Collaborative Research: Fostering conceptual understanding and skill with an intelligent tutoring system for equation solving

这个教育和人力资源核心研究项目解决了设计有效数学教学的一个持续挑战：如何促进概念和技能的深度学习，它们之间有很强的联系。这一挑战尤其在初中和高中代数中出现；通常情况下，代数课程的学生对程序（例如，解方程）只有中等水平的技能，对这些程序的工作原理几乎没有概念上的理解。这种缺乏深刻理解是不幸的，因为代数是通往高等数学以及未来教育和就业机会的重要垫脚石。因此，我们非常需要代数教学，帮助学生掌握关键概念和解决问题过程的完整知识。本项目利用基于软件的智能辅导系统作为平台来解决这一问题，该系统将扩展现有的代数辅导系统。新的，增强的辅导系统将支持在解决方程和新的概念为导向的代数活动的实践。它将提供详细和有针对性的指导，以适应个别学生的错误，策略和发展代数知识。基于实验室和课堂的研究将探讨三个关键问题：(1)什么样的概念性和程序性活动组合最能帮助学生获得代数的深度技能和概念性知识？(2)学生在这两种类型的活动之间来回转换的频率是多少？(3)与可汗学院等常用软件或仅提供程序性练习的标准智能辅导系统相比，新辅导系统的效果如何？该项目将带来关于如何为初中和高中代数创建有效教学的新知识，以及体现这些知识的新型智能辅导软件。根据过去的研究，人们普遍认为，在许多领域，概念性知识的增加会导致程序性知识的增加，反之亦然。此外，有证据表明，在以概念为导向的活动和以程序为导向的活动之间来回转换的教学特别有效。然而，目前尚不清楚哪种活动最能支持概念性知识的发展，或者如何将它们与以学习程序为目标的活动结合起来，以便学生在两者之间建立联系。为了解决这些问题，该项目将扩展现有的方程求解辅导系统（称为Lynnette）与概念活动，基于过去的研究表明，工作实例，自我解释，视觉表征（即条形图）和激活先验知识可以有效地促进概念知识的获得。在这样做的过程中，这将利用现有的技术基础设施进行智能辅导系统的研究和开发，称为CTAT/Tutorshop。该项目将进行两项实验室研究和两项课堂研究，以调查以概念为导向的活动组合是该系统当前以程序为导向的活动的最有效补充，学生应该在程序和概念为导向的活动之间来回转换的频率，以及由此产生的辅导系统的有效性。与学生使用代数学习常用软件（可汗学院和Lynnette）的两种控制条件相比。该项目将产生关于如何设计有效教学的新知识，帮助学生获得在实际教育环境中有效的概念和程序知识的良好整合。这些知识可能会导致更好、更以概念为导向的代数教学，并有可能改善其他STEM学习环境中的教学。该项目还将创建一个新的智能代数辅导系统，比现有系统更有效。该系统将免费提供给教师和学校。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This Education and Human Resources Core Research project addresses a persistent challenge in designing effective mathematics instruction: how to foster deep learning of concepts and skills, with strong connections among them. This challenge arises especially in middle school and high school algebra; far too often, students in algebra courses come away with only a moderate level of skill at procedures (e.g., equation solving) with almost no conceptual understanding of why these procedures work. This lack of deep understanding is unfortunate, because algebra is an important stepping stone to advanced mathematics as well as to future educational and employment opportunities. Thus, there is a great need for algebra instruction that helps students acquire well-integrated knowledge of key concepts and problem-solving procedures. This project tackles this issue using, as a platform, a software-based intelligent tutoring system that will extend an existing tutoring system for algebra. The new, enhanced tutoring system will support both practice in solving equations and new conceptually-oriented algebra activities. It will provide detailed and targeted guidance that adapts to individual students' errors, strategies, and developing algebra knowledge. Lab-based and classroom studies will investigate three key questions: (1) What mix of conceptual and procedural activities best helps students acquire deep skill and conceptual knowledge of algebra? (2) How frequently should students shift back and forth between the two types of activities? (3) How effective is the new tutoring system compared to commonly-used software, namely, Khan Academy, or a standard intelligent tutoring system that provides procedural practice only? The project will lead to new knowledge about how to create effective instruction for middle school and high school algebra, together with new intelligent tutoring software that embodies this knowledge.Based on past research, it is commonly accepted that in many domains, gains in conceptual knowledge can lead to gains in procedural knowledge, and vice versa. Further, there is some evidence that instruction that shifts back and forth between conceptually-oriented activities and procedurally-oriented activities is especially effective. However, it is not yet known what sorts of activities best support the development of conceptual knowledge, or how they should be integrated with activities that target learning of procedures so that students might make connections between the two. To address these issues, the project will extend existing tutoring system for equation solving (called Lynnette) with conceptual activities, based on past research that suggests that worked examples, self-explanation, visual representations (namely, bar diagrams), and activation of prior knowledge can be effective for promoting gains in conceptual knowledge. In doing so, this will leverage an established technical infrastructure for intelligent tutoring systems research and development, called CTAT/Tutorshop. The project will conduct two lab studies and two classroom studies to investigate what mix of conceptually-oriented activities is the most effective complement to the system's current set of procedurally-oriented activities, how frequently students should shift back and forth between procedurally- and conceptually-oriented activities, and how effective the resulting tutoring system is, compared to two control conditions in which students work with commonly used types of software for algebra learning (Khan Academy and Lynnette). The project will generate new knowledge about how to design effective instruction that helps students acquire well-integrated conceptual and procedural knowledge that is effective in real educational settings. This knowledge could lead to better, more conceptually-oriented instruction in algebra, and it has the potential to improve instruction in other STEM learning environments. The project will also create a new intelligent tutoring system for algebra that is much more effective than current systems. The system will be made available to teachers and schools free of charge.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.