Erik S. Tillema , Andrew M. Gatza , Weverton Ataide Pinheiro
{"title":"Stage 3 high school students’ generalization of a cubic identity","authors":"Erik S. Tillema , Andrew M. Gatza , Weverton Ataide Pinheiro","doi":"10.1016/j.jmathb.2025.101283","DOIUrl":null,"url":null,"abstract":"<div><div>This paper reports on one study in a series of design research studies that have taken as a guiding design principle that combinatorial and quantitative reasoning can serve as a constructive resource for high school students to establish algebraic structure between a polynomial and its factors. Within this framing, we report on an interview study with eight 10th-12th grade students whose purpose was to investigate their progress towards generalization of the cubic identity <span><math><mrow><msup><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>3</mn></mrow></msup><mo>=</mo><mspace></mspace><msup><mrow><mn>1</mn><mo>∙</mo><mo>(</mo><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo><mo>+</mo><mn>3</mn><mo>∙</mo><mrow><mfenced><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>∙</mo><mn>1</mn></mrow></mfenced></mrow><mo>+</mo><mn>3</mn><mo>∙</mo><mrow><mfenced><mrow><msup><mrow><mi>x</mi><mo>∙</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow><mo>+</mo><mn>1</mn><mo>∙</mo><mo>(</mo><msup><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></mrow></math></span>. The students worked on this generalization by solving cases of a 3-D combinatorics problem and representing their solutions using 3-D arrays. Findings include the identification of how differences in students’ combinatorial reasoning impacted their reasoning with 3-dimensional arrays and their progress towards a general statement of the cubic identity.</div></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":"81 ","pages":"Article 101283"},"PeriodicalIF":1.7000,"publicationDate":"2025-09-06","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/S0732312325000471","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 one study in a series of design research studies that have taken as a guiding design principle that combinatorial and quantitative reasoning can serve as a constructive resource for high school students to establish algebraic structure between a polynomial and its factors. Within this framing, we report on an interview study with eight 10th-12th grade students whose purpose was to investigate their progress towards generalization of the cubic identity . The students worked on this generalization by solving cases of a 3-D combinatorics problem and representing their solutions using 3-D arrays. Findings include the identification of how differences in students’ combinatorial reasoning impacted their reasoning with 3-dimensional arrays and their progress towards a general statement of the cubic identity.
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.