用图形来显示知识 ——思维导图和概念图在数学教育中充当着有效的工具这一重要角色外文翻译资料

用图形来显示知识

——思维导图和概念图在数学教育中充当着有效的工具这一重要角色

原文作者 Astrid Brinkmann 单位

摘要：在教学讨论中，通常会有这样一种广泛的共识，那就是数学应该让学生感受到一系列相互关联的概念和程序，而不是一系列孤立的规则和事实。这个事实应该被形式网络中的数学知识用图表来展现出来。

在这篇文章中，两种关于数学网络，思维导图和概念图的特殊图形表示方法被呈现出来。这两种方法都是为了呈现与主体有关的思想和概念。在数学教育中，作为一种教育学的工具，他们被认为是合适的。并且，思维导图和概念图的可能的应用情况和他们的优点和缺点都在被人们广泛地讨论。后来大家发现，原来思维导图和概念图两个都可能是提高数学教育质量的有效工具。

1. 前言

数学知识的特征是知识之间是以网状结构相互连接的。正如数学中的各种事物，即概念、定义、定理、证明、算法、规则和学说，它们不仅互相之间相互关联并且也是我们现实生活中不可或缺的一部分。因此，在教学讨论中通常有一个广泛的共识，那就是数学应该让学生感受到一系列相互关联的概念和程序，而不是一系列孤立的规则和事实。在最近的国际学生评估项目学习中，概念的重要性也越来越被凸显出来。其中，在国际学生评估项目中，互相关联和相似的观点是最主要的元素。

想要去感受数学知识之间相互关联的特性，其中一个办法就是去把这个特性表现出来。两种特别适合用图形来表示数学网状结构特性的方法是思维导图和概念图。这两种技巧在下文中将会被提到；作为一种为了数学教育而被提倡的教育学工具，它们的适应性和优缺点正在被学者们热烈地讨论着。

2. 思维导图

2.1 背景

托尼·布赞是思维导图的发明者。它是一位数学家，心理学家和大脑研究者。作为一种特殊的技能，思维导图使得记笔记变得更加的简略，但与此同时，你会更加关注你眼睛所看到的东西。从那时以后，思维导图在很多不同的领域中都被证明是行之有效的，并且效果远远超过知识简单地记笔记。尽管思维导图在数学中很少出现出人意料地有效性，但思维导图仍然被应用到教育中去。

思维导图这种方式考虑过这样一个问题，那就是人的左右两个大脑分别操作着不同的任务。在左脑主要负责逻辑，词汇，艺术，线性，序列，分析，举例这些内容的同时，右脑正在管理多重空间，想象力，情感，颜色，节奏，形状，几何，合成这些任务。思维导图就是要让左右脑，并且因此提高了生产力和记忆力。这是个人通过把逻辑结构和艺术空间想象结合起来所完成的。因此思维导图连接了结构的想象力和图形的逻辑性。

2.2 制作思维导图的规则

思维导图的结构是分层的。他们通过以下的规则被创造出来：

• 准备一张很大的没有任何线条和图案的白纸。
• 把思维导图的主体放置在纸的中心。（这个思维导图的主体应当以一种引人注目的方式展现出来，最好可以配上一幅色彩丰富的图像。如果这幅画显得不那么恰当，这个主题应当换成一个更好的关键词。）
• 从主题一端引申出一条和每个重要信息相关的主要的分支。在线条上直接写下表示主要思想的关键词。使用印刷体字母。（分支的次序不是那么的重要。如果为了理解主题，需要一个特殊的顺序，分支可以被标上数字或者按顺时针方向排列。尽可能在一条线上只写一个单词，最好是一个名词。由于文章中90%的词汇是不需要的，因此使用一些有意义的关键词可以使人足以记得整个上下文。
• 你可以在你的主要分支上继续画分支，这样，你的二次想法就可以表现在子分支上了。画分支的顺序依照下面的原则：从抽象到具体，从一般到特殊。
• 在你的思维导图中尽可能的使用色彩。
• 为你的思维导图添加图片、草图、符号，比如说小箭头，几何数据，感叹号或问号，以及有定义的符号。

图1：关于三角形的思维导图

2.3 数学中的思维导图

思维导图的结构和制作技巧同时决定了数学问题的有效性，就像主题对思维导图来说很重要一样。

图2：一个思维导图的结构

思维导图的结构类似于从顶端开始观察一棵树一样：中间的主干代表思维导图的主题。主题附近的分支线就像是树干附近的小枝桠。因此思维导图和数学的内容真的非常像：“数学往往被描述成一棵有牢固的树根的数，数学的分类就像是树干，树枝和一些有标记的枝桠。这就像是一棵正在生长的树。”数学对象之间的关系可能就这样被思维导图以结构化的方法表现出来了。

思维导图可以同时使用左右脑并使它们一起工作的这种特殊技能对数学思维的开发是很有帮助的。左脑适合分析推理和算术，右脑适合进行空间想象，例如几何。数学教育通常以强调规则和算法为主，这可能会抑制人们的创造力的发展并且削弱人们的空间想象能力。因此，“把握好逻辑和创造力之间的平衡是非常重要的。如果你过多的强调逻辑教育，那么创造力将会被削减。那个在逻辑上完胜的人将失去创造力，反之亦然”。 所以，戴维斯和赫斯表明，在数学学习中，左右脑应该合作，互补并且互相扶持对方而不是互相干涉造成冲突。

2.4 如何在数学教育中使用思维导图

在数学教育中存在着一些非常重要的思维导图使用方法，他们在下文中将被提及。

• 思维导图有助于整理信息。

思维导图的层次结构符合一般的假设，那就是知识是有层次的。根据这个知识的心理表征，数学知识因此能够以思维导图的形式组织起来。数学知识之间的连通性也因此被建立起来，并在人们看来是明确而且简洁的。色彩与图片也可以在其中使用。

此外，思维导图以随机和非线性的方式支持自然思维过程。由于思维导图有一个开放的结构，你可以让任何想法都参与其中；每一个想法都将可能与此思维导图有关，并且不会对你的精神造成任何压力。

• 思维导图可以作为记忆的援助。

每一个思维导图都有一个独一无二的外形和强大的视觉吸引力。因此，信息可以很快的被记住和回忆起，而学习的效率也增加了，信息的生命力更加久远了。

• 思维导图有利于复习和总结。

在一个单元即将结束的时候，可以使用思维导图把这个单元的知识做一个小结；这个思维导图将会作为唤醒你记忆的良师益友。

• 思维导图可以总结几个学生的想法。

思维导图可以成为全班共同的任务：老师可以在黑板中央写下主题，询问学生与之相关的中心思想。教师为每一个想法画出一条主分支。然后，要求学生谈谈与这些中心思想相关的其他想法。由于思维导图的结构是开放的，每个同学的想法都能被融合进去。一个完整的思维导图应该囊括每个学生的个人风格。

• 思维导图有利于连接新的知识。

新的信息可以被组合到现有的思维导图中并与先前学习的概念产生关联。这样的学生活动一开始必须由教师来组织，因此教师已经知道怎样制作思维导图并且知道怎样让新概念融合进过去的话题中。

• 思维导图可以介绍新的概念。

Entrekin 报告说，她在数学课堂中用思维导图来介绍新的概念。这个新的概念将被写在黑板上。在以后的课堂中，随着这个新概念的深入人心，教师可能会增加一些新的组件并且扩展思维导图的形式。这种视觉化的表现可能帮助学生把未知概念和已知概念连接起来。

新概念可能是由思维导图引入。

• 思维导图让学生的认知结构变得可见。

学生画的思维导图提供了学生所了解的知识。大致而言，不管是教师还是学生，思维导图都可以让他们的思维结构变得可视化。

• 学生提高了自身对知识的组织能力.

这个过程可能通过学生在小组内构建思维导图来加快步伐。学生需要讨论将要使用的概念和连接的方式。

• 学生知识的错误连接变得可视化，因此教师可以很快帮助学生纠正问题。建议教师首先问问学生为什么要这样连接；学生自身的解释可能会让学生自己意识到错误所在，这比教师直接告诉他原因要强多了。
• 教师可以通过要求学生画检查前的思维导图和检查之后的思维导图来观察学生对 主题的理解是否加深。教师可能会以一种有意义的方式中看见学生这样写，比如说，如果这个概念被补充到主题中，那就更完美了。

外文文献出处：Astrid Brikmann. Graphical Knowledge Display-Mind mapping and Concept Mapping as Efficient Tools in Mathematics Education[J]. Mathematics Education Review, No 16, April,2003:35-48

附外文文献原文

Graphical Knowledge Display – Mind Mapping and Concept Mapping as Efficient Tools in Mathematics Education

Astrid Brinkmann

University of Duisburg, Germany

In didactical discussion there is a widespread consensus that mathematics should be experienced by students as a network of interrelated concepts and procedures rather than a collection of isolated rules and facts. This experience may be supported by representing mathematical knowledge graphically in the form of networks.

In this paper, two special graphical representations of mathematical networks, mind maps and concept maps, are presented. Both are means to show ideas and concepts connected with a topic. Their suitability as a pedagogical tool for mathematics education is considered and the possible applications of mind mapping and concept mapping in mathematics education together with their

advantages and limits are discussed. It turns out that both, mind mapping and concept mapping, may be efficient tools to improve mathematics achievement.

1. Introduction

Mathematical knowledge has the character of a network, as mathematical objects, i.e. concepts, definitions, theorems, proofs, algorithms, rules, theories, are interrelated but also connected with components of the external world. Accordingly, there is a widespread consensus in the didactical discussion that mathematics should be experienced by students in its interrelatedness (see e.g.

NCTM Yearbook 1995, Preface, or NCTM Principles and Standards for School Mathematics 2000, p.64). The importance of this notion also becomes apparent in the recent –Study, where interconnections and common ideas are central Elements (OECD, 1999, p.48).

One means to experience the network character of mathematics is by visualizing it. Two methods especially suited for representing graphically a mathematical （35Mathematics Education Review, No 16, April, 2003） network around a topic are mind mapping and concept mapping. These two

techn

剩余内容已隐藏，支付完成后下载完整资料

Graphical Knowledge Display – Mind Mapping and Concept Mapping as Efficient Tools in Mathematics Education

Astrid Brinkmann

University of Duisburg, Germany

In didactical discussion there is a widespread consensus that mathematics should be experienced by students as a network of interrelated concepts and procedures rather than a collection of isolated rules and facts. This experience may be supported by representing mathematical knowledge graphically in the form of networks.

In this paper, two special graphical representations of mathematical networks, mind maps and concept maps, are presented. Both are means to show ideas and concepts connected with a topic. Their suitability as a pedagogical tool for mathematics education is considered and the possible applications of mind mapping and concept mapping in mathematics education together with their

advantages and limits are discussed. It turns out that both, mind mapping and concept mapping, may be efficient tools to improve mathematics achievement.

1. Introduction

Mathematical knowledge has the character of a network, as mathematical objects, i.e. concepts, definitions, theorems, proofs, algorithms, rules, theories, are interrelated but also connected with components of the external world. Accordingly, there is a widespread consensus in the didactical discussion that mathematics should be experienced by students in its interrelatedness (see e.g.

NCTM Yearbook 1995, Preface, or NCTM Principles and Standards for School Mathematics 2000, p.64). The importance of this notion also becomes apparent in the recent –Study, where interconnections and common ideas are central Elements (OECD, 1999, p.48).

One means to experience the network character of mathematics is by visualizing it. Two methods especially suited for representing graphically a mathematical （35Mathematics Education Review, No 16, April, 2003） network around a topic are mind mapping and concept mapping. These two

techniques are presented below; their suitability as a pedagogical tool for mathematics education is considered and the possible applications of mind mapping and concept mapping in mathematics education together with their advantages and limits are discussed.

2.Mind mapping

2.1 Background

Mind mapping was firstly developed by Tony Buzan, a mathematician, psychologist and brain researcher, as a special technique for taking notes as briefly as possible whilst being interesting to the eye as possible. Since then, mind mapping turned out to be usable in many different ways other than just simple note taking. Mind maps have, among other things, been used in education, but despite their usefulness (see 2.2) are surprisingly rarely used in mathematics.

The method of mind mapping takes into account that the two halves of the human brain are performing different tasks. While the left side is mainly Responsible for logic, words, arithmetic, linearity, sequences, analysis, lists, the right side of the brain mainly performs tasks like multidimensionality, imagination, emotion, colour, rhythm, shapes, geometry, synthesis. Mind

mapping uses both sides of the brain (Buzan, 1976), letting them work together and thus increases productivity and memory retention. This is accomplished by representing logical structures using an artistic spatial image that the individual creates. Thus mind mapping connects imagination with structure and pictures with logic (Svantesson, 1992, p. 44; Beyer, 1996).

2.2 Rules for making mind maps

Mind maps are hierarchically structured. They are produced following the rules given below (see e.g. Beyer, 1993; T. Buzan amp; B. Buzan, 1993; Hemmerich et al., 1994; Hugl, 1995, p. 182; Svantesson, 1992, p. 55-56):

• Use a large sheet of paper without lines in landscape format.
• Place the topic of the mind map in the centre of the paper. (The topic of the mind map should be displayed in an eye-catching way, preferably by a coloured image. If a picture does not seem appropriate, the topic should be named by a well-chosen keyword.)
• From the topic draw a main branch for each of the main ideas linked to the topic. Write keywords denoting the main ideas directly on the lines. Use printed letters. (The order of the branches is not important. If a special order is needed for understanding the topics, the branches may be 36Mathematics Education Review, No 16, April, 2003 numbered or ordered clockwise. If possible, only one word per line, preferably a noun, should be written down. As 90% of the words in texts are unnecessary, using a few meaningful keywords will be sufficient to remember the entire context.
• Starting from the main branches you may draw further lines (sub-Branches for secondary ideas (sub-topics) and so on. The order follows the principle: from the abstract to the concrete, from the general to the Special.
• Use colours when drawing a mind map.
• Add images, sketches, symbols, such as little arrows, geometric figures, exclamation marks or question marks, as well as self-defined symbols to your mind map.

Figure 1: Mind map on the topic of triangles

2.3 Mathematical mind maps

Both, the structure of a mind map and the technique of mind mapping emphasise the usefulness of mathematical issues as topics for mind maps (Brinkmann, 2000, 2001b, 2002, in press).

Figure 2: Structure of a mind map

The structure of a mind map resembles a tree seen from the top (figure 2): from the trunk in the middle, representing the topic of the mind map, the lines for the ideas linked to the topic branch off like tree branches. Thus a mind map is structured similarly to mathematics: 'Mathematics is often depicted as a mighty tree with its roots, trunk, branches, and twigs labelled according

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