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[.]
Trang 1V 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)
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 khi 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 trong
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 trong 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 trong TSC chi
hieu dugc chinh xdc khi su dyng nhung cdng cy
mgnh cua todn hgc hien dgi Ngodi ra, cdc bdi
TSC tuy hinh thuc phdt bieu hay phuang phap
gidi khdc nhau 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 trong mdt chinh the
thdng nhdt cua todn hgc, cdc gido trinh cdn phdi
ren luyen dugc tu duy cho SV, tgo tien de de sau
nay 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 trong 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 do thu
hai Id vj tri cua bdi hgc trong 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
2 Chuang trinh TSC a khoa Todn cdc
TDHSP hien nay
- 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
do dgc trong hinh hgc (do dai, dien tich, the tich);
Cac phep bien hinh trong 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 trong 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 trong mat phang Theo chung tdi, nhung kien thuc nay nen duqc gidng dgy trong Hinh hoc Euclide trong khdng gian E2 Nhu the se vua tranh duqc sy trung lap vua cd the dua ra cdc each chung minh ngdn gqn dya tren dang chudn Jordan cua ma trdn vudng cap 2 Myc tieu chinh trong 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 trong vd cd y nghTa trong chuang trinh hinh hoc sa cap
- Hinh loi Id mdt trong nhung ddi tugng quen thudc vd quan trong nhdt ciia Hinh hoc Euclide
Da gidc loi vd hinh trdn duqc gidng day trong 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 2 6 3 (k 1 6 / 2 0 1 n
Trang 2hinh cau dugc giang dgy trong 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 khi la rdt hien
nhien) Vi the, sy chinh xac hoa khai niem ve
hinh loi de tu do dua ra 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 do can den nhung cong cy nhu: Tdpd,
Li thuyet do do, Giai tich ham, Li thuyet nhom va
tac dong cua nhom Giao trinh Hinh hgc so cap
hien nay trinh bay ve phan hinh loi con 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 ra duqc
tdm quan trong cua cdc hinh ldi trong Todn hoc
vd trong 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 do 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
trong 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 trong cdc trudng phd thdng
3 Td chuc day hoc TSC d khoa Todn cdc
TDHSP
Hien nay, viec dgy hoc TSC d khoa Todn cdc
TDHSP thudng theo mdt trong hai hinh thuc:
1) Hoan todn do 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 trong 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 nay ddi hdi ddi ngu gidng vien thudc bd
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 nay 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 nay 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 nay 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
4 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 khi
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: 4 1 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)
Tap chi Giao due so 2 6 3 (k i - 6/201 n
Trang 3nghTde 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 trong
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 trong
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 3 (GV dua ra de todn): Cho phuong
trinh bgc hai: x2+ 2.(m - 2).x - 2m + 1 = 0 (1) (m
Id tham sd)
Cdc em hay hoan thien de todn tren
HS cd the dua ra 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 2 nghiem trdi ddu; 3) Tim gid tri cua m de phuang trinh (1) cd
2 nghiem cung ddu; 4) Trong trudng hqp phuang
trinh (1) cd nghiem x,, x^; tinh td'ng vd tich 2
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
1 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
2 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
V e v i e c d a y h o c
(Tiep theo trang 37)
loqi hinh da dien deu; Chuong 6: Dien tich vd
the tich: 6.1 Dien tich da gidc ldi; 6.2 The tich
khdi da dien; 6.3 Dien tich mat cua hinh da
dien; 6.4 Xdp xi tap ldi compact bdi da dien;
6.5 Cdc bdt ddng thuc ddng chu; 6.6 Vdn de
thu ba cua Hilbert; Chuong 7: Cdc phep bien
hinh trong mat phang: 7.1 Nhung kien thuc
chudn bj; 7.2 Cdc phep ddi hinh cua mat phang;
7.3 Hinh cd tdm ddi xung Ddi xung bgc n;
7.4 Ddi xung true vd ddi xung trugt; 7.5 Phep
dong dgng; 7.6 Phep nghich ddo; Chuong 8:
Hinh hoc phi Euclid
3) Nqi dung chi tiet cua chuong trinh Dqi sd
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
nguyen li ca bdn cho cdc bdi todn dem; 1.5
Nhin vdn de theo quan diem cue trj; Chuong 2:
Nhung van de sa cap ve day sd': 2.1 Nhung
td'ng huu hgn khdng the bieu dien duqc qua cdc
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 3: Mdt vdi trgng diem ve gidi tich trong
chuang trinh phd thdng: 3.1 Cdc bdi todn ve tiep tuyen; 3.2 Cdc ung dyng cua djnh li gid trj trung binh; 3.3 Cdc phuang phap tim gid trj
ldn nhdt vd gid tri nhd nhdt; Chuong 4: Dong
nhdt thuc vd bdt ddng thuc: 4 1 Mdt sd ddng nhdt thuc cd dien; 4.2 Mdt sd phuang phap chung minh bdt ddng thuc; 4.3 Ham ldi vd bdt
ddng thuc Jensen; Chuong 5: Phuang trinh vd
bd't phuang trinh: 5.1 Nhung khdi niem co bdn; 5.2 Nhung dgng phuong trinh ca bdn; 5.3 Bdt phuang trinh vd he bdt phuang trinh; 5.4 Ket thuc vd biet thuc •
Tai lieu tham khao
1 Dinh Xuan Son - Nguyen Anh Tuan Giao trinh m6n Nghiep vu supham Dai hoc Thai Nguyen, 2002
2 Dai hoc Thai Nguyen Ki yeu hoi thdo khoa hoc nghiep vu su pham todn.qudc Thai Nguyen, 2004
Tap c h i Giao due so 2 6 3 (k i 6/2011)