Local hardness: a critical account

This paper gives a critical account on the present status of the local hardness. We analyze the behavior of hardness related global, local, and nonlocal properties, paying particular attention to local hardness. Although this reactivity index has become very useful in predicting the regioselectivity of chemical reactions, the lack of a rigorous definition in the literature has shown that different approximations can yield different and sometimes contradictory conclusions. The present paper tries to provide insights into this controversial issue, starting from an approximate model of the hardness kernel and analyzing the features of the different models to evaluate the local and global hardnesses.     

Is Believing at Will ‘Conceptually Impossible’?

In this paper I discuss the claim that believing at will is ‘conceptually impossible’ or, to use a formulation encountered in the debate, “that nothing could be a belief and be willed directly”. I argue that such a claim is only plausible if directed against the claim that believing itself is an action-type. However, in the debate, the claim has been univocally directed against the position that forming a belief is an action-type. I argue that the many arguments offered in favor of the ‘conceptual impossibility’ of performing such actions fail without exception. If we are to argue against doxastic voluntarism we are better off by resorting to more modest means.

增进化学学科理解的“反应热”单元整体教学设计

秉承2017年版课标的理念和要求,按照鲁科版新教材的编排体系,系统梳理了对“反应热”的学科理解,包括理解知识本体与其中蕴含的思维方式方法。为增进学生的学科理解,将反应热单元分为3课时:基于体系理解反应热概念,借助内能将反应热与焓变建立联系,焓变的计算。精心设计了每一课时的驱动性问题(任务)以及课中与课后作业评价并进行了教学实践。对比上一轮此内容的单元整体教学,从教学侧重点、教学方法、教学效果等角度进行了反思。     

创新产品目标设计阶段的需求分析和研究

本针对创新产品目标设计阶段的顾客需求进行研究，分析了需求的层次、需求的类型以及需求的特点，探讨了识别需求的方法、途径和所需注意的问题，论述了需求分析的方法、手段和重要性。最后，给出了需求的产品设计规范所包含的内容及一般的表达形式。     

Fifth graders’ understanding of fractions on the number line

The goal of this research was to examine fifth graders’ understanding of fractions on the number line. This case‐study design focused on the various ways that students represented fractions on number lines. Students responded to task‐based interview questions by identifying fractions as a number on the number line as well as equivalency and problem solving. The tasks were administered individually to 26 fifth‐grade students over a 15‐minute time frame in their respective schools. The two groups of 10‐year‐old students answered most questions in written form with pencil and paper and were often asked to explain how they arrived at an answer. Student performance was highest when instructed to plot ½ on a number line of 0 to 1 as well as naming a fraction less than ½. The students performed lowest when they attempted to plot ½, ¼, and 1 on a number line with a predetermined unit 0 to 1/3. Other low performing concepts consisted of plotting ¼ on a number line from 0‐3, identifying ¼ on a non‐routine number line, and plotting a unit fraction with an equivalent fraction as well as an improper fraction on a common number line.     

Procedural and relational understanding of pre-service mathematics teachers regarding spatial perception of angles in pyramids

In this study, we aim to explore the extent of mathematics pre-service teachers’ ability to apply their procedural understanding combined with spatial perception for drawing conceptual conclusions related to angles in a pyramid. The participants are 16 pre-service high school mathematics teachers. They have studied solid geometry during one academic year, solving problems with various 3-D geometric figures including pyramids and engaging in activities designed to develop spatial perception. At the end of the year, they have taken a final test which examines procedural understanding of 3-D geometric figures as well as relational understanding and spatial perception regarding angles in pyramids. The results illustrate that attainments of the majority of the pre-service teachers in problems requiring only procedural understanding are higher than the attainments in problems which require relational understanding. The results also lead to the assumption that relational understanding of learned material requires application of special teaching methods. Hence, we recommend integrating in the syllabus appropriate courses that focus on the development of this type of understanding.     

Skewing (n+1) tensor indices where the index range is n

It is known that any totally skew quantity with (n?+?1) indices, each of which ranges over n values, vanishes identically. The aim of this short note is to show that this is equivalent to the simple fact that any (n?+?1) vectors in an n-dimensional vector space are linearly dependent.     

对工科高等数学的研究性教学的一些认识

当前"研究性教学"是比较提倡的教学方式,它有利于培养学生的数学思想和数学素养.为了更好地进行"研究性教学",教师应研究数学史培养学生数学思想,应研究数学教学的认知规律培养学生"研究性学习"习惯,以及进行应用数学知识能力的研究,完美的逻辑体系与学生接受能力的研究,支持这种教学方式的考核方式的研究.