[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-588582-105":59,"doc-detail-588582-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","procedural-and-conceptual-knowledge-adults-reviewing-fractions","Procedural and Conceptual Knowledge: Adults Reviewing Fractions","","In the United States, many community-college entrants need pre-college mathematics review, and in pre-algebra fractions are the hardest topic with high dropout risk. The study builds on Kieren’s five sub-construct model, focusing on adults and testing an alternate route to operator and measure via multiplication and addition, grounded in Piaget’s procedure-based conceptual development. Mixed quantitative and qualitative analysis also examines whether measure belongs to fractions’ multiplicative structure.",{"@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/procedural-and-conceptual-knowledge-adults-reviewing-fractions/588582/",{"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/procedural-and-conceptual-knowledge-adults-reviewing-fractions/588582.png","ImageObject",300,407,{"name":92,"@type":93},"Rizky","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-29","2026-10-15",true,{"@type":102,"interactionType":103,"userInteractionCount":24},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"Why are fractions a major challenge for community-college students?","Question",{"text":112,"@type":113},"Fractions are identified as the most difficult topic in pre-algebra, a course with a high dropout rate, making them a persistent barrier for adult learners.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What is Kieren’s framework for understanding the fraction concept?",{"text":117,"@type":113},"Kieren separates the fraction concept into five related sub-constructs: the foundational part-whole construct and four secondary constructs—ratio, operator, quotient, and measure.",{"name":119,"@type":110,"acceptedAnswer":120},"What does this study test about operator and measure for adults?",{"text":121,"@type":113},"It tests whether conceptual knowledge of operator and measure serves as a goal of instruction rather than a pathway to procedures, comparing pathways based on part-whole extension, procedural multiplication/addition, or their combination.","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},588582,1792171931,{"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":24,"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","Procedural and Conceptual Knowledge: Adults Reviewing Fractions  \nWilliam J. Baker, Bronislaw Czarnocha, Olen Dias, Kathleen Doyle  \nand James R. Kennis  \nHostos Community College, CUNY, New York City  \nVrunda Prabhu  \nBronx Community College, CUNY, New York City  \nAbstract  \nIn the United States a majority of the students who enroll in community colleges require a review of secondary math before they are eligible for college level mathematics. In the pre-algebra course, that has a high drop-out rate, the most difficult topic for students is fractions. In order to better understand the fraction concept Kieren separated it into five related sub-constructs. In Kieren’s model the foundational sub-construct of part-whole provides a path to the sub-constructs of ratio, operator, quotient and measure. This model was previously investigated and found effective by Charalambous & Pitta Pantazi (2007) with children and Baker, Dias, Doyle, Czarnocha & Prabhu (2009) with adults. This article extends the earlier study of Baker et al with adults by considering an alternate pathway to competency with operator and measure through multiplication and addition of fractions. This path is based upon Piaget's theories in which concepts develop through reflection upon corresponding procedures. The goal is to show that the path through direct extension of part-whole knowledge and the path through procedural knowledge work together and complement one another in explaining students’ competency with operator and measure. An open question of whether measure is part of the multiplicative structure of fractions is also addressed using a mixed methodology of quantitative and qualitative analysis.  \nKeywords: Fractions, adult education, sub-construct, operator, measure  \nIntroduction  \nIt is increasingly common for prospective college students in the United States, especially those entering community colleges with open admission, to take entrance exams in mathematics.  \nIf they fail they must enroll in remedial courses that review middle and high school level mathematics. These courses have a high dropout rate and thus serve as a barrier for adult students exploring the option of a college education (Hagedorn, Siadat, Fogelo, Nora & Pascarella, 1999) . In these remedial courses fractions are typically the cause of more student angst than any other topic. One explanation for students' difficulties with fractions is the multifaceted nature of this concept. Inan effort to clarify the different aspects of the fraction construct, Kieren (1976) maps out a fraction schema composed of five sub-constructs, the foundational sub-construct of part-whole knowledge and four secondary sub-constructs: ratio, operator, quotient and measure.  \nAn extension of this model by Behr, Harel, Post & Lesh (1993) that relates the four secondary subconstructs to corresponding procedural knowledge is often used to support an educational approach based upon the direct extension of students' part-whole knowledge to the four secondary sub  \nVolume 7(2)– September 2012 39  \nconstructs in order to give meaning to the corresponding procedural knowledge. Reform minded educators follow this approach because they believe the traditional fraction curriculum has too much focus on procedural algorithms and prefer an emphasis on Kieren's sub-constructs as the starting point for instruction.  \nThe quantitative analyses by Charalambous & Pitta-Pantazi (2007) with children, and Baker, Dias, Doyle, Czarnocha & Prabhu (2009) using an identical set of exercises with adults, both confirm Kieren's underlying hypothesis that part-whole knowledge should be used as a basis for competency with the remaining fraction sub-constructs. However, with regard to the Behr et al (1993) extension of the Kieren model to procedures (Kieren-Behr model) the work of Charalambous & Pitta-Pantazi (2007) and Baker et al (2009) highlight cognitive differences between children and adults. Forchildren the approach of operator and m","cbCaidLxf6JrxbwG","https://ap.wps.com/l/cbCaidLxf6JrxbwG","pdf",1255722,27,"English","# Introduction\n## Fractions as a barrier in remedial education\n## Kieren’s five sub-constructs and related models\n# Study objective and methodology\n## Pathways to competency with operator and measure\n## Quantitative and qualitative analysis of sub-constructs","[{\"question\":\"Why are fractions a major challenge for community-college students?\",\"answer\":\"Fractions are identified as the most difficult topic in pre-algebra, a course with a high dropout rate, making them a persistent barrier for adult learners.\"},{\"question\":\"What is Kieren’s framework for understanding the fraction concept?\",\"answer\":\"Kieren separates the fraction concept into five related sub-constructs: the foundational part-whole construct and four secondary constructs—ratio, operator, quotient, and measure.\"},{\"question\":\"What does this study test about operator and measure for adults?\",\"answer\":\"It tests whether conceptual knowledge of operator and measure serves as a goal of instruction rather than a pathway to procedures, comparing pathways based on part-whole extension, procedural multiplication/addition, or their combination.\"}]","Procedural and Conceptual Knowledge: Adults Reviewing Fractions | PDF",1792065536,68]