Abstract

The present study is motivated by a significant body of research documenting teachers’ perennial difficulties with a critical swath of topics related to multiplication. In response, we track how Nina, a future middle grades mathematics teacher, made progress constructing explanations across topics by reasoning with measurement-based definitions of multiplication and of fractions and by coordinating symbolic representations with math drawings. The dataset spans 1 semester of Nina’s in-class work during a content course—explicitly designed to foster coherence within the multiplicative conceptual field—as well as her written assignments for the course and her moment-to-moment reasoning during three interviews conducted near the beginning, middle, and end of the semester. A main result is that constructs from coordination class theory, a strand of theory within the knowledge-in-pieces epistemological perspective, were particularly useful for tracking and explaining Nina’s piecemeal progress. The broad contribution of the article is two-fold—(a) a shift in focus from research on reasoning about one or two topics toward reasoning across a wider range of topics related to multiplication and (b) highlighting refinement and coordination of knowledge resources as basic processes by which future teachers can progress toward coherent understandings of critical school mathematics content.

摘要

本研究的动机是大量研究记录了教师常年遇到的困难，其中包含与乘法相关的一系列关键主题。作为回应，我们追踪了未来的初中数学老师 Nina 如何通过基于测量的乘法和分数定义进行推理，以及通过将符号表示与数学绘图协调起来，在构建跨主题的解释方面取得进展。该数据集涵盖了 Nina 在内容课程中的 1 个学期的课堂作业——明确旨在促进乘法概念领域内的连贯性——以及她在课程中的书面作业和她在附近进行的三个采访中的即时推理学期的开始、中期和结束。一个主要结果是从协调类理论构建，知识论认识论视角下的一系列理论，对于追踪和解释 Nina 的零碎进步特别有用。这篇文章的广泛贡献有两方面——(a) 将重点从对一两个主题的推理研究转向对与乘法相关的更广泛主题的推理；(b) 强调知识资源的改进和协调是未来教师可以通过这些基本过程对关键的学校数学内容有一致的理解。