Collaborative Research: Applying and Refining a Model for Dynamic, Discussion-Based Professional Development for Middle School Teachers about Fractions, Ratios and Proportions

本项目探讨了两种不同版本的专业发展（PD）的有效性，旨在提高中学数学教师对分数和比例的理解，以及他们向学生教授这些数学概念。PD采用一种方法，让教师参与基于网络的应用程序，允许他们测试和实验他们的数学思想。这些应用程序与指导性问题相结合，挑战了教师对分数和比例的思考，既促进了对概念的批判性思考，又进一步发展了他们对概念的理解。研究人员将使用一种创新的方法，主题建模，来检查每个版本的PD的有效性。主题建模将使研究人员能够识别教师参与PD时使用的语言模式，从而更好地了解教师理解的发展。该项目不仅有望提高教师对学生传统上难以掌握的主题的指导，而且还证明了主题模型作为衡量使用自然语言学习的一种手段的可行性——这种方法有可能取代大规模的多项选择测试，作为衡量学生在课堂上学习活动的学习情况的一种方法。该项目采用基于设计的研究方法，重点关注三个主要目标：扩展试点PD模型，包括分数和比例；PD模型的一个猜想的检验侧重于支持教师以新颖的方式在学习内容之间建立联系，从而从他们的经验中发展教学内容知识（PCK）；实施主题建模作为衡量教师参与PD变化的评估模型。主题模型将被用作主要评估，而不是铅笔和纸的评估，因为它们与内容和PD方法一致，PD方法是高度互动的，专注于猜想测试。主题建模的使用为定性数据作为评估工具提供了机会，而不是依赖于可能与指令不一致的评估工具。这项工作的建议成果包括：为教师提供一系列数学任务，支持他们以有趣的方式参与重要的数学思想；一个指南概述了连贯的30小时的经验，教师从事开发内容知识和PCK相关的分数，比例和比例；PD项目有效性的证据，它依赖于主题建模来衡量教师参与和学习的变化；支持教师将内容知识发展与PCK发展联系起来的两种不同方法的数据；还有一个编码框架，定义了对与参考单位和不变性相关的分数的“健壮”理解。PD的所有方面都将以使用动态的、基于网络的“玩具”（应用程序）为基础，这些“玩具”旨在支持对比例和分数概念的强大理解的发展。对该领域的贡献包括PD模型、健壮的理解框架和特定于主题模型使用的发现。这样的贡献之一将是进一步了解使用主题模型进行评估所需的样本量，这将有助于该领域更好地了解如何在相对较小的定性数据集上使用这种定量方法。探索研究preK-12项目（DRK-12）旨在通过研究和开发创新资源、模型和工具，显著提高preK-12学生和教师对科学、技术、工程和数学（STEM）的学习和教学。DRK-12计划中的项目建立在STEM教育的基础研究和先前的研究和开发工作的基础上，为拟议的项目提供了理论和实证依据。 该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project explores the effectiveness of two different versions of professional development (PD) designed to enhance middle school mathematics teachers’ understanding of fractions and proportions, and their teaching of these mathematical concepts to students. The PD uses an approach that engages teachers with web-based apps that allow them to test and experiment with their mathematical ideas. The apps, combined with guiding questions that challenge teachers’ thinking about fractions and proportions, serve both to promote critical thinking about the concepts and to further developing their understandings of the concepts. The researchers will use an innovative approach, topic modeling, to examine the effectiveness of each of version of the PD. Topic modeling will allow the researchers to identify patterns in the language teachers use as they participate in the PD, and subsequently, to better understand the development of the understanding teachers develop. The project has promise to not only enhance teachers’ instruction on topics with which students have traditionally struggled, but also to demonstrate the viability of Topic Models as a means to measure learning using natural language—an approach that could potentially replace large-scale, multiple-choice tests as a measure of student learning with learning activities in the classroom. The project uses a design-based research approach to focus on three main goals: expansion of a pilot PD model to include fractions and proportions; testing of a conjecture about the PD model focused on supporting teachers in making connections between learning content in novel ways to developing pedagogical content knowledge (PCK) from their experiences; and implementing topic modeling as an assessment model for measuring changes in teachers’ participation in the PD. Topic models will be used as the primary assessment, rather than pencil and paper assessments, because of their alignment to the content and the PD approach, which is highly interactive and focused on conjecture testing. The use of topic modeling opens an opportunity for qualitative data to be used as an assessment tool rather than relying on assessment instruments that may not be aligned to the instruction. Proposed outcomes of this work include: a pool of mathematical tasks for teachers that support playful engagement with important mathematical ideas; a guide outlining a coherent 30-hour experience for teachers that engages in developing both content knowledge and PCK related to fractions, ratios, and proportions; evidence of the PD program’s effectiveness that relies on topic modeling to measure change in teachers’ participation and learning in the workshops; data on two different approaches to supporting teachers in linking content knowledge development to PCK development; and a coding framework that defines “robust” understanding of fractions related to referent units and invariance. All aspects of the PD will be grounded in the use of dynamic, web-based “toys” (apps) that have been designed to support the development of robust understandings of proportion and fraction concepts. Contributions to the field include the PD model, the robust understandings framework, and findings specific to the use of topic models. One such contribution will be further insight into the sample size necessary to use topic models for assessment, which will help the field better understand how to use this quantitative approach with relatively small qualitative data sets. The Discovery Research preK-12 program (DRK-12) seeks to significantly enhance the learning and teaching of science, technology, engineering and mathematics (STEM) by preK-12 students and teachers, through research and development of innovative resources, models and tools. Projects in the DRK-12 program build on fundamental research in STEM education and prior research and development efforts that provide theoretical and empirical justification for proposed projects. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.