[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-592361-105":59,"doc-detail-592361-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","mathematical-ability-relies-on-knowledge-too","Mathematical Ability Relies on Knowledge, Too","","Problem solving is central to mathematics, yet many commonly held beliefs mischaracterize how problem-solving skill develops. The article argues that teaching general problem-solving strategies is not supported by rigorous randomized controlled evidence and that “reform” curricula misunderstand cognition. Evidence from research, including chess expertise studies, shows skill depends on long-term memory storing domain-specific configurations and best moves. Effective instruction therefore emphasizes worked examples and explicit guidance in mathematics.",{"@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":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/mathematical-ability-relies-on-knowledge-too/592361/",{"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/mathematical-ability-relies-on-knowledge-too/592361.png","ImageObject",300,407,{"name":92,"@type":93},"McGucket","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-15",true,{"@type":102,"interactionType":103,"userInteractionCount":19},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"Why do educators’ beliefs about teaching general problem-solving strategies lack support?","Question",{"text":112,"@type":113},"The article notes no body of research from randomized, controlled experiments shows that teaching general strategies improves problem solving in mathematics. It also describes many findings as counterintuitive and contrary to widely held views.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What does chess research suggest about how experts solve problems?",{"text":117,"@type":113},"Dutch psychologist Adriaan de Groot found masters outperform weekend players for real-game configurations, but equal accuracy appears for random configurations. This indicates expertise comes from long-term memory of many domain-specific board patterns and best moves.",{"name":119,"@type":110,"acceptedAnswer":120},"How should mathematics instruction be designed according to the article?",{"text":121,"@type":113},"Instruction should not rely on loosely derived “reform” curricula that reduce the importance of mathematical content. Instead, it should emphasize worked examples and direct, explicit guidance so students build integrated knowledge that supports problem solving.","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},592361,1793116927,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":19,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":14,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":143,"read_time":24,"preview_page_count":4},1236954412713,"https://us-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","Mathematical Ability Relies on Knowledge, Too  \nBy JOHN SWELLER, RICHARD E. CLARK, AND PAUL A. KIRSCHNER  \nProblem solving is central to mathematics. Yet problem-solving skill is not what it seems. Indeed, the field of problem solving has recently undergone a surge in research interest and insight, but many of the results of this research are both counterintuitive and contrary to many widely held views. For example, many educators assume that general problemsolving strategies are not only learnable and teachable but are a critical adjunct to mathematical knowledge. The bestknown exposition of this view was provided by the mathematician George Pólya.1 He discussed a range of general problem-solving strategies, such as encouraging mathematics students to think of a related problem and then solve the current problem by analogy, or to think of a simpler problem and then extrapolate to the current problem. The examples Pólya used to demonstrate his problem-solving strategies are fascinating, and his influence probably can be sourced, at least in part, to those examples. Nevertheless, in over a half century, no systematic body of evidence demonstrating the effectiveness of any  \nJohn Sweller is an emeritus professor of education atthe School of Education at the University of New South Wales. Richard E. Clark is a professor of educational psychology, clinical research professor of surgery, and director of the Center for Cognitive Technology at the University of Southern California. PaulA. Kirschner is a professor of educational psychology at the Centre for Learning Sciences and Technologies at the Open University of the Netherlands. This article is adapted, with permission from the American Mathematical Society, from John Sweller, Richard Clark, and Paul Kirschner,“Teaching General Problem-Solving Skills Is Not a Substitute for, or a Viable Addition to, Teaching Mathematics,”which appeared in the Notices of the American Mathematical Society 57, no. 10 (November 2010), 1303–1304, © copyright 2010.  \ngeneral problem-solving strategies has emerged. It is possible to teach learners to use general strategies such as those suggested by Pólya,2 but that is insufficient. There is no body of research based on randomized, controlled experiments indicating that such teaching leads to better problem solving.  \nRecent “reform” curricula both ignore the absence of supporting data and completely misunderstand the role of problem solving in cognition. If, the argument goes, we are not really teaching people mathematics but rather are teaching them some form of general problem solving, then mathematical content can be reduced in importance. According to this argument, we can teach students how to solve problems in general, and that will make them good mathematicians able to discover novel solutions irrespective of the content.  \nWe believe this argument ignores all the empirical evidence about mathematics learning. Although some mathematicians, in the absence of adequate instruction, may have learned to solve mathematics problems by discovering solutions without explicit guidance, this approach has never been the most effective or efficient way to learn mathematics.  \nThe alternative route to acquiring problem-solving skill in mathematics derives from the work of a Dutch psychologist, Adriaan de Groot,3 investigating the source of skill in chess. Researching why chess masters always defeated weekend players, de Groot managed to find only one difference. He showed masters and weekend players aboard configuration from a real game, removed it after five seconds, and asked  \nthem to reproduce the board. Masters could do so with an accuracy rate of about 70 percent compared with 30 percent for weekend players. Other researchers replicated these results and additionally demonstrated that when the experiment was repeated with random configurations, rather than real-game configurations, masters and weekend players had equal accuracy (roughly 30 percent) .4 M","cbCaioBoY9kEgtXf","https://ap.wps.com/l/cbCaioBoY9kEgtXf","pdf",316185,"English","# Mathematical problem solving and common assumptions\n## Limits of general strategy teaching\n## Evidence from chess expertise research\n## Domain-specific skill and long-term memory\n## Implications for mathematics instruction","[{\"question\":\"Why do educators’ beliefs about teaching general problem-solving strategies lack support?\",\"answer\":\"The article notes no body of research from randomized, controlled experiments shows that teaching general strategies improves problem solving in mathematics. It also describes many findings as counterintuitive and contrary to widely held views.\"},{\"question\":\"What does chess research suggest about how experts solve problems?\",\"answer\":\"Dutch psychologist Adriaan de Groot found masters outperform weekend players for real-game configurations, but equal accuracy appears for random configurations. This indicates expertise comes from long-term memory of many domain-specific board patterns and best moves.\"},{\"question\":\"How should mathematics instruction be designed according to the article?\",\"answer\":\"Instruction should not rely on loosely derived “reform” curricula that reduce the importance of mathematical content. Instead, it should emphasize worked examples and direct, explicit guidance so students build integrated knowledge that supports problem solving.\"}]","Mathematical Ability Relies on Knowledge, Too | PDF",1792074409]