{"id":"https://openalex.org/W2163300350","doi":null,"title":"An investigation of grade 11 learners' understanding of the cosine function with Sketchpad.","display_name":"An investigation of grade 11 learners' understanding of the cosine function with Sketchpad.","publication_year":2010,"publication_date":"2010-01-01","ids":{"openalex":"https://openalex.org/W2163300350","mag":"2163300350"},"language":"en","primary_location":{"id":"pmh:oai:researchspace.ukzn.ac.za:10413/10581","doi":null,"is_oa":false,"landing_page_url":"http://hdl.handle.net/10413/10581","pdf_url":null,"source":{"id":"https://openalex.org/S4306402416","display_name":"ResearchSpace (University of KwaZulu-Natal)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I95023434","host_organization_name":"University of KwaZulu-Natal","host_organization_lineage":["https://openalex.org/I95023434"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Thesis"},"volume":null,"type":"dissertation","is_traditional":true,"indexed_in":[],"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/A5042124584","display_name":"Calisto Majengwa","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Calisto Majengwa","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5042124584"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10130","display_name":"Mathematics Education and Teaching Techniques","score":0.9804999828338623,"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":0.9804999828338623,"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":0.012000000104308128,"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/T13412","display_name":"Education and Technology Integration","score":0.0008999999845400453,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/cosine-function","display_name":"cosine function","score":0.8209999799728394},{"id":"https://openalex.org/keywords/trigonometry","display_name":"trigonometry","score":0.8679999709129333},{"id":"https://openalex.org/keywords/geometer-s-sketchpad","display_name":"Geometer's Sketchpad","score":0.9449999928474426},{"id":"https://openalex.org/keywords/dynamic-geometry-software","display_name":"dynamic geometry software","score":0.972000002861023},{"id":"https://openalex.org/keywords/learner-understanding","display_name":"learner comprehension","score":0.5659999847412109},{"id":"https://openalex.org/keywords/eleventh-grade-students","display_name":"eleventh grade students","score":0.7699999809265137},{"id":"https://openalex.org/keywords/mathematics-education","display_name":"mathematics education","score":0.46799999475479126},{"id":"https://openalex.org/keywords/computer-assisted-instruction","display_name":"computer-assisted instruction","score":0.675000011920929},{"id":"https://openalex.org/keywords/african-schools","display_name":"African schools","score":0.42100000381469727},{"id":"https://openalex.org/keywords/task-based-interviews","display_name":"task-based interviews","score":0.5109999775886536},{"id":"https://openalex.org/keywords/memorization","display_name":"memorization","score":0.3889999985694885},{"id":"https://openalex.org/keywords/classroom-interaction-patterns","display_name":"classroom interaction patterns","score":0.531000018119812},{"id":"https://openalex.org/keywords/triangulation","display_name":"triangulation","score":0.4000000059604645},{"id":"https://openalex.org/keywords/student-motivation","display_name":"student motivation","score":0.3479999899864197},{"id":"https://openalex.org/keywords/student-self-efficacy","display_name":"student self-efficacy","score":0.375}],"concepts":[{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.49901866912841797},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4616469442844391},{"id":"https://openalex.org/C15744967","wikidata":"https://www.wikidata.org/wiki/Q9418","display_name":"Psychology","level":0,"score":0.3467065393924713},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:researchspace.ukzn.ac.za:10413/10581","doi":null,"is_oa":false,"landing_page_url":"http://hdl.handle.net/10413/10581","pdf_url":null,"source":{"id":"https://openalex.org/S4306402416","display_name":"ResearchSpace (University of KwaZulu-Natal)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I95023434","host_organization_name":"University of KwaZulu-Natal","host_organization_lineage":["https://openalex.org/I95023434"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Thesis"},{"id":"mag:2163300350","doi":null,"is_oa":false,"landing_page_url":"https://researchspace.ukzn.ac.za/handle/10413/10581","pdf_url":null,"source":null,"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":null}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/4","score":0.975600004196167,"display_name":"Quality education"}],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":72,"referenced_works":["https://openalex.org/W27474068","https://openalex.org/W44225559","https://openalex.org/W79323211","https://openalex.org/W82428586","https://openalex.org/W105330412","https://openalex.org/W111908685","https://openalex.org/W157203826","https://openalex.org/W161214126","https://openalex.org/W167647892","https://openalex.org/W232458980","https://openalex.org/W244701492","https://openalex.org/W278739031","https://openalex.org/W283356536","https://openalex.org/W343257020","https://openalex.org/W345589860","https://openalex.org/W373431233","https://openalex.org/W592373331","https://openalex.org/W634988253","https://openalex.org/W859240336","https://openalex.org/W1519144258","https://openalex.org/W1524337223","https://openalex.org/W1539318451","https://openalex.org/W1542932561","https://openalex.org/W1544375721","https://openalex.org/W1574026255","https://openalex.org/W1584475005","https://openalex.org/W1584697775","https://openalex.org/W1591721817","https://openalex.org/W1597874749","https://openalex.org/W1601189769","https://openalex.org/W1627058018","https://openalex.org/W1668767679","https://openalex.org/W1780382453","https://openalex.org/W1969865542","https://openalex.org/W1975711826","https://openalex.org/W1985743219","https://openalex.org/W1992274553","https://openalex.org/W1996962164","https://openalex.org/W2022371018","https://openalex.org/W2022481558","https://openalex.org/W2023707110","https://openalex.org/W2026701456","https://openalex.org/W2027658208","https://openalex.org/W2028783879","https://openalex.org/W2034478886","https://openalex.org/W2041452239","https://openalex.org/W2072267223","https://openalex.org/W2073628681","https://openalex.org/W2079553224","https://openalex.org/W2099597561","https://openalex.org/W2115041347","https://openalex.org/W2116199508","https://openalex.org/W2121546132","https://openalex.org/W2122120800","https://openalex.org/W2122349951","https://openalex.org/W2159938354","https://openalex.org/W2166688909","https://openalex.org/W2171138899","https://openalex.org/W2238397456","https://openalex.org/W2354157941","https://openalex.org/W2528083366","https://openalex.org/W2606330123","https://openalex.org/W2616371189","https://openalex.org/W2730267981","https://openalex.org/W2769416443","https://openalex.org/W2795522246","https://openalex.org/W2795615519","https://openalex.org/W2899573755","https://openalex.org/W2970264147","https://openalex.org/W2992473810","https://openalex.org/W3021970622","https://openalex.org/W3119955191"],"related_works":["https://openalex.org/W2146857655","https://openalex.org/W1601189769","https://openalex.org/W7038396844","https://openalex.org/W3015545505","https://openalex.org/W3015612388","https://openalex.org/W2401307151","https://openalex.org/W7206649572","https://openalex.org/W7153690768","https://openalex.org/W3109889652","https://openalex.org/W3001211935"],"abstract_inverted_index":{"This":[0,27,81,160,254],"study":[1,384,495,507],"investigated":[2],"how":[3,452,773],"Grade":[4,39,517,544,555],"11":[5],"learners":[6,91,120,147,227,249,294,531,552,559,612,689,766,782],"from":[7,119,164,515],"a":[8,191,306,499,538,561,575,580,586,606,630,675,721],"school":[9,127,156],"in":[10,14,38,52,66,99,136,155,170,182,209,252,317,346,410,520,543,554,597,636,649,653,677,685],"KwaNdengezi,":[11],"near":[12],"\\nPinetown,":[13],"Durban,":[15],"understood":[16,719],"the":[17,31,49,79,103,132,178,185,199,202,235,238,285,292,321,329,436,444,448,459,478,494,506,512,534,565,623,643,654,660,665,672,692,699,702,764,788,791],"cosine":[18,513],"function":[19,514,536,539,567],"with":[20,102,730],"software":[21],"known":[22],"as":[23,303,427,441,537,595,629,657,720,796,798],"The":[24,61,126,212,310,463,503,688],"\\nGeometer\u2019s":[25],"Sketchpad.":[26],"was":[28,43,128,143,488,496,603,646,664],"done":[29],"on":[30,262,744],"basis":[32],"of":[33,131,134,187,201,243,291,308,312,328,381,407,474,480,511,533,540,564,577,579,601,651,674,694,701,724,790],"what":[34],"they":[35,46,707],"had":[36,47,690],"learnt":[37],"10.":[40],"\\nThe":[41,56,599],"timing":[42],"just":[44],"before":[45],"covered":[48],"topic":[50],"again":[51],"their":[53,100,232,263,516,731,801,816],"current":[54],"grade.":[55],"researcher":[57],"hoped,":[58],"by":[59,246,385],"using":[60,340,569],"Geometer\u2019s":[62],"Sketchpad,":[63,505],"to":[64,70,78,92,94,111,152,176,194,218,231,260,280,282,319,323,353,356,401,417,423,446,458,468,524,616,659,696,727,735,762,783,814],"contribute":[65],"some":[67,681,808],"small":[68],"\\nway":[69],"teaching":[71,203],"and":[72,86,90,108,124,140,157,204,220,265,368,373,387,450,549,591,594,608,714,774,780],"learning":[73,205],"methods":[74],"that":[75,207,225,256,266,287,391,472,493,757,811],"are":[76,258,268,338,375,415,421,473,716,740],"applicable":[77],"subject.":[80],"may":[82],"also,":[83,169],"\\nhopefully,":[84],"assist":[85],"motivate":[87],"both":[88,154],"teachers":[89,247,758],"attempt":[93],"recreate":[95],"similar":[96,106],"\\nlearning":[97,166,631],"experiences":[98],"schools":[101,139],"same":[104],"or":[105,404,746],"content":[107],"concepts":[109],"\\nappropriate":[110],"them.":[112],"\\nIn":[113,571],"this":[114,440,490,678],"research":[115,679],"project,":[116],"data":[117],"came":[118],"through":[121],"task-based":[122],"interviews":[123],"\\nquestionnaires.":[125],"chosen":[129],"because":[130,141],"uniqueness":[133],"activities":[135,638],"most":[137,296,330,396,670],"\\nAfrican":[138],"it":[142,174,298,304,399,442],"easily":[144],"accessible.":[145],"Most":[146],"do":[148],"not":[149,604,749],"have":[150,435],"access":[151],"\\ncomputers":[153],"at":[158],"home.":[159],"somehow":[161],"alienates":[162],"them":[163],"modern":[165],"trends.":[167],"They":[168,278,770],"many":[171],"occasions,":[172],"find":[173,295,398],"difficult":[175,400],"grasp":[177],"knowledge":[179],"\\nthey":[180],"receive":[181],"class":[183],"since":[184],"medium":[186],"instruction":[188],"is":[189,198,289,299,315,326,343,394,456,465],"English,":[190],"second":[192],"language":[193],"\\nthem.":[195],"\\nAnother":[196],"reason":[197],"nature":[200],"process":[206],"prevails":[208],"such":[210,426,785],"\\nschools.":[211],"Primary":[213],"Education":[214],"Upgrading":[215],"Programme,":[216],"according":[217],"Taylor":[219],"\\nVinjevold":[221],"(1999),":[222],"found":[223],"out":[224,406],"African":[226],"would":[228,767],"mostly":[229],"listen":[230],"teacher":[233],"\\nthrough-out":[234],"lesson.":[236],"Predominantly,":[237],"classroom":[239],"interaction":[240],"pattern":[241,492],"consists":[242],"oral":[244],"\\ninput":[245],"where":[248],"occasionally":[250],"chant":[251],"response.":[253],"shows":[255],"\\nquestions":[257],"asked":[259],"check":[261,525],"attentiveness":[264],"tasks":[267],"oriented":[269],"towards":[270],"\\ninformation":[271],"acquisition":[272],"rather":[273],"than":[274],"higher":[275],"cognitive":[276,809],"skills.":[277],"tend":[279],"resort":[281],"\\nmemorisation.":[283],"\\nDespite":[284],"fact":[286],"trigonometry":[288,393],"one":[290,327,662],"topics":[293],"challenging,":[297],"\\nnonetheless":[300],"very":[301],"important":[302],"has":[305,351],"lot":[307],"applications.":[309],"technique":[311],"triangulation,":[313],"\\nwhich":[314],"used":[316,345],"astronomy":[318],"measure":[320],"distance":[322],"nearby":[324],"stars,":[325],"\\nimportant":[331],"ones.":[332],"In":[333,710,787],"geography,":[334],"distances":[335],"between":[336,589],"landmarks":[337],"measured":[339],"\\ntrigonometry.":[341],"It":[342,487,626],"also":[344,614,627,755,806],"satellite":[347],"navigation":[348],"systems.":[349,359],"Trigonometry":[350],"proved":[352],"\\nbe":[354],"valuable":[355],"global":[357],"positioning":[358],"Besides":[360],"astronomy,":[361],"financial":[362],"markets":[363],"analysis,":[364],"\\nelectronics,":[365],"probability":[366],"theory,":[367],"medical":[369],"imaging":[370],"(CAT":[371],"scans":[372],"ultrasound),":[374],"\\nother":[376],"fields":[377],"which":[378,663],"make":[379,402,777],"use":[380,600,673],"trigonometry.":[382],"\\nA":[383],"Blackett":[386],"Tall":[388],"(1991),":[389],"states":[390],"when":[392,568,698],"introduced,":[395],"\\nlearners":[397,776,813],"head":[403],"tail":[405],"it.":[408],"Typically,":[409],"trigonometry,":[411],"pictures":[412],"\\nof":[413],"triangles":[414],"aligned":[416],"numerical":[418,461],"relationships.":[419],"Learners":[420],"expected":[422],"understand":[424],"\\nratios":[425],"Cos":[428],"A=":[429],"adjacent/hypotenuse.":[430],"A":[431],"dynamic":[432],"approach":[433],"might":[434],"potential":[437],"\\nto":[438],"change":[439,697],"allows":[443],"learner":[445,464],"manipulate":[447],"diagram":[449],"see":[451],"its":[453],"changing":[454],"\\nstate":[455],"related":[457],"corresponding":[460],"concepts.":[462],"thus":[466],"free":[467],"focus":[469],"\\non":[470],"relationships":[471],"prime":[475],"importance,":[476],"called":[477],"principle":[479],"selective":[481],"\\nconstruction":[482],"(Blackett":[483],"&":[484],"Tall,":[485],"1991).":[486],"along":[489],"thought":[491],"\\ncarried-out.":[497],"\\nGiven":[498],"self-exploration":[500],"opportunity":[501],"within":[502],"Geometers'":[504],"\\ninvestigated":[508],"learners'":[509,792],"understanding":[510,529,563],"10":[518],"work":[519],"\\nall":[521],"four":[522],"quadrants":[523],"on:":[526],"\\n*":[527,546,557,573,584],"What":[528,547],"did":[530,551,705],"develop":[532],"Cosine":[535,566],"\\nan":[541],"angle":[542],"10?":[545,556],"intuitions":[548],"misconceptions":[550,684,715,799],"acquire":[553],"Do":[558],"display":[560],"greater":[562],"\\nSketchpad?":[570],"particular,":[572],"As":[574,585],"ratio":[576],"sides":[578],"right-angled":[581],"triangle?":[582],"\\n":[583],"functional":[587],"relationship":[588],"input":[590],"output":[592],"values":[593],"depicted":[596],"\\ngraphs?":[598],"Sketchpad":[602],"only":[605],"successful":[607],"useful":[609],"activity":[610],"for":[611,621],"but":[613,706],"\\nproved":[615],"be":[617,718,760],"an":[618],"appropriate":[619],"tool":[620,632],"answering":[622],"above":[624],"questions.":[625],"served":[628],"besides":[633],"being":[634],"time-saving":[635],"time-consuming":[637],"like":[639],"sketching":[640],"\\ngraphs.":[641],"At":[642],"end,":[644],"there":[645],"great":[647],"improvement":[648],"terms":[650],"marks":[652],"final":[655],"test":[656],"\\ncompared":[658],"initial":[661],"control":[666],"yard":[667],"stick.":[668],"\\nHowever,":[669],"importantly,":[671],"computer":[676],"revealed":[680],"errors":[682,713,765,779],"\\nand":[683],"learners\u2019":[686],"mathematics.":[687],"anticipated":[691],"ratios":[693],"\\nsides":[695],"radius":[700],"unit":[703],"circle":[704],"discovered":[708],"otherwise.":[709],"\\nany":[711],"case,":[712],"can":[717],"spontaneous":[722],"result":[723],"\\nlearner's":[725],"efforts":[726],"come":[728],"up":[729],"own":[732],"knowledge.":[733,752],"According":[734],"Olivier":[736,753],"(1989),":[737],"these":[738,778],"\\nmisconceptions":[739],"intelligent":[741],"constructions":[742],"based":[743],"correct":[745,784,794],"incomplete":[747],"(but":[748],"\\nwrong)":[750],"previous":[751],"(1989)":[754],"argues":[756],"should":[759,771],"able":[761],"\\npredict":[763],"typically":[768],"make.":[769],"explain":[772],"why":[775],"help":[781],"misconceptions.":[786],"\\nanalysis":[789],"understanding,":[793],"understandings,":[795],"well":[797],"\\nin":[800],"mathematics":[802],"were":[803],"exposed.":[804],"There":[805],"arose":[807],"conflicts":[810],"helped":[812],"reconstruct":[815],"conceptions.":[817]},"counts_by_year":[],"updated_date":"2026-11-09T13:22:03.289127","created_date":"2016-06-24T00:00:00"}
