ROLE: Knowledge of Algebra for Teaching

我们建议就这一项目的两个组成部分开展工作，该项目最初被提议分为三个主要组成部分。我们将这项工作重新设计为一个为期24个月的项目，主要目的是框架开发和评估工具的初步工作；所附预算将足以支持这项工作。我们最初的建议包括两个研究问题：问题1。中学代数教师利用什么代数知识(数学知识、信念、性格和数学概念)来支持他们的教学？问题2。预科教师在数学和教师教育方面的经验如何促进用于教学的代数知识的发展？对于这个缩小版的项目，我们将只回答第一个研究问题。我们已经确定了与此有关的几个分问题；它们包括在下面。我们将作为一个项目团队开会，以完善和缩小这个清单，并期望专注于其中的一些子集：1.教师的代数知识是如何组织和组块的，这与他们的教学有什么关系？2.教师如何阐明代数是关于什么的？他们如何描述构成他们教授的课程的“大创意”？他们能与其他人比较和对比他们所采取的方法吗？这与他们的教学有什么关系？3.教师在回应学生的想法时利用了什么知识？4.更一般的数学知识类型(推理，表示，问题解决)与教学代数知识有关？5.教师使用什么知识来着眼于更长远的数学目标或“大想法”，同时也回应学生的概念和兴趣？6.教师利用什么资源和策略来加深他们的教学代数知识？在这个项目的24个月里，我们将通过以下几项工作来开发“代数教学知识”的框架：1.综合文献，包括：学校代数的理论分析，关于代数教与学的研究，关于教学中使用的知识的研究，关于代数教师如何使用课程的调查。检查代数课程材料和教师手册，并对这些材料进行分析。采访当代中学代数材料的开发人员4。通过现有的录像带检查教学5。从选定的跨各种课程方法的在职教室收集数据，以支持框架的制定。此外，我们将开始开发和试行评估工具，以评估和描述教师的教学代数知识。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，可能在代数中相对较窄的内容范围内，并进行有限的试点工作。这项工作的产品将是一个分析和经验派生的框架，描述用于教学的代数知识，以及一组原型任务，可以用作后续项目中更广泛的任务设计的基础。此外，我们将使用该框架作为分析我们教师样本中的代数教学的基础，并期望产生一份关于教师的代数教学知识的论文，因为它与他们在不同课程取向的课堂实践有关。

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.