Generalization Across Multiple Mathematical Areas

将概括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建和表达正确的数学概括方面的困难，以及教师在支持学生概括能力方面面临的挑战。此外，尽管学生的困难得到了很好的记录，但培养泛化所需的教学条件并没有得到很好的理解，特别是在中学和本科阶段。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年级数学概括的特点，并确定了能够支持正确概括的教学干预措施。该项目涉及从中学到本科的多个数学领域，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在不同的数学环境中培养学生概括能力的具体任务和活动，这将对教师、学区、教师教育工作者和大学教师有用。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个年级数学教学的核心组成部分，共同核心国家标准在内容和实践标准中都将泛化作为主要目标。然而，考虑到学生在创建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，学生在创建正确的概括性陈述方面只有20%的成功率。学生在数学概括方面的挑战也导致了在许多领域取得数学成绩的困难，包括代数、几何和组合学。这个项目将通过调查学生如何有效地概括和教师如何支持更有效的数学概括来应对这些挑战。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。不同的学生年龄和数学领域将有助于建立一个稳健的模型，描述学生在8-16年级如何概括。这些发现还将确定在许多不同的环境中能够更好地支持泛化的教学活动，这将对教师、学区、教师教育工作者和大学教师有用。该项目产生的知识将支持提高学生在本科数学关键领域的表现，从而有助于培养一支多元化和具有全球竞争力的STEM劳动力队伍。

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.