Generalization Across Multiple Mathematical Areas

跨多个数学领域的泛化

• 批准号: 1736156
• 负责人: Amy Ellis
• 金额: $ 87.72万
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-08-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1736156&HistoricalAwards=false
• 关键词: Generalization; Across; Multiple; Mathematical; Areas

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.

鉴于研究基础的研究基础，这一建议使小学通过本科数学的数学指导成为数学教学的核心组成部分，这一建议构成了学生在创造和表达正确的数学概括方面的困难以及教师在支持学生能力方面面临的挑战的困难。此外，尽管学生的困难已得到充分记录，但培养概括所需的教学条件尚不清楚，尤其是在二级和本科水平上。该项目通过开发一个综合框架来解决这些挑战，该框架表征了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。该项目发生在从中学到本科数学的多个数学领域内，包括代数，几何，微积分和组合学。项目调查人员将利用学生的访谈，教授实验和设计实验方法，以表征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促进学生在多种数学环境中推广的特定任务和活动，这将对教师，学区，教师教育者和大学教师进行实际使用。数学概括，创建一般规则，公式和策略的能力是进行数学的关键方面。政策制定者建议将概括作为从小学到本科数学的每个年级级别的数学指导的核心组成部分，共同的核心州标准强调了概括是内容和实践标准的主要目标。但是，这些建议构成了严重的挑战，鉴于学生在创造和表达概括方面存在普遍的困难。在一份有关60,000多名中学生绩效评估的报告中，发现仅显示学生创建正确的一般性陈述的成功率20％。学生对数学概括的挑战也有助于许多领域的数学成就困难，包括代数，几何和组合学。该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。学生参与者的范围从中学生到本科生。学生年龄和数学领域的各种范围将有助于一个强大的模型，以表征学生在8 - 16年级中的推广方式。这些发现还将确定可以在许多不同的环境中更好地支持概括的教学活动，这将用于教师，学区，教师教育工作者和大学教师。该项目产生的知识将支持在本科数学的关键领域提高学生表现，从而有助于多样化且具有全球竞争性的STEM劳动力。