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.     

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.     

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...     

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.     

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.     

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.     

Mathematically modeling munitions prepositioning and movement

The Air Force's ability to deploy, employ, and sustain operations in forward operating locations is a key to mission success. An integral part of this new strategy involving forward operating locations is equipment prepositioning, to include: vehicles, aircraft support, consumable inventory, and munitions. Proper prepositioning strategies provide a means to deploy forces rapidly without resorting to an increased overseas presence. This research focuses on defining and developing a mathematical model to aid decision makers with a strategy for positioning and configuring prepositioned assets. This research places particular emphasis on the strategic, global prepositioning of the afloat prepositioning fleet (APF), the configuration of these ships with respect to precision guided weaponry, the development of a transportation plan in response to modeled contingencies, and a port selection and distribution strategy once the APF ship is tasked to support a contingency. In addition to the APF assets, the model considers U.S.-based supply points used to augment on-hand and APF-provided munitions assets. The primary objective is to minimize the overall response time involved with offloading these ships and transporting their cargo (the munitions) to the intended point of use.The Pre-Po model developed is a mixed integer program, implemented using the general algebraic modeling system (GAMS), solved using the XA solver package, and tested against a realistically stressing planning scenario. Analytical results and insights are presented along with avenues for further work.     

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 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.     

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.     

Mathematical problem solving: an evolving research and practice domain

Research programs in mathematical problem solving have evolved with the development and availability of computational tools. I review and discuss research programs that have influenced and shaped the development of mathematical education in Mexico and elsewhere. An overarching principle that distinguishes the problem solving approach to develop and learn mathematics is to conceptualize the discipline as a set of dilemmas or problems that need to be explored and solved in terms of mathematical resources and strategies. In this context, relevant questions that help structure and organize this paper include: What does it mean to learn mathematics in terms of problem solving? To what extent do research programs in problem solving orient curricular proposals? What types of instructional scenarios promote the students’ development of mathematical thinking based on problem solving? What type of reasoning do students develop as a result of using distinct computational tools in mathematical problem solving?     

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.     

Mathematical Working Spaces in schooling: an introduction

ZDM – Mathematics Education - The theoretical and methodological model of Mathematical Working Space (MWS) is introduced in this paper. For over 10 years, the model has been the...