Bridging Arithmatic and Algebra: Exploring How Children Understand Changes on Equal Amounts

连接算术和代数：探索孩子如何理解等量变化

• 批准号: 9722732
• 负责人: Analucia Schliemann
• 金额: $ 9.77万
• 依托单位: Tufts University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-15 至 1999-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9722732&HistoricalAwards=false
• 关键词: Bridging; Arithmatic; Algebra; Exploring; How

PROJECT SUMMARY Bridging Arithmetic to Algebra: Exploring how children understand equal changes to equal amounts When, after years of arithmetic problem solving, students are introduced to algebra, the meaning of equivalence, operations, and equations undergoes a paradigm shift. Instead of one-way transformations to find an answer, mathematical operations are meant to describe logical relations among elements (quantities or variables). Given the gulf between arithmetic and algebra, it is no surprise that students have enmrmous difficulties in algebraic problem solving. Such difficulties seem to justify that instruction on algebra should only start when children are about 12 years old. Only then are they required to leave aside the direct arithmetical path and to focus, instead, on the extraction and representation of relevant mathematical relations. The contrast between children's difficulties with algebra in high school and some successful attempts to teach algebra at earlier grades suggest that it is time to consider deep changes in the elementary mathematics curriculum including the possibility of having children discussing, understanding, and dealing with algebraic concepts and relations much earlier than usual. But such changes demand careful analysis of children's understanding about the logico- mathematical relations implicit in algebraic rules, of their own ways of approaching and representing algebra problems in different contexts, and of the most adequate instructional models for initiating algebra instruction. This project is a preliminary investigation of seven to eleven year-old children's understanding of basic relations involved in dealing with equations in different contexts. A total of 120 upper middle-class 7 to 11 year-old Brazilian children participated in the study. Each child individually participated in two of a total of four tasks, each one containing 16 different items. Each task referred to one of the four following contexts: (a) equivalence between weights on a two-pan balance scale, (b) equivalence between quantities of discrete concrete objects, (c) equivalence between quantities expressed in verbal problems, and (d) equivalence between the two sides of written equations. For each item an equality was established and the child was asked whether or not the equality would remain if similar or dissimilar transformations were to take place on each of the two compared amounts. They were also asked to justifty their answers and to use whatever tools and representations they judged necessary to solve the problems. The analysis of data already collected will provide a starting point for a future analysis of how American children from different backgrounds understand and deal with similar problems. With this study we hope raise new questions and to frame new hypotheses for research concerning the development of algebra activities in the elementary mathematics curriculum. NSF FORM 1358 (1/94)

项目摘要 连接算术和代数：探索孩子们如何理解等量变化到等量 经过多年的算术问题解决后，当学生们开始接触代数时，等价、运算和方程的含义发生了范式转变。数学运算不是为了寻找答案而进行单向变换，而是旨在描述元素（数量或变量）之间的逻辑关系。鉴于算术和代数之间的鸿沟，学生在解决代数问题时遇到巨大困难也就不足为奇了。 这些困难似乎证明代数教学应该从孩子 12 岁左右开始。只有这样，他们才需要抛开直接的算术路径，而专注于相关数学关系的提取和表示。 儿童在高中学习代数时遇到的困难与低年级代数教学的一些成功尝试之间的对比表明，是时候考虑对基础数学课程进行深刻的改变了，包括让孩子们比平常更早地讨论、理解和处理代数概念和关系的可能性。但这种变化需要仔细分析儿童对代数规则中隐含的逻辑数学关系的理解，对他们在不同背景下处理和表示代数问题的方式的理解，以及对启动代数教学的最适当的教学模型的理解。 该项目是对七至十一岁儿童对不同背景下处理方程所涉及的基本关系的理解的初步调查。共有 120 名 7 至 11 岁的巴西中上阶层儿童参与了这项研究。每个孩子单独参与总共四项任务中的两项，每项任务包含 16 个不同的项目。每项任务都涉及以下四种情况之一：（a）两盘天平秤上的重量之间的等价，（b）离散具体物体的数量之间的等价，（c）言语问题中表达的数量之间的等价，以及（d）书面方程两侧之间的等价。对于每个项目都建立了平等，并询问孩子如果两个比较金额中的每一个都发生相似或不相似的转换，平等是否会保持不变。他们还被要求证明自己的答案是合理的，并使用他们认为解决问题所必需的任何工具和表述。 对已收集数据的分析将为未来分析来自不同背景的美国儿童如何理解和处理类似问题提供一个起点。通过这项研究，我们希望提出新的问题，并为有关基础数学课程中代数活动发展的研究提出新的问题并提出新的假设。 NSF 表格 1358 (1/94)