基于数学问题解决的模式识别研究述评

数学问题解决中的模式识别的研究视角,可以分为基于数学解题认知过程与解题策略角度、基于"归类"的视角、基于数学问题解决中模式识别与其他因素的关系的视角等,具体研究领域涉及几何解题中的视觉模式识别、几何问题解决中的模式识别、解代数应用题的认知模式、数学建模中的模式识别等.由于在知觉领域与问题解决领域"模式识别"的表述存在一定的混乱性,将基于数学问题解决的模式识别界定为:当主体接触到数学问题后,与自己认知结构中的某数学问题图式相匹配的思维与认知过程.并进一步通过其与"归类"的区别与联系、与"化归"的区别与联系使"基于数学问题解决的模式识别"的概念得以澄清.在范围上,把问题解决中的模式识别界定为一种思维过程的阶段或者思维策略,认为它是解题的重要组成部分,但并不是解题的全部.对于未来的展望,期望系统的理论研究、期望对学生问题解决中模式识别的认知过程与机理的实质性的研究以及对学生问题解决中模式识别的教学实验研究.     

再论中小学"数学情境与提出问题"的数学学习

辩证唯物主义认识论，现代数学观和建构主义学习观指导下的“设置数学情境与提出数学问题”教学实验，旨在培养学生的数学问题意识，提高学生的提出数学问题和解决数学问题能力，增强学生的创新意识和实践能力，创设数学情境，就是呈现给学生刺激性数学信息，引起学生学习数学的兴趣，启迪思维，激起学生的好奇心，发现欲，产生认知冲突，诱发质疑猜想，唤醒强烈的问题意识，从而使其发现和提出数学问题，解决数学问题。     

三论中小学"数学情境与提出问题"的数学学习

中小学“情境-问题”数学学习是指中小学生在教师的引导下，从熟悉的或感兴趣的数学情境出发，通过积极思考、主动探究、提出问题、分析问题和解决问题，从而获取数学知识、思想方法和技能技巧并应用数学知识的过程。中小学“数学情境与提出问题”的数学学习活动，既重视学生数学问题意识的培养，又重视学生数学应用能力的培养；既关注数学知识的发现过程，又关注数学问题的解决过程；既强调学习内容的开放性，又强调学习过程的探究性。     

What Makes a Sign a Mathematical Sign? – An Epistemological Perspective on Mathematical Interaction

Mathematical signs and symbols have a decisive role for coding, constructing and communicating mathematical knowledge. Nevertheless these mathematical signs do not already contain mathematical meaning and conceptual ideas themselves. The contribution will present basic elements of an epistemology of mathematical knowledge and then apply these theoretical ideas for analyzing case studies of two teaching episodes from elementary mathematics teaching. In this way different roles of mathematical signs as means of communication (oral function), of indicating (deictic function) and of writing (symbolic function) will be elaborated.     

对数学情境及其性质、作用的探讨

对小学、初中、高中共六个年级的学生解决数学真实性问题的调查表明,虽然当下的课堂教学重视数学情境的创设,但是学生用数学的意识与能力却未发生人们期望中的变化。案例分析显示,为了适应课标要求,着意于为数学问题罩上一个未必恰当的情境外套的现象普遍存在。对情境在联结学校数学与校外数学作用的分析表明,数学情境是校外数学走向学校数学的中介,并在此过程中起着导向功能。     

Mathematical Modeling in the Primary School: Children's Construction of a Consumer Guide

This paper examines 6th grade children's local conceptual development and mathematization processes as they worked a comprehensive mathematical modeling problem (creating a consumer guide for deciding the best snack chip) over several class periods. The children and their teachers were participating in a 3-year longitudinal teaching experiment in which sequences of mathematical modeling problems were implemented from the 5th grade (10 years of age) though to the 7th grade. In contrast to traditional problem solving, mathematical modeling requires children to generate and develop their own mathematical ideas and processes, and to form systems of relationships that are generalizable and reusable. Reported here is a detailed analysis of the iterative cycles of development of one group of children as they worked the problem, followed by a summary of the mathematization processes displayed by all groups. Children's critical reflections on their models are also reported. The results show how children can independently develop constructs and processes through meaningful problem solving. Children's development included creating systems for operationally defining constructs; selecting, categorizing, and ranking factors; quantifying quantitative and qualitative data; and transforming quantities.     

把脉解题方向 对接思维通道

文章从数学解题的前提、核心和抓手三个方面剖析数学解题过程中的思维活动和心理变化,为数学解题思路的获得和教学提供思想上的指导.     

小学生数学表述状况调查及数学表述教学模式研究

数学表述是学生掌握数学知识与技能的一种手段,也是教师评价学生数学学习能力和水平的一个重要标准.目前,小学生问题解决中数学表述的状况是:不同年级学生采用表述类型的倾向性不同,数学语言的掌握程度是影响学生数学表述的根本原因.而小学数学表述教学方面存在的问题包括:教师对数学表述的重视程度一般,数学表述教学质量不高;学生存在数学语言理解和转换障碍,数学交流能力有待提高.认清数学表述教学的价值、理清数学表述与数学语言和数学交流的关系、构建小学数学表述教学模式,对培养小学生数学表述能力具有重要意义.     

数学问题解决的实证研究述评

数学问题解决的心理学实证研究主要集中在数学应用问题、平面几何问题、解题中的迁移、解题中的元认知等方面。就目前的研究状况来看，存在研究选题面窄、研究层面较低、研究起点单一等问题。因而，开展深层次的研究，是数学解题心理研究的发展方向。     

儿童早期数学问题解决的生态观

数学问题解决是儿童早期数学教育的基本目标。从数学问题解决的生态观来看,儿童早期数学问题解决具有显著的文化特征,其数学问题解决的过程是认知加工与情感态度交互作用的过程,也是一个知识提取与知识建构的共生过程,同时还是一个开放式的循环渐进过程。     

作为数学教育任务的数学解题

作为数学教育任务的数学解题与数学家的解题既有联系又有区别.它触及数学教育的3个基本矛盾,需要回答两个基本问题:怎样解题?怎样学会解题?解题理论建设成为一个独立分支有3个标志.解题研究已初步积累有题、解题、解题过程、解题程序、解题力量、解题方法、解题策略、数学问题解决的基本框架等成果.学会解题需要经历4个阶段:简单模仿、变式练习、自发领悟和自觉分析.     

试论数学问题教学中常用的数学思维方式

数学思维是学习数学知识过程中难得的、实用的方法,它同时又是教师在讲授教学理论过程中应当注意加以提炼并引导学生持续掌握、运用进而解决实际问题的一种能力。本文试图通过一些数学问题的分析解答,凸显数学思维及其作用,以达到引起读者关注并试图体验之目的。     

基于PISA2021数学素养的数学推理与问题解决

PISA测试结果的每一次公布都会引起世界的瞩目,各国政府及相关教育政策决策者会依据其结果对其相关教育政策作出调整。在正式实施测试之前,OECD会提前公布相关测试框架,这会在一定程度上影响未来的教学与评价走向。PISA2021测试框架最为显著的一个变化体现在数学素养定义中的数学推理,侧重在数学推理的介绍及其与问题解决的关系。通过对PISA2021的分析发现,数学推理包括演绎推理和归纳推理,贯穿问题解决的全过程,所有数学活动的展开都围绕数学推理而进行。     

关于数学问题提出的若干思考

问题提出是数学活动的显著特点，是提高学生问题解决能力和改进学生对数学的态度的有效手段，是促进学生理解数学的一个窗口．对问题提出的研究可基于以下几个方面：作为促进问题解决的一种手段，问题提出与问题解决的互动，作为一种相对独立的数学活动。培养学生问题提出能力，应关注影响该能力的认知和情感的因素，鼓励学生大胆地给出自己的问题，培养他们提出问题的兴趣与自信心．     

Mathematical knowledge for teaching teachers: knowledge used and developed by mathematics teacher educators in learning to teach via problem solving

In this study, we report on what types of mathematical knowledge for teaching teachers (MKTT) mathematics teacher educators (MTEs) use and develop when they work together and reflect on their teaching in a Community of Practice while helping prospective primary teachers (PTs) generate their own mathematical knowledge for teaching in learning mathematics via problem solving. Two novice MTEs worked with an experienced MTE and reflected on the process of learning to teach via problem solving and supporting PTs in developing deep understandings of foundational mathematical ideas. Taking a position of inquiry as stance, we examined our experiences teaching mathematics content courses for PTs via problem solving. We found that all of the MTEs used and developed some MKTT through (a) understanding and deciding on the mathematical goals of both the individual lessons and the two-course sequence as a whole, (b) choosing and facilitating tasks, and (c) using questions to scaffold PTs learning and engage them in mathematical processes such as making conjectures, justifying their reasoning, and proving or disproving conjectures.     

Mathematical Problem Solving with Technology: the Techno-Mathematical Fluency of a Student-with-GeoGebra

This study offers a view on students’ technology-based problem solving activity through the lens of a theoretical model which accounts for the relationship between mathematical and technological knowledge in successful problem solving. This study takes a qualitative approach building on the work of a 13-year-old girl as an exemplary case of the nature of young students’ spontaneous mathematical problem solving with technology. The empirical data comprise digital records of her approaches to two problems from a web-based mathematical competition where she resorted to GeoGebra and an interview where she explains and describes her usual problem solving activity with this tool. Based on a proposed model for describing the processes of mathematical problem solving with technologies (MPST), the main results show that this student’s solving and expressing the solution are held from the early and continuing interplay between mathematical skills and the perception of the affordances of the tool. The analytical model offers a clear picture of the type of actions that lead to the solution of each problem, revealing the student’s ability to deal with mathematics and technology in problem solving. By acknowledging this as a case of a human-with-media in solving mathematical problems, the students’ efficient way of merging technological and mathematical knowledge is portrayed in terms of her techno-mathematical fluency.     

“Accepting Emotional Complexity”: A Socio-Constructivist Perspective on the Role of Emotions in the Mathematics Classroom

A socio-constructivist account of learning and emotions stresses the situatedness of every learning activity and points to the close interactions between cognitive, conative and affective factors in students’ learning and problem solving. Emotions are perceived as being constituted by the dynamic interplay of cognitive, physiological, and motivational processes in a specific context. Understanding the role of emotions in the mathematics classroom then implies understanding the nature of these situated processes and the way they relate to students’ problem-solving behaviour. We will present data from a multiple-case study of 16 students out of 4 different junior high classes that aimed to investigate students’ emotional processes when solving a mathematical problem in their classrooms. After identifying the different emotions and analyzing their relations to motivational and cognitive processes, the relation with students’ mathematics-related beliefs will be examined. We will specifically use Frank’s case to illustrate how the use of a thoughtful combination of a variety of different research instruments enabled us to gather insightful data on the role of emotions in mathematical problem solving.     

数模思想在大学数学教学过程中的应用探讨

针对数学及数学建模的重要性,探讨在大学数学教学过程中融入数模思想的原因和原则,并说明了如何在教学过程中融入数学建模,最后列举主干课程中较为适合的实际问题模型。     

An Onto-Semiotic Analysis of Combinatorial Problems and the Solving Processes by University Students

In this paper we describe an ontological and semiotic model for mathematical knowledge, using elementary combinatorics as an example. We then apply this model to analyze the solving process of some combinatorial problems by students with high mathematical training, and show its utility in providing a semiotic explanation for the difficulty of combinatorial reasoning. We finally analyze the implications of the theoretical model and type of analysis presented for mathematics education research and practice.     

浅谈应用型人才数学素质的培养

文章按照应用型本科院校的定位,以培养应用型本科人才的数学素质为目标,构建公共数学课四个教学模块的课程体系,实现教学内容与专业需要相结合,理论与实践相结合,必修课与选修课相结合,从而达到培养学生自主学习的能力和解决实际问题的应用能力。