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This study investigates teachers’ argumentation aiming to convince students about the invalidity of their mathematical claims in the context of calculus. 18 secondary school mathematics teachers were given three hypothetical scenarios of a student's proof that included an invalid algebraic claim. The teachers were asked to identify possible mistakes and explain how they would refute the student's invalid claims. Two of them were also interviewed. The data were analysed in terms of the content and structure of argumentation and the types of counterexamples the teachers generated. The findings show that teachers used two main approaches to refute students’ invalid claims, the use of theory and the use of counterexamples. The role of these approaches in the argumentation process was analysed by Toulmin's model and three types of reasoning emerged that indicate the structure of argumentation in the case of refutation. Concerning the counterexamples, the study shows that few teachers use them in their argumentation and in general they underestimate their value as a proof method.

This paper investigates the role of tools in the formation of mathematical practices and the construction of mathematical meanings in the setting of a telecommunication organization through the actions undertaken by a group of technicians in their working activity. The theoretical and analytical framework is guided by the first-generation activity theory model and Leont’ev’s work on the three-tiered explanation of activity. Having conducted a 1-year ethnographic research study, we identified, classified, and correlated the tools that mediated the technicians’ activity, and we studied the mathematical meanings that emerged. A systemic network was generated, presenting the categories of tools such as mathematical (communicative, processes, and concepts) and non-mathematical (physical and written texts). This classification was grounded on data from three central actions of the technicians’ activity, while the constant interrelation and association of these tools during the working process addressed the mathematical practices and supported the construction of mathematical meanings that this group developed from the researchers’ perspective. Technicians’ emerging mathematical meanings referred to place value, spatial, and algebraic relations and were expressed through personal algorithms and metaphorical and metonymic reasoning. Finally, the educational implications of the findings are discussed.

This paper is methodologically based, addressing the study of mathematics teaching by linking micro- and macro-perspectives. Considering teaching as activity, it uses Activity Theory and, in particular, the Expanded Mediational Triangle (EMT) to consider the role of the broader social frame in which classroom teaching is situated. Theoretical and methodological approaches are illustrated through episodes from a study of the mathematics teaching and learning in a Year-10 class in a UK secondary school where students were considered as “lower achievers” in their year group. We show how a number of questions about mathematics teaching and learning emerging from microanalysis were investigated by the use of the EMT. This framework provided a way to address complexity in the activity of teaching and its development based on recognition of central social factors in mathematics teaching–learning.

This article refers to a longitudinal case study of a primary school teacher over a period of 4 years. The focus is on the development of the teacher’s beliefs regarding mathematics teaching and learning from the last year of her university studies up to the third year of teaching mathematics in school. This development has been investigated within three different contexts, which have been distinguished in terms of the kind of support provided to this teacher. Two dominant beliefs emerged which have been traced through the period of the study from both the teacher’s reflections and actions. The first belief drew on the idea that what was considered an easy mathematical task by an adult could also be easily understood by children, while the second was that children learn mathematics through their actual involvement in a variety of teaching activities. The results indicate the way that teacher’s experiences from her university studies, actual classroom practice and inservice education interact and influence her beliefs and professional development.

Mathematical and pedagogical awareness is crucial for a teacher to notice, articulate, interpret aspects of classroom practice, and make on the spot decisions. Awareness is related to teacher knowledge and is rooted in the context of the actual classroom practice. Mason (2002) talks about different levels of awareness both in mathematics and in mathematics teaching and relates them to the process of noticing that involves systematic reflection on acts or issues. A number of studies at both pre-service and in-service levels that have been published in this Journal of Mathematics Teacher Education [JMTE] focus on what the teachers ‘‘notice’’ in a classroom while teaching or reflecting on it and investigate ways that can support them in the process of developing this ability (e.g., Mellon 2011; Scherer and Steinbring 2007). Approaches that seem to be effective in teacher education and professional development for supporting the development of teacher awareness include teacher interaction and collaboration and the use of theoretical tools in analyzing and studying mathematics teaching. This theme of teacher awareness is reflected in the articles of this issue of JMTE. The three articles in the research category and one article in the Mathematics Teacher Education around the World category provide further insights into the nature and development of mathematics teachers’ awareness or noticing during practice. The article by Jimmy Sherrer and Mary Kay Stein and the article by Shari Stockero and Laura Van Zoest focus on teachers’ noticing of critical classroom interactions that involves students’ contributions and teachers’ reactions. Sherrer and Stein propose and evaluate an intervention where a coding scheme was given to the teachers to analyze teaching while Stockero and Van Zoest examine teachers’ actions and decisions in the context of actual teaching when pivotal teaching moments emerge. The article by Jennifer Tobias focuses on the development of prospective teachers mathematical awareness related to the fraction concept in the context of a teaching experiment that allowed mathematical communication and argumentation. Finally, the article by Su Liang, Sarah Glaz, Thomas DeFranco, Charles Vinsonhaler, Robin Grenier, and Fabiana Gardetti reports on mathematics teacher education and professional development in China and indicates ways that mathematical and pedagogical awareness of

This paper reports research that attempts tomake sense of the complexity of mathematicsteaching and its development at secondaryschool level. The research was conducted inpartnership between two teachers and twoeducator/researchers over one school term intwo U.K. schools. A theoretical construct, theteaching triad, was used as an analyticaldevice (by the researchers) and as a reflectiveagent for teaching development (by theteachers). The focus of analysis was theinteractions between teacher and students atwhole class and small group level. Both micro-and macro-analyses were undertaken. We presentdetails of the processes involved in examplesfrom the teaching of one teacher as shetranslated theoretical aims into classroompractice. The use of the triad allowed accessto complexity, involving both psychological andsociological elements, and to the position of asincere teacher with respect to competingforces in the educational system. Thepotential of the triad for teacher and teachingdevelopment is discussed.

This study focuses on the role of tools, provided by a computer microworld (C.AR.ME), on the strategies developed by 14-year-old students for the area measurement of a non-convex polygon. Students' strategies on a transformation and a comparison task were interpreted and classified into categories in terms of the tools used for their development. The analysis of the data shows that an environment providing the students with the opportunity to select various tools and asking them to produce solutions `in any possible way' can stimulate them to construct a plurality of solution strategies. The students selected tools appropriate for their cognitive development and expressed their own individual approaches regarding the concept of area measurement. The nature of tools used affected the nature of solution strategies that the students constructed. Moreover, all students were involved in the tasks and succeeded in completing them with more than one correct solution strategy thereby developing a broader view of the concept, although not all of them realized the same strategies. Three different approaches to area measurement emerged from the strategies which were constructed by the students in this microworld: automatic area measurement, provided by the environment, the operation of area measurement using spatial units and the use of area formulae. Almost all the students experienced qualitative aspects of area measurement through being involved in the process of covering areas using spatial units. Students also managed to use the area formulae meaningfully by studying it in relation to automatic area measurement and to area measurement using spatial units. Through these strategies, the concepts of conservation of area and its measurement as well as area formulae were viewed by the students as interrelated. Finally, some basic difficulties regarding area measurement were overcome in this computer environment.

The four research reports published in this issue of JMTE study teacher knowledge focusing mainly on its mathematical dimension. Issues concerning the characteristics of teacher knowledge for mathematics teaching especially at secondary level as well as the interplay between mathematics and pedagogical knowledge emerge. In the first article, Sarah Bleiher, Denisse Thompson and Mile Krajcevski study prospective secondary mathematics teachers’ (PSMT) knowledge of mathematical proof focusing on how they validate students’ arguments and the feedback they provide. In particular, 34 prospective secondary mathematics teachers validated high school students’ arguments before and after implementation of a set of activities in a mathematics methods course. The authors adopt Stylianides’ (2007) position of proof as a mathematical argument that is a connected sequence of assertions for or against a mathematical claim with certain characteristics such as accepted statements, modes of argumentation and modes of argument representation. In their design of the instructional activities, they took into account two difficulties reported in the literature about proof validation: the focus on local specifics of an argument overlooking the global logical structure and the tendency to use an empirical inductive proof scheme. The research questions are related to the effectiveness of the instructional activities on PSMT’s validation skills and to the type of errors they identify. The results indicate that PSMT were successful at judging the validity of an argument in the preand posttest. Concerning the errors to which they attend, it appeared that they did not any longer hold an empirical proof scheme. Although the intervention seemed to have an impact on PSMT’s understanding of proof, using indirect methods seemed to be difficult. Prospective secondary mathematics teachers are also the focus of the second article by Jill Adler, Sarmin Hossain, Mary Stevenson, John Clarke, Rosa Archer and Barry Grantham who study how PSMT experience mathematics in a course (Mathematics Enhancement Course—MEC) addressing mathematics in ‘depth.’ The course is offered in a range of institutions in England and provides an alternative route to mathematics teaching for

This paper focuses on two main issues concerning the mathematics education of prospective primary school teachers in Greece: the integration of mathematics and pedagogy and the relation of theory to practice. In particular, specific teaching approaches are discussed concerning the problem of integration both in mathematics and in mathematics education courses. The problem of "theory – practice" is examined through an analysis of the kind of teaching practice in which the prospective teachers are involved. Finally, the constraints that the mathematics educators face and the impact of their work on the professional life of the teachers in the future are discussed.

In this paper, we present the design and implementation of a microworld as a possible learning environment for the concept of conservation of area and its measurement. This microworld is the result of the synthesis of three models. The first is the model of the subject matter. The second is a model of children's sensory-motor actions while they face specific measurement tasks. The third is a model of learning viewed as an active, subjective and constructive process. All these models are enriched by the features of the electronic media which have an impact on the children's cognitive development. The modelling is the result of the study of past research in the above areas but also reflects the designers' beliefs about the nature of mathematics, its teaching and learning, and about the ways that computers can be used in the teaching and learning of mathematics. The requirements for the design of the software emerged from the educational requirements defined as a result of the modelling. The aim of this microworld is to provide the students with a set of tools to create their own objects (shapes) and transform or compare them using measurement or conservation concepts. The rationale of the design of this computer environment and an analysis of its main features are presented. Finally, the design of the pilot study of evaluation of the microworld is presented and the results of this study are discussed.

This study focuses on 12 years old children's approaches to area measurement in a project environment. These approaches are not explored through a specific set of mathematical tasks. The tasks, here, are defined through researchers' and children's interactions in a classroom. The children by working in small groups are asked to make a proposal about the location and the form of an area which would be given to them for their leisure activities. This environment defines different contexts where the children act and consider different aspects of the area measurement. These aspects are identified and compared among the three groups of children. The study has shown that the concept of area measurement carries different cultural dimensions for the children. Moreover, the children use those elements of the concept which fit in with their personal experience and the tasks they have to face.

This chapter is a response to the three chapters of Fermandez & Zilliox, Murata & Pothen, and Yu based on Lesson Study at pre-service teacher education. The chapter is structured around three dimensions: the way that Lesson Study has been implemented in the three studies; the prospective teachers’ learning about teaching mathematics; and the main factors that framed learning. Common approaches but also different perspectives and ways of implementation were identified. Moreover, a number of issues emerged related to the relation of theory and practice, the role of teacher educator and group collaboration. Finally, some ideas about future research and development in the context of Lesson study are discussed.

Around a quarter of WG17 papers focused on mathematical content knowledge needed for teaching (MCKT), trying to define it in relation to characteristics of the mathematical knowledge of other profe...

The reorganization and reform of mathematics courses in Scottish Colleges of Further Education is described. The teaching and learning approaches for the new courses are discussed and an evaluation is made of their implementation in one such course.

This study explores the arguments used by 14-year-old students in making decisions about the design of a road in their area. The whole activity was based on an actual problem at the time the research took place. The procedure was a sequence, where students worked first individually, then in groups and finally they had to take a class decision. Then they had to realize one of the final proposals and design the construction of a bridge that their planning had involved. This paper first elaborates the general perspective of such an approach; it then describes the process of argumentation and analyses the nature of the students' arguments, which are discussed on the basis of a specifically constructed network.

The research reported is an attempt to explore 11-year-old children's approaches to the concept of volume. Six tasks have been devised to identify any commonalities in children's responses. Results suggest that children hold and use different conceptions in their effort to explain and compare aspects of volume. However, these conceptions fall in certain categories which are met in almost all the tasks: volume is the space occupied, the capacity, related to the material substance, related to the geometrical properties, related to the weight and indefinable. Children's comparing actions (justifications and strategies) have also been analyzed and characterized as tautological, phenomenological, experimental, or as referred to the properties of the object. Common patterns in children's conceptions and justifications across the different tasks showed a variety of possible approaches to volume which will probably be met in other contexts and are presented in the form of a systemic network. As our study indicates, the concept of volume is related to a number of factors like the shape and geometrical properties, the nature of the matter, the mass and the weight, the hollowness, and the openness of the object. This complexity is not appreciated in teaching where the concept of volume is approached in a fragmented way and without paying attention to the referent. It is suggested that appropriate integrated activities could facilitate children to construct and unify a plurality of ideas in order to form a “mature concept” of volume. © 1996 John Wiley & Sons, Inc.

International audience

The process of interpretation and assessment of students’ mathematical productions represents a crucial aspect of teachers’ practices. In such processes, teachers rely on the so-called interpretative knowledge, which includes particular aspects of their mathematical and pedagogical knowledge, their view of mathematics, and their values. In this paper, we analyze and discuss prospective primary teachers’ interpretative knowledge gained through their assessment of different subtraction algorithms.