{"title":"Investigating two teachers’ development of combinatorial meaning for algebraic structure","authors":"Lori J. Burch","doi":"10.1016/j.jmathb.2023.101039","DOIUrl":null,"url":null,"abstract":"<div><p>This paper reports on the results of a four-day teaching experiment that supported two algebra teachers to develop a <em>combinatorial meaning</em> for algebraic structure. The purpose of the teaching episodes was to support the teachers (a) to establish a <em>combinatorial understanding</em> for algebraic structure (Tillema & Burch, 2022) by generalizing the cubic identity, <span><math><mrow><msup><mrow><mfenced><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced></mrow><mrow><mn>3</mn></mrow></msup><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mn>3</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>b</mi><mo>+</mo><mn>3</mn><mi>a</mi><msup><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>b</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></math></span>, as a symbolization of quantitative and combinatorial relationships out of a contextualized problem (Tillema & Gatza, 2016) and (b) to develop a <em>combinatorial meaning</em> as a mobilization of their understanding through a series of algebraic tasks (cf. Thompson et al., 2014). The findings from this study contribute to research literature on teachers’ mathematical meanings within secondary algebra by investigating how teachers’ combinatorial meanings developed and how differences in their combinatorial meanings impacted their algebraic reasoning. The findings demonstrate a combinatorial pathway for supporting the development of expanding and factoring as reversible polynomial operations (cf. Sangwin & Jones, 2017).</p></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2023-06-01","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/S0732312323000093","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
This paper reports on the results of a four-day teaching experiment that supported two algebra teachers to develop a combinatorial meaning for algebraic structure. The purpose of the teaching episodes was to support the teachers (a) to establish a combinatorial understanding for algebraic structure (Tillema & Burch, 2022) by generalizing the cubic identity, , as a symbolization of quantitative and combinatorial relationships out of a contextualized problem (Tillema & Gatza, 2016) and (b) to develop a combinatorial meaning as a mobilization of their understanding through a series of algebraic tasks (cf. Thompson et al., 2014). The findings from this study contribute to research literature on teachers’ mathematical meanings within secondary algebra by investigating how teachers’ combinatorial meanings developed and how differences in their combinatorial meanings impacted their algebraic reasoning. The findings demonstrate a combinatorial pathway for supporting the development of expanding and factoring as reversible polynomial operations (cf. Sangwin & Jones, 2017).
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.