Improving Formative Assessment Practices: Using Learning Trajectories to Develop Resources that Support Teacher Instructional Practice and Student Learning in CMP2

该项目的总体目标是开发创新的教学资源和专业发展，以支持初中教师应对学生学习代数的大学和职业标准所带来的挑战。这个为期四年的项目包括三个主要部分：(1)开发线性函数和线性方程的学习轨迹并进行实证测试；(2)与关联数学项目2 （CMP2）的教师合作，创建并测试一套侧重于形成性评估过程的教学资源；(3)迭代改进专业发展模型，使教师以优化学生代数学习的方式参与教学资源。专业发展活动为教师提供了发展线性函数和代数方程学习轨迹的专业知识的机会，解释学生在这些轨迹上的表现的过程，并根据学生相对于整体学习轨迹的“位置”提供反馈和额外的教学活动。教师知识和实践的这种变化有望提高学生对代数关键概念和过程的学习效果。教师实施有效的形成性评估实践的主要障碍之一是监控和解释学生表现所需的信息的绝对数量和管理。该项目通过使用ASSISTments平台解决了这一障碍。ASSISTments是一个基于网络的在线系统，用于提供数学问题集，并能够根据学生的实时表现调整问题的呈现。数学学习轨迹的研究主要集中在小学阶段的概念。虽然这一水平的研究在数学教育的多个方面已经取得了丰硕的成果和丰富的信息，但我们对数学的几个领域，特别是代数的学习轨迹的理解存在重大的知识差距。事实上，越来越需要新的研究和开发项目来填补这些关键的知识空白。这个项目集中在数学的两个关键领域：学生对线性函数和线性方程的理解，以及学生使用它们解决问题的能力。经验验证的学习轨迹将支持这些领域的课程开发。此外，本研究为中学生内容标准选择的假设轨迹提供了经验证据，为完善课程标准提供了研究基础。最后，通过在线管理系统增强CMP2的使用，增加了针对教师形成性评估实践的专业发展模式产生广泛影响的可能性。虽然我们使用的是一个特定的课程计划，但CMP中对线性函数和方程主题的处理与美国其他基于函数的课程是一致的。因此，在本项目背景下所做的工作将有助于检查其他教学计划中的学习轨迹和形成性评估。

The overarching goal of this project is to develop innovative instructional resources and professional development to support middle grades teachers in meeting the challenges set by college- and career-ready standards for students' learning of algebra. This 4-year project includes three major components: (1) development and empirical testing of learning trajectories for linear functions and linear equations, (2) collaborations with teachers of Connected Mathematics Project 2 (CMP2) to create and test a set of instructional resources focused on formative assessment processes, and (3) iterative refinement of a professional development model for engaging teachers with the instructional resources in ways that optimize students' learning of algebra. The professional development activities provide opportunities for teachers to develop specialized content knowledge of learning trajectories for linear functions and equations in algebra, processes for interpreting students' performances with respect to those trajectories and providing feedback and additional instructional activities based on "where" the student is with respect to the overall learning trajectory. Such changes in teacher knowledge and practice are anticipated to produce improved student learning outcomes for key concepts and procedures in algebra. One of the major stumbling blocks to teachers' implementation of effective formative assessment practice is the sheer volume and management of the information needed to monitor and interpret student performance. The project addresses this impediment by employing the ASSISTments platform, a web-based online system for delivering mathematics problem sets and capable of adapting problem presentation to student performance in real time. Research on learning trajectories in mathematics has mostly centered on concepts at the primary school level. While research at this level has been prolific and informative in multiple aspects of mathematics education, there are major knowledge gaps in our understanding of learning trajectories in several domains of mathematics, specifically in algebra. Indeed, there is a growing need for new research and development projects to fill these critical knowledge gaps. This project focuses on two critical areas in mathematics: students' understanding of linear functions and linear equations, and students' ability to use them to solve problems. Empirically validated learning trajectories will support curriculum development in these areas. In addition, this project contributes to the research base to improve the curriculum standards by providing empirical evidence for hypothesized trajectories for selected content standards for middle school students. Finally, the use of CMP2 augmented by the online management system increases the probability of widespread impact of the professional development model targeted at teachers' formative assessment practices. Although we are using a specific curriculum program, the treatment of linear functions and equations topics in CMP is consistent with other functions-based curricula in the U.S. Thus, the work done in the context of this project will be useful in examining learning trajectories and formative assessment in other instructional programs.