三聚氰胺金属(II)配合物的结构、紫外-可见光谱和反应活性

三聚氰胺是婴幼儿“肾结石事件”的重要前体. 本文选取几个典型的二价金属离子与三聚氰胺(L)形成的三聚氰胺金属配合物ML2(OH)2(M=Ca, Mg, Zn, Cu, Ni, Fe), 使用密度泛函理论(DFT)、含时DFT和概念DFT等工具, 系统地计算和比较了ML2(OH)2的结构、紫外-可见光谱和反应性质的异同. 模拟结果揭示了ML2(OH)2的结构、光谱及其反应性质是一类不同于其前体L, 形成ML2(OH)2配合物后, 将有较高的亲电指数和较低的化学硬度以及呈现红外吸收峰红移; 在这些典型的二价金属配合物中, 金属M离子电荷与配体O和N原子之间的电荷、以及与金属M离子和配体原子之间的二级微扰相互作用能, 配合物最低空轨道能级与其亲电反应指数、最低空轨道能级与化学硬度指数等方面, 存在着一系列定量的相关关系, 相关系数(R2)达0.889-0.997; 前线分子轨道模拟结果表明, ML2(OH)2体系反应活性的差异源于金属离子对前线轨道贡献有所不同, FeL2(OH)2、CuL2(OH)2、NiL2(OH)2等过渡金属离子的配合物中, 金属离子贡献较多, 共价性成分较多. 这些结果将为进一步理解人体内三聚氰胺致结石的成因提供有益的启示.     

Characterizing the growth of one student’s mathematical understanding in a multi-representational learning environment

The purpose of this study was to characterize the growth of one student’s mathematical understanding and use of different representations about a geometric transformation, dilation. We accomplished this purpose by using the Pirie-Kieren model jointly with the Semiotic Representation Theory as a lens. Elif, a 10^th- grade student, was purposefully chosen as the case for this study because of the growth of mathematical understanding about dilation she exhibited over time. Elif participated in task-based interviews before, during and after participating in a variety of transformation lessons where she used multiple representations, including physical and virtual manipulatives. The results revealed that Elif was able to progress in her mathematical understanding from informal levels to the formal levels in the Pirie-Kieren model as she performed treatments and conversions, movements involving different registers of representations. The results also showed numerous examples of Elif’s mathematical understanding based on folding back activities, complementary aspects of acting and expressing, and interventions. Using the two theories together provides a powerful and holistic approach to a deeper understanding of mathematical learning by characterizing and articulating the growth of mathematical understanding and the way of mathematical thinking.     

The analysis of the understanding of the three-dimensional (Euclidian) space and the two-variable function concept by university students

Conceptual understanding is being emphasized in mathematics education. Students often have difficulty understanding the multi-variable function, a key concept. Based on the APOS theory, which analyzes the cognitive structures formed by individuals in learning a mathematical concept and produces components related to that learning, this study analyzes the conceptual understanding of three-dimensional spaces and two-variable functions by university students. The genetic decomposition of these concepts proposed by Trigueros and Martinez-Planell is also considered. The analyzes results revealed that only one student constructed the concept of three-dimensional space as an object within the framework of genetic decomposition. Some students could not relate the concepts of two-variable function and three-dimensional space. Students who could perform algebraic operations had problems related to geometric representation. This study suggests the refinement of genetic decomposition to include, e.g., mental construction steps for writing algebraic equations of special surfaces whose graphs are given in R³.     

利用三段式诊断法探查化学平衡概念的理解

采用三段式诊断测试的方法,以浙江省某二级重点中学的高三化学选考生为研究对象,从“化学平衡状态的判定”“平衡常数与转化率”“平衡移动的影响因素”等3个维度,探查选考生在化学平衡概念理解上的表现。结果显示,选考生化学平衡概念理解的整体水平良好,但对平衡状态的速率判据理解不够全面,相异构想严重;转化率的计算能力和自信表现稍有不足;对催化剂、压强等平衡移动的影响因素存在认知缺陷,特别是压强因素,还未真正掌握压强影响平衡移动的本质机理。     

A model of students’ combinatorial thinking

Combinatorial topics have become increasingly prevalent in K-12 and undergraduate curricula, yet research on combinatorics education indicates that students face difficulties when solving counting problems. The research community has not yet addressed students’ ways of thinking at a level that facilitates deeper understanding of how students conceptualize counting problems. To this end, a model of students’ combinatorial thinking was empirically and theoretically developed; it represents a conceptual analysis of students’ thinking related to counting and has been refined through analyzing students’ counting activity. In this paper, the model is presented, and relationships between formulas/expressions, counting processes, and sets of outcomes are elaborated. Additionally, the usefulness and potential explanatory power of the model are demonstrated through examining data both from a study the author conducted, and from existing literature on combinatorics education.     

The Gibbs' phenomenon and harmonic oscillators with periodic forcing

We present an interesting nonlinear equation originating from a geometrical problem. We show how it can be solved numerically, how algorithmic differentiation is useful, and how the problem can also be solved analytically using the computer algebra system Maple.     

The numerical evaluation of the error term in Gaussian quadrature rules

A method for the numerical evaluation of the error term in Gaussian quadrature rules is derived by means of Chebyshev polynomials of the first kind.     

Developing teacher content understanding by integrating pH and logarithms concepts

Logarithms are notorious for being a difficult concept to understand and teach. Research suggests that learners can be supported in understanding logarithms by making connections between mathematics and science concepts such as pH. This study investigated how an integrated chemistry and mathematics lesson impacted 29 teachers’ understanding of the logarithmic relationship and pH. Pre- and post-test data indicated 23 teachers improved their understanding of logarithms and 28 improved their understanding of pH, suggesting that teacher educators in both science and mathematics context can use this approach to foster better understanding with their teachers and ultimately school students. Our analysis also identified professional development components and teacher characteristics associated with gains in understanding of pH and logarithms, which mathematics and science teacher educators can use to strategically adapt and implement the lesson within other teacher education settings.     

大概念的内涵解析及大概念教学设计与实施策略

将大概念作为一项重要的化学课程内容,是《义务教育化学课程标准(2022年版)》的重大变化之一。对大概念内涵从抽象知识到上位概念再到学科观念的发展过程进行了梳理,认为主题大概念对发展学生核心素养更具有独特价值;基于学科理解理论阐释了大概念的idea功能,认为大概念是对解决学科本原性问题的一系列idea的概括;从大概念素养发展功能的定位,基于大概念idea功能的教学内容的组织与呈现,以及探究式、建构式教学方式的运用等3个方面,探讨了大概念教学的重要实施策略。