[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-582235-105":3,"detail-sidebar-cat-0-en-105":80,"doc-detail-582235-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":73,"head_meta":75,"extra_data":77,"updated_unix":79},105,"en","the-mathematics-educator-vol-33-no-1-a-framework-for-time-and-covariational-reasoning","The Mathematics Educator - Vol. 33 - No. 1 - A Framework for Time and Covariational Reasoning","","A Framework for Time and Covariational Reasoning develops a theoretical account of how time functions within students’ covariational reasoning. Grounded in Piaget’s perspective on time as constructed through mental operations, the framework distinguishes experiential and conceptual time and clarifies multiplicative objects that underlie students’ covariational relationships. The author illustrates the framework through a task informed by second-order models from teaching experiments and connects task engagement to the emergence of covariational reasoning.",{"@graph":14,"@context":72},[15,34,55],{"@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/the-mathematics-educator-vol-33-no-1-a-framework-for-time-and-covariational-reasoning/582235/",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/the-mathematics-educator-vol-33-no-1-a-framework-for-time-and-covariational-reasoning/582235.png","ImageObject",300,407,{"name":42,"@type":43},"Ophelia","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-11-10","2026-10-15",true,{"@type":52,"interactionType":53,"userInteractionCount":30},"InteractionCounter",{"@type":54},"ViewAction",{"@type":56,"mainEntity":57},"FAQPage",[58,64,68],{"name":59,"@type":60,"acceptedAnswer":61},"What is the main purpose of the framework developed in the article?","Question",{"text":62,"@type":63},"To characterize the role of time in students’ covariational reasoning and explain how multiplicative objects underlie their covariational relationships.","Answer",{"name":65,"@type":60,"acceptedAnswer":66},"How does the article use Piaget’s perspective on time?",{"text":67,"@type":63},"It treats time as a constructed concept arising from mental operations that coordinate motions of at least two objects, and then uses that view to structure the framework for covariational reasoning.",{"name":69,"@type":60,"acceptedAnswer":70},"What provides empirical support for the framework’s connection to students’ reasoning?",{"text":71,"@type":63},"A task designed to reveal insights into time’s role, informed by second-order models from a series of teaching experiments.","https://schema.org",{"og:url":32,"og:type":74,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":76,"canonical":32},"index,follow",{"doc_id":78,"site_id":7},582235,1792057219,{"code":4,"msg":81,"data":82},"success",[83,87,91,95,100,105,110,114,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":84,"show_sort_weight":85,"slug":86},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":88,"show_sort_weight":89,"slug":90},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":92,"show_sort_weight":93,"slug":94},"Exam",70,"exam",{"id":96,"doc_module":4,"doc_module_name":25,"category_name":97,"show_sort_weight":98,"slug":99},5,"Comic",60,"comic",{"id":101,"doc_module":4,"doc_module_name":25,"category_name":102,"show_sort_weight":103,"slug":104},6,"Technology",50,"technology",{"id":106,"doc_module":4,"doc_module_name":25,"category_name":107,"show_sort_weight":108,"slug":109},7,"Healthcare",40,"healthcare",{"id":111,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":112,"slug":113},8,30,"research-report",{"id":115,"doc_module":4,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":25,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":25,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":25,"category_name":128,"show_sort_weight":96,"slug":129},19,"General","general",{"code":4,"msg":81,"data":131},{"doc_id":78,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":4,"category_id":111,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":30,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":112,"language":139,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":12,"update_tm":143,"read_time":144,"preview_page_count":4},7971461741311,"https://ap-avatar.wpscdn.com/avatar/74000253aff267980c6?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779345379180704826","The Mathematics Educator 2025 Vol. 33, No. 1, pp. 62-90  \nA Framework for Time and Covariational Reasoning  \nKevin C. Moore  \nCovariational reasoning has emerged as a productive construct to characterize students’ mathematical development. Researchers have illustrated its importance for major middle, secondary, and undergraduate mathematical concepts, including rate of change, accumulation, and modeling. Within this line of work, several researchers have indicated differences between experiential and conceptual time with respect to the covariational relationships students construct. I draw on this body of literature and return to Piaget’s perspective of time to provide a frameworkfor the role of time in students’ covariational reasoning. The framework also clarifies the nature of the multiplicative objects underlying students’ covariational relationships. To illustrate the framework and capture its emergence from second-order models of students’ mathematics, I describe the framework as it relates to students’ engagement in a task.  \n“We are far too readily tempted to speak of intuitive ideas of time, as if time, or for that matter space, could be perceived and conceived apart from the entities or the events that fill it”(Piaget, 1970, p. 1) .  \nTime has been a topic of contemplation for researchers and philosophers for centuries, and ontological and epistemological considerations of time are certainly not restricted to the academy. The mere question of“What is time?” can provoke a lively conversation among most groups of individuals, and numerous pop culture pieces have built their plotlines around the fabric of time and its manipulations. The prevalence of time as a subject of rumination is unsurprising, given its inseparable role from existence and experience. Kant (1781/2003) considered time and its properties to be so ubiquitous as to be given a priori.  \n[Kevin C. Moore] is a professor of mathematics education at the University of Georgia. His primary research focuses on student and teacher meanings, particularly as related to quantitative reasoning, covariation, and re-presentation.  \nA Framework for Time and Covariational Reasoning  \nCountering Kant’s perspective, Piaget considered time to be a constructed concept subject to the mental operations defining a person’s experience (Piaget, 1954; von Glasersfeld, 1995, 1997) . He dedicated several studies to developing conceptual models of that construction (e.g., Piaget, 1954, 1970), ultimately determining that the mental operations involved in coordinating at least two objects’ motions are fundamental to an individual’s construction of time. Piaget concluded that concepts of time are inseparable from the mental operations defining space, motion, and objects; not only is time a constructed concept, but it is also the product of coordinating mental operations involved in the construction of other concepts (Piaget, 1970; von Glasersfeld, 1984) . Ludwig and Luciano expressed as much, “We cannot compare a process with the ‘passage of time’ – there is no such thing – but only with another process”(2021, p. 240) .  \nBuilding on the work of researchers who have alluded to covariational reasoning—which involves the variation and motion of objects—being connected to conceptions of time, I return to Piaget’s (1970) conceptual models for time to further develop the role of time in students’ covariational reasoning. In doing so, I elaborate on the constructs of experiential time and conceptual time (Castillo-Garsow, 2012; Thompson & Carlson, 2017) to provide a framework for characterizing students’covariational reasoning in relation to concepts of time. Reflecting its empirical roots, I illustrate the framework by describing a task designed to provide insights into the role of time with respect to students’ covariational reasoning. The task and its description are informed by second-order models of students’ mathematics developed during a series of teaching experiments (Steffe & Thomps","cbCait7vRevBWdU6","https://ap.wps.com/l/cbCait7vRevBWdU6","pdf",742734,"English","# A Framework for Time and Covariational Reasoning\n## Covariational Reasoning and Time","[{\"question\":\"What is the main purpose of the framework developed in the article?\",\"answer\":\"To characterize the role of time in students’ covariational reasoning and explain how multiplicative objects underlie their covariational relationships.\"},{\"question\":\"How does the article use Piaget’s perspective on time?\",\"answer\":\"It treats time as a constructed concept arising from mental operations that coordinate motions of at least two objects, and then uses that view to structure the framework for covariational reasoning.\"},{\"question\":\"What provides empirical support for the framework’s connection to students’ reasoning?\",\"answer\":\"A task designed to reveal insights into time’s role, informed by second-order models from a series of teaching experiments.\"}]","The Mathematics Educator - Vol. 33 - No. 1 - A Framework for Time and Covariational Reasoning | PDF",1792042875,76]