strategies suitable for problem solving in mathematics education. (Novotná et al., 2014) focused on strategies Analogy, Guess – check – revise, Systematic experimentation, Problem reformulation, Solution drawing, Working backwards and Use of graphs of functions. This paper focuses on two other heuristic strategies convenient for improvement of pupils’ culture of problem solving: Introduction of an auxiliary element and Omitting a condition. In the first part, the strategies Guess – Check – Revise, Working backwards, Introduction of an auxiliary element and Omitting a condition are characterized in detail and illustrated by examples of their use in order to capture their characteristics. In the second part we focus on the newly introduced strategies and analyse work with them in lessons using the tools from (Novotná et al., 2014). The analysis of results of the experiment indicates that, unlike in case of the strategy Introduction of an auxiliary element, successful use of the strategy Omitting a condition requires longer teacher’s work with the pupils. The following analysis works with the strategy Systematic experimentation, which seemed to be the easiest to master in (Novotná et al., 2014); we focus on the dangers it bears when it is used by pupils. The conclusion from (Novotná et al., 2014), which showed that if pupils are introduced to an environment that supports their creativity, their attitude towards problem solving changes in a positive way already after the period of four months, is confirmed.

The paper describes one of the ways of developing pupils’ creative approach to problem solving. The described experiment is a part of a longitudinal research focusing on improvement of culture of problem solving by pupils. It deals with solving of problems using the following heuristic strategies: Analogy, Guess – check – revise, Systematic experimentation, Problem reformulation, Solution drawing, Way back and Use of graphs of functions. Most attention is paid to the question whether short-term work, in this case only over the period of three months, can result in improvement of pupils’ abilities to solve problems whose solving algorithms are easily accessible. It also answers the question which strategies pupils will prefer and with what results. The experiment shows that even short-term work can bear positive results as far as pupils’ approach to problem solving is concerned.

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.

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.

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.

The paper describes one problem solving strategy -- the Use of false assumption. The objective of the paper is to show, in accordance with Phylogenesis and Ontogenesis Theory, that it is worthwhile to reiterate the process of development of the concept of a variable and thus provide to pupils one of the ways helping them to eliminate usual difficulties when solving word problems using linear equations, namely construction of the equations. The paper presents the outcomes of a study conducted on three lower secondary schools in the Czech Republic with 147 14-15-year-old pupils. Pupils from the experimental group were, unlike pupils from the control group, taught the strategy the Use of false assumption before being taught the topic Solving word problems. The tool for the study was a test of four problems that was sat by all the involved pupils three weeks after finishing the topic "Solving word problems" and whose results were evaluated statistically. The experiment confirmed the research hypothesis that the introduction of the strategy the Use of false assumption into 8th grade mathematics lessons (14-15-year-old pupils) helps pupils construct equations more successfully when solving word problems.

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.

The paper investigates how the order of numerical data in word problems on division of a whole into unequal parts affects achievement and reasoning of 14-16-year old pupils. The variable was altered in two word problems, in one of which also the context was changed (psychological variable) and in the latter "if-clause" is or is not used (linguistic variable). 182 pupils were involved in the experiment. The solutions were analysed quantitatively using Item Response Theory as well as qualitatively. The data suggest that pupils' success is affected by the order of numerical data in the statement in an unfamiliar context. The presence of "if-clause" in the statement was studied in a two-level problem. The order of numerical data played its role in case of formulations without "if". The results of the experiment are of interest for mathematics education as well as for construction of tests. The paper is an extended version of the paper by Novotná (2018).

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.

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.

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 (2004) and Schoenfeld (1985). 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.

Aim. The aim of this paper is to present the results of a pilot validation of an artificial intelligence chatbot for mathematics tutoring. The subject of the analysis is the behaviour of the students in the course, in particular their approach to solving problems without external motivation through scoring or other assessment. Methods. The analysis of student behaviour was carried out on the basis of records in the application database. The subject of the analysis was not only the pupils' responses, but also the time taken to produce them and the overall trajectory through the course. Results. The results of the analysis show that students who choose to join th course, work intensively in the lessons. Most pupils solve more than one basic task. A significant proportion of pupils also use the prepared help and instructional videos. The analysis of the solution time allowed identification of a group of pupils who do not even read the problems and only confirm the results. Conclusion. The pilot validation confirmed that the chat-bot format is close to the learners, and they are able to use the courses created in this environment. Solution time analysis suggests that this parameter can be used to analyse learner behaviour and system control. Further analyses in this area are needed. [ABSTRACT FROM AUTHOR]