The cultural dimension of beliefs: an investigation of future primary teachers’ epistemological beliefs concerning the nature of mathematics in 15 countries

Beliefs constitute a central part of a person’s professional competencies and are crucial to the perception of situations as they influence our choice of actions. This paper focuses on epistemological beliefs about the nature of mathematics of future primary teachers from an international perspective. The data reported are part of a larger sample originating from the TEDS-M study which compares primary mathematics teacher education in 15 countries. In this paper we examine the pattern of beliefs of future teachers aiming to teach mathematics at primary level. We explore whether and to what extent beliefs concerning the nature of mathematics are influenced by cultural factors, in our case the extent to which a country’s culture can be characterized by an individualistic versus collectivistic orientation according to Hofstede’s terminology. In the first part of the paper, the literature on epistemological beliefs is reviewed and the role of culture and individualism/collectivism on the formation of beliefs concerning the nature of mathematics will be discussed. In the empirical part, means and distributions of belief ratings will be reported. Finally, multilevel analyses explore how much of the variation of belief preferences between countries can be explained by the individualistic orientation of a country.     

Polish teachers’ conceptions of and approaches to the teaching of linear equations to grade six students: an exploratory case study

In this article we present an exploratory case study of six Polish teachers’ perspectives on the teaching of linear equations to grade six students. Data, which derived from semi-structured interviews, were analysed against an extant framework and yielded a number of commonly held beliefs about what teachers aimed to achieve and how they would achieve them. In general, teachers’ aims were procedural fluency founded on students understanding the equals sign as a relational rather than an operational entity and the balance scale as a representation supportive of students’ understanding of an equation as the equivalence of two expressions. The analyses also indicated that the ways teachers proposed to conduct their lessons, whereby they pose single problems for individual work before inviting whole class sharing of solutions, resonates with the didactical traditions found in other East and Central European countries previously influenced by the Soviet Union.     

Carpenter,tractors and microbes for the development of logical–mathematical thinking – the way 10th graders and pre-service teachers solve thinking challenges

The objective of this case study was to investigate the ability of 10th graders and pre-service teachers to solve logical–mathematical thinking challenges. The challenges do not require mathematical knowledge beyond that of primary school but rather an informed use of the problem representation. The percentage of correct answers given by the 10th graders was higher than that of the pre-service teachers. Unlike the 10th graders, some of whom used various strategies for representing the problem, most of the pre-service teachers’ answers were based on a technical algorithm, without using control processes. The obvious conclusion drawn from the findings supports and recommends expanding and enhancing the development of logical–mathematical thinking, both in specific lessons and as an integral part of other lessons in pre-service frameworks.     

Declining participation in post-compulsory secondary school mathematics: students’ views of and solutions to the problem

We analysed multivariable calculus students' meanings for domain and range and their generalisation of that meaning as they reasoned about the domain and range of multivariable functions. We found that students' thinking about domain and range fell into three broad categories: input/output, independence/dependence, and/or as attached to specific variables. We used Ellis' actor-oriented generalisations framework to characterise how students generalised their meanings for domain and range from single-variable to multivariable functions. This framework focuses on the process of generalisation – what students see as similar between ideas in multiple contexts. We found that students generalised their meanings for domain and range by relating objects, extending their meanings, using general principles and rules, and using/modifying previous ideas. Our findings suggest that the domain and range of multivariable functions is a topic instructors should explicitly address.     

From design-based research to re-sourcing ‘in the wild’: reflections on studies of the co-evolution of mathematics teaching resources and practices

This commentary paper looks across the studies of the design and use of mathematics teaching resources included in this issue. It analyses everyday and educational notions of resource; particularly how usage of that term varies across the papers. Key characteristics of each study are identified and the studies are organized into five broad groups representing different lines of investigation of resource design and use. Across the studies, the broad notion of appropriation is influential in conceptualizing user/tool relations, with the instrumental/documentational approach particularly prominent, although close inspection shows that researchers draw on this in different ways. A majority of the studies relate to professional development interventions in a context of large-scale reform efforts, offering insights into a range of approaches to providing practical support for teacher appropriation of resources. One study investigates the resource systems established by ordinary teachers: alongside other research, this highlights the challenges of a re-sourcing approach in which teachers collaborate to curate a localized resource system, and points to conditions which are conducive to the success of such efforts.     

Existence,uniqueness and asymptotic behavior of the solutions to the fully parabolic Keller–Segel system in the plane

In the present article we consider several issues concerning the doubly parabolic Keller–Segel system  and  in the plane, when the initial data belong to critical scaling-invariant Lebesgue spaces. More specifically, we analyze the global existence of integral solutions, their optimal time decay, uniqueness and positivity, together with the uniqueness of self-similar solutions. In particular, we prove that there exist integral solutions of any mass, provided that ε>0$\epsilon >0$ is sufficiently large. With those results at hand, we are then able to study the large time behavior of global solutions and prove that in the absence of the degradation term (α=0)$(\alpha =0)$ the solutions behave like self-similar solutions, while in the presence of the degradation term (α>0)$(\alpha >0)$ the global solutions behave like the heat kernel.     

Spreading the gospel of management science: operational research in Iron and Steel, 1950–1970

This paper builds on the author's earlier work on the history of operational research by presenting an analysis of the development of the discipline in Iron and Steel, an industry long regarded as one of the outstanding pioneers in the application of management science to decision-making processes. The contribution of Sir Charles Goodeve and BISRA to the diffusion and practice of operational research is well-documented. Less well known is the reaction to operational research within the managerial hierarchies of the private sector iron and steel companies. In the light of the development of dedicated operational research groups by the leading companies after 1950, it might be assumed that the industry was highly receptive to the discipline in terms of its perceived benefits. The present paper questions this assumption by highlighting the problems encountered by operational researchers in two of the largest Iron and Steel Companies which gave every appearance of being at the forefront of the practice of management science in British industry. Where appropriate, the paper draws contrasts and comparisons with the development of operational research within the nationalised Coal Industry.     

Nash bargaining and leader–follower models in water allocation: Application to the Zarrinehrud River basin,Iran

Mathematical models for conflict resolution are very important in integrated water resources and environmental management. This study proposes a new methodology to resolve conflicts among different water users and water suppliers while considering environmental requirements and the system’s constraints. A two-level leader–follower model is applied to maximize the net benefit with the Iran Water Resources Management Company as the leader and agricultural, domestic, and industrial users as followers subject to the system’s constraints. As a comparison, the Nash bargaining solution is also used to find a solution when simultaneous moves are assumed by the participants. The suggested method is then applied to the real case of the Zarrinehrud River basin that is one of the areas facing water shortages in Iran. For the actual optimization, Genetic Algorithm is used in order to avoid local optimum. As the contribution of this study, the results show that benefits for the leader in the leader–follower model increased in comparison with the Nash bargaining solutions.     

Building teachers’ expertise in understanding, assessing and developing children’s mathematical thinking: the power of task-based, one-to-one assessment interviews

In this paper, we outline the benefits to teachers’ expertise of the use of research-based, one-to-one assessment interviews in mathematics. Drawing upon our research and professional development work with teachers and students in primary and middle years in Australia and the research of others, we argue that the use of the interviews builds teacher expertise through enhancing teachers’ knowledge of individual and group understanding of mathematics, and also provides an understanding of typical learning paths in various mathematical domains. The use of such interviews also provides a model for teachers’ interactions and discussions with children, building both their pedagogical content knowledge and their subject matter knowledge.     

Note: Alternative treatments of congestion in DEA – a response to the Cherchye,Kuosmanen and Post critique

As its title suggests, this paper constitutes a response to the critique by Cherchye, Kuosmanen and Post which is leveled at the paper “Comparisons and evaluations of alternative approaches to the treatment of congestion in DEA” by Cooper, Gu and Li. All three papers are in this same issue of the European Journal of Operational Research where the just-cited paper by Cooper, Gu and Li appears.     

Gas Near a Wall: Shortened Mean Free Path,Reduced Viscosity,and the Manifestation of the Knudsen Layer in the Navier–Stokes Solution of a Shear Flow

For the gas near a solid planar wall, we propose a scaling formula for the mean free path of a molecule as a function of the distance from the wall, under the assumption of a uniform distribution of the incident directions of the molecular free flight. We subsequently impose the same scaling onto the viscosity of the gas near the wall and compute the Navier–Stokes solution of the velocity of a shear flow parallel to the wall. Under the simplifying assumption of constant temperature of the gas, the velocity profile becomes an explicit nonlinear function of the distance from the wall and exhibits a Knudsen boundary layer near the wall. To verify the validity of the obtained formula, we perform the Direct Simulation Monte Carlo computations for the shear flow of argon and nitrogen at normal density and temperature. We find excellent agreement between our velocity approximation and the computed DSMC velocity profiles both within the Knudsen boundary layer and away from it.