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Tap chl KHOA HQC OHSP TPHCM 86 34 ndm 2012 CAC CACH TIEP CAN CUA KHAI NIf M PHAN S6 TRONG LICH SU VA SACH GIAO KHOA TOAN 6 TIEU HQC DL/QNG H O U TONG T6M TAT Lich sic mang lgi nhieng cdch liip cgn khd[.]
86 34 ndm 2012 Tap chl KHOA HQC OHSP TPHCM CAC CACH TIEP CAN CUA KHAI NIf M PHAN S6 TRONG LICH SU VA SACH GIAO KHOA TOAN TIEU HQC DL/QNG H O U T O N G T6M TAT Lich sic mang lgi nhieng cdch liip cgn khdc cho mqt khdi ni?m todn hqc Cdc nhd li ludn dgy hqc, ldc gid sach gido khoa Iga chqn cde cdch tiip can phit hqp vdi trinh nhdn ihicc vd dgc diim ciia hqc sinh Do do, mqt khdi ni^m todn hqc sdeh gido khoa cd Ihi duqc tiip can khdng luang dong nhu Irong lieh sir Trong bdi bdo ndy, cdc cdch liip can cua khdi ni^m phdn so lich sic vd sdeh gido khoa Todn a tieu hqc si duqc ldm rd Tukltda: cdch tilp cgn, phan so, ljch sir todn, khdi nifm todn ABSTRACT The approaches to fractional concept in history and mathematics textbooks in primary school History brought different approaches to a mathematical concept Learning theorists, textbook authors chose the approaches in accordance with pupils 'cognitive levels and characteristics Therefore, a mathematical concept in the textbook could not be approached as the same in the history In this article, the approaches to fractional concept in history and mathematics textbooks in primary school are clarified Keywords: approach, fractions, mathematical history, mathematics concept Ddt van de Mdt nhiing khdi nifm loan hgc ma hgc sinh (HS) tiiu hgc thudng gap khd khan nhgn thiic la khdi nifm phan sd Phdn so cd vj tri, vai trd quan trgng cac mgch kiln thirc loan d lidu hgc, ddng thdi nd la ca sd dl md rgng cac logi so khdc: hon so, so thgp phan, sd hthi ti, Do do, nhifm vy ddt ddi vdi giao vidn (GV) tiiu hgc la phai lam cho HS cd nhu'ng hiiu bilt dung din vl khai nifm phdn so, d^c bift la hinh lhanh khdi nifm ban diu vl phdn so mdt each chinh xdc Cdc cdch tiep c^n khdi nifm phan so lich sir NCS, Tru'dng Ogi hpc Su phgm TPHCM 68 Nghidn ciiu cdc tdi lifu ljch su, chung tdi nhgn thiy vifc md rgng he thdng so tir so ty nhidn sang sd biiu dien bdi phdn so dugc tiin hdnh theo hai each: ' y xuat phdt tir nhu cdu cua cugc sdng va xuit phdt tir ngi bg todn hgc Thiir nhat, phan sd ddi dd gidi quydt cdc van de thyc tl: nhu ciu dgc (nhiiu ta gap cd nhumg dgi lugng khdng chiia dyng mot sd ty nhidn lin dom vj do) va nhu cau chia nhihig vgt nbilu phin bdng Thii hai, tgp hgp so biiu dien bdi phan s6 ddi xuit phdt tir ngi bg todn hgc: dl f cho phep chia cdc sd nguyen cho mdt s6 khdc ludn ludn thyc hifn dugc, hoac cdc phuomg trinh dgng bxx = a (b khae 0) ludn ludn cd nghifm Trong qud trinh Tap chi KHOA HOC OHSP TPHCM Duang Hitu Tong md rdng nhu tren, phdn sd dugc tilp can theo each nhu sau: 2.1 Cdch tiip can dua tren s6 phhn eua cdi todn thi f Caeh tiep can cd lien quan den bai loan: "Tim mdt so phin ciia mgt ddi tugng dugc chia cac phin bdng nhau" Trong ljch sii, khdi nifm vl dgi lugng phan sd phat triin tir thdi co dgi "phan sd" dd dugc quan nifm nhu "khdng chia dugc va khdng chia hit" (Klein, 1968) Mdt dgi lugng phan so khdng dugc xem nhu la mgt sd nhiiu thi ki; diing hom, nd da dugc sir dyng nhu mdt dom vj mdi biiu didn cho mdt phdn hodc cdc phdn ciia mgt sd cho din Stevin (1548-1620) tuydn bd ring dgi lupmg la mdt sd bdng cdch djnh nghia phdn sd nhu la "mdt phin cua cac bd phdn cua cai toan thd" (Klein, 1968), 2.2 Cdch tiip can dua trin lu&ng Ngudi ta tim thiy cdc phdn sd tir cdc sd ty nhidn qua cdc sd vd ti If, gidi quydt nhu cdu tim mdt dom vi ludng ehung ddi vdi hai dgi lugng Trong ljch sir, thudt ngii bao gdm sd dgi lugng va ti If Id "tinh ed thi so sdnh dupe" dugc djnh nghia bdi nhd todn hgc Hi Lgp, Euclide (thi ki III, tmdc edng nguyen) nhu sau: "Nhumg ldn dugc cho la cd the so sanh dugc vdi ndu dugc ludng bdi cimg dom vj do, va chiing khdng thd so sdnh dugc nlu chiing khdng cd dom vj ludng chung" (Heath, 1956) Theo y nghia hifn dgi, nlu A vd B (khdc 0) la hai sd cd thi so sdnh dugc vdi ndu tdn tgi dgi lugng C cho A = mC vd B = nC vdi m, n la cde so nguyen va n^O Euclide khdng xem dgi lugng C nhu la mgt sd, nhung nhu Id "mgt phin hay cdc phin cua mgt sd" (Klein, 1968), 2.3 Cdch tiep cgn dua trin phip chia Cdch tilp can ndy sinh Iiic ngudi la di tim nghifm cho phuang trinh bxx = a vdi a, b Id cac sd nguyen, b khdc Cy thi, nd dugc tim thiy djnh nghia thdng thudng cua mdt tmdng, dugc hinh thdnh diu lidn bdi Galois vao diu thi ki XIX va dugc thill lap cy thi bdi Dedekind vdo ndm 1871 (Baumgart, 1966), Chung tdi ggi day la each tilp can dya trdn phep chia vi nhu cdu phdi ed phdn so — la kit qud ciia sy can thidt de cd mdt tap hgp sd phep chia la ddng kin (tiic la tdn tgi phin tii nghich ddo vd thda mdn cac tidn dd cua tmdng) nhim gidi quydt cdc vin dd dgi sd, 2.4 Cdch tiep can dua trin li thuyet tap hpp Theo cdch tilp can nay, ngudi ta djnh nghia cdc phdn sd nhu la tap hgp cdc cap sd nguyen cd thii ty Cy thi, cae nhd loan hgc tilp can nhu sau: Liy tap hgp S gdm cdc cap sd nguydn cd thii ty (a, b), vdi b khdc Phdn chia tap S cdc tap hgp vdi quy tic: hai cdp (a, b) vd (c, d) ndm cung mdt tap hgp nlu ti so - bdng b vdi ti sd —; tiic la, nlu va chi nlu d ad = bc (Childs, 1995) Cdch tilp can cd thi dugc tim thiy thi ki XIX vd thi ki XX Bdng sy nd lyc dk phat triin mdt nin tang todn hgc chat che 69 86 34 ndm 2012 Tgp chl KHOA HQC OHSP TPHCM Chuang trinh Todn gidi thifu cdc mgt so nhd todn hgc chuyin sang s6 hgc nhu Id nguin goe cho nin tdng nhu vgy phdn so: ]-, - , - , - Trong do, Vdo cuoi thi ki XIX, Cantor phdt triin li SGK Todn cho HS lam quen vdi cdc thuyit tgp hgp, md cuoi ciing din den vifc hinh thdnh cdc djnh nghTa li thuyit phdn s6 dom vj - vdi w < 10, tgp hgp vl so hthi ti, Diiu ndy rit rd rdng n phong trdo "todn hgc mdi" cua Trong bdi "Phdp chia", cdc tac gia nhirng ndm 60, dya vdo tde phim cua SGK Todn trinh bdy khdi nifm "phin Bourbaki, bing nhau" cua mgt dan vj, Cdc cdch tiep c|n khdi nifm phdn n n n so sdeh gido khoa ldp 2, ldp vd d chia phin bdng nhau, ldp Dgy hgc phan so d tiiu hgc nhdm moi phin cd d, CJ ddy, ngudi ta chi cung cip cho hgc sinh mgt logi so mdi, ngim in gidi thifu vl khdi nifm "phin bidu didn duge thuang diing ciia hai so ty bing nhau" chii khdng gidi thifu trye tilp nhidn, ciing nhim ddp iing nhu ciu bidu vl phdn sd SGK cung dua thdm nhiiu didn chinh xdc cdc so dgi lugng bdi tap theo kidu tidp can so sdnh so ddi sdng thuc tidn Phdn so dugc chinh lugng ciia mgt bg phgn cua tgp so vdi thiic dua vdo gidng dgy mft cdch tuomg todn tgp hgp dd Chinh vi Id do, chung ta ddi diy dil d chuang trinh Todn ldp cd the ggi tdn cdch tiep can la "tidp Dgy hgc phdn sd Todn Id sy tilp cgn kiiu tap hgp" Ldp mang lgi cho HS cdch tidp noi mgch kiln thiic vl phdn so d ldp vd ldp 3, dong thdi ldm ca sd vumg chic dl cgn phdn so dom v) theo difn tich cua mft dgy hgc vl phdn so thdp phdn, hdn so d so hinh ca bdn nhu hinh vudng, hinh chii ldp Tir dd, SGK hf thong hda vd hoan nhdt Cdc hinh dugc chia thdnh cac chinh todn bg ngi dung dgy hgc phdn so d phan bdng nhau, ngudi ta tde ddng den tidu hgc, chuan bi cho dgy hgc so thgp mgt sd phan ndo dd, tir dd ldm ndy sinh khdi nifm phdn so Ching hgn, mgt bai phdn 3.1 Cdch tiip c^n phdn s6 & l&p vd tgp dugc dua SGK Todn nhu sau: l&p3 ^ mt © Da td vdo - hinh ndo? Hinhl 70 Hinh Hinh T9P chl KHOA HOC OHSP TPHCM Duang Hiru Tong Tdm Igi, SGK Toan va chi dl 9 cap den cac phan sd dom vj, Tuy nhidn, cdc tac gia khdng ndu tdn phan sd md chi dl cap mdt each an tang thdng qua khai nifm "phin bdng nhau" Phan sd dugc xem nhu la "cdng cy ngim an" dd giai quydt dgng loan "Tim mdl cdc phin bing ciia mdt sd" 3.2 Cdch tiip can phSn so SGK Todn a) Cdch hinh thdnh khdi niem phdn sd SGK SGK Todn hinh khai nifm phan sd nhu sau: Chia hinh trdn thdnh phdn bdng nhau, td mdu vdo phdn Ta ndi: Dd td mdu vdo ndm phdn sdu hirdi trdn , Ta viet: —, doc Id ndm phdn sdu chiing tdi cd the phat bieu nhu sau: Phan sd la cdp sd thir ty (a, b) dd a, b la / r ' Id sd tu nhien viet tren ggeh ngang Tir sd cho biit phdn bdng dd duac td mdu SGK gidi thifu khai nifm phdn sd qua vifc chia cdi loan the thdnh b phdn bing nhau, Sau dd, liy a phin tdng , a sd b phdn dd Nhu vdy, cd duac phan so — b Cdch trinh bdy ndy phu hgp vdi each tilp can dya tren sd phin cua cdi loan thi lich sir ciia phdn sd SGK khdng dua dinh nghia chinh thiic ciia phdn sd theo cdch tilp can ndy CJ day \ cdc sd ty nhien va ?t 0, b ehi sd phan bdng md dom vj trgn v?n duge chia vd a chi sd phin bing dd liy Djnh nghTa ndy dugc cy thi nhu sau: I chia cho b, ta dugc - (mdt phin b b cua dan vj) Tilp din, liy a lin so hang - , tiic b I I l a - + - + + - = b b b b * V ' a sd hgng Thdm vdo dd, SGK cdn ndu len cdch vill mdu sd, tii sd vd diiu kifn ciia mdu sd thdng qua nhgn xet sau: "Mdi phdn sd ed tir sd vd mdu sd " Tie sd la sd 9 6 Id 5, mdu sd Id Mdu sd Id so tir nhien viet duoi ddu ggeh ngang Mdu sd eho biet hinh trdn duac ehia thdnh phdn bdng Tir sd t 9V 9 Mr ^ tu nhien viet tren ggeh ngang Mdu sd Id sd tu nhien khde viit dudi ggeh ngang" Ngoai ra, SGK Todn cdn tilp cgn phdn sd nhu la kit qua ciia phep chia ciia hai sd ty nhidn ma sd chia khdc thdng qua bdi "PHAN SO VA PHEP CHIA SO TV NHIEN": "Cd edi bdnh, chia diu cho em Hdi mdi em duac bao nhieu phdn ciia edi bdnh " SGK trinh bdy: 3:4 = Hodc "Cd edi bdnh, ehia diu cho em Hdi mdi em dupe bao nhieu phdn cita cdi bdnh " SGK trinh bay: 5:4 = - Din ddy, ta thiy dugc each gidi thifu phan sd cd sy phdi hgp cua each ma da dugc dl cap tmdc dd: xuit phdt tir 71 86 34 ndm 2012 Tap chl KHOA HOC OHSP TPHCM nhu ciu thyc le va nhu cau ciia ngi bg toan hgc Nhu cau thuc tl d chd: SGK dua linh hudng nhu tren cd tir thyc lien cugc sdng Dd Id kit qua cua nhiing phep chia khdng hll Chimg td, thyc ll cd nhirng linh huong cho phep lam sinh khai niem so mdi - phan so, Nhu ciu ndi bg loan hgc d cho: Khai nifm phan so ddi cho phep thyc hifn mgi phdp chia thdng qua nhgn xet sau SGK: "Thteang eiia phep ehia SO tu nhien eho sd tu nhien (khde 0) ed the viet thdnh mdt phdn sd, tie sd Id sd bi muiiii N '*> * ' m* 9 \ -1 *•.• f z: Ta dugc — = - , vdi nhdn xet ring: t ehia vd mdu sd Id sd ehia " Ngam an sau do, phan sd ddi cdn cd mdt y nghTa khdc, Nd cho phep mgi phuomg trinh dgi sd dgng bxx = a (b^O) ludn cd nghifm, Vgy phan so la thuang cua phep chia mgt sd ty nhidn a cho sd ty nhidn b, b^O Trdn tap hgp sd mdi (Q") phep f chia sd ty nhidn a cho sd ty nhidn b, bjtO ludn ludn thyc hifn dugc (ddng kin ddi vdi phep chia) vd tap hgp Q" chiia mdt bd phan ddng ciu vdi N, Hom niia, each tilp can phan sd dya trdn phep chia td hifu qua ban each tilp tmdc dd vi gidi thidu them phdn sd khdng thyc sy (phan sd tir sd ldn ban mau sd), Bdn cgnh dd, tde gia cung neu ldn mdi quan hf cua mgt phin tOr ciia lap N vdi tap sd Q*: "Mqi sd tu nhien cd thi viet thdnh mdt phdn sd ed tu sd Id so tir nhien dd vd ed mdu sd bdng 1" Mdi quan hf se td rit hthi dyng thyc cdc phdp tinh sau nay, Tilp dd, cin dgy HS tinh chit ca ban ciia phdn sd, SGK trinh bay chu dl: 72 phan so bdng Kiln thiic rit cin thill cho vifc hgc quy ddng mau sd cac phan so, so sanh hai phan so, lam tinh vdi cac phan so Phan s6 bing dugc tac gia gidi thifu qua md hinh trye quan: Chia hai bdng giiy bing nhau, Bdng giiy thii nhal dugc chia thdnh phan, liy phin Bdng giiy thir hai dugc chia thdnh phin, liy phin 3x2 4x2 6:2 8:2 Rut kdt luan: ~8 Bai "Phdn so bdng nhau" ddnh dau each tiep can phan sd dya trdn li thuyit lgp hgp (da dugc de cap phin ljch sir) mgt each khdng tudng minh, Chiing tdi nhgn thiy SGK chua de cap each tidp can bdn dudi ddy ma dugc nhic ddn rdt nhidu dgy hgc so ty nhidn, Vidt tidp phdn sd thich hgp vao chl chdm: 1 ii^ 10 10 Cdch tilp cgn cd thi dugc ggi la cdch tilp can tia sd Nd cd hidu qua cdc bdi tap so sanh cdc phdn so Ngodi ra, no cho thiy tgp hgp Q* la t|p hgp so trii mat, kbac vdi tap hgp so rai rgc N, tire trdn [O, l] khdng tdn tai sd tu nhidn nao nhung cd rit nbilu phan s6 ^H Dirang Hieu Tdng Tap chi KHOA HOC OHSP TPHCM Ngoai ra, SGK cdn dk xuit thdm each tilp can ti sd qua bai "Gidi thieu ti SO : "Mqt ddi xe ed xe tdi vd xe khdeh Ta ndi: Ti sd ciia sd xe tdi vd sd xe 9 » khdch la 5:7 hay — Ti sd ciia sd xe khdch vd sd xe tdi Id 7:5 hay — " Gidng nhu each tidp can dya tren phep chia, cdch tiep can ti sd cho phep 9 9 giai quyit cac linh huong Trong dd, d Idp phan so dugc nghien ciiu nhu la mgt "ddi tugng" tudng minh, HS chinh thiic dugc tim hiiu nd qua cdch hinh lhanh khai nifm, nghidn ciru cdc tinh chit CO ban, cac phep linh Tir dd, phan so trd "cdng cy tudng minh" dl giai quyit cdc kiiu nhifm vy cd lien quan Kit ludn Nhumg kit qua cua vifc phan tich d trdn cho thiy cac lac gia SGK dd cd sy chgn lya ddi vdi cac each tilp can khai nifm phan so Phan sd dugc tilp can tren tu tudng sd phan/cai loan thi, theo phep chia hai sd ly nhidn, tia sd, li sd,,,, SGK ciing chuyin tai dugc sy cin thill phai cd phan sd: xuit phat tir nhu ciu thyc tl eudc sdng va nhu ciu ciia ndi bd loan hgc, Cde each tilp can khai nifm phan sd ljch sii soi sang dugc cac each tilp cua ddi tugng SGK Toan d tidu hoc gidi thifu ca hai loai phan sd: phan sd thyc sy va phan so khdng thyc sy Tuy nhien, ddi nd dan din hiiu nhim cua HS khdng phdn bift dugc phdn sd va ti sd Nhgn xet: Phdn sd dugc nghidn curu d Idp 2, ldp d gdc in tang Khi dd nd chi dugc xem nhu la "cdng cy ngim in" de TAI LIEU THAM K H A O 1, Vu Quoc Chung (chu bien) (2007), Phuang phdp dgy hqc Todn d tiiu hqc, Nxb Gido due, Nxb Dgi hgc Su phgm 2, Do Trung Hifu, Do Dinh Hoan, Vu Duong Thuy, Vu Qude Chung (2004), Gido trinh Phuang phdp dgy hqc mdn Todn d tiiu hqc, Nxb Dai hpc Su pham 3, Do Dinh Hoan (2006), Todn 2, 3, 4, Nxb Gido due, (SGK hifn hanh), 4, Do Dinh Hoan (2006), Todn 2, 3, 4, Nxb Giao due, (SGV hifn hdnh) 5, Nguyen Thanh Hung (2008), Phuang phdp dgy hqc mdn Todn d tiiu hqc, Nxb Gido due 6, Nguyen Phii Ldc (2008), Lich sic todn hqc, Nxb Giao due, 7, Ddo Tam, Phgm Thanh Thdng, Hoang Bd Thinh, Thuc hdnh phuang phdp dgy hqc Todn d tiiu hqc, Nxb Da Ndng, 8, Pham Dinh Thuc (2003), Phuang phdp dgy hoc Todn bdc tiiu hqc, Nxb Dai hpc Su pham (Ngiy Tia soan nh$n dugc bit: 14-9-2011; ngiy chip nh$n ding: 04-10-2011) 73 ... 34 ndm 2012 Tgp chl KHOA HQC OHSP TPHCM Chuang trinh Todn gidi thifu cdc mgt so nhd todn hgc chuyin sang s6 hgc nhu Id nguin goe cho nin tdng nhu vgy phdn so: ]-, - , - , - Trong do, Vdo cuoi... can cd lien quan den bai loan: "Tim mdt so phin ciia mgt ddi tugng dugc chia cac phin bdng nhau" Trong ljch sii, khdi nifm vl dgi lugng phan sd phat triin tir thdi co dgi "phan sd" dd dugc quan... nhidn qua cdc sd vd ti If, gidi quydt nhu cdu tim mdt dom vi ludng ehung ddi vdi hai dgi lugng Trong ljch sir, thudt ngii bao gdm sd dgi lugng va ti If Id "tinh ed thi so sdnh dupe" dugc djnh