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V E V I E C D A Y H O C T O A N S O C A P 6 K H O A T O A N C A C T R U O N G D A I H O C S O P H A M • • • O GS TSKH D 6 BUG THAI TS NGUYEN ANH TUAN* 1 Vai tro cua viec day hoc Toan sa cap (TSC) Tron[.]
V E V I E C D K H O A T O A N A Y H O C T O A N S O CAC T R U O N G D A I H O C S O • • O C A P P H A M • GS TSKH D BUG THAI - TS NGUYEN ANH TUAN* Vai tro cua viec day hoc Toan sa cap (TSC) Trong chuong trinh ddo tgo sinh vien (SV) a khoa Todn cdc trudng dgi hgc su phgm (TDHSP), viec nam vung kien thuc TSC se giup SV hieu rd han cdc bd mdn Todn khdc Thyc te dgy hgc cho thdy, SV gap nhieu khd khan hgc tap cdc mdn ca sd cua dgi so truu tugng (Dgi so dgi cuang, Li thuyet module, Li thuyet Galois ), vdn de ndm vung vd hieu bdn chdt cdu true todn hgc dgi sd truu tugng cdn hgn che Viec hieu sdu sdc cdc khdi niem, chdng hgn nhu khdi niem tap hgp sd, tap hgp da thuc chuang trinh todn phd thdng se cd «tdc dung ngugc", giup SV hgc tap tdt han cdc mdn hgc ve dgi sd truu tugng Mat khdc, cd nhieu khdi niem TSC chi hieu dugc chinh xdc su dyng nhung cdng cy mgnh cua todn hgc hien dgi Ngodi ra, cdc bdi TSC hinh thuc phdt bieu hay phuang phap gidi khdc nhung cd cung bdn chdt todn hoc Do vdy, cd each nhin tu todn hgc hien dgi se giup SV hieu rd han chuang trinh todn phd thdng Can cu vdo chuang trinh todn d phd thdng ke tu bgc tieu hgc den trung hgc phd thdng, viec dgy hgc TSC d cdc TDHSP cdn ddm bdo cho SV ndm dugc todn bd chuang trinh todn mdt each chinh xdc, dung bdn chdt mdt chinh the thdng nhdt cua todn hgc, cdc gido trinh cdn phdi ren luyen dugc tu cho SV, tgo tien de de sau SV ndm dugc phuang phap dgy hgc mdn Todn d phd thdng Ben cgnh dd, viec dgy hgc TSC cung phdi ddm bdo cho SV biet each xdc djnh bdi hgc chuang trinh todn phd thdng thdng gua bd ba toa dd: - Toa dd thu nhdt Id vi tri cua bdi hgc tren tryc sd, md td tien trinh xdy dyng chuang trinh todn phd thdng; - Toa thu hai Id vj tri cua bdi hgc todn hgc hien dgi; - Toa dd thu ba Id vi tri cua bdi hgc tren tryc sd trinh bay Ijch su hinh thdnh he thdng tri thuc todn hgc cua lodi ngudi Dgy hgc TSC d cdc TDHSP Id viec Idm rdt cdn thiet, khdng nhung giup SV hieu sdu kien thuc todn phd thdng md cdn biet van dyng vdo kien thuc todn hoc hien dgi dugc gidng dgy tgi khoa Todn cdc TDHSP Tuy nhien, chung tdi cho rdng, viec dgy hgc TSC d cdc TDHSP cdn dugc tien hdnh theo each nghien cuu vd van dyng nhung cdng cy mgnh cua Todn hgc hien dgi Ve mat khoa hgc, cdc ket qua cua TSC dd mang lgi nhung cdng cy huu ich cho toan hoc hien dqi Chuang trinh TSC a khoa Todn cdc TDHSP hien - Ndi dung chuang trinh mdn Hinh hoc sacdp chu yeu gdm: Cdc he tien de cua hinh hgc sa cap, xay dyng hinh hgc bdng phuang phap tien de; Hinh da dien va hinh loi; Mot vai vdn de ve dgc hinh hgc (do dai, dien tich, the tich); Cac phep bien hinh mat phang Viec gidi thieu cho SV he tien de Hilbert vd mdt vdi he tien de khdc cua mdn Hinh hgc so cap Id het sue cdn thiet, dqc biet Id ddi vdi SV su pham todn Day Id dieu thdnh cdng nhdt chuong trinh hinh hoc sa cap - Phdn li thuyet ve cdc phep bien hinh ddnh cho viec gidi thieu phep bien hinh phang quen thudc, dang chinh tdc cua cdc phep ddi hinh vd phdn chieu mat phang Theo chung tdi, nhung kien thuc nen duqc gidng dgy Hinh hoc Euclide khdng gian E Nhu the se vua tranh duqc sy trung lap vua cd the dua cdc each chung minh ngdn gqn dya tren dang chudn Jordan cua ma trdn vudng cap Myc tieu chinh chu de ve cdc phep bien hinh phdng Id ren luyen cho ngudi hoc kT ndng gidi cdc bdi todn hinh hoc bdng phuong phap bien hinh Day cung Id mot ndi dung quan vd cd y nghTa chuang trinh hinh hoc sa cap - Hinh loi Id mdt nhung ddi tugng quen thudc vd quan nhdt ciia Hinh hoc Euclide Da gidc loi vd hinh trdn duqc gidng day chuang trinh todn trung hoc ca sd; da dien ldi vd * Tnfdng Dai hoc sir pham Ha Npi Tap c h i Giao due so (k / n hinh cau dugc giang dgy chuang trinh toan trung hgc phd thong Tuy nhien, co nhieu tinh chdt hinh hgc cua hinh loi dugc thua nhan hoac chung minh dya vao tryc gidc (nhieu la rdt hien nhien) Vi the, sy chinh xac hoa khai niem ve hinh loi de tu dua chung minh (ve mat toan hoc thuan tuy) cac tinh chdt hinh hgc cua no la mot viec lam can thiet nhung khong don gian Cong viec can den nhung cong cy nhu: Tdpd, Li thuyet do, Giai tich ham, Li thuyet nhom va tac dong cua nhom Giao trinh Hinh hgc so cap hien trinh bay ve phan hinh loi sa luge, chua tqo duqc sy ket ndi vdi kien thuc toan hoc hien dqi Theo chung toi, day la chu de can duqc bien soqn Iqi ve ca hai phuang dien: neu duqc tdm quan cua cdc hinh ldi Todn hoc vd Khoa hoc may tinh (Computer science); Id «cdu ndi" giua todn hoc hien dqi vdi todn hoc phd thdng - Ndi dung chuang trinh mdn Dgi so so cap chu yeu bao gom kien thuc ve: Ddng thdc vd bdt dang thdc; Dgi cuang ve ham sdso cap vd thi cua cdc ham sd; Da thuc tren cdc vdnh sd vd phdn thuc huu ti; Ham sd lugng gidc vd da thuc lugng gidc; Phuong trinh vd bdt phuang trinh Uu diem cua chuang trinh mdn Dqi sd sa cap Id dd gdn chdt vdi ndi dung dgi sd vd md dau ve gidi tich chuang trinh todn phd thdng He thdng bdi tap dd gdp phdn ren luyen kT ndng gidi TSC cho SV su pham Tuy nhien, cd the nhdn thdy, ndi dung chuang trinh mdn Dgi sd so cap cd phdn cd dien vd thieu sy gdn ket vdi cdc mdn hoc cua dqi sd hien dqi Vi the, chua tqo duqc cho SV cdi nhin tu Dqi sd hien dqi xudng Dgi sd sa cap duqc gidng dgy cdc trudng phd thdng Td chuc day hoc TSC d khoa Todn cdc TDHSP Hien nay, viec dgy hoc TSC d khoa Todn cdc TDHSP thudng theo mdt hai hinh thuc: 1) Hoan todn bd mdn Li ludn vd phuong phap dgy hgc todn ddm nhdn; 2) Bd mdn Hinh hoc gidng dgy Hinh hoc sa cap, bd mdn Dqi sd day Dqi sd sa cap, bd mdn Li ludn vd phuong phap dgy hgc todn ddm nhdn day hoc nhung phdn li ludn vd phuang phap dgy hoc nhung ndi dung cy the chuong trinh todn phd thdng Hinh thuc thu nhdt cd uu diem Id thdng nhdt duqc viec day hoc TSC thdnh mdt khdi thdng nhdt cd ve ndi dung vd phuang phap dgy hoc Hinh thuc ddi hdi ddi ngu gidng vien thudc bd Tap chi Giao due so (k i - 6/201 n mdn Li ludn vd phuang phap day hoc todn phdi nam vung todn bd chuang trinh duqc gidng day d khoa Todn vd biet van dyng vdo day hoc TSC Thyc te cho thdy dieu Id khd thyc hien Hinh thuc thu hai cd uu diem Id phdt huy duqc chuyen mdn cua ddi ngu gidng vien d cdc bd mdn Todn ca bdn, nhung ddi ngu lgi khdng ndm vung nhung van de ve li ludn vd phuang phap day hoc Vi vdy, dan den tinh trang TSC chua dap ung duqc myc tieu ren luyen nghiep vy su pham, ho tra cdng tdc gidng day d phd thdng sau cho SV Tu nhung phdn tich d tren cho thdy, chung ta cdn tim mdt hinh thuc td chuc day hoc TSC that hieu qua d khoa Todn cdc TDHSP De xudt xdy dyng chuang trinh day hoc TSC Chung tdi de xudt chuang trinh TSC d khoa Todn cdc TDHSP nhu sau: I) Myc tieu mdn hgc: Ve kien thuc: Sau hoc xong mdn hoc nay, SV cdn ndm duqc: - Ca sd todn hoc hien dqi cua chuang trinh todn phd thdng; - Cd duqc cdi nhin sdng rd tu todn hoc hien dqi xudng TSC Qua dd, hieu sdu sdc chuang trinh todn phd thdng; - Tqo duqc sy gdn ket giua Todn hoc hien dqi vdi TSC; - Tang cudng ddo tgo nghe cho SV Ve kT nang: SV can hinh thdnh cdc kTndng gidi todn sacd'p, biet ung dyng todn hoc vdo gidi quyet van de thyc tiin; kT ndng nhin nhdn sy kien todn hoc theo ljch su phdt trien cua todn hoc 2) Ngi dung chi tiet cua chuong trinh Hinh hgc so cap: Chuong I: Tong quan ve hinh hoc Euclid; Chuong 2: Gidi thieu mdt so he tien de xdy dyng hinh hgc Euclid: 2.1 He tien de Hinbe; 2.2 He tien de Pogorelov; 2.3 He tien de Weyl, Chuang 3: Dyng hinh vd md rdng trudng: 3.1 Dyng hinh bdng thudc vd compa; 3.2 Da gidc deu 17 cgnh; 3.3 Ba bdi todn ndi tieng ve dyng hinh; Chuong 4: Hinh loi: Djnh nghTa vd vi dy; 4.2 Phep cdng cdc tap hqp ldi; 4.3 Khodng each Hausdorff giua cdc tap compact; 4.4 Ddi xung hod theo Steiner; 4.5 Tap hqp eye cua tap hqp ldi; 4.6 Tieu chudn cua tap hqp ldi; 4.7 Bao ldi; 4.8 Td pd vd chieu cua tap hqp ldi; 4.9 Djnh li Helli vd ung dyng; 4.10 Li thuyet Brunn-Minkowski; Chuong 5: Da dien ldi: 5.1 Dinh nghTa vd vi dy; 5.2 Cdc tinh chat cua da dien ldi; 5.3 Cdng thuc Euler; 5.4 Djnh li Cd si; 5.5 Da gidc deu; 5.6 Da dien deu; 5.7 Phdn (Xem tiep trang 40) nghTde tim loi giai cho bai toan Neu chua giai dugc thi co the xet mot bdi todn tuang ty khdc nhung dan gian han, ho trg cho cdc em viec tim ldi gidi bdi todn ban ddu Nhu vdy, thdng qua viec gidi mdt sd bdi todn cu the se giup HS tim duqc ldi gidi cua cdc bdi todn khdc nhung tinh hudng mdi De gidi mdt bdi todn, HS thudng tien hdnh theo bdn budc sau: - Tim hieu de bdi; - Xdy dyng chuang trinh gidi; - Thyc hien gidi bdi todn; - Kiem tra vd nghien cuu ldi gidi dd tim duqc 5) Ren luyen HS kT nang phat hien va giai quyet vdn de GV dua HS vdo cdc tinh hudng cd vdn de, HS ty nghien cuu, chu ddng khdm phd de chiem hnh tri thuc vd phdt trien tu duy, van dung kien thuc dd biet vdo cdc tinh hudng mdi GV cd the gqi y cho HS cdc hudng gidi quyet vdn de GV ddng vai Id ngudi cung cap thdng tin, tgo tinh hudng, giup HS gidi duqc cdc bdi todn Vi dy (GV dua de todn): Cho phuong trinh bgc hai: x + 2.(m - 2).x - 2m + = (1) (m Id tham sd) Cdc em hay hoan thien de todn tren HS cd the dua cdc yeu nhdm hoan thien bdi todn tren, chdng hgn nhu: 1) Vdi gid trj ndo cua m thi phuang trinh (1) cd nghiem; 2) Tim gid trj cua m de phuong trinh (1) cd nghiem trdi ddu; 3) Tim gid tri cua m de phuang trinh (1) cd nghiem cung ddu; 4) Trong trudng hqp phuang trinh (1) cd nghiem x,, x^; tinh td'ng vd tich nghiem theo m *** Viec td chuc cho HS hoc tap theo nhdm cd tdc dung tgo mdi trudng Idp hoc sdi ndi, cdc em co ca hdi duqc the hien khd ndng cua minh trudc thdy cd vd ban be; tqo mdi trudng hoc tap than thien, cd sy hqp tdc, giup dd, tuang tdc giua thdy vd trd, trd vd trd Trong dgy hoc todn theo PPDH «HDCN phdi hqp HD nhdm nhd" d THCS, viec GV td chuc cdc tinh hudng hoc tap da dang, phong phu se tqo dieu kien cho moi HS chu ddng phdt hien vd gidi quyet van de, chiem hnh tri thuc, gdp phdn ndng cao hieu qua day hoc • Tai lieu tham khao Nguyen Hai Chau - Pham Due Quang - NguySn The' Thach Nhung van de chung ve ddi mdi phuong phap day hoc toan trung hoc cusd NXB Gido due, H 2007 T6n Than - Pham Thj Luyen - D2ng Thi Thu Thuy Mot so van de ddi mdi phuong phap day hoc mdn Toan NXB Gido due, H 2008 ham dqi sd; 2.2 Mdt vdi loqi day truy hoi; 2.3 Cdc phuang phap xdc djnh cdc td'ng huu han; 2.4 Phuang phap su dyng ham sinh; Chuong loqi hinh da dien deu; Chuong 6: Dien tich vd 3: Mdt v d i trgng diem ve g i d i tich the tich: 6.1 Dien tich da gidc ldi; 6.2 The tich chuang trinh phd thdng: 3.1 Cdc bdi todn ve khdi da dien; 6.3 Dien tich mat cua hinh da tiep tuyen; 3.2 Cdc ung dyng cua djnh li gid trj dien; 6.4 Xdp xi tap ldi compact bdi da dien; trung binh; 3.3 Cdc phuang phap tim gid trj 6.5 Cdc bdt ddng thuc ddng chu; 6.6 Vdn de ldn nhdt vd gid tri nhd nhdt; Chuong 4: Dong thu ba cua Hilbert; Chuong 7: Cdc phep bien nhdt thuc vd bdt ddng thuc: Mdt sd ddng hinh mat phang: 7.1 Nhung kien thuc nhdt thuc cd dien; 4.2 Mdt sd phuang phap chudn bj; 7.2 Cdc phep ddi hinh cua mat phang; chung minh bdt ddng thuc; 4.3 Ham ldi vd bdt 7.3 Hinh cd tdm ddi xung Ddi xung bgc n; ddng thuc Jensen; Chuong 5: Phuang trinh vd 7.4 Ddi xung true vd ddi xung trugt; 7.5 Phep bd't phuang trinh: 5.1 Nhung khdi niem co dong dgng; 7.6 Phep nghich ddo; Chuong 8: bdn; 5.2 Nhung dgng phuong trinh ca bdn; 5.3 Bdt phuang trinh vd he bdt phuang trinh; Hinh hoc phi Euclid 3) Nqi dung chi tiet cua chuong trinh Dqi sd 5.4 Ket thuc vd biet thuc • so cap: Chuong I: Mdt vdi nguyen li ca bdn: 1.1 Nguyen li Dirichlet; 1.2 Nguyen li cue trj rdi rgc; 1.3 Nguyen li xudng thang; 1.4 Cdc Tai lieu tham khao nguyen li ca bdn cho cdc bdi todn dem; 1.5 Dinh Xuan Son - Nguyen Anh Tuan Giao trinh Nhin vdn de theo quan diem cue trj; Chuong 2: m6n Nghiep vu supham Dai hoc Thai Nguyen, 2002 Nhung van de sa cap ve day sd': 2.1 Nhung Dai hoc Thai Nguyen Ki yeu hoi thdo khoa hoc td'ng huu hgn khdng the bieu dien duqc qua cdc nghiep vu su pham todn.qudc Thai Nguyen, 2004 Ve v i e c d a y hoc (Tiep theo trang 37) Tap c h i Giao due so (k i 6/2011) ... Dgi sd sa cap duqc gidng dgy cdc trudng phd thdng Td chuc day hoc TSC d khoa Todn cdc TDHSP Hien nay, viec dgy hoc TSC d khoa Todn cdc TDHSP thudng theo mdt hai hinh thuc: 1) Hoan todn bd mdn... hinh thuc td chuc day hoc TSC that hieu qua d khoa Todn cdc TDHSP De xudt xdy dyng chuang trinh day hoc TSC Chung tdi de xudt chuang trinh TSC d khoa Todn cdc TDHSP nhu sau: I) Myc tieu mdn hgc:... 6/201 n mdn Li ludn vd phuang phap day hoc todn phdi nam vung todn bd chuang trinh duqc gidng day d khoa Todn vd biet van dyng vdo day hoc TSC Thyc te cho thdy dieu Id khd thyc hien Hinh thuc thu