MAKING LINKS BETWEEN ARITHMETIC AND ALGEBRAIC THINKING

In the Purposeful Algebraic Activity project we have designed and implemented a series of six spreadsheet based tasks for year 7 pupils aimed at developing early algebraic competence. Our data collection and analysis have been designed to collect evidence about specific features of both the spreadsheet environment and of task design, which influence the ways in which pupils engage with the notion of variable. In this paper we present a case study of one pair of pupils using data from the teaching programme and interviews. We use this case study to identify elements of the task design which have influenced the development of the pupils' thinking.     

A framework and rubric for guiding the training of mathematics tutors in third-level education

This paper uses a framework established by one of the author's in a previous study and refines it to develop a rubric for mathematics tutor training in third-level education. The paper further opens the discussion on the necessity of, and practical methods for, providing high quality, efficient tutor training at third-level. Models of mathematics teacher knowledge are unanimous in stating that solely possessing strong subject matter knowledge (SMK) is not sufficient for one to be deemed to be a good teacher. Despite the necessity of high quality tutoring at third-level, past research indicates that though tutors’ SMK is usually of high quality, their ability to transform their knowledge so that students will understand the content, as well as their general teaching skills (e.g. board work, speech), often has room to improve. This paper discusses the development of the rubric and its use in a mathematics tutor training programme at third-level. The tutor training programme was found to lead to improvements in multiple elements of the tutors’ teaching.     

A column generation algorithm for tactical timber transportation planning

We present a tactical wood flow model that appears in the context of the Canadian forestry industry, and describe the implementation of a decision support system created for use by an industrial partner. In this problem, mill demands and harvested volumes of a heterogeneous set of log types are given over a multi-period planning horizon. Wood can be stored at the forest roadside before delivery at a financial cost. Rather than solve this as a network linear programme on the basis of out-and-back deliveries, we choose to model this problem as a generalization of a log-truck scheduling problem. By routing and scheduling the trucks in the resolution, this allows us to both anticipate potential backhaul opportunities for cost and fuel savings, and also minimize queuing times at log-loaders, management of which is a major concern in the industry. We model this problem as a mixed integer linear programme and solve it via column generation. The methodology is tested on several case studies.     

A Case Study on the Development of the Home Defence Training Game HOT SEAT

Home defence is concerned with enhancing the prospects for survival and recovery of this country in the eventuality of nuclear war. Contingency plans exist for the creation of a system of wartime regional government which includes elements from peacetime local and central governments. Senior officers in local government with designated wartime roles receive training at the Home Defence College. The Home Office decided that a crisis management game should form part of their overall programme of indoctrination. The resulting game, which is known as HOT SEAT, has been successfully implemented and is now in regular use at the College. This paper describes how the need for the game arose, the structure of the resulting game, the problems that arose during its development, and should be of interest to those concerned with the development of management and crisis games.     

Developing teachers' models for assessing students' competence in mathematical modelling through lesson study

Applications and modelling have gained a prominent role in mathematics education reform documents and curricula. Thus, there is a growing need for studies focusing on the effective use of mathematical modelling in classrooms. Assessment is an integral part of using modelling activities in classrooms, since it allows teachers to identify and manage problems that arise in various stages of the modelling process. However, teachers’ difficulties in assessing student modelling work are a challenge to be considered when implementing modelling in the classroom. Thus, the purpose of this study was to investigate how teachers’ knowledge on generating assessment criteria for assessing student competence in mathematical modelling evolved through a professional development programme, which is based on a lesson study approach and modelling perspective. The data was collected with four teachers from two public high schools over a five-month period. The professional development programme included a cyclical process, with each cycle consisting of an introductory meeting, the implementation of a model-eliciting activity with students, and a follow-up meeting. The results showed that the professional development programme contributed to teachers’ knowledge for generating assessment criteria on the products, and the observable actions that affect the modelling cycle.     

Ways of promoting the sustainability of mathematics teachers’ professional development

This paper deals with the sustainable effectiveness of professional development programmes. Based on a review of literature and research findings, the following questions are raised: What is regarded as an effective way of promoting mathematics teachers’ sustainable professional development? Which levels of impacts are aimed at? What are the factors promoting the effectiveness of professional development programmes? Regarding these questions, the article links theoretical considerations with research findings from a case study. A secondary mathematics teacher, taking part in a teacher professional development programme in 2002, was revisited in 2005 and 2010 to gather data regarding the sustainable impact of the programme. The case study’s results provide information about the teacher’s professional growth and lead to a discussion of implications for mathematics teachers’ professional development and teacher education in general.     

A Mixed Linear Programme for Short-term Irrigation Scheduling

For farmers in the UK who use hosereel-raingun irrigation systems, irrigation scheduling requires that three questions be answered: (i) when should each field be irrigated; (ii) how much water should be applied; and (iii) which field should be irrigated first on a particular day? A mixed linear programme has been built to produce optimum irrigation schedules. The objective function maximizes the financial return from irrigation over the short term (typically 7 days). The constraints are equipment, labour and water availability. Use of the mixed linear programme is illustrated with a typical situation faced by a UK farmer.     

Conceptualizing inquiry-based education in mathematics

The terms inquiry-based learning and inquiry-based education have appeared with increasing frequency in educational policy and curriculum documents related to mathematics and science education over the past decade, indicating a major educational trend. We go back to the origin of inquiry as a pedagogical concept in the work of Dewey (e.g. 1916, 1938) to analyse and discuss its migration to science and mathematics education. For conceptualizing inquiry-based mathematics education (IBME) it is important to analyse how this concept resonates with already well-established theoretical frameworks in mathematics education. Six such frameworks are analysed from the perspective of inquiry: the problem-solving tradition, the theory of didactical situations, the realistic mathematics education programme, the mathematical modelling perspective, the anthropological theory of didactics, and the dialogical and critical approach to mathematics education. In an appendix these frameworks are illustrated with paradigmatic examples of teaching activities with inquiry elements. The paper is rounded off with a list of ten concerns for the development and implementation of IBME.     

Optimal Control of Some Simple Deterministic Epidemic Models

Using control variables such as the level of medicare programme effort and the level of inoculation programme effort, three simple mathematical models of epidemics are transformed into optimal control problems. These are multi-state problems with the state variables as numbers of infectives and susceptibles and with the objective function being minimization of the total present value of the social or economic costs of infectives and medical controls. The problems are analysed with the framework of the maximum principle to obtain or, at least, partially characterize the optimal policies over time.     

The impact of loyalty and promotion effects on retention rate

Researching customer retention rates is made difficult because of the lack of suitable systems with which to measure it. From a first-order Markov brand-switching model, serial equations are derived to estimate the retention rate using market share and loyalty. To explore the impact of loyalty and promotion programme on retention rate, data from five leading brands for three product categories were obtained to fit the developed model. Findings suggest that: (1) when the loyalty effect remains unchanged over a period of time, the greater the effect of a promotion programme, the greater the retention rate; (2) when the promotion effects among brands are similar, the retention rate varies depending on level of loyalty; and (3) when the proportion of loyal customers is high, the proportion of potential switchers affected by promotion programmes is low. In this case, the impact of a promotion programme on the retention rate is not significant. This model successfully demonstrates how the Markov brand-switching model can be employed to estimate a measure of retention rate from consumer panel data as well as to better understand how promotion affects loyalty, switching behaviour, and retention rate.     

Using Multicriteria Methods in the Early Stages of New Product Development

The R&D manager is commonly faced with the problem of deciding which projects to fund to meet overall corporate and technical goals. Because outcomes can rarely be predicted with certainty, decisions aimed at striking a balance between cost and risk are likely to involve some amount of redundancy at the project level. The intent of this paper is to examine the difficulties that arise when trying to pursue a parallel strategy in the presence of multiple objectives. The basic elements of the problem include a set of projects, a set of objectives, the associated probability measures relating effort to success, budgetary and performance constraints, and a utility function defined on the range of outcomes. In the model it will be assumed that each project contributes to one or more objectives, and that the selection criterion is based on expected utility maximization. With this in mind, the problem is formulated as a probabilistic goal programme and solved with a heuristic that computes the K best funding schemes. Results are presented for a case involving the development of a non-petroleum-powered vehicle which demonstrate the robustness of the algorithm and the implications of the underlying decision rules.     

Education for Operational Research. A Practitioner's Viewpoint

University education for operational research in Britain is making great and welcome strides forward, but there has been little open discussion of its aims in relation to the advance of the profession. Progress in operational research, and in its application to the really worth-while problems of industry in particular, depends primarily on strengthening the industrial operational research group in its “front line” role. We need a research study of the teaching and practice of industrial operational research, in relation to effectiveness. We need “collaborative” programmes of operational research education in which the university and the industrial operational research group each contribute, in a man's training over a period of years, in the ways in which each is most competent. We need emphasis on “breadth” as well as “depth” in operational research education, and greater attention to the mutual contributions of operational research, teaching and practice, and the teaching and development of management. An impressionist sketch of a programme which might meet these needs is put forward for discussion of principles and feasibility.     

A multidisciplinary graduate level project-based programme in applied statistics

The use of a project-based approach for designing a one-year graduate level programme in applied statistics is addressed. The pedagogical approach, academic setting, and learning activities are described in a multidisciplinary context. Some comments on implementation of such a programme based on the results from five successive graduating classes at the University of Veracruz (1994–2000) are included.     

ILP approaches to the blockmodel problem

Blockmodelling is a method for identifying structural similarities or equivalences between elements which has applications in a variety of contexts, including multiattribute performance assessment. One criterion for forming blocks results in a difficult non-linear integer programme. We give several integer linear programming formulations of this problem and provide comparative computational results. We show that methods of reducing symmetry proposed by Sherali and Smith are not effective in this case and propose an iterative approach in which the size of the problem is reduced.     

Online paths in mathematics teacher training: new resources and new skills for teacher educators

This paper considers the work carried out by online teacher educators and their professional development. We use the theoretical perspective of the documentational approach that focuses, in this case, on the interaction between teacher educators and the resources they use for their online training work. We thus study the following issues: (1) What kinds of resources do online teacher educators need, and how are such resources modified according to the educators?? specific skills and needs? (2) What specific skills are needed for setting up online training for mathematics teachers and how do these skills evolve as teacher education resources are used? We consider both questions simultaneously, while presenting results from a study within a specific teacher training programme in France that proposes ??training paths?? on a national platform. These ??paths?? are resources designed for teacher educators. We follow the appropriation of two training paths by two educator teams. The ways in which these educator teams were able to appropriate the paths give insights into the teacher educators?? skills and, as well, into the resources they need. By looking at their use of resources (as online mathematics teacher educators), we observe and analyse professional geneses, leading to the development of new skills.     

High level radiosonde ascents at hyderabad during January to April, 1961

The paper describes a new radiosonde developed at Poona for high altitude balloon soundings and the results of the high level radiosonde ascents taken at Hyderabad as part of the Indo-US Balloon Flight Programme—1961. The temperature data obtained during the programme are presented and compared with the results of the sounding balloon ascents made over Poona and Hyderabad during 1928–35.     

Finite element Galerkin methods for multi-phase stefan problems

The modelling of heat conduction with phase transitions in terms of integral indentities in Galerkin form is actually based directly on physical concepts like conservation of heat, the conduction law etc. In this way we obtain a formulation which apparently covers general physical situations, for instance with melting and freezing of layered materials. Choosing finite elements in a fixed grid we can still track the moving boundaries by an accurate treatment of the nonlinear enthalpy terms. This makes it possible to use standard finite element packages to build an efficient programme for Stefan-like problems.     

LP modelling of the finishing hot rolling programme for low carbon steel coils

In this paper a mathematical model was developed to optimize the finishing rolling of hot rolled coils by increasing the productivity of the rolling programme, to help achieve the required level of quality assurance and to facilitate production planning and control in the hot rolling mill. A brief account of the technological and planning aspects of the hot rolling processes and mills relevant to strip steel is given. Linear (mixed integer) programming is used to formulate the objective function and the various types of constraints of the model. The model takes into consideration, the general aspects pertinent to hot rolling of low carbon steel and the characteristics of the hot rolling mills as stipulated by the operational codes and guidelines of the relevant establishments. Owing to the flexibility offered by linear programming the model can incorporate any modifications and/or additional requirements, if any, in case of other types of steel and/or other types of mills. The full modelling of the problem required the incorporation of some zero/one variable constraints. Owing to the complexity involved and the need to keep the model as simple as possible, it was decided to exclude these constraints and deal with them externally. HYPER LINDO PC was used to solve the programme. Using available data, in the case under consideration the model showed astonishing results in achieving the objectives. Taking into account the effect on the overall productivity as well as quality improvement, the investigation showed that a net improvement in conforming output to the effect of around 43%, could have been obtained had the model been used in the case under consideration.     

Stochastic Approaches of Nonlinear Model-Based Simulations for Vibratory Compaction Process

The paper presents a basic approach regarding nonlinear models intended for computational dynamics of vibratory compaction process in order to evaluate the consolidation level. Taking into account the complex rheology of terrains and the relative variable dynamic load in time for each point inside the working area, a stochastic approach both of the material characteristics, and of the excitatory charges have been appropriate. Numerical simulations has developed based on nonlinear models containing conservative and complex dissipative rheological elements, which include two main groups of stochastic data as follows: terrain characteristics from preconditioning experimental tests, and vibratory actions from analytical approaches supposing the soil-drum contact area and the number of dynamic load transitions over this area. Simulation results dignify small variances between the real and computed terrain responses, comparative with the case supposing full determinist parameters. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Independent learning and operational research in the classroom

While Operational Research and Education are common keywords in OR journals, most work focuses on course content. Relatively little has been written about processes of education and learning in OR. This paper adds to this important area by considering the use of more Independent Learning (IL) in Operational Research education. Included is an explanation of IL and justification for using it. Activities which encourage IL are described and supported by three case studies which give examples of how an academic might introduce IL into a range of OR education settings. Other topics include IL and assessment, arguments against the use of IL, and critical success factors.