[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-588112-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-588112-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","on-teaching-of-generalized-catalan-numbers-with-the-maples-help","ON TEACHING OF GENERALIZED CATALAN NUMBERS WITH THE MAPLE'S HELP","","Work on teaching generalized Catalan numbers through Didactic Engineering assumptions, emphasizing historical and epistemological evolution relevant to Brazilian classroom reality. It presents matrices and combinatorial properties that students can explore using CAS Maple, aiming to engage learners with an approach that links historical, mathematical, and evolutionary perspectives. The study also highlights that effective instruction requires broad knowledge of “numbers,” not only differential and integral calculus foundations.",{"@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/exam/","Exam",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/on-teaching-of-generalized-catalan-numbers-with-the-maples-help/588112/",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/on-teaching-of-generalized-catalan-numbers-with-the-maples-help/588112.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",6,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What is the main teaching focus of the work?","Question",{"text":63,"@type":64},"The work focuses on teaching generalized Catalan numbers, including their generalization process and epistemological aspects.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"How is Maple used in the proposed teaching approach?",{"text":68,"@type":64},"It uses CAS Maple to explore matrices and combinatorial properties tied to generalized Catalan numbers, supporting a more engaging classroom experience.",{"name":70,"@type":61,"acceptedAnswer":71},"Why is historical context considered important for mathematics teachers in Brazil?",{"text":72,"@type":64},"The work argues that many Mathematics History books emphasize infinitesimal calculus foundations, while other historical knowledge relevant to classroom teaching is often disregarded and should be included.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},588112,1792230839,{"code":4,"msg":82,"data":83},"success",[84,88,92,95,100,104,109,114,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":93,"slug":94},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":55,"doc_module":4,"doc_module_name":25,"category_name":101,"show_sort_weight":102,"slug":103},"Technology",50,"technology",{"id":105,"doc_module":4,"doc_module_name":25,"category_name":106,"show_sort_weight":107,"slug":108},7,"Healthcare",40,"healthcare",{"id":110,"doc_module":4,"doc_module_name":25,"category_name":111,"show_sort_weight":112,"slug":113},8,"Research & Report",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":82,"data":131},{"doc_id":79,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":4,"category_id":33,"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":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":139,"language":140,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":12,"update_tm":144,"read_time":107,"preview_page_count":4},962085564549,"https://ap-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45","ON TEACHING OF GENERALIZED CATALAN NUMBERS WITH  \nTHE MAPLE´ S HELP  \nFrancisco Regis Vieira Alves  \nAbstract: In Brazil we have identified a predilection of the authors of Mathematical History books for the discussion of the fundamentals of Differential and Integral Calculus. On the other hand, when we consider the teaching of Mathematics in the school context, it is essential to know the teaching of the historical and dynamic evolution of the concepts, moreover, mathematical objects more close to the Brazilian school reality. Thus, the present work discusses the notion of Catalan numbers, including its generalization process and epistemological aspects, with the adoption of some assumptions of Didactic Engineering. The work also presents a series of matrices and combinatorial properties that can be explored with the CAS Maple wih the scope to attract a larger number of students to a teaching that shows the historical, mathematical and evolutionary epistemological aspects of the generalized Catalan numbers. In this way, it is observed that the teacher must present an extensive knowledge about the notion of “numbers” and not only of the fundamentals of Differential and Integral Calculus.  \nKey words: Numbers of Catalan, Professor of Mathematics, Teaching, History of Mathematics.  \n1. Introduction  \nIn Brazil we record historical and epistemological knowledge as especially important for mathematics teachers. However, in a hegemonic way, the books of History of Mathematics (Boyer, 1968; Eves, 1976; Roque, 2012) usually devote greater space to the discussion of the foundations of the Infinitesimal Calculus, originating from the thought of Leibinitz and Newton in the 17th century. On the other hand, certain historical knowledge remains disregarded and not divulged among the professors of Mathematics in the schools relative to the current advance of Mathematics (Alves, 2016) .  \nMoreover, in our researches (Alves, 2016; 2017) directed to the field of initial mathematics teacher education developed in Brazil, we have developed an approach design that adopts the presuppositions of Didactics of Mathematics (Brousseau, 1986), according to the tradition of the french research, aiming the initial formation of teachers with attention to the context of the classroom. In particular, in teaching in the context of the History of Mathematics, the research methodology named Didactic Engineering can contribute in the sense of describing a way to a new approach of certain historical contents that, in general, are not available in the History books of ordinary mathematics.  \nBesides that, a field of research that acquires greater visibility provides the teaching of Mathematics, within the context of the historical context, from the aid of technology. In our work, we have exposed the use of GeoGebra and CAS Maple softwares with the aim of exploring certain mathematical properties neglected by the History of Mathematics books (Alves, 2017) . Particularly, in the present work, we will explore certain matrices involving representations of Catalan numbers and some resulting properties en virtue some generalization and it´s evolutionary process (Bobrowski; 2015; Koshy & Salmassi; 2006; Ribenboin, 1996; Varadarajan, 2006; Shapiro, 1976; Stanley, 2015) .  \nNevertheless, in order to describe certain structured didactic situations, in view of the teaching and training of Mathematics teachers in Brazil, we used the perspective originated from Didactic Engineering (Margolinas & Drijvers, 2015), with emphasis on the character of teacher training and that aims at proposing a didactic transposition aiming at the contents ofthe history of Mathematics. Thus, in  \nthe following section we will describe some elements of a didactic engineering and, aiming at the formation of Mathematics teachers for the teaching in a historical context.  \n2. Some elements of didactic engineering  \nDidactic engineering emerged in France in the context of major educational reforms in t","cbCaiiqupk7L5G0v","https://ap.wps.com/l/cbCaiiqupk7L5G0v","pdf",1057921,16,"English","# Introduction\n## Some elements of didactic engineering","[{\"question\":\"What is the main teaching focus of the work?\",\"answer\":\"The work focuses on teaching generalized Catalan numbers, including their generalization process and epistemological aspects.\"},{\"question\":\"How is Maple used in the proposed teaching approach?\",\"answer\":\"It uses CAS Maple to explore matrices and combinatorial properties tied to generalized Catalan numbers, supporting a more engaging classroom experience.\"},{\"question\":\"Why is historical context considered important for mathematics teachers in Brazil?\",\"answer\":\"The work argues that many Mathematics History books emphasize infinitesimal calculus foundations, while other historical knowledge relevant to classroom teaching is often disregarded and should be included.\"}]","ON TEACHING OF GENERALIZED CATALAN NUMBERS WITH THE MAPLE'S HELP | PDF",1792064550]