{"id":"https://openalex.org/W2070388735","doi":"https://doi.org/10.1207/s15516709cog1403_4","title":"Algebra Mal\u2010Rules and Cognitive Accounts of Error","display_name":"Algebra Mal\u2010Rules and Cognitive Accounts of Error","publication_year":1990,"publication_date":"1990-07-01","ids":{"openalex":"https://openalex.org/W2070388735","doi":"https://doi.org/10.1207/s15516709cog1403_4","mag":"2070388735"},"language":"en","primary_location":{"id":"doi:10.1207/s15516709cog1403_4","is_oa":false,"landing_page_url":"https://doi.org/10.1207/s15516709cog1403_4","pdf_url":null,"source":{"id":"https://openalex.org/S78735424","display_name":"Cognitive Science","issn_l":"0364-0213","issn":["0364-0213","1551-6709"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Cognitive Science","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5104063842","display_name":"Stephen J. Payne","orcid":null},"institutions":[{"id":"https://openalex.org/I4210106992","display_name":"Lancaster University Ghana","ror":"https://ror.org/01g3dya06","country_code":"GH","type":"education","lineage":["https://openalex.org/I4210106992"]}],"countries":["GH"],"is_corresponding":true,"raw_author_name":"Stephen J. Payne","raw_affiliation_strings":["University of Lancaster"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Lancaster","institution_ids":["https://openalex.org/I4210106992"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5018868103","display_name":"Helen R. Squibb","orcid":null},"institutions":[{"id":"https://openalex.org/I4210106992","display_name":"Lancaster University Ghana","ror":"https://ror.org/01g3dya06","country_code":"GH","type":"education","lineage":["https://openalex.org/I4210106992"]}],"countries":["GH"],"is_corresponding":false,"raw_author_name":"Helen R. Squibb","raw_affiliation_strings":["University of Lancaster"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Lancaster","institution_ids":["https://openalex.org/I4210106992"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5104063842"],"corresponding_institution_ids":["https://openalex.org/I4210106992"],"apc_list":{"value":3810,"currency":"USD","value_usd":3810},"apc_paid":null,"fwci":8.6484,"has_fulltext":false,"cited_by_count":113,"citation_normalized_percentile":{"value":0.97531476,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"14","issue":"3","first_page":"445","last_page":"481"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11902","display_name":"Intelligent Tutoring Systems and Adaptive Learning","score":0.9973999857902527,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11902","display_name":"Intelligent Tutoring Systems and Adaptive Learning","score":0.9973999857902527,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10906","display_name":"AI-based Problem Solving and Planning","score":0.9939000010490417,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11516","display_name":"Visual and Cognitive Learning Processes","score":0.9717000126838684,"subfield":{"id":"https://openalex.org/subfields/3205","display_name":"Experimental and Cognitive Psychology"},"field":{"id":"https://openalex.org/fields/32","display_name":"Psychology"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/cognition","display_name":"Cognition","score":0.544002115726471},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.5390585660934448},{"id":"https://openalex.org/keywords/partition","display_name":"Partition (number theory)","score":0.4558176100254059},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.40870606899261475},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3998086750507355},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3846033215522766},{"id":"https://openalex.org/keywords/psychology","display_name":"Psychology","score":0.2644594609737396},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.2018783688545227},{"id":"https://openalex.org/keywords/programming-language","display_name":"Programming language","score":0.12597283720970154},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.10033351182937622}],"concepts":[{"id":"https://openalex.org/C169900460","wikidata":"https://www.wikidata.org/wiki/Q2200417","display_name":"Cognition","level":2,"score":0.544002115726471},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.5390585660934448},{"id":"https://openalex.org/C42812","wikidata":"https://www.wikidata.org/wiki/Q1082910","display_name":"Partition (number theory)","level":2,"score":0.4558176100254059},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.40870606899261475},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3998086750507355},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3846033215522766},{"id":"https://openalex.org/C15744967","wikidata":"https://www.wikidata.org/wiki/Q9418","display_name":"Psychology","level":0,"score":0.2644594609737396},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2018783688545227},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.12597283720970154},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.10033351182937622},{"id":"https://openalex.org/C169760540","wikidata":"https://www.wikidata.org/wiki/Q207011","display_name":"Neuroscience","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1207/s15516709cog1403_4","is_oa":false,"landing_page_url":"https://doi.org/10.1207/s15516709cog1403_4","pdf_url":null,"source":{"id":"https://openalex.org/S78735424","display_name":"Cognitive Science","issn_l":"0364-0213","issn":["0364-0213","1551-6709"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Cognitive Science","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/4","score":0.8600000143051147,"display_name":"Quality Education"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":40,"referenced_works":["https://openalex.org/W12681578","https://openalex.org/W28411281","https://openalex.org/W31157647","https://openalex.org/W43336110","https://openalex.org/W49036840","https://openalex.org/W69470806","https://openalex.org/W125481431","https://openalex.org/W136639002","https://openalex.org/W197426493","https://openalex.org/W227649490","https://openalex.org/W230233429","https://openalex.org/W573044666","https://openalex.org/W626922200","https://openalex.org/W1492518088","https://openalex.org/W1708874574","https://openalex.org/W1966028617","https://openalex.org/W1968201198","https://openalex.org/W1991691398","https://openalex.org/W1998696323","https://openalex.org/W2013910226","https://openalex.org/W2016900109","https://openalex.org/W2047806569","https://openalex.org/W2064430731","https://openalex.org/W2143418418","https://openalex.org/W2161829306","https://openalex.org/W2171389263","https://openalex.org/W2180809782","https://openalex.org/W2315761891","https://openalex.org/W2517319411","https://openalex.org/W2521152576","https://openalex.org/W2571532437","https://openalex.org/W2593934623","https://openalex.org/W2732714540","https://openalex.org/W2798445359","https://openalex.org/W3207342693","https://openalex.org/W3215907522","https://openalex.org/W4242555830","https://openalex.org/W4243199530","https://openalex.org/W4243772987","https://openalex.org/W4300402905"],"related_works":["https://openalex.org/W1979597421","https://openalex.org/W2007980826","https://openalex.org/W4245490552","https://openalex.org/W4225152035","https://openalex.org/W2061531152","https://openalex.org/W3002753104","https://openalex.org/W2077600819","https://openalex.org/W1587224694","https://openalex.org/W2911598644","https://openalex.org/W2042127053"],"abstract_inverted_index":{"We":[0],"report":[1],"an":[2],"empirical":[3],"study":[4],"of":[5,32,37,70,96,115,121,150],"elementary":[6],"algebra":[7,33],"errors,":[8],"conducted":[9],"in":[10,66],"three":[11],"separate":[12],"schools.":[13],"The":[14,35,113],"errors":[15,146],"are":[16,43,52,75,124],"diagnosed":[17],"using":[18],"mal\u2010rules,":[19,97],"as":[20],"proposed":[21],"by":[22,108],"Sleeman":[23],"(1984,":[24],"1,985).":[25],"Our":[26],"analysis":[27],"uncovers":[28],"the":[29,93,151,155],"following":[30],"properties":[31,149],"mal\u2010rules:":[34],"frequency":[36],"mal\u2010rules":[38,46,51,58,62,74],"is":[39,102,134],"severely":[40],"skewed,":[41],"there":[42],"many":[44],"infrequent":[45],"and":[47,143,154],"few":[48],"frequent":[49],"ones;":[50],"very":[53,59],"unstable,":[54],"students":[55],"typically":[56],"use":[57],"irregularly;":[60],"different":[61,67],"have":[63],"explanatory":[64],"power":[65],"schools":[68],"(many":[69],"our":[71,116],"most":[72,145],"powerful":[73],"previously":[76],"unreported);":[77],"mal\u2010rule":[78,99],"diagnosis":[79],"Is":[80],"more":[81,84],"successful":[82],"with":[83,104],"skilled":[85],"students;":[86],"students'":[87],"confidence":[88,106],"ratings":[89,107],"do":[90],"not":[91],"partition":[92],"total":[94],"set":[95],"every":[98],"we":[100,130],"find":[101],"associated":[103],"high":[105],"at":[109],"least":[110],"one":[111],"student.":[112],"Implications":[114],"data":[117],"for":[118],"cognitive":[119,156],"theories":[120],"error":[122],"generation":[123],"discussed.":[125],"Contrary":[126],"to":[127,136,166],"commonplace":[128],"assumptions,":[129],"argue":[131],"that":[132],"It":[133],"impossible":[135],"make":[137],"a":[138,160],"clear":[139],"distinction":[140],"between":[141],"slips":[142],"mistakes;":[144],"depend":[147],"on":[148],"knowledge":[152],"base":[153],"architecture.":[157],"Errors":[158],"In":[159,170],"procedural":[161],"skill":[162],"cannot":[163],"be":[164,167],"assumed":[165],"purely":[168],"syntactic":[169],"orgin.":[171]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":6},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}