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It argues that the core issue is ownership of mathematics, where students may perceive it as externally imposed. The chapter proposes challenge as a pathway to more resilient understanding, linking puzzle instincts and cultural curiosity to cooperation and competition. It outlines the study’s aims: define challenge, identify sources and contexts for instruction, and evaluate whether challenges enhance learning.",{"@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":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/challenging-mathematics-in-and-beyond-the-classroom-chapter-0-introduction/265165/",{"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/challenging-mathematics-in-and-beyond-the-classroom-chapter-0-introduction/265165.png","ImageObject",300,407,{"name":92,"@type":93},"Valentina","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-02","2026-09-14",true,{"@type":102,"interactionType":103,"userInteractionCount":52},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"Why is school mathematics often criticized according to this chapter?","Question",{"text":112,"@type":113},"The chapter points to rigid syllabi and assessment imperatives, which can make students uncomfortable and reduce appreciation of mathematics’ structure and significance.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What does the chapter identify as the key problem in students’ relationship with mathematics?",{"text":117,"@type":113},"It argues the problem is one of ownership—whether mathematics feels like it is imposed by “the system” or “the teacher” rather than something learners can take responsibility for.",{"name":119,"@type":110,"acceptedAnswer":120},"How does the chapter connect challenge to learning and understanding?",{"text":121,"@type":113},"It proposes that aversion to the aridity of school mathematics can be counteracted by exploiting challenge, and it aims to examine sources of challenge and whether challenges enhance learning.","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},265165,1789392927,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":52,"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},13056703020460,"https://ap-avatar.wpscdn.com/avatar/be000253dac470eee5d?_k=1778207105932848923","Chapter 0 Introduction  \nEd Barbeau  \n0.1 Challenging: a human activity  \nIn modern society, mathematics is a prominent part of the school syllabus. It is praised for its utility and regarded as a foundation of our modern technological society. Yet school mathematics is the locus of much concern and criticism. Many leave school uncomfortable with, if not disdainful of, the subject.  \nEven those who get good grades may lack fluency and appreciation of its structure and significance. Mathematicians themselves may see little that reflects the character of mathematics as they experience it.  \nThe blame for this situation is often laid at the door of the demands ofa rigid syllabus and the imperatives of assessment.  \nWe need to analyze these complaints. Mathematics is a highly structured subject. It is hard to see how one can proceed very far without orchestrating topics and assessing the mastery ofits students from time to time to make sure they are prepared to make further progress. But does this mean that mathematical instruction should embrace so much mechanical learning and rely on recall and stock situations?  \nThe issue is really one of ownership—who owns the mathematics?  \nToo often, the answer is ‘‘the system’’ or ‘‘the teacher’’. From the pupils’perspective, it seems imposed from without to achieve extrinsic goals. For many, it makes little sense. To be sure, mathematics can be difficult, but is it a difficulty one would want to surmount?  \nIn the third book of the ‘‘Divine Comedy’’, Dante’s pilgrim is advised by Beatrice that  \nconvienti ancor sedere un poco a mensa  \nper`o che ’l cibo rigido c’hai preso,  \nrichiedi ancora aiuto a tua dispensa.  \nEd Barbeau (*)  \nDepartment of Mathematics, University of Toronto, Canada e-mail: [barbeau@math.utoronto.ca](barbeau@math.utoronto.ca)  \nE.J. Barbeau, P.J. Taylor (eds.), Challenging Mathematics In and Beyond the Classroom, New ICMI Study Series 12, DOI 10 . 1007/978-0-387-09603-2_ 1,􀀂 Springer ScienceþBusiness Media, LLC 2009  \nChallenging Mathematics In and Beyond the Classroom  \nApri la merte a quel ch’io ti paleso  \na fermalvi entro; ch´e non fa scienza  \nsanza lo ritenere, avere inteso.  \nOr, in English  \nyou must stay longer at the table  \nbecause the food you have eaten is tough  \nand needs time for digestion.  \nOpen your mind to what I shall reveal  \nand keep it there; whoever has heard  \nand not retained, knows nothing.  \nParadise, Canto V, 37–42  \nThis could be applied to the student of mathematics. She must learn tobe patient, open the mind, and seek resilient rather than transient understanding.  \nThese are qualities that require the learner to draw on internal resources. If the student is permitted to be passive, she may become alienated and resist accepting this responsibility.  \nThe task of the teacher and expositor is to present the subject in such a way as to awaken these resources.  \nDanesi (2000) suggests the key to this conundrum. He notes that ‘‘puzzles have been around since the dawn of history’’ and that people have ‘‘been so fascinated by seemingly trivial posers, which nonetheless require substantial time and mental effort to solve, for no apparent reward other than the simple satisfaction of solving them’’. He asks,  \nIs there a puzzle instinct in the human species,  \ndeveloped and refined by the forces of natural selection  \nfor some survival function? Or is this instinctual love  \nof puzzles the product of some metaphysical force buried  \ndeep within the psyche, impelling people to behave in ways  \nthat defy rational explanation?  \nDanesi sees this propensity for puzzles as tightly bound up in human culture. There is a natural curiosity to be found in every human civilization; the environment poses wonders, threats and opportunities that must be understood and exploited by beings lacking many of the physical advantages of the beasts but endowed with a powerful mind. However, it is not just necessity but choice that leads people to accept challenges, a","cbCaiboCGbKrCwap","https://ap.wps.com/l/cbCaiboCGbKrCwap","pdf",12154224,329,"English","# Chapter 0 Introduction\n## 0.1 Challenging: a human activity\n## 0.2 Challenges and education","[{\"question\":\"Why is school mathematics often criticized according to this chapter?\",\"answer\":\"The chapter points to rigid syllabi and assessment imperatives, which can make students uncomfortable and reduce appreciation of mathematics’ structure and significance.\"},{\"question\":\"What does the chapter identify as the key problem in students’ relationship with mathematics?\",\"answer\":\"It argues the problem is one of ownership—whether mathematics feels like it is imposed by “the system” or “the teacher” rather than something learners can take responsibility for.\"},{\"question\":\"How does the chapter connect challenge to learning and understanding?\",\"answer\":\"It proposes that aversion to the aridity of school mathematics can be counteracted by exploiting challenge, and it aims to examine sources of challenge and whether challenges enhance learning.\"}]","Challenging Mathematics In and Beyond the Classroom - Chapter 0 - Introduction | PDF",829]