Cross-curricular activities within one subject?

The structural organization, of the Danish Gymnasium greatly hinders cross-curricular activities. However, it is possible to integrate other subjects in the mathematics curriculum, not the least due to the existence of the so-called “aspects” I will discuss a particular course on modeling ozone depletion which was framed by the “model aspect”. The organization and outcome of the course are linked to three types of competencies mathematical. technological and reflective. I will focus on the reflective competency, in particular the criticla evaluation of mathematical models and their use. One conclusion is that modeling furthers all three competencies, and thus should be given more emphasis in mathematics instruction. However, if the reflective competency is to be furthered, the topic must be seen in a broader societal context, and this would be better supported by cross-curricular activities.     

数学建模课内容和教学方法的探讨

拟就以下内容进行了探讨 .(i)该课程究竟应该讲什么内容、怎样讲 ,才能使学生在较短的时间内 ,掌握数学建模的基本知识和基本方法 ;(ii)该课程怎样与数学实验更好地结合起来 ,以培养学生的动手能力 ;(iii)该课程应采用什么样的教学手段和教学方法 ,才能加大课堂信息量 ,加强直观性和趣味性等 .我们的解决方法是 :(i)以介绍建立数学模型为主 ,按数学知识内容的不同来选取数学模型的典型案例 ,通过案例介绍 ,使学生学会怎样建立模型 .(ii)适当介绍数学软件包 ,让学生掌握运用软件包来求解模型能力 .(iii)做大作业 ,教员给出题目 ,学生自己收集资料、讨论、上机求解 ,最后写出报告 .(iv)开展多媒体教学 ,对主要的教学内容进行模块化教学 ,将建模分成 1 4个专题 ,做成 1 4个多媒体课件     

研究生数学建模教学方法分析

面向研究生开设数学建模课程,可以提高研究生的培养质量,增强研究生开展高水平科研工作的能力．为了实现这些目标,数学建模教学需要兼顾基础方法的介绍和提供足够的深度、广度．本文主要介绍我们对研究生数学建模教学思路的一些认识,并给出了两个教学案例．     

模糊数学在数学建模中的应用

主要讨论模糊数学在数学建模课程和数学建模竞赛中的应用问题.在数学建模中应当引进模糊数学方法,可以得到一些计算较为简单,更符合实际情况的数学模型.     

From the principle of bijection to the isomorphism of structures: an analysis of some teaching paradigms in discrete mathematics

This paper is concerned with the teaching of Discrete Mathematics to university undergraduate students. Two to three decades ago this course became a requirement for math and computer science students in most universities world wide. Today this course is taken by students in many other disciplines as well. The paper begins with a discussion of a few topics that we feel should be included in the syllabus for any course in Discrete Mathematics, independent of the audience. We then discuss several potential models for teaching the course, depending upon the interests and mathematical background of the audience. We also investigate various educational links with other components of the curriculum, consider pedagogical issues associated with the teaching of discrete mathematics, and discuss some logistical and psychological difficulties that must be overcome. A special emphasis is placed on the role of textbooks.     

Recent advances of variational model in medical imaging and applications to computer aided surgery

In this paper, we review some mathematical models in medical image processing. Due to the superiority in modeling and computation, variational methods have been proven to be powerful techniques, which have been extremely popular and dramatically improved in the past two decades. On one hand, many models have been proposed for nearly all kinds of applications. On the other hand, a lot of models can be globally optimized and also many computation tools have been introduced. Under the variational framework, we focus on two basic problems in medical imaging: image restoration and segmentation, which are core components for kinds of specific tasks. For image restoration, we discuss some models on both additive and multiplicative noises. For image segmentation, we review some models on both whole image segmentation and specific target delineation, with the later being a key step in computer aided surgery. Additionally, we present some models on liver delineation and give their applications to living donor liver transplantation.     

Some aspects of studying an optimization or decision problem in different computational models

In this paper we want to discuss some of the features coming up when analyzing a problem in different complexity theoretic frameworks. The focus will be on two problems. The first is related to mathematical optimization. We consider the quadratic programming problem of minimizing a quadratic polynomial on a polyhedron. We discuss how the complexity of this problem might change if we consider real data together with an algebraic model of computation (the Blum–Shub–Smale model) instead of rational inputs together with the Turing machine model. The results obtained will lead us to the second problem; it deals with the intrinsic structure of complexity classes in models over real- or algebraically closed fields. A classical theorem by Ladner for the Turing model is examined in these different frameworks. Both examples serve well for working out in how far different approaches to the same problem might shed light upon each other. In some cases this will lead to quite diverse results with respect to the different models. On the other hand, for some problems the more general approach can also give a unifying idea why results hold true in several frameworks.The paper is of tutorial character in that it collects some results into the above direction obtained previously.     

Teachers' ways of listening and responding to students' emerging mathematical models

In this paper, I present the results of a case study of the practices of four experienced secondary teachers as they engaged their students in the initial development of mathematical models for exponential growth. The study focuses on two related aspects of their practices: (a) when, how and to what extent they saw and interpreted students' ways of thinking about exponential functions and (b) how they responded to the students' thinking in their classroom practice. Through an analysis of the teachers' actions in the classroom, I describe the teachers' developing knowledge when using modeling tasks with secondary students. The analysis suggests that there is considerable variation in the approaches that teachers take in listening to and responding to students' emerging mathematical models. Having a well-developed schema for how students might approach the task enabled one teacher to press students to express, evaluate, and revise their emerging models of exponential growth. Implications for the knowledge needed to teach mathematics through modeling are discussed.     

Ryle and Intentionality

After some opening comments on how I think one should approach the philosophy of mind, I look at what relatively little Gilbert Ryle had to say explicitly about intentionality, that occurring almost exclusively in his several papers on phenomenology. Then, I discuss the notion of intentionality with respect to the doctrines of The Concept of Mind, although neither the word nor the idea, strictly speaking, appears anywhere in the book. Following more exposition of my own views, including an argument I have made for a certain specific theory of intentionality, I close with some reflections on Ryle as a modern-day Aristotelian.     

Mathematical Modeling for Preservice Teachers: A Problem from Anesthesiology

The study reported in this article deals with the observed actions of prospective Swedish mathematics teachers as they were working with a modeling situation. These prospective teachers were preparing to teach in Grades 4 to9 or in the gymnasium (Grades 10 to 12) and were students in a course in mathematical modeling. The larger study of which this study was a part focused on these students' understanding of modeling and how they related mathematical models to the real world. This article also attempts to illustrate how mathematics is intertwined with many other subjects, in this case medicine. This revised version was published online in July 2006 with corrections to the Cover Date.     

数学建模对高中生构建数学底层思维的作用及其教学实施建议

数学底层思维即用数学的眼光观察世界、用数学的思维分析世界以及用数学的语言表达世界,是人们面对自然和社会中纷繁多样的现象和问题时,所展现的自发的、不依赖监督的、融汇数学学科核心素养的思维方式.作为国家高中新课程标准中数学六大核心素养之一的数学建模,是培养学生数学底层思维的良好载体,对人才培养和社会发展均起到良好的促进作用.本文主要阐述了数学建模对高中生构建数学底层思维的作用,并结合教学实例给出教学实施建议.     

The Role of Metanetworks in Network Evolution

The question of what structures of relations between actors emerge in the evolution of social networks is of fundamental sociological interest. The present research proposes that processes of network evolution can be usefully conceptualized in terms of a network of networks, or “metanetwork,” wherein networks that are one link manipulation away from one another are connected. Moreover, the geography of metanetworks has real effects on the course of network evolution. Specifically, both equilibrium and non-equilibrium networks located in more desirable regions of the metanetwork are found to be more probable. These effects of metanetwork geography are illustrated by two dynamic network models: one in which actors pursue access to unique information through “structural holes,” and the other in which actors pursue access to valid information by minimizing path length. Finally, I discuss future directions for modeling network dynamics in terms of metanetworks.     

Book Review

An Introduction to Difference Equations. Second Edition by Saber N. Elaydi, New York: Springer—Verlag, 1999. ISBN 0-387-98830-0. $54.95. Gone are the days when difference equations arose mainly in the context of sections of flows or as finite difference approximations to PDE's. Today difference equations have come into their own, both as objects of intrinsic mathematical interest and as dynamical models in their own right. Discrete models form an important part of dynamical systems theory independently from their continuous cousins. In Saber Elaydi's book dynamicists have the long awaited discrete counterpart to standard textbooks such as Hirsch and Smale (“Differential Equations, Dynamical Systems, and Linear Algebra”). The first edition of this book appeared in 1996. The second edition includes substantial new material including appendices on global stability and periodic solutions, a section on applications to mathematical biology, and a new chapter entitled “Applications to Continued Fractions and Orthogonal Polynomials”. Additional material on Birkhoff's theory now appears in the chapter on asymptotic behavior. The initial chapter covers first order equations, including equilibria, cobwebbing, stability, cycles, and the bifurcations of the discrete logistic equation. Chapter 2 moves on to higher order linear equations and briefly treats the difference calculus (for an in—depth treatment, see “Difference Equations: Theory and Applications. Second Edition” by Ronald E. Mickens, New York: Van Nostrand Reinhold, 1990). The subsequent chapters include systems of difference equations, stability theory, Z—transforms, control theory, oscillation theory, asymptotic behavior, and applications to continued fractions and orthogonal polynomials.

The chapters are composed of short sections, each of which ends with a nice selection of exercises. Answers to the odd—numbered problems appear in the back of the book. The core chapters include sections of applications to various fields such as population biology, economics, and physics. Several famous examples and topics are treated in the applications, including Gambler's Ruin, the Nicholson—Bailey host/parasitoid model, the heat equation, and Markov chains. Many discrete models are noninvertible, yet as many frustrated modelers know, most of the old standard treatments of linearization and the Stable Manifold Theorem., coming as they do from the context of sections of flows, require invertibility. Commendably, Elaydi avoids the needless assumption of invertibility in his stability theorems, and also in the Stable Manifold Theorem. However, invertibility is assumed in the Hartman—Grobman Theorem, where indeed it is necessary to establish conjugacy between the map and its linearization (see “An Introduction to Structured Population Dynamics”, CBMS—NSF Regional Conference Series in Applied Mathematics, Vol. 71, SIAM, Philadelphia, 1998 by J. M. Gushing, for an example of a noninvertible map for which the conjugacy fails. Readers may be interested to know that in this reference a weaker version of the Hartman—Grobman Theorem is proved that does not require invertibility but does establish the desired correspondence between types of hyperbolic equilibria in maps and their linearizations.)

This book is in Springer's Undergraduate Texts in Mathematics series and is indeed a very readable and appropriate text for advanced undergraduates or beginning graduate students. According to the author, the main prerequisites for such a course are calculus and linear algebra, with basic advanced calculus and complex analysis needed only for some topics in the later chapters. This is true; however in most situations the book would be best appreciated by students with a bit more mathematical maturity than is engendered by today's calculus and beginning linear algebra courses.Elaydi's book is a valuable reference for anyone who models discrete systems. It is so well written and well designed, and the content is so interesting to me, that I had a difficult time putting it down. I am pleased to own a copy for reference purposes, and am looking forward to using it to teach a senior topics course in difference equations.     

Multiagent Temporal Logics with Multivaluations

We study multiagent logics and use temporal relational models with multivaluations. The key distinction from the standard relational models is the introduction of a particular valuation for each agent and the computation of the global valuation using all agents’ valuations. We discuss this approach, illustrate it with examples, and demonstrate that this is not a mechanical combination of standard models, but a much more subtle and sophisticated modeling of the computation of truth values in multiagent environments. To express the properties of these models we define a logical language with temporal formulas and introduce the logics based at classes of such models. The main mathematical problem under study is the satisfiability problem. We solve it and find deciding algorithms. Also we discuss some interesting open problems and trends of possible further investigations.     

From generalized kinetic theory to discrete velocity modeling of vehicular traffic. A stochastic game approach

This work reports on vehicular traffic modeling by methods of the discrete kinetic theory. The purpose is to detail a reference mathematical framework for some discrete velocity kinetic models recently introduced in the literature, which proved capable of reproducing interesting traffic phenomena without using experimental information as modeling assumptions. To this end, we firstly derive a general discrete velocity kinetic framework with binary nonlocal interactions. Then, resorting to some ideas of stochastic game theory, we outline specific modeling guidelines for vehicular traffic, and finally we discuss the derivation of the above-mentioned vehicular traffic models from these mathematical structures.     

Neuansätze und eine andere Sichtweise des mathematischen und naturwissenschaftlichen Unterrichts

In this paper, we discuss the question of how mathematics (in a typical manner) can contribute to general abilities aimed at at school, to general education and to the “Allgemeinbildung” of the pupils (especially of higher ages and in secondary schools). Our discussion concerns contributions of mathematics education in addition to providing mathematical literacy, technological aspects and all those concrete mathematical abilities necessary for “modern life”. Amongst others, the paper was motivated by the results of the international TIMS-studies (TIMSS) and—as well—by the discussions caused by the book of H. W. Heymann (1996) in Germany which, in many cases, had been held in a wrong way. Of course, the questions as well as some of our results are old ones, but they have to be discussed under new aspects from time to time, and they should be illustrated by concrete examples.     

Mathematical Theory of Fluids in Motion

The goal of this paper is to present the recent development of mathematical fluid dynamics in the framework of classical continuum mechanics phenomenological models. In particular, we discuss the Navier–Stokes (viscous) and the Euler (inviscid) systems modeling the motion of a compressible fluid. The theory is developed from fundamental physical principles, the necessary mathematical tools introduced at the moment when needed. In particular, we discuss various concepts of solutions and their relevance in applications. Particular interest is devoted to well-posedness of the initial-value problems and their approximations including possibly certain numerical schemes.     

Describing cognitive orientation of Calculus I tasks across different types of coursework

We discuss the findings of an analysis of cognitive orientation of 4,953 mathematical tasks (representing all bookwork, worksheets, and exams) used by five instructors teaching Calculus I in a two-year college in the United States over a one-semester period. This study uses data from one of 18 cases from the Characteristics of Successful Programs in College Calculus. We found differences in the cognitive orientation by type of course work assigned (graded vs. ungraded) and differences by the instructors who assigned the course work. We discuss implications for practice and propose some areas for further exploration.     

Improving PRETREE's predictive capabilities

Compared to compensatory type choice models (e.g., logit), hierarchical choice models have been used much less frequently in the real world. This has happened even though there is a lot of evidence to suggest that the hierarchical structure is closer to the actual process by which individuals make choices. In this paper, we have considered one of the original hierarchical models (PRETREE). The PRETREE model has been heavily cited in the marketing and psychometric literatures. We wish to examine PRETREE at the aggregate level making n-way (market share) predictions.A primary strength of the PRETREE model was its apparent ability to capture violations of the property of the “independence of irrelevant alternatives” (IIA). We prove that there are limitations to this argument. We suggest a method to overcome these limitations; this change resulted in substantial improvement in the predictive abilities of the model, when the modified PRETREE is applied to real world data. The predictions are at an aggregate level on a hold-out sample.The above modification to PRETREE will hopefully not only renew but also increase interest in applying a hierarchical approach to modeling real-world problems on choice.     

The contribution of mathematical modelling to the practice of project management

This paper looks at the contribution that mathematical modellinghas made to project management over the past 50 years, and thecontribution it is currently making and can make in the future.Project Management started with well-defined foundations posingprecise, well-defined problems. In its growing phase, modellersplayed an essential role in taking the problems defined by theproject-management world and offering solutions, from the originalPERT, through resource allocation and levelling procedures,Monte Carlo simulation models, criticality analyses and so on.Since then, however, while the project management field itselfhas tried to establish its procedures, keeping to its philosophicalstance, much of the mathematical-modelling world has continuedalong its trajectory, producing ever more complex solutionsto ever more complex models, motivated by mathematical impressivenessrather than the need to solve real-world problems. This paperoutlines much of this work, some of which does find its wayinto project-network software but much of which languishes injournals. However, over the last decade or so, Operational Researchershave begun to build models of projects that are systemic anddynamic and explain many of the behaviours of projects thatconventional decomposition models do not; and at the same time,some of the Project Management world has started to realizethe limitations of its philosophical stance and started lookingto build new theory for modern, complex, dynamic projects. Asthese two trends come together, it is essential that modellersare at the forefront of building this new theory.