Reasoning productively across algebraic contexts: Students develop coordinated notions of inverse

IF 1 Q3 EDUCATION & EDUCATIONAL RESEARCH
John Paul Cook , Kathleen Melhuish , Rosaura Uscanga
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引用次数: 0

Abstract

The concept of inverse is threaded throughout K-16 mathematics. Scholars frequently advocate for students to understand the underlying structure: combining an element and its inverse through the binary operations yields the relevant identity element. This ‘coordinated’ way of reasoning is challenging for students to employ; however, little is known about how students might reason en route to developing it. In this study, we analyze a teaching experiment with two beginning abstract algebra students through the lens of three ways of reasoning about inverse: inverse as an undoing, inverse as a manipulated element, and inverse as a coordination of the binary operation, identity, and set. In particular, we examine the implications of these ways of reasoning as students work to develop inverse as a coordination. We identify pedagogical tools and facets of instructional design that appeared to support students’ development of inverse as a coordination. We further suggest that all three ways of reasoning can support productive activity with inverses.

在代数背景下有效地推理:学生发展反的协调概念
逆的概念贯穿于K-16数学。学者们经常提倡学生理解其底层结构:通过二进制运算将一个元素和它的逆元素结合起来,产生相关的单位元素。这种“协调”的推理方式对学生来说是具有挑战性的;然而,对于学生如何在发展思维的过程中进行推理却知之甚少。在这项研究中,我们分析了一个教学实验与两个初学抽象代数的学生通过三种方式推理的镜头:逆作为一个撤消,逆作为一个被操纵的元素,逆作为二进制操作,单位和集合的协调。特别是,我们检查这些推理方式的含义,因为学生工作,以发展逆作为一个协调。我们确定了教学工具和教学设计的各个方面,这些工具和方面似乎支持学生逆的发展作为一种协调。我们进一步建议,所有三种推理方式都可以用逆来支持生产活动。
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来源期刊
Journal of Mathematical Behavior
Journal of Mathematical Behavior EDUCATION & EDUCATIONAL RESEARCH-
CiteScore
2.70
自引率
17.60%
发文量
69
期刊介绍: The Journal of Mathematical Behavior solicits original research on the learning and teaching of mathematics. We are interested especially in basic research, research that aims to clarify, in detail and depth, how mathematical ideas develop in learners. Over three decades, our experience confirms a founding premise of this journal: that mathematical thinking, hence mathematics learning as a social enterprise, is special. It is special because mathematics is special, both logically and psychologically. Logically, through the way that mathematical ideas and methods have been built, refined and organized for centuries across a range of cultures; and psychologically, through the variety of ways people today, in many walks of life, make sense of mathematics, develop it, make it their own.
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