数学知识的两种形态与数学教学

教学实践表明，数字知识有学术形态和教育形态2种，它们之间的差异表现在：过程的有无不同，表述的顺序不同，语言的详略不同，确定性与非确定性的不同等。在数学教学中，教师必须吃透教材，深知2种形态的差异，并把数学知识的学术形态向教育形态转化，这一转化大致要经过教学设计和课堂教学实施2个阶段。     

数学理解的三种基本形态及教学启示

数学理解包括三种基本形态,即:记忆性理解、解释性理解和探究性理解,这三种数学理解分别对应着“记得、晓得和明得”三种不同的状态。三种数学理解对数学学习都是有价值的,但仅有记忆性和解释性理解是不够的,探究性理解才是数学教学的最终目标。实践中,不少水平不高的教师常常只能让学生达到记忆性理解,有一定水平的教师能让学生达到解释性理解,真正让学生达到探究性理解的教师并不是很多。教师要不失时机地促进学生数学理解层级的迭代升级,促使学生最终达到探究性理解,吴文俊院士数学学习的经验对把握数学理解的三种基本形态有借鉴和启迪意义。在课堂教学中引导学生从事生动活泼的数学探索性活动常常是一个相当艰难的过程,对教师的数学探究素质提出了较高的要求,教师应努力引导学生去探求数学知识的意义和发现的过程,促使学生数学探究性理解方式的养成。     

Negative Knowledge: Understanding Professional Learning and Expertise

In this paper, we critically analyze how the concept of negative knowledge contributes to the understanding of professionals’ expert practice and learning. Negative knowledge is experientially acquired knowledge about what is wrong and what is to be avoided during performance in a given work situation. In terms of its theoretical foundation, the concept relates to constructivist theorization and metacognition. Building on existing conceptions of negative knowledge, we systematically relate the concept to research on expertise and learning from errors. The concept of negative knowledge augments existing theories of professional knowledge by emphasizing knowing about what to avoid as part of experts’ effective actions. During routine actions, negative knowledge enhances professionals’ certainty of how to proceed and increases the efficacy through the avoidance of impasses and suboptimal problem-solving strategies. Quality and depth of reflective processes after actions are related to the development of negative knowledge. The potential of negative knowledge for the investigation of professional learning is discussed through reference to recent empirical work. Re-submitted for publication in: Vocations and Learning: Studies in Vocational and Professional Education.     

教与学的数学模型

本文试图用敷学的方法定义教学过程中概念和推理论证中所包含的知识量.以便于根据学生的智力水平和知识水平估算其接受知识的能力,并据此选用合适的教材,合理安排教学内容和进度.     

Understanding Learning for Teaching

Educational researchers are concerned about the ways in which researcher identity can influence practice and findings for better or worse effect. However, writings which offer narratives, intended to instruct others in the ways in which the positioning and reflexivity of the researcher operates for better or worse, often present a view of identity as singular, fixed and stable. In this paper we trouble this view of identity as represented in the notion of inside/outside researchers. We reconsider a project in which we worked with a group of pupil researchers to investigate bullying in their school. We show how our identities multiplied and shifted throughout the project in ways that we can see more clearly in hindsight. We mobilize Zygmunt Bauman’s notion of ‘fluid identities’ and argue that the inside/outside binary may be politically helpful but also limiting of understandings of the real politic and experience of messy research practice in and with schools.     

Three Forms of the Knowledge Economy: Learning,Creativity and Openness

ABSTRACT

This paper outlines and reviews three forms and associated discourses of the ‘knowledge economy’: the ‘learning economy’, based on the work of Bengt-Åke Lundvall; the ‘creative economy’ based on the work of Charles Landry, John Howkins and Richard Florida; and the ‘open knowledge economy’ based on the work of Yochai Benkler and others. Arguably, these three forms and discourses represent three recent related but different conceptions of the knowledge economy, each with clear significance and implications for education and education policy. The last provides a model of radically non-propertarian form that incorporates both ‘open education’ and ‘open science’ economies.     

建构学习中知识的动态变化:以数学为例

在建构学习过程中．知识发生着复杂的动态变化。本文以数学的建构学习为例，分析了知识的一般变化、表征媒介的变化、知识的内化与外化、每种知识类型自身的变化、知识类型之间的相互转化等一系列问题，详细勾画了建构过程中知识变化的轮廓。这对教师理解建构学习过程、合理设计教学有重要启发意义。     

论理解教学的实现形式

作为一种新的教学形态,理解教学已引起国内外教育学者、一线教师的关注, 并开始进入课堂教学领域。理解教学主要有反思型理解和对话型理解两种实现形式。反思型理解是以反思为主要手段的教学理解,主要表现是师生的理解自我;对话型理解是以对话为主要手段的教学理解,表现为师生对话式理解、生生对话式理解和生本对话式理解。     

论数学活动的过程知识

就当今数学课堂教学而言，过程教学异化为结果教学的现象相当普遍，主要原因之一就是：在过程教学的理念下，仍以获取结果知识为最终目的而设计教学，从教学活动的动态特征来看，“过程知识”是普遍存在的，它与提高学生的数学素养，培养学生的创新能力密切相关。因此，以获取过程知识为目的而设计教学应作为数学教学改革的重要取向之一。     

理解词汇知识的三个阶段对大学英语词汇教学的启示

从认知的角度探讨词汇习得中词汇知识理解的三个阶段。通过分析词的类别及词汇知识的特点, 探讨理解词汇知识的三个阶段对词汇学习的重要性, 并且提出了几种有效的英语词汇教学策略。     

论知识教学与兴趣学习

通过对知识教学与兴趣学习的分析与比较,对如何从过去过分注重对知识的传授转向注重对学生科学学习方法的形成及能力的培养,从传统的教学模式转向兴趣学习这种双向型的教学模式进行探讨。     

数学知识在地理教学中的应用

在地理教学中借用数学知识去解决地理问题,非常方便,且能提高学习、研究乃至培养地理综合素质的能力。     

为理解而学:人工智能时代的知识学习

人工智能教育应用背景下,面对巨量的信息洪流与便捷的获取方式,为获得而学正面临着极大的挑战。为理解而学的提出旨在回应深度学习的真实发生、高阶思维的发展及个人价值的实现。为理解而学旨在发展学习者能够基于移情理解开展意义性联系与反思性运用,强调以迁移与运用为本的知识学习过程,进而实现知识与理解间的转化与生成。     

Learning Fractions by Splitting: Using Learning Analytics to Illuminate the Development of Mathematical Understanding

The struggle with fraction learning in kindergarten through Grade 12 in the United States is a persistent problem and one of the major stumbling blocks to succeeding in higher mathematics. Research into this problem has identified several areas where students commonly struggle with fractions. While there are many theories of fraction learning, none of the research on these theories employs samples large enough to test theories at scale or nuanced enough to demonstrate how learning unfolds over time during instructional activities based on these theories. The work reported here uses learning analytics methods with fine-grained log data from an online fraction game to unpack how splitting (i.e. partitioning a whole into equal-sized parts) impacts learning. Study 1 demonstrated that playing the game significantly improved students’ fraction understanding. In addition, a cluster analysis suggested that exploring splitting was beneficial. Study 2 replicated the learning results, and a cluster analysis showed that compared to early game play, later game play showed more optimal splitting strategies. In addition, in looking at the types of transitions that were possible between a student’s early cluster categorization and later cluster categorization, we found that some types of transitions were more beneficial for learning than others.     

论数学思想方法的学与教

数学思想方法是学生数学认知结构中最积极、最活跃的因素，处于数学认知结构的顶端，对学生数学认知结构的建构具有重要的意义。本文主要论述了数学思想方法学习的涵义、数学思想方法学习的心理学意义、促进数学思想方法学习的教学原则（意识性、渗透性、化隐为显、循序渐进等原则）及教学途径等，旨在帮助学生正确理解和掌握数学思想方法，促进学生数学认知结构的建构和完善，使学生真正懂得数学的价值，建立科学的数学观，从而积极主动地学习数学、发展数学和卓有成效地应用数学。     

“教学做合一“的实践性特征与对知识的透彻理解

“教学做合一”的显著特征是它的实践性。“做”是教学的中心 ,理解“做”的含义是正确理解“教学做合一”的前提。陶行知赋予“做”如此重要地位源于对知识的透彻理解 ,他对知识的分类和对“亲知”的重视表明他对知识的研究与当代理论前沿相近。他在知识论领域的探索对今天的教学改革至少有三点启示 :纠正“几乎以‘闻知’概括一切知识’ ,学校只是传授知识场所的错误观点 ,正视和肯定默会知识在教学中的重要价值 ;克服教学只是教师“一言堂”的习惯 ,而应是师生及同学之间“教学做合一”的生动过程 ;改变课堂教学与实践教学相脱离的错误做法。“教学做合一”正是实现课堂教学与实践教学紧密结合 ,培养学生创新精神和实践能力的有效途径。     

数学教与学、数学动机及知识熟悉程度对数学核心素养的影响

选取湖北省16个市州的17,592名高中生为调查对象,运用SPSS 26.0、AMOS 22.0及Mplus 8.3统计软件,对教师的教及学生的学、数学动机、知识熟悉程度与数学核心素养之间的关系进行研究.结果表明:学生的学、教师的教、数学动机、知识熟悉程度对数学核心素养均有显著的正向影响;教师的教、学生的学、数学动机对知识熟悉程度均有显著的正向影响;学生的学、教师的教对数学动机均有显著正向影响;数学动机在学生的学与数学核心素养之间,以及教师的教与数学核心素养之间存在显著的中介作用;知识熟悉程度在学生的学与数学核心素养之间,以及教师的教与数学核心素养之间存在显著的中介作用;数学动机、知识熟悉程度在学生的学与数学核心素养之间,以及教师的教与数学核心素养之间存在显著的链式中介作用.     

Forms of Knowledge in Art Education and the Corollary of Authenticity in the Teaching and Assessment of Such Forms of Knowledge

This article explores why art education after modernism needs to engage with and assess two forms of knowledge. It distinguishes procedural knowledge or ‘knowing how’ from declarative knowledge or ‘knowing that’, and argues that current classroom practice and more general thinking in art education in the UK confuses evidence of procedural knowledge with evidence for declarative knowledge. A corollary is that assessment evidence for ‘knowing how’, which is shown or demonstrated, is confused with assessment evidence for ‘knowing that’, which requires spoken or written forms of reporting. This conceptual confusion is currently embedded in the national, flagship examination known as the General Certificate of Secondary Education, taken by students at the age of 16, resulting in that examination requiring evidence of understanding the meaning of art in its socio‐historical context while at the same time denying the necessity of written or spoken work to reveal such knowledge. The article advocates a Wittgensteinian, socio‐cultural solution to the confusions of current practice.