Problem-Solving Methodology Design on the Run

This paper discusses a risk analysis study which considered the best way to cross a river with a large diameter gas pipeline, one possibility being a bridge carrying a large diameter oil line. About 30 ‘riskateers’ were involved, over a 3-month period, most of whom were engineers or economists with no risk analysis, O.R. or systems experience. A large number of interested parties were involved, with different perspectives. The paper stresses methodology design as an organizational vehicle, and the implications of designing problem-solving methodology in the midst of a problem-solving process.     

Critical systems thinking and practice

Critical systems thinking and the methodologies associated with it were developed precisely to allow analysis of complex societal problems and intervention to resolve such problems. Early approaches employing systems ideas, such as operational research, systems analysis and systems engineering, were suitable for tackling certain well-defined problems, but were found to have limitations when faced with complex problems involving people with a variety of viewpoints and frequently at odds with one another. Systems thinkers responded with approaches such as system dynamics and organisational cybernetics to tackle complexity; soft systems methodology (SSM) and interactive planning to handle subjectivity; and critical systems heuristics to help the disadvantaged in situations involving conflict. There was a corresponding enlargement of the range of problem contexts in which they felt competent to intervene. It has been critical systems thinking, however, which has supplied the bigger picture, has allowed systems thinking to mature as a discipline and has set out how the variety of methodologies now available can be used together in a coherent manner to promote successful intervention in complex societal problem situations. This paper outlines, at the request of the editor of this special issue, my involvement in developing critical systems thinking and practice, describes its origins, nature and use, and sets out a programme for future research in the area.     

Visual interactive simulation: A methodological perspective

The ideas and methods of Visual Interactive Simulation (VIS) are nearly fifteen years old, yet are still often misunderstood or misrepresented. It is argued that VIS is primarily a methodology, a way of solving problems with simulation modeling, not necessarily a way of building simulation models. Two distinct methodologies are identified, namely active and passive VIS. It is shown that simulation languages and packages tend to implicitly employ a particular methodology, and that each approach requires a different problem-solving life cycle. Present research, and necessary future research, are reviewed from our methodological perspective.     

Revealing hidden mathematics curriculum to pre-teachers using technology: the case of partitions

Motivated by work done with pre-teachers of mathematics in a problem-solving course, this paper shows how computing technologies, including a spreadsheet and Maple, facilitate an informal journey into a hidden aspect of the formal content of the pre-college curriculum dealing with the arithmetic of partitions. By using three problems from different grade levels within a state curriculum as an example, the paper suggests that a deeper perspective on seemingly disconnected problem-solving contexts may serve as a powerful didactical tool in helping teachers to appreciate mathematics and its pedagogy as an integrated whole. The connection of the hidden aspect of the curriculum to the concept of mathematical play is also explored.     

Beyond a System of Systems Methodologies

This paper describes a research programme which used the idea that there is a ‘system of systems methodologies’ to analyse the relationship between OR and other systems-based, problem-solving approaches, and to relate different systems methodologies to appropriate problem-contexts. The results are promising for those who wish to broaden the theoretical basis of OR in order to extend its domain of effective application. Further progress, however, depends upon reconstituting the research programme on surer theoretical foundations. Unless this is done, both the potential of the original work and its initial intent risk being lost. It is now necessary to go beyond ‘a system of systems methodologies’ and to embed that work in an explicit proposal for a critical management science.     

How do students’ behaviors relate to the growth of their mathematical ideas?

The purpose of this study is to analyze the relationship between student behaviors and the growth of mathematical ideas (using the Pirie-Kieren model). This analysis was accomplished through a series of case studies, involving middle school students of varying ability levels, who were investigating a combinatorics problem in after-school problem-solving sessions. The results suggest that certain types of student behaviors appear to be associated with the growth of ideas and emerge in specific patterns. More specifically, as understanding grows, there is a general shift from behaviors such as students questioning each other, explaining and using their own and others’ ideas toward behaviors involving the setting up of hypothetical situations, linking of representations and connecting of contexts. Recognizing that certain types of student behaviors tend to emerge in specific layers of the Pirie-Kieren model can be important in helping us to understand the development of mathematical ideas in children.     

An investigation of statistical thinking in two different contexts: Detecting a signal in a noisy process and determining a typical value

The study describes students’ patterns of thinking for statistical problems set in two different contexts. Fifteen students representing a wide range of experiences with high school mathematics participated in problem-solving clinical interview sessions. At one point during the interviews, each solved a problem that involved determining the typical value within a set of incomes. At another point, they solved a problem set in a signal-versus-noise context [Konold, C., & Pollatsek, A. (2002). Data analysis as the search for signals in noisy processes. Journal for Research in Mathematics Education, 33, 259-289]. Several patterns of thinking emerged in the responses to each task. In responding to the two tasks, some students attempted to incorporate formal measures, while others used informal estimating strategies. The different types of thinking employed in using formal measures and informal estimates are described. The types of thinking exhibited in the signal-versus-noise context are then compared against those in the typical value context. Students displayed varying amounts of attention to both data and context in formulating responses to both problems. Suggestions for teachers in regard to helping students attend to both data and context when analyzing statistical data are given.     

Multimethodology in series and parallel: strategic planning using hard and soft OR

This paper examines two distinct ways in which hard and soft operational research (OR) methodologies can be combined, in series and in parallel. Multimethodology in series is acknowledged as the simpler and more common approach. Multimethodology in parallel is identified as having the potential to provide significant benefits to projects in political, changing, or ‘wicked’ contexts that multimethodology in series cannot. Observations regarding these approaches to multimethodology are examined in light of an information systems strategic planning project in the Australian public sector. Two distinct methodologies were combined in the project: soft systems methodology and project management. These methodologies are based on the soft and hard paradigms, respectively. However, findings in this paper have the potential to be transferred to combinations of other hard and soft OR methodologies.     

A naturalistic study of executive function and mathematical problem-solving

Our goal in this research was to understand the specific challenges middle-school students face when engaging in mathematical problem-solving by using executive function (i.e., shifting, updating, and inhibiting) of working memory as a functional construct for the analysis. Using modified talk-aloud protocols, real-time naturalistic analysis of eighth-grade students’ mathematical problem-solving were conducted. A fine-grained coding of the students’ talking-aloud during problem-solving in mathematics involved isolating the challenges students faced in each one of the four problem-solving phases, and then making a functional link to one of the executive functions of shifting, updating, and inhibiting. In total, 344 episodes were analyzed. Our results show that updating proved to be most challenging during the understanding the problem phase, inhibiting during the carrying out the plan phase, and shifting during the looking back and evaluation phase. Furthermore, students are more likely to make progress with the problem-solving if they are able to engage in a conscious appraisal of the problem at the onset of the problem-solving. Assisting students in establishing what the problem requires through the cognitive clues presented in the problem may necessitate explicit instructional on behalf of the teacher.     

The Role of Visual Representations in Advanced Mathematical Problem Solving: An Examination of Expert-Novice Similarities and Differences

Expert mathematicians are contrasted with undergraduate students through a two-part analysis of the potential and actual use of visual representations in problem solving. In the first part, a classification task is used to indicate the extent to which visual representations are perceived as having potential utility for advanced mathematical problem solving. The analysis reveals that both experts and novices perceive visual representation use as a viable strategy. However, the two groups judge visual representations likely to be useful with different sets of problems. Novices generally indicate that visual representations would likely be useful mostly for geometry problems, whereas the experts indicate potential application to a wider variety of problems. In the second part, written solutions to problems and verbal protocols of problem-solving episodes are analyzed to determine the frequency, nature, and function of the visual representations actually used during problem solving. Experts construct visual representations more frequently than do novices and use them as dynamic objects to explore the problem space qualitatively, to develop a better understanding of the problem situation, and to guide their solution planning and enactment of problem-solving activity. In contrast, novices typically make little use of visual representations.     

Using SSM to rethink the analysis of energy efficiency initiatives

This paper reflects an attempt to rethink the process of analysis of energy efficiency initiatives using soft systems methodology (SSM) as a problem structuring tool. The aim of the work is to provide public and private initiative promoters or evaluators with a structured support for a more informed decision regarding the implementation of energy efficiency measures. The SSM approach contributed with the identification of all market players and their relations, as well as the insight into the deficiencies of current methodologies. Some future work directions are also proposed.     

The Role of Visual Representations in Advanced Mathematical Problem Solving: An Examination of Expert-Novice Similarities and Differences

Expert mathematicians are contrasted with undergraduate students through a two-part analysis of the potential and actual use of visual representations in problem solving. In the first part, a classification task is used to indicate the extent to which visual representations are perceived as having potential utility for advanced mathematical problem solving. The analysis reveals that both experts and novices perceive visual representation use as a viable strategy. However, the two groups judge visual representations likely to be useful with different sets of problems. Novices generally indicate that visual representations would likely be useful mostly for geometry problems, whereas the experts indicate potential application to a wider variety of problems. In the second part, written solutions to problems and verbal protocols of problem-solving episodes are analyzed to determine the frequency, nature, and function of the visual representations actually used during problem solving. Experts construct visual representations more frequently than do novices and use them as dynamic objects to explore the problem space qualitatively, to develop a better understanding of the problem situation, and to guide their solution planning and enactment of problem-solving activity. In contrast, novices typically make little use of visual representations.     

A distance-based measure of super efficiency in data envelopment analysis: an application to gas companies

Conventional data envelopment analysis (DEA) assists decision makers in distinguishing between efficient and inefficient decision making units (DMUs) in a homogeneous group. Standard DEA models can not provide more information about efficient units. Super-efficiency DEA models can be used in ranking the performance of efficient DMUs and overcome this obstacle. Because of the possible infeasibility, the use of super efficiency models has been restricted. This research proposes a methodology to determine a distance-based measure of super-efficiency. The proposed methodology overcomes the infeasibility problem of the existing ranking methodologies. The applicability of the proposed model is illustrated in the context of the analysis of gas companies?? performance.     

“We definitely wouldn't be able to solve it all by ourselves,but together…”: group synergy in tertiary students' problem-solving practices

The ability to address and solve problems in minimally familiar contexts is the core business of research mathematicians. Recent studies have identified key traits and techniques that individuals exhibit while problem solving, and revealed strategies and behaviours that are frequently invoked in the process. We studied advanced calculus students working in groups to identify what strategies they employed and how, including what encouraged the opportunities to invoke them. The study revealed behaviours not included in the original taxonomy, including one that we termed ‘group synergy’. We propose extensions to Carlson and Bloom's original taxonomy to encompass group behaviour and identify the importance of these behaviours in developing problem-solving skills. Finally, we suggest improvements for future problem-solving session iterations, with the goal of promoting opportunity for more expert performance.     

Modelling impacts of cropping systems: Demands and solutions for DEX methodology

Decision modelling of diverse groups of problems makes different requirements to the modelling methodologies and software. We present an actual decision problem and the required characteristics of corresponding decision models. The problem is from agronomy and addresses the ecological and economic impacts of cropping systems, with the focus on the differences between cropping systems with conventional crops and the ones with genetically modified crops. We describe the extensions of an existing DEX qualitative multi-attribute modelling methodology, which were made to cope with the challenges of the problem. The extensions address general hierarchical structures, probabilistic utility functions and numerical values of basic attributes. A new, freely available software tool called proDEX was implemented to support the extended methodology. In this paper we describe the problem of cropping system assessment, propose methodological extensions to DEX, and present the implementation of proDEX.     

The relationship between students' mathematical thinking types and representation preferences in definite integral problems

ABSTRACT

Given that curriculum materials serve as cultural artefacts, this study addresses the need for more research on curriculum materials in different contexts. Most studies concerning curriculum materials have been conducted in US and, therefore little is known about the nature of materials in other cultural-educational contexts. The aim of this paper is to identify the underlying cultural norms of potentially constructed classrooms, by analysing recurrent activities in the most commonly used Finnish teacher guides at primary-school level. We identified three norms embedded in them: (1) creating opportunities for learning through a variety of activities and communication; (2) keeping the class gathered around a specific mathematical topic; and (3) concurrent active involvement of teachers and students. The results add to knowledge about both teacher guides and the Finnish educational context. Moreover, it adds to the growing body of methodologies, as our analytical approach is novel in the context of textual analysis.     

Rethinking tradeoffs and OR/MS methodology

The purpose of this paper is to explore the role of a process known as the ‘evaporating cloud’ (EC) as a way of enhancing OR/MS modelling. While other authors have described the use of the EC process to explore the tradeoffs in a traditional Economic Order Quantity model, this paper demonstrates the use of the EC with a facilities location problem, which was originally solved via mixed integer programming. We relate our findings to OR/MS modelling in general, and argue that the EC could contribute effectively to different phases of the problem-solving process, for example, aiding the problem-structuring phase and helping to find better solutions in many OR/MS modelling situations. We demonstrate that the EC is a valuable tool to use in modelling practice as a problem-structuring tool, as a way of dealing specifically with tradeoffs, and that it can therefore be a useful complement to OR/MS methodology, and contribute to OR/MS practice and teaching.     

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.     

The technical elements of the system dynamics approach

The paper examines the general nature of problem-solving, and it is argued that System Dynamics is appropriate, for certain problem types, both for enhancing understanding and for more formal analysis. This leads to a discussion of some of the more technical modelling issues related to the use of SD as a method of problem analysis in dynamical systems.The technical steps described include influence diagramming, model formulation, validation, and solution procedures, illustrated by a practical application of the SD approach to improving the controllability of a manufacturing system.The paper ends with an examination of the characteristics of, and requirements for, three distinctly different types of SD models.     

浅谈高等数学教学实践

培养学生分析解决问外的能力和培养学生自学能力,是高等数学教学的基本任务之一.本文结合从事高等数学教学的实践,阐述如何培养提高大学生的分析问题、解决问题的能力及自学能力,以及如何用科学家的坚忍不拔的精神激励鞭策大学生学习.