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An Unexpected Outcome: Students’ Focus on Order in the Multiplication Principle
International Journal of Research in Undergraduate Mathematics Education Pub Date : 2019-12-16 , DOI: 10.1007/s40753-019-00107-3
Elise Lockwood , Branwen Purdy

The multiplication principle (MP) is a fundamental aspect of combinatorial enumeration. In an effort to better understand students’ reasoning about the MP, we had two undergraduate students reinvent a statement of the MP in a teaching experiment. In this paper, we adopt an actor-oriented perspective (Lobato, Educational Researcher , 32 (1), 17–20, 2003 , Educational Psychologist, 47 (3), 1–16 2012 ) and draw on Lockwood’s, Journal of Mathematical Behavior, 32 , 251–265 ( 2013 ) model of students’ combinatorial thinking to trace the students’ development of a certain conception about order – that a statement of the MP should allow for any ordering of the stages of a counting process. Notably, this conception differs from how order is treated in textbook statements of the MP. We report on how this conception persisted for the students, and we explain how the researchers (interviewers) and students managed to proceed through the teaching experiment while maintaining different conceptions of order. The students’ reasoning about order sheds light on ways in which students may think about order and highlights potential subtleties about multiplication in counting that deserve attention.

中文翻译:

意外的结果:学生专注于乘法原理中的顺序

乘法原理(MP)是组合枚举的基本方面。为了更好地理解学生对MP的推理,我们让两名本科生在教学实验中重塑了MP的陈述。在本文中,我们采用了行为者导向的观点(Lobato,教育研究人员,32(1),17–20,2003,教育心理学家,47(3),1–16 2012),并借鉴了洛克伍德的《数学行为杂志》 ,32,251–265(2013)的学生组合思维模型可追溯学生对某种秩序概念的发展-MP的声明应允许计数过程各个阶段的任何秩序。值得注意的是,该概念与MP的教科书中的顺序处理方式不同。我们报告了这种观念对学生的影响,并且我们解释了研究人员(访谈者)和学生如何在保持不同顺序概念的同时,成功地进行了教学实验。学生对顺序的推理揭示了学生思考顺序的方式,并强调了计数中潜在的细微之处值得关注。
更新日期:2019-12-16
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