John Paul Cook , Kathleen Melhuish , Rosaura Uscanga
{"title":"Reasoning productively across algebraic contexts: Students develop coordinated notions of inverse","authors":"John Paul Cook , Kathleen Melhuish , Rosaura Uscanga","doi":"10.1016/j.jmathb.2023.101099","DOIUrl":null,"url":null,"abstract":"<div><p>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.</p></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Behavior","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S073231232300069X","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"EDUCATION & EDUCATIONAL RESEARCH","Score":null,"Total":0}
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.
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.