The Role and Use of Examples in Learning to Prove

本研究项目是关于范例在帮助学习者熟练证明数学猜想方面所起的作用的调查。威斯康星大学和纽约大学的研究人员正在构建一个框架，该框架描述了学生从中学进入大专阶段时榜样使用的发展。利用这些发展信息，研究人员正在制定教学策略，帮助学生思考证明的性质和价值，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数学家，以了解他们在创建证明或演绎论点的过程中如何使用例子。他们正在利用教学实验在学生身上测试各种策略，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更近了一步。对学生来说，创建和理解数学证明一直是困难的，但它是发展数学理解的关键基础。学生们常常认为，一大堆例子就足以证明一个猜想是正确的。这种误解经常阻碍教师在教学生证明猜想时使用例子。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据。该项目旨在促进对学生如何学习证明的理解，并提供经过测试的教学策略，教师可以使用这些策略来帮助学生。

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.