Reviewers for Volume 10

This article compares first-year university students' development of the concept of limits to mathematicians' historical development of the concept. The aim was to find out if students perceive the notion as mathematicians of the past did, as understandings of the concept evolved. The results imply that there are some similarities—for example, the struggle with rigor and attainability. Knowledge of such critical areas can be used to improve students' opportunities of learning limits of functions. Some teaching aspects related to the study are also discussed.     

关于极限的讲法

众所周知,极限是高等数学所有重要概念的基础,又是教学的公认难点,特别是用“ε-δ”语言证明极限的题目．本文是作者在新疆大学讲授这一内容时所做的一些探索性工作,主要方法是：提前引入无穷小概念,以特殊无穷小为标准尺对极限证明题进行论证,使得证明思路清晰、论述简单、易于掌握．     

关于极限的讲法

众所周知,极限是高等数学所有重要概念的基础,又是教学的公认难点,特别是用"ε-δ"语言证明极限的题目.本文是作者在新疆大学讲授这一内容时所做的一些探索性工作,主要方法是:提前引入无穷小概念,以特殊无穷小为标准尺对极限证明题进行论证,使得证明思路清晰、论述简单、易于掌握.     

COGARCH as a continuous-time limit of GARCH(1,1)

COGARCH is an extension of the GARCH time series concept to continuous time, which has been suggested by Klüppelberg, Lindner and Maller [C. Klüppelberg, A. Lindner, R. Maller, A continuous-time GARCH process driven by a Lévy process: Stationarity and second order behaviour, Journal of Applied Probability 41 (2004) 601–622]. We show that any COGARCH process can be represented as the limit in law of a sequence of GARCH(1,1) processes. As a by-product we derive the infinitesimal generator of the bivariate Markov process representation of COGARCH. Moreover, we argue heuristically that COGARCH and the classical bivariate diffusion limit of Nelson [D. Nelson, ARCH models as diffusion approximations, Journal of Econometrics 45 (1990) 7–38] are probably the only continuous-time limits of GARCH.     

Middle School Mathematics Teachers' Knowledge of Students' Understanding of Core Algebraic Concepts: Equal Sign and Variable

This article reports results from a study focused on teachers' knowledge of students' understanding of core algebraic concepts. In particular, the study examined middle school mathematics teachers' knowledge of students' understanding of the equal sign and variable, and students' success applying their understanding of these concepts. Interview data were collected from 20 middle school teachers regarding their predictions of student responses to written assessment items focusing on the equal sign and variable. Teachers' predictions of students' understanding of variable aligned to a large extent with students' actual responses to corresponding items. In contrast, teachers' predictions of students' understanding of the equal sign did not correspond with actual student responses. Further, teachers rarely identified misconceptions about either variable or the equal sign as an obstacle to solving problems that required application of these concepts. Implications for teacher professional development are discussed.     

浅谈高等数学教学中的概念引入模式

本文介绍了几何图示等五种概念引入模式．提出高等数学教学中应高度重视概念引入模式的观点,试图以此作为提高教学水平与学生数学素质的一个新的突破口.     

Ecological accounting and evaluation of urban economy: Taking Beijing city as the case

Urban economy is confronted with increasing biophysical limitations derived from the exhaustion of natural resources and the depletion of environmental capacity, and human cultural diversity has been declining during the fast urbanization. The conventional anthropocentric economics, regarding the natural environment as the ‘exterior’ of human economy, is invalid in the scientific evaluation on the contribution of natural resources and environment as well as human culture when facing the current urban crises. The theory of embodied cosmic exergy, as the latest development of ecological economics and ecological thermodynamics, is introduced in this study to construct an ecological evaluation framework of urban economy. The advantage of embodied cosmic exergy dedicated to ecological economics has been discussed in comparison with other ecological evaluation alternatives. Transformities describing hierarchies and manifesting quality are systematically calculated and tabulated. A new framework of embodied cosmic exergy based on network accounting (EmexNA) is sketched out in this study, taking not only diversity flows but also ecological stocks into consideration. The stock based concept of ‘ecological wealth’ and the flow based concept of ‘ecological cost’ as well as related evaluation indicators are developed based on EmexNA. Taking Beijing city as the case, the network accounting and related ecological evaluation of a practical urban economy are carried out in this study in the light of the basic social, economic and environmental data available from 1990 to 2005 of Beijing. The system construction and the ecological mechanism of the development of Beijing economy are correspondingly illuminated and discussed.     

Simulation as a pedagogical tool for managerial decision-making in a transition economy

Eastern European countries are undergoing a transition from centralized economic planning to more open economic systems. A team of Bulgarian and US researchers collaborated to study this problem, using a Bulgarian winery as the focus of their research. The study resulted in development of a business simulation of the winery, with the purpose of generating a pedagogical tool for knowledge acquisition by winery management as well as Bulgarian business students. This paper discusses the concept of simulation as a pedagogical tool, outlines the purposes of this simulation, reviews the development of this model using soft systems approaches, and suggests its applicability for any pedagogical learning situation but more specifically to the general operations of the firm at the microeconomic level of decision-making.     

Struggles With Developing the Concept of Angle: Comparing Sixth-Grade Students' Discourse to the History of the Angle Concept

This article highlights the similarities between sixth-grade students' developing notions of angle and mathematicians' struggles to define this complex concept. Students learning from a reformed curriculum had the opportunity to voice their ideas concerning the definition of angle. These conversations were compared with past discussions and debates concerning the meaning of angle. The data were classified into three major categories: (a) What exactly is being measured when referring to the size of angles? (b) Can angles contain curves?, and (c) Difficulties with conceiving of 0°, 180°, and 360° angles. The data suggest that using multiple representations such as those examples from history would seem to be more comprehensive and a better preparation as students study the concept of angle.     

Ranked Highly-Arc-Transitive Digraphs as Colimits

This paper builds on work characterising digraphs with rich transitivity properties. When such digraphs are equipped with a rank function onto a colexicographic power of ? they arise as direct limits of better understood structures. These structures, it will be seen, contain sufficient information to describe the cycles of the full structure.     

U.S. and Chinese Teachers' Constructing, Knowing, and Evaluating Representations to Teach Mathematics

This study examined U.S. and Chinese teachers' constructing, knowing, and evaluating representations to teach mathematics. All Chinese lesson plans are very similar, because they are all based on the Chinese national unified curriculum in mathematics. However, the U.S. lesson plans are extremely varied, even for those teachers from the same school. The Chinese teachers' lessons are very detailed; the U.S. teachers' lesson plans have exclusively adopted the “outline and worksheet” format. In the Chinese lesson plans, concrete representations are used exclusively to mediate students' understanding of the concept of average. In U.S. lessons, concrete representations are not only used to model the averaging processes to foster students' understanding of the concept, but they are also used to generate data. The U.S. teachers are much more likely than the Chinese teachers to predict drawing and guess-and-check strategies. For some problems, the Chinese teachers are much more likely than are the U.S. teachers to predict algebraic approaches. For the responses using conventional strategies, both the U.S. and Chinese teachers gave them high and almost identical scores. If a response involved a drawing or an estimate of an answer, the Chinese teachers usually gave a relatively lower score, even though the strategy is appropriate for the correct answer, because it is less generalizable. This study contributed to our understanding of the cross-national differences between U.S. and Chinese students' mathematical thinking. It also contributed to our understanding about teachers' beliefs from a cross-cultural perspective.     

微言要义之数列的上下极限

本文分别从语义,数学和教学三个方面讨论上下极限与极限概念之间的差别和联系,以求厘清差别,透彻理解上下极限概念的本质.     

Robustness and Optimality as Criteria for Strategic Decisions

The use of "optimality" as an operational research criterion is insufficiently discriminating. Ample evidence exists that for many problems simple optimization (particularly profit maximization) does not represent the aims of management. In this paper we discuss the nature of the problem situations for which alternative decision criteria are more appropriate. In particular the structure of strategic planning problems is analysed. The provisional commitment involved in a plan (in contrast to the irrevocable commitment of a decision) leads to the development of a particular criterion, robustness—a measure of the flexibility which an initial decision of a plan maintains for achieving near-optimal states in conditions of uncertainty. The robustness concept is developed through the case study of a sequential factory location problem.     

PRELOCALIZATIONS AND NATURAL NUMBERS

Abstract

A natural concept of pointed endofunctors which are close to localizations with respect to both, the reflection property and the preservation of finite limits, is introduced and characterized in terms of factorization systems. These results are applied to two categories of natural numbers.     

A Framework for Monitoring Progress and Planning Teaching Towards the Effective Use of Computer Algebra Systems

This article suggests a framework to organise a cluster of variables that are associated with students' effective use of computer algebra systems (CAS) in mathematics learning. Based on a review of the literature and from the authors' own teaching experience, the framework identifies the main characteristics of students' interactions with CAS technology and how these may be used to monitor students' developing use of CAS; from this, the framework may be used to plan teaching in order to gain greater benefit from the availability of CAS. Four case studies describing students' development over a semester are reported. These demonstrate a variety of combinations of technical competencies and personal attributes. They indicate the importance of both the technical and personal aspects but suggest that negative attitudes rather than technical difficulties can limit the effective use of CAS. Finally practical suggestions are given for teaching strategies which may promote effective use of CAS.This revised version was published online in September 2005 with corrections to the Cover Date.     

非参数秩检验在宿舍因素对学风影响中的应用

学风建设是大学教育教学管理中的重要环节,对学生的学习成绩、顺利毕业和就业等都有至关重要的影响。宿舍是学生学习和生活的重要场所,本文应用非参数统计秩检验和中位数检验研究了宿舍对学生学业成绩的影响,得到目前大学宿舍因素对学生学业成绩的影响高度显著,进而建议高等学校应加强学生宿舍管理,把宿舍作为学风建设的重要环节。     

Short selling and replaning as tools to enhance revenues

Short selling and replaning have emerged from different sources as tools to enhance revenues. This paper presents these tools in the context of revenue management, and highlights their similarities and differences. Conditions under which the use of short selling and replanning enhance revenues are derived, and the impact of these tools on overbooking limits is investigated.     

Weight 2 CM newforms as p-adic limits

The Ramanujan Journal - Previous works have shown that certain weight 2 newforms are p-adic limits of weakly holomorphic modular forms under repeated application of the U-operator. The proofs of...     

Qualia Qua Qualitons: Mental Qualities as Abstract Particulars

In this paper we advocate the thesis that qualia are tropes (or qualitons), and not (universal) properties. The main advantage of the thesis is that we can accept both the Wittgensteinian and Sellarsian assault on the given and the claim that only subjective and private states can do justice to the qualitative character of experience. We hint that if we take qualia to be tropes, we dissolve the problem of inverted qualia. We develop an account of sensory concept acquisition that takes the presence of qualia as an enabling condition for learning. We argue that qualia taken to be qualitons are part of our mechanism of sensory concept acquisition.