R&D: Diagnosing Teachers' Multiplicative Reasoning

诊断教师的乘法推理（DTMR）解决了DR-K12情境挑战链的评估组成部分。调查教师使学生能够学习所需的知识，并对这些知识进行评估，是数学教育的核心挑战。一种方法强调教师的知识量和学生的成绩之间的相互关系，以达到问责的目的。另一个以数学思维研究为基础，经常使用案例研究来调查教师在学生解决问题的策略中发现和利用机会的能力。这些方法之间存在的紧张关系，因为工具方便评估大量的教师是不敏感的推理能力，而案例研究方法用于调查教师的推理是不实用的大样本。DTMR项目正在建立一个演示工具，既适合用于大量的教师样本，又能提供有关他们以支持学生思维的方式对内容进行推理的能力的信息。具体而言，DTMR项目正在开发和评估一种测试形式，以诊断教师在两个密切相关的推理数量之间乘法关系的案例中的能力。第一个是在测量上，当划分给定的数量时，通常依赖于乘法嵌套的单位水平。第二个是协变，它通常依赖于不同数量之间的乘法关系。该项目的重点是这些推理的各个方面，这些方面是相互关联的，并且是数量的加减，数量的乘法，数量的引用和分割以及数量的比率的基础。该项目将考虑分数，小数和比率。该项目的一个主要目标是解决内容和结构有效性的演示形式在足够的深度，以便更大规模的工作和预测有效性的研究可能会遵循。该项目正在开发一种新的心理测量模型，称为认知诊断模型（CDM）。使用CDMs涉及指定特定领域的推理组件，然后系统地构建测试问题（通常是多项选择），以便每个选择对应于这些组件的不同组合的推理。该项目借鉴了对学生和教师乘法推理的研究，正在开发一种30至40个项目的测试形式。清洁发展机制估计和等值方法的模拟研究也是该项目的一个重要组成部分。

Diagnosing Teachers' Multiplicative Reasoning (DTMR) addresses the assessment component of the DR-K12 Contextual Challenges strand. Investigating knowledge that teachers need to enable students' learning and developing assessments of that knowledge are central challenges for mathematics education. One approach emphasizes correlations between amounts of teachers' knowledge and students' achievement for accountability purposes. Another, grounded in research on mathematical thinking, often uses case studies to investigate teachers' capacities for identifying and building upon opportunities in students' problem-solving strategies. Tensions exist between these approaches because instruments convenient for assessing large numbers of teachers are insensitive to capacities for reasoning, while case study methods used to investigate teachers' reasoning are not practical with large samples. The DTMR project is building a demonstration instrument both suitable for use with large samples of teachers and informative about their capacities to reason about content in ways that support students' thinking. In particular, the DTMR project is developing and evaluating a test form that diagnoses teachers' capacities in two closely connected cases of reasoning about multiplicative relations among quantities. The first is on on measurement that often relies on multiplicatively nested levels of units when partitioning a given quantity. The second is on covariation that often relies on multiplicative relations between distinct quantities. The project is focusing on aspects of such reasoning that are interconnected and fundamental to addition and subtraction of quantities, multiplication of quantities, quotative and partitive division of quantities, and ratios of quantities. The project will consider fractions, decimals, and ratios. A main goal of the project is to address content and construct validity of the demonstration form in sufficient depth so that larger scale work and predictive validity studies may follow. The project is developing instruments using a new class of psychometric models called cognitive diagnosis models (CDMs). Using CDMs involves specifying components of reasoning in a particular domain and then constructing test questions (typically multiple-choice) systematically so that each choice corresponds to reasoning with a different combination of those components. Drawing on the research on students' and teachers' multiplicative reasoning, the project is developing one test form of 30 to 40 items. CDM simulation studies of estimation and equating methods are also an important component of the project.