Learning Trajectories in Grades K-2 Children's Understanding of Algebraic Relationships

代数是学校数学教育的中心问题。它在限制学生的职业和生活选择方面的历史守门人角色是有据可查的。近年来，对此的回应是将代数重新定义为K-12课程。为此，对3-5年级儿童代数思维的研究表明，学生在小学阶段就可以开始理解代数概念，之后他们会更正式地探索这些概念。然而，关于K-2年级的孩子如何理解适合他们年龄的代数概念，还有很多未知的地方。本项目旨在了解K-2年级儿童开始进行代数思考的具体方式。它将确定儿童如何理解数学关系，他们如何表示他们注意到的关系，以及他们如何使用这些关系作为更复杂思考的基石。该项目将以课堂为基础，教授孩子们重要的代数概念，并仔细探索孩子们如何理解这些概念。主要目标是通过教学确定儿童思维发展的复杂程度。了解儿童的思维发展方式将为设计课程、制定内容标准和制定教育政策提供重要基础，所有这些都可以帮助儿童在代数方面取得成功，并有更广泛的机会从事与stem相关的职业。虽然大学和职业准备标准指出代数在幼儿园开始的作用，但K-2年级有限的研究基础限制了代数在K-2课堂上的潜力。这个项目将发展关于儿童如何学习用代数关系进行概括、表示和推理的认知基础。这些发现将为设计新的干预措施和资源提供信息，以加强K-2年级的代数学习，并改善教育政策、实践和资源。该项目将使用设计研究来确定：(1)学习轨迹作为K-2年级儿童如何学习概括、表示和推理这些实践可能发生的内容维度内的代数关系的认知模型（例如，广义算术）；(2)这些轨迹发展的关键节点；(3)促进沿轨迹运动的任务和教学特征。项目的设计将包括使用课堂教学实验，包括：(1)教学设计和规划；(2)对课堂事件的持续分析；(3)对实验过程中产生的所有数据源进行回顾性分析。这将允许学生对代数关系的理解中假设轨迹的发展和经验验证。这一探索性研究将为K-12教学和代数学习提供关键的早期认知基础，有助于使被传统代数教学方法边缘化的学生群体民主化。此外，该项目将在人口结构多样化的学区开展，从而增加了调查结果在不同背景下的普遍性。

Algebra is a central concern in school mathematics education. Its historical gatekeeper role in limiting students' career and life choices is well documented. In recent years, the response has been to reframe algebra as a K-12 endeavor. To this end, research on children's algebraic thinking in grades 3-5 shows that students can begin to understand algebraic concepts in elementary grades that they will later explore more formally. However, there is much that is unknown about how children in grades K-2 make sense of algebraic concepts appropriate for their age. This project aims to understand specific ways in which grades K-2 children begin to think algebraically. It will identify how children understand mathematical relationships, how they represent the relationships they notice, and how they use these relationships as building blocks for more sophisticated thinking. The project will use classroom-based research to teach children about important algebraic concepts and to carefully explore how children come to understand these concepts. The primary goal is to identify levels of sophistication in children's thinking as it develops through instruction. Understanding how children's thinking develops will provide a critical foundation for designing curricula, developing content standards, and informing educational policies, all in ways that can help children become successful in algebra and have wider access to STEM-related careers.While college and career readiness standards point to the role of algebra beginning in kindergarten, the limited research base in grades K-2 restricts algebra's potential in K-2 classrooms. This project will develop cognitive foundations regarding how children learn to generalize, represent, and reason with algebraic relationships. Such findings will inform both the design of new interventions and resources to strengthen algebra learning in grades K-2 and the improvement of educational policies, practices, and resources. The project will use design research to identify: (1) learning trajectories as cognitive models of how grades K-2 children learn to generalize, represent, and reason with algebraic relationships within content dimensions where these practices can occur (e.g., generalized arithmetic); (2) critical junctures in the development of these trajectories; and (3) characteristics of tasks and instruction that facilitate movement along the trajectories. The project's design will include the use of classroom teaching experiments that incorporate: (1) instructional design and planning; (2) ongoing analysis of classroom events; and (3) retrospective analysis of all data sources generated in the course of the experiment. This will allow for the development and empirical validation of hypothesized trajectories in students' understanding of algebraic relationships. This exploratory research will contribute critical early-grade cognitive foundations of K-12 teaching and learning algebra that can help democratize access to student populations historically marginalized by a traditional approach to teaching algebra. Moreover, the project will occur in demographically diverse school districts, thereby increasing the generalizability of findings across settings.