Changed views on mathematical knowledge in the course of didactical theory development—independent corpus of scientific knowledge or result of social constructions?

The study tries to show one line of how the German didactical tradition has evolved in response to new theoretical ideas and new—empirical—research approaches in mathematics education. First, the classical mathematical didactics, notably ‘stoffdidaktik’ as one (besides other) specific German tradition are described. The critiques raised against ‘stoffdidaktik’ concepts [for example, forms of ‘progressive mathematisation’, ‘actively discovering learning processes’ and ‘guided reinvention’ (cf. Freudenthal, Wittmann)] changed the basic views on the roles that ‘mathematical knowledge’, ‘teacher’ and ‘student’ have to play in teaching–learning processes; this conceptual change was supported by empirical studies on the professional knowledge and activities of mathematics teachers [for example, empirical studies of teacher thinking (cf. Bromme)] and of students’ conceptions and misconceptions (for example, psychological research on students’ mathematical thinking). With the interpretative empirical research on everyday mathematical teaching–learning situations (for example, the work of the research group around Bauersfeld) a new research paradigm for mathematics education was constituted: the cultural system of mathematical interaction (for instance, in the classroom) between teacher and students.     

Networking strategies and methods for connecting theoretical approaches: first steps towards a conceptual framework

The article contributes to the ongoing discussion on ways to deal with the diversity of theories in mathematics education research. It introduces and systematizes a collection of case studies using different strategies and methods for networking theoretical approaches which all frame (qualitative) empirical research. The term ‘networking strategies’ is used to conceptualize those connecting strategies, which aim at reducing the number of unconnected theoretical approaches while respecting their specificity. The article starts with some clarifications on the character and role of theories in general, before proposing first steps towards a conceptual framework for networking strategies. Their application by different methods as well as their contribution to the development of theories in mathematics education are discussed with respect to the case studies in the ZDM-issue.     

How is cultural influence manifested in the formation of mathematics textbooks? A comparative case study of resource book series between Shanghai and England

Focusing on issues about the development of mathematics textbooks from a cultural perspective, this study examined a widely-used curriculum resource series, One Lesson One Exercise, published in China, and its adapted English series, published in the UK, to explore how cultural influence is manifested in the two series of resource books. For the study we established a conceptual framework classifying culture into six types in relation to people’s beliefs, values and ways of interacting about them, for data collection and analysis. The results indicate that there exist considerable differences between the Chinese and the English series that are related to cultural factors. It appears that, to a large extent, culture plays an essential role in the development of mathematics textbooks. Concerning the different types of culture, the results show that most adaptations between the Chinese series and the English series are related to ‘ways of behaving and customs’ and ‘artifacts, flora and fauna’, followed by ‘identities’ and ‘geography’, and the least are related to ‘organisations’ and ‘history’. Based on the study, we argue that the relevance and importance of culture to the development of mathematics textbooks must not be underestimated, and more research in this direction is needed.     

‘Re’-presenting qualitative data from multilingual mathematics classrooms

In this paper I consider what it means to ‘re’-present qualitative data from multilingual mathematics classrooms. I draw from a recent study that focused on language practices in multilingual mathematics classrooms to explore the different levels involved in the ‘re’-presentation of multilingual data. The purpose of the paper is not to discuss the details of the study but to use data from the study to raise the awareness of the conceptual underpinnings of data re-presentation in mathematics education research. I use the data to show one perspective to ‘re’-presentation of multilingual data. The main argument of the paper is that ‘re’-presentation of multilingual data is not just talk written down, it is inevitably a process of selection and is informed by theory, research questions, tools of analysis and the purposes of re-presenting the data.     

Mexican high school students' social representations of mathematics,its teaching and learning

This paper reports a qualitative research that identifies Mexican high school students’ social representations of mathematics. For this purpose, the social representations of ‘mathematics’, ‘learning mathematics’ and ‘teaching mathematics’ were identified in a group of 50 students. Focus group interviews were carried out in order to obtain the data. The constant comparative style was the strategy used for the data analysis because it allowed the categories to emerge from the data. The students’ social representations are: (A) Mathematics is…(1) important for daily life, (2) important for careers and for life, (3) important because it is in everything that surrounds us, (4) a way to solve problems of daily life, (5) calculations and operations with numbers, (6) complex and difficult, (7) exact and (6) a subject that develops thinking skills; (B) To learn mathematics is…(1) to possess knowledge to solve problems, (2) to be able to solve everyday problems, (3) to be able to make calculations and operations, and (4) to think logically to be able to solve problems; and (C) To teach mathematics is…(1) to transmit knowledge, (2) to know to share it, (3) to transmit the reasoning ability, and (4) to show how to solve problems.     

Dragging as a conceptual tool in dynamic geometry environments

Some ‘drag-to-fit’ solutions given by student teachers to three geometric construction problems in a dynamic geometry environment (DGE) are analysed. The responses of a group of experienced mathematics teachers to the question whether or not such solutions can be considered ‘legitimate’ are then discussed. This raises fundamental questions concerning the concept of legitimacy, the relationship between DGEs and Formal Axiomatic Euclidean Geometry, the nature of ‘conceptual tools’ in different geometric environments, and the functions of dragging in DGEs. The authors argue that, if dragging is viewed as a conceptual tool, then certain drag-to-fit solutions, although soft constructions, may still be considered as conceptually legitimate and therefore valid. Finally, some important questions are raised concerning the impact that teachers’ different attitudes towards legitimacy might have on students’ learning through DGEs.     

A Case Study in Community Care Using Systems Thinking

This paper uses a recent application of System Dynamics in community care to contribute to a clarification of the role of System Dynamics as a framework for strategic debate. A revised framework for System Dynamics within a philosophy of ‘modelling as learning’ is summarized. This framework is now referred to as Systems Thinking and combines knowledge acquisition and both qualitative and quantitative modelling, supported by microworlds and archetypes to aid dissemination of insights. The emphasis of the paper is on using qualitative models interactively with different groups of participants in the problem domain. The purpose is to assist the development of a shared understanding of how culture, power and politics combine to affect the behaviour of a process, when subjected to externally superimposed changes in responsibility.     

Intervening in the Cultural Constitution of Systems—Methodological Complementarism and other Visions for Systems Research

This paper uses a culture perspective to examine the complementarist idea of combining hard, soft and critical methodologies in systems research. The culture perspective is adopted because it reminds us that all research is underpinned by socially constructed meanings that often severely constrain the range of options that are possible. It is particularly appropriate, therefore, in the context of a situation where systems researchers are being urged to develop their research capabilities in distinctive ways. The future prospects for complementarism are examined in relation to two major factors—its cultural ‘content’, and the overall level of receptiveness to culture change in the systems community.     

Quantifying vagueness in decision models

When modelling specific decision situations the decision maker often feels overstrained when he is asked for precise numerical quotations concerning the objectives or the constraints, whereas qualitative statements are easily given. In the recent past the theory of fuzzy sets has proven to be very useful for representing this type of information. Though it is quite advanced formally, the practical determination of its core elements, i.e. membership functions and operators, has only been explored to a very limited extent. This paper presents results of empirical research which focused on the problem of modelling vagueness, i.e. determining membership functions of fuzzy sets which are considered as quantitative representations of vague concepts such as ‘young man’, ‘long sticks’, ‘high profits’, etc.     

Pupils’ attitudes towards mathematics: a comparative study of Norwegian and English secondary students

Comparing English and Norwegian pupils’ attitude towards mathematics, in this article I develop a deeper understanding of the factors that may shape and influence ‘pupil attitude towards mathematics’, and argue for it as a socio-cultural construct embedded in and shaped by students’ environment and context in which they learn mathematics. The theoretical framework leans on work by Zan and Di Martino (The Montana Mathematics Enthusiast, Monograph 3, pp. 157–168, 2007) to elicit Norwegian and English pupils’ attitude of mathematics as they experience it in their respective environments. Whilst there were differences which could be seen to be accounted for by differently ‘figured’ environments, there are also many similarities. It was interesting to see that, albeit based on a small statistical sample, in both countries students had a positive attitude towards mathematics in year 7/8, which dropped in year 9, and increased again in years 10/11. This result could be explained and compared with other larger scale studies (e.g. Hodgen et al. in Proceedings of the British Society for Research into Learning Mathematics. 29(3), 2009). The analysis of pupils’ qualitative comments (and classroom observations) suggested seven factors that appeared to influence pupil attitude most, and these had ‘superficial’ commonalities, but the perceptions that appeared to underpin these mentions were different, and could be linked to the environments of learning mathematics in their respective classrooms. In summary, it is claimed that it is not enough to identify the factors that may shape and influence pupil attitude, but more importantly, to study how these are ‘lived’ by pupils, what meanings are made in classrooms and in different contexts, and how the factors interrelate and can be understood.     

Can we have confidence in generic structures?

The system dynamics concept of ‘generic structure’ is dividable into three sub-types. This paper analyses the validity of these three, using both practical and theoretical perspectives. Firstly, a new set of measures is developed for generating validity—‘confidence’—amongst a group using generic structures in a practical modelling situation. It is concluded that different confidence criteria are implicitly employed; there is an argument for trading-off model precision and analytical quality for simplicity and ease of use and future research is needed to combine these ‘process’ and ‘content’ aspects of confidence. From a theoretical stance it is shown that with two of the sub-types a scientific notion of confidence is achievable whereas the third (‘archetypes’) involves merely metaphorical thinking. It is concluded that the theoretical status of archetypes requires further development, whilst ensuring that its benefits are retained.     

Causal study of low stakeholder engagement in healthcare simulation projects

Stakeholder engagement plays a fundamental role in the success of ‘operational research’ initiatives including simulation projects. However, there is little empirical evidence of real engagement in the context of healthcare simulation. This paper principally examines this issue and aims to provide insights into the possible causes. The paper reports on the results of a literature review and 10 field studies within the UK healthcare settings, supplemented with the authors’ experience in order to arrive at an initial list of the causes, which will then be tested through a survey of expert opinions. Twelve primary and 26 secondary causal factors, which received statistically significant level of agreement from the experts, are presented in a fish-bone diagram. The findings indicate that communication gap between simulation and stakeholder groups is the top primary factor contributing the most to the poor stakeholder engagement in healthcare simulation projects, followed by ‘poor management support’, ‘clinician’s high workload’ and ‘failure in producing tangible and quick results’.     

Understanding kindergarten teachers’ perspectives of teaching basic geometric shapes: a phenomenographic research

Geometry is one of the disciplines children involve within early years of their lives. However, there is not much information about geometry education in Turkish kindergarten classes. The current study aims to examine teachers’ perspectives on teaching geometry in kindergarten classes. The researchers inquired about teachers’ in-class experiences in geometry and asked a series of questions such as “what are the benchmarks in your kindergarten class?”; “what kind of tools and materials you use to teach geometry in your class?”; “what shape do you teach first in your kindergarten class?”; “what do you expect to hear when you asked your students ‘what is square’?”; “how do you teach rectangular?”. The study utilized one of the qualitative research methods, namely phenomenography, to collect the data and analyze the data. The study involved with eight kindergarten teachers who work in different schools in central Kutahya, Turkey. The researchers collected data by conducting face-to-face half-structured interviews. The findings of this phenomenographic research showed that kindergarten teachers have some difficulties in teaching geometry and have lack of knowledge and skills in teaching geometry in kindergarten classes.     

Recent advances in automatic classification

Traditional classification methods are divided into two broad types: hierarchical methods and non-hierarchical methods in which the number of classes has to be fixed in advance. Both methods can handle both quantitative and qualitative data. A third type finds a partition optimizing a linear classification criterion (e.g. the Condorcet criterion) in which the number of classes does not have to be fixed in advance, but the data must be qualitative. A recent generalization, the ‘S theory’ can handle simultaneously both quantitative and qualitative data, and both linear and non-linear classification criteria (in the space of paired comparisons of elements). With this ‘S theory’ the partition is obtained in order n (in terms of memory space and elementary operations), n being the number of elements to classify.     

基于质量安全投入的乳制品企业与经销商合作沟通机制形成的演化博弈分析

从我国乳制品企业和经销商尽快建立规范、高效的沟通机制的客观要求出发,本文探讨了乳制品企业和经销商之间的演化博弈关系。在阐述乳制品企业和经销商合作策略的基础上,基于有限理性,应用演化博弈模型分析基于质量安全投入的乳制品企业和经销商合作沟通机制形成的长期演化趋势,并运用Matlab 6.0软件对研究结果进行仿真分析。研究发现,降低经销商的“质量安全投入”成本、降低乳制品企业采取“建立沟通机制”所支付的成本、提高经销商的销售额外收益、增加在“质量安全不投入”策略下乳制品企业采取“不建立沟通机制”所多付出的损失成本、增加在“建立沟通机制”策略下经销商采取“质量安全不投入”所需缴纳的罚金、增加在“不建立沟通机制”策略下经销商给乳制品企业带来的损失,有助于乳制品企业和经销商合作沟通机制的形成。     

CHILDREN'S PROPORTIONAL REASONING AND TENDENCY FOR AN ADDITIVE STRATEGY: THE ROLE OF MODELS

In this paper we report on 10 –14 year old children's strategies while solving two versions of ratio and proportion tasks: one ‘with models’ thought to facilitate proportional reasoning and one ‘without’. Rasch methodology was used to develop ‘with’ and ‘without models’ test versions which were given to a linked sample involving 673 children. We examine the pupils’ additive errors, their effect on ratio reasoning and how contingent on ‘model’ presentation this is. First, we provide a single scale on which pupils, item-difficulty and additive errors can be located. We then provide a new scale constructed from the error prone items, which we name the ‘tendency for additive strategy’. The measurement data is supported by qualitative data showing that the presence of ‘models’ can sometimes affect children's strategies, both positively and negatively but rarely makes a significant measurement difference on this, untutored, sample.     

On the importance of mathematical play

A number of existing theories and proposals for the meaning and characteristics of ‘play’ are considered before the authors suggest six characteristics of mathematical play, including the idea that it is not confined to childhood. Previous studies provide evidence for relating play to cognitive gain while the place of mathematical play in research activities is illustrated by describing a mathematician's approach to a number investigation from the classroom-The Six Circles. The problem-solving process for the Six Circles and observations of students solving calculator and integration problems are analysed in relation to theories of play and cognitive gain and also considered from the perspective of the students' experience. Piaget's theory for the assimilation and accommodation of new information and Davis's view of play as ‘space to support learning’ are reflected in the authors' rationale for suggesting that open questions and mathematical play provide opportunities for students to develop their own conjectures, with no threat of failure, and provide a foundation for mathematical learning. Some difficulties of implementing a ‘play’ approach in the classroom are discussed and further research questions proposed.     

A networking method to compare theories: metacognition in problem solving reformulated within the Anthropological Theory of the Didactic

An important role of theory in research is to provide new ways of conceptualizing practical questions, essentially by transforming them into scientific problems that can be more easily delimited, typified and approached. In mathematics education, theoretical developments around ‘metacognition’ initially appeared in the research domain of Problem Solving closely related to the practical question of how to learn (and teach) to solve non-routine problems. This paper presents a networking method to approach a notion as ‘metacognition’ within a different theoretical perspective, as the one provided by the Anthropological Theory of the Didactic. Instead of trying to directly ‘translate’ this notion from one perspective to another, the strategy used consists in going back to the practical question that is at the origin of ‘metacognition’ and show how the new perspective relates this initial question to a very different kind of phenomena. The analysis is supported by an empirical study focused on a teaching proposal in grade 10 concerning the problem of comparing mobile phone tariffs.     

In Defence of Powerful Qualities

The ontology of ‘powerful qualities’ is gaining an increasing amount of attention in the literature on properties. This is the view that the so-called categorical or qualitative properties are identical with ‘dispositional’ properties. The position is associated with C.B. Martin, John Heil, Galen Strawson and Jonathan Jacobs. Robert Schroer (2012) has recently mounted a number of criticisms against the powerful qualities view as conceived by these main adherents, and has also advanced his own (radically different) version of the view. In this paper I have three main aims: firstly, I shall defend the ontology from his critique, arguing that his criticisms do not damage the position. Secondly, I shall argue that Schroer’s own version of the view is untenable. Thirdly, the paper shall serve to clear up some conceptual confusions that often bedevil the powerful qualities view.