The Role and Use of Examples in Learning to Prove

例子在学习证明中的作用和使用

• 批准号: 1220623
• 负责人: Eric Knuth
• 金额: $ 99.6万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-10-01 至 2017-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1220623&HistoricalAwards=false
• 关键词: Role; Use; Examples; Learning; Prove

This research project is an investigation of the role that examples play in helping learners become proficient in proving mathematical conjectures. Researchers at the University of Wisconsin and New York University are building a framework that characterizes the development of example use as students advance from middle school into post secondary school. Using this developmental information, the researchers are creating instructional strategies that help students think about the nature and value of proof as well as how to construct a mathematical proof.The researchers are interviewing middle school students, high school students, undergraduate mathematics majors in college, and practicing mathematicians in order to learn how they use examples in the process of creating a proof or a deductive argument. They are using teaching experiments to test various strategies on individual students and to learn more about students' thinking about the process of proving. They are also using small group instruction to experiment with instructional strategies and move closer to a future goal of preparing materials for full class instruction.Creating and understanding mathematical proofs has always been difficult for students, and yet it is a critical foundation for developing mathematical understanding. Students often cling to the idea that a large set of examples is sufficient for proving a conjecture true. This misconception has often discouraged instructors from using examples in teaching students to prove conjectures. However, examples can stimulate thinking that guides a student to construct a proof through valid, deductive reasoning. This project is advancing understanding of how students learn to prove and providing tested, instructional strategies that teachers can use to help students.

本研究旨在探讨范例在帮助学习者熟练证明数学命题中所扮演的角色。 威斯康星州大学和纽约大学的研究人员正在建立一个框架，描述学生从中学到中学后的发展过程。 利用这些发展信息，研究人员正在创建教学策略，帮助学生思考证明的本质和价值，以及如何构建数学证明。研究人员采访了中学生、高中生、大学数学专业的本科生和实践数学家，以了解他们如何在创建证明或演绎论证的过程中使用示例。 他们正在使用教学实验来测试学生个人的各种策略，并更多地了解学生对证明过程的思考。 他们还使用小组教学来试验教学策略，并向为全班教学准备材料的未来目标迈进。创建和理解数学证明对学生来说一直很困难，但它是发展数学理解的关键基础。 学生们经常坚持这样的想法，即大量的例子足以证明一个猜想是正确的。 这种误解常常使教师不愿用例子来教学生证明自己的能力。 然而，例子可以激发思维，引导学生通过有效的演绎推理来构建证明。 该项目正在推进对学生如何学习证明的理解，并提供教师可以用来帮助学生的经过测试的教学策略。