[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-588665-105":59,"doc-detail-588665-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","the-rational-number-sub-constructs-as-a-foundation-for-problem-solving","The Rational Number Sub-Constructs as a Foundation for Problem Solving","","Community colleges in the United States help students bridge from secondary school into college mathematics when graduates enter with weak mathematics skills. Remedial or developmental courses often target pre-algebra and algebra skills, where fractions are widely viewed as difficult due to their abstract nature. This study with adult pre-algebra learners uses Kieren’s fraction sub-constructs as foundational concept knowledge during problem solving, combining quantitative and qualitative analyses to examine transitions toward proportional reasoning.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/the-rational-number-sub-constructs-as-a-foundation-for-problem-solving/588665/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/the-rational-number-sub-constructs-as-a-foundation-for-problem-solving/588665.png","ImageObject",300,407,{"name":92,"@type":93},"Rizky","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-15",true,{"@type":102,"interactionType":103,"userInteractionCount":39},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What foundational concepts does the study use for adult pre-algebra learners?","Question",{"text":112,"@type":113},"The study uses Kieren’s fraction sub-constructs—part-whole, ratio, operator, quotient, measure, and fractional equivalence—as foundational concept knowledge during problem solving.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How does the research evaluate students’ problem-solving competency?",{"text":117,"@type":113},"In the first quantitative part, students’ proficiency with the rational number sub-constructs serves as independent variables in a multiple linear regression model to predict or explain competency in formal problem solving.",{"name":119,"@type":110,"acceptedAnswer":120},"What does the qualitative part focus on?",{"text":121,"@type":113},"The qualitative part reviews classroom lecture transcripts from the teaching research experiment to understand how students used these rational number concepts during informal reasoning with ratio, quotient, proportion, and percent problems.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},588665,1792171892,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":39,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":145,"preview_page_count":4},962085564807,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","Doyle, K.M., Dias, O., Kennis, J.R., Czarnocha, B. & Baker, W. (2015) . The rational number subconstructs as a foundation for problem solving. Adults Learning Mathematics: An International Journal, 11(1), 21-42  \nThe Rational Number Sub-Constructs as a Foundation for Problem  \nSolving  \nKathleen M. Doyle  \nCity University of New York, USA  \n\u003C[kdoyle@hostos.cuny.edu](kdoyle@hostos.cuny.edu)>  \nOlen Dias  \nCity University of New York, USA  \n\u003C[odias@hostos.cuny.edu](odias@hostos.cuny.edu)>  \nJames R. Kennis  \nCity University of New York, USA  \n\u003C[jkennis@ hostos.cuny.edu](jkennis@ hostos.cuny.edu) >  \nBronislaw Czarnocha  \nCity University of New York, USA  \n\u003C[bczarnocha@ hostos.cuny.edu](bczarnocha@ hostos.cuny.edu) >  \nWilliam Baker  \nCity University of New York, USA  \n\u003C[wbaker@ hostos.cuny.edu](wbaker@ hostos.cuny.edu) >  \nAbstract  \nOne of the many roles of two year community colleges in the United States is to bridge the gap between secondary school and college for students who graduate from high school with weak mathematics skills that prevent them from enrolling in college level mathematics courses. At community colleges remedial or developmental mathematics courses review the pre-algebra and/or algebra skills required for college level mathematics. Fractions are often cited as the most difficult topic for students due to their abstract nature (Wilensky, 1991) . This study with adult pre-algebra students is based upon a teaching research experiment in which the Kieren’s fraction sub-constructs of part-whole, ratio, operator, quotient, measure and the fractional equivalence were used as foundational concept knowledge during problem solving. In the first quantitative part of this study, students’proficiency with Kieren's rational number sub-constructs are used as independent variables in a multiple linear regression model to  \nVolume 11(1)– January 2016  \nCopyright © 2016 Adults Learning Mathematics International Journal. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license ([http://creativecommons.org/licenses/by/4.0/](http://creativecommons.org/licenses/by/4.0/)).  \n21  \npredict or explain students' competency in formal problem solving. This part of the study supplies hypothetical or statistical suggested pathways for students learning and transition from fraction concepts to proportional reasoning. Then in the second qualitative part of this study, transcripts from classroom lectures during the teaching research experiment are reviewed in order to understand how students used these rational number sub-constructs during problem solving with ratio, quotient, proportion, and percent.  \nKeywords: Adult remedial mathematics, fractions, sub-construct, ratio, operator, quotient, measure, informal and formal proportional reasoning  \nIntroduction  \nProportional reasoning is often cited as a critical component in the transition from informal to formal mathematical thought. In the pre-algebra curriculum proportion typically come after fraction and ratio, however many educators believe it should be introduced earlier and the connections between these topics should be emphasized (Streefland, 1984) . The claim that instruction in proportions should be based upon and connected to students’understanding of fractions puts more emphasis on this important concept. Fractions represent a difficult concept for many students. Almost every instructor has heard a student proclaim,“I hate fractions.” Inan effort to clarify the relationships between various fraction concepts the Kieren (1976) model of fraction and the extension of this model by Behr, Lesh, Post and Silver (1983) was studied using quantitative analyses by Charalambous & Pitta-Pantazi (2007), with children, and Baker, Czarnocha, Dias, Doyle and Prabhu (2009) with adults. The Behr et al. (1985) extension of Kieren’s work was used as a theoretical foundation to study the relationship between procedural and conceptua","cbCaiieFthjlQGuN","https://ap.wps.com/l/cbCaiieFthjlQGuN","pdf",818482,22,"English","# Abstract\n# Introduction\n## Proportional reasoning and fractions\n# Literature Review\n## Problem-Solving","[{\"question\":\"What foundational concepts does the study use for adult pre-algebra learners?\",\"answer\":\"The study uses Kieren’s fraction sub-constructs—part-whole, ratio, operator, quotient, measure, and fractional equivalence—as foundational concept knowledge during problem solving.\"},{\"question\":\"How does the research evaluate students’ problem-solving competency?\",\"answer\":\"In the first quantitative part, students’ proficiency with the rational number sub-constructs serves as independent variables in a multiple linear regression model to predict or explain competency in formal problem solving.\"},{\"question\":\"What does the qualitative part focus on?\",\"answer\":\"The qualitative part reviews classroom lecture transcripts from the teaching research experiment to understand how students used these rational number concepts during informal reasoning with ratio, quotient, proportion, and percent problems.\"}]","The Rational Number Sub-Constructs as a Foundation for Problem Solving | PDF",1792065701,55]