[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-592676-105":59,"doc-detail-592676-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","dimensions-of-knowledge-and-ways-of-thinking-of-irrational-numbers","Dimensions of Knowledge and Ways of Thinking of Irrational Numbers","","Explores how 157 Lebanese students across different grades define and identify irrational numbers and how these choices relate to their performance. Focuses on dimensions of knowledge and on procedural or proceptual ways of thinking, including how students place irrational numbers on the number line and respond to different representations. Results indicate inconsistencies in knowledge dimensions, a reliance on procedural thinking, and grade-dependent performance changes, except for identifying the nature of given numbers.",{"@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/dimensions-of-knowledge-and-ways-of-thinking-of-irrational-numbers/592676/",{"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/dimensions-of-knowledge-and-ways-of-thinking-of-irrational-numbers/592676.png","ImageObject",300,407,{"name":92,"@type":93},"Mimi","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-09","2026-10-15",true,{"@type":102,"interactionType":103,"userInteractionCount":14},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What is the main goal of this study about irrational numbers?","Question",{"text":112,"@type":113},"To examine the dimensions of knowledge and the ways of thinking used by 157 students of different grades when defining and identifying irrational numbers.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How are students expected to define irrational numbers in this study?",{"text":117,"@type":113},"They are expected to provide one of two definitions: irrational numbers as non-ratio numbers of two integers, or numbers with an unlimited, non-periodic decimal part.",{"name":119,"@type":110,"acceptedAnswer":120},"What key findings are reported about students’ thinking?",{"text":121,"@type":113},"Students show inconsistency in the dimensions of knowledge they use and a dependency on procedural ways of thinking, with performance changing across grades except when identifying the nature of the given numbers.","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},592676,1792075077,{"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":14,"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":129,"read_time":144,"preview_page_count":4},2336477974920,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Dimensions of Knowledge and Ways of Thinking of Irrational Numbers  \nBy Nina Hayfa􀀍  \nLayal Saikaly†  \nThis paper explores the dimensions of knowledge and the ways of thinking (procedural or proceptual thinking) adopted by 157 Lebanese students indifferent grades when defining and identifying irrational numbers when locating them on the number line. It searches for a relation between the performance of the participants and their different grades. The results show the participants’ inconsistency in the dimensions of knowledge used and a dependency on the procedural ways of thinking is detected. A change in the performance within different grades is also found except when identifying the nature of the given numbers.  \nKeywords: dimensions of knowledge, irrational numbers, procedural and proceptual thinking  \nIntroduction  \nBeing essential to the completeness of the set of real numbers, irrational numbers are defined by Dedekind (1901) as the cut which is created in order to fill the gaps produced by rational numbers. In this report, participants are expected to give one of two definitions, either as the numbers that cannot be written as a ratio of two integers a and b ( b 􀁺 0), or as the numbers with an unlimited, un-periodic decimal part. In school mathematics education, irrational numbers arise to solve problems that the elements of the set of rational numbers cannot. Students learn the natural numbers from primary school, a knowledge which expands to a set of integers, then the rational numbers, to finally reach the set of the real numbers. The irrational numbers first introduced in school mathematics education are the solutions of the equations in the form x 2 􀀐 p 􀀠 0 , where p is a non-perfect square positive number. This research investigates what kind of knowledge do the students use when dealing with irrational numbers? Do they encapsulate these numbers into concepts or are they mostly regarded as processes? It seeks to examine the ways of thinking and the dimensions of knowledge used by a group of 157 students of different grades. The purpose of this study is to provide a record of the students’ conception of irrational numbers. It explains how difficulties in the understanding of irrational numbers occur and examines whether these difficulties disappear moving up the grades. This report could be considered as  \n􀀍  \n†  \nAssistant Professor, Lebanese University, Lebanon. Master Student, Lebanese University, Lebanon.  \n[https://doi.org/10.30958/aje.3-2-3](https://doi.org/10.30958/aje.3-2-3) doi=10.30958/aje.3-2-3  \nan account of the participants’ progression in the cognition, intuition and perception of irrational numbers as a part of a set of real numbers. It can be perceived as a way to describe the extent to which the participants depend on the same way of thinking when defining irrational numbers and when they have to use this definition in a different context.  \nBackgroud  \nIn their ongoing study done on 46 prospective secondary mathematics teachers, Zazkis and Sirotic (2010) claimed that the minority of the wrong definitions of the irrational numbers and the heavy dependence on the calculator to decide the irrationality of these numbers is \"troublesome\". This study was to investigate the participants ’ understanding of different representations of irrational numbers, how their answers are influenced when given different representations, and about the teachers’ awareness of the different definitions of the irrationals and their flexibility in using these definitions when needed. The researchers explored how the \"missing link\"between the two definitions of irrational numbers influenced the prospective teachers’ overall understanding of irrational numbers. They showed that the same definitions used in school to define irrational numbers were also used by these prospective teachers although they have reached a high level of maturity. These results were similar to those given by Fischbein, Jehiam and Cohen (1995), that t","cbCaiuFYgIRS8IaL","https://ap.wps.com/l/cbCaiuFYgIRS8IaL","pdf",846668,18,"English","# Introduction\n## Background","[{\"question\":\"What is the main goal of this study about irrational numbers?\",\"answer\":\"To examine the dimensions of knowledge and the ways of thinking used by 157 students of different grades when defining and identifying irrational numbers.\"},{\"question\":\"How are students expected to define irrational numbers in this study?\",\"answer\":\"They are expected to provide one of two definitions: irrational numbers as non-ratio numbers of two integers, or numbers with an unlimited, non-periodic decimal part.\"},{\"question\":\"What key findings are reported about students’ thinking?\",\"answer\":\"Students show inconsistency in the dimensions of knowledge they use and a dependency on procedural ways of thinking, with performance changing across grades except when identifying the nature of the given numbers.\"}]","Dimensions of Knowledge and Ways of Thinking of Irrational Numbers | PDF",45]