{"id":"https://openalex.org/W2071025864","doi":"https://doi.org/10.1080/10986065.2012.625074","title":"What Kinds of Numbers Do Students Assign to Literal Symbols? Aspects of the Transition from Arithmetic to Algebra","display_name":"What Kinds of Numbers Do Students Assign to Literal Symbols? Aspects of the Transition from Arithmetic to Algebra","publication_year":2012,"publication_date":"2012-01-01","ids":{"openalex":"https://openalex.org/W2071025864","doi":"https://doi.org/10.1080/10986065.2012.625074","mag":"2071025864"},"language":"en","primary_location":{"id":"doi:10.1080/10986065.2012.625074","doi":"https://doi.org/10.1080/10986065.2012.625074","is_oa":false,"landing_page_url":"https://doi.org/10.1080/10986065.2012.625074","pdf_url":null,"source":{"id":"https://openalex.org/S74814605","display_name":"Mathematical Thinking and Learning","issn_l":"1098-6065","issn":["1098-6065","1532-7833"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","erih-plus","jufo-2","ki-jl-2","norway-2","poland-100","russia-white-list-2","vabb-shw"],"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547","https://openalex.org/P4310320449"],"host_organization_lineage_names":["Taylor & Francis","Informa"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Thinking and Learning","raw_type":"journal-article"},"volume":null,"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/A5011347120","display_name":"Konstantinos P. Christou","orcid":"https://orcid.org/0000-0002-7615-4968","observed_orcids":["https://orcid.org/0000-0002-7615-4968"]},"institutions":[{"id":"https://openalex.org/I200777214","display_name":"National and Kapodistrian University of Athens","ror":"https://ror.org/04gnjpq42","country_code":"GR","type":"education","lineage":["https://openalex.org/I200777214"]}],"countries":["GR"],"is_corresponding":false,"raw_author_name":"Konstantinos P. Christou","raw_affiliation_strings":["Department of Philosophy and History of Science, National and Kapodistrian University of Athens"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Philosophy and History of Science, National and Kapodistrian University of Athens","institution_ids":["https://openalex.org/I200777214"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5003422657","display_name":"Stella Vosniadou","orcid":"https://orcid.org/0000-0003-4850-8578","observed_orcids":["https://orcid.org/0000-0003-4850-8578"]},"institutions":[{"id":"https://openalex.org/I200777214","display_name":"National and Kapodistrian University of Athens","ror":"https://ror.org/04gnjpq42","country_code":"GR","type":"education","lineage":["https://openalex.org/I200777214"]}],"countries":["GR"],"is_corresponding":true,"raw_author_name":"Stella Vosniadou","raw_affiliation_strings":["Department of Philosophy and History of Science, National and Kapodistrian University of Athens"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Philosophy and History of Science, National and Kapodistrian University of Athens","institution_ids":["https://openalex.org/I200777214"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5003422657"],"corresponding_institution_ids":["https://openalex.org/I200777214"],"apc_list":null,"apc_paid":null,"fwci":20.1287,"has_fulltext":false,"cited_by_count":55,"citation_normalized_percentile":{"value":0.99018391,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"14","issue":"1","first_page":"1","last_page":"27"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10130","display_name":"Mathematics Education and Teaching Techniques","score":0.6890000104904175,"subfield":{"id":"https://openalex.org/subfields/3304","display_name":"Education"},"field":{"id":"https://openalex.org/fields/33","display_name":"Social Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T10130","display_name":"Mathematics Education and Teaching Techniques","score":0.6890000104904175,"subfield":{"id":"https://openalex.org/subfields/3304","display_name":"Education"},"field":{"id":"https://openalex.org/fields/33","display_name":"Social Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T12522","display_name":"Mathematics Education and Pedagogy","score":0.11659999936819077,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11345","display_name":"Cognitive and developmental aspects of mathematical skills","score":0.07039999961853027,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/real-variables","display_name":"real variables","score":0.421999990940094},{"id":"https://openalex.org/keywords/natural-number-bias","display_name":"natural number bias","score":0.8270000219345093},{"id":"https://openalex.org/keywords/transition-from-arithmetic-to-algebra","display_name":"transition from arithmetic to algebra","score":0.7099999785423279},{"id":"https://openalex.org/keywords/secondary-school-students","display_name":"high school students","score":0.75},{"id":"https://openalex.org/keywords/conceptual-change","display_name":"conceptual change","score":0.367000013589859},{"id":"https://openalex.org/keywords/algebraic-expressions","display_name":"algebraic expressions","score":0.6570000052452087}],"concepts":[{"id":"https://openalex.org/C2780882242","wikidata":"https://www.wikidata.org/wiki/Q14235582","display_name":"Literal (mathematical logic)","level":2,"score":0.8829420804977417},{"id":"https://openalex.org/C139676723","wikidata":"https://www.wikidata.org/wiki/Q1193832","display_name":"Sign (mathematics)","level":2,"score":0.752436637878418},{"id":"https://openalex.org/C184264201","wikidata":"https://www.wikidata.org/wiki/Q21199","display_name":"Natural number","level":2,"score":0.6720702648162842},{"id":"https://openalex.org/C145420912","wikidata":"https://www.wikidata.org/wiki/Q853077","display_name":"Mathematics education","level":1,"score":0.5810665488243103},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.5714461803436279},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5595077276229858},{"id":"https://openalex.org/C2776608160","wikidata":"https://www.wikidata.org/wiki/Q4785462","display_name":"Natural (archaeology)","level":2,"score":0.5460517406463623},{"id":"https://openalex.org/C182365436","wikidata":"https://www.wikidata.org/wiki/Q50701","display_name":"Variable (mathematics)","level":2,"score":0.463032603263855},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.4586160182952881},{"id":"https://openalex.org/C527412718","wikidata":"https://www.wikidata.org/wiki/Q855395","display_name":"Interpretation (philosophy)","level":2,"score":0.4579443335533142},{"id":"https://openalex.org/C18912844","wikidata":"https://www.wikidata.org/wiki/Q4724001","display_name":"Algebraic expression","level":3,"score":0.44246381521224976},{"id":"https://openalex.org/C194232998","wikidata":"https://www.wikidata.org/wiki/Q1606712","display_name":"Transition (genetics)","level":3,"score":0.4309682846069336},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.401008665561676},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.25086870789527893},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.1694202423095703},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.14107802510261536},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.08799144625663757},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C95457728","wikidata":"https://www.wikidata.org/wiki/Q309","display_name":"History","level":0,"score":0.0},{"id":"https://openalex.org/C166957645","wikidata":"https://www.wikidata.org/wiki/Q23498","display_name":"Archaeology","level":1,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1080/10986065.2012.625074","doi":"https://doi.org/10.1080/10986065.2012.625074","is_oa":false,"landing_page_url":"https://doi.org/10.1080/10986065.2012.625074","pdf_url":null,"source":{"id":"https://openalex.org/S74814605","display_name":"Mathematical Thinking and Learning","issn_l":"1098-6065","issn":["1098-6065","1532-7833"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","erih-plus","jufo-2","ki-jl-2","norway-2","poland-100","russia-white-list-2","vabb-shw"],"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547","https://openalex.org/P4310320449"],"host_organization_lineage_names":["Taylor & Francis","Informa"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Thinking and Learning","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"score":0.6164000034332275,"id":"https://metadata.un.org/sdg/4","display_name":"Quality education"}],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":81,"referenced_works":["https://openalex.org/W82428586","https://openalex.org/W85730625","https://openalex.org/W86993336","https://openalex.org/W99792822","https://openalex.org/W159386515","https://openalex.org/W166014084","https://openalex.org/W283356536","https://openalex.org/W406344374","https://openalex.org/W629040682","https://openalex.org/W1498617146","https://openalex.org/W1502862628","https://openalex.org/W1514187077","https://openalex.org/W1524047387","https://openalex.org/W1545352087","https://openalex.org/W1555409383","https://openalex.org/W1579408594","https://openalex.org/W1590089318","https://openalex.org/W1590808534","https://openalex.org/W1591879720","https://openalex.org/W1592894700","https://openalex.org/W1597878878","https://openalex.org/W1606328991","https://openalex.org/W1762139154","https://openalex.org/W1949020749","https://openalex.org/W1964413906","https://openalex.org/W1966861531","https://openalex.org/W1970706444","https://openalex.org/W1971139377","https://openalex.org/W1974658064","https://openalex.org/W1984655437","https://openalex.org/W1989757574","https://openalex.org/W2001728985","https://openalex.org/W2002749349","https://openalex.org/W2006045181","https://openalex.org/W2011447619","https://openalex.org/W2011512932","https://openalex.org/W2013701626","https://openalex.org/W2019917174","https://openalex.org/W2023330715","https://openalex.org/W2043724524","https://openalex.org/W2044989106","https://openalex.org/W2049926613","https://openalex.org/W2058346042","https://openalex.org/W2070437222","https://openalex.org/W2071127407","https://openalex.org/W2072267223","https://openalex.org/W2080620213","https://openalex.org/W2080678496","https://openalex.org/W2080791986","https://openalex.org/W2101922032","https://openalex.org/W2122165184","https://openalex.org/W2123407405","https://openalex.org/W2171138899","https://openalex.org/W2187527228","https://openalex.org/W2241643353","https://openalex.org/W2246717614","https://openalex.org/W2323190809","https://openalex.org/W2323749923","https://openalex.org/W2329434402","https://openalex.org/W2479578231","https://openalex.org/W2492401337","https://openalex.org/W2525116681","https://openalex.org/W2557546946","https://openalex.org/W2580787246","https://openalex.org/W2614751831","https://openalex.org/W2635475831","https://openalex.org/W2735402199","https://openalex.org/W2798734824","https://openalex.org/W2904285923","https://openalex.org/W2944390027","https://openalex.org/W2990292436","https://openalex.org/W3038328679","https://openalex.org/W3150111307","https://openalex.org/W3193637440","https://openalex.org/W3196493418","https://openalex.org/W3200916490","https://openalex.org/W4210644278","https://openalex.org/W4239613992","https://openalex.org/W4253865845","https://openalex.org/W4255465331","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W4297376198","https://openalex.org/W2591324967","https://openalex.org/W2525116681","https://openalex.org/W2523983326","https://openalex.org/W2332161582","https://openalex.org/W2526153679","https://openalex.org/W2520228774","https://openalex.org/W2181531217","https://openalex.org/W2423358081","https://openalex.org/W263266182"],"abstract_inverted_index":{"Three":[0],"experiments":[1,54],"used":[2,78],"multiple":[3],"methods\u2014open-ended":[4],"assessments,":[5],"multiple-choice":[6],"questionnaires,":[7],"and":[8,110],"interviews\u2014to":[9],"investigate":[10],"the":[11,14,20,45,51,96,99,116,120,148],"hypothesis":[12],"that":[13,72,89,150],"development":[15],"of":[16,19,22,37,47,98,119,161],"students'":[17,159],"understanding":[18],"concept":[21,36],"real":[23],"variable":[24],"in":[25,30,60,83,131,164],"algebra":[26],"may":[27],"be":[28,42,77],"influenced":[29],"fundamental":[31],"ways":[32],"by":[33],"their":[34,124],"initial":[35],"number,":[38],"which":[39],"seems":[40],"to":[41,64,69,79,101,105,114,146],"organized":[43],"around":[44],"notion":[46],"natural":[48,108,155],"number.":[49],"In":[50],"first":[52],"two":[53],"91":[55],"secondary":[56],"school":[57],"students":[58,100],"(ranging":[59],"age":[61],"from":[62],"12.5":[63],"14.5":[65],"years)":[66],"were":[67,129,144],"asked":[68],"indicate":[70],"numbers":[71,109],"could":[73,75],"or":[74],"not":[76],"substitute":[80],"literal":[81,103,162],"symbols":[82,104,163],"algebraic":[84,121],"expressions.":[85],"The":[86,142],"results":[87,143],"showed":[88],"there":[90,151],"was":[91],"a":[92,111,132,153],"strong":[93],"tendency":[94,113],"on":[95,158],"part":[97],"interpret":[102],"stand":[106],"for":[107],"related":[112],"consider":[115],"phenomenal":[117],"sign":[118],"expressions":[122],"as":[123],"\u201creal\u201d":[125],"sign.":[126],"Similar":[127],"findings":[128],"obtained":[130],"third,":[133],"individual":[134],"interview":[135],"study,":[136],"conducted":[137],"with":[138],"tenth":[139],"grade":[140],"students.":[141],"interpreted":[145],"support":[147],"interpretation":[149],"is":[152],"systematic":[154],"number":[156],"bias":[157],"substitutions":[160],"algebra.":[165]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":6},{"year":2018,"cited_by_count":7},{"year":2017,"cited_by_count":6},{"year":2016,"cited_by_count":5},{"year":2015,"cited_by_count":6},{"year":2014,"cited_by_count":6},{"year":2013,"cited_by_count":1}],"updated_date":"2026-10-30T12:39:25.321706","created_date":"2016-06-24T00:00:00"}