解决归纳推理问题 数学中不可小觑的追求外文翻译资料

本科毕业设计（论文）

外文翻译

解决归纳推理问题

数学中不可小觑的追求

作者： LISA A HAVERTY, KENNETH R KOEDINGER, DAVID KLAHR,

MARTHA W ALIBALI

国籍：USA

出处：COGNITIVE SCIENCE

摘要：本研究探讨的是归纳推理的认知过程. 名大学生通过解决二次函数的求解问题一致认为，归纳推理的三个基本策略为:数据收集、发现关系和生成假设. 在研究函数的过程中，这三个策略的应用是十分普遍的. 现有的研究表明，发现关系在研究中的作用显著. 其中，最常见方式是追踪策略，即实验者从和中定义新的变量，并检测这些变量间的关系，然后用表示与这些变量的关系. 最后构建成完整的假设. 此策略中涉及的流程是基于模型对成功和失败的方案模拟. 该策略和模型表明，数值对关系的检测是至关重要的，在一些高层次问题的求解中也是如此.

关键词：归纳推理；数学；函数

1.介绍

有一位老师想从一天的课程中得到一个喘息的机会，于是布置了任务：每位同学在课上把前一百个数加起来. 他想这肯定会占用孩子们很长一段时间，于是就安静的等待孩子们算出来. 在其他孩子只艰难的算到 “”的时候，高斯突然走上前把答案放在老师的桌子上. 我们可以想象，老师最初对高斯的行为是充满怀疑和沮丧的，但当他看到高斯的答案时，却惊讶的发现结果竟是完全正确的. 那么高斯究竟是怎么做到的呢？实际上，高斯是通过归纳推理解决这个问题. 归纳推理是指通过观察和分析具体实例来推断一般规律. 高斯通过观察得到以下规律:数字从到，若将两端相加则恒等于. 由此他推断会有对这样的结果，于是，他将乘以得到答案，但我们的高斯并没有就此打住. 他进一步猜想从到的数仍可以这样表示，因此，他推导得到从加到的数字之和公式：.

2.归纳推理在问题解决和数学中的应用

高斯通过使用归纳推理将一个繁重的计算任务变成了一个有趣且相对快速的解决过程. 归纳推理在解决许多问题时都十分便利，因此被数学家们广泛使用. 通过研究已经确定了归纳推理对解决问题、学习和获得专业知识的重要性. Pellegrino和Glaser指出“归纳推理因素hellip;hellip;，可以从大多数的能力发展和智力测试中提取出来，是对学习成绩和能力发展的最好预测. ” Klauer指出“解决问题需要发现规律，即利用归纳推理'，并引用Simon和Lea的规律归纳工作，Egan和Greeno对概念学习、序列模式和问题解决的总结作为证据. 即使在表面上看起来是演绎的问题领域，实际解决问题的方法仍是通过归纳学习获得，而不是直接通过抽象规则获得的. Wason的选择任务研究表明，该任务需要演绎推理的知识，而人们要么使用基于具体心理模型的归纳方法来解决这些问题，要么使用从经验中归纳的半概括半推理模式.

Zhu and Simon在从例题学习中阐述了归纳推理对学习的重要性. 当学生们看到一个例题时，他们必须从这些例题中归纳出解决方法以及应用策略，并进行转化迁移. Klauer为归纳推理对学习的影响提供了更直接的证据. 他的研究结论表明，通过归纳推理的训练，学生学习陈述性知识的能力得到了提高. Koedinger和Anderson证明了归纳推理在数学学习中的作用. 研究结果表明，让学生从算术过程中归纳代数表达式的教学方法比基于教科书的教学方法更有效.

最后，以上研究都表明了归纳推理对专业知识发展的重要性. 除了Chi等人在这方面的工作外，Cummins的工作表明，在处理方程时，通过归纳问题之间的结构，能有助于概念的形成. 即使在几何证明方面，专家也表示让学生通过图表归纳得到结论，比教科书直接传授定义和定理来得有效. 因此，归纳推理可以说对于解决问题、学习和专业知识的发展都十分有效. 它是数学学习和表现的基础，因此，研究归纳推理是加深对数学认知理解的重要过程.

3.确定函数关系是归纳推理的代表问题

回顾一下我们对归纳推理的定义，归纳推理是通过观察和分析具体实例来推断一般规则的过程. 以下这些文献涵盖了各种各样的归纳推理任务: 序列完成问题、瑞文矩阵问题、分类问题、类比问题. 这些不同的任务由Klauer根据它们所需要的归纳过程来组织分类(见表1). 在Klauer的系统中，确定了几个归纳过程，并将每一个过程都与特定的认知操作配对，探究属性和关系中的相似性、差异性.

表1 Klauer归纳推理任务分类系统

Klauer将“比较关系”定义为需要“仔细检查至少两对关系”的对象，例如“理解系列就需要理解关系和之间的关系”. 因此，他认为根据这个系统，可将文献中的问题归为“泛化”问题. 类似地，因为序列完成问题需要注意在实例之间的相同之处，所以它们被归为“识别关系”问题. 由于矩阵问题需要检测单元与单元之间的相同和不同之处，因此将它们分类为“系统构建”问题. 我们还将数字类比问题也归为“系统构建”问题. 像年轻的高斯提出的问题就不属于任何一种其他文献中研究的问题类别，但在Klauer的系统中可以被归类为“认识关系”的问题.

在这项研究中，我们的目标是检验归纳推理在数学中的特殊作用. 因此，我们在寻找数值时不仅仅针对一个问题，而应该能够适用于更多基本的数学问题. 我们选择函数问题为例，这需要观察和归纳连续对数字之间的相似性和差异性. 因此，它被归类为Klauer系统中的一个“系统构建”问题. 例如通过观察、归纳得到与图1数据相匹配的函数为：.

图1 函数的数据样本集

从数据中归纳函数关系的问题对于数学来说是基础问题，我们将在下一节中对此进行说明. 此外，作为一个归纳推理任务，它包含了在Klauer系统中确定的几个归纳过程. 因此，无论是从表示归纳推理问题的角度，还是从选择数学的代表性问题角度来看，函数问题都是理想的研究问题.

4.函数问题在数学中很普遍

数学中的许多归纳推理问题，以及科学中的许多归纳推理问题，都可以归结为从一组数字中归纳出一个函数关系的基本问题. 函数也被广泛应用于代数、几何、微积分、数论、组合学等学科中. 从几何角度考虑这个例子:假设你知道的度数，在图中，并且你试图用来表示的度数. 然而，你还不知道这个事实(或者你已经忘记了)直线上两个角之和为. 你可以用量角器测量几组这样的角，并将这些测量的度量值记录在表中. 假设你已经收集了部分实际数据，如表所示.

图1 给定，求出

表2 和的测量值

从这些数据实例中可以推断出的度数等于减去的度数. 此时，你就成功的找到了符合这些数据的函数关系. 全国数学教师委员会()和一些几何教科书都曾提倡，通过建立和解决函数问题来学习几何问题.

下面这个例子说明了求解函数问题还能运用在其他不同的数学领域，如组合学中的:要确定一个含个元素的集合中有多少个子集. 有些人能直接计算这个答案而不需要计算出实际的集合，然而，其他人可能会采取别的策略. 下以元素个数较少的例子进行研究. 首先研究只有三个元素的集合中有多少个子集. 此时，很容易通过实际计算来得到子集，然后计算总数. 类似的，给出一些示例，我们可获得一些数据. 对于只有两个元素的集合，有个可能的子集. 对于一个包含三个元素的集合，有个可能的子集. 对于一个包含个元素的集合，则有个可能的子集，见表.

表3 一组“”个元素集合中可能的子集的数目

我们现在可以归纳出，对于一个由个元素组成的集合，有个可能的子集. 因此每一个含“”个元素的集合，其子集的数目似乎等于的“”次. 如果是这样，那么我们可以用的次来确定一组包含个元素集合的子集数目. 事实上，这个问题的答案是或. 刚才描述的过程就是一个确定函数关系的过程：先研究小的集合，生成一些数据，从中推断出一个通用表达式，最后便可应用于感兴趣的实例中.

这些例子说明了如何将其他问题转换成函数问题(例如，由个元素组成的集合可以包含多少子集)，以帮助它解决. 同时也表明，函数不仅对发现问题有价值，而且对问题的解决和回顾具有启发作用. 通过函数问题的解决可以促进数学学习:Koedinger和Anderson表明，在教学中使用函数作为支架，可以促进将问题转化为代数表达式的学习. 因此，函数在数学中扮演着多重角色:在发现中解决问题、学习新知以及巩固旧知. 确定函数关系除了与解决数学问题密切相关，也是归纳推理的代表.

附：外文原文（原文可直接复印附后）

Solving Inductive Reasoning Problems in

Mathematics: Not-so-Trivial Pursuit

Abstract

This study investigated the cognitive processes involved in inductive reasoning. Sixteen undergraduates solved quadratic function-finding problems and provided concurrent verbal protocols. Three fundamental areas of inductive activity were identified: Data Gathering, Pattern Finding, and Hypothesis Generation. These activities are evident in three different strategies that they used to successfully find functions. In all three strategies, Pattern Finding played a critical role not previously identified in the literature. In the most common strategy, called the Pursuit strategy, participants created new quantities from x and y, detected patterns in these quantities, and expressed these patterns in terms of x. These expressions were then built into full hypothes

剩余内容已隐藏，支付完成后下载完整资料

本科毕业设计（论文）

外文翻译

Solving Inductive Reasoning Problems in

Mathematics: Not-so-Trivial Pursuit

Abstract

This study investigated the cognitive processes involved in inductive reasoning. Sixteen undergraduates solved quadratic function-finding problems and provided concurrent verbal protocols. Three fundamental areas of inductive activity were identified: Data Gathering, Pattern Finding, and Hypothesis Generation. These activities are evident in three different strategies that they used to successfully find functions. In all three strategies, Pattern Finding played a critical role not previously identified in the literature. In the most common strategy, called the Pursuit strategy, participants created new quantities from x and y, detected patterns in these quantities, and expressed these patterns in terms of x. These expressions were then built into full hypotheses. The processes involved in this strategy are instantiated in an ACT-based model that simulates both successful and unsuccessful performance. The protocols and the model suggest that numerical knowledge is essential to the detection of

patterns and, therefore, to higher-order problem solving.

1. Introduction

One of his teachers, apparently eager for a respite from the days lessons, asked the class to work quietly at their desks and add up the first hundred whole numbers. Surely this would occupy the little tykes for a good long time. Yet the teacher had barely spoken, and the other children had hardly proceeded pastwhen Carl walked up and placed the answer on the teachers desk. One imagines that the teacher registered a combination of incredulity and frustration at this unexpected turn of events, but a quick look at Gausss answer showed it to be

perfectly correct. How did he do it?

He did it by inductive reasoning. Inductive reasoning is defined as the process of inferring a general rule by observation and analysis of specific instances. Gauss recognized a pattern: that the numbers from 1 to 100, when added together from end to end always equal 101. He inferred that there would be such pairs, and thus, he multiplied 101 by 50 to reach the answer that But our Gauss did not stop there. He realized that the sum of the numbers from to n would always be expressible in this way: times. Thus, he induced the equals the sum of the numbers from

1 to n.

2. Reasoning in Problem Solving and Mathematics Gauss turned a potentially

Gauss turned a potentially onerous computational task into an interesting and relatively Speedy process of discovery by using inductive reasoning. Inductive reasoning can be useful in many problem-solving situations and is used commonly by practitioners of mathematics. Research has established the importance of inductive reasoning for problem solving, for learning, and for gaining expertise. Indeed, Pellegrino and Glaser noted that “the inductive reasoning factor, which can be extracted from most aptitude and intelligence tests, is the single best predictor of academic performance and achievement test scores.”. Klauer notes that “problem solving requires one to induce rules, i.e., to make use of inductive reasoning” and cites as evidence the rule induction work conducted by Simon and Lea, the review of concept learning, serial patterns, and problem solving by Egan and Greeno, and the investigation of expertise and problem solving in physics by Chi, Glaser, and Rees. Even in problem domains that appear deductive on the surface, it seems that problem solving knowledge is acquired primarily through inductive learning methods rather than through abstract rule following. Research on the Wason selection task, which nominally requires deductive knowledge of modus ponens and modus tollens, has shown that people solve such problems using either inductive methods based on concrete mental models or by applying semigeneral reasoning schemas induced from

experience.

The importance of inductive reasoning to learning is illustrated in work by Zhu and Simon about learning from worked-out examples. Students learned and were able to transfer what they learned when presented with worked-out examples from which they had to induce how and when to apply each problem-solving method. Klauer provides more direct evidence of the effect of inductive reasoning on learning. In his work, acquisition of declarative knowledge was improved after training in inductive reasoning. The role of inductive reasoning in mathematics learning was demonstrated by Koedinger and Anderson. They showed that an instructional approach based on helping students induce algebraic expressions from arithmetic procedures led to

greater learning than a textbook-based instructional approach.

Finally, research has demonstrated the importance of inductive reasoning to the development of expertise. In addition to the work by Chi et al. in this area, work by demonstrates that induction of structural similarities between problems leads to expert- level conceptual performance when working with equations. Even in the domain of geometry theorem proving, expert representations seem to reflect inductive experience with diagrams rather than command of textbook definitions and theorems. Thus, inductive reasoning facilitates problem solving, learning, and the development of expertise. It is fundamental to the learning and performance of mathematics, and is, therefore, an important process to investigate to gain a deeper understanding of

mathematical cognition.

<stro

剩余内容已隐藏，支付完成后下载完整资料</stro

资料编号：[272117]，资料为PDF文档或Word文档，PDF文档可免费转换为Word