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T^ chi Khoa hgc Trudng Dgi hgc Can Tho Phdn C Khoa hgc Xa hot Shan van vd Giao due 34 (2014) 27 S'''' Tap chl Khoa hoc Tn/dng €)ai hpc Can Thd website sj ctu edu vn raDABpc LOI CUA HOC SINH TRONG GIAI TO[.]
Phdn C: Khoa hgc Xa hot Shan van vd Giao due: 34 (2014): 27-S' T^ chi Khoa hgc Trudng Dgi hgc Can Tho raDABpc Tap chl Khoa hoc Tn/dng €)ai hpc Can Thd website: sj.ctu.edu.vn LOI CUA HOC SINH TRONG GIAI TOAN GIAI TICH: NGHIEN CUtJ DIEU TRA HOC SINH VA GIAO \TEN d THI XA TAN CHAU - TiNH AN GIANG Trdn Cong Thai Hoc' va Nguyen Phii Lgc' ' Ldp Cao hgc KI9 Chuyin ngdnh LL&PPDH bd mon Todn ^ Khoa Suphgm, Truong Dgi hgc Cdn Tha Thong tin chung: Ngdy nhdn: 05/05/2014 Ngdy chdp nhdn: 31/10/2014 Title: Students' errors in solving calculus problem: A survey of students and teachers in Tan Chau town - An Giang Province Tdkhda: Ldi phdn lich ldi, dgy hgc gidi rich, gidi todn, gido due todn hgc Keywords: Error, error analysis, teaching calculus, solving problem, mathematics education ABSTRACT The article reporied the surveyed results of students ' errors in calculus problem solving Sw^eyed subjects were 12 ff-ade students and mathematics teachers in grades The results showed that the students committed several different error types and teachers, also, said that the violations of students ' errors have been frequent Hie results obtained were also compatible with the opinions of domestic and foreign experts in mathematics education TOM TAT Bdi bdo tudng thudt kit qud khdo sdt ldi cda hgc sinh gidi todn gfdi tich Ddi tugng khdo sdt Id hgc sinh ldp 12, va gido viin dgy todn d cdc ldp dugc khdo sdt Kit qud cho thdy Id hgc sinh phgm nhiiu logi ldi khdc gidi todn ffoi rich, gido vien cOng cho rdng viec phgm cdc ldi trin cua hgc sinh Id thudng xuyin Kit qud thu dugc cung tuang hgp vai cdc nhgn dinh cOa cdc chuyin gia vd ngodi nttdc B A T VAN B E N ^ a i cftu tdi cua hgc smh Id mgt cdng vide cdn thidt cua gjdo vidn day hgc d trucmg phd thdng R Marzano (1992) xem phdn tich ldi efta hgc sinh Id mdt bidn phap dd m d rdng (extend) vd tinh lgc kidn tiiftc (refine), jrfidn Uch loi cdn chft y : phdi xdc djnh dd la Idi gj, nguydn nhan ndo ddn ddn loi \ d each ngan ngua Ciing ban v e loi cua hgc sinh, tdc ^ d N ^ i y f e Phu Lgc (2008) ddc bidt chft y d y dodn vd i ^ a n n ^ ldi cua bgc sinh qud trinh day bgc toan Ve thai cua giao vidn ddi VOT loi, tdc gia M L a ^ n k o (2008) quan niem rdng: (1) ^ d o vien thfta nhaii quydn bi ldi cua hgc smh; (2) giao vien phdi cd gang h i i u bilt loi cua hgc suih da xay ra; (3) trcM^ qua trinh day hgc can day cbo hgc sinh cdc chidn luge han c b l ldi Iam bdi nhu k i l m tra lai ddp sd, kidm tta Igi cac budc biln ddi, k i l m tra lai vide tinh todn, lidn h e v ^ bdi canh thyc tidn, sft dyng dd thj, giai bai todn bang cdc cdch khac Vd hgc tap mdn Toan, tdc ^ d Legutko cung cho rang vide hgc sinh pham Idi la didu khdng thd trdnh khdi Dac bidt ddi vdi m d n Gidi n'ch, theo tdc gia N g u y i n Phft Lgc (2010) m d n cd tinh phftc t ^ ndi tgi cao vd thudng lien quan den qua trinh vo ban; vgy, hgc sinh h g c t ^ m d n Giai tt'ch sd g ^ nhidu khd khdn va c h u d n g nggi, v a se pham nhidu Idi gidi todn £>l tim h i ^ nhftng ldi md bQC smh da gap phdi ttong giai toan Gidi tidi sao, chung tdi thyc hidn n g h i a i c u u tmdng hq> (Nguyen Phft Lgc, 2014) d huyen Tan Chdu, tinh An Giang vdi hai cau bdi D ^ ^ c u u sau day: Trong giai todn m d n Gidi tfch, hgc sinh cudi cdp thudng phgm nhung ldi ^ ? Tgp ehi Khoa hgc Trudng Dgi hgc Cdn Tho Pfidn C: Khoa hge Xd hgi Nhdn vdn vd Gido due- 34 (2014): 27-33 Y kidn cua gido vidn vl mftc dd thudng xuydn cua cac iSi cua bgc smh sao? Djnh nghia vd khdi nidm Theo tii diln tiing Vidt phd thdng (Chu Bich Thu vd ctv, 2013) thi ldi cd nghia la: "Chd sai sdt khdng tiiyc hidn dung quy tdc; Didu sai sdt, khdng ndn, khdng ph^ ttong each cu xft, ttong hdnh dOng; Cd chd sai sdt vl mgt 1^tiiugt;Cd didu sai, trdi, khdng dung dgo Ii" Trdn ca sd djnh nghia ttdn day, ttong bdi bdo ndy chung tdi dinh nghia l6i ttong Idi giai mdt bai todn nhu sau: Ldi ldi gidi mOt bai todn la chd sai sdt thye hidn khdng dung quy tdc, khdng dp dung dung cdng tiiftc, dinh ly hogc hiiu sai khai nidm, djnh Iy, hilti sai di bdi, hoSe ldi cd tiid tinh todn nhdm Idn, khdng chinh xac ttong sft dung ngdn ngu vd suy ludn PHU'ONG PHAP NGHIEN CtitI VA B6I TU'ONG KHAO SAT Phdn tich ngi dung (NguyIn Phft LOc, 2014) Phdn tich bdi ldm ciia hgc sinh ttong cdc ky kidm tta ttong nam hgc 2013-2014 cua hgc sinh Idp 12 de tim va phan loai cde Id! eua hgc sinh dd gap phdi giai cac bai toan giai tich Diiu tra bdng bdng cdu hdi: Sau phan logi cdc loi efta hgc sinh, chiing tdi dCmg Bang cau hdi di tim hiiu ^ kidn cua gido vidn vd mftc dO thudng xuydn ve cdc ldi cua hgc sinh Ddi tugng khdo sdt: Hgc sinh: Hgc sinh Idp 12 ndm hgc 20132014 thuOe bon trudng tmng hge thdng thugc Thi xd Tdn Chdu, tinh An Giang Cy thi nhu sau (xem Bdng 1) Gido vien: 28 gido vidn todn efta bdn tmdng tmng hgc phd thdng cd hgc sinh dugc khdo sat Bang 1: Sd bai ldm cua hgc sinh dirge phan tich Tnrcmg Lop So bai THPT Tan CMu 12A1, 12A2, 12A3, 12A4, 12E5 499 THPT Nguyin Quang DiSu 12A1, 12A4 225 THPT CMu Phong 12A3, 12A4, I2A6 291 THPT Diic Tri 12A1, 12A2,12A3,12A4 285 trdn dja bdn thi xS Tan Chau thuOc tinh An Giang KET QUA KHAO SAT VA BAN LUAN nhu da ndu d tten chftng tdi nhin thdy ring: 3.1 Ket qua nhftng loi md hgc sinh mdc phdi hgc tdp Ve hgc sinh giai tich la rdt da dang Bdng cho chftng ta thdy cdc loai iSi cua hgc sinh vd ti Id pham iSi efta Qua qud ttinh didu tra khao sat thye td bdi vidt tftng tmdng eua hge sinh d mOt s6 tmdng trtmg hgc phd thdng Bang 2: Ket qua phieu dieu tra hgc sinh :hau Nguyen Quang Di€jiiK:Fhoiigi^STM Cdc loai ldi (%)41,3 (%) 14,1 51,5 31,6 Ldi tinh todn sai Ldi thidu didu kien h o ^ dgt didu kidn klidng dftng 26,3 17,3 61,9 63,2 Ldi hidu sai khdi ni?m 6,5 25,3 5,2 10,5 Ldi hidu sai djnh ti, hodc cdng thftc 23,7 10,6 56,7 17,9 Ldi nhd sai cdng thdc, quy tdc vd ky hieu 12,8 58,7 46,4 54,7 Ldi khdng thdnh thgo dp dung cac kytiiudtca bdn, 19,2 21,3 27,8 35,1 gidi cdc dang toan ca ban 11,5 18,7 13,4 Ldi khdng bidt dien dgt chinh xdc ttinh bdy ldi giai 21,6 Ldi ghi sai de, khdng chu;? gid thidt ctia de bdi 26,7 10,3 12,6 1,3 9,6 5,3 8,2 13,7 Ldi ngO nhdn kidn thftc 10 Ldi xet titilu tmdng hgp 11.5 17,9 20,6 25,3 dd thudng xuydn cua hgc sinh tftng loai l5i Vi gido vien Bdng tudng thudt y kiln cua giao vien vd muc %) (%) Tcp ehi Khoa hgc Trudng Dm hgc Cdn Tho Phdn C: Khoa hge Xd hgi Nhdn van vd Gido d^tc: 34 (2014): 27-33 Bang 3: Ket qsa phien didB tra giao vien (N=28) Thirdng xuyen Cac loi ciia hgc sinh hoc tdp giai tich Mncda Thtih thoang (%) f%) Hau Dhii kbongcA oql 71,4 28,6 Ldi tinh todn sai 21,4 78,6 Loi dotiiidudidu kien hoSc dgt dilu kidn khdng dung 35,7 60,8 3,6 Loi hieu sai khdi nidm 32,1 67,9 Loi hiiu sai ^nh li, bo|k: cdi^ thftc 39,3 60,7 Loi nhd sai cdngtiiuc,quy tdc vd ky hidu Ldi khdng thdnh thao ^ dung cdc I^ thuat ca ban, gidi 28,6 71,4 cdc dgng todn ca bdn 57,1 42,9 Ldi khdng bilt dien dat chinh xdc trinh bay Idi gjdi 14.3 67,8 Loi ghi sai dd, khdng chu y gid thidt cua di bdi 17,9 28,6 67,8 Ldi ngd nhgn kidn thftc 3,6 10 Loi xet thidu trudng hgp 35,7 60,7 3,6 0 Cdc nguydn nhan khac 3.2 Bdnlaln Trong muc ndy, di thdy rd bdn didt ldi cua hoc sinh ttong gidi todn gidi tt'ch chftng tdi ghi lai mudi Thdng gua kdt qud efta hai bdng khdo sdt, Idi gidi (cd tinh dm didn) ciia hgc sinh cd chung ta thdy Idi hgc sinh ttong gjdi toan gidi tich pham Idi mdt ttong mudi Idi ma chftng tdi liet ke Id khd da dgng va nhieu nguydn nhdn khdc ttong Bang Cac gido vien dugc khdo sdt cung ddng tinh cho rdng cdc l5i nhu vgy xdy gan nhu thudng xuydn 4.1 Ldi tinb toan sai d hgc sinh Ngoai ra, kdt qud thu dugc cho thdy De bdi: Tim mOt nguydn ham F^x) cua ham thyc tien pham loi ciia hgc sinh tuong hgp vdi nhgn dinh vd dac didm ciia Gidi tich (Nguydn Phu Lgc, 2010) vd quan didm vl loi cua M Legutko (2008) sd /(jc) = cos3xcos5JC,bidt F\ — \ = -2 Do vdy, dd ndng cao hidu qud dgy hgc mdn Giai tich d trudng phd thdng, ttong qud trinh dgy hgc (trich tft de kidm tra tap thd 45 phut chuong giao vidn can chu y ngan ngira vd kjp thdi sua ldi Nguydn hdm - Tich phdn vd iTng dung efta tmdng cho hgc sinh, eung nhu hudng ddn hgc sinh cdc Trung hgc Phd thdng Due Tri) cdch han chl bj ldi gidi todn ^di tfch fi r Ldi gidi (P.T.K.T., Idp 12AI, ttudng Tmng hoc Phd tiidng E)ftc Tri) BAN CHlTNG CAC TRUdNG H(;rP LOI CUA HOC SINH Nguyen ham F(x) efta ham / ( x ) cd dgng: f^M = i fMdx = lcos3xcos5xdx = -l(cos2x 1 1-i + cos8x)dx ^ ~\-sm2ji: + —sinSx 2l2 J (i 2x 8jr 1 Ma F | _ | = - nen + ~sin— +C = - o - + C = - o C = itj — sm ^ ) 1,2 4.2 Loi thilu dat di£n Iden boac d^t diei Vay f(jr) = -sin2j:+—sinSjr— kien khong dung •* 16 , Di bai: Viet phuong trinh tifip tuy&i cua dd thi Mi; h(ic smh dJ sai tmh C = - , Icet qua dung ta C = — (C):y = x^ -3x + , biet tijp tuyin song song voiduongthlng 3' = 9x+18 Tgp chi Khoa hgc Trudng Dgi hgc Cdn Tho Phdn C: Khoa hgc Xa hgi, Nhdn vdn vd Gido due: 34 (2014): 7-33 _ (trich td de kilm tra tgp thd 45 phut chuang Ung dung dao hdm dl khdo sdt vd ve dd thi hdm sd cua ttudng Trung hgc Phd tiidng Dftc Tri) Ldi gjdi ( H.V.L., Idp 12AI, tmdng Trung hge Phd thdng Nguyin Quang Dieu) Ldi giai ( L.M.S., Idp 12A1, tmdng Trung hgc Phd tiidng Dfte Tri) t =3x +l=>2tih = 6xdx=i>xbc=-U Ggi d Id tidp cua dd thi (C) song song vdi dudng tiidng y = 9x + I Doi can: x = 0=> Phuang trinh tidp tuyln d cd dang: y = 9x + i» Khi do, ta co: xdx _ tdt 1? |2 = - / * = - « , +C = -i 31 3 " d la tidp tuydn eua (C) vd chi h? phuang ttinh sau cd nghidm: J9jc +fc= ;c^-3;c + 2(l) [9 = 3^:^-3 (2) VJy: Gidi phuang Uinh (2), ta dugc x = hogc x = -l Thay x = vao phuang trinh (I), «, duac & = ~14 Ldi: hgc sinh khdng dat didu kien b^\% ndn da khdng loai dudng thdng y - J: +18 4.3 Hidu sai khai ni^m \ De bdi: Tinh tich phdn sau: \ ! xdx o'JiJTi' Ldi: hgc sinli hieu khdng chinli xdc dinh nghia tich phan vd nguyen ham ndn dd nhdm Idn giiia tieh phdn vd nguydn hdm 4.4 Loi hidu sai djnh ly, cdng thuc Thay x = -2 vao phuang ttinli (1), ta dugc fo = 18 Vdy cae phuang ttinh tidp tuyen can tim td y = 9x-14,y = 9;c + 18 x=\ Di bdi: Tim gia tti Idn nhdt vd gid tti nhd nhdt cua ham sd y = J: In JC ttdn dogn (ttich tft di thi hgc ky I eua tmdng Tmng hgc Phd thdng Tdn Chau) Ldi giai (C.A.N., ldp !2A1, tmdng Trung hgc Phd thdng Tdn Chdu) xdx oJ? Hdm sd y = t" In JC lidn tuc ttdn dogn -1 (ttich tft dl thi hgc kJ I ctia tmdng Trung hgc Pho thdng Nguydn Quang Didu) 2 y' = (x y\nx + x ([.nxy = 2x.\nx + x —= \n\e =0 (logi) =e (nhgn) _i y' = « X In JC + ;c = O X (2 In JC +1J = 0,Vx e Vay m-i Maxy = y(c) = e , Miny = y\ e y(c) = e Ldi: Sai; e - e^, d day hgc sinh cho I I Vay Maxy =y(e) = e ,Mny =y\ — = 4- h i 2- rding (a«f=aa''(fl>0) (dftng Id: aP (a>0)) 4.S Loi nhd khdng chinh xdc quy tdc, cong thftc, ky hi^u Ldi: hgc sinh khdng thuOc quy tSe tinh dgo hdm (u.v)' = u'.v+u.v" (vdi u = u{x),v = v(,x) la cdc hdm sd cd dao hdm tai diem x thuOc khodng xdc djnh) ndn tu vide tfnh dao hdm sai dan den kdt qud sai 4.6 Ldi khdng thanb thao dp dung cdc ky thuat cu ban, giai cdc dang todn ca ban Di bdi: Tim gid trj Idn nhdt va gid tri nhd nhdt Di bdi: Tinh tich phdn sau: cua hdm sd 3- = J: hi X trdn dogn J vI + 3cosj:.sin«ic (trich tft dd thi hgc ky I efta trudng Tmng hgc phd Thdng Tdn Chdu) Ldi gidi (H.M.L., Idp I2AI, tnrcmg Trung hgc Phd tiidng Tdn Chdu) (dd kidm tta 45 phut ldp 12A4 ciia tmdng Tmng hgc Phd thdng Nguydn Quang Didu) Ldi giai ( N.T.K.T., Idp I2A4, ttudng Trung hoc Phd thdng Nguydn Quang Didu) D$t t = yl\ +3 cos X => t =1 + cos J: => sin xdx = Khi do: J vl + 3cosjc.sinjait = 2? 30 Ldi: hgc sinh dung phuang phdp ddi bidn sd nhung Igi qudn ddi cdn din den kdt qua sai 4.7 Ldi khdng biet dien dat chinh xdc trinh bdy ldi gidi Di bdi Tinh tich phdn sau: K = J In—dx 2 TT 36 (trich tii di kidm tra tap thi 45 phut chuong Nguydn hdm - Tich phan vd LTng dyng efta tmdng Tnmg hgc Phd thdng Dfte Tri) Ldi gidi (T T.N., lap UAI, trudng Trung hgc Phd thdng Due Tri) Phdn C: Khoa hoe Xd hgi Nhdn van vd Gido due- 34 (2014): 27-33 Tgp ehi Khoa hgc Trudng Dgi hgc Cdn Tha 2e hi—.2e2 0-(2e-2) ioj: hgc sinh khdng bidt cdch ttinh bay ldi giai 4.8 Ldi ghi sai dl, khdng sft dung gia thiet cua dd bai Taduoc 2e „ , X 2« In—,x - / x.-dx ^ 2 2e 2e , In —.J: - f IJx 2 2e , X |2e m — x 2 Di bdi Tinh tich phdn sau: J (JC -1) xdx (ttich tu dl kidm tta 45 phut Idp 12A1 cua tmdng Trung hoc Phd thdng Nguydn Quang Didu) Ldi gidi ( P.T.M.D., Idp 12A1, tmdng Trung hgc Phd thdng Nguydn Quang Didu) Taco: I(x-l)2(fa= f(j:2-2:t + l)A I Ldi: hgc sinh chdp sai dd ndn ddn ddn sai 4.9 Loi ngd nhan kien thftc Di bdi Tim gid tti ldn nhdt vd gia tti nlid nhdt cua hdm so y = x \nx ttdn doan _ 3 (ttich tft dd thi hgc ky I cua tmdng Trung hgc Phd tiidng Tdn Chau) Ldi gidi ( D.Q.K., ldp 12A2, tmdng Tmng hoc Phd thdng Tan Chau) [l Ham sd y = JC hix lien tuc ttdn doan 2 y' = (x )'\nx + x (]nxy^2x.]nx + x —= 2jc.lnx + Ji: JC - y' = 0«2jc.Injc + jc^0