N. T The. T. T. H. Lam/ Ren luyen ndng lUc giai toan cho smh vien ngd,nh suphgrn, toon...
R E N L U Y E N N A N G L l J C G I A I T O A N C H O S I N K V I E N N G A N H STJ P H A M T O A N H O C T R O N G DAY H O C K I E N THLfC X A C SUAT
Nguyin Thi The, Thai Thi Hdng Lam Khoa Sii pham Toan hoc, Trudng Dai hoc Vinh Ngay nhan bai 28/7/2016, ngay nhan dang 19/11/2016
Tdm tat: Bai bao dg cap den viec ren luygn nang luc giai toan cho sinh vien nganh Su pham Toan hoc trong day hpc mon Xac suat th6ng qua viec to cbiic, budng dan sinb vien khai thac kien thiic vh xac suat dg giai mot so bai toan pho tbong.
1 Mc( dau
Trong Taxa K hpc, nang lUc la dac digm tam If cua ca nhan phd hop vdi nhiing yeu cau cua mOt boat dong nbat dinh, dam bao cbo lioat d6ng do co k^t qua [6]. Do do co the hieu nang luc giai toan (NLGT) la dSc diim tam 11 ca nhan ciia con ngUcii dap ilng dupe yeu cau cua boat dgng giai toan va la dieu kien cS,n thilt dg boan thanh t6t boat dgng giai toan.
Mot ngUcfi duoc coi la c6 nang lUc giai toan ngu ngubi do nam viing tri thiic toan hoc, c6 ki nang, ki xao trong cac boat dgng giai toan. Nhu v|Ly, NLGT la mot bS phan quan trong tao nen nang luc toan hpc cho mOi sinh viSn sU pham va dUOc the hien thong qua be thong cac kl nang trong boat dgng giai toan.
Ren luyen NLGT chigm vi tri quan trpng trong d^y bpc cac hpc phan m6n Toan ca vg phuong dien dac thu va khoi lupng thdi gian. Ren luyen NLGT cho sinh vien sU pham Toan se giiip sinh vien nam viing kign thiic va bigt van dimg kien tbiic da hpc vao giai toan, dong thSi gop phan ren luyen nang luc nghe nghigp cho cac em, giup cac em sau khi ra trudng day toan t6t hon. Tim bilu nhiing kho khan cua giao vign Toan pb6 thong khi van dung cac kiln thiic xac suat da dupe hpc d trUcfng Dai hoc vao day bpc xac sudt ciing nbu giai cac bai toan co be ngoai khong lien quan d6n xac suat, chung toi nhan thay: kha nhigu giao vign toan chua higu thUc sU cac khai niem cd ban va cac tinh chat cua xac suat dupe hpc d trudng Dai hpc, nhat la ap dung H thuygt dg giai cac bai tap; kbo khan trong viec van dung cac phudng phap giai cac bai toan xac suat d Dai bpc vao day hgc giai bai tap phan xac suat d THPT; Dac biet, nhilu giao vign ph6 tbong chua biet hoac chUa co y thiic khai thac kien thiic xac suat de giai cac bai toan Dai s6, Hinh hgc, Giai tich. TtC do, khong dam bao duoc tinh tich hop npi m6n trong day bpc mon Toan 6 trucing phd thong, Nhu vay, viec ren luygn NLGT cbo sinh vign Su pham Toan bgc thong qua day hgc xac suat d trUcfng Dai bpc c^n dupe quan tam. Bai vilt nay dg c$.p den viec ren luyen NLGT cho sinh vien Su pbam Toan trong hoc tap mSn xac suit, cii the la khai tbac m6t so kiln thiic ve xac suat de giai cac bai toan trong npi bo xac suit ciing nbu nhflng bai toan co ve be ngoai khong lien quan den xac suat. Pham vi ren luyen NLGT tbeo ba trpng tam: ki
' Email: [email protected] (N T. The)
Tntdng Dai hoc Vmli Tcip chi khoa hoc, Tap 45, Sd 3A (2016), tr 9-2
nang dinb budng tim phUdng phap giai; ki nang thuc bign Idi giai; ki nang khai thac, ling dung kit qua phUdng phap giai. D&i vdi mOi kl nSng can ren luyen cho sinh vign nhiing thao tac ki thuat cu the trong khai thac bai toan.
2 Day hoc xac suit cho sinh vien nganh Sit pham Toan hoc theo hudng gan vdi viec nang cao tay nghe day hoc
Viec doi mdi phUdng phap bign nay xac dinh giao vien co vai tro la ngUdi thigt kg, iiy tbac, dilu khign va thi chi hoa qua trinh nhan thiic cua hgc sinh. Van de quan trgng la trong qua trinh dieu khign giao vign phai bilt dinh hudng giup hpc sinb phat hien each thiic giai quylt vin dg, tim toi kiln thiic mdi, ciing nhu danh gia va thg chi boa kien thiic. Dl giai quygt duoc van dl do, giao vign phai diing tren tam cao dg nhin nhan vin dg pho thSng, ngoai viec khai tbac trigt dl tai lieu sach giao khoa con phai bilt sii dung cac kiln tbiic toan da dupc bpc d trudng dai hpc vao viec nhm nhan kien tbiic toan lien quan d pbo tbdng mot each toan dien hOn. Chinh vi vay. cac muc tigu cbinh cua viec day hpc hgc phan xac suat cho sinh vien sU pham nganh Toan d trudng dai bpc la:
+) Trang bi cbo sinh vign cac kiln thiic cd scf, nang cao vg xac suat;
-I-) Ren luyen cho sinh vien ki nang van dung cac kiln tbiic da hgc vao giai cac bai tap.
Cling nhu giai quylt cac van dg thiic tiln;
-i-) Lam cho sinh vign bilt dua vao cac kiln tbiic xac suat d trudng Dai hgc dl nhin nhan cac kiln thiic xac suit d php thong mot each day du va sau sic bdn;
+) Lam cbo sinh vign bilt van dung cac kiln thiic xac suat d trirdng Dai hoc dl dinb hudng tim toi Idi giai cho cac bai toan xac suit d pho tbong, cung nbU cac bai toan trong cac phan mon khac nhu dai so, hinh hoc, giai tich;
+) Lam cho sinb vien bilt van dung cac kiln thiic xac suit da hoc 6 trudug Dai hpc dl sang tao cac bai toan d pho thong.
3 Ren luyen NLGT cho sinh vien nganh su* pham Toan hoc
Ren luyen NLGT la viec giao vign thUc hien chii dong viec luyen tap cac ki nang giai toan cho sinh vign dat din sU thanh thao co tinh sang tao. Trong ren luyen NLGT, sinh vign can thuc bign cac giai doan khac nbau gOm: xac dinh hufing giai, tim kiem Idi giai, thuc hien giai va ngbien ciiu, khai tbac ling dung ket qak giai. Nhan thiic ve xac dmh bitdng giai phai gdn v6i viec nam vflng cac dang, cac loai bai toan, pham vi kiln thiic trUc tilp va gian tilp co lign quan. Tim Idi giai la xac dinh cac pbUdng phap giai khac nhau theo cac dinh hudng giai. Viec tim dudc phUdng hudng giai din viec giai hoan cbinh bai toan la ca mot qua trinh, tit sii dung chinh xac cac kiln thiic cd ban vl npi dung ly thuylt din viec ap dung, chuyln dOi tbanh thao cac quy trinh va cac thao tac co tinh ki tbuat Nhung dinh hudng tim Idi giai co tinh chat quyet dinh trong toan b5 cdng viec giai toan, dinh hUdng chinh xac va hieu qua thg hien kba nang nam bat kiln thiic va kha nang tu duy. Nghien cdu, khai tbac kit qua giai se hinh thanh cbo sinb vien y thiic nhin nhan lai qua trinh giai, danh gia phudng phap giai va phat hien kit qua mdi. Ren luyen NLGT la ren luygn kha
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A^. T. The, T. T. H. Lam/ Ren luyen nang lUc giai toan cho sinh vien nganh sit pham toan. ,
nang sang tao trong tim Idi giai va ren luygn cac kl nang, thao tac, ki thuat giai. Vigc ren luyen NLGT cho sinh vien ng^nh sU pham Toan khOng chi la kit qua giai bai toan do ma con doi boi tiiib sang tao trong viec tao ra cac bai toan mdi tii bai toSn ban dau, ciing nhu the hien d vigc tim each giai doc dao cho bai toan.
4 Ren luyen NLGT cho sinh vien nganh sU pham Toan hoc trong day hoc kiln thiic xac suat
4.1. Nhiera vu cua giao vien
Cung mot noi dung day bpc, cac giao vien c6 thi thilt kl cac tinh huSng day hpc khac nhau. Tuy nhien, theo chiing toi dl ren luygn NLGT cho sinh vign sU pbam Toan thdng qua day hpc mot ngi dung cu thg vl xac suat, giao vien cin phai:
- Xac dinh ro kiln thiic, Id nang, thai do va tu duy sinh vign cin dat trong qua trinh day hpc npi dung do;
- Can cii vao ndi dung day hpc, xay dung cac bai toan trong ndi bp mdn Xac suat va cac bai toan thudc cac phan mdn khac co thi giai dUpc bang viec sii dung cac kiln thiic vl xac suit viia dUdc hpc, giao vign cin quan tam hai dang bai toan: dang bai toan thuan tuy chuong trinh dai hgc va dang bai toan gan giii vdi chUdng trinh xac suit d pbo thdng;
- Dieu kbidn boat dong giai toan cua sinh vien;
- Lap kl hoach danh gia ket qua giai toan cua sinh vign.
4.2. Nhtem vu cua sm/i vien
Dg giii dupe bai toan do giao vign yeu ciu, qua do dat dUdc muc tieu day hgc ma giao vien da dat ra khi hpc mdt noi dung nao dd, sinh vien cin thuc liign theo quy trinh sau:
Budc 1: Phan tich cac diu hieu cua bai toan cd lign quan din cac ddi tUdng trong khdng gian xac suit,
Budc 2 Xay dung md hinh khong gian xac suit va ddi tUdng tbich bdp cbo bai toan;
Budc 3: Giai bai toan trong ndi bd kiln thiJc xac suit;
Budc 4: Ngbign ciiu sau kit qua (tim each giai khac cua bai toan; tim bai toan tong quat, bai toan dac biet, bai toan tUdng tu, . . . ) .
Luu y ring, nlu bai toan la bai toan thuan tuy xac suit thi sinh vien kbdng cin thdc hien Budc 1 va Budc 2.
4-3. Mot sd vi du mm/t hoa cho viec ren luyen NLGT cho sinh vien nganh Sit pham Toan HQC trong day hoc hin thiic xac suit
Muc nay nham neu ra mdt so bai toan minh hoa cbo quy trinh da neu trgn. Chang ban, sau khi day cbo sinh vign cac kiln thiic cO ban vl biln c6; bg diy d i cac biln co; . . . , giao vien CO the yen cau sinb vign giai bai toan sau
Chiing minh ring vdi cac sd tu nhien m,A;,ni,.. .,nm,{m,}z > 1), ta cd 2 ^ ^ni^^.^ . . . C^2 = ^ni+n2+-+nm- (.Tt.ri,. .,rm]eN"',Ti+r2+-+r„=k
Bai toan nay nlu khai triln true tilp di giai thi se kha khd khan. Tii dd budc sinh vien
Trudng Dai hoc Vmh Tap chi khoa hoc, Tap 45, Sd SA (2016), tr. 9-i
phai suy nghi din vigc biln ddi bai tofin thanh bai toan tUdng duong sau
E ^ ^ ^ ^ ^ = 1. (1)
{rur2,. ,rm)eAf"',J-i+T-2+-+7-m=fe ^"1+^24- •+1m Giao vign cd thg td chlic hudng din sinh vign giai bai toan nhU sau:
Budc 1: Phan tich cac diu hieu cua bai toan co lien quan din cac doi tUdng trong khong gian xac suat.
Ta thiy vl trai ciia (1) la tdng cua cac sd hang khdng am va cd cic ciii sd vet hit cac klia nang cd thd. Tdng cac s6 do lai bang 1. Tii dd gpi cbo sinb vien lien tUdng din xac suit cua nhdm day dii cac bien co. Van de dat ra la xay dung md hinh xac suit nhu thi nao cho phii hop?
Budc 2: Xay duug md hinh khdng gian xac suit vh, ddi tUdng thich hop cho bai toan.
Dua vao sU xuat hien ciia cac sd hang cd dang — | —, cbiing ta xem man sd chinh la sd each cbgn k phan tii trong tap hop co ni -I- n2 H h Jim phan tii, xem tii so la s5 each chpn lien tilp ri phan tii tif tap rzi phin tii, . . . , chon Vk phan tii tii tap rifc phin tii.
Do do ta cd thg xay duog mo hinh phep thii nhu sau: Xet mdt bdp chiia ni qua ciu loai 1, 7^2 qua ciu loai 2, . ., Ji„, qua ciu loai m. Tii h6p lay ra k qua ciu.
Budc 3: Giai bai toan trong npi bd kiln thiic xac suit:
Ung vdi mdi bp k so nguyen khdng am
(ri,r2,.. ,rniy,ri+r2-\ \-rm = k, ta cd mdt biln co vdi xac suit la
^ni+n2+- +nm Cac bien cd tren lap thanh ho diy dii cac biln cd, do dd
Z^ nk ^•
{ri,r2,...,T„)€N"';ri+T2+- +T^=k "l+«2+ +nm
Do dd ta CO dieu pbai chiing minh. Nhu vay, dl giai bai toan nay sinh vign pbai bilt chon phfip thii va kbai tbac tinh chat ciia nhdm diy dh. cac bien cd.
Budc 4 '• Ngbien cilu sau kit qua.
Bang each xet trudng hpp dac biet ciia bai toan tren khi chi cd bai chi so n = jzi, m :^
712, ta cd bai toan sau:
Bai toan 1.1: Chiing minh rang vdi 0 < m < A; < TI thi
clfil + clfil-^ + • • • + czc^"^ = cl^^.
Nlu tCr Bai toan 1.1, xet cbo trUdng hdp m ^ n ^ fc, ta cd bai toan sau:
N. T. Thi, T. T. H. Lam/ Ren luyen nang lUc gidi toan cho smh vi^n nganh siipham toan...
Bai toan 1.2: Chiing minh rang vdi mgi sd tu nhien n thi
Trf Bai toan 1.2, nlu cho n bing mdt gii tri ci;i thi nao dd, ta dUdc cac bai toan dac biet cua Bai toan 1.2.
Cung tii Bai toan 1.1, ting vdi m = 1, ta cd bai toan quen thudc sau day Bai toan 1.3: Chiing minb rang vdi cac sd tu nhien 1 < fc < n, ta co
Day chinh la dang thiic Pascal quen thuoc d6i vdi hpc sinh trong sach giao khoa Dai s6 va Giai tich 11. Tuy nhien d day viec chiing minb dUdc xay dung theo each khac. Dilu nay it nhilu tao nen si;r hiing tbu cbo sinh vign trong qua trinh bpc tap mdn xac suit d trudng dai bpc.
Nhu vay, tif mdt bai toan tdng quat dUdc giai bang kiln thulc xac suit dupe hpc d triidng dai hoc, sinh vien da dl xuit dUdc mot sd bai toan cbo hoc sinh d trudng phd thdng. Qua dd, sinh vign se phan nao bilt each tao ra bai toan mdi vdi nguon goc xuit phat cua no, tvf do dinh hudng dupe each giai cho hpc sinb.
Vdi bai toan sau day, sinh vign cin bilt xay dUng md binh xac suit phii hdp va bilt van dung tinh chit cua cac biln co dpc lap, biln cd ddi cua mot bien cd.
Cho x,y la cac sd thuc khong am co tdng bing 1 va m,Ti la cac so tu nhien. Chiing minh rang
( i - i " r + ( i - ! / ' " r > i . (2) Cd thg md ta qua trinh suy nghi va. each giai bai toan nay nhU sau:
- Dua vao sir xuit bign ciia hai sd x, y khdng am vk cd tdng bang 1 gdi cbo sinh vien lign tudng din x va y la xac suit ciia hai bien cd ddi.
- Mat khac, cac so hang d vl trai cua bieu thiic (2) xuat hien tich cua cac so kbdng Sm va khong Idn hdn 1 gpi cho sinh vien nghi din xac suit cua tich cac biln cd doc lap.
- Hon niia, cin chiing minb t6ng ciia hai so hang 6 ve trai ciia biiu thiic (2) Idn hdn hoac bang 1. Do do, sinh vien se xem mdt so hang la xac xuit cua biln co A, va sd hang con lai la xac suit cua biln cd D nao do sao cho D D A.
Van de dat ra la cd xay dung dupc mo binh phep thii phii hdp hay khdng. Vdi each phan tich bai toan nhU tren, sinh vign cd the xay dUng mo hinh xac suit tUOng thich VA giai bai toan nhu sau:
Giasiici,C2,...,Cn la cac ddng xusao cbo vdi moi i = l,...,n, xac suit dl cac ddngxu c, ra mat sip la x. Tung cac ddng xu nay mot each ddc lap m lin.
• Goi A la biln co "mCi mot trong m lin tung, co it nbit mdt ddng xu ra mat ngiia"
Khidd
P(A) = ( l - r ' ' ) ' " .
That vay, goi Aj la bien cd lin tung thii j cd it nhat mdt ddng xu ra m5t ngiia (j ==
1,2,..., m). Kbi do cac A_, la dpc lap va
A = A1A2 ...A^.
Tmdng Dat hoc Vinh Tcip chi khoa hoc. Tap 45, S6 3A (2016), tr. 9-20
Suy ra,
^{A) = V{Ai)^{A2)...f{A^).
Rd rang Aj la biln co trong Iln tung thii j ca n ddng xu diu sip. Do dd IP(ij) = r".
Nhu vay,
P(A) {\-^Ai))...{\-f{A,n))
• Gpi B la biln co mdi mot dong xu xuit bien it nbit mdt lin sip trong m lin tung. Gpi Di la biln cd ddng xu thii i xuit bign it nhat mot lan sap trong m lan tung (t = 1,2, .. ,n) Khi dd cac biln co D^, dgc lap, va
D = DlD2...Dn.
Ma^t khac, ta cd D^ la biln cd ddng xu thii i xuit hign mat ngiia trong ca m lan tung. Do dd,
P(A) = ( l - x ) ( l - i ) . . . ( l - x ) - ( l - 3 ; ) " ^ = y'^.
Suy ra,
n c ) = (1 - j / ^ j a - D ••• (1-»"•) = (1 - »")"•
Xet biln c6 doi cua biln cd D: dd la biln cd tdn tai it nbit mdt dong xu xuit bign mat ngiia trong ca TTI lin tung. Nhu vay nlu D xay ra thi A xay ra, tiic la
DC A.
Theo tinh chit ciia xac suit ta cd
1P(£J) > P(A).
Tii dd ta cd
( l - 3 : X + ( l - r r ¥{A)+F{D)
> ¥{A) + F{A) 1.
Diu bang xay ra khi va cbi kbi ¥{D) - W(A). Khi dd n = 1.
Bing vide nhin lai each giai bai toan, ta thiy cdc ddng xu dupc ebon cd xac suit xuit hiSn mat sip la nhu nhau (bang x). Tit dd, giao vign cd the dat van dl cbo sinh vign xem xet: nlu xac suit cua cac ddng xu xuit hign mat sip la khac nhau thi sao? Co the xay dung dUdc bai toan t6ng quat nao hay khdng? Bai toan sau day cbinb la mot bai toan tong quat cua bai toan trgn.
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N. T. Thi, T. T. H. Lam/ R&n luyen n&ng lUc gidi todn cho smh viSn nganh sUphctm toan...
(IMOBungari, 19S4) Chox^,y^,{^ = 1,2,...,n) Ia2ns6 thuckhdngamsaocboa;,+yi = 1. Chiing minh rang vdi mgi so ti; nhign m, ta deu co
(1 - iiX2-x„r+(1 - !/r)(i - »2"')-(i - y") > 1-
Viec giai bai toan thuan tuy xac suit ciing chinh la da thuc hien theo qui trinh trgn nhung da bd qua Budc 1 va Budc 2 (va cd tbi ca bUdc 4). Cu thi ta xet bai toan sau
Cho A la tap con cua tap 5 ^ {1,2,..., ri}. Hay tinh xkc suit cua biln co E ma tap A xuit hign la doan dau trong mdt boan vi cua 5. Giai. O day phep tbii la viec chon mdt hoan vi cua S. Do do so kha nang cd thg la TI!.
Bay gid ta tim so trudng bpp thuan Idi cho biln cd E. Gia sii (? = {xi, X2,..., Xn) la mdt hoan vi ciia tap {1,2,..., n}. Ky higu a la s6 phan tii cua A. Kbi dd do E la biln c6 ma A xuit hien nbu doan diu ciia a, nen {3;i,a;2,.. • ,Xa} = A. Vl vay, so cac trudng hdp thuan loi cbo bien co E la ai{n — a)\. Do dd xac suit cua E la
' ( - ) - ^ ^ = ^-
Cuoi ciing ta md rong bai toan tren dl chiing minh bit ding thiic Lubell-Yamamoto- Meshalkin bing phudng phap xac suit.
Cho .41,^2,..., As la cac tap con cua {1,2,..., n}, va gia sii ring kbdng cd tap Aj nao la tap con cua cac tap con lai. Ky higu a, la sd phin tii cQa tap A,. Cbiing minh ring
U<--
Gidi. Doi vdi bai toan nay, ta nghi den vl trai la xac suit cua hdp cac biln cd xung khac.
Ta xay dUng md binh xac suit nhU sau: dd la khdng gian xac suit rdi rac vdi khong gian biln co sd cap gom cac hoan vi a = {xi,X2,--.,Xn) ciia tap {1,2,...,7z}.
Ky higu El labilncdma Aj xuit Men nhU doan diu cua c, tiic la {3;i,X2,... i-^a,} = Ai.
Tit Bai toan 4, ta co
ne,) = ^ .
Do khdng cd tap Ai nao la tap con cua cac tap con lai ngn cac biln co Ei,E2,---,Es la xung khac ddi mot. Suy ra
J'(U'=.) = E ^
Mat khac, xac suit cua mot biln co kbdng vUdt qua 1. Vay ta cd
Tmdng Dai hoc Vmh Tg,p chi khoa hoc, Tdp 45, SS 3A (2016), tr. 9-2
TAI LIEU THAM KHAO [1] Thdng tin toan hpc, tap 16, Sd 4, 2012.
[2] Thdng tin toan bpc, tap 17, So 1, 2013.
[3] Thdng tin toan hpc, tap 17, Sd 2, 2013.
[4] Nguyin Van Quang, Xac su&t nang cao, NXB Dai hoc Qudc gia Ha Ndi, 2008.
[5] Noga Alon, Joel Spencer, The Probabilistic Method, Wiley, 2000.
[6] Nguygn Quang Uln, TVan TVgng Tbuy, Tam li hoc dai cuang, NXB Dai hpc Su pbam.
Ha Ndi, 2004.
SUMMARY
TRAINING THE COMPETENCE OF SOLVING MATHEMATICAL PROBLEMS FOR STUDENTS OF MATHEMATICAL PEDAGOGY MAJOR IN TEACHING AND LEARNING
THE KNOWLEDGE OP PROBABILITY THEORY
The article refers to train the competence of solving mathematics problems for students of Mathematics Pedagogy major in teaching and learning the knowledge of Probability through organizing, instructing students to explore the knowledge of Probability in order to solve some hight school mathematical exercises.
