Generalization Across Multiple Mathematical Areas

将泛化作为从小学到本科数学教学的核心组成部分的建议提出了严峻的挑战，因为研究基础确定了学生在创造和表达正确的数学泛化方面的困难，以及教师在支持学生泛化能力方面面临的挑战。此外，尽管学生的困难有很好的记录，但促进泛化的必要教学条件并没有得到很好的理解，特别是在中学和本科阶段。本项目通过开发一个全面的框架来描述8-16年级的生产性数学概括，并确定可以支持正确概括的教学干预措施，解决了这些挑战。该项目涉及从中学到本科数学的多个数学领域，包括代数、几何、微积分和组合数学。项目调查人员将利用学生访谈、教学实验和设计实验方法来描述概括的过程，并确定支持生产性概括的教学条件。该项目的结果将确定具体的任务和活动，以促进学生在各种数学环境下的泛化，这将对教师、学区、教师教育工作者和大学教师有实际用处。数学泛化，即建立一般规则、公式和策略的能力，是数学研究的一个关键方面。政策制定者建议将泛化作为从小学到本科数学的每个年级数学教学的核心组成部分，共同核心州标准强调泛化是内容和实践标准的主要目标。然而，考虑到学生在创造和表达概括方面普遍存在的困难，这些建议构成了严峻的挑战。在一份对6万多名中学生进行的成绩评估报告中，调查结果显示，学生们正确概括陈述的成功率只有20%。学生在数学推广方面的挑战也导致了在许多领域数学成就的困难，包括代数、几何和组合。该项目将通过调查学生如何有效地进行归纳以及教师如何支持更有效的数学归纳来解决这些挑战。通过学生访谈和教学实验，项目研究者将探索代数、几何、微积分和组合学中的这些问题。学生参与者的范围从中学生到大学生。学生年龄和数学领域的不同范围将有助于建立一个稳健的模型，以表征8-16年级学生如何进行概括。这些发现还将确定能够在许多不同环境中更好地支持泛化的教学活动，这将对教师、学区、教师教育者和大学教师有用。该项目产生的知识将有助于提高学生在本科数学关键领域的表现，从而为多样化和具有全球竞争力的STEM劳动力做出贡献。

The recommendation to make generalization a central component of mathematics instruction from elementary school through undergraduate mathematics poses serious challenges in light of the research base that identifies students' difficulties in creating and expressing correct mathematical generalizations and the challenges teachers face in supporting students' abilities to generalize. Furthermore, although student difficulties are well documented, the instructional conditions necessary for fostering generalization are not well understood, particularly at the secondary and undergraduate levels. This project addresses these challenges by developing a comprehensive framework characterizing productive mathematical generalization in Grades 8-16 and identifying instructional interventions that can support correct generalizing. The project occurs within multiple mathematical domains extending from middle school to undergraduate mathematics, including algebra, geometry, calculus, and combinatorics. The project investigators will leverage student interviews, teaching experiments, and design experiment methodologies in order to characterize the processes of generalizing and to identify the instructional conditions that support productive generalization. The results of the project will identify specific tasks and activities fostering student generalizing in a diversity of mathematical settings, which will be of practical use to teachers, school districts, teacher educators, and university instructors.Mathematical generalization, the ability to create general rules, formulas, and strategies, is a key aspect of doing mathematics. Policy makers recommend making generalization a central component of mathematics instruction at every grade level from elementary school through undergraduate mathematics, with the Common Core State Standards highlighting generalization as a major goal in both the content and the practice standards. However, these recommendations pose serious challenges given students' pervasive difficulties in creating and expressing generalizations. In a report on performance assessments from more than 60,000 secondary students, findings revealed only a 20% success rate in students' creation of correct general statements. Students' challenges with mathematical generalizations also contribute to difficulties in mathematics achievement in many domains, including algebra, geometry, and combinatorics. This project will address these challenges by investigating how students generalize productively and how teachers can support more effective mathematical generalization. Through student interviews and teaching experiments, the project investigators will explore these issues in algebra, geometry, calculus, and combinatorics. Student participants will range from middle school students through undergraduates. The diverse range of student ages and mathematical domains will contribute to a robust model characterizing how students generalize in Grades 8-16. These findings will also identify instructional activities that can better support generalization in many different settings, which will be of use to teachers, school districts, teacher educators, and university instructors. The knowledge generated from the project will support improved student performance in critical areas of undergraduate mathematics, thus contributing to a diverse and globally competitive STEM workforce.