Preservice teachers learning to teach proof through classroom implementation: Successes and challenges

Proof and reasoning are central to learning mathematics with understanding. Yet proof is seen as challenging to teach and to learn. In a capstone course for preservice teachers, we developed instructional modules that guided prospective secondary mathematics teachers (PSTs) through a cycle of learning about the logical aspects of proof, then planning and implementing lessons in secondary classrooms that integrate these aspects with traditional mathematics curriculum in the United States. In this paper we highlight our framework on mathematical knowledge for teaching proof and focus on some of the logical aspects of proof that are seen as particularly challenging (four proof themes). We analyze 60 lesson plans, video recordings of a subset of 13 enacted lessons, and the PSTs’ self- reported data to shed light on how the PSTs planned and enacted lessons that integrate these proof themes. The results provide insights into successes and challenges the PSTs encountered in this process and illustrate potential pathways for preparing PSTs to enact reasoning and proof in secondary classrooms. We also highlight the design principles for supporting the development of PSTs’ mathematical knowledge for teaching proof.     

Probing for reasons: Presentations, questions, phases

This paper reports on a research study based on data from experimental teaching. Undergraduate dance majors were invited, through real-world problem tasks that raised central conceptual issues, to invent major ideas of calculus. This study focuses on work and thinking by these students, as they sought to build key ideas, representations and compelling lines of reasoning. Speiser and Walter's psychological and logical perspectives (see Speiser, Walter, & Sullivan, 2007) provide opportunities to focus not just on the students’ thinking, but perhaps most especially, through detailed examination of important choices, on their exercise of agency as learners. Close analysis of student data through these lenses triggered the development of two new analytic categories—logic of agency and logic of proof. The analysis presented here treats students as active shapers of their own experience and understanding, whose choices open opportunities for continued growth and learning, not just for themselves but also for each other.     

基于关系数据库语义解析的信息推理研究

研究支持关系数据库语义模型,引入领域主词关系网描述工程基本概念和关系,设计规范化描述与关联规则,完成工程逻辑概念与数据关系模式的映射。根据现有数据模型针对需要生产数据信息,在关系数据内部搭建关联模式,设计并建立语义解析推理机制。采用元词汇、工程事实及决策规则三层知识结构,将SQL操纵与主词关系网、关系数据库关联关系相结合,实现所描述的生产信息的获取与分析,给出相应油田动态开发数据解析结果。     

一种基于模糊推理的因素权重确定方法

在模糊综合评价中,如何确定因素的权重是非常重要的,也是非常困难的,它直接关系到评价结果的合理性与真实性.利用模糊推理技术,提出了一种基于模糊推理的因素权重确定方法,其利用人们处理复杂问题的逻辑思维方式,建立评价空间和单位评价空间的映射,得到模糊关系方程并用于确定因素的权重.实验结果表明,利用该模型得到的权重能反映各因素的重要性,用其对待评对象进行模糊综合,其结果也是合理的.     

两种含可信度的推理

本文旨在建立两种单一结论,多个条件,且含有可信度的推理机制,这种推理机制是非单调的,和不精确的。推理的目的是在计算可信度,从而得到结论成立的可能性大小。这种推理机制有两个特点,特点1是在证据间引入取值于[-1,1]二元函数ξ（A1,A2）反映证据间的相互支持度,有正支持、零支持与负支持之分。在此基础上,建立了可信度计算公式。另一个特点为,当条件是模糊时,引入取值于[0,1]的二元R算子,用来耦合条件模糊隶属度以及规则的可信度,其结果恰好是结论成立的可信度。     

一种新的三角模糊数算子在加权模糊推理中的应用

针对基于模糊逻辑的加权模糊推理,Chen Shy i-M ing提出了两种计算合取式前件整体真值的方法。由于所用模糊数算子的影响,两种方法的求取结果在准确性和合理性上都存在一定的缺陷。这种缺陷将直接影响推理的性能。因此,为改善这种缺陷,提高推理性能,本文提出了一种新的三角模糊数算子。它的应用可以提高推理的准确性和合理性。     

基于S蕴涵算子的区间值模糊推理的三I算法

针对S蕴涵算子讨论了FM P模型及FM T模型的区间值模糊集三I推理,借助S蕴涵算子的性质,给出相应的推理算法,并讨论了算法的还原性。     

模糊蚁群优化算法在交通导航系统中的应用

为了能处理交通导航系统中的模糊信息,并且能快速的综合多种信息求解最优导航路径,将模糊逻辑推理技术与改进的蚁群算法相结合提出了一种新的算法——模糊蚁群混合优化算法。实验表明,该算法不仅能够处理导航系统中的各种模糊信息,并且能利用改进的蚁群算法快速求解最优导航路径。     

Lukasiewicz三值命题逻辑中命题的真度理论

利用势为3的均匀概率空间的无穷乘积在Lukasiewicz三值命题逻辑中引入了公式的真度概念，证明了全体公式的真度值之集在[0，1]上是稠密的，并给出真度的表达式；利用真度定义公式问的相似度，进而导出全体公式集上的一种伪距离，为三值命题的近似推理理论提供一种可能的框架。     

关于GMP和GMT问题的最优解

进一步研究模糊推理的非模糊形式，在几个重要的逻辑系统中形式地讨论GMP(广义取式)和GMT(广义拒取式)问题的最优解。结果表明，GMP和GMT问题的三I解和一种新的三I解都是某种意义下的最优解。还讨论所给算法的还原性问题。