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Instructional Supports for Representational Fluency in Solving Linear Equations with Computer Algebra Systems and Paper-and-Pencil

机译：使用计算机代数系统和纸和铅笔求解线性方程组中表示流利性的教学支持

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This research addresses the issue of how to support students' representational fluency-the ability to create, move within, translate across, and derive meaning from external representations of mathematical ideas. The context of solving linear equations in a combined computer algebra system (CAS) and paper-and-pencil classroom environment is targeted as a rich and pressing context to study this issue. We report results of a collaborative teaching experiment in which we designed for and tested a functions approach to solving equations with ninth-grade algebra students, and link to results of semi-structured interviews with students before and after the experiment. Results of analyzing the five-week experiment include instructional supports for students' representational fluency in solving linear equations: (a) sequencing the use of graphs, tables, and CAS feedback prior to formal symbolic transpositions, (b) connecting solutions to equations across representations, and (c) encouraging understanding of equations as equivalence relations that are sometimes, always, or never true. While some students' change in sophistication of representational fluency helps substantiate the productive nature of these supports, other students' persistent struggles raise questions of how to address the diverse needs of learners in complex learning environments involving multiple tool-based representations.
机译：这项研究解决了如何支持学生的表征流利性的问题-能够在数学思想的外部表征中创建，移动，翻译以及从中获得含义。在组合计算机代数系统（CAS）和纸笔教室环境中求解线性方程组的上下文是研究此问题的丰富而紧迫的上下文。我们报告了一项合作教学实验的结果，在该实验中，我们设计并测试了用于解决九年级代数学生方程式的函数方法，并链接到实验前后对学生进行的半结构式访谈的结果。为期五周的实验分析结果包括为学生解决线性方程式的表达流利性提供的教学支持：（a）在正式符号换位之前对图形，表格和CAS反馈的使用进行排序，（b）将解决方案与跨表示的方程式联系起来；（c）鼓励将等式理解为有时，总是或永远不正确的等价关系。尽管有些学生对表述流利程度的复杂程度的改变有助于证实这些支持的生产性，但其他学生的不懈努力提出了如何解决复杂学习环境中学习者多样化需求的问题，这些学习环境涉及多个基于工具的表示形式。

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