Generalization Across Multiple Mathematical Areas

从小学到本科数学教学的核心组成部分的建议提出了严重的挑战，根据研究基础，确定学生的困难，创造和表达正确的数学概括和教师面临的挑战，支持学生的能力，概括。此外，虽然学生的困难是有据可查的，但促进普遍化所需的教学条件并没有得到很好的理解，特别是在中学和大学一级。该项目通过开发一个综合框架来应对这些挑战，该框架描述了8-16年级的生产性数学概括，并确定了可以支持正确概括的教学干预措施。该项目发生在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，以描述概括的过程，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，培养学生在各种数学环境中的概括能力，这将对教师、学区、教师教育工作者和大学教师有实际用途。数学概括，即创造一般规则、公式和策略的能力，是做数学的一个关键方面。政策制定者建议将泛化作为从小学到本科数学的每一年级数学教学的核心组成部分，共同核心国家标准强调泛化作为内容和实践标准的主要目标。然而，鉴于学生在创造和表达概括方面普遍存在的困难，这些建议构成了严重的挑战。在一份对60，000多名中学生进行的成绩评估报告中，结果显示，学生在创造正确的一般陈述方面的成功率只有20%。学生在数学概括方面的挑战也导致了许多领域的数学成就困难，包括代数，几何和组合学。本项目将通过调查学生如何有效地概括以及教师如何支持更有效的数学概括来解决这些挑战。通过学生访谈和教学实验，项目调查人员将在代数，几何，微积分和组合学中探索这些问题。学生参与者将从中学生到本科生不等。不同的学生年龄和数学领域将有助于一个强大的模型，描述学生如何在8-16年级概括。这些研究结果还将确定教学活动，可以更好地支持在许多不同的设置，这将是有用的教师，学区，教师教育工作者和大学讲师的泛化。从该项目产生的知识将支持提高学生在本科数学的关键领域的表现，从而有助于多样化和全球竞争力的干劳动力。

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.