代数在历史各阶段的教学意义
原文作者 VICTOR J. KATZ 单位 华盛顿大学的哥伦比亚区
摘要:在这篇文章中,我们将迅速浏览代数的历史的征程,并指出在这段历史中的教学中的重要性的重要发展和反映出代数教学在中学或大学中的重要性。经常,代数被认为有三个阶段,其历史发展:直观形象阶段,形象抽象阶段,初步本质阶段。但是,除了表达代数思想这三个阶段,有四个沿着这些变化在表达式中的发生更多改变的概念性的阶段。这些阶段是几何阶段,大多数的代数的概念是几何的;静态方程解决的阶段,其目标是找到某些满足确定关系;动态功能的阶段,其中,动态似乎是一个基础的构思,最终,抽象的阶段,其中数学结构起着核心作用。该代数阶段,当然不完全是相互脱节的;总有一些重叠的部分。我们在这里讨论的重点,是代数在这些阶段的发展以及反映在教学中的应用。
关键词:代数;抽象阶段;方程求解阶段;功能阶段;几何现阶段;直观形象阶段;形象抽象阶段;初步本质阶段
这篇文章是关于代数的。所以,第一个问题是,我们通过代数表达了什么?最近一些中学教科书给出定义,但两个世纪前事实并非如此。例如,科林麦克劳林在他1748代数的文章写道,“代数是一个通过某些标志和符号计算的一般的方法,并被发现很方便。这就是所谓的通用算法,并通过运营收益,规则类似于通用算术,建立在相同的原则上。”(麦克劳林,1748年,第1页)欧拉,在1770年他自己的代数文章中写道,“代数已经被定义,通过那些已知的方法教导如何确定未知量的科学。”(欧拉,1984年,第186页)。即,在18世纪,代数用标志和符号处理决定未知符号以及为这些方法作出明确的定义。代数仍然是这样的吗?这是很难说的。有时,代数被定义为“广义算术”不论是什么意思。但一看一个典型的中学教科书揭示了各种各样的主题。这些措施包括符号数,线性方程,二次方程的解决方案,以及直链和/或二次方程的系统的算法,并操纵多项式,包括保理和指数的规则。该文字也可能涵盖矩阵,功能和图形,圆锥曲线,以及其他议题。 当然,如果你去一个抽象代数文章,你会发现许多其他主题,包括组,环和域的方面。所以很显然,“代数”现在涵盖了很多方面。
在这篇文章中,我们将采取代数的历史了快速之旅首先,它在18世纪被定义,然后现在被我们理解。我们想看看那里代数来自哪里又是为什么?什么是原来的目的是什么?是如何表达的思想? 现在又是如何运用的?最后,我们要考虑代数的历史如何给代数的教学一些启示,在任何级别做到这一点。
在许多历史文本中,代数被认为具有三个阶段中其历史发展:直观形象阶段,形象抽象阶段,初步本质阶段。通过直接观察,我们的意思是哪里都是陈述阶段和参数的单词和句子进行。在形象抽象阶段,当处理代数表达式时使用一些缩写。最后,在初步本质阶段,是总符号,所有的数字,操作,关系通过一组容易识别的表达和符号象征,操作需要用容易理解的规则进行代换。
这三个阶段是肯定看代数的历史的一种方式。但我想说,除了这三个阶段的表达代数的想法,有四概念阶段,发生了一起在表达式中的这些变化。概念阶段的几何阶段,其中大部分代数的概念是几何;求解阶段的静力平衡方程,其目的是要找到满足一定关系的数量;动态功能阶段,运动似乎是一个潜在的想法;最后抽象阶段,其中结构是目标。当然,没有这些阶段和早期的三是不相交的一个;总有一些重叠。我会考虑这两种阶段看到他们有时是互相独立的,有时在一起工作。但由于阶段第一组是众所周知的,详细讨论了路易斯在代数最近ICMI研究(Puig,2004),我将集中讨论概念的。我们开始在代数的开始,不管它是什么。这似乎是最早的代数–思想涉及Euler和麦克劳林–第十八世纪的定义来自美索不达米亚大约4000年前开始。美索不达米亚的数学(通常称为巴比伦数学)有两个根源–是一个会计问题,从开始了最早的美索不达米亚时期的官僚体系的重要组成部分,第二是“剪切和粘贴”几何可能由师他们想了解土地划分方法。这主要是从剪切和粘贴的几何体,我们称之为巴比伦的代数增长。特别是,许多古巴比伦粘土片从2000年代–公元前1700年,包含了我们现在所称的二次规划问题的广泛的名单,其中的目标是找到这样的几何量为一个矩形的长度和宽度。为了达到这个目标,他们充分利用了测量师的“剪切和粘贴”几何。
我要强调,这是一个现代的公式。巴比伦人没有类似的在他们的平板电脑。他们所描述的,完全的话,是一个过程,一个算法。我刚刚翻译了这对我国便利–因为我们习惯于这样做的。但巴比伦人肯定在“修辞”的阶段,第一个阶段提到。因此,这个问题和其他类似的问题,换言之,但使用几何思想表达的问题,我们有什么我称之为代数开始,按照固定的规则,通过对原始数据进行操作的解决数值问题过程的开始。这个例子仅仅是许多在粘土片要求确定几何量发现,所以不知怎么处理那些问题是这一代数对象。在希腊,当然,数学几何。然而,我们所认为的代数概念肯定是在欧几里德和阿波罗神的工作。有无数的命题,特别是在欧几里德的元素第二册(欧几里得,2002),展示如何直接操纵的长方形和正方形。然后还有命题欧几里得解决了“什么是代数几何研究”的问题,如线路上的某一点的位置。欧几里得的解决了这些问题,通过操纵的几何图形,但是,不像巴比伦人,他基于操作的明确的公理。伊斯兰数学家,然而,援引这一结果几个世纪后为自己的二次方程组的求解算法。然而,欧几里得自己也在189阶段教学启示代数史解决我们所谓的方程。这些都是一些在书六要素,但他们更清晰的在另一个工作。
命题1 如果两条直线,包含一个给定的区域在一个特定的角度,如果他们的总和,是给定的,然后将它们给予(即,确定)。如果我们把给定角度
命题2 若给定区域被应用到一个给定的直线下降,由图给定的物种,对不足的方面给出了。
在巴比伦的代数表示几何和希腊的“几何代数之间的许多相似之处,所以自然要问的问题是希腊的材料是一种适应材料:希腊人从巴比伦人。有论据支持这个问题的两面,但是答案仍然是未知的。是否有传输,它是明确的,我们认识到“代数”在希腊几何的作品是基于几何操作,就像在巴比伦片代数。当然,尽管巴比伦方程求解的理论基础是几何,巴比伦人还在开发的算法,或程序,求解方程。最后,该算法开始取代几何。代数的历史开始移动到“方程求解”阶段。我们看到这个二次方程求解丢番图中的知识的算法依据,完全基于数字,在第三世纪。在印度,二次公式似乎也没有任何几何基础早在第六世纪。
一个新的符号进入的地方,在第十七世纪,一个观点的重大变化也发生在代数本身。数学家们开始要求比其他“找到那个问题表示为一个方程的解的问题。“这有可能是多方面的,但其中一个原因是天文学和物理学的兴趣增加。开普勒在行星的轨道有兴趣。伽利略是在弹丸的路感兴趣。在这两种情况下,它是不是一个“数”,被通缉,但整个曲线。开普勒和伽利略发现他们问题的解决方案是圆锥曲线,和他们唯一知道如何处理这些被他们所学到的东西从阿波罗尼奥斯。他的数学在很大程度上是“静态”的,他不关心移动点–只与一个特定的片锥。然而,开普勒和伽利略能拉出他的工作所需的代表运动的思想。开普勒和伽利略都有用的符号代表运动。他们没有使用代数,但依赖于希腊的模型,包括详细的使用比例理论。例如,伽利略,在他的两个新的科学1638,描述了一个可移动的物体投影在水平面上的最终端的路径。他写道:“移动,。..驱动这架飞机最后的进一步深入,加上其先前的稳定运动,向下的趋势已经从自己的沉重和难以磨灭的。因此出现了一定的运动,由均匀水平和自然加速向下运动,我称之为“投影”。(伽利略,1974,p. 217)然后他表明,由一个纯粹的几何参数的使用,没有在所有的代数符号。
外文文献出处:Commentary from a Mathematics Educator Bill Barton
附外文文献原文
见附录
剩余内容已隐藏,支付完成后下载完整资料
VICTOR J. KATZ
STAGES IN THE HISTORY OF ALGEBRA WITH IMPLICATIONS
FOR TEACHINGlowast;
ABSTRACT. In this article, we take a rapid journey through the history of algebra, noting the important developments and reflecting on the importance of this history in the teaching of algebra in secondary school or university. Frequently, algebra is considered to have three stages in its historical development: the rhetorical stage, the syncopated stage, and the symbolic stage. But besides these three stages of expressing algebraic ideas, there are four more conceptual stages which have happened along side of these changes in expressions. These stages are the geometric stage, where most of the concepts of algebra are geometric ones; the static equation-solving stage, where the goal is to find numbers satisfying certain relationships; the dynamic function stage, where motion seems to be an underlying idea, and finally, the abstract stage, where mathematical structure plays the central role. The stages of algebra are, of course not entirely disjoint from one another; there is always some overlap. We discuss here high points of the development of these stages and reflect on the use of these historical stages in the teaching of algebra.
KEY WORDS: algebra, abstract stage, equation-solving stage, function stage, geometric stage, rhetorical stage, symbolic stage, syncopated stage
This article is about algebra. So the first question is, what do we mean by algebra? Few secondary textbooks these days give a definition of the subject, but that was not the case two centuries ago. For example, Colin Maclaurin wrote, in his 1748 algebra text, “Algebra is a general Method of Computation by certain Signs and Symbols which have been contrived for this Purpose, and found convenient. It is called an Universal Arithmetic, and proceeds by Operations and Rules similar to those in Common Arithmetic, founded upon the same Principles.” (Maclaurin, 1748, p. 1) Leonhard Euler, in his own algebra text of 1770 wrote, “Algebra has been defined, The science which teaches how to determine unknown quantities by means of those that are known.” (Euler, 1984, p. 186) That is, in the 18th century, algebra dealt with determining unknowns by using signs and symbols and certain well-defined methods of manipulation of these. Is that what algebra still is? It is hard to say. Sometimes, algebra is defined as“generalized arithmetic,” whatever that means. But a look at a typical secondary algebra textbook reveals a wide variety of topics. These include the arithmetic of signed numbers, solutions of linear equations, quadratic equations, and systems of linear and/or quadratic equations, and the manipulation of polynomials, including factoring and rules of exponents. The text might also cover matrices, functions and graphs, conic sections, and other topics. And, of course, if you go to an abstract algebra text, you will find many other topics, including groups, rings, and fields. So evidently,“algebra” today covers a lot of ground.
In this article, we will take a rapid tour of the history of algebra, first as it was defined in the eighteenth century and then as it is understood today. We want to look at where algebra came from and why? What was its original purpose? How were the ideas expressed? And how did it get to where it is today? Finally, we want to consider how the history of algebra might have some implications for the teaching of algebra, at whatever level this is done. In many history texts, algebra is considered to have three stages in its historical development: the rhetorical stage, the syncopated stage, and the symbolic stage. By the rhetorical, we mean the stage where are all statements and arguments are made in words and sentences. In the syncopated stage, some abbreviations are used when dealing with algebraic expressions. And finally, in the symbolic stage, there is total symbolization – all numbers, operations, relationships are expressed through a set of easily recognized symbols, and manipulations on the symbols take place according to well-understood rules.
These three stages are certainly one way of looking at the history of algebra. But I want to argue that, besides these three stages of expressing algebraic ideas, there are four conceptual stages that have happened along side of these changes in expressions. The conceptual stages are the geometric stage, where most of the concepts of algebra are geometric; the static equation-solving stage, where the goal is to find numbers satisfying certain relationships; the dynamic function stage, where motion seems to be an underlying idea; and finally the abstract stage, where structure is the goal. Naturally, neither these stages nor the earlier three are disjoint from one another; there is always some overlap. I will consider both of these sets of stages to see how they are sometimes independent of one another and at other times work together. But because the first set of stages is well known and discussed in detail by Luis Puig in the recent ICMI Study on Algebra (Puig, 2004), I will concentrate on the conceptual ones. We begin at the beginning of algebra, whatever it is. It would seem that the earliest algebra – ideas which relate to the eighteenth century definitions by Euler and Maclaurin – comes from Mesopotamia starting about 4000 years ago. Mesopotamian mathematics (often called Babylonian mathematics) had two roots – one is accountancy problems, which from the
beginning were an important part of the bureaucratic system of the earliest Mesopotamian dynasties, and the second is a “cut and paste” geometry probably developed by surveyors as they figured out ways to understand the division of land. It is chiefly out of this cut and paste geometry that what we call Babylonian algebra grew. In particular, many old-Babylonian
clay tablets dating from 2000–1700 BCE contain extensive lists of what we now call quadratic problems, whe
剩余内容已隐藏,支付完成后下载完整资料
资料编号:[287097],资料为PDF文档或Word文档,PDF文档可免费转换为Word
课题毕业论文、外文翻译、任务书、文献综述、开题报告、程序设计、图纸设计等资料可联系客服协助查找。