Approach to mathematics in textbooks at tertiary level – exploring authors’ views about their texts

The aim of this article is to present and discuss some results from an inquiry into mathematics textbooks authors’ visions about their texts and approaches they choose when new concepts are introduced. Authors’ responses are discussed in relation to results about students’ difficulties with approaching calculus reported by previous research. A questionnaire has been designed and sent to seven authors of the most used calculus textbooks in Norway and four authors have responded. The responses show that the authors mainly view teaching in terms of transmission so they focus mainly on getting the mathematical content correct and ‘clear’. The dominant view is that the textbook is intended to help the students to learn by explaining and clarifying. The authors prefer the approach to introduce new concepts based on the traditional way of perceiving mathematics as a system of definitions, examples and exercises. The results of this study may enhance our understanding of the role of the textbook at tertiary level. They may also form a foundation for further research.     

Heuristics for convex mixed integer nonlinear programs

In this paper, we describe the implementation of some heuristics for convex mixed integer nonlinear programs. The work focuses on three families of heuristics that have been successfully used for mixed integer linear programs: diving heuristics, the Feasibility Pump, and Relaxation Induced Neighborhood Search (RINS). We show how these heuristics can be adapted in the context of mixed integer nonlinear programming. We present results from computational experiments on a set of instances that show how the heuristics implemented help finding feasible solutions faster than the traditional branch-and-bound algorithm and how they help in reducing the total solution time of the branch-and-bound algorithm.     

A profile of OR research and practice published in the <Emphasis Type="Italic">Journal of the Operational Research Society</Emphasis>

In this paper we reflect on the last 10 years of the Journal of the Operational Research Society (JORS). We use metadata and citation analysis to profile OR research and practice published in this prestigious journal. The analysis of the published material includes examining variables such as the most productive authors, the papers having the highest number of citations, the universities and organisations associated with the most publications and their geographic diversity, OR techniques and their application areas, the number of authors per paper, the background of the authors, etc. Moreover, this work includes variables from a previously published study of JORS that profiled research from 1981 to 1999. Therefore, the analysis allows a comparison to be conducted between some of the findings of the two studies. This research has implications for researchers, journal editors and research institutions.     

Approximation of smooth functions by S-splines

The paper contains a method for recovering a function from a table of its values. The method ensures smoothing of measured data, can be easily transformed into an algorithm and used in automatic information processing. The reconstruction method described in the paper is related to the approximation of smooth functions by means of some special splines, called S-splines, introduced by one of the authors. Conditions for stability and convergence of S-splines are obtained.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 10, pp. 197–206, 1984.The authors would like to thank Prof. V. M. Tikhomirov for advice and help with this work; thanks are also due to V. A. Gruzdev and E. V. Logacheva who provided their programs and computations.     

The prime number theorem and fragments ofP A

Summary We show that versions of the prime number theorem as well as equivalent statements hold in an arbitrary model ofI ₀+exp.A part of the preparation of this paper was done while the authors were visiting the University of Oxford in the spring of 1992; their visit was partially supported by the University of Crete (grant no. 185) and the British Council. The authors thank Drs. R. Kaye and A. Wilkie for useful discussions and the Mathematical Institute of the University of Oxford for their hospitality; they also thank Dr. K. Skandalis for pointing out an error in an earlier proof of Theorem 3.3 and the two referees, for suggesting Lemma 1.13 as a common generalization of some earlier Lemmas and for many other comments, which helped to improve this paper     

Complexity management in practice: A Viable System Model intervention in an Irish eco-community

While soft OR tools offer specific solutions to manage complexity in organisations, little is known concerning soft OR tools to deal with self-organisation in communities. This paper describes an action research project where the authors facilitated a process of self-organisation in a developing Irish eco-community whose members operate in a non-hierarchical, and cooperative fashion. We used the Viable System Model as a hermeneutical enabler of the community learning process concerning their self-organisation: by embedding VSM distinctions, they redesigned their primary tasks and developed meta-systemic management tools to deal with the complexity they were facing. Observations of the dynamics of the self-organising process over a period of 3 years show the community designed their roles and tasks more effectively, improved the connectivity of roles, and in general, their viability and sustainability. We reflect on the distinctiveness of the methods used, and their contributions to research in soft OR in community projects.     

Analyzing Student Work as a Professional Development Activity

When worthwhile mathematical tasks are used in classrooms, they should also become a crucial element of assessment. For teachers, using these tasks in classrooms requires a different way to analyze student thinking than the traditional assessment model. Looking carefully at students' written work on worthwhile mathematical tasks and listening carefully while students explore these worthwhile tasks can contribute to a teacher's professional development. This paper reports on a professional development activity in which teachers analyzed mathematical tasks, predicted students' achievement on tasks, evaluated students' written work, listened to students' reasoning, and assessed students' understanding. Teachers' engagement in this way can help them develop flexibility and proficiency in the evaluation of their own students' work. These experiences allow teachers the opportunity to recognize students' potential, strengthen their own mathematical understanding, and engage in conversations with peers about assessment and instruction.     

Some properties of efficient solutions for vector optimization

In this paper, we discuss the optimality conditions for vector optimization problems. Properties of efficient and weakly efficient solutions are studied, and some new necessary conditions are obtained. Most of them are related to the mapping properties of the derivative operatorf(x) of the objective functionf. Almost all of our results are based on the methods of functional analysis and the theory of degree.The authors would like to thank Professor Y. D. Hu, Deputy General Secretary of the Chinese Operations Research Society, for his help and directions. Also, the authors would like to thank Professors T. K. Sung and Y. J. Chang, Chairmen of the authors' present department, for their sincere concern and encouragement. Finally, the authors are grateful to Professor G. Leitmann for his valuable comments, suggestions, and his careful editing of an earlier version of this paper.     

Systemic evaluation: a participative,multi-method approach

This paper presents some new ideas on systemic evaluation developed in the context of a project to support capacity building for the evaluation of community health services. Emphases are placed on the need for stakeholder participation; dialogue on the boundaries of evaluations; considering multiple values; and ensuring that marginalized people and issues are properly accounted for. Further developing the work of previous authors, three different approaches to evaluation are outlined, each of which can be applied participatively: goal-based (where goals are set and their achievement is measured); stakeholder (where there are no pre-set goals, and different people's experiences and stories are surveyed to reveal significant issues); and organizational (where organizational processes are compared with models of good practice). There is a logical relationship between these: a stakeholder evaluation can lead to the setting of community-sensitive goals, the achievement of which can be measured through goal-based evaluation, and pursuit of the goals can be enhanced by organizational evaluation. A participative, flexible and responsive evaluation practice will often need to draw upon aspects of all three approaches. Initial feedback from health practitioners suggests that this way of thinking about systemic evaluation may prove useful in a range of situations faced by people in the statutory, voluntary and community sectors.     

An algorithm for minimizing a class of locally Lipschitz functions

In this paper, we present an algorithm for minimizing a class of locally Lipschitz functions. The method generalized the -smeared method to a class of functions whose Clarke generalized gradients are singleton at almost all differentiable poinst. We analyze the global convergence of the method and report some numerical results.This work was supported by the National Science Foundation of China. The authors would like to thank the referees for their valuable comments, which led to significant improvements in the presentation.     

Some Methodologies for Community Operational Research

As experience of conducting community OR projects grows and learning accrues, new approaches will be developed offering guidance on how best to proceed in alleviating the problems of community organizations. In the interim, it is sensible to consider which, if any, among the existing OR and systems methodologies might prove useful for this kind of work. In this paper it is argued that there already exist, within what might be described as ‘enhanced OR’, a number of approaches which are suitable for guiding community OR practice. This is demonstrated by outlining some of these methodologies, giving examples of their use with community organizations and showing their particular strengths in the community context. Some conclusions are drawn about the community OR enterprise.     

The stories we tell: Why unpacking narratives of mathematics is important for teacher conocimiento

This article articulates how mathematics (e.g., what they are, how they can be used, who they are by and for, who is able to do and understand them) are a social construct, just like racial or gender identities are social constructs. The authors describe how Political Conocimiento in Teaching Mathematics–a relational knowing that involves the entanglement of mathematics, pedagogies, students, and politics–can be used as a lens to reveal narratives about mathematics re(told) through stories (e.g., “Mathematics is culture-free, objective, and universal”). Drawing on their work with teachers, the authors offer an example scenario/activity and a teacher discussion that unpacks where mathematical stories are being told, from a dominant perspective, as well as how focusing on healthier narratives can help teachers work toward liberatory futures. Implications for teaching, teacher education, and future research are described.     

On flipping first-semester calculus: a case study

High failure rates in calculus have plagued students, teachers, and administrators for decades, while science, technology, engineering, and mathematics programmes continue to suffer from low enrollments and high attrition. In an effort to affect this reality, some educators are ‘flipping’ (or inverting) their classrooms. By flipping, we mean administering course content outside of the classroom and replacing the traditional in-class lectures with discussion, practice, group work, and other elements of active learning. This paper presents the major results from a three-year study of a flipped, first-semester calculus course at a small, comprehensive, American university with a well-known engineering programme. The data we have collected help quantify the positive and substantial effects of our flipped calculus course on failure rates, scores on the common final exam, student opinion of calculus, teacher impact on measurable outcomes, and success in second-semester calculus. While flipping may not be suitable for every teacher, every student, and in every situation, this report provides some evidence that it may be a viable option for those seeking an alternative to the traditional lecture model.     

Numerical Godeaux surfaces with an involution

Minimal algebraic surfaces of general type with the smallest possible invariants have geometric genus zero and and are usually called numerical Godeaux surfaces. Although they have been studied by several authors, their complete classification is not known.

In this paper we classify numerical Godeaux surfaces with an involution, i.e. an automorphism of order 2. We prove that they are birationally equivalent either to double covers of Enriques surfaces or to double planes of two different types: the branch curve either has degree 10 and suitable singularities, originally suggested by Campedelli, or is the union of two lines and a curve of degree 12 with certain singularities. The latter type of double planes are degenerations of examples described by Du Val, and their existence was previously unknown; we show some examples of this new type, also computing their torsion group.

     

广义连续流场论中新的功能及功率能率原理

提出极性和非局部极性连续统场论中具有交叉项的新的功能及功率能率原理,并据此和广义的Piola定理一次性地而且无需其它要求地推导出所有相应的运动方程和边界条件以及新的能量和能率均衡方程。同时,建立起广义连续统力学中的新的能量和能率均衡原理。给出的新的功能及能率原理纠正了现有文献中所有有关不带交叉项的能量和能率原理的不完整性。     

Fractional programming

The paper reviews some of the major developments in fractional programming. The emphasis is on applications and algorithms. Theoretical results are only outlined to the extent as they are needed to describe the algorithms.The survey is subjective in nature. It tries to critically evaluate developments in fractional programming after more than 20 years of research with over 500 articles. The authors hope that this somewhat critical report will serve as a guide for meaningful future work in this growing area of nonlinear programming.     

New identities involving sums of the tails related to real quadratic fields

In previous work, the authors discovered new examples of q-hypergeometric series related to the arithmetic of $\mathbb {Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt{3})$ . Building on this work, we construct in this paper sum of the tails identities for which some which some of these functions occur as error terms. As an application, we obtain formulas for the generating function of a certain zeta functions for real quadratic fields at negative integers.     

Testing optimality for quadratic 0–1 unconstrained problems

This paper analyses a necessary and sufficient optimality condition for quadratic pseudo-Boolean unconstrained problems. It is proved that in general testing any necessary and sufficient optimality condition is a difficult task for anyNP-hard problem. An-optimality condition is derived together with an approximation scheme to test it.This work has been supported by the Progetto Finalizzato Trasporti 2, 93.01799.PF74.Professor Paolo Carraresi died unexpectedly on March 5, 1994. At the time of his death this paper had been completed. While undertaking the final revision, the other two authors were reminded just how much they were indebted to Professor Carraresi after many years of common work together.     

The foundation of the grey matrix and the grey input–output analysis

The grey systems theory aims at the objects that their information is inadequate and this situation is general in reality. It has been urgent work to study the uncertain problems using the missing information. With the help of the simple introduction of grey systems theory, we further study the covered operation and get some calculation rules about grey number. The definition of grey matrix (GM) and its covered operation are proposed. Particularly, some results of the inverse grey matrix are obtained. Also with the help of the proposed grey matrix theory and the traditional input–output analysis, we propose the grey input–output analysis. The most important results are the computational formulas and their rigorous proofs of the matrix-covered set of the inverse grey Leontief coefficient’s matrix. It provides an effective tool to study an economic system by the input–output analysis under the uncertain situation. The modified case verifies the effectiveness of our methodology.