数学教学中的想象与联想漫谈

数学特别是现代数学的表述址抽象的,但是人们在创作和教学的思维活动中,需要借助于形象.正是想象和联想,使得抽象的数学具有现实生活的韵律和色彩,以致有人说数学是美学.富有想象力,才能使数学教学充满活力,具有魅力,才能使抽象的表述与人们已有的可感知的形象相联系.这是教学活动取得成功的基本前提.     

形象思维与数学教学

“形象”的特点是直观。无疑,它是相对于一般人的感官的一般感知限度而言的。类此于形象之直观,“抽象”的特点是什么?迄今为止,还没有一个约定俗称的名字,不妨暂以“理想”相称。 经验证明,直观的信息易于识记、提取,理想的信息则难于识记、提取,尚未引起人们特别关注的一个特殊事实是,某些形象相当理想,而某些抽象则相当直观.前者如中国京剧艺术中不同个性     

浅谈初中数学建模教学

数学建模教学是让学生学会从现实原型中抽象出形象的数学表达式(模型),再将其应用到现实生活中去解决实际问题的数学思维教学.其过程是让学生亲身经历将实际问题抽象成数学模型并进行解释与应用的过程,进而使学生获得对数学理解的同时,在思维能力、情感态度与价值观等多方面得到进步和发展.几年来,笔者根据这一新的教学理念,在初中数学建模教学中,遵循学生学习数学的心理规律,根据教学内容、学生生活环境、学习经验和认知水平创设问题情境,以激发学生     

让数学教学“动”起来——以勾股定理及其逆定理的教学为例

勾股定理和逆定理是初中数学中最为常见的数学定理,也是解决数学问题的有效工具.课堂教学实践证明,通过勾股定理以及逆定理的应用,可将原本复杂、抽象的数学问题进行转化,使其形象化、具体化、简单化,以便于学生迅速形成解题思路.本论文就以此出发,结合例题,对勾股定理以及逆定理的应用,以及应用中注意事项进行了详细地研究和分析,具备一定的参考价值.     

以形助数,把握概念本质——“认识小数”教学实践与思考

从整数到小数,是数系的一次扩充,也标志着学生的数概念从具有离散性的整数向具有稠密性的有理数发展.充分挖掘小数中的直观成分,利用生动、形象的图形能将抽象的数学知识直观化,进而引导学生进行探索,把握概念的本质.     

在三角课的教学中我对运用直观性原则的点滴体会

(一)对直观性原则在教学中的重要意义的认识直观是感觉和知觉的源泉,人的认识都是由感性认识而到理性认识;思维的发展都是从形象(具体)的思维而到抽象的思维.所以直观性在教学过程中是具有重要意义的.我们应予以重视和充分运用.     

珠心算与小学数学整合教学的三种方法

小学数学课本上认识IOI,R内的数时,是从“实物→计数器物像→抽象数”这一过程来进行的。比如认识7时．从同上有7个人到计数器上0000000再到数7,这虽然符合学生从形象到抽象的学习规律,但物像对学生来说不常用到,往往记忆不深。     

抽象函数处理方法

抽象函数一般地缺乏具体的表达式,因此解决此类问题主要是变抽象为具体、形象.通过充分挖掘题目提供的信息结合已有知识进行联想、归纳、推理、类比.用特殊值代入,根据函数性质特征,作出合乎题意的示意图等方法找到问题的突破口,以最终解决问题.     

形象思维的整合特征

对思维的整合特征进行研究 ,有助于人们加深对各种思维形式的认知 ,从而能在教学中更好地去培养学生的思维品质 .本文将对形象思维的整合特征作一探讨 .研究表明 :形象思维的过程 ,要经历形象感受、形象储存、形象判断、形象创造和形象描述等五个环节 .这五个环节 ,环环相扣 ,前后联系 ,但又各自独立 .形象思维的整合主要借助想象有机地运用这五个环节 ,进行形象的思维 ,解决某一问题 ,达到某种目的 .它借助于具体形象、图式形象、模式形象、概念形象、抽象形象展开思维 .我们研究形象思维的整合 ,其目的就是要研究找出解决问题的“形象” ,…     

马氏链平稳分布应用案例的教学设计

马氏链的平稳分布在实际中应用非常广泛,本文介绍了它的一个经典应用案例——市场占有率预测问题的教学设计.本设计从实际生活中的问题引入,以问题为导向,引发同学们的好奇心,从提出问题、分析问题、介绍解决问题的基本原理和方法、再拓展问题,从形象的描述引出抽象的概念,讲解过程中适当融入思政元素,让随机过程教学起到"润物无声"教书...     

Methods for the visualization of clustered climate data

Summary  Increasing amounts of large climate data require new analysis techniques. The area of data mining investigates new paradigms and methods including factors like scalability, flexibility and problem abstraction for large data sets. The field of visual data mining in particular offers valuable methods for analyzing large amounts of data intuitively. In this paper we describe our approach of integrating cluster analysis and visualization methods for the exploration of climate data. We integrated cluster algorithms, appropriate visualization techniques and sophisticated interaction paradigms into a general framework.     

Hierarchical visual data mining for large-scale data

Summary  An increasingly important problem in exploratory data analysis and visualization is that of scale; more and more data sets are much too large to analyze using traditional techniques, either in terms of the number of variables or the number of records. One approach to addressing this problem is the development and use of multiresolution strategies, where we represent the data at different levels of abstraction or detail through aggregation and summarization. In this paper we present an overview of our recent and current activities in the development of a multiresolution exploratory visualization environment for large-scale multivariate data. We have developed visualization, interaction, and data management techniques for effectively dealing with data sets that contain millions of records and/or hundreds of dimensions, and propose methods for applying similar approaches to extend the system to handle nominal as well as ordinal data.     

Flexible isosurfaces: Simplifying and displaying scalar topology using the contour tree

The contour tree is an abstraction of a scalar field that encodes the nesting relationships of isosurfaces. We show how to use the contour tree to represent individual contours of a scalar field, how to simplify both the contour tree and the topology of the scalar field, how to compute and store geometric properties for all possible contours in the contour tree, and how to use the simplified contour tree as an interface for exploratory visualization.     

Modelling and hierarchical diagnosis of timed discrete-event systems

A new diagnostic method for hierarchically structured discrete-event systems is presented. The efficiency of this method results from the fact that the complexity of the diagnostic task is reduced by first detecting a faulty component using a coarse process model on a high level of abstraction, and subsequently refining the result by investigating the faulty component with the help of a detailed component model in order to identify the fault with sufficient precision. On both abstraction levels, the method uses a timed discrete-event model of the appropriate part of the system. On the higher abstraction level a timed event graph is used that describes how the temporal distance of the events is changed by component faults. On the lower level of abstraction, timed automata are used to cope with the non-determinism of the event sequence generated by the faulty and faultless components. The approach is illustrated by the diagnosis of a batch process.     

Designing for Mathematical Abstraction

Our focus is on the design of systems (pedagogical, technical, social) that encourage mathematical abstraction, a process we refer to as designing for abstraction. In this paper, we draw on detailed design experiments from our research on children’s understanding about chance and distribution to re-present this work as a case study in designing for abstraction. Through the case study, we elaborate a number of design heuristics that we claim are also identifiable in the broader literature on designing for mathematical abstraction. Our previous work on the micro-evolution of mathematical knowledge indicated that new mathematical abstractions are routinely forged in activity with available tools and representations, coordinated with relatively na?ve unstructured knowledge. In this paper, we identify the role of design in steering the micro-evolution of knowledge towards the focus of the designer’s aspirations. A significant finding from the current analysis is the identification of a heuristic in designing for abstraction that requires the intentional blurring of the key mathematical concepts with the tools whose use might foster the construction of that abstraction. It is commonly recognized that meaningful design constructs emerge from careful analysis of children’s activity in relation to the designer’s own framework for mathematical abstraction. The case study in this paper emphasizes the insufficiency of such a model for the relationship between epistemology and design. In fact, the case study characterises the dialectic relationship between epistemological analysis and design, in which the theoretical foundations of designing for abstraction and for the micro-evolution of mathematical knowledge can co-emerge.     

The physics of categorization

In the context of cognition, categorization is the process through which several elements (i.e., words) are grouped into a single set which by naming becomes an abstraction of its elements. For example, tiger, kitty, and max can be categorized as Cats. In this article, we aim to show how the physical, biological and cognitive dimensions are related in the process of categorization or abstraction through the physics of computation. Drawing on Landauer's principle, we show that the price paid in terms of entropy is higher when grouping elements of low ranking (high probability) than when grouping elements of high ranking (low probability). Therefore, the logic of the cognitive process of abstraction is explained through constraints imposed by memory on the computation of categories. © 2015 Wiley Periodicals, Inc. Complexity 21: 269–274, 2016     

Reflective abstraction in computational thinking

Computational thinking has become an increasingly popular notion in K-12 and college level education. Although researchers have accepted that abstraction is a central concept in computational thinking, they are quick to disagree on the meaning of it. A focus on reflective abstraction has led to the development of APOS Theory in Mathematics education. This has resulted in many cases of improved student learning in Mathematics (Arnon et al., 2013). Our main aim in this paper is to construct a theoretical bridge between computational thinking and APOS Theory and show that reflective abstraction can be used in the context of computational thinking.     

数学抽象度概念与抽象度分析法

§1 引言 本文将给出“数学抽象度”的一般概念并论述“抽象度分析法”的基本概要。 如所熟知,抽象是认识事物本质、掌握事物内在规律的方法。凡科学中的一切概念都是抽象过程的产物,而且都有不同程度的抽象性。数学中的许多概念的抽象性更是明显地经过一系列阶段而产生的。例如,整数、有理数、无理数、复数、函数、微分、积分、变分、泛函、范畴等这些概念的抽象性几乎是一个高于一个。这说明数学内部各个概念的抽象程度是不一样的。在本文中我们要引进抽象度概念,用以刻划一个概念的抽象性层次;     

The reductionist blind spot

Can there be higher level laws of nature even though everything is reducible to the fundamental laws of physics? The computer science notion of level of abstraction explains how there can be. The key relationship between elements on different levels of abstraction is not the is‐composed‐of relationship but the implements relationship. I take a scientific realist position with respect to (material) levels of abstraction and their instantiation as (material) entities. They exist as objective elements of nature. Reducing them away to lower order phenomena produces a reductionist blind spot and is bad science. © 2009 Wiley Periodicals, Inc. Complexity, 2009.     

OPTIMIZING THE PERFORMANCE OF IRRIGATED AGRICULTURE IN EASTERN ENGLAND UNDER DIFFERENT WATER PRICING AND REGULATION STRATEGIES

Irrigation water shortage is becoming an increasingly serious problem in agricultural production. Growing pressure on water resources is leading to increasing restrictions on abstraction for irrigation and consideration of the use of economic instruments, such as increased abstraction charges and/or tradable licences, to restrict demand and encourage wiser use of water. We evaluate irrigation using selected economic, social, and environmental indicators of performance, including the value of water used for irrigation. A linear programming model is developed and used to simulate possible responses by irrigators and the impact on irrigation performance of intervention measures, namely abstraction quota restrictions and volumetric pricing that might be used to ration water and/or increase water use efficiency. Through the use of parametric programming a scenario analysis is performed to a case study in eastern England with regard to perturbations of irrigation water under alternative policy instruments.