AI 中文总结
本研究探究5至11年级学生能否通过15拼图学习置换循环，经简短指导后多数学生可构建循环表示并用于分类，发现构建表示与用其得出结论为独立要求。
AI 中文摘要
置换循环通常与本科抽象代数相关联。本探索性研究调查5至11年级的学生是否能在15拼图的情境中构建并使用循环表示法。在一节45分钟的教师指导课（从拼图操作过渡到箭头图和循环记法）后，313名学生分析两种新构型中的一种——一种可解、一种不可解——并使用提供的规则对其进行分类。这些学生中，78.3%构建了正确的循环表示，67.7%既正确构建了表示又得出了正确分类。两种构型的循环构建准确率相近，但不可解构型的分类正确率更低。研究结果涉及有支持的即时表现，而非对置换循环或群论的完全理解。不过，它们表明许多学生在简短指导后就能使用循环表示，且构建表示与用其得出结论是两个独立要求。教授置换的一个原因是，作为有限、离散、非公式化的函数，它们能让学生的函数经验超出熟悉的公式和连续图像。
英文摘要
Permutation cycles are generally associated with undergraduate abstract algebra. This exploratory study examined whether students in Grades 5-11 could construct and use cycle representations in the context of the 15-puzzle. After a 45-minute teacher-guided lesson moving from puzzle manipulation to arrow diagrams and cycle notation, 313 students analyzed one of two new configurations - one solvable and one unsolvable - and used a supplied rule to classify it. Of these students, $78.3\%$ constructed a correct cycle representation, and $67.7\%$ both constructed the representation correctly and reached the correct classification. Cycle-construction accuracy was similar for the two configurations, but classification was less often correct for the unsolvable configuration. The findings concern immediate, supported performance rather than full understanding of permutation cycles or group theory. Nevertheless, they show that many students could use cycle representations after brief instruction and that constructing the representation and using it to reach a conclusion were separate demands. One reason to teach permutations is that, as finite, discrete, non-formulaic functions, they can extend students' experience of functions beyond familiar formulae and continuous graphs.
Comments26 pages, 12 figures
