自动推理及其在数学教育中的应用

自动推理是人工智能中最成功的部分.自动推理的基本思想就是希望对一类一类问题分别给出一个一个确定的、能够机械地执行的解决方案.凡是用到计算机的地方,或多或少都要用到自动推理.自动推理在数学教育中的应用主要表现在几何作图、符号运算、几何证明、动画设计和机器学习等方面.我国在数学机械化的应用上有一些先进成果,但是在自动推理平台方面,还远远落后于世界先进水平,智能软件的开发需要长期努力.     

基于超级画板的平面几何自动推理

如何利用计算机进行几何定理的自动证明,是学术界长期研究的课题。本文从直接启动自动推理、增加条件或结论的推理以及设置推理规则的推理三个方面对张景中院士领衔开发的Z+Z智能教育平台——超级画板软件的功能展开论述,并举例介绍三种方法的操作过程。     

培养七年级学生几何推理能力的有效途径

推理是数学学习活动的核心内容,几何教学则是发展学生逻辑推理能力的重要途径。在七年级几何课程中,要重视几何推理起始教学,注意培养学生学习几何的兴趣,巩固基础知识和能力,关注探究的过程,帮助学生积累学习活动经验,在发展合情推理能力的同时,适当渗透演绎推理的要求,以逐步培养学生的几何推理能力。     

基于代数推理的功能谈初中代数推理的基本类型及教学建议

推理是逻辑思维的一种基本形式,代数推理是构成数学推理的重要组成部分,是代数思维的一部分.代数推理与几何推理是分别建立在代数知识与几何知识基础上的推理,由于代数与几何的研究对象及其推理的功能存在差异,因此代数推理与几何推理的主要类型应有所差异.笔者基于代数推理的功能将初中代数推理的基本类型分为证实类推理、推测(演)类推理、应用类推理,开展初中代数推理教学要重视要素结构化、证据显性化、问题模型化等,促进代数推理核心素养落地.     

关于初中代数推理的认识与思考——基于初中代数推理的文献分析

随着新时期学科育人的新要求,初中阶段,数学教育要发展的学生核心素养,主要包括抽象能力、运算能力、几何直观、空间观念、推理能力、数据观念、模型观念、应用意识、创新意识,而在发展“推理能力”中我们过于依赖几何推理,而忽视代数推理.本文将梳理与追问代数推理的含义及其分类,代数推理的特点和代数推理教学的现状,并提出加强代数推理教学的建议.     

打开几何证明思路的四种常见方法

新课程标准对七一九年级空间与图形提出了具体要求：即在探索图形性质与他人合作交流等活动过程中，发展合情推理。但在平时几何教学过程中学生的推理能力发展不快，遇到几何推理证明题时往往是一脸愁容，究其原因还是没有掌握好打开几何证明思路的方法。现结合具体实例，谈谈打开几何证明思路的四种常见方法：     

初中数学几何推理与图形证明对策

初中数学中的几何推理与图形证明有很多的技巧和规律,本文从几何的重要推理过程、基本图形的利用、题目要素的分析与辅助线的应用四个方面入手,分析了其在几何推理与图形证明过程中的作用,以期为初中数学几何推理与图形证明提供良好的对策。     

创新平面几何试题并能自动推理的《数学实验室—平面几何》

本文主要论述了《数学实验室》系列软件之一平面几何的主要功能与自动推理这一重大研究成果在教育教学中的应用，使数学教师能够深入研究几何问题并能辅助几何教学，创新试题启发与调动学生发散思维，创作自主学习的环境。     

初中学生几何推理能力的培养策略

逻辑推理贯穿初中几何教学的全过程,但初中生几何推理能力相对薄弱,主要表现在推理意识、推理严谨性不足。结合教学实际,从"培养学生几何推理意识和习惯、增强学生几何推理的严谨性、合情推理和演绎推理并重提升"三个方面来提升初中生几何推理能力。     

一种基于机器学习的推理系统的设计与应用

通用推理和验证系统利用正则匹配算法并引入机器学习算法,实现了智能化形式推理和验证平台,为构建智能系统提供了基本推理模型.将该系统应用于特定领域是重要的应用目标.通过扩充领域推理规则,该平台能够自动检查复杂的应用领域推理过程的完善性和正确性,为特定领域的推理提供了重要的自动检查手段.     

Insights into Students’ Geometric Reasoning Relating to Prisms

Whilst spatial reasoning skills have been found to predict mathematical achievement, little is known about how primary (elementary) students’ conceptual understanding of three-dimensional objects develops. In this article, we report a qualitative study and the impact of rich learning experiences on 48 Years 3–6 students’ geometric reasoning relating to prisms. A one-to-one task-based interview, refined by the researchers, was used to assess student learning. Coding and data analysis were informed by our previous research. The findings reveal noticeable shifts in students’ knowledge of and reasoning about prisms, their ability to construct and describe prisms with geometric language, and their visualisation and spatial structuring skills. The implications of these findings highlight the importance of teachers’ choice of tasks that require students to compose and decompose three-dimensional (3D) objects; compare 3D objects through physical and mental transformations; take different perspectives; and visualise and reason geometrically.

     

Designing and Validating Assessments of Complex Thinking in Science

Typical assessment systems often measure isolated ideas rather than the coherent understanding valued in current science classrooms. Such assessments may motivate students to memorize, rather than to use new ideas to solve complex problems. To meet the requirements of the Next Generation Science Standards, instruction needs to emphasize sustained investigations, and assessments need to create a detailed picture of students’ conceptual understanding and reasoning processes.

This article describes the design process and potential for automated scoring of 2 forms of inquiry assessment: Energy Stories and MySystem. To design these assessments, we formed a partnership of teachers, discipline experts, researchers, technologists, and psychometricians to align curriculum, assessments, and rubrics. We illustrate how these items document middle school students’ reasoning about energy flow in life science. We used evidence from review by science teachers and experts in the discipline; classroom experiments; and psychometric analysis to validate the assessments, rubrics, and automated scoring.     

Designing,evaluating, and deploying automated scoring systems with validity in mind: Methodological design decisions

ABSTRACT

This article discusses critical methodological design decisions for collecting, interpreting, and synthesizing empirical evidence during the design, deployment, and operational quality-control phases for automated scoring systems. The discussion is inspired by work on operational large-scale systems for automated essay scoring but many of the principles have implications for principled reasoning and workflow management for other use contexts. The overall workflow is described as a series of five phases, each one having two critical sub-phases with a large number of associated methodological design decisions. These phases involve assessment design, linguistic component design, model design, model validation, and operational deployment. Through brief examples, the various considerations for these design decisions are illustrated, which have to be carefully weighed in the overall decision-making process for the system in order to unveil the complexities that underlie this work. The article closes with reflections on resource demands as well as recommendations for best practices of interdisciplinary teams who engage in this work, underscoring how this work is a blend of scientific rigor and artful practice.     

A Review of Automatically Scorable Constructed-Response Item Types for Large-Scale Assessment

The use of automated scanning of test sheets, beginning in the 1930s, led to widespread use of the multiple-choice format in standardized testing. New forms of automated scoring now hold out the possibility of making a wide range of constructed-response item formats feasible for use on a large-scale basis. We describe new developments in five domains: mathematical reasoning, algebra problem solving, computer science, architecture, and natural language. For each one, we describe the task as presented to the examinee, the methods used to score the response, and the psychometric properties of the item responses. We then highlight general challenges and issues spanning these technologies. We conclude by offering our views on the ways in which such technologies are likely to shape the future of testing.     

Eco-cultural influences upon students' concept attainment in science

It is becoming increasingly evident that the nature of the environment (ecology) influences the culture of a people. The prediction that such eco-cultural variables could exert influence on students' concept attainment in science was tested in this study using a 2 (general environment) × 2 (reasoning pattern) × 2 (nature of home) × 2 (goal structure) fixed-effect ANOVA design. The results showed that (1) students who live in a predominantly automated environment did better than those in a predominantly manual environment; (2) students whose reasoning patterns were predominantly magical and superstitious performed significantly lower than those who were empirical in reasoning; (3) rural dwellers were predominantly cooperative in outlook; (4) students who expressed preference for cooperative learning did significantly better than those who expressed preference for competitive and individual work; and (5) students from authoritarian homes achieved less well on the science concept test when compared with those from permissive homes. A number of important implications from these findings are drawn.     

Realizing the full potential of individualizing learning

The potential of individualization to transform learning that new technology makes possible has generated wide interest. We ask here whether individualization has been exploited to its maximum advantage. We explore its potential to provide individualized scaffolding at the meta-level of students’ reflection on their own thinking as they engaged in inquiry activity to support their reasoning about a multivariable causal system – a capability central to scientific thinking and higher-order thinking more broadly. In Study 1, middle-school pairs’ self-paced inquiry was individually guided by an adult who prompted them to question their assertions and strategies. Study 2 investigated how such scaffolding might be automated to provide individualization at scale. Delayed posttests for both studies involving new scenarios showed that gains in both inquiry and multivariable causal inference skills transferred to new content. Delayed far-transfer assessments showed that the intervention achieved its learning goals most effectively when an adult worked with a pair of students, compared to students working as a whole class (Study 1); students also learned effectively with an automated agent, but only when a human adult was also involved (Study 2). Implications are considered for developing and deploying technology that individualizes and supports self-directed, reflective meta-level thinking and learning, while remaining mindful of human social context.     

几何直观的教育价值及其教学建议

在分析几何直观的教育价值的基础上,通过教学实践经验的总结,提出发展几何直观的6条教学建议：（1）重视几何直观教学与学生生活的实际相联系;（2）重视学生对几何对象的观察与操作;（3）重视几何教学与其他知识教学之间的联系;（4）重视学生用自己的语言表述对几何问题的直观感受;（5）重视几何直观的合情推理教学;（6）重视现代信息技术在几何直观教学中的应用．     

托勒玫《至大论》的方法论基础

在对托勒玫《至大论》英译本全面解读的基础上,通过一些比较典型的案例,详细分析和讨论了托勒玫宇宙论的逻辑推理基础、对自然假说的数学化、理想假设和严密性在托勒玫天文学中的重要性以及他对于几何模型的一般假设的考虑,认为这些内容反映了他的方法论基础。     

Geometry in engineering education

In this paper, a few examples are given of the importance of mathematical geometric education for engineers. The newly established information civilization, utilizing new information media, requires an abbreviated form of delivering enormous amounts of information at incredible speed. We use and relay models and involve them in our communication in almost all branches of science, culture, education, or social life. Models are strictly based on some graphical information. They are independent and almost perfectly understood at all levels of human literacy. Different scientific branches invented different models with respect to different needs and applied techniques, while the basic original idea of creating an'image' remained. Engineering education today is a complex knowledge base composed of several separate parts of different scientific and professional technical branches of the complex human knowledge, where geometry plays its definite important role. It is the basic platform to get involved with the creation of models in the form of images. Geometry can offer an unprecedented basis for logical reasoning supported by practical applications. Good three-dimensional visualization, spatial capabilities and geometric understanding are invaluable components of creative work in all kinds of human activities. An information society using communication techniques concentrates nowadays on creating models of an object in the form of computer images. Creative processes of modelling objects in scenes are related basically to the synthetic geometric reasoning and depend on the skills and geometric knowledge of the users. We must take this fact into consideration when speaking about teaching geometry to engineers.

Geometry is not just a simple science, nor the way to knowledge. It is an intellectual arrangement. From the arrangement, all European culture originated.