CAREER: Developing a Mathematical Knowledge Base for Teaching and Learning Generalization in Basic Algebra at the Middle-Grades in Urban Contexts

本研究为城市背景下中学基础代数泛化教学开发了一个数学知识库。它包括分析儿童基于个人和社会活动的概括过程。课堂教学实验的开发、实施、测试和修订应以阐明儿童泛化模型为目标。该项目还应探讨相关的教学策略和因素，使儿童能够了解泛化过程，包括如何将这些因素适当地协调起来，以便教师能够为学生提供成功的经验。研究将从加州一所学校的一个五年级班级开始，他们的思维过程将被密切监测一个学期。如果有可能，将对同一组5年级学生进行为期4年的纵向调查。本研究的整体理论框架是从涌现的角度出发的，其中儿童的泛化策略被视为从个人和社会过程中构建的。此外，现实数学教育的教学理论（RME），由弗罗伊登塔尔研究所开发，将用于课堂教学实验的设计。这项研究的重点是发展研究。孩子们将进行一些活动，这些活动将使他们能够参与复杂的推理链和意义链，从而形成泛化的模型。课堂教学实验采用开发周期进行开发。这个循环假设教学理论和实际实践之间的共生关系。数据来源包括所有课堂和小组讨论的录像、项目会议和参与教师汇报会议的录音带、学生书面作业的副本和个人临床访谈。课堂社区中个人和集体学习的数据分析应使用Cobb、Stephan、McClain和Gravemeijer（2001）开发的方法。综合研究和教育计划涉及发展性研究活动。第一项活动涉及城市中学儿童概括结构的发展，包括在这种情况下中学教师可能能够在自己的课堂上适应的可用教学理论。这个为期五年的项目以学校为基础，将与同意参与研究的教师及其学生合作进行。课堂教学实验将被设计、测试和修订，特别强调如何协调教学资源和课堂社会环境，使学生成功完成与代数相关的任务，包括泛化。中学数学课程为代数提供了重要的形式基础，至少在表述概括方面需要加强，以便学生能够成功地过渡到高中代数。在第二个活动中，为了建立发展模型，将对儿童用于概括的策略进行描述、分析和测试的持续循环过程。以研究为基础和经过课堂测试的教学材料，包括从这两项活动中汲取的教训和分析，应通过在各种研究和教师中进行介绍的方式编写和传播，供公众使用。会议;城市中学教师在职项目报告将研究成果纳入大学各类数学教育课程和专业在职课程，以培养从研究中汲取教学实践的城市教师；教师和研究人员的期刊文章；cd;一个网站将包括活动、思考片段和相关出版物，以便为所有教师、学生和相关利益相关者提供帮助所有中学生培养有效泛化技能的方法。拟议的项目还将受益于一个多样化的专家小组，他们的专业知识来自于在地方、全州和国际范围内与儿童广泛合作的经验。

Intellectual Merit The proposed study develops a mathematical knowledge base for teaching and learning generalization in basic algebra at the middle school level in an urban context. It involves analyzing children's generalizing processes based on individual and social activity. Classroom teaching experiments shall be developed, implemented, tested, and revised with the goal of articulating models of generalization in children. The project shall also address relevant instructional strategies and factors that will enable children to learn about the generalizing process, including ways the factors can all be coordinated properly so that teachers are able to provide their students with successful experiences. The study shall start with a class of 5th graders in a school in California whose thinking processes will be closely monitored for one semester. If it is possible, the same group of 5th graders will be investigated longitudinally for a period of four years. The overall theoretical framework of the study is drawn from an emergent perspective in which children's strategies of generalization are viewed as being constructed from individual and social processes. Further, the instructional theory of realistic mathematics education (RME), developed at the Freudenthal Institute, will be used in the design of classroom teaching experiments. The study has a developmental research focus. Children will perform activities that will allow them to engage in complex chains of reasoning and signification leading to the emergence of models for generalization. Also, classroom teaching experiments are developed using a development cycle. This cycle assumes a symbiotic relationship between instructional theory and actual practice. Data sources include video recordings of all classroom and small group discussions, audiotape recordings of project meetings and debriefing sessions with participating teachers, copies of student's written work, and individual clinical interviews. Data analyses of individual and collective learning in a classroom community shall use a method developed by Cobb, Stephan, McClain, and Gravemeijer (2001). Broader Impact The integrated research and education plan involves developmental research activities. The first activity concerns the development of structures of generalization among urban middle school children, including usable instructional theories that middle school teachers in such contexts may be able to adapt in their own classrooms. The five-year project is school-based and will be conducted in a collaborative partnership with teachers and their students who agree to participate in the study. Classroom teaching experiments will be designed, tested, and revised with a particular emphasis on how instructional resources and the classroom social context can be coordinated so that students succeed in algebra-related tasks involving generalization. The middle school mathematics curriculum provides the important formal foundation in algebra that needs to be strengthened at least in the area of formulating generalizations so that students are able to successfully transition to high school algebra. In the second activity, an ongoing circular process of describing, analyzing, and testing of strategies that children employ to generalize will be conducted in order to establish developmental models. Research-based and classroom-tested instructional materials that contain lessons and analyses drawn from the two activities shall be prepared and disseminated for public use by way of: presentations in various research and teachers. conferences; presentations in faculty in-service programs in urban middle schools; incorporation of research findings in various mathematics education courses in the university and in professional in-service programs with the aim of developing urban teachers whose instructional practices have been drawn from research; journal articles for teachers and researchers; CDs; and a website that will include activities, think pieces, and relevant publications in order to provide all teachers, students, and concerned stakeholders with ways in which to assist all middle-school students develop effective generalization skills. The proposed project shall also benefit from a diverse panel of experts whose expert knowledge has been drawn from having extensively worked with children in local, statewide, and international contexts.