Problem solving in the borderland between mathematics and physics

The article addresses the problématique of where mathematization is taught in the educational system, and who teaches it. Mathematization is usually not a part of mathematics programs at the upper secondary level, but we argue that physics teaching has something to offer in this respect, if it focuses on solving so-called unformalized problems, where a major challenge is to formalize the problems in mathematics and physics terms. We analyse four concrete examples of unformalized problems for which the formalization involves different order of mathematization and applying physics to the problem, but all require mathematization. The analysis leads to the formulation of a model by which we attempt to capture the important steps of the process of solving unformalized problems by means of mathematization and physicalization.     

Problem solving in the mathematics classroom: the German perspective

In Germany, problem solving has important roots that date back at least to the beginning of the twentieth century. However, problem solving was not primarily an aspect of mathematics education but was particularly influenced by cognitive psychologists. Above all, the Gestalt psychology developed by researchers such as Köhler (Intelligenzprüfungen an Anthropoiden. Verlag der Königlichen Akademie des Wissens, Berlin, 1917; English translation: The mentality of apes. Harcourt, Brace, New York, 1925), Duncker (Zur Psychologie des produktiven Denkens. Springer, Berlin, 1935), Wertheimer (Productive thinking. Harper, New York, 1945), and Metzger (Schöpferische Freiheit. Waldemar Kramer, Frankfurt, 1962) made extensive use of mathematical problems in order to describe their specific problem-solving theories. However, this research had hardly any influence on mathematics education—neither as a scientific discipline nor as a foundation for mathematics instruction. In the German mathematics classroom, problem solving, which is according to Halmos (in Am Math Mon 87:519–524, 1980) the “heart of mathematics,” did not attract the interest it deserved as a genuine mathematical topic. There is some evidence that this situation may change. In the past few years, nationwide standards for school mathematics have been introduced in Germany. In these standards, problem solving is specifically addressed as a process-oriented standard that should be part of the mathematics classroom through all grades. This article provides an overview on problem solving in Germany with reference to psychology, mathematics, and mathematics education. It starts with a presentation of the historical roots but gives also insights into contemporary developments and the classroom practice.     

Manpower Planning in the United Kingdom: An Historical Review

The Manpower Planning Study Group of the Operational Research Society was formed on 16th November 1967. This paper traces the growth of manpower planning in the United Kingdom from its origins in the Second World War to the present day. It identifies a series of stages, starting with the beginnings and growing awareness of the 1950s and early 1960s, which led on to an explosive growth between 1965 and 1970, and the lengthy period of consolidation thereafter. It places on record the interest and contributions of many individuals and organizations who laid the foundations of manpower planning in this country, and review trends in methodology and the growing role of computers. The paper concludes with some notes on possible future developments.     

A Survey of Mathematics in Engineering Degree Courses in the United Kingdom

Following the publication of the Organization for Economic Co‐operation and Development Report on the Mathematical Education of Engineers, the Council of Engineering Institutions and the Joint Mathematical Council of the United Kingdom set up a Committee on Mathematics in Engineering. This survey, carried out by the authors at the request of the Committee, is an analysis of the replies to a questionnaire sent out under the auspices of the Committee relating to the time spent on mathematics in engineering degree courses. Recommendations of the Committee, based on the analysis, are included at the end of the survey, which makes frequent comparisons between the United Kingdom and the OECD recommendations.     

Problem solving in France: didactic and curricular perspectives

In this paper, we address the issue of problem solving in classrooms in France through two different and complementary approaches: didactic research and curricular choices. These two approaches correspond to two different, but not independent perspectives on problem solving and we investigate the existing links between them. We show that in France, the solving of problems is given a central role both in didactic research and curricular choices and that problem solving, as generally understood, is an object taking controversial positions, and we try to elucidate the rationale behind such positions. This paper is structured into two main parts: the first part devoted to didactic research, the second part to curricula, and one last part, discussion, wherein we question didactic research and curricula as regards their potential for influencing the reality of problem solving in classroom practices.     

A meta-heuristic approach for solving the Urban Network Design Problem

This paper proposes an optimisation model and a meta-heuristic algorithm for solving the urban network design problem. The problem consists in optimising the layout of an urban road network by designing directions of existing roads and signal settings at intersections. A non-linear constrained optimisation model for solving this problem is formulated, adopting a bi-level approach in order to reduce the complexity of solution methods and the computation times. A Scatter Search algorithm based on a random descent method is proposed and tested on a real dimension network. Initial results show that the proposed approach allows local optimal solutions to be obtained in reasonable computation times.     

A branch-and-cut algorithm for solving the Non-Preemptive Capacitated Swapping Problem

This paper models and solves a capacitated version of the Non-Preemptive Swapping Problem. This problem is defined on a complete digraph G=(V,A), at every vertex of which there may be one unit of supply of an item, one unit of demand, or both. The objective is to determine a minimum cost capacitated vehicle route for transporting the items in such a way that all demands are satisfied. The vehicle can carry more than one item at a time. Three mathematical programming formulations of the problem are provided. Several classes of valid inequalities are derived and incorporated within a branch-and-cut algorithm, and extensive computational experiments are performed on instances adapted from TSPLIB.     

Problem solving: a personal perspective from Brazil

In this paper, I do a historical review of the concept of Problem Solving, and make some considerations about the State of the Art nowadays. A very brief notice of the art of Problem Solving in Latin America is also presented. Finally, I present some reflections on the future of the Art of Problem Solving.     

Modeling and solving a Crew Assignment Problem in air transportation

A typical problem arising in airline crew management consists in optimally assigning the required crew members to each flight segment of a given time period, while complying with a variety of work regulations and collective agreements. This problem called the Crew Assignment Problem (CAP) is currently decomposed into two independent sub-problems which are modeled and solved sequentially: (a) the well-known Crew Pairing Problem followed by (b) the Working Schedules Construction Problem. In the first sub-problem, a set of legal minimum-cost pairings is constructed, covering all the planned flight segments. In the second sub-problem, pairings, rest periods, training periods, annual leaves, etc. are combined to form working schedules which are then assigned to crew members.In this paper, we present a new approach to the Crew Assignment Problem arising in the context of airline companies operating short and medium haul flights. Contrary to most previously published work on the subject, our approach is not based on the concept of crew-pairings, though it is capable of handling many of the constraints present in crew-pairing-based models. Moreover, contrary to crew-pairing-based approaches, one of its distinctive features is that it formulates and solves the two sub-problems (a) and (b) simultaneously for the technical crew members (pilots and officers) with specific constraints. We show how this problem can be formulated as a large scale integer linear program with a general structure combining different types of constraints and not exclusively partitioning or covering constraints as usually suggested in previous papers. We introduce then, a formulation enhancement phase where we replace a large number of binary exclusion constraints by stronger and less numerous ones: the clique constraints. Using data provided by the Tunisian airline company TunisAir, we demonstrate that thanks to this new formulation, the Crew Assignment Problem can be solved by currently available integer linear programming technology. Finally, we propose an efficient heuristic method based on a rounding strategy embedded in a partial tree search procedure.The implementation of these methods (both exact and heuristic ones) provides good solutions in reasonable computation times using CPLEX 6.0.2: guaranteed exact solutions are obtained for 60% of the test instances and solutions within 5% of the lower bound for the others.     

Problem solving in Chinese mathematics education: research and practice

This paper is an attempt to paint a picture of problem solving in Chinese mathematics education, where problem solving has been viewed both as an instructional goal and as an instructional approach. In discussing problem-solving research from four perspectives, it is found that the research in China has been much more content and experience-based than cognitive and empirical-based. We also describe several problem-solving activities in the Chinese classroom, including “one problem multiple solutions,” “multiple problems one solution,” and “one problem multiple changes.” Unfortunately, there are no empirical investigations that document the actual effectiveness and reasons for the effectiveness of those problem-solving activities. Nevertheless, these problem-solving activities should be useful references for helping students make sense of mathematics.     

Operational Research Projects in Health and Welfare Services in the United Kingdom and Ireland

The paper presents a comparison of Health and Welfare O.R. in the U.K. and Ireland as represented by two registers of current projects, compiled in 1972 and 1977 respectively. The survey shows that

1. i)
   There appear to be very few O.R. studies of social services;
    
2. ii)
   The major focus has apparently moved away from hospital-based tactical studies, towards more broadly-based strategic studies;
    
3. iii)
   Related to this, the number of projects labelled as concerned with "planning" has increased;
    
4. iv)
   The number of projects involving "optimisation" has decreased, whilst those involving simulation have shown some increase; and
    
5. v)
   The majority of projects in both years were described as involving "applied common sense", rather than any specific technique.
    

     

Problem solving in mathematics education in Italy: dreams and reality

Drawing on personal experiences in in-service teacher training and curricular innovation in Italy, this paper addresses some questions relevant to mathematics education in the specific area of problem solving research. What are the effects of research results on national programs and curricula? To what extent, and how, are these assimilated in the school system? The analysis of some specific aspects of the evolution of the Italian situation in the last 30 years will suggest some possible answers suitable for comparison with other countries.     

Problem solving and Working Mathematically: an Australian perspective

This article reviews “problem solving” in mathematics in Australia and how it has evolved in recent years. In particular, problem solving is examined from the perspectives of research, curricula and instructional practice, and assessment. We identify three key themes underlying observed changes in the research agenda in Australia in relation to problem solving: Obliteration, Maturation and Generalisation. Within state mathematics curricula in Australia, changes in the language and construction of the curriculum and in related policy documents have subsumed problem solving within the broader category of Working Mathematically. In relation to assessment, research in Australia has demonstrated the need for alignment of curriculum, instruction and assessment, particularly in the case of complex performances such as mathematical problem solving. Within the category of Working Mathematically, recent Australian curriculum documents appear to accept an obligation to provide both standards for mathematical problem solving and student work samples that illustrate such complex performances and how they might be assessed.     

Problem solving heuristics,affect, and discrete mathematics

It has been suggested that activities in discrete mathematics allow a kind of new beginning for students and teachers. Students who have been “turned off” by traditional school mathematics, and teachers who have long ago routinized their instruction, can find in the domain of discrete mathematics opportunities for mathematical discovery and interesting, nonroutine problem solving. Sometimes formerly low-achieving students demonstrate mathematical abilities their teachers did not know they had. To take maximum advantage of these possibilities, it is important to know what kinds of thinking during problem solving can be naturally evoked by discrete mathematical situations—so that in developing a curriculum, the objectives can include pathways to desired mathematical reasoning processes. This article discusses some of these ways of thinking, with special attention to the idea of “modeling the general on the particular.” Some comments are also offered about students' possible affective pathways and structures.     

Problem solving as a challenge for mathematics education in The Netherlands

This paper deals with the challenge to establish problem solving as a living domain in mathematics education in The Netherlands. While serious attempts are made to implement a problem-oriented curriculum based on principles of realistic mathematics education with room for modelling and with integrated use of technology, the PISA 2003 results suggest that this has been successful in educational practice only to a limited extent. The main difficulties encountered include institutional factors such as national examinations and textbooks, and issues concerning design and training. One of the main challenges is the design of good problem solving tasks that are original, non-routine and new to the students. It is recommended to pay attention to problem solving in primary education and in textbook series, to exploit the benefits of technology for problem solving activities and to use the schools’ freedom to organize school-based examinations for types of assessment that are more appropriate for problem solving.     

Strategic Level MS/OR Tool Usage in the United Kingdom: An Empirical Survey

A survey of the practitioner members of the United Kingdom Operational Research Society was conducted to identify current levels of MS/OR tool usage at the strategic level. In this paper details of the survey will be outlined, the framework of strategic tasks will be described, and the MS/OR tool usage results for these strategic tasks will be presented and discussed.     

An Econometric Model for Supply and Demand of Undergraduate University Places in the United Kingdom

This paper presents a model describing the demand for Undergraduate University places and the way in which universities satisfy this demand, and respond to changes in it.The model comprises a number of equations examining not only the whole academic field but also the individual subjects of accountancy and economics; in addition they deal with both the total figures for all Universities and those for Southampton University only.     

Problem solving and the use of digital technologies within the Mathematical Working Space framework

ZDM – Mathematics Education - The aim of this study is to analyze and document the extent to which high school teachers rely on a set of technology affordances to articulate epistemological...