[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-173490-en":3,"doc-seo-173490-105":30,"detail-sidebar-cat-1-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":11,"category_id":12,"category_name":13,"doc_title":14,"doc_description":15,"doc_content":16,"file_id":17,"file_url":18,"file_type":19,"file_size":20,"view_count":11,"is_deleted":4,"is_public":11,"is_downloadable":11,"audit_status":11,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":15,"update_tm":28,"read_time":29},173490,1099523885336,"Taylor Morgan","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",1,11,"Presentations","Appreciating Mathematical Structure For All - Theoretical Frame - Document Draft","The paper develops a view of mathematical structure as the identification of general properties instantiated in particular situations, expressed through relationships among elements or subsets. It argues that structural attention is a necessary component of mathematics teaching and learning and that students must be guided to engage in structural thinking. The theoretical framework defines structural thinking as using, explicating, and connecting properties, explaining how procedural mastery becomes effective when anchored in properties and reconstruction rather than rote memory.","Appreciating Mathematical Structure For All\nJohn Mason, Max Stephens and Anne Watson\u000bOpen University and University of Oxford\u000bUniversity of Melbourne\u000bUniversity of Oxford\nWe take mathematical structure to mean the identification of general properties which are instantiated in particular situations as relationships between elements or subsets of elements of a set. We put forward the view that attention to structure is an essential part of mathematical teaching and learning. This is not to be confused with teaching children mathematical structure. We observe that children from quite early ages are able to appreciate structure to a greater extent than some authors have imagined. Initiating students to appreciate structure implies, of course, that their appreciation of it needs to be cultivated in order to deepen and to become more mature. We first consider some recent research that supports this view and then go on to argue that unless students are encouraged to attend to structure and to engage in structural thinking they will be blocked from thinking productively and deeply about mathematics. We provide several illustrative cases in which structural thinking helps to bridge the mythical chasm between conceptual and procedural approaches to teaching and learning mathematics. Finally we place our proposals in the context of how several writers in the past have attempted to explore the importance of structure in mathematics teaching and learning.\nTheoretical Frame\nWe take mathematical structure to mean the identification of general properties which are instantiated in particular situations as relationships between elements.  These elements can be mathematical objects like numbers and triangles, sets with functions between them, relations on sets, even relations between relations in an ongoing hierarchy. Usually it is helpful to think of structure in terms of an agreed list of properties which are taken as axioms and from which other properties can be deduced. Mathematically, the definition of a relation derives from set theory as a subset of a Cartesian product of sets. Psychologically, a relationship is some connection or association between elements or subsets which have been themselves been discerned. When the relationship is seen as instantiation of a property, the relation becomes (part of) a structure. For example\nThe relation between a whole number and its double can be denoted by {[n, 2n]: n = 1, 2, 3, …} or by n –> 2n among other ways; being a pair consisting of a number and its double is a property instantiated here among the whole numbers, but other numbers are possible.\nThe set {[1, 2], [2, 5], [3, 2], [1, 4], [4, 4]} is also a relation, though not one that deserves a special name, and unlikely to be an instantiation of a specific property other than that it specifies this relation!\nRecognising a relationship amongst two or more objects is not in itself structural or relational thinking, which, for us, involves making use of relationships as instantiations of properties.  Awareness of the use of properties lies at the core of structural thinking. We define structural thinking as a disposition to use, explicate and connect these properties in one’s mathematical thinking.\nWe adopt a phenomenological approach to studying opportunities for stimulating learners to appreciate mathematical structure, at every age and stage of their exposure to mathematics and to mathematical thinking.  We start from the position that mastering procedures is an important component of taking advantage of opportunities to make mathematical sense, but that it is of little value to learners if it is simply a procedure, because as the number of procedures increases, the load on memory and retrieval becomes more and more burdensome. When procedures are accompanied by even a minimal appreciation of the mathematical structures which make them effective and which provide criteria for appropriateness, learning shifts to focusing on re-construction based","cbCaiq8XlyWx13cp","https://ap.wps.com/l/cbCaiq8XlyWx13cp","docx",80844,20,"English","en",105,"# Appreciating Mathematical Structure For All\n# Theoretical Frame\n## Structural thinking and properties\n## Relations as instantiations\n## Learning shifts from procedures to reconstruction\n## Historical roots of structural reasoning\n## Attention states in mathematical reasoning","[{\"question\":\"What does the paper mean by “mathematical structure”?\",\"answer\":\"Mathematical structure is defined as the identification of general properties that are instantiated in particular situations as relationships between elements or subsets of elements.\"},{\"question\":\"Why is attention to structure considered essential for learning mathematics?\",\"answer\":\"Students need encouragement to attend to structure and engage in structural thinking; otherwise they can be blocked from thinking productively and deeply about mathematics.\"},{\"question\":\"How does the theoretical frame explain the role of procedures?\",\"answer\":\"Procedures are important for taking advantage of learning opportunities, but they have limited value if treated as standalone routines; when paired with appreciation of structures and criteria for appropriateness, learning shifts toward reconstruction based on remembering.\"}]","Appreciating Mathematical Structure For All - Theoretical Frame - Document Draft | DOCX",1788303927,7,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":14,"keywords":34,"description":15,"schema_data":35,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":28},"appreciating-mathematical-structure-for-all-theoretical-frame-document-draft","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":11},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/template/","Template",2,{"item":49,"name":13,"@type":43,"position":50},"https://docshare.wps.com/template/presentations/",3,{"item":52,"name":14,"@type":43,"position":53},"https://docshare.wps.com/template/appreciating-mathematical-structure-for-all-theoretical-frame-document-draft/173490/",4,{"url":52,"name":14,"@type":55,"author":56,"headline":14,"publisher":58,"fileFormat":61,"inLanguage":23,"description":15,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/vnd.openxmlformats-officedocument.wordprocessingml.document","2026-09-06","2026-09-01",true,{"@type":66,"interactionType":67,"userInteractionCount":11},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What does the paper mean by “mathematical structure”?","Question",{"text":76,"@type":77},"Mathematical structure is defined as the identification of general properties that are instantiated in particular situations as relationships between elements or subsets of elements.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"Why is attention to structure considered essential for learning mathematics?",{"text":81,"@type":77},"Students need encouragement to attend to structure and engage in structural thinking; otherwise they can be blocked from thinking productively and deeply about mathematics.",{"name":83,"@type":74,"acceptedAnswer":84},"How does the theoretical frame explain the role of procedures?",{"text":85,"@type":77},"Procedures are important for taking advantage of learning opportunities, but they have limited value if treated as standalone routines; when paired with appreciation of structures and criteria for appropriateness, learning shifts toward reconstruction based on remembering.","https://schema.org",{"og:url":52,"og:type":88,"og:title":14,"og:site_name":59,"og:description":15},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":93},[94,97,102,107,112,117,122,127,132],{"id":12,"doc_module":11,"doc_module_name":46,"category_name":13,"show_sort_weight":95,"slug":96},90,"presentations",{"id":98,"doc_module":11,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},12,"Resumes",80,"resumes",{"id":103,"doc_module":11,"doc_module_name":46,"category_name":104,"show_sort_weight":105,"slug":106},14,"Invoices",70,"invoices",{"id":108,"doc_module":11,"doc_module_name":46,"category_name":109,"show_sort_weight":110,"slug":111},15,"Posters",60,"posters",{"id":113,"doc_module":11,"doc_module_name":46,"category_name":114,"show_sort_weight":115,"slug":116},16,"Social Media",50,"social-media",{"id":118,"doc_module":11,"doc_module_name":46,"category_name":119,"show_sort_weight":120,"slug":121},17,"Forms",40,"forms",{"id":123,"doc_module":11,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},18,"Letters",30,"letters",{"id":128,"doc_module":11,"doc_module_name":46,"category_name":129,"show_sort_weight":130,"slug":131},21,"Paper Templates",5,"papers-templates",{"id":133,"doc_module":11,"doc_module_name":46,"category_name":134,"show_sort_weight":4,"slug":135},158,"General","general-158"]