高等数学在线教学的设计与实践

2020年春天一场突发的新冠疫情让全国大中小学的教师面临全新的在线教学,在“停课不停教、停课不停学”、确保人才培养“不断线、不降质”的思想指导下,根据高等数学在线教学的全过程,归纳总结了课前、课中、课后各个实施环节设计与实践过程,并给出了在线教学的若干教学案例.     

“线上教学”背景下的中考数学复习

“线上教学”,顾名思义,就是采用线上的方式讲授教学内容,学生在线上和教师沟通疑虑,教师解惑答疑.对九年级学生来说,线上教学很新鲜,但也无奈.据部分班级任课教师统计的反馈结果,80%以上的学生认为学得不满意,或觉得没有在教室面对面听教师讲课效果好.     

疫情背景下初中数学“线上教学”效度提升的策略研究

在新冠肺炎疫情影响下,教育部提出了“停课不停学”的教学指导方针.如何在疫情防控期间,提升线上教学的效度是各年级各科教师密切关注的问题.本文基于这一现实背景,在结合个人多年初中数学教学经验的基础上,对疫情背景下初中数学开展“线上教学”的必要性、原则及策略展开了详细探讨.     

单元教学设计浅析——以“分式”为例

数学单元教学设计是在整体思维的指导下,从培养学生的综合素质出发,根据学生的认知发展特点、数学学习需要等因素对课程知识进行重新整合的教学设计.教师在教学实施过程中,需要讲解的不是单一的知识点,而是一个单元的知识理论内容.要求教师在对教材单元整体分析的基础上,把握单元中数学知识的各个元素,并确定其中的核心要素,再借助所具备的教学能力,经由学生心理上发生知识的环节,将知识自然地传授给学生.本文以“分式”一章为例,进行“单元结构教学设计”,以更好地发挥数学知识的教育教学价值。     

基于动态平衡的课堂“五环节”设计

教学是一个有预设、有生成的过程。华东师范大学教授叶澜曾说过：“对教师而言，如果将其教学工作任务进行高度的概括，我们就会抽取出两个最核心的要素--‘教什么’和‘怎么教’，即教学预设和课堂生成。”在新课程改革中，更需要把握“生成”与“预设”之间的平衡，在平衡中寻求教学效果的最优化。如何在课堂教学中实现“预设”和“生成”的动态平衡（这里说的“平衡”不是0。5，是0。618，是“生成”与“预设”之间的黄金分割点）？关键是要从课堂的五个主要环节入手，即从新课引入、问题设计、解题过程、反思小结和巩固训练等环节入手，从学生的最近发展区出发，关注学生的认知差异，关注学生的发展，让“生成”为每个学生创造主动投身教学活动的机会，用教师已有的实践智慧对课程资源灵活机智地加以激活、捕捉和运用，促使课堂向多角度、全方位、高效率的目标发展。     

创新出版模式 助力大学数学教学改革

当前,国内外在线开放课程资源日渐丰富,各类优质资源为组织"翻转课堂"等混合式教学模式改革提供了可能和便利.文章着眼于如何利用现有资源,通过新形态教材的出版和数字课程的出版与定制应用,服务教师进行线上、线下相结合的教学实践,帮助学生自主学习、提高学习兴趣的同时,实现教师自身教学能力的提升,从而提高大学数学课程质量.     

精心构思板书,传递研究方法——以“图形的旋转”新授课为例

网络教研时代,我们足不出户就可以收看各地的线上教研活动的直播或视频回放.比如,近期我们观摩学习了某地级市优秀课展评活动,课题是“图形的旋转(第1课时)”,来自各县区的上课教师共21人参加了展评活动,笔者在线上观摩学习了全部21人的课堂教学,展评课教师的教学基本功、课堂组织与驾驭能力都是非常优秀的,最让笔者印象深刻的是,有好几位教师的板书设计比较新颖,值得学习.本文摘选3个有代表性的板书案例,进行简要评析,并进一步思考板书设计对提升教学品质的价值,与同行研讨.     

“336”混合式教学在初中数学试卷讲评课中的运用——以《整式乘法与因式分解》为例

随着信息化不断发展,知识获取方式和传授方式、教和学关系都发生了革命性变化.“336”混合式教学模式,充分发挥“线上教学”与“面授教学”各自的优势,是提高教学成效的有效途径.结合教学实践探讨“336”混合式教学在初中数学试卷讲评课中的运用,以引起一线数学教师的关注与思考.     

消除解题教学中“懂而不会”现象的例说

数学的解题教学是一种特殊的教学形式,在整个学习活动中是十分重要的环节,灵活运用所学的数学知识与技能解决问题,是“会”数学知识的表现．但在实际教学中,许多教师追求浅层次的“懂操作”,或是深层次的“是什么”与“为什么”,长期这样,就会造成解题教学中的“懂而不会”现象．相反,如果教师能够引导学生反思解题教学,使学生正确理解运算求解的含义、发展学生的思维能力、完善学生的知识结构、加强学生对算理和算法的认识,就会消除解题教学中的“懂而不会”现象．     

“题组诱导复习法”一例

“题组诱导复习法”主要是通过题组把教学的重心从教师转移到学生。通过教师的启发诱导,分析解题思路,使学生悟其原理,明其方法,晓其变化。“题组”的选题必须注意针对性、典型性、灵活性和变通性:教师点拨的适时,启发的恰当是上好课的关键。在教学中,我们以启发式为主线,通过“启”、“讲”、“诱”、“练”、“议”相结合的五步教学法,完成整个复习教学的任务。下面是我们的“一元二次方程”复习课的教案。教学目的:使学生能系统的掌握一元二次方程的各种基本解法,突出各种解法的适用范围。教学重点和难点。重点:一元二次方程首项系数不为零。难点:一元二次方程有实根,必须判别式△≥0. 教学过程:     

On the residual properties of link groups

We study the residual properties of finitely generated linear groups. Using the methods under consideration, we prove the residual 2-finiteness of the groups of the Whitehead link, the Borromean links (answering a question of Cochran), and some other links. We show also that each link is a sublink of some link whose group is residually 2-finite.     

Pedogogical perspectives on the method of Wilmer III and Costa for solving second-order differential equations

Recently, Wilmer III and Costa introduced a method into the mathematics education research literature which they employed to construct solutions to certain classes of ordinary differential equations. In this article, we build on their ideas in the following ways. We establish a link between their approach and the method of successive approximations. We show how applying the method of approximations naturally leads to the constructed approximation of Wilmer III and Costa. The new link is important because it enables us to respond to several challenges posed by Wilmer III and Costa. This includes addressing issues raised therein with convergence of their recursively constructed sequence of functions, and responding to their call regarding more mathematical rigour when relaxing the polynomial condition on the coefficients in the differential equation. Furthermore, the new link is pedagogically significant because it also opens up new pedagogical points of view. For example, the results in this paper provide potentially alternate, but overlapping, perspectives that are suitable for, and jointly inform, the learning and teaching of solution methods to differential equations. The value of this is supported by the presumption of Tisdell that teaching multiple ways to solve the same problem has academic and social value.     

高等数学中的“例证法”思想

介绍了几何定理机器证明中的“例证法”思想,并挖掘了高等数学中几个“例证法”实例,目的在于建议教师在教学过程中要善于发现并向学生输送新的思想方法,以提高教学质量.     

Some families of -cyclic codes

We introduce and solve several problems on -cyclic codes.We study the link between -linear cyclic codes and -cyclic codes (not necessarily linear) obtained by using two binary linear cyclic codes. We use these results to present a family of -self-dual linear cyclic codes.     

Reforming and Assessing Undergraduate Science Instruction Using Collaborative Action‐Based Research Teams

Faculty members at Purdue University in the departments of Earth and Atmospheric Sciences, Biological Sciences, and Chemistry conducted a reform effort for the undergraduate curriculum utilizing action‐based research teams. These action‐based research teams developed, implemented, and assessed constructivist approaches to teaching undergraduate science content in each department. This effort utilized a partnership of scientists, science educators, master teachers, graduate students, and undergraduate students. Results indicated that the project partners were able to (a) implement more inquiry‐based teaching that emphasized conceptual understanding, (b) provide opportunities for cooperative learning experiences, (c) use models as an ongoing theme, (d) link concepts and models to real‐world situations, e.g., field trips, (e) provide a more diverse range of assessment strategies, and (f) have students present their understandings in a variety of different forms. Further, we found that we were able to (a) involve graduate and undergraduate students, classroom teachers, scientists, and science educators together to work on the reform in a collaborative manner, (b) bring multiple perspectives for teaching and for science to support instruction and, (c) provide scientists and graduate science students (who will become university professors) with more effective teaching models. We also found that the collaborative action‐based research process was effective for contributing to the reform of undergraduate teaching.     

The Transparent and the Invisible in Professional Pedagogical Vision for Science Teaching

Science teacher educators use examples of practice to support teacher candidates (TCs) learning to engage in new forms of science teaching. However, interpretation of these examples assumes a level of expertise about practice TCs lack. This article describes a study designed to determine some of the differences between expert teachers' and TCs' professional pedagogical vision for science teaching. Specifically, the study examines what each group attends to (highlights) in examples of science teaching and how they interpret the events they attend to (codes). Both groups were asked to analyze video of classroom science teaching. Differences were found between TCs and expert teachers in terms of both highlighting and coding of science teaching practice. Four key areas of difference are described in detail: actor focus, questions, grain size, and enactment. The implications for science teacher education are discussed, in particular a set of tools and teacher education practices to support acculturating TCs into more a sophisticated professional pedagogical vision for science teaching. This article features a Research to Practice Companion Article . Please click on the supporting information link below to access.     

Multi-objective OLSR for proactive routing in MANET with delay,energy, and link lifetime predictions

In this paper, we develop a multi-objective approach for proactive routing in a Mobile Ad Hoc Network (MANET). We consider three routing objectives: minimizing average end-to-end delay, maximizing network energy lifetime, and maximizing packet delivery ratio. Accordingly, we develop three routing metrics: mean queuing delay on each node, energy cost on each node, and link stability on each link. For the proposed multi-objective approach, we develop efficient prediction methods: (a) predicting queuing delay and energy consumption using double exponential smoothing, and (b) predicting residual link lifetime using a heuristic of the distributions of the link lifetimes in MANET. Extensive simulation (by using ns2) is performed for the comparison of this multi-objective OLSR with existing OLSRs. The results show that the multi-objective OLSR is effective in finding optimal routing by tradeoffs among proposed objectives.     

数学实验课程的教学与数学素质的培养

分析了现行工科数学教学中存在的主要问题 ,认为数学实验课程的教学是培养数学素质不可缺少的环节 ,并就如何更好地发挥数学实验课程教学的作用提出了一些具体的建议 .     

Heuristics for Two-Level Hierarchical Transportation Networks

Given a set of points, we wish to design a network consisting of a primary link and a set of secondary links connecting the points to the primary link. The objective of the problem is to find the location and length of the primary link in order to minimize the sum of its weighted length and the weighted lengths of all secondary links. We assume that the weight of the secondary link from any point varies depending on the location of that point. In this paper, we describe efficient algorithms and their computer implementation for two scenarios of this problem. In the first scenario, only direct secondary links are allowed from each point to the primary link. In the second scenario, the secondary link from a point is allowed to pass through other points before reaching the primary link.