C Q NGHIEN COU VAN DUNG THUYET UEN T U A N G VAO QUA TRINH DAY HOC MON TOAN A TRUfilNGPHd THONG PGS TS CAO THI HA Tnrdng Bai boc Sif pham Bai hoc V\il ttwiit ThS PHAN THANH HAI TRTftmi THPT TnfAng Chl[.]
C Q NGHIEN COU - VAN DUNG THUYET UEN T U A N G VAO QUA TRINH DAY HOC MON TOAN A TRUfilNGPHd THONG PGS.TS CAO THI HA - Tnrdng Bai boc Sif pham - Bai hoc V\il ttwiit ThS PHAN THANH HAI - TRTftmi THPT TnfAng Chlnh - Bak NSng Dat van &e Nghien cdu v l ning lUc nhan thdc, cac cO ehl cila qua trinh nhin thdc, cac dudng nhan thdc cua loai ngudi vin dang la van de thu hut dUpc sU quan tam cila nhieu tae gia Cac gia t h u y l t li thuyet ve qua trinh nhan thdc ndi chung v l qua trinh hpc tap cua hpe sinh (HS) ndi rieng van lien tuc dUpc cac nha khoa hoe tren the gidi kilm nghiim IPiaget cho rang: "Qua trinh nhin thdc l i qui trinh ngUdi hpc tao dun^ v i biln ddi cac sO tri thdc thdng qua hoat ddng dong hoa v i dieu Qng cac kiln thdc kT nang da cd eho phii hop vdi tinh hudng mdi." "Nhin thdc l i mdt qua trinh ngudi hpc thieh nghj v i t d chde lai the gidi quan cua ehinh ngUdihpc, "[1].Dl thich nghi va to chde lai the gidi quan eda mlnh, ngUdi hpc phii thyc hifn eac hoat ddng tim tdi tri tue, cae liin tudng tuduy cung nhU li^n ket hoat ddng vdi ngUdi khac Trong bai vilt nay, chiing tdi de cap din mpt sd van d l cOa thuyft lien tu'dng va van dung nd qua trinh day hpc mdn Toan d trUdng thdng Thuylt liSn tudng 2.1 Khdi niim lien tiTdng Theo Ta^en Tiing Viit li^n tudng cd nghTa l i nhan sU vile, hifn tupng nao ma nghT tdi sy vifc, hien tiiong khac cd lien quan [2].Tuy nhien, ndi d i n lien tudng chung ta deu hiiu rang dd la mdt hoat ddng tu eda eon ngUdi Do vay, Tim ti lipc, trUdng phii tiep can lien tUdng van df tU cho ring; TU l i qua trinh thay ddi ty t i p hdp eae hinh Inh, la sU lien tu'dng cac bieu tUOng [3] K K l^lantdndv xem tU nhU la mot qui trinh gdm nhieu giai doan k l tilp nhau, ma hai sd d c giai doan ay la: xudi hien cdc lien ti/dng, sang Igccdc liin tudng va hinh cac gii thuylt Moi quan t i m cua cac nha nghidn edu ta toe dp va mi/cdd liin ket cac hinh inh, d e bieu tupng da cd, tdc la quan tam chu ylu den each tai tad d c mdi li^n tudng Theo cac tac gii, cd bdn loai lien tiifmg la: lien ti/dng giong nhau, liin tudng tuong phan, lien tudng gdn vi khong gian vd thdi gian, lien tudng nhdn qud Tac g i i BCii Van Huf chia lien tudng bdn loai: lien tudng gan ve khong gian vd thdi gian, lien tudng gidng vi hinh thii hogc ngi dung, lien tiidng trai ngUOc Hen tudng nhan quo Tac gia cho rang lien tudng ed vai trd rat quan trong ghi nhd va nhd lai Nha tam li hpc PA Sevarev da nghien cdu rat ti mi nhdng moi Hen tudng khdi qudt ddc ddo vd vai tro eda chiing dgy hgc Ong chi rang, nhdng moi lien tudng khai quit bao gdm ba kilu cO ban: nhiing lien tudng dupe biln doi mdt nda, nhOng lien tudng - bifn thifn va nhutig li#n tu'dng cy the - bien thi§n 2.2 Ndi dung chinh eda tiiuy& lidn tiidng Cd the neu tdm luocdc lu^n diem ehinh eda thuyft lif n tucmg nhu sau: -Tdm/('(hieu theo nghTa la ylu t o y thdc) dupe cau td eac d m giie Cic d u eao hon nhu bleu tupng, y nghT, tinh dm, xult hifn nhd lien tudng cac elm giac Ndi each khac dudng hlnh tam li eon ngudi la lien kit cic eim giae v i eic •y tudng 4-KHOAHKGUoni - Diiu kiin di hinh d c lif n tudng l i sy gan gui eda cic q u i trinh t i m li - Sullen kit d c cim giac va 5? tUdng d l hinh y tudmg mdi khdng phai la sy k i t hpp gian ddn d c eim giae hoac ^ tudng da cd, ma gidng nhu su kit hpp c£ia cae nguyen to hoa hpc de tao hop chit mdi, nhu Oxy v i Hidro kit hdp vdi di tao nudc - Cdc moi liin tudng bi quy ^ n h bdi su linh hoat cQa d c d m giac v i cac y tUdng phin dUdc lif n iisdng va t i n sd nhic lai cua ehiing kinh nghifm NghTa la cac d m giac hay yi tudng song ddng hdn, thudng xuyen hdn thi tao cac d m giae va d c y tUdng manh hdn - Cdc lien tUdng dupe tilnh theo mdt sd quy luit sau: Quy ludt tuong iii V thdc cua chdng ta de dang di td mpt y tuc^g niy sang mpt -^ tUdng khac tUdng tU vdi no; Quy ludt tt/Ong can: Khi nghT d i n mdt vat, ta c6 khuynh hudng nhd lai nhflng vat khac da trii qua cfing mdt nPi va cCing mdt thdi gian Vf du: NghT dfn mdn qua, nhd d i n ngudi tang qui Cd the dien tUong cin theo khdng gian, th£^ gian va theo tUong phin gifla d c d m giac va y tudng; Quy ludt nhan qua: Khi cd mdt j? tUdng vf kit qua thudng xuat hien d c ; / tudng la nguyen nhan d i n den k i t qui do- Trong cic quy luat tren, quy luat nhin q u i cd vai trd dac biet quan trpng qua trinh nhan thdc va phat trien trf tuf Su phat triin nhin thdc la q u i trinh tieh luy cac mdi lien tUdng Sy khac bietvl trinh dp nhan thdc dupe quy vf sd lUdng cic mdi lien tudng, ve tde dp hoat hoa d c mdi liln tu&ig - Cac lien tudrig ddpe giii thich v l phUcffig dien sinh li than kinh la sy hinh v l khdi phuc d c dUdng mdn than kmh nhd cie kich thieh Nha t i m If hpe K.K Platdndp da tdm tat cac^iai doan ciia q u i trinh tU duoc the hifn bing so (Xem sd 1) Sd cho thay, tie gia K.K Platdndp xem tU nhu la mdt q u i trinh gom nhifu giai doan k l tiep C I ^ Xuit h i ^ cac lien tinhig Sang I9C cac lien tucmg v i hlnh ihanli gia IhuySt Sodo 1: Cdc giai dogn cOa qud trinh tuduy NGHllNCOuD m i hai so d c glal doan l y la: xult hien lien tUdng, sang Ipc lif n tucmg v i hinh thinh g i i t h u y l t Nhii vly, vife xuat hi&n cac liln bidng la giai doan mau chdt quan trpng de ngudi co the hoan thien dupc q u i trinh tu nhat dinh Moi quan he giOa sii lien tiT^g - qua trinh tU v l k i t noi tri thiJc troiig qua trinh day hoc mon loan d tntdngphS thong Khi xet ve mdi lifn he gifla tri thfle va tU duy, M.Crugliac da nhan manh vai trd cua ti/ viec kit noi tri thdc dd cd vdi tri thdc mdi cdn tim t i c g i i etio ring: "Dya vao cii da b i l t va nhd tu duy, HS suy dupc tri thdc mdi" Tri thflc va tu g i n bd vdi nhu la sin pham di ddi vdi q u i trinh, ITnh hdi tri thflc v l mdt ddi tflcffig n i o thi la san pham, la Icit qua cua qua trinh trien khai logic hifn tupng ay tfl duy, tri thflc dupc bpc 16 v l hinh tfl Mat khac, nhiing tri thflc da ITnh hdi dupe lal tham gia vao qua trinh tU nhfl I I mpt y l u td eda tfl de t i l p thu nhung tri thflc mdi khac [4] Nhfl vay, tfl dfi tfl he thdng tri thflc da bilt den d c tri thflc mdi can tim Ndi each khac, tfl kit ndi he thdng tri thflc da bilt d i n d c tri thflc d n bilt Khi dung trudc mdt v i n de (khai niem, djnh li, quy tac, bai toan, ), n l u HS cd nang Iflc lien tfldng tdt thi sf lien tudn^ dflpe nhieu khai niem, djnh if, quy tie, bai toin phu de giii quyet vin d l don giin, nhung neu cac em khdng lifn tfldng dupe hay chi lien tudng dupe it khii nifm, mfnh df, dinh li, quy tie, bai toan phy thi v i n df sf bi be tie Cd the minh hpa ndi dung tren thdng qua mdt sd vi du sau day: Vidu 7;Cho hlnh hdp ABCD.A,B C,D, Dyng dudng thing MhJ cho M thude AC , N thuoc B D va MN// A,D ^ n Vdi b i i t o i n nay, HS cd the lien tudng den nhung tri thflc sau: ' - Hi qud: Nlu hat mat phing phan bift cung song song vdi mdt dfldng thing \-*['— :•'—'\-:::!f^ f'\ th'i giao tuyen cua chung (neu ed) cung song song vdl dudng thing Nhfl vay, de dyng dflpc MN, HS se tim giao tuyln ITinh efla hai mat phang song song vdi A^D va lan IflcJt ehfla AC,, B,D, " - Quy fdc/i/nh/idp.Tflgil t h i l t HS cd the lien tfldng: MeAC^.NsB^D^ suy AM ^xAC[,B{N^yB{Di:MNIIA^D^, hay w^ = XAp v i i p dung quy t i c hinh hop bieu thi ba N chinh la I n h cila M qua phep chilu song song dd Nang Iflc Hen tfldng va huy ddng kiln thfle cd vai trd quan trong tien trinh nhan thflc v i phat trien tri tue cua HS Nlu ngfldi hoc khdng cd nang lyc thi kha riang giii quyet van df se bj han che, each nhin v l toin hpc sf cue bp, rcH rae Tuy nhien, dflng trfldc mpt v i n d l cu the, khdng phai luc nao mpi su lifn tfldng va huy ddng deu cd ich cho viec giii quylt van de, d n chpn Ipc thdng qua phep thfl sai d l tien tdi mdt sfl lien tfldng v l huy ddng phu hop nhat V/du2:Cho hlnh chdp S.ABC cd day ABC la tam giac vudng tai C Canh huyen AB=c Cic canh ben nghieng d i u tren day mdt gdc a Hay tfnh b i n kinh mat d u ngoai tiep hinh ehdp theo c v l a Trong vi dy nay, nhflng tri thflc ma HS dflpe chuan bi la: -True dfldng trdn ngoai tiep tam giac ABC ia dfldng thing vudng gdc vdi mat phang ABC va di qua tam H cfla dudng trdn ngoai tiep tam giic ABC -Trong bai toan cac canh ben SA, SB, SC nghieng deu tren diy nen hlnh ehieu cua S Ifn mat phang dly ABC trung vdi tam dfldng trdn ngoai tiep tam giac ABC, hay ndi d c h khic SH la trye dfldng tron - Do tam O cua mat d u phii thude true dfldng trdn nen mat phang (SAB) se ehfla tam O cua mat d u va mat phSng niy d t mat d u theo mdt dfldng trdn Idn, ehinh la dflcmg trdn ngoai tifp tam giic SAB Tfl he thong tri thflc ndi tren, nhd hoat ddn^ tfl duy, cy the la nhd thflc hien eic thao tac phin tich, tong hpp, so sanh HS biet chuyen viec tim ban kinh mat d u v l tim ban kinh dfldng trdn ldn Dd chfnh la ban kinh dudng trdn ngoai tilp tam giic d n SAB ed gde d day bang a va canh AB=c S Hinh 1.2 Hmh 1.3 Do tam giac SAB ein dinh S va gdc d dly bang a Suy 'w(_,r-2a) vecto ^AM,~B[N.mi qua ba vecto khdng dong phang 2sm(!r-2a) Isinla Ket ndi tri thdc theo quan ^ I m li thuyet hoat vdi AB = a,AD = biAAi = c , ta co; dpng Theo Nguyin Ba Kim [5], tri thflc vfla l i dilu kien lM=x'ACi=x{^+b+Tj vfla la ket q u i cfla hoat dpng Ta ed the hiiu nhin djnh theo nghTa fri thdc dong vol tro Id yiu to de dieu chinh va djnh hUdng hogt ddng Hoat ddng d dly dflpc bpc Id ehu ylu l i hoat dpng ciia chCi t h i xim nhap vao Viec xic djnh M, N chinh la viec tim dflpc gia trj x, y ddi tticmg d l lam bdc 16 doi tflpng Ddi tfldng hoat dpng day hpe toan l i cac mdi li^n he toan hpc, cic - Phip chiiu song song: Vi UN I IA^D.M e AC^,N e B^D^ quy luit toan hoc NhU vly, hoat dpng tao nhflng sin nen cd the sfl dung phep chieu song song theo phflOng pham la d c tri thflc mdi, bao gdm cac ddi tflpng t o i n DA len mat phing (A,B,C,D,) de tim dilm N Chu y rang hoc mdi, cac quy luat toan hpc mdi dot vdi HS Nhfl vly StflZB-THJ^S/2016>5 C Q NGHl£NCaU_ tfnh chat cua d c bp phin phing thich hpp, tfl dd huy ddng eae kiln thflc da b i l t hinh hoc phing d l giii quyet van di cOa hinh khdng gian Vai trd ^ nghTa cfla phflong thflc 2: Thflc hien phflong phip niy tao sfl k i t ndi gifla vile hpc tap, nghiin eflu hinh hpc khdng gian vdi eic kiln thflc hlhli hpe phang da biet Tfl tranh dflpc sy dflt quang vl phflong diln t i m li cua HS HS nghien eflu hinh hpc theo phfldng pliip II co hdi d l t i l p can phit hiln cldi giii quyet van de khdng gian nhd viec ehuyin hda cich gill bli toan khdng gian v l b l i toin phing Ching hgn: - Chuyin b i i toan tim t i m v i b i n kinh mit c3u v^ bai t o i n tim t i m v l b i n kinh ctla dudng trdn Idri, d mdt mat phing xac djnh di qua t i m - Chuyen hda b i i t o i n quy tich khdng gian v l bai toan quy tich mat phing - Chflng minh mdt djnh If eila hlnh hpc khdng gian dflpc chuyen ve van dung djnh li da bilt mat phing PhiAfng thdc 3: Luyen t i p eho HS ehuyin hda cic lifn tflc^g tfl ddi tflpng niy sang ddi tflpng khic nhlm d u true lai hlnh thflc va ndi dung cOa van de d n nghifn eflu de de dang xic lap mdi lifn hf vdi d c kiln thflc da cd, tfl huy ddng dting dan d c kiln thfle d l gill quylt van di ? nghTa v i mijc dich cfla phfldng thflc la: Thtic hifn bifn phap nhlm giup HS khac phyc khd khin ehiia xle lip dupe mdi llin hf giu'a ddi tfldng can kham p h i vdi cic kiln thflc va phfldng phip da cd cCia HS Thuc hifn phflong thflc niy nhlm giiip HS biln doi Vfdu 3: Cho hlnh lip phfldng ABCD.A,B,C,D| Got thdng tin In ehfla v i n d l d n gill quylt t?o co hdi cic dilm M;N;P lan Ifldt 11 cac trung dilm cua d c canh d l HS xem xft nhin nhin d c van d l eCia hlnh hpc khdng AD,BB,,C,D, Hay dUng thilt dien cOa hlnh lip phflong gian theo nhilu d c h khac nhau, khai thae cie mil lifn tao bdt mit phing (MNP) hf phd bifn day hpc hinh hpc Phuong thflc nayse Khi giii bai toan niy, HS gap khd khan II xic dinh tao cd hdi de giio dye tfl linh hoat vifc phat giao cila mat phing (MNP) vdi mpt m i t cua hinh lip hi§n each giii quylt van de phUdng d l tfl xac dinh giao cua m i t phing vdi Phuang thdc 4: Khio s i t nghifn eflu cic dang sai d c m i t cdn lai Tri thflc cdi ngudn d i ehuan bj cho HS lim cCia HS day hoc hinh hpc khdng gian va tao co bao gdm: hdt sfla ehfla cie sal lam cho HS ho hoat ddng huy - Quy trinh tim giao tuyen ciia hai mit phing ddng kiln thflc giii quylt eic vin de dat [6] - Quy trinh tim giao eda mdt dudng thing vdi mdt Cie li thuylt day hpc hifn dai die bift nhin manh mat phing day hpe di ngfldi hpc tfl mlnh pliit hifn kiln thflc rncS - Cieh xic djnh mat phing: Mat phing di qua ba thdng qua hoat ddng tim tdi tri tuf: Phit hif n miu thuan, dilm; mat phang xac dinh bdi hai dudng thing d t nhau; xac dinh nhifm vu nhin thflc, d l xuat phin doin, gii mat phSng xic dinh bdi hai dUdng thing song song; m i t thuylt hoat ddng xim nhap vio vin d l , xam nhip v&o phing ehfla mot dfldng thing va mdt dilm khdng thude ddi tfldng thdrig qua sfl dung cic thao tac tfl duy, hojtt dfldng thing dd Tfl do, ed t h i giiip HS huy ddng kiln ddng ddng hda, dieu flng di thfeh nghi vdi bdi cinh mdi, thflc ndi tren thdng qua he thdng cac cau hdi hoic d l mdi trfldng mdi v l tfl dd t i l p nhin tri thflc mdi d c yf u clu ddi vdi HS, eu t h i nhfl sau: Cae hoat ddng tim tdi tri tuf neu tren dflOc dieu - N§u quy trinh dflng giao cua dudng thing MN vdi chinh bdi hf thdiig tri thflc v i kinh nghifm d i cd dflpc to mat phing A,B,C^p,? ' ehflc tuin tliCi theo eae quy luat nhin thflc Hoat dfng tim - Xic djnh mat phing (a) ehfla MN, dd l i mat phing tdi tri tuf eung cd cd sd tfl hoat ddng tu di tfl cii (15 ehfla hai dfldng song song BB, v i MM,, dd M, l i bift den d i ehfla biet v i t i n bilt Tie gii Dio Tam da quan trung dilm efla eanh A,D nifm: "Tim tdi tri tuf II hoat ddng eila HS hfldng suy nghi - Xic cnnh giao tuyen cua mat phing (a) vdi mat cfla minh vio ddi tflpng dang tim hieu; hp ed nhu clu tim phing diy (A,Bj^C,D), l i dfldng thing B,M^ hiiu ben chinh doi tflpng giy nfn, hfldng ho - Trong mat phing (a) xae ^ n h giao dilm S cua vio hoat ddng tieh cflc nghifn eflu phan tfeti ddi tUpng dfldng thing MN v i B M,? d l tim each giit quyft van d l HS tilp thu tri thflc khdng Tfl dd y§u d u HS xic djnh giao cfla mat phing phii true tilp qua truyen dat giin dcm m l thdng qua (MNP) v l mat A,B C|D, cfla hlnh lap phuong Nhd d e duemg vdng nhd nhflng hinh ddng tri tuf v l ^ao tie t\i kiln thfle d i biet HS tim dflpc giao II doan PQ (xem hinh can thilt, dupc dieu chinh bdi tri tiifle v i kinh nghifm 1.4) Cic giao diem cdn lai HS cd t h i tfl xic djnh nhd hf da ed nhlm tim difu ehfla bllt"[6] thdng d c kiln thflc da bilt neu tren v i tim duoc thiet TTieo G Polya:'Tat c l nhOng tfl lifu, ylu t l phu cac difn l i luc giie dfu MTNQPR, Q;R;T Iln Iflpt la djnh li sfldung q u i trinh giii bai toin dflpc lly ttf trung dilm cua d e eanh B C,; D,D v i AB dau? Ngfldi giii da dflpe tich liiy kiln thdc dd in Phmmg fhut2; Chuyen vife nghifn eflu, tim tdi cic tinh chit cila hinh khdng gian qua vifc nghten eflu d c (Xem tiep trang 1%': quy lu|t hoat ddng cua chu t h i l i nhin td kit ndi tri thflc da cd vdi tri'thfle d n tim Theo Jean Piaget: Cic tri thflc phat sinh tfl hoat ddng, khdng theo nghTa d i p lai lien tUdng gidn dan m i theo nghTa siu sic hon nhilu, nghTa ddng hda thue tai vao nhflng phdi hdp d n thilt v i tong quit cua hoat ddng Bilt mdt ddi tupng la t i c ddng Ifn nd va thay doi nd de nim bit nhflng co c h l cua biln ddi gin liln vdi chinh nhflng hoat ddng bien doi Vay bilt la ddng hda thflc tai vio d u true cua nhflng biln doi, v l chfnh tri t u i xiy diing nhCffig d u trdc nhu la sfl keo d l i true tiep cua hoat ddng 111Tie gii Dio Tam d l quan tam dfn hoat ddng k i t not tri thflc da ed va tri thflc d n tim qua d c phuong thflc hoat ddng biln doi doi tUtffig sau day: PhMmg thiic 1: Luyfn tap cho HS xic djnh tri thflc cdi ngudn efla tri thflc cin tim an ehfla eac v i n de dfla nhlm dinh hfldng dflng cho hoat ddng xam nhip ddi tflpng nghifn eflu Vife xac djnh cie tri thflc cdi ngudn d l lim sang td quan hf nhan q u i gifla tri thflc da bift v l tri thflc d n tim; cie tri thflc cdi ngudn ddng vai trd dinh hfldng, dilu chinh hoat ddng xim nhap vio ddi tflong lam sing td eac ddi tfldng toin hpe, d c quan hf can khim phi Dd la cd sd de HS hoat ddng chilm linh tri thflc Phflong thflc niy cd thf thyc hien day hpc cac djnh If, quy tac v i day hpc giii bai t i p toin Giao vifn ed t h i xay dUng hf thdng d c d u hdi sU pham, giup H^5 huy ddng kiln thflc va ^ n h hfldng hoat ddng giii quyet vin d l 6*KH0AH0CGlA0DUC NGHIEN CDU U I den t o i n hoc xfl If dflpc thong tin va nhd cic khii nifm, Hoing Doan Huy - D i o Thj Oanh - My Giang Sdn, (2015), cdng thflc djnh li, quy tic mdn Toin); NL tinh Ddo tgo nghiep vu sii phgm theo dinh hudng hinh thdnh toin, gill t o i n (thflc hifn cle phep t o i n bing sd va c l ndng li/e nghi cho sinh viin cdc trudng dgi hge sfl biln dfii d e bilu thflc dai sd); NL tfl toin hpc (khi pham, NXB Oai hoe Sfl pham Ha Ndi nang phin tich, tfing hop, lip luin logic, phin bifn v i [2] Vu Qude Chung, (2015), Bdi dudng ndng lucphdt sang t^o); NL giao tiep toan: NL t h i hifn quan diern efla triin ehuang trinh Idp hgc cua gido vien tieu hgc day HS q u i trinh hpe toin, bao gom NL giao t l l p v l hgc mdn Todn Kt ylu Hdi thio khoa hpc qufic gia Dao toin, NL giao tifp toin, NL giao tifp vdi toin; NL tao v i phit trien ngufin nhin Iflc giio due tiiu hoc, NXB van dyng toan hpe vio thflc tien (van dung toan vio diH Hdng Dflc tr 195-203 sdng, giii quylt d c b i i tctan, van de thifc tiln, cd nhilu [3] Bd Giio dye va D i o tao, (2012), PISA vd cdc dgng ngfldi gpi la NL md hinh h o l t o i n hpc); NL sing tao toan hpc (cd d HS^gidi toin, cae n h i toin hpe, la kh'l nang cdu hdi, NXB Gilo due Vift Nam [4] Hoang Hoa Binh, (2015), Wdn§/flcvdc