Recursive and explicit rules: How can we build student algebraic understanding?

Differing perspectives have been offered about student use of recursive and explicit rules. These include: (a) promoting the use of explicit rules over the use of recursive rules, and (b) encouraging student use of both recursive and explicit rules. This study sought to explore students’ use of recursive and explicit rules by examining the reasoning of 25 sixth-grade students, including a focus on four target students, as they approached tasks in which they were required to develop generalizations while using computer spreadsheets as an instructional tool. The results demonstrate the difficulty that students had moving from the successful use of recursive rules toward explicit rules. In particular, two students abandoned general reasoning, instead focusing on particular values in an attempt to construct explicit rules. It is recommended that students be encouraged to connect recursive and explicit rules as a potential means for constructing successful generalizations.     

Feynman's Operational Calculi for Noncommuting Systems of Operators: Tensors,Ordered Supports,and Disentangling an Exponential Factor

In a recent paper, the authors presented the key ideas involved in their approach to Feynman's operational calculi for systems of not necessarily commuting bounded linear operators acting on a Banach space. The central objects of the theory are the disentangling algebra, a commutative Banach algebra, and the disentangling map which carries this commutative structure into the noncommutative algebra of operators. The study of properties of these disentangling maps will be pursued in this paper with an emphasis on (i) Feynman's formula for disentangling an exponential factor, and (ii) the effect on the disentangling map of ordered supports of some or all of the measures which govern the disentangling.     

The analysis of the understanding of the three-dimensional (Euclidian) space and the two-variable function concept by university students

Conceptual understanding is being emphasized in mathematics education. Students often have difficulty understanding the multi-variable function, a key concept. Based on the APOS theory, which analyzes the cognitive structures formed by individuals in learning a mathematical concept and produces components related to that learning, this study analyzes the conceptual understanding of three-dimensional spaces and two-variable functions by university students. The genetic decomposition of these concepts proposed by Trigueros and Martinez-Planell is also considered. The analyzes results revealed that only one student constructed the concept of three-dimensional space as an object within the framework of genetic decomposition. Some students could not relate the concepts of two-variable function and three-dimensional space. Students who could perform algebraic operations had problems related to geometric representation. This study suggests the refinement of genetic decomposition to include, e.g., mental construction steps for writing algebraic equations of special surfaces whose graphs are given in R³.     

Teaching ideas and activities for classroom: integrating technology into the pedagogy of integral calculus and the approximation of definite integrals

The purpose of this article is to offer teaching ideas in the treatment of the definite integral concept and the Riemann sums in a technology-supported environment. Specifically, the article offers teaching ideas and activities for classroom for the numerical methods of approximating a definite integral via left- and right-hand Riemann sums, along with midpoint and trapezoidal Riemann sums. The activities demonstrate innovative and creative ways of integrating technology, in particular, GeoGebra dynamic software, into the pedagogy of college-level integral calculus. It also provides, among other things, interesting and original teaching ideas incorporating technology, and an evaluation of these activities by the students themselves who experienced these activities with the GeoGebra dynamic software.     

Inspecting the foundations of claims about cognitive demand and student learning: A citation analysis of Stein and Lane (1996)

Educational research communities bear responsibility for establishing a substantial body of evidence to support claims that drive the field. For example, one commonly accepted claim is that there is a relationship between the cognitive demand of mathematical task enactments and students’ learning. One study that is often cited in association with this claim is Stein and Lane (1996), and in 44% of those citations, Stein and Lane (1996) is the sole reference. Citation analysis reveals that many of these claims go beyond the warrants provided by the Stein and Lane study, either by granting more confidence in the relationship than the study design allows or by phrasing the claim as a causal relationship between cognitive demand and student learning. A few other studies are occasionally cited in conjunction with Stein and Lane (1996) and are summarized in this article, but there remains a need for replication studies to provide better empirical support for claims about cognitive demand and student learning and to refine our shared understanding.     

Understanding and representing the arithmetic averaging algorithm: an analysis and comparison of US and Chinese students' responses

Analysing the responses of 311 sixth-grade Chinese students and 232 sixth-grade US students to two problems involving arithmetic average, this study explored students' understanding and representation of the averaging algorithm from a cross-national perspective. Results of the study show that Chinese students were more successful than US students in obtaining correct numerical answers to each of the problems, but US and Chinese students had similar cognitive difficulties in solving the second task. The difficulties were not due to their lack of procedural knowledge of the averaging algorithm, rather due to their lack of conceptual understanding of the algorithm. There were significant differences between the US and Chinese students in their solution representations of the two average problems. Chinese students were more likely to use algebraic representations than US students; while US students were more likely to use pictorial or verbal representations. US and Chinese students' use of representations are related to their mathematical problem-solving performance. Students who used more advanced representations were better problem solvers. The findings of the study suggest that Chinese students' superior performance on the averaging problems is partly due to their use of advanced representations (e.g. algebraic).     

A typological analysis: understanding pre-service teacher beliefs and how they are transformed

This article describes the beliefs and their transformations of members of a cohort of early-childhood, elementary and middle-level pre-service teachers (PSTs) as they professionally develop. A typological analysis of both quantitative and qualitative data collected between August 2011 and May 2013 was utilized to categorize how 40 PSTs’ beliefs transformed throughout their formal teacher preparation. Five typologies were identified, showing variation in how PST beliefs transform or remain static.

Among the findings, strong support related to the development of innovative beliefs during coursework coupled with at least one transformative experience where innovation was observed ‘working’ in the field were sufficient for the transformation to innovative beliefs, despite potential constraints by supervisors, cooperating teachers and/or mandated curricula (Typology 3). Another finding revealed disguised growth toward innovation among those in Typology 5, who reported being innovative and having productive beliefs but described extremely traditional practices. Implications call for improved connections between mathematics methods professors and field supervisors, particularly during clinical internships when PSTs are no longer enrolled in methods courses, to enhance PSTs’ productive struggle in their development of innovative beliefs (T3) and to increase opportunities for disconnects between innovative beliefs and traditional practices to be made explicit and negotiated (T5).     

Education campaign and family understanding affect stability and qualitative behavior of an online game addiction model for children and youth in Thailand

Online game addiction has become a large problem worldwide, and it could give negative impacts on children in many ways such as physical health, learning, emotion, and behavior. Online game addiction is mostly found in children and youth at the age of 15 to 24 years. Therefore, in this paper, we have developed a mathematical model of online game addiction to explore the effects of education campaign and family understanding on online game addiction in Thailand. Analysis of this model reveals two main equilibria, addiction‐free (AFE) and addiction‐present (APE) ones. Results show that the AFE is locally asymptotically stable when the value of basic reproduction number (R₀) is less than the unity; otherwise, there is a unique endemic equilibrium point, and it is locally stable when satisfies the Routh‐Hurwitz criterion. Further, the conditions of AFE to be globally stable are demonstrated. Finally, the results of sensitivity analysis and numerical simulation show that the effectiveness of both education campaign and family understanding is an important factor in reducing the value of R₀ and holds great promise for lowering the number of children and youth who addict to online game in Thailand.     

Analysis of multiserver retrial queueing system: A martingale approach and an algorithm of solution

The paper studies a multiserver retrial queueing system withm servers. Arrival process is a point process with strictly stationary and ergodic increments. A customer arriving to the system occupies one of the free servers. If upon arrival all servers are busy, then the customer goes to the secondary queue, orbit, and after some random time retries more and more to occupy a server. A service time of each customer is exponentially distributed random variable with parameter μ₁. A time between retrials is exponentially distributed with parameter μ₂ for each customer. Using a martingale approach the paper provides an analysis of this system. The paper establishes the stability condition and studies a behavior of the limiting queue-length distributions as μ₂ increases to infinity. As μ₂→∞, the paper also proves the convergence of appropriate queue-length distributions to those of the associated “usual” multiserver queueing system without retrials. An algorithm for numerical solution of the equations, associated with the limiting queue-length distribution of retrial systems, is provided. AMS 2000 Subject classifications: 60K25 60H30.     

Trajectories of efficiency measurement: A bibliometric analysis of DEA and SFA

This study surveys the increasing research field of performance measurement by making use of a bibliometric literature analysis. We concentrate on two approaches, namely Data Envelopment Analysis (DEA) and Stochastic Frontier Analysis (SFA) as the most important methods to evaluate the efficiency of individual and organizational performance. It is the first literature survey that analyses DEA and SFA publications jointly, covering contributions published in journals, indexed by the Web of Science database from 1978 to 2012. Our aim is to identify seminal papers, playing a major role in DEA and SFA development and to determine areas of adoption. We recognized a constant growth of publications during the years identifying DEA as a standard technique in Operations Research, whereas SFA is mainly adopted in Economic research fields. Making use of document co-citation analysis we identify Airports and Supplier Selection (DEA) as well as Banking and Agriculture (SFA) as most influential application areas. Furthermore, Sensitivity and Fuzzy Set Theory (DEA) as well as Bayesian Analysis and Heterogeneity (SFA) are found to be most influential research areas and seem to be methodological trends. By developing an adoption rate of knowledge we identify that research, in terms of citations, is more focusing on relatively old and recent research at the expenses of middle-aged contributions, which is a typical phenomenon of a fast developing discipline.     

A constant enclosure method for validating existence and uniqueness of the solution of an initial value problem for a fractional differential equation

This paper presents a new method for validating existence and uniqueness of the solution of an initial value problems for fractional differential equations. An algorithm selecting a stepsize and computing a priori constant enclosure of the solution is proposed. Several illustrative examples, with linear and nonlinear fractional differential equations, are given to demonstrate the effectiveness of the method.     

A receding horizon control approach for integrated vector management of Aedes aegypti using chemical and biological control: A mono and a multiobjective approach

This paper addresses an integrated vector management (IVM) approach for combating Aedes aegypti, the transmission vector of dengue, zika, and chikungunya diseases, some of the most important viral epidemics worldwide. In order to tackle this problem, a receding horizon control (RHC) strategy is adopted, considering a mono-objective and a multiobjective version of the optimal control model of combating the mosquito using chemical and biological control. RHC is essentially a suboptimal scheme of classical optimal control strategies considering discrete-time approximations. The integrated vector control actions used in this work consist in applying insecticides and inserting sterile males produced by irradiation in the population of mosquitoes. The cost function is defined in terms of social and economic costs, in order to quantify the effectiveness of the proposed epidemiological control throughout a time window of 4 months. Numerical simulations show that the obtained results are better than those from the optimal control strategies found in literature. Furthermore, through the application of the multiobjetive approach, varying the scenarios in the mono-objective formulation is no longer necessary and a set of optimal strategies can be obtained at once. Finally, in order to help health authorities in the choice of the best solution of the Pareto-optimal set to be implemented in practice, a cost-effectiveness analysis is performed and a strategy representing the most cost-effective control policy is obtained.     

Estimating technical and allocative efficiency in the public sector: A nonparametric analysis of Dutch schools

Public sector output provision is influenced not only by discretionary inputs but also by exogenous environmental factors. In this paper, we extended the literature by developing a conditional DEA estimator of allocative efficiency that allows a decomposition of overall cost efficiency into allocative and technical components while simultaneously controlling for the environment. We apply the model to analyze technical and allocative efficiency of Dutch secondary schools. The results reveal that allocative efficiency represents a significant 37 percent of overall cost efficiency on average, although technical inefficiency is still the dominant part. Furthermore, the results show that the impact of environment largely differs between schools and that having a more unfavorable environment is very expensive to schools. These results highlight the importance of including environmental variables in both technical and allocative efficiency analysis.     

Multi-criteria analysis and the resolution of sustainable development dilemmas: A stakeholder management approach

We demonstrate that stakeholder-oriented multi-criteria analysis (MCA) can adequately address a variety of sustainable development dilemmas in decision-making, especially when applied to complex project evaluations involving multiple objectives and multiple stakeholder groups. Such evaluations are typically geared towards satisfying simultaneously private economic goals, broader social objectives and environmental targets. We show that, under specific conditions, a variety of stakeholder-oriented MCA approaches may be able to contribute substantively to the resolution or improved governance of societal conflicts and the pursuit of the public good in the form of sustainable development. We contrast the potential usefulness of these stakeholder-oriented approaches – in terms of their ability to contribute to sustainable development – with more conventional MCA approaches and social cost–benefit analysis.     

International trade and biological invasions: A queuing theoretic analysis of the prevention problem

We propose and develop a new framework for studying the problem of preventing biological invasions caused by ships transporting internationally traded goods between countries and continents. Specifically, we apply the methods of queuing theory to analyze the problem of preventing a biological invasion from a long run perspective. First, we characterize two simple regulatory regimes as two different kinds of queues. Second, we show how to pose a publically owned port manager’s decision problem as an optimization problem using queuing theoretic techniques. Third, we compare and contrast the optimality conditions emanating from our analysis of the M/M/I/U and the M/M/I/I inspection regimes. We conclude by discussing possible extensions to our basic models.     

The European regional policy and the socio-economic diversity of European regions: A multivariate analysis

There are significant differences among the European Union regions, which have been heightened due to the most recent enlargement in 2004. This paper aims to analyse this diversity and to propose a classification of European regions that is adjusted to the different axes of socio-economic development and, simultaneously, is useful for European regional policy purposes. Multivariate statistical techniques allow the identification of clusters of socio-economic similarity, which are contrasted with the classes considered in the financial proposal of the European Commission (EC) for the period 2007–2013.     

A functional analysis of NOx levels: location and scale estimation and outlier detection

This paper analyzes the NOx levels measured by a control station near a power plant by using techniques for functional data. First, we test for differences between the levels on working and non working days. Second, we obtain several location estimators and confidence sets of the center of the functional distribution. Third, we provide scale estimators and confidence sets of the dispersion of the functional distribution. Finally, a distance based procedure provides a criterion to determinate the presence of outlying observations, which allows to detect relevant NOx levels.