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Proof and proving in school mathematics instruction: Making the elementary grades part of the equation.

机译：学校数学教学中的证明和证明：使小学成绩成为方程式的一部分。

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摘要

The concept of proof and the practice of proving have not typically been a focus of attention by those involved in the teaching and learning of mathematics in the elementary grades. This constitutes a serious lack in the way we conceptualize elementary school mathematics. Because proof and proving are at the core of doing and knowing mathematics, we cannot have a viable elementary school mathematics curriculum, or opportunities for students to learn it that have integrity, without having a way to incorporate proof and proving into a coherent conception of mathematics instruction in the early grades. This study investigates (1) how proof and proving might be conceptualized in the context of elementary mathematics instruction, and (2) how this conceptualization can inform the work that elementary teachers would need to do, and the knowledge that they would need to have, to promote proof and proving in their classrooms.;The study is structured around three interrelated strands of work. The first strand illuminates the nature of proof in the early grades by identifying and exploring parameters potentially determinant of which arguments qualify as proofs. The second strand capitalizes on the first and sets forth a framework about the meaning of proof in K-12 mathematics instruction. Additionally, it uses this framework as a tool to examine instruction in the early grades in order to clarify aspects of teachers' role in fostering proof and proving. As a result, the second strand also develops a framework about instructional practices for cultivating proof and proving that make sense even in the early grades. The third strand investigates the knowledge of proof needed for cultivating proof and proving in elementary mathematics instruction. It advances a classification of different kinds of knowledge of proof that elementary teachers might need, paying particular attention to what is involved for teachers as they mobilize opportunities for their students to engage in proving. The study pursues the three strands of work both conceptually, using scholarly work on proof (including work on mathematical practice), and empirically, using data from the teaching practice of an elementary teacher who was trying to cultivate her students' reasoning skills.;The products of the study offer insight into what it might mean, and what it would take, to make proof and proving central to elementary children's learning of mathematics. The conceptual analytic tools developed by the study contribute to theory building in the teaching and learning of proof and proving in the early grades but also more broadly. Furthermore, these tools can support the design of materials for the professional education of elementary teachers, and provide guidance for mathematics teacher educators who might implement them.
机译：证明的概念和证明的实践通常不是那些参与小学年级数学教学的人们关注的焦点。这严重缺乏我们对小学数学概念化的方式。因为证明和证明是完成和了解数学的核心，所以我们无法拥有可行的小学数学课程，也无法为学生提供具有完整性的学习机会，而没有办法将证明和证明融入数学的连贯概念中早期的教学。这项研究调查（1）在基础数学教学的背景下如何将证明和证明概念化，以及（2）这种概念化如何为基础教师需要做的工作以及他们需要具备的知识提供信息，这项研究是围绕三个相互关联的工作结构进行的。第一条通过识别和探索可能决定哪些论点有资格作为证明的参数，阐明了早期成绩的证明性质。第二部分借鉴了第一部分，并提出了有关K-12数学教学中证明意义的框架。此外，它使用此框架作为检查早期教学的工具，以阐明教师在促进证明和证明方面的作用。结果，第二阶段也建立了一个关于教学实践的框架，该框架即使在早期的时候也可以培养证明和证明。第三部分研究在基础数学教学中培养证明和证明所需的证明知识。它对基础教师可能需要的各种证明知识进行了分类，并特别注意教师在调动学生参与证明的机会时所涉及的知识。该研究从理论上讲这三方面的工作，既使用了证明方面的学术著作（包括数学实践方面的著作），又是根据经验，利用了一位初等教师的教学实践中的数据，该教师正在努力培养学生的推理能力。该研究的产品提供了对进行证明和证明对小学儿童数学学习的核心意义的理解，以及它将需要什么。该研究开发的概念分析工具有助于早期阶段的证据和证明的教与学，但在更广泛的范围内也有助于理论构建。此外，这些工具可以支持基础教师专业教育的材料设计，并为可能实施它们的数学教师教育者提供指导。

著录项

  • 作者

    Stylianides, Andreas J.;

  • 作者单位

    University of Michigan.;

  • 授予单位 University of Michigan.;
  • 学科 Education Mathematics.;Education Elementary.
  • 学位 Ph.D.
  • 年度 2005
  • 页码 232 p.
  • 总页数 232
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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