JOURNAL OF SCIENCE OF HNUE Vol 56, No 5, pp 109 116 SLf D U N G BIEU DIEN T R U C Q U A N D O N G H 6 T R d S U Y L U A N QUI N A P VA N G O A I S U Y C U A HOC SINH T R O N G QUA T R I N H K H A M P[.]
JOURNAL OF SCIENCE OF HNUE Vol 56, No 5, pp 109-116 SLf D U N G B I E U D I E N TRUC QUAN DONG H T R d S U Y L U A N QUI N A P VA N G O A I S U Y C U A H O C SINH T R O N G QUA TRINH K H A M P H A TOAN HOC TnJdng Thi Khanh Pliirdng Trudng Dgi hgc Y Dugc Dgi h.gc Hue E-mail: phuongspli'oyahoo.com Tom tat .Nhflng bii'ni (lii-'ii trUc (juan khong chi mang tinh minh hija nia (•(In (tong vai trd cong cu cho eiiui trinh suy luan toan IK.)C Biii bao cho lliiiy bieu di("Mi fti/c quan dgng vdi sU h6 trd ciia cong nghe thong tin co the giup hoc sinh dua dit doiin va tim kielin cilc ly giai hdp ly tien hiinh say ludn quy ngp va ngogi say Tfl k6t qua nay, chung toi de xuat quy trinh su dung suy ludn quy ngp vd ngogi 'iuy vdi sU ho tra cua cdc bieu dien trUc quan dgng de khdm phd todn hoc Mof dau Suy luan la sir kit ndi nhflng kinh ughicnn vii kiin thflc da cd eua mgt ca nhan vdi viec sfl di.mg cac cpiy tiic, cdc bdng chflng d i di din kit luan, dua cac dfl dddn hay d i giai thich ddnh gid mdt ldi gidi Theo Peirce [1;24], toan hgc cd ba loai suy ludn cd ban: su>- diin, quy nap vk ngoai suy Trong suy ludn suy diin giup hgc sinh ren bn^en tu logic thi suy ludn quy ngp vk ngogi suy lii tiin trinh lam nin tang cho sfl tham eld va khdm phd cac- y tudng mdi .Xghien cflu dudng tiip cdn tri thflc ine3i eua hgc sinh la melt nhflng ehii dk chinh ciia giao di.ic todn hgc nhflng ndm gdn dd>' Ddc biet, ngudi ta thudng xem xe^t nd bdi ciinh sfl di.mg biiu dien trflc quan A'di sfl ho trg eua cdng nghe thdng tin da vk dang cd nhflng tac ddng tich e-flc rd nc'-t din epid trinh khdm phd toan hgc cua hgc sinh Trong bai bdo tdc gid se lam rd hai vdn di sau: Thfl nhdt, suy ludn quy ngp va suy ludn ngogi suy cd eiuan he nhu t h i ndo vdi viec khdm pha tri thflc mdi?; Thfl hai, cae biiu diin tiiie cjuan ddng cd dnh hfldng nhfl t h i nao din qua trinh khdm phd toan hgc ciia hgc sinh bdng suy ludn quy ngp va suy ludn ngogi suy'! 2.1 N o i d u n g nghien cu'u Suy luan quy nap Suy ludn quy ngp dflgc gidi thieu ddu tign bdi Francis Bacon (1501-1020), la 109 IVudiig 4"hi Khiinh Phirdiig eiiui trinh suy luiln nhruii dUii iiidl kit cjuii tc5ng qudt tfl hflu hi-in cac kit qua tUdiig tu e-d dudc- vdi c-iie tnrdng bgi) cbu- bic:-! Khong (-d gi dam bdo gia thu.vcH quy uiip lii c-liilc- (-hiui dung nhirng gia ihu.vet s( dUpc- (iing ed hdn e:d tbcnn nhiiu kit eiuii thi.re- nghicMu dugc- khiing cliiili vii lii|) tfle- bi bib- bd ed mdt phan vi du dugc ciu Til Reid (|2,3|) md lii ciua Irinh my ludn qug ngp gdm ndm buc'Jc tlic>o thfl tfl sau: Thflc- himh vdi c-iie- Iririfng hdp diie bi(;'t: C,)iiaii sat eiii.\- bui I; Dfl dodn rilng (piy luat dd ed the d]) dung c-ho tnrdng hgp tong eiuat; Ki(~Mti tra dit doiin: 41)iig (|iiiit hdii cbr doiiii Suy ludn quy ngp tiiii iigirg(- vdi say ludn ntiy dien Trcnig su\- eli(''u sfl dung c|uy tile tdng eiuiit dii ditoe khdng duih tfl trudc d i dp dung c-ho mgt truelng he.Jp cu t h i thi epiy nap k i kiim e:liflng sfl dung dilu vdi mgt ,s6 trudng hdp eu rlu' ren dfl docin rang cd t h i dp di.ing kit, epid nig- e-licj mgt nhgm ddi tugng Idu hdn Do dd kit luan c-iia suy ludn quy ngp mang tuih md rgng Dfl cloan ve mgt hi(;>n tm.iiig phat triin tlux) epiy luiit t i n g qudt hda mgt sfl kic^n di.ra tren e-ae trfldng hdp dae biet hay di xuat eae giii thuvit khoa hoc tfl nhflng kit eiud thfle nghiem diu la nhflng khilm phd mang liii tri thfle mdi cbo eon ngudi di.ra tren suy ludn ipig nap Vi du Vdi Ciie- kit ciua tuih toiin c-i.i thi: -|-3 - -F3 t -)-3H 5-F7 = 16 ^'il sfl dung suy ludn quy ngp hgc sinh cb the di.r doan coug thflc tinh tdng n sd tir nhien le diui tien la: S.„ - f -1- -I- -I- 4- (2// - 1) — n- 2.2 Suy luan ngoai suy Suy ludn ngogi suy dugc gidi thieu dau tien bdi triit gia ngudi My Charles Saunders Peire(^ (1839 1914) Id eiua trinh suy liuln nlu^m tim kiim hoac Xily di.mg mgt giii thu.\et phu hgp nhat de giai t liic-li e-lio nhflng gi eiuan siit etueJc Theo Magnani l4;19| CO hai locii suy ludn ugoi.ii suy cd biin Ngogi suy "chgn hio nhdm chgn mgt quy tdc i)liu hgp nhdt so cile- eiii\' tdc- cd siin de giiii thie-li cbo trUdng hdp dang e-d Khi khdng cd quy tile- niio \dii kiin thflc liic^n tiii eiia eon ngUdi co the giai thieh dude, mot quy trie- mdi duge- hinh la siin philm cua ngogi suy "sdng tgo^' Nhu vky iigocii suy tcio lii (iic y tudng mdi va giup md rdng tri thflc .1 .losejibson \'ii S .losephson tong kc'^t md hinh vi plie^'p iigoiii suy nhu sau [3,5]: D la melt tdp eile dfl lie^ui (sir kie~ii, eiuan sat ciii da e-ho); 11 giiii thich 1) (niu H dung, sc^^ giiii thie-h D): Khong ed giii thu\et khde- ed t h i giiii thieh D tdt hdn H; Xhu \'il>- H ed Ic'' la clung Can phdn biet ph.cp ngogi suy vk phep quy ngp Ngoiii suy giong vdi quy nap d chg cii hai diu lien quan din \ic;'e- kham phd cdc kit qua mdi Tuy nhign luc epiy iii-ip phat hic^n nhflng quy luat khuynh hudng va thu dugc cac dfl doan thi 110 Sii dimg bieu diin trU.c quan ddng hd tra suy ludn qui ngp vd ngogi suy ngoiii suy khclm pha nhflng sfl kieu mdi va thu dugc eae giai thich bgn la cae- dfl doan, bdi vi kit luan ciia no la kbdng the biet dUdc mgt c-ach true tiep, Cdn luu y rdng, chung ta md rdng kit ludn eua quy nap vuiit qiul xa so vdi gidi han nhflng gi quan sat dugc thi kit ludn dd ed phdu nao mang ban chat eua iigoiu suy Suy ludn ngogi suy ciia bgc sinb ldp hgc todn gan liiu vdi Cile boat ddng nhfl: khao sat toan, lam viee \'di e-iie bdi todn kit thuc md, tdng quat hda, phdt biiu eac bai toan md rgng luiy giiii ciuyit cile viui d i thfle ii Trenig cpid trinh diiy toan, giao vien ddi kbdng de v r.lug nhflng suy ludn xay hoc sinh khdo sdt tren Ccic md binb toclu lu.)e dgng nham plult bic^^u i;i e;le dinh l.y vii coug thflc mdi, dfl dodn ve quy tich ba\- elienu (d dinh, thdm ebi dUa cile ly Ibiiyit dk giai thieh cbo sfl xukt hien mdt khai niem mdi khdng phdi la suy ludn suy die'n, ma chu yiu la ngoai suy Dae- bieM., suy ludn ngogi suy ditae sfl clung melt eacli thirc5ng xuyc'>n cdc boat dgng hdng ugii>- ciia eon ngudi can tim kiim nguyen nhdn, ly giai eae su kien, hien tfldng Sau dil\- la mgt nhiem \'i.i thUc t i can sfl trd giup cua suy ludn ngoai suy 17 du -Mgt ten cudp biin mudn ehdn kho ban tren mdt hdn dao Tren hdn diio cd mot tang da c0t mdc I Idn, mdt cay eg va mgt cdy soi Tdn kho bau cudp bien d i m so bfldc chan di cOt moc thdng tfl tang da den cay eg Ngay d vi tri cav eg, ten cudp di theo hfldng tao vdi hfldng nhin ve phia tang da mgt gde 90" quay ve phia phdi, vdi so cay SOI cay CQ bfldc chan dung bdng sd bfldc chan Hinh Scf dd chon kho hdu da dim thi dflng lai va cdm mot cot moc thfl nhat d \ i tri na\' Ten cu'dp quay trd lai cho tang da va dim so bude ehdn di thiing tir eld d i n cay sdi, sau eld tfl vi tri c;ay sdi di tbex) buc'Jng tao vdi hfldng nhin vi phia tang da mgt gdc 90° quay vi phia trai, vdi so bfldc chan dung bang so budc chan vfla d i m thi dflng lai va cam cdt moc thfl hai d i ddnh dau Kho ban dfldc ehdn chinh gifla vi tri cua hai cdt mcjc (Hinh 1) Mgt thdi gian sau, mdt nhdm nghien cflu tim din ndi ehdn kho ban Cay eg va cay soi van cdn, nhUng tang da va cac cot moc da biin mat va kbdng dk bii dau vit Du vay hg van tim kho bau Hg da lam the nao? 2.3 B i i u d i i n tru'c quan va bi§u dien tri^c quan dong Theo Arcavi [2;217], true quan hda la kha nang, qua trinh va san phdm eua su sang tao, giai thich, sfl dung va phan anh dfla trdn cac hinh ve, hinh anh, sd dau chung ta, tren giay hay tren cac cdng cu khoa hgc cdng nghe, vdi muc dich md ta va giao tiip thdng tin, tfl va phat triin cac y tfldng chfla biit trfldc dd di den vice bieu Cac bieu dien trflc quan nhfl binb vc, dd tin, sd dd, bieu 111 TrUdng Tin Khdnh PhUdng bang dfldc xem hi edug eu dk tn.re c|Uiui hda nhdm giup hgc sinh hic'Mi dirge: cae; doi tugng toilii hoe tifm tu'diig Hiiu dii^'n trUe (|iuin ngiiy luiy kbdiig edu dfldc xem nhfl (hi diuih e-bc) mue- dich minh hga mii edn dfle.K: tbua nhan nhfl bl md1 phan ehiiih eua suy hum, cbie biet la uliUiig suy Imln - eg \-a eay soi mil khdng phi.i thugc vao vi tri tang da mnh \'ic^c kiiuu ngbic^nn giii thuvit vdi biiu diin trflc quan dgng Id d i dimg hgc sinh chi Ccdn keo rg diim C y trgn mat phang vii nhan thiy trung diim G cua doan DE vdn khdng di chuvin Vi tri ehdn kho ban la diim cd dinh G ma dudng thang DE ludn di qua Hgc sinb tiep ti.ie thao tae- tren mdy tinh vdi vie^c keo rg diim C din cac vi tri ddc biet d i thu thdp dfl lieu cho suy ludn quy nap Quan sat biiu diin trudng hgp C trung vdi A (Hinh 4), C trung vdi B, hay C trung vdi trung diim H cua doan AB (Hinh 5) cdc em dua gia thuyit: G ndm trgn trung tri.rc cua AB vk HG = HA = HB G/ JD G E c ~A H B Hinh 4- C triing J 4 H B m nh C trUng Niee kiim nghiem gid thuyit dugc thi.rc hien d i dang vdi cae- cdng cu do dai cua GSP Cd t h i thay, \'di cdc bdi todn hinh hgc sfl dung biiu diin true quan ddng thao tdc dugc trgn may tinh, chi bdng mdt vdi thao tdc keo re ddn gian hgc sinh da cd thi tiin hanh mdt sd lugng Idn cdc thfl nghicfui todn hgc vd thu dugc cdc phan bdi nhanh chdng va chinh xdc, ngn \dgc dodn nhan quy ludt hay dfla eac gia thuyit trd ngn dg dang va cd cd sd hdn nhiiu Sd vdi viec chi kiim tra dugc mdt si lugng hiin ehi cdc trudng hdp trgn gidy but Cung vdi chuyin ddng lign tue cua cdc doi tfldng trgn mdn binh, cdc cdng cii hd trd dfldc tich hgp cdc phdn mim hinh hgc ddng cho phep hgc sinh thirc hien cac phep dac, tinh todn vd nhdn dugc 113 TrUdng Tin Khdnh PhUdng nhiiu hdn cdc dfl lieu bdng so, qua dd phdt hien dfldc tinh bat bien ciia cac dai lugng e-d t h i khdng dflgc: c-hu ,v din ldm vien: mdi trfldng gidy but Nhflng phat hicni niiy la eiuiui Irong va lam ea sd dk hgc- sinh thflc hien qua trinh suy ludn quy ngp vk ngogi suy Khong nhflng t h i cac- bieu die^n l.ruc quan dgng cdn hd trg bcK- sinh dua lii ciic- phan vi du, cung e-i1p ciic v tfldng chflng minh, kiim chflng cac giii thuyit khdng c-d ciln e-fl biu' kham phd c-ilc- kit luan ed \y xem tliiTf nd c:d dang gia cbo c-clc- chflng minh, d i cucii (uiig di din vie(- kiin tcio tri thfl(- todn me3i cho rieing nnnh Thfl vk sai, dfl dodn lug' ciie phan chflng minh, phan vi du la (-a(- veu to ciia khdm phii loan hgc Can dii.v, chung ta da su dung qua trinh niiy nhfl mdt each tu khdng suy eliin md lie'-n epian nhiiu hdn din phep ngogi sug vk quy ngp .Mgt elieu qmm trgng la ngogi suy vii quy ngp khdng tach rdi ma ho trg lan (iiul triiih kham plui todn hgc- Ngogi suy dk xiu'il mgt gia thuvit Quy ngp kiim tra gia thuvit duge ugoiii suy thdng cpia e-ac- thire- nghiem vk tang mflc dg thdnh cdng cdc- pbelj) thfl, nghia la tang mflc- tin cay cua gia thuve'^t -^ Quy nap Ngo^i suy 1 1 1 • Quan sat Thao tac 1 • Gia thuyet H Kiem tra gia thuyet H • L _ Bieu dieu true Do dac tinh toan' quan dong ThLrn^hiem i Dieu chinh H : ; < • ' , > • Ket qua cua qua trinh kbani pba A 1 1 C hap nhan/f • Bac bo H Hinh Quy trinh svC dung suy ludn quy ngp vd ngogi suy vdi sii ho trd cua bieu dien triic quan dpng de khdm phd todn hgc Sd dd trgn minh hga chd quy trinh sir diing kit hgp suy ludn quy ngp vk ngagi suy dk khdm pha toan hgc vdi sfl hd trd eua biiu dien fn/c quan dgng Quy trinh trgn se"- duge minh hga cu t h i hdn trdng mgt vi di.i sau ddy Vi du Trgn cdc e-cinh eua mdt tam gidc ABC bdt ki vi phia ngoai ta dflng eac hinh vudng ABFG, BCDE ACKH Cd nhflng nhan xe'-t gi vi dien tich cua cdc tam gidc BEF, CDK .\GH'! Nde- minh nhdn xet do? (Hinh 7) Ldm vi(;'e trgn md hinh biiu dien tri.re quan eua bai todn vdi phan m i m trg giup GSP, hgc sinh da ludn phien sfl dung suy luan ngoai suy va epiy nap d i khdm phd nhflng mdi eiuan he \e dien tich ciie tam gidc BEE CDK, AGH Cu t h i : Giai doan 1: Ngoai suy 1: Dien tich eae- tam giac BEE CDK, AGH bdng Quy nap 1: Gia thuyit tren dung nhflng trfldng hdp kiim chflng hgc sinh dich chuyin cac diim A, B, C cho tam giac ABC deu Kiim tra lai dfl doan 114 Sii dung bieu dien true quan dgng ho trg suy ludn qui v,gp vd ngogi suy Hinh Hinh bang each dich c-liu>-eii cac diim A.B^C v va sfl dung cong eu tinh dien tich tam giac cua GSP vdi cdc tam gidc BEE CDK, AGH Giai dogn 2: Ngoai suy 2: Dien tich tam gidc BEE bc^ng dien tieh tam gidc ABC Quy nap 2: Gid thuyit tien rd rang dung dich ciiuyin cdc diim A,B,C cho tam gidc ABC \mong tai B Kiim tra liii di.r dodn bdng cong cu tinh dien tich tam gidc cua GSP Giai dog.n 3: Ngoai suy 3: Dien tich cdc tam gidc BEE CDK, AGH khong chi bdng ma bdng dien tich tam gidc ABC Quy Uirip 3: Kich boat cho cdc dieiji A, B, C ehcxy tfl mat phing va tinh dien tich cdc tam gidc ABC BEE CDK, AGH dk cd cdc s6 lieu minh chflng Gia.i dogn, 4-' NgOiii suy 4: Dg xdc minh nhan xe;t tren chi can so sdnh dien tich cua hai tam gidc ABC vk BEE (tUdng tfl cho cdc trfldng hdp edn lai) Dd day BC vd BE cua hai tam gidc na.\' bdng nen cd the"- nghi din viec chflng minh chiiu cad tfl A vk E cua chung cung bdng (Hinh 8) Nhdn tlun' rdng cae- tam gidc BE J vk BAI dflng trgn cdc dfldng cad tfldng flng bdng cd t h i la mgt phflpng an hdp ly, diiu ndy thuc ddy hgc sinh huy ddng kiin thfle da hgc vi cdc tie^u chudn bdng cua hai tam giac vd chgn lfla mdt tieu chudn phu hgp dg gidi ciuyit Ket luan Kiin thflc todn hgc dflgc bgc sinh tiip nlmn theo dfldng tfl tri.rc quan sinh ddng din tfl trflu tugng Sfl di.ing biiu diin tri.rc quan lam can ndi cho cdc biiu diin thi.rc t i quen thugc vd eae biiu diin ki hieu trflu tugng da dugc thfla nhdn la mot each lam mang lai hieu qua day vd hgc mdn Todn Ddc biet, mdt biiu diin true quan ddng tdt cho hgc sinh nhiiu cd hdi hdn d i quan sdt, phdng doan 115 TiUdng Thi Khdnh PhUdng ddt gia thuyit vd kiim chflng, hd trd tich circ cho qua trinh suy ludn quy ngp va ngoai suy Chfldng trinh todn bile: tiung hgc thong eua chung ta hicm dang ddt tdm vao kiin thfle- toiin dfla tren nhflng bieu dign ky hieu, gdn bin vdi suy luan suy dii^ii Ngudi gido vien can tich cvr(- tim kiem nhflng biiu diin triic quan ddng tdt d(' thuc dcay hgc sinh sfl dung vii phiit trien cd hieu ciua ndng Iflc suy luan quy ngp vk ngogi suy cnui trinh hgc toan TAI LIEU THAM KHAO [1] Atocha, A., 1997 Seeking Explanations: Abduction in Logic, Philosophy of Science and Artifical Intelligence ILLC Dissertations Series, Vol 4, Mexico |2| A.rcavi, A., 2003 The role of visual representations in the learning of mathematics Educational Studies in Mathematics, Kluwer Academic Publishers |3| Canadas .M.C & Castro, K 2007 A proposal of categorisation for anlysing inductive reasoning PN.A 1(2) [4| Magnani, L., 2001 Abduction, Reason and Science Procc.s.scs of Discovery and Explanation Kluwer Academic/Plenum Publishers, Nevi' \'ork [5] Rivera, F k Becker, J., 2007 Abduction in pattern generalization Proceedings of the 31st conference of the IME, Vol 4, Seoul, Korea [6] Truong Thi Khdnh PhUdng, 2009 Sit dung bdi todn tim kiim qui lugt cd bieu diin hinh hgc di ndng cao ndng liic suy ludn qui ngp vd ngocii suy cim hgc sinh Trung hgc phd thdng Tap ehi Khoa hgc va Gido due, Trfldng Dai bgc Su pham Dai hgc Hui, so 2(10), tr 108-1 IG ABSTRACT Using dynamic visual representations to support abductive and inductive reasoning of students in the process of exploring Mathematics Visual representations are not used only for the pnrpo.se of illustration, but also play an important role as tools for mathematical reasoning processes This paper shows that the dynamic \isual repre>sentations with the power of information technology have significantly supporte^l students in generating conjectures and finding out reasonable explanations when using inductiAc and abductive reasoning From these results, we propose \ procedure of using inductive and abductive reasoning with the aid of dynamic visual presentations to explore mathematics 116 ... d i i n tru''c quan dong h6 trd suy luan quy nap va ngoai suy cua hoc sinh qua trinh kham pha toan Trong mdi trudng bgc tap vdi sU hd trg eiiii eac bic''Mi dien tri/c quan ddng hgc sinh cd diiu... 112 Sd dung bieu didn triic quan dgng hd tra suy ludn qui ngp vd ngogi suy chflng minh dai sd Bign dign true quan dgng dfldi ddy dugc thiit ke tren GSP nhdm giup hgcsinh giai qu.^•it nhiem vu... giai thieh cbo sfl xukt hien mdt khai niem mdi khdng phdi la suy ludn suy die''n, ma chu yiu la ngoai suy Dae- bieM., suy ludn ngogi suy ditae sfl clung melt eacli thirc5ng xuyc''>n cdc boat dgng