Rational number and fractions concepts are inherently difficult; and lack of mastery often holds students back from success in subsequent mathematics courses. This paper describes design characteristics of a software based, adaptive, rational number tutor with virtual manipulatives, realistic contexts, and procedural feedback. Then, the paper presents findings from a randomized crossover trial designed to evaluate the impact of the described intervention on student fraction achievement. The trial was conducted in two districts in two states with 17 elementary school teachers and 297 4th and 5th grade students. Half of the participating classrooms replaced worksheet practice time (over 10 instructional hours) with Woot Math Adaptive Learning (WMAL), a digital tutor that incorporates dynamic digital manipulatives, video tutorials and automatic feedback to students and teachers. After 10 instructional hours the groups switched conditions for another 10 instructional hours so each student would have exposure to the treatment and control conditions. Analysis shows that regardless of treatment order, student performance gains on assessment items (mostly from NAEP and TIMMS) were statistically higher, (d = 0.39, F(1,589) = 5.73, p < 0.05), after 10 days of instruction with WMAL than they were after 10 days of business-as-usual instruction. This difference of 0.39 standard units corresponds to an improvement index of 10.9 (percentile point gain compared to non-treatment). The results demonstrate how the described approach to software and instructional design can work together to support significant increases in student achievement in rational number domains by combining best practices in feedback, curricular design, and adaptive instructional sequences.

中文翻译：

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基于软件的数字操作干预，以支持学生对分数的概念性理解

有理数和分数概念本质上是困难的；缺乏精通常常阻碍学生在随后的数学课程中取得成功。本文描述了基于软件的、自适应的、有理数导师的设计特征，具有虚拟操作、现实环境和程序反馈。然后，本文介绍了一项随机交叉试验的结果，该试验旨在评估所描述的干预措施对学生分数成绩的影响。该试验在两个州的两个地区进行，有 17 名小学教师和 297 名四年级和五年级学生。一半的参与教室用 Woot Math Adaptive Learning (WMAL) 取代了工作表练习时间（超过 10 个教学小时），这是一种结合动态数字操作的数字导师，视频教程和自动反馈给学生和教师。在 10 个教学小时后，小组将条件转换为另外 10 个教学小时，这样每个学生都将接触治疗和控制条件。分析表明，无论治疗顺序如何，学生在评估项目（主要来自 NAEP 和 TIMMS）上的表现在统计上更高，（d = 0.39, F (1,589) = 5.73, p  < 0.05)，在接受 WMAL 指导 10 天后比在接受 10 天照常指导之后。这种 0.39 标准单位的差异对应于 10.9 的改善指数（与未治疗相比的百分位数增益）。结果表明，所描述的软件和教学设计方法如何通过结合反馈、课程设计和自适应教学序列中的最佳实践，共同支持学生在有理数领域的成绩显着提高。