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Learning progression toward a measurement concept of fractions

机译：学习分数测量概念的进展

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Background Fractions continue to pose a critical challenge for students and their teachers alike. Mathematics education research indicates that the challenge with fractions may stem from the limitations of part-whole concepts of fractions, which is the central focus of fractions curriculum and instruction in the USA. Students’ development of more sophisticated concepts of fractions, beyond the part-whole concept, lays the groundwork for the later study of important mathematical topics, such as algebra, ratios, and proportions, which are foundational understandings for most STEM-related fields. In particular, the Common Core State Standards for Mathematics call for students to develop measurement concepts of fractions. In order to support such concepts, it is important to understand the underlying mental actions that undergird them so that teachers can design appropriate instructional opportunities. In this study, we propose a learning progression for the measurement concept of fractions—one that focuses on students’ mental actions and informs instructional design. Results A hierarchy of fraction schemes is charted outlining a progression from part-whole concepts to measurement concepts of fractions: (a) part-whole scheme (PWS), (b) measurement scheme for unit fractions (MSUF), (c) measurement scheme for proper fractions (MSPF), and (d) generalized measurement scheme for fractions (GMSF). These schemes describe concepts with explicit attention to the mental actions that undergird them. A synthesis of previous studies provides empirical evidence to support this learning progression. Conclusions Evidence from the synthesis of a series of research studies suggests that children’s measurement concept of fractions develops through several distinct developmental stages characterized by the construction of distinct schemes. The mental actions associated with these schemes provide a guide for teachers to design instructional opportunities for children to advance their construction of a measurement concept of fractions. Specifically, the collection of quantitative studies suggest that students need opportunities to engage in activities that support two kinds of coordinations—the coordination of partitioning and iterating, and the coordination of three levels of units inherent in fractions. Instructional implications are discussed with example tasks and activities designed to provoke these coordinations.
机译：背景技术分数对学生及其老师而言仍然构成严峻的挑战。数学教育研究表明，分数的挑战可能源于分数的整体概念的局限性，分数概念是美国分数课程和教学的重点。学生逐渐发展出更复杂的分数概念，而不是整个部分的概念，这为以后学习重要的数学主题（如代数，比率和比例）奠定了基础，这些数学主题是大多数STEM相关领域的基础知识。特别是，《数学通用核心州标准》要求学生发展分数的测量概念。为了支持这样的概念，重要的是要理解支撑它们的潜在心理行为，以便教师可以设计适当的教学机会。在这项研究中，我们针对分数的测量概念提出了一种学习进展，即关注学生的心理行为并指导教学设计的过程。结果绘制了分数方案的层次结构，概述了从整体概念到分数测量概念的进展：（a）整体方案（PWS），（b）单位分数测量方案（MSUF），（c）测量方案适当的分数（MSPF），以及（d）分数的通用测量方案（GMSF）。这些方案在描述概念时特别注意其背后的心理行为。先前研究的综合提供了支持这种学习进展的经验证据。结论从一系列研究综合中得出的证据表明，儿童分数测量的概念是通过几个不同的发展阶段发展起来的，这些阶段的特点是构造不同的方案。与这些计划相关的精神行为为教师提供了指导，为孩子们设计教学机会，以提高他们对分数的测量概念的建构。具体来说，定量研究的集合表明，学生需要机会参与支持两种协调的活动：分区和迭代的协调以及分数中固有的三个级别单元的协调。通过旨在激发这些协调的示例任务和活动来讨论教学含义。

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