NGHIEN CUU U LUAN m KHAI QUAT HOA VA TRUfU TlTOlNG HOA TRONG TOAN CAO CAP NGUYEN CHIEN THANG Trucfng Bai hoc Vinh Email ncthang200g@gmail com LE TH! NGOC THUY Tnfong Cao ding Su pham Nghe An Email; Ui[.]
NGHIEN CUU U LUAN m KHAI QUAT HOA VA TRUfU TlTOlNG HOA TRONG TOAN CAO CAP NGUYEN CHIEN THANG - Trucfng Bai hoc Vinh Email: ncthang200g@gmail.com LE TH! NGOC THUY - Tnfong Cao ding Su pham Nghe An Email; Uiuy76sp@gmail.com Tdm tdt: Bdi viet de cdp den van de khdi qudt hoa vd trtfu tuang hda Todn cao cap Trong bdi, tde gid phdn tieh: 1/Cdc qud trinh khdi qudt hoa vd trtfu tugng hoa hmh thdnh kiin thtfc noi chung, khdi niem ndi neng Todn cao cap, 2/Nhtfng tro ngai vd edeh thuc vuat qua tien hdnh cdc qud tnnh do; 3/ Cach tiip cdn suphdn logi khdi qudt hoa, trdu tugng hoa ve mdt nhdn thtfc vd mot sd phuang hudng nhdm phdt then ttfng loai khdi qudt hda day hoc kiin thtfe Todn eao cap d bde dai hoe Theo tdc gid bdi viet, khdi qudt hda vd trdu tuang hoa duac xem Id hai thao tde quan nhdt qud trinh hinh thdnh cdc khdi niem Todn cao cap Chung lien quan mdt thiet vdi vd bo sung cho qud trinh phdt smh, phdt trien tri tue cua nguoi hoc, ddc biet doi vdi sinh vien ngdnh SUphqm Todn hoc a bdc dai hoe Cd the xem khdi qudt hda Id chdng dudng md ddu vatrau tuang hda la chdng dudng kit thuc hdnh tnnh tdi chdn li cua mot kien thtfc Todn cao cdp noi chung hay mot khdi niem Todn hoc eao cap ndi neng Ttfkhod: Khdi qudt hod; truu tugng hod; Todn cao cdp, sinh vien, dai hoc (Nhan bdi 30/03/2014; Nhdn kit qua phdn bien vd chinh sua 15 O^QO 17; Duyet ddng 25/04/2017) Dat van de Toan cao d p la linh vUc kien thuc thudc khoa hoc eo ban dao tao giao vien toan Cac kien thdc Toan cao cap ndi chung v l cac khai mem chiing thudng duoc hinh dya tren cae thao tac khai quat hoa {generalization) va trdu tupng hda (abstraction), vdn chua thao ddi vdi hau het sinh vien nhung nam dau bac dai hoc Vi vay, ho thudng nlm cac khai niem dd mdt each hinh thde, de quen va khd ap dung Mat khae, tinh he thdng cua d c kien thde dd se khdng duoc xac lap hoac de bi pha vd neu cac khai niem ca bin chiing khdng duoc gan ket bdi cac qua tnnh khai quat hda va trdu tuong hda O nudc va tren the gidi, da eo nhieu cdng trinh nghien eUu ve khai quat hda va trdu tuong hoa qua tnnh phat sinh va phat tnen tn tue eiia ngudl ndi chung nhu cac cdng trinh cua J Piaget, VV Davydov.,, hay giao due toan hoc nhu cac cdng trinh cua Nguyen Ba Kim, Hoang Chung,., Tuy nhien, nhdng cdng trinh nghien euu van de ITnh hdi Toan hoc cao cap chua nhieu, VI vly, ben canh viec tim hieu nhdng elch tiep can eae khai niem khai quat hoa va trdu tuong hoa ve mat h luan, chung tdi phat tnen va cung cd cac y tudng thu nhan duoc bdi nhieu vi du riit tu thuc ti^n giang day Toan cao cap dao tao smh vien nganh Supham Toan hoc Khai quat hoa va trdu tucTng hda Khai quat hoa va trUu tuong hoa la hai thao tac quan trong tU ndi chung, tU toan hoc ndi rieng cua ngUoi Chung cd mdi hen he mat thiet vdi Nho trdu tuong hda ta cd the khai quat hoa rdng hon va do nhan thuc sU vat, hien tuong sau sac hon Oay la co sd de hmh ele khai niem bac eao, Trong qua trinh ITnh hdi toan hoe, dac biet la linh vUeToan cao d p , cac thao tic ddng vai trd chii yeu xay dung cae khai niem toan hoc Theo Nguyen Ba Kim 111- khdi qudt hod la chuyen tu mdt tap hop ddi tuong sang mdt tap hap Idn hon chda tap hop ban dau bang each neu bat mdt sd dac diem chung ciia cac phan tu tap hop xuat phat Trtfu tuong hod la tach nhdng dac diem bin chat khdi nhdng dac diem khdng bin chat, Tat nhien, su phan biet ban chat vdi khdng ban chat d day mang y nghTa tuong ddi, nd phu thudc mue dich hanh ddng Theo Hoang Chung [2], khai qudt hod la dung tn dc tach cai chung cac doi tuang hoac hien tUong, su kien Mudn khai quat hoa, thuang phai so sanh nhieu ddi tuong, hien tuong, su kien vdi Trong qua tnnh dd, chung ta gat bd nhdng thudc tinh khae cua ehung, nhdng thudc tfnh phan biet cac ddi tuong hay hien tuong ay voi nhau, khdng nghT tdi nhdng thudc tmh dd ma chi khao sat neng nhdng eai duoc tach ra; la qua tnnh truu tuang hoa ChIng han, tu eae trudng hop r i e n g : ^ - , = 3-^7;16 = - 11,20 = 7-^ 13, ta cd the khai quat hoa rang "mdi sd chin bing tdng ciia hai sd le' hoac so chin Idn han bao gio cung viet duoe duoi dang tdng cua hai sd nguyen td le" Theo N Hashemi 13;, khai quat hoa cd th^ duoc xac dinh td mdt quan diem toan hoe vot y nghTa la tim kiem mot hinh anh Idn hon", hay sU xem xet tren cae nhom han che chuyen len su xem xet tren cac nhom lan Hon hoac md rdng khai ni^m den pham vi Ion hon de kham pha nc ','ason Burton and Stace/ 20" Tneo quan diem tan- 'i hoc '•'hd-quC-hoa ca naW\c'a dJa'a cuna SD 139-TWANG VZ017 - 3 NBHIEN COU Li LUAN clu tra ldi cho nhdng su kich thich gidng nhau" (Olson and Herghenhan, 2010) Cung ed the hieu khdi qudt hod la tim kiem d c dang mau, cac moi quan he Idn hon va tao ket ndi cle mde khlc cua tU toln hoc (Stacey, 2006; Mason, Burton and Stacey, 2010) Theo M Mitehelmore [4], bin chat eiia tinh trtfu tUdngtrong toan hoe chfnh la dac trUng khep kin cua nd, tach biet khdi the gidi vat chat va xa hdi Mot ddi tuong toan hoc trdu tuang ehi cd y nghia d he thdng ma nd dupe x-ae dmh DT nhien, nhu Sierpmska nhan manh thi trtfu tuang hod toan hoc - xet d moi cap - bao ham viee Id di cae die diem nao dd v l ilm noi bat cac dac diem khlc NhUng dieu quan trpng II d c ddi tUOng mdl cd hen he vdi mdt he thdng nhat quIn cd the hoat ddng ma khdng lien quan den nghia tfUdc ddciia chung, ductoanhpcsddynghaithuatngd/rh(i/gudt/)dtivl trdu tuang hda theo nhdng sic thai y nghia khae nhau, Moi thuat ngd dd deu bleu thi hai mIt: vda la mdt qua trinh, vda 11 ket qua ciia q u i trinh dd Vi du 1: Td cac khdng gian vecto hai chieu R^ va ba chieu R' quen thudc, chung ta khai quat hoa thinh khdng gian vecto n chieu R" va trdu tUOng hda khai niem khdng gian vecta Khi lam nhU vly, hai doi tuong tinh than khlc dUOc sinh ra: Cli khli quat hda R" v l d i trdu tuong hda V, mdt khdng gian vecto tren trudng Ftuyy, d day can Idu y ring, ele nha t o l n hoe thudng xem khdng gian vecto V tren trUdng F y vda la d l trdu tuong hda vda la cai khli quit hda eiia khdng gian vecto hai chieu R^ va dd viec sd dyng cac thuat ngd eho phii hpp viec sd dung chung toln hoc la Theo Nguyen Phu Lde [5], trtfu tuang hoa II qua dieu quan trong,Trong dd, eae nhl giao dye toan hoc trinh dung tri dc de gat bd nhdng mat, nhdng thudc lai chu xem xet cac q u i trinh nhan thdc lien quan tinh, nhdng hen he, quan he thdyeu, khdng cln thiet va Khai quit hda khli niem R" tdviec mdrdngR' den chi gid lai nhdng yeu td can thiet cho tU R^ den R^.„ cd the dUoe md t l bing elch ap dung cac Nha tam li hoc J, Piaget phan trdu tUong ba quy trinh sd hpe thdng thudng ddi vdi moi toa dd Trifu mde; frdu tuang thUc nghiem; trtfu tuang gid thUe nghiem tupng hda thinh khdng gian vecta V tren trUdng F tiiy va trtfu tugng phdn chieu [6], y la mdt doi tuong tinh thin (mental) khae han, nd dUOc - Trtfu tuang thUc nghiem {empirical abstraction) la xac dmh bdi mdt he tien de Trong d trUdng hop thiJ qui trinh trdu tUong dUa tren cic ddng tie thUc de gay nhat, edng viee don gian chi gdm mdt sU md rdng cac hien tinh {vat tinh va hoat tfnh) So sinh cle hien ciia cac qua tnnh quen thudc, thi d trUdng hop sau yeu tinh dd, lien ket chiing de tao cic suy If Su trdu tuong cau phli cd mdt su to chdc lai kha Idn ve mat tinh than duoc thUc hien khdng phli sd vat hay ddng tie Nhu vay, qua trinh khai quit hda mang dac tinh tu bdc Id ma la qua trinh chu the ddng hda chung tiem tien cdn qua trinh trUu tupng hda mang dac tinh Tuy nhien, sU trdu tUOng mdi chi dien rad ben ngoai, nhay vot qua trinh phit trien nhan thde cua ngUdi chUa dl vao ndi dung sU vat hay ddng tac, hoc Ta se rd hon nda ve dieu qua vi du sau: - TrUu tUOng gid thUc nghiem (pseudo-empirical Wdu2:Xetkhli niem tich Descartes X >cY ciiahai abstraction) la trdu tUpng khdng edn phli dUa true tiep vao nhdng ddng tac tren vat thuc ciia chu the, ma dUa tap hop A'va }', mdi phan td ciia A ' x l ' la mdt cap cd vao cac gil ddng tac dd Nhdng ddng tac nay, mdt mat thiiXU (a.b) ri e X.b& Y Td day de ding khai van phai dua vao d c vat cu the, nen nd van mang tinh quat hda tfch Descartes cua hdu han gdm n tap chat thuc nghiem; mat khae, nhdng tinh chat eiia thao hop A'l A'; , X„ Mdl phln td cua A', x ATn x ^X„ tae khdng phai cd sin suvat, ma hoat ddng ciia ehil the tao Vi vay, nd ed tinh chat ciia sU trdu tuong la mdt bd (•V|,,t2- ,-v„) dd x, e A ' , , / - 1,2 ,,,i., phln chieu J Piaget cho rang loai trdu tUang dien Khi dd, t i p hop / = {1.2.,, 'i} duoc goi la tap cac chi manh nha't giai doan so thao tac cu the - Trdu tuong phdn chieu (reflective abstraction) cd sd Rd rang / cd luc luong hdu han Khi / ed lUc luong nghia la thu nhan duoc tinh trang, y nghia ciia sU phdi vd han dem duoc, tich Descartes ciia ho d c tap hop tri (coordination) thao tac (hinh thdc va ndi dung), |A'^, | a e / | danh sd bdi / cung khdng khd hinh Khai mem phln chieu cd hai ndi dung bd sung cho nhau* chuyen hda de chieu len mdc cao hon nhdng phan td {nhdng thu nhan) riit duoc td eae mdc trdu tUong dung Mdl phan td ciia tieh Descartes T T X, ed dang trudc, hen ket cac phan td da dUOc dua len, hda nhap ehung voi cac phan td d cap ay da cd td trudc Do sU chuyen hda va lien ket dan den cai td lai, td chdc lai, suy xet lai toan bdsoddcac phan tddmdecao, S u d i to da tao "trUu tuang suy li" Ddng tac da trd thao tac la mdt tap hop vd han khdng dem dUOc thi viec dmh Theo G Harel [7], cle nha toan hoc va eae nha giao phli sd dung khli niem dly trUde het nhln xet rang 4 • KHOA HOC GIAO DUC ('^i ':- •'•>,•] ,v, e A',,/= 1.2 , Gil sd / nghTa tich Descartes P I X se khd hon nhieu Ta NGH|£N CUU Li LUAN I moi phan td j ; = (;ci,X2) cua tich Descartes Xj x ATj cd the xem la mdt anh xa x:{\,2] -> X^uXj dd :v(l) = J : , , X { ) = :V, DO do, ta cd the trdu tUOng hda khai niem tieh Descartes I I X, nhU la je/ tap hap cac Inh xa v: / -> I I A', thda man dieu kien x(j) E A'l- Khli niem cung diing cho trUdng hop / cd hdu han hay / cd lde luong vd han dem duoc, Chu y ring / la mdt tap hop vd han khdng dem dUde thi thi each viet phln td ciia tich Descartes Y\^, bdi {^,),^, II khdng chinh xae vi / khdng danh sd dupc Nlm duoc khai niem tich Descartes n A", vdi / II mpt tap hpp vd han khdng dem duoc mdi cd the xay dung dupc khai niem khdng gian vecto vd ban chieu NhOrng t r d ngai qua trinh khai quat hoa va trCru tUtfng hoa Qua trinh dinh nghia mdt khai mem Toan hoc cao cap phli duoc ndi tiep bdi mdt chuoi eae dinh li suy luan cic tinh chit ciia khli mem dd td cac tien de Xet ve mat nhan thde, day khdng chi la mdt qui trinh suy dien ma cdn la mdt qui trinh kien thiet, dd ngudi hoc xay dung cac tinh chat ciia ddi tuang trdu tuang, chIng han nhu ele tien de xac dinh khdng gian vecta phai dim blo cac tinh chat "thdng thudng" cita phep cdng cac vecto va phep nhan vd hUdng dUoc thda man, Tuy nhien, de sau khdng gap mau thuIn vdi co sd va sd chieu cua khdng gian vecto thi he tien de phli dim bao cac vecto ddc lip tuyen tinh tdi dai cua cung mdt khdng gian deu ed cung lUe luong Do dd, khai niem khdng gian vecto V tren trUdng F tiiy y gdm mdt he tien de, dd tien de dau lien quan den phep edng vecto cho V cung vdi phep cdng ay trd mdt nhdm giao hoan, tien de tiep theo neu len mdi lien he gida phep cdng vecto va phep nhan vd hUdng; tien de thdbly chi lien quan ndi bd phep nhan vd hUdng, tien de cudi ciing ndi vetinh chat unita ciia V Thue tien cho thIy ring, vdi he tien de dd de xay dUng khai niem khdng gian vecta tdng quat vda dim blo tinh khdng mau thuan ve mat Idgic toan hoc, vda dam bao khdng gian vecto tdng quatcd day dii cae tfnh chat quen thudc ciia R^ va R^ ma smh vien da biet gilo trtnh hinh hoc phd thdng, Tuy nhien, ddi yeu clu sU pham, sd tien de cd the duoe neu "ddi ra" chang han tien de giao hoan cua phep edng vecto ed the duae suy tU bIy tien de edn lai Mat khlc, qua trinh xay dung cac tinh chat eiia ddi tdang toan hoe trdu tdang nay, ele trudng hop ey the da biet se ddng vai trd vda la nhan td ho trd vda 11 nhln td xung dot tiem tang Chung ho trpvi chung gOi y nhdng tinh chat ed the dung, nhUng mat khae chiing lai gay khd khan tiem tang ngUdi hoe d m thi'y phli chdng mmh nhdng dieu ma dudng nhuhien nhien dung td d c trudng hap cu the dd, hole tin ring cac tinh chat chung cho cac trUdng hpp eu the dupe xet cd the diing mpt each tdng quat cho khai niem trdu tupng Nhu vay, sudt qua trinh xay dUng nly, ed mdt sU xung ddt gida cac tinh chat ciia cac trddng hap cy the ma ngUdi hoc biet v l cac tinh chat cua khli niem trdu tdang mdi xly dUng can dUOC suy luan Idgic tddjnh nghia, Ngoai ra, vi sinh vien da biet qua nhieu tinh chat ciia khli niem cu, nen khai quit hda hay trdu tupng hda chiing, hp khdng biet nen gid lai nhdng tfnh chat nlo dac trung nhat Do dd, khli quat hda hay trdu tuong hda, bude dIu ho cd the gap lung tiing va dan den that bai, Vi vay, viee phai xay ddng lai khai mem hay dieu chinh sd tien de khai mem ban dau da dUa la khdng the tranh khdi Trong nhieu trUdng hop, gilng vien phai cd nhdng goi y can thiet va dung luc, Chang han, sinh vien da rl't quen thudc vdi tinh chat giao holn ciia vi nhdm edng N hay nhdm cdng d c sd nguyen Z, dd, khai quat hda va trdu tupng hda thinh d c khli niem vi nhdm hay nhdm tong quit, sinh vien khdng the "bd qua" tinh chit giao hoan Vi vly, gilng vien phli dua nhdng phep toan khdng giao hoan ma hoc vien da biet, chIng han phep nhan cac phep the bac n (nen xet tap hop S,) mdi hi vpng ngUdi hoc khdng dUa tfnh chit giao holn vao khai mem mdi, Cung vay, khai quat hda va trdu tupng hda vanh cac sd nguyen Z thinh khai niem vanh tiiy y, ho khdng the "bd qua" tinh chl't giao hoan cua cac vinh nay, vi cac vanh ho biet nhuZ,Q, R hay tham ehi cac vanh da thdc Z[x], Q[x], R[x] deu giao holn, Trong trUdng hap nay, gilng vien budc phai goi y ve phep nhan eae ma tran cua cae ma tran vudng cung cap ndi chung khdng cd tinh chat giao holn Ngay ca khai mem mdi da duoc thiet lap, quan tinh qua trinh nhan thde, ngUdi hoc van chua the xda bd duoc nhdng dau hieu khdng bin chit eua khai mem, Gilng vien cln cho hoe sinh mdt sd luong phan vf du de xda bd chung Wdu.3:6tren, ta da xet khli mem vanh tdng quit dua tren su khai quit hda va trdu tUOng hda vanh cac sd nguyen Z Nhung td nhd, smh vien da dUOc khlc sau ket qui neu tfch cua hai sd nguyen bing thi it nhat mdt hai sd dd phai bing Do dd, ho cho rang vdi mdt vanh I'tuy y, neu ab ^ vdi aeV, beVse suy a ^ hole b = 0, Giang vien can giup ho chi ring vinh Z,^ chIng han thi 2.3 = nhung T^ ?^ De eho ho khdng thIy vi du tren dae biet, d n xay dUng khai niem ude eiia khdng Sau dd tim dieu kien ciia n de vanh Z„ co $dl3g-THANG4/2B17>45 Q NGHIEN CUU Li LUAN ude eua khdng, hoac chdng td rang vanh M^(R) cac ma trln vudng cap hai vdi phan td thue ed Ude ciia khdng (chu y ring hV-h 0 0) (0 0' 0 ) Nhung cd le phai den khai quat hda v l trdu tuong hda Z khai niem mien nguyen may mdi xda bd dupe ket luan sai lam "neu tieh ciia hai phan td bang thi it nhlt mdt hai phan dd phai bing O"cho mdt vanh y Mat khae, nhieu khli niem mdi duoc hinh thinh nhd qua trinh khli quit hda va trdu tuong hda, nhung hoac d c khli niem q u i trdu tuong, hoac d c sinh vien khdng nlm duoc cac khai mem hen quan nen hp hieu cle khai niem mdi mdt d c h hinh thdc Do dd, ho thudng de quen va khd van dung chiing de giai d c bli toan lien quan Tuy nhien, nhd d c khai mem duoc tiep thu sau dd, d c khli mem mdi se dupc lam sang td Vi du 4: Khli niem gidi han ciia day: "Day |.v„} dupe gpi la/vd/ru vea neu Vi~ > nhd y ton tai sd nguyen duang n^ cho vdi moi sd tU nhien n > n^ ta cd |-v-«j < s "; hoac khai mem lien tuc "Him sd f- R —• R duoc gpi la lien tuc taix, V f > nhdtuy y tdn tai sd thuc duong eho vdi moi x m l j, S ta cd [/•(.v)-/(.V||) < E" RO rang CIC khai niem rat hinh thde De nam duoe ehung d n hieu rang \a ~h\ bieu thi khoang elch gida hai diem tren true sd, ThUc te phai sau hoc khai mem khoing each ciia "Khdng gian metric" smh vien mdi nam vdng ban chat eiia cac khai niem nay, Cung can nhan manh ring "san pham" cua khai quat hda thudng la cac khai mem toln hoc dUOc xay dung theo phuang phdp kien thiet, edn cac "san pham" cua trdu tUong hda duoc xay dung theo phuang phdp tiende, dd Li thuyet tap hop ddng vai trd cdng cu then chdt viec xay dung khai niem theo quy trinh trdu tuong hda Su phan loai l3) theo cae dinh thdc bac nhd hOn, Thue tien gilng day d mdt trudng dai hoc da nganh cho thay phuang phap thd nhat ehl thich hop cho sinh vien cac Idp Su pham hoac Cd nhan Toan hoc, vi qua tnnh xay dung phai sd dung cdng cu phep the vdn rat xa la vol cac smh vien he ki thuat hay kinh tetai ehinh Vi vay, ddi vdi cac Idp thudc he khdng chuyen sau ve toan, tdt nhat 11 dung phuong phap thd hai Tuy nhien, chi can thdng qua mdt trudng hop dac biet la n - 3, smh vien ed the nam duoc quy trinh xay dung khai niem dinh thdc bacn voin - , n = va n tuyy Idn hon Qua hai vi du tren, ta thay loai quy tnnh dien hmh bdi mdt truong hop dac biet thuong cd hieu qua ddi vol ddi tuong duoc xet den ed tinh hdu han, d day la mdt gia tn n cu the Ddi vdi cac ddi tUong dUOC xet ed lUe luong vd han, can sddung mdt quy trinh cao hon thien vetrUu tuong hoa Mdt su tiep can tong quit nhu vay vda tao phuang phIp khli quit hoa de dang, bdi vi nd ap dung d c quy trinh da biet mdt bdi canh rdng han, vda la budc khdi dIu hudng tdi trtfu tuang hoa hinh thtfc (formal abstraction), nd khdng lien quan den mdt sU tai tao ve mat nhln thdc chii yeu nao Thuc ra, mdt sinh vien da suy nghi ve q u i trinh khai quat hda va xem nd nhu mdt hinh ddng mdrdng tfnh kha dyng ciia phuang phap cu the mdt each cd y thde, thi nd ed the duoc xem nhu mdt hinh thdc trdu tuong hda G, Harel va D, Tall [8] da dung thuat ngd trtfu tUgng hda tong qudt (generic abstraction) de ehi hinh thde trdu tupng hda Nd trang bi mdt phuong phap tiep can cd gia tn dac biet ddi vdi cac sinh vien quan tam den toan hoc dng dyng ban toan hoc hmh thdc (thuan If thuyet), Nd ciing cd the cung d p mdt pha chuyen tiep phu hop giup cho cac smh vien vuor qua trdu tuong hda hinh thde, nhien dieu nly se ddi hdi smh vien phai fd chtfc iai mdt nhdn thtfc mae du dUOc chuan bi tdt han Ngay nay, sy phat tnen v l tinh hdu fch cua cdng nghe thdng tm va truyen thdng giang vien cln khuyen khich sinh vien viet cac chuong trinh bang ngdn ngd may tfnh, bdi vi ngdn ngd cd nhieu cau true tuong tu vdi eae eau true ciia tU toln hoc- t i p hop, day, cap sap thdtu, quan he, ham, ChIng han, ma may tinh the hien quy trinh thue hien mdt qua trinh hdm, bit dau bang viec kiem tra xem dau vao cd thol man cle dieu kien xlc dmh ham hay khdng; dd, duac yeu cau Viet ma may tinh, sinh vien bude phai suy nghi ve cac quy tic cua qua trinh hdm Co the thay hanh ddng lap trinh la mdt qua trinh long quit vi nd tien hanh xay dung mdt cau triic tdng quat hon tU d c trudng hop neng de tien tdi mdt su trdu tupng hoi tdng quat ciia khai niem hdm Ketluan Cd the ndi, khdi qudt hda v l frdu tuang hoa la hai thao tac quan nhat qua trinh hinh cac khai niem Toan eao d p Chung lien quan mat thiet vdi va bd sung eho qua trinh phat smh va phat tnen tri tue eua ngUdi hoc, dae biet ddi vdi cle smh vien nganh Su pham Toan hoc bac dai hoc Cd the xem khai quat hda la chang dudng md dau va trUu tuong hda la chang duong ket thiic hanh trinh toi chan It ciia mdt kien thue Toan cao cap ndi chung hay mdt khai mem Toan hoc eao d p ndi neng Tren hanh trinh nhan thUc dd, sinh vien se gap nhieu trd ngai Nhung neu giang vien biet phln loai va de nhdng bien phIp thfch hap, hose vuat qua cac trd ngai va linh hdi duoc cac tn thue Toan cao d p mdt each ban chat, sau sac va ben vdng TAILIEUTHAMKHAO [1j, Nguyen Ba Kim, (2015), Phuong phdp day hoc mon Todn, NXB Oai hoc Supham, Ha Ndi SD 139-THAHG 4/2017.47 Q HGHIEN Clhl Ll LUAN [2] Hoing Chung, {1978), Phuang phdp dgy hge todn hge, NXB Gitio due, Hh Npi [3] N Hashemi, M S Abu, H Kashefi, K RahimI, (2013), Genelization in the Learning of Mathematics, 2^ International Seminar on Quality and Affordable Education [4] M Mitehelmore, R White, (2004), Abstraction in Mathematics and Mathematics Learning, Proceedings of the 28^ Conference of the International Group for the Psychology of l\Aathematics Education, Vol 3, p 329-336 [5] Nguyen Phu L6c (2010), Dgy hoc hieu qud mdn Gidi tich trang trUdng thdng, NXB Giio due Viet Nam [6] Phan Trpng Ngp (chO bi&n), Nguyen Qtfc Hudng, (2003), Cdc li thuyet phdt triin tdm II ngddi NXB Dai hoc Supham, Ha N6i E7].G Harel, D.Tall, (1991), TheGeneral, theAbstraa and the Generic in Advanced Mathematical Thinking, Fa the Learning of Mathematics, Vol 11 Num 1, p.38-42 [8].D.Tall,(2002),^dvoneedmofhemat/ca/Th/nA/ng, Kluwer Academic Publishers [9] Nguyen HOu Vi&t Hifng, (2001), Dgi so tuyin tinh NXB Oai hoc Quoc gia H i Npi [10] Hoing Xuan Sinh, (2003), Dai sd dgi euang NXB Giao due Ha Ndi GENERALIZATION AND ABSTRACTION IN ADVANCED MATHEMATICS Nguyen Chien Thang - Wnh Unh/ersity Emaih ncthang2009@gmail.com Le Thi Ngoc Thuy- Nghe An College of Education Emaih thuy76sp@gmail.eom Abstract: The article addresses the generaiization and abstraction in advanced Maths The author analyses: Processes of generalization and abstraction when forming knowledge and concepts in advanced Maths; 2/Analysi difficulties and how to overcome the difficulties when implementing the above mentioned processes; 3/Approach in classifying generalization and abstraction in terms of perception, and orientations for developing each types o generalization in teaching advanced Maths at universities In author's opinions, generalization and abstraction are the two most important actions in the process of forming advanced Maths concepts They are closely connected supplemented to each other in the process of intellectual development by learners, particularly by Maths pedagog students at universities It can be seen that generalization is the first stage and abstraction as the final stage in the way the truth of an advanced Maths knowledge in general or an advanced Maths concept in particular Keywords: Generalization; abstraction: advanced mathematics; student; university •KHOA HOC GlAO DUC ... d p them cle thdng tin chap ndi ma khdng giup smh vien nang cao duoc viec nlm cac him y trdu tuong Idn hon Trong qua trtnh hoc Toan cao d p , smh vien td minh tien hanh qua tnnh khai quat hda... sdc quan trong ITnh hdi cae kien thde toan cao cap, nd la ddng luc chii yeu de phat trien nang lue tu trdu tuqng cho smh vien, Nhung qua trinh khai quit hda md rdng eiing het sdc quan trong, vi... hpp viec sd dung chung toln hoc la Theo Nguyen Phu Lde [5], trtfu tuang hoa II qua dieu quan trong, Trong dd, eae nhl giao dye toan hoc trinh dung tri dc de gat bd nhdng mat, nhdng thudc lai chu