Exploring K-2 Children Understandings of Visual Representations in Algebraic Reasoning

代数是数学和STEM的基础课题。由于研究已经证明了算术是如何与代数联系在一起的，因此更需要探索K-2儿童如何理解和使用图形和表格等视觉表征。本项目将探讨K-2年级的儿童如何理解代数概念的视觉表征。例如，孩子们可能会创建表格或图表来组织关于两个数量之间关系的信息。他们可能会使用图形和图表来解释他们的数学思维，并发展他们对数字和运算关系的理解。该项目将使用在K-2教室收集的数据，并通过对儿童的采访来描述他们对视觉表征的使用。这个探索性项目旨在开发学习轨迹作为K-2年级儿童如何理解代数关系的视觉表征的认知模型。广义算术包括对算术关系进行泛化，并利用这些泛化进行明确的推理。这些关系包括数的域性质和运算（例如，加法的交换性质），以及数类上的关系，例如偶数和奇数。它需要对算术表达式和方程的结构进行推理，而不是它们的计算值。该项目有三个指导性研究问题。首先，K-2年级儿童如何理解表格、图形和代数关系图等视觉表征的学习轨迹是什么？第二，任务或教学的哪些特征有助于学生在轨迹内的思维运动？第三，儿童在功能思维和广义算术的内容维度上理解代数关系的视觉表征有什么异同？本研究将采用基于设计的研究方法框架，通过课堂教学实验、半临床访谈、课堂观察和儿童书面作业来组织数据收集。这项研究将产生认知模型，可用于设计K-2年级代数学习的进一步干预措施。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。

Algebra is a foundational topic in mathematics and STEM. As research has demonstrated how arithmetic is connected to algebra, there is more need to explore how K-2 children understand and use visual representations such as graphs and tables. This project will explore how children in grades K-2 understand visual representations of algebraic concepts. For instance, children might create tables or graphs to organize information about the relationship between two quantities. They might use graphs and diagrams to explain their mathematical thinking and develop their understanding of relationships in numbers and operations. The project will use data gathered in K-2 classrooms and via interviews with children to describe their use of the visual representations. This exploratory project aims to develop learning trajectories as cognitive models of how children in grades K–2 understand visual representations for algebraic relationships. Generalized arithmetic involves generalizing arithmetic relationships and reasoning explicitly with these generalizations. These relationships include field properties of number and operations (e.g., the Commutative Property of Addition) as well as relationships on classes of numbers, such as evens and odds. It entails reasoning about the structure of arithmetic expressions and equations rather than their computational value. The project has three guiding research questions. First, what are trajectories of learning in how grades K–2 children understand visual representations such as tables, graphs, and diagrams of algebraic relationships? Second, what features of tasks or instruction facilitate movement in students’ thinking within the trajectories? Third, what are similarities and differences in how children understand visual representations of algebraic relationships across the content dimensions of functional thinking and generalized arithmetic? The study will use a design-based research methodological framework to organize data collection via classroom teaching experiments, semi-clinical interviews, classroom observations, and children’s written work. The research will produce cognitive models that can be used to design further interventions for algebra learning in grades K-2. 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.