[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-588235-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-588235-en":130},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","an-investigation-of-prospective-elementary-mathematics-teachers-understanding-of-the-formal-definition-of-the-limit-of-a-sequence","AN INVESTIGATION OF PROSPECTIVE ELEMENTARY MATHEMATICS TEACHERS UNDERSTANDING OF THE FORMAL DEFINITION OF THE LIMIT OF A SEQUENCE","","This study investigates how prospective elementary mathematics teachers understand the formal definition of the limit of a sequence. Thirty-two third-year prospective teachers from Istanbul University-Cerrahpasa in Türkiye participated, and data were collected using a test developed by the authors. Results show persistent difficulty defining limit formally, representing graphical variables and inequalities, and establishing the relationship between variables in the definition. Teachers often memorize the definition without understanding its meaning, limiting their ability to reason about contained variables.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & Report",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/an-investigation-of-prospective-elementary-mathematics-teachers-understanding-of-the-formal-definition-of-the-limit-of-a-sequence/588235/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/an-investigation-of-prospective-elementary-mathematics-teachers-understanding-of-the-formal-definition-of-the-limit-of-a-sequence/588235.png","ImageObject",300,407,{"name":42,"@type":43},"Aditya","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-30","2026-10-15",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",5,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"Who participated in the study and how was the data collected?","Question",{"text":63,"@type":64},"The study included 32 prospective elementary mathematics teachers in their third year of a four-year mathematics teacher preparation programme. 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The participants were 32 prospective mathematics teachers (8 male and 24 female) who were in their third year of study in a four-year mathematics teacher preparation programme Istanbul University-Cerrahpasa in Türkiye.  \nThe data were collected via a test that was developed by the authors. The findings of the study demonstrate that the prospective elementary mathematics teachers had difficulty in defining the concept of limit of a sequence and in determining the graphical representations ofthe variables and the inequalities in the formal definition. Moreover, they also had difficulty comprehending the relationship between the variables of 􀟝 and 􀝊0 , and hence, when a numerical value was assigned to one of these variables, they were not able to find an appropriate value for the other. It was concluded that the prospective mathematics teachers generally tended to memorize the formal definition of the limit of sequences but lacked conceptual understanding of its meaning. Therefore, they were notable to reason about the variables it contained.  \nKey words: limit, sequence, 􀟝-N definition, calculus instruction, prospective mathematics teachers  \n1. Introduction  \nThe concept of limits is an essential aspect of calculus and constitutes a substantial portion of the mathematics taught at the undergraduate level. According to Tall (1992), this concept serves as a gateway for the transition to advanced mathematical thinking. In this regard, some of the primary concepts covered in calculus are special types of limits, although they are not always referred to by this name. For instance, derivatives are the limit of a difference quotient, and the definite integral is the limit of a Riemann sum; both of these concepts are fundamental to calculus, thus underscoring Tall’s view.  \nGenerally, in the introductory calculus courses offered at the undergraduate level, the concept of limits is addressed in two different contexts: namely, the investigation of the limits of single-variable real valued functions and the investigation of the limits of real number sequences. There is a huge body of research that concentrates on the different aspects of teaching the concept of limits in the former context. Some of these studies have focused on producing conceptual frameworks to describe the learning process (Cottrill et al., 1996; Przenioslo, 2004; Roh, 2007; Williams, 2001) . Some studies have also aimed to document the difficulties that students encounter in learning the subject (Bezuidenhout, 2001; Cornu, 2002; Sierpińska, 1987; Tall & Vinner, 1981); while others have comprised investigations of students’ understanding of the formal definition of limits (Baki & Çekmez, 2012; Doruk et al., 2018; Kabael et al., 2015; Oktaviyanthi et al., 2018; Swinyard & Larsen, 2012) . The studies that focused on the formal definition of the concept have reported various difficulties and misunderstandings on the part of students. For instance, Baki and Çekmez (2012) revealed that prospective mathematics teachers had problems understanding and representing the variables and inequalities within the formal definition in the Cartesian coordinate system. Their findings also revealed that students tended to memorize the definition without comprehending the logical structure. Similarly, Tall and Vinner (1981) reported that students could not interpret the roles of the quantifiers “∀” and “∃” in the formal definition, and hence were not successful in proving the existence of a limit. In this regard, Kabael et al. (2015) suggested that the main sources of difficulty for students were the inability to use quantifiers included in the formal  \n","cbCaijbNDILKSzwa","https://ap.wps.com/l/cbCaijbNDILKSzwa","pdf",395743,13,"English","# Abstract\n# Introduction\n## Limits in calculus and undergraduate learning\n## Prior research on difficulties with formal definitions\n## Symbolic and verbal descriptions of the formal definition","[{\"question\":\"Who participated in the study and how was the data collected?\",\"answer\":\"The study included 32 prospective elementary mathematics teachers in their third year of a four-year mathematics teacher preparation programme. Data were collected using a test developed by the authors.\"},{\"question\":\"What difficulties did prospective teachers show with the formal definition?\",\"answer\":\"They had difficulty defining the limit concept formally, determining graphical representations of variables and inequalities, and comprehending the relationship between variables in the definition.\"},{\"question\":\"What was the overall conclusion about their understanding?\",\"answer\":\"Prospective teachers generally tended to memorize the formal definition of limits but lacked conceptual understanding of its meaning, so they struggled to reason about variables involved.\"}]","AN INVESTIGATION OF PROSPECTIVE ELEMENTARY MATHEMATICS TEACHERS UNDERSTANDING OF THE FORMAL DEFINITION OF THE LIMIT OF A SEQUENCE | PDF",1792064811,33]