EAGER: Collaborative Research: Technology to Support Mathematical Argumentation

这个项目要解决的大问题是，“技术如何支持数学论证的教与学？”这项热切的建议侧重于迈向帮助青少年学习数学/代数论证这一愿景的第一步。该项目有两个相辅相成的目标：更好地定义支持小学生学习证明和辩论所需的技术工具，以及更好地理解小学生在有这些工具的情况下如何学习提出和辩护数学主张。PI设想并正在开发一种在平板电脑上实施的动画工具(这样每个孩子都有他/她自己的电子笔记本)，为小组和整个班级分享和提炼论点以及促进具体的数学讨论提供基础设施。一个巨大的挑战是提供正确的表达工具；要进行分享，学习者首先需要能够表达他们的理解，而且许多研究表明，与简单的口头理解表达相比，具体的“书面”(可分享的)理解表达为促进具体讨论提供了更好的基础。这个项目在表达方面取得的特别进步是为动态表示提供了工具；也就是说，当执行数学运算时，它们能够以动画形式呈现所发生的事情。然后可以回放、暂停这些理解的具体表达，等等。学生通过绘制、擦除、复制、移动和分组对象来创建动画。他们可以编辑彼此的动画，老师还可以创建动画供学生查看和编辑。他们记录并保存他们的口头解释与动画一起播放，从而使他们的口头描述具体、可审查和可分享。准备和提出数学论证的能力是学生在小学高年级(3-5年级)需要达到的数学能力的关键组成部分。许多学生很难理解这一主题，因此也很难进行数学论证。PI设想了一个计算工具集，年轻的学生将使用它来构建和分享数学论点，以帮助学习成为称职的代数推理者。在这个热切的项目中，他们专注于开发工具集的第一步，并调查年轻学习者在拥有这样的工具时如何提出和辩护数学论点。这个项目代表了一个未经检验但可能具有变革性的想法的早期工作，并可能在帮助年轻学习者成为数学推理者方面催化快速和创新的进展。

The big question being addressed in this project is, "How can technology support the teaching and learning of mathematical argumentation?" This EAGER proposal focuses on the first steps in moving towards the vision of helping youngsters learn mathematical/algebraic argumentation. The project has two complementary goals: to better define the technological tools needed to support elementary students learning about proof and argumentation and to understand better how elementary school students learn to make and defend mathematical claims when such tools are available. The PIs envision and are developing an animation tool to be implemented on tablet computers (so each child has his/her own electronic notebook) that provides infrastructure for sharing and refining arguments in small groups and across the class and for promoting concrete math discussions. A big challenge is providing the right tools for expression; for sharing to happen, learners need to first be able to express their understanding, and much research shows that a concrete "written" (sharable) expression of understanding provides better foundations for promoting concrete discussion than do expressions of understanding that are simply verbal. The particular advance in expression that is being made in this project is providing tools for dynamic representations; that is, they are able to animate what happens when mathematical operations are carried out. These concrete expressions of understanding can then be played back, paused, and so on. Students create animations by drawing, erasing, duplicating, moving, and grouping objects. They can edit each others' animations, and the teacher will also be able to create animations for students to view and edit. They record and save their oral explanations along with the animations and play them back together, thus making their verbal descriptions concrete, examinable, and sharable. The ability to prepare and present a mathematical argument is a key component of the mathematical competence students need to achieve in upper elementary school (grades 3 through 5). Many students struggle with the subject matter and therefore struggle as well with making mathematical arguments. The PIs envision a computational tool set that young students will use to construct and share mathematical arguments, in the service of learning to be competent algebraic reasoners. In this EAGER project, they focus on first steps in developing that tool set and on investigating how young learners make and defend mathematical arguments when they have such tools available. This project represents work in its early states on an untested but potentially transformative idea and is likely to catalyze rapid and innovative advances in helping young learners become mathematical reasoners.