ROLE: Knowledge of Algebra for Teaching

我们建议对这个项目的两个组成部分进行工作，最初提议有三个主要组成部分。我们将这项工作重新设计为一个为期24个月的项目，主要针对框架开发和评估工具的初步工作；所附预算将充分支持这项工作。我们最初的提案包括两个研究问题：问题1。中学代数教师会利用哪些代数知识（数学知识、信念、性格和数学概念）来支持他们的教学？问题2。职前教师在数学和教师教育方面的经验如何促进代数教学知识的发展？对于这个缩减版的项目，我们将只解决第一个研究问题。我们已经确定了与此相关的几个子问题；它们包括在下面。我们将作为一个项目团队开会来细化和缩小这个列表，并期望将重点放在其中的一些子集上：教师的代数知识是如何组织和分块的，这与他们的教学有什么关系？老师是如何阐述代数的内容的？他们如何描述构成他们所教课程的“大思想”？他们能将自己采取的方法与他人进行比较和对比吗？这和他们的教学有什么关系？教师在回应学生的想法时借鉴了哪些知识？4. 更一般的数学知识类型（推理、表示、解决问题）如何与教学中的代数知识联系起来？教师在关注长期数学目标或“大想法”的同时，也对学生的概念和兴趣做出反应，他们使用了哪些知识？教师利用什么资源和策略来加深教学中的代数知识？在本项目为期24个月的时间里，我们将承担“教学用代数知识”框架的开发工作，具体工作如下：综合文献，包括：学校代数的理论分析，代数的教与学的研究，教学中知识运用的研究，代数教师如何使用课程的调查。检查代数教材和教师手册，并对这些教材进行分析。访谈当代中学代数教材的开发者通过现有录像带检查教学。从服务教室中收集数据，以跨越一系列课程方法，以支持框架的开发。此外，我们将开始开发和试点一种评估工具，以评估和描述教师的代数知识。资源将无法用于全面的项目开发、试验和分析。相反，我们将设计原型任务，可能在代数中相对狭窄的内容范围内，并进行有限的试点工作。这项工作的成果将是一个分析和经验推导的框架，描述教学中的代数知识，以及一组原型任务，可作为后续项目中更扩展任务设计的基础。此外，我们将使用该框架作为分析教师样本代数教学的基础，并期望撰写一篇关于教师代数教学知识的论文，因为这与他们在各种课程导向下的课堂实践有关。

We propose to undertake work on two components of this project, which was originally proposed as having three major components. We have redesigned this effort as a 24-month project primarily aimed at framework development and initial work on assessment tools; the attached budget would adequately support this work.Our original proposal included two research questions:Question 1. What knowledge of algebra for teaching (mathematical knowledge, beliefs, dispositions and conceptions of mathematics) do secondary school teachers of algebra draw upon to support their instruction?Question 2. How do preservice teachers' experiences in mathematics and in teacher education foster the development of knowledge of algebra for teaching?For this scaled-back version of the project we will address only the first research question. We had identified the several subquestions related to this; they are included below. We will meet as a project team to refine and narrow this list, and expect to focus on some subset of these:1. How is teachers' knowledge of algebra organized and chunked, and how does this relate to their teaching?2. How do teachers articulate what algebra is about? How do they describe the "big ideas" that structure the courses they teach? Can they compare and contrast the approach they take with others? How does this relate to their teaching?3. What kind of knowledge do teachers draw upon in responding to students' ideas? 4. How do more general mathematical types of knowledge (reasoning, representing, problem solving) relate to knowledge of algebra for teaching?5. What knowledge do teachers use in keeping their sights on longer-term mathematical goals or "big ideas" while also responding to student conceptions and interests?6. What resources and strategies do teachers draw on for deepening their knowledge of algebra for teaching?In the 24 months of this project we will undertake the development of a framework for "knowledge of algebra for teaching" by doing the following:1. Synthesizing literature, including: theoretical analyses of school algebra, research about the teaching and learning of algebra, research about knowledge used in teaching, investigations of how algebra teachers are using curricula.2. Examining algebra curriculum materials and teachers' manuals, and analyses of those materials3. Interviewing developers of contemporary secondary school algebra materials4. Examining teaching through existing videotapes5. Gathering data from inservice classrooms selected to span a range of curricular approaches, to support the framework development. In addition, we will begin the development and piloting of an assessment tool to assess and describe teachers' knowledge of algebra for teaching. Resources will not be available for full scale item development, piloting, and analysis. Rather we will design prototype tasks, probably in a relatively narrow content band within algebra, and conduct limited piloting efforts.The products of this work will be an analytic and empirically-derived framework describing algebra knowledge for teaching, and a set of prototype tasks that can be used as a basis for more extended task design in a subsequent project. In addition, we will use the framework as a basis for analysis of the algebra teaching in our teacher sample, and expect to produce a paper about teachers' knowledge of algebra for teaching as it relates to their classroom practice in a variety of curricular orientations.