Differentiation of sets – The general case

In recent work by Khmaladze and Weil (2008) and by Einmahl and Khmaladze (2011), limit theorems were established for local empirical processes near the boundary of compact convex sets K   in R^d${\mathbb{R}}^d$. The limit processes were shown to live on the normal cylinder Σ of K, respectively on a class of set-valued derivatives in Σ. The latter result was based on the concept of differentiation of sets at the boundary ∂K of K, which was developed in Khmaladze (2007). Here, we extend the theory of set-valued derivatives to boundaries ∂F   of rather general closed sets F⊂R^d$F\subset {\mathbb{R}}^d$, making use of a local Steiner formula for closed sets, established in Hug, Last and Weil (2004).     

The intangible task – a revelatory case of teaching mathematical thinking in Japanese elementary schools

This paper deals with the nature of teaching mathematical thinking and presents a case study of a single Japanese lesson where the characteristics of mathematical thinking and the teaching thereof are identified in relation to multiplication. The raison d’être for this teaching is questioned and investigated by looking at how multiplication is described in the curriculum and representative textbook material. It is seen how Japanese teachers are institutionally conditioned to incorporate mathematical thinking in the context of multiplication, something which may appear in contrast to other countries. The lesson is analysed using the notion of praxeologies and didactic co-determination conceptualised in the Anthropological Theory of the Didactic.     

The use of simulation and case study†

Although statistics is a rational subject, it does not follow that students can or do learn statistics in a rational way. Students learn statistics in the pattern‐forming mode more than in the rational mode. The paper describes and compares the two modes, and identifies three conditions that prevent or discourage learning in the rational mode. The diagnosis of pattern‐forming de‐emphasizes mathematical derivations and proofs as a means of allowing students to understand statistical concepts. The primary recommendation is that the teacher should deliberately manipulate the pattern‐forming process. The underlying idea is for the teacher to view the process for what it means to the student.     

Validating a concept inventory for measuring students’ probabilistic reasoning: The case of reasoning within the context of a raffle

Assessing students’ conceptions related to independence of events and determining probabilities from a sample space has been the focus of research in probability education for over 40 years. While we know a lot from past studies about predictable ways students may reason with well-known tasks, developing a diagnostic assessment that can be used by teachers to inform instruction demands the use of familiar and unfamiliar contexts. This paper presents the current work of a research team whose aim is to create a formative concept inventory with strong evidence of validity that uses a psychometric model to confidently predict whether a student exhibits one or more misconception across many items. We illustrate this process in this paper using a particular item with a context of a raffle aimed to measure whether a student reasons with misconceptions related to independence or equiprobability. The results of two aspects of the validity process: cognitive interviews to assess response processes on individual items, and a large-scale administration to examine internal structure of the concept inventory revealed difficulties in assessing students’ reasoning about these key probability concepts and trends in the prevalence of misconceptions across grades. Results can provide guidance for others aiming to develop assessments in mathematics education and also support further possibilities for research into understanding students’ reasoning about independence and sample space.     

Application of three controls optimally in a vector-borne disease – a mathematical study

We have proposed and analyzed a vector-borne disease model with three types of controls for the eradication of the disease. Four different classes for the human population namely susceptible, infected, recovered and vaccinated and two different classes for the vector populations namely susceptible and infected are considered. In the first part of our analysis the disease dynamics are described for fixed controls and some inferences have been drawn regarding the spread of the disease. Next the optimal control problem is formulated and solved considering control parameters as time dependent. Different possible combination of controls are used and their effectiveness are compared by numerical simulation.     

‘Experimental pedagogy’ in Germany,elaborated for mathematics – a case study in searching the roots of PME

PME, the International Group for the Psychology of Mathematics Education, was founded in 1976, at the Third International Congress on Mathematical Education in Karlsruhe, organised by the International Commission on Mathematics Instruction (ICMI). While PME is thus beyond coming of age and is reflecting its further orientation – due to the present “social turn” – the origins of investigating psychological aspects of mathematics learning have not yet been systematically studied. I am undertaking here a first such approach, concentrating on Germany, where the first pertinent monographs were published in 1913 and 1916. Different endeavours, focussing in particular on the notion of error, merged into the characteristic approach of ‘experimental pedagogy’. Given the key function of ICMI for founding PME, an additional aspect is whether the forerunner of ICMI: the Internationale Mathematische Unterrichtskommission (IMUK), founded in 1908, had an impact upon promoting research into the psychology of mathematics education. The pertinent research was effected by psychology; doing research themselves was still outside the horizon of mathematics educators. Perspectives of future research, in particular comparative ones, are outlined.     

Correlation of experimental flap-lag-torsion damping—a case study

In 1982, carefully done experiments measured the flap-lag-torsion damping of hingeless rotor at various RPM's and for several combinations of flexure geometry and stiffness. The best elastic-blade stability codes of that time were unable to predict this damping and typically gave errors of the order of 100% (as percentage of the largest measured damping, 5 rad/s). Through the past ten years, various unsuccessful efforts have been made to improve those predictions. Now, improved correlation has been performed with errors of 0% to 20% of 5 rad/s. The results show that most of the former discrepancies can be attributed to aerodynamic effects, the largest of which is the dynamics of the three-dimensional wake.     

Interactive policy-making – a model of management for public works

In this article a management model for interactive policy-making is proposed. Interactive policy-making is a process whereby multiple parties play an active role and jointly arrive at a decision. The management model consists of six stages: exploration, initiative, common perception, joint problem-solving, decision-making, and implementation. The activities assigned to each stage are examined in detail. Finally, the last section of this article reviews the criteria that can be used to assess interactive policy-making. Three perspectives are relevant. The first perspective is the course of the process; the second is democratic legitimacy; and the third is problem resolution.     

Changes of Working Styles in a Computer Algebra Environment – The Case of Functions

This study is an empirical investigation of 11th graders at a German high school (Gymnasium). Working over a 24-hour period in a computer lab, we investigated students' use of quadratic functions with `Derive', and trigonometric functions with `Mathplus' (or `Theorist' for Macintosh). We were particularly interested in the working styles of students while they solved problems and looked for changes in these styles, as compared to traditional paper and pencil activities. While students worked on the computer, their activities (such as inputs from the keyboard, menu choices or mouse movements) were saved by a special program, which ran in the `background'. We are interested in the possibilities of developing a research method based on these `computer protocols'. The study should be seen as an exploratory study for developing hypotheses for further empirical investigations.This revised version was published online in September 2005 with corrections to the Cover Date.     

Comparison of various concepts of a random attractor:¶ A case study

Several non-equivalent definitions of an attractor of a random dynamical system have been proposed in the literature. We consider a rather simple special case: random dynamical systems with state space [0, ￥) [0, \infty) which fix 0. We examine conditions under which the set {0} is an attractor for three different notions of an attractor. It turns out that even in this simple case the various concepts are quite different. The purpose of this note is to highlight these differences and thus provide a basis for discussion about the "correct" concept of a random attractor.     

Hypoellipticity in spaces of ultradistributions—Study of a model case

In this work we study C ^??-hypoellipticity in spaces of ultradistributions for analytic linear partial differential operators. Our main tool is a new a-priori inequality, which is stated in terms of the behaviour of holomorphic functions on appropriate wedges. In particular, for sum of squares operators satisfying H?rmander??s condition, we thus obtain a new method for studying analytic hypoellipticity for such a class. We also show how this method can be explicitly applied by studying a model operator, which is constructed as a perturbation of the so-called Baouendi-Goulaouic operator.     

The role of innovations in secondary school performance – Evidence from a conditional efficiency model

This paper studies the influence of educational innovations on school performance. We apply a tailored, fully nonparametric conditional efficiency model to study secondary school efficiency in the Netherlands. The application uses official school data and a self-collected questionnaire on recent innovations in schools. In the nonparametric model, it is assumed that schools aim to maximize educational attainments of students under a budget constraint. The results suggest that innovations are positively related to efficiency. We find that profiling, pedagogic, process and education chain innovations are significantly related to school efficiency, whereas innovations in the professionalization of teachers are insignificantly related to school efficiency. Furthermore, the number of locations per school and the number of schools per governing body are negatively and significantly related to school efficiency. School type and region significantly influence school efficiency, whereas share of disadvantaged students, degree of urbanization and student/teacher ratio do not have significant influence.     

The value of an integral in Gradshteyn and Ryzhik’s table

The Ramanujan Journal - In 2010, V. H. Moll observed that entry 3.248.5 in the sixth edition of Gradshteyn and Ryzhik’s table of integrals was incorrect, and he asked for...