Calculus and Analysis in Early 19th-Century Britain: The Work of William Wallace

Scottish-born William Wallace (1768–1843) was an early exponent of the differential calculus in Britain and translator of French mathematical works. Encyclopaedias published during the early 19th century provided a valuable educational resource, to which Wallace and his colleague, James Ivory, contributed. Wallace's encyclopaedia articles on “Fluxions” and his other analytical writings are examined here, as are his relations with James Ivory, John Herschel, and others. Wallace's long 1815 article on “Fluxions” in the Edinburgh Encyclopaedia was the first complete account of calculus, using differential notation, to be published in English. There, he attempted an original and rigorous “doctrine of limits,” which deserved more attention than it received. In 1832, while professor of mathematics in Edinburgh, he applied analysis to support the reform of taxation proposed in the Reform Bill. It is suggested that the later neglect of Wallace's achievements is attributable to a mix of personal, institutional, political, and national rivalries. Copyright 1999 Academic Press.William Wallace né en Écosse (1768–1843) fut un interprète précoce du calcul différentiel en Grande Bretagne et traduisit des ouvrages mathématiques de langue française. Les encyclopédies publiées au début du XIXe siècle fournirent une précieuse ressource pédagogique à laquelle contribuèrent Wallace et son collègue James Ivory. Les articles encyclopédiques de Wallace sur les “Fluxions” et d'autres écrits analytiques font l'objet du présent article; et aussi ses relations avec James Ivory, John Herschel et d'autres mathématiciens. Le long article de Wallace sur les “Fluxions” paru dans l'Edinburgh Encyclopaedia de 1815 fut le premier exposé complet du calcul, utilisant la notation différentielle, à être publié en anglais. Dans cet article, il mit en oeuvre une “doctrine des limites” originale et rigoureuse, qui méritait plus d'attention qu'elle ne reçut. En 1832, alors qu'il était professeur de mathématiques à Edimbourg, il utilisa l'analyse pour soutenir la réforme fiscale proposée dans le Projet de loi de Réforme. Cet article suggère que l'oubli postérieur de l'oeuvre accomplie par Wallace doit être attribué à un mélange de rivalités personnelles, institutionnelles, politiques et nationales.     

Mathematics Teaching in Italy: A Cross-Cultural Video Analysis

This study investigates the cultural nature of teaching. It compares a sample of 39 videotaped Italian mathematics lessons to German, Japanese, and U.S. lessons videotaped in TIMSS. This study expands on earlier work that was based on a smaller sample; analysis is also extended to the nature of the mathematical content presented. The results confirm the existence of an Italian cultural pattern for mathematics teaching, whose features we outline here. Italian teachers prefer whole-class instruction to individual seatwork; they engage in teacher talk/demonstration to transmit information; and they often call on students to solve problems at the board before the rest of the class. Italian lessons are characterized by the inclusion of a large number of mathematical principles and properties. These are explained 50% of the time, and simply stated the rest of the time. This study adds yet another perspective from which mathematics teaching can be studied, and, by acknowledging the difficulty to change cultural practices, it offers practical implications for teacher learning.     

A Comparison of Preservice Elementary Teachers' Beliefs About Mathematics and Teaching Mathematics: 1968 and 1998

The study replicates Collier's (1972) work. It focuses on the beliefs of a large sample of elementary education students at four stages of teacher preparation, about both the nature of and the teaching of mathematics. The instrument measures what Collier termed a “formal‐informal” dimension of belief. The data suggest that initially the 1998 students held significantly more informal (constructivist) beliefs than did their 1968 counterparts. In both years, students moved toward more informal beliefs during the course of their programs, with the most significant changes occurring in their beliefs about how mathematics should be taught. However, apparent contradictions in belief structures were observed both at the start and at the end of their programs. Thus, it appears that though many students acquired new, more informal beliefs during the course of their programs, they did not develop robust, consistent philosophies of mathematics education.     

Mathematics Teaching and Learning as a Mediating Process: The Case of Tape Diagrams

This article examines how visual representations may mediate the teaching and learning of mathematics over time in Japanese elementary classrooms. Using the Zone of Proximal Development Mathematical Learning Model (Murata & Fuson, 2006 Murata, A. and Fuson, K. C. 2006. Teaching as assisting individual constructive paths within an interdependent class learning zone: Japanese first graders learning to add using ten. Journal for Research in Mathematics Education., 37(5): 421–456. [Web of Science ®] , [Google Scholar]; Fuson & Murata, 2007 Fuson, K. C. and Murata, A. 2007. Integrating the NRC principles and the NCTM process standards: Cognitively guided teaching to individualize instruction within whole-class activities and move all students within their learning path. National Council of Supervisors of Mathematics Journal, 10(1): 72–91.  [Google Scholar]), the process of mediation is explicated. The tape diagram, a central visual representation used in Japanese mathematics curriculum, is explored for its roles and the student learning that is intended to be mediated over time, illuminating aspects of the process. The study argues that the consistent and coherent use of one representation can bridge student understanding over time, focusing on mathematical relationships and problem-solving processes. The study also suggests different instructional approaches between U.S. and Japanese curricula that are reflected in the uses of representations.     

Connected graphs as subgraphs of Cayley graphs: Conditions on Hamiltonicity

Let Γ be a connected simple graph, let V(Γ) and E(Γ) denote the vertex-set and the edge-set of Γ, respectively, and let n=|V(Γ)|. For 1≤i≤n, let e_i be the element of elementary abelian group which has 1 in the ith coordinate, and 0 in all other coordinates. Assume that V(Γ)={e_i∣1≤i≤n}. We define a set Ω by Ω={e_i+e_j∣{e_i,e_j}∈E(Γ)}, and let Cay_Γ denote the Cayley graph over with respect to Ω. It turns out that Cay_Γ contains Γ as an isometric subgraph. In this paper, the relations between the spectra of Γ and Cay_Γ are discussed. Some conditions on the existence of Hamilton paths and cycles in Γ are obtained.     

关于大学数学的创造性思维教学模式的探讨

传统的注入式大学数学教学已无法适应现代社会的发展,培养学生创造性思维的能力,建立全新的大学数学教学模式,已成为大学数学教学的首要任务.本文对此做了一些初步的探讨.     

Cayley graphs as classifiers for data mining: The influence of asymmetries

The endomorphism monoids of graphs have been actively investigated. They are convenient tools expressing asymmetries of the graphs. One of the most important classes of graphs considered in this framework is that of Cayley graphs. Our paper proposes a new method of using Cayley graphs for classification of data. We give a survey of recent results devoted to the Cayley graphs also involving their endomorphism monoids.     

一本适用于线上线下相结合的混合式教学的高等数学教材

随着以MOOC为代表的国内外在线开放课程的飞速发展,高校在课程建设、教学方法和教学模式等方面都在进行相应的改革.高等教育出版社紧密配合高校的教学改革,结合当前线上线下相结合的混合式教学的新趋势,在数字化教材方面也进行了很多有益的探索,推出了纸质教材+数字课程的教材出版新模式.文中以高等教育出版社出版的朱士信、唐烁主编的《高等数学(上、下)》为例,介绍了该书在教材建设、教学资源建设等方面的工作,并对学生使用平台学习的数据进行了简要的分析.     

Math Autobiographies: A Window into Teachers' Identities as Mathematics Learners

Mathematics autobiographies have the potential to help teachers reflect on their identities as mathematics learners and to understand their role in the development of their students' mathematics identities. This paper reports on a professional development project for K‐2 teachers (n = 41), in which participants were asked to write mathematics autobiographies. Using an adaptation of an existing framework for characterizing teachers' mathematics stories, we describe the consistencies among the participants' experiences as mathematics learners and the events that are identified as being the impetus for a transition from a negative to a positive attitude toward mathematics. Implications for both teachers and teacher educators are presented.     

Music as Embodied Mathematics: A Study of a Mutually Informing Affinity

The argument examined in this paper is that music – when approached through making and responding to coherent musical structures,facilitated by multiple, intuitively accessible representations – can become a learning context in which basic mathematical ideas can be elicited and perceived as relevant and important. Students' inquiry into the bases for their perceptions of musical coherence provides a path into the mathematics of ratio,proportion, fractions, and common multiples. Ina similar manner, we conjecture that other topics in mathematics – patterns of change,transformations and invariants – might also expose, illuminate and account for more general organizing structures in music. Drawing on experience with 11–12 year old students working in a software music/math environment, we illustrate the role of multiple representations, multi-media, and the use of multiple sensory modalities in eliciting and developing students' initially implicit knowledge of music and its inherent mathematics. This revised version was published online in July 2006 with corrections to the Cover Date.     

新形势下高等数学分级教学模式的探索

突破以前按照不同专业分层次教学的模式,根据学生不同的学习目的,采用一种新的分级教学模式,针对相同专业学生不同的学习要求,给予不同层次的培养.通过对教育观念的新认识和培养目标的再确定,在高等数学的课程体系,课程内容,教学方法,教学手段等方面进行改革和调整,因材施教,使各层次的学生真正受益,提高整体教学质量,也使基础课教育更有针对性.     

Practitioner's Research: A Study in Changing Preservice Teachers’ Conceptions About Mathematics and Mathematics Teaching and Learning

Educators involved in the current reform movement in mathematics education recommend that students be actively involved in constructing their own knowledge and developing powerful mathematical concepts. This study suggests how ideas based on constructivist learning theory can be put into practice in a preservice mathematics education class.     

浅议独立学院高等数学教学——以情促学,以需导学

立足笔者教学实践,在分析独立学院学生学习心理及需求基础之上,针对高等数学学习现状与呈现的问题,探索"以情感交流促进教学,以需求引导提升学习"的高等数学教学模式.