Eliminating counterexamples: A Grade 8 student’s learning trajectory for contrapositive proving

This study uses a teaching experiment and retrospective analysis to develop a learning trajectory for improving a Grade 8 student’s ability to construct, critique, and validate contrapositive arguments. The study is predicated on the hypothesis that adolescents perform poorly on contrapositive reasoning tasks because they lack sufficient ways of justifying contrapositive argumentation as a viable mode of argumentation. By studying a student’s actions and comments as she develops, critiques, and validates not-the-conclusion-implies-the-conditions-are-impossible arguments for conditional claims, a promising learning trajectory for contrapositive argumentation is developed. The student’s learning trajectory demonstrates how a conception of contrapositive proving as eliminating counterexamples can be useful in developing, critiquing, and validating contrapositive arguments.     

A counterexample to Sorenson’s conjecture: The finitely generated case

It is shown that there exists a finitely generated, left amenable semigroup which is right cancellative but not left cancellative. This resolves negatively a question raised by M. Klawe.     

Allocating nursing staff to hospital wards— A case study

In this case study a system for estimating daily nursing workloads on hospital wards is described and its potential is investigated. The system is to be adopted by the local health authority, and some of the preliminary results have already been used. A key feature of the work has been the active involvement of nursing staff. This style of work is proposed as one approach to improving efficiency within the Public Sector whilst maintaining acceptable standards of care.     

Integrated Fuzzy AHP–TOPSIS for selecting the best plastic recycling method: A case study

Due to the rapid depletion of natural resources and undesired environmental changes in a global scale, it is necessary to conserve the natural resources and protect the environment. Industries which manufacture plastic based products have the necessity to recycle plastics. There are number of methods to recycle plastics. Since the selection of the best recycling method involves complex decision variables, it is considered to be a multiple criteria decision-making (MCDM) problem. This article develops an evaluation model based on the fuzzy Analytic Hierarchy Process (AHP) and the technique for order performance by similarity to ideal solution (TOPSIS) to enable the industry practitioners to perform performance evaluation in a fuzzy environment. The purpose of the study is to determine the best method for recycling plastics among the various plastic recycling processes. By observing the results, it is identified that mechanical recycling process is found to be the best plastic recycling process using the integrated approach.     

Aggregate–disaggregate approach to an airfreight forwarder’s planning under uncertainty: a case study

Resource planning of airfreight forwarders is a complex endeavor, requiring decisions to be made in a dynamic and uncertain environment. Airfreight forwarders acquire airfreight spaces from three sources: allotment from carriers, retail from carriers and subcontracting to partners, all of which differ in terms of cost, flexibility and timing of booking. This real-life problem has many planning decisions (bookings in terms of carriers, route, time, ULDs, etc.). In this case study, we propose an aggregate–disaggregate approach and focus on the most critical decisions. A two-stage stochastic dynamic program first determines, in aggregates, the amount of allotment bookings, retail resources, and subcontracting or surplus co-loading. Then, a heuristic is used to disaggregate resource requirements into specific bookings. An analysis is provided to examine the relevant managerial issues. Based on real-life data, we show several patterns of aggregate resource bookings with respect to different levels of demand uncertainty and cost parameters. We show that resource disaggregation has to balance cost-effectiveness, capacity flexibility and routing flexibility of a resource plan.     

A case study: assessment of preservice secondary mathematics teachers’ metacognitive behaviour in the problem-solving process

The purpose of the present study was to investigate preservice secondary mathematics teachers’ metacognitive behaviour in the mathematical problem-solving process. The case study methodology was employed with six preservice mathematics teachers, enrolled at one university in Ankara, Turkey. We collected data by using the think aloud method, which lasted for two sessions. It was found that there was no relationship between academic achievement and frequencies of metacognitive behaviour. However, the types of problems could affect these frequencies. Furthermore, there was no pattern in metacognitive behaviour with respect to achievement and type of problem.     

Assessing science students’ attitudes to mathematics: a case study on a modelling project with mathematical software

This article reports the attitudes of students towards mathematics after they had participated in an applied mathematical modelling project that was part of an Applied Mathematics course. The students were majoring in Earth Science at the National Taiwan Normal University. Twenty-six students took part in the project. It was the first time a mathematical modelling project had been incorporated into the Applied Mathematics course for such students at this University. This was also the first time the students experienced applied mathematical modelling and used the mathematical software. The main aim of this modelling project was to assess whether the students’ attitudes toward mathematics changed after participating in the project. We used two questionnaires and interviews to assess the students. The results were encouraging especially the attitude of enjoyment. Hence the approach of the modelling project seems to be an effective method for Earth Science students.     

A remark on Jeśmanowicz’ conjecture for the non-coprimality case

Let a, b, c be relatively prime positive integers such that a² + b² = c². Je?manowicz' conjecture on Pythagorean numbers states that for any positive integer N, the Diophantine equation (aN)^x + (bN)^y = (cN)^z has no positive solution (x, y, z) other than x = y = z = 2. In this paper, we prove this conjecture for the case that a or b is a power of 2.     

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.     

A “direct” method to prove the generalized Itô–Venttsel’ formula for a generalized stochastic differential equation

For the first time we present a complete proof (from the standpoint of stochastic analysis) of the generalized Itô–Venttsel’ formula whose ideas were adduced in [8]. The proposed proof is an approach to construct the generalized Itô–Venttsel’ formula based on the direct application of the generalized Itô formula and the theory of stochastic approximation in contrast to the proof presented in [17] and based on the method of the integral invariants of a stochastic differential equation.     

Application of Lyapunov’s direct method to the study of the stability of solutions to systems of impulsive differential equations

Classical theorems on the stability of the solutions of impulsive differential equations are further developed.     

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.     

A counterexample to a ‘simplex algorithm’ for convex bodies

A counterexample, in E ³, is given to the following conjecture. Suppose f ^* is a linear functional, and e an exposed point of a convex body K such that f ^* does not attain its maximum on K at e; then there is an f ^*-strictly increasing path in the one-skeleton of K emanating from e. The counterexample shows that a certain generalized simplex algorithm fails. Furthermore for a different linear functional f, there are no three disjoint f-strictly increasing paths in the one-skeleton of K leading to e.     

Limiting factor analysis and regulation for urban ecosystems—A case study of Ningbo,China

Considering the importance of supporting system for the urban ecosystem development, the urban ecological carrying capacity (UECC) representing the supporting functions of biophysical environment for urban development is introduced into the analysis of urban ecosystem degradation. It is proposed in this paper that the urban ecosystem degradation can be attributed to the imbalance between supply and demand of UECC, i.e. the conflict between the infinite demand of urban development and finite supply of biophysical supporting system, which can be mainly described by the dominant limiting factors of UECC. Through the shortage indices of various factors that denoted by the ratio of the gap between the demand and supply to the demand, the limiting factors of UECC can be identified and graded with different levels. The urban ecosystem degradation can thus be regulated by adjusting the relationship of supply and demand in terms of limiting factors from both the layers of technical practice and goal-based government management. As for Ningbo City, the shortage of water resources quantity, the pressure of SO₂ emission, insufficiency of sewage treatment and green area are the prominent limiting factors and corresponding regulation scheme is thereafter suggested to adjust the relationship between supply and demand of UECC.     

Erratum to: A Variational Approach to Givental’s Nonlinear Maslov Index

In this article we consider a variant of Rabinowitz Floer homology in order to define a homological count of discriminant points for paths of contactomorphisms. The growth rate of this count can be seen as an analogue of Givental’s nonlinear Maslov index. As an application we prove a Bott–Samelson type obstruction theorem for positive loops of contactomorphisms.     

Algorithmic contexts and learning potentiality: a case study of students’ understanding of calculus

The study explores the nature of students’ conceptual understanding of calculus. Twenty students of engineering were asked to reflect in writing on the meaning of the concepts of limit and integral. A sub-sample of four students was selected for subsequent interviews, which explored in detail the students’ understandings of the two concepts. Intentional analysis of the students’ written and oral accounts revealed that the students were expressing their understanding of limit and integral within an algorithmic context, in which the very ‘operations’ of these concepts were seen as crucial. The students also displayed great confidence in their ability to deal with these concepts. Implications for the development of a conceptual understanding of calculus are discussed, and it is argued that developing understanding within an algorithmic context can be seen as a stepping stone towards a more complete conceptual understanding of calculus.