SU" DUNG MO HINH H O A TOAN HOC
TRONG VIEC DAY HOC T O A N
O TRAN DONG - NGUYEN THI TAN AN*
3n hqe Id eon dudng tu duy cd he thdng, sdn sinh ro nhi/ng gidi phdp cho nhung vdn de bdng cdch md hinh hda cdc tinh hudng trong eude sdng. M d hinh hda todn hqe duqc hieu Id sy chuyen ddi mdt vdn de gdn gui sang nhCrng khdi niem todn hqc, sy md td nd bdng ngdn ngCr todn hqc nhd viec xem todn hqe nhu Id edng cy de gidi quyet vdn de. Tren thye te thi nhung khdi niem todn hqe deu bdt ngudn tu thye hen (1). Sudyng md hinh hda todn hqe, md hinh todn trong viec dqy hqc gdp phdn gidi quyet nhung khd khdn trong viee dqy hqc Todn. Sou ddy, chung tdi se trinh bdy cdc khdi niem lien quan den md hinh hda todn hqe, md hinh cung vdi mdt vi dy de phdn tich tinh hieu qud trong viee trq giup cua md hinh.
1. Cdc khdi niem
Md hinh Id mdt mdu, mdt ke hoqch, mdt dqi dien, mdt minh hoq duqc thiet ke de md td cdu true, cdch vdn hdnh cuo mot ddi tuqng, mdt he thdng hoy mdt khdi niem. Md hinh theo y nghla vdt li cua nd, dd Id bdn sao, thudng thi nhd hon cuo mdt ddi tuqng. Md hinh dd cd cung nhieu tinh ehdt vdi ddi tuqng gdc: nd cd cung nhung diem dqe trung, ed the Id mdu sdc thdm ehi cd chuc ndng vdi ddi tuqng md md hinh dd bieu d i l n . Mot md hinh If thuyet cuo mdt sy vdt hien tuqng Id mdt tqp hqp cdc quy tde bieu d i l n chinh xdc sy vdt hien tuqng dd trong ddu cuo ngudi quan sdt (2).
Md hinh todn hqc Id mdt edu true todn hqc md td gdn dung dqe trung cuo hien tuqng dd (2).
Sou ddy Id hai vf dy ve md hinh todn hqe:
Vidu 1: Md hinh do (MHA) Id nhung md hinh chu yeu duqc xdy dyng bdng cdc phdn mem tren mdy tfnh nhdm md phdng nhung md hinh trong thye te md ngudi su dyng ed the thao tde, sua ddi. MHA ve todn duqc xdy dyng de ho trq ngudi hqe kien tqo tri thuc todn.
Chdng hqn khi dqy khdi niem mat trdn xoay:
«Trong khdng gian, cho hinh (H) vd dudng thdng
Tap ehi Giao due so 2 1 9 (ki i • 8/2009)
d. Hinh gam tdt cd cdc dudng trdn (CJ vdi M thudc (H) dugc ggi Id hinh trdn xoay sinh bdi (H) khi quay quanh d. Khi hinh (H) Id dudng hinh
mdt thi trdn sinh xoay
bdi nd edn
ggi Id mat Hinh 3 trdn xoay"
(Hinh hqc 1 2 - ndng cao), ed the tqo ra nhi/ng MHA bdng phdn mem Cobri 3D de giup HS tiep thu duqc khdi niem mong tfnh truu tuqng ndy.
Trude het, to dyng trong khdng gion mdt dudng thdng d , dudng cong (H), vd dudng trdn (C^^) vdi M Id diem chuyen dqng tren (H) (hinh l);vd\ chuc ndng hogt ndo vd quy tich cuo phdn mem to ed the cho HS quan sdt duqc hinh dnh cuo mdt mat trdn xoay duqc tqo ro theo edeh SGK dmh nghla (hinh 2, hinh 3) vd cdc em cung ed the quan sdt mdt trdn xoay ndy d nhieu vj tri, gdc do khde nhau vdi chuc ndng hinh cdu kinh cuo phdn mem.
Vi du 2: Md hinh thao tde ddng Id cdc MHA tqo ra mdi trudng hqe todn trong dd HS thyc hien duqc cdc thao tde keo re, nhdy ehudt, su dyng bdn phim mdy tfnh de thay ddi cdc yeu td cua md hinh nhdm quan sdt, phdng dodn, khdm phd kien thue. M d hinh thao tde ddng cung Id mdt trong so nhi/ng dd dung dqy hqc thao tde duqc. Dd dung dqy hqe do thao tde duqc trong todn hqc Id mdt sy bieu dien cd tinh tuong tde
* Tnrang Dai lipc sir pham - Dai lipc Hue
# >
Hinh 4
Hinh 6
cua mdt md hinh todn hqc tren mdy tinh cho phep ngudi hqc xdy dung hoqc cung c d viec hieu kien thuc t o d n hqc ( M o y e r , Bolyard & Spikell, (3).
Bdi todn d y n g thiet dien sou Id mdt vf dy cho md hinh ndy: «Cho hinh lap phuang ABCD.
A'B'C'D' vd dudng cheo AC. Dung thiet dien cua .,^ ••, hinh lap phuong edt bai '/••'t*^ mat phdng qua diem M f','/ thugc AC vd vudng gdc vdi AC". Vdi phdn mem Cabri 3D, chung ta cd the tqo ra mot md hinh thao tde dqng nhu hinh ben (hinh 4, hinh 5), vd HS cd the djch chuyen diem M trong doqn A C , quan sdt de thdy duqc cdc trudng hqp cua thiet dien, tu dd cd mdt Idi gidi ddy du cho bdi todn. Ngodi ra, nhd cdc chuc ndng dyng hinh cua phdn mem (song song, vudng gdc...) HS cd the kiem tra Iqi tinh dung ddn cdch dyng cua minh, chdng hqn, dyng mdt phdng qua M song song vdi (CB'D') hoqc (BDA') thi se thdy rdng nd trung vdi mdt phdng (P) (qua M vudng gdc vdi A C ) (hinh 6).
Md hinh hod todn hgc: quy trinh tqo ra mdt md hinh todn hqc duqc gqi Id md hinh hod todn hqc. Mot vdi cdu true todn hqc co bdn cd the dung de md hinh hod Id cdc dd thj, phuong trinh (cdng thue) hoqe he phuong trinh hay bdt phuong trinh, chi sd, bdng sd hay cdc thudt todn (2).
Vidu: Khdo sdt mdy bom nudc (4): «Mdt ngudi ndng ddn edn mua mdt mdy bom nude. Sou khi di khdo sdt thj trudng dng ta quan tdm den ba loqi mdy cua hdng National (Nhdt) vdi luu luong nudc 45 lit/ phut. Notional A-130 JTC: 1.750.000 ddng; A - 1 2 5 JBC-1: 1.400.000 dong; GP-125 JB: 800.000 ddng. Vd tuong ung vdi mdi loqi tren, tien dien tieu thy Idn luqt Id: 1000 ddng,
1200 ddng vd 1800 ddng cho moi gid bom nude.
Trong trudng hqp ndy, bqn tu vdn cho ngudi ndng ddn d d nen mua loqi ndo vd gidi thfch cho ngudi ndng ddn ve sy lya ehqn cuo bqn".
Kien thuc, kl ndng edn su dyng: Hdm so bde nhdt vd dd thj, kl ndng dqe dd thj; cdch gidi he phuong trinh bde nhdt hai dn so. ;
Cd the gidi bdi todn nhu sou: Neu chi can cu vdo gid thdnh ban ddu de ehqn mua mdy thi ta cd the ehqn ngay loqi mdy thu ba cd gid 800.000 ddng. Tuy nhien, so tien that su bao gdm cd sd tien mua mdy ban ddu vd so tien tieu thy dien.
Gid su f(x), g(x), h(x) Idn luqt Id sd tien (tfnh bdng nghin ddng) phdi trd khi su dyng mdy bom loqi thu nhdt, thu hai vd thu ba trong x ngdy (bao gdm tien dien vd tien mua mdy bam). Luc d d ta cd phuong trinh vd d d thi cua ba hdm so tren mdt he true too d d :
f(x) = 1 7 5 0 - I - 2 4 x g(x) = 1.400 -I- 28,8x
h(x) = 800 + 43,2x
Hodnh do giao diem cua 66 thj f(x) vd g(x) Id 7 3 , d d thi g(x) vd h(x) Id 4 2 vd dd thj h(x) vd f(x) Id 4 9 (hinh 7).
D y o v d o cdc d d thj ta thdy: neu thdi gian su dyng trong vdng 4 2 ngdy (cd the khdng lien tyc) ngudi ndng ddn nen mua loqi mdy bom gid 8 0 0 . 0 0 0 d , neu su dyng k h o d n g 4 2 d e n 7 3 n g d y
ngudi ndng ddn nen mua loqi mdy gid 1.400.000 d vd neu thdi gian su dyng tren 73 ngdy thi loqi mdy gid 1.750.000d Id lya chon tdt nhdt. DT nhien, ngodi hai chi so quan trqng tren, thyc te ngudi ta edn quan tdm den do ben cue sdn phdm, thdi gian bdo hdnh...
2. Quy trinh mo hinh hdo toan hoc Kaiser vd Blum d d de xudt quy trinh sou (4):
Hinh 7
M6 hinh thuc te
A
(a) Tinh huong thuc te
(b)
(d)
Mo hinh toan hoc (c) Ket qua toan hoc
#
Ndi chung, quy trinh md hinh hod todn hoc thudng bao gdm cdc budc: Mdt tinh hudng thyc te Id diem xudt phdt cho qud trinh; tinh hudng do duqc li tudng hod, h>c Id don gidn hod vd cdu true hod de cd duqc mdt md hinh thyc te. Tiep theo, md hinh ndy duqc todn hqc hod, h>e Id duqc thdng djch sang ngdn ngu todn hqc de dan den mot mo hinh todn hqc cua tinh hudng ban ddu. Sy xem xet todn hqc trong md hinh todn hqc de tqo ra ket qud todn hqe; ket qud se duqc thdng djch Iqi qua tinh hudng thyc te. Sy hodn thien cua ket qud phdi duqc
Tap ehi Giao due so 2 1 9 (ki i . 8/2009)
kiem h-o, h/c Id duqc edng nhdn tinh dung ddn.
Trong h-udng hqp Idi gidi khdng duqc thod ddng qud trinh ndy phdi duqc lap Iqi.
Cd hoi hudng khde nhau de su dyng nhi/ng md hinh trong mdi trudng hqc tdp: «hoe de md hinh hdo" vd «hqe vdi nhung md hinh". Hudng
«hqc de md hinh hda" tdn thdnh viee dqy cdch md hinh hdo nhi/ng vdn de thye tien. Ngudi hqe duqc trdng dqi xdy dyng nhi/ng md hinh cuo rieng minh vd su dyng nd nhu Id phuang tien giao tiep trung gion de dien dqt kien thuc cuo ngudi hqe. Nguqe Iqi, hudng «hqc vdi nhi/ng md hinh" khuyen khich ngudi hqc gidi quyet nhirng vdn de vdi sy ho trq cua nhCrng md hinh dd duqc tqo sdn. O ddy, ngudi hqe duqc cung cdp nhCfng md hinh tqo sdn tu nhirng tinh hudng hoqc vdn de cy the vd ddi hdi ngudi hqc thay ddi nhi/ng tham bien xdc dinh trong md hinh de ed the gidi quyet nhCrng vdn de lien quan. Ngudi hqe duqc trdng dqi se nhin thdy nhi/ng mdi quan he giCfo nhCfng ddi tuqng ben trong md hinh vd tu dd xdy dyng nhung khdi niem todn hqc thdng qua «Sy truu tuqng cuo todn hqc". Cdch tiep edn ndy tdn thdnh viee tqo ro nhirng md hinh dqe trung, tich eye vd thao tde duqc (1).
3. Md hinh ddng vd viec dqy hqc todn Heddes khdng dinh rdng, dd dung dqy hqe do thao tde duqc trong todn hqe se giup HS hqc cdch hqp tde cung nhau de gidi quyet vdn de, thdo ludn cdc y tudng vd khdi niem todn hqc, dien dqt tu duy todn hqe cuo minh, trinh bdy y tudng trudc nhdm Idm viec. Trong cdc nghien euu cuo minh, Spieer dd nhdn djnh dd dung dqy hqc do thao tde duqc cd the tqo ra sue mqnh hiru hinh cho nhi/ng gi khdng the nhin thdy hoqe tudng tuqng duqc. Bolyard, Steen, Brooks, & Lyon cung de cap den vol trd cua dd dung dqy hqe do thao tde duqc: nd Idm cho nhirng ddi tuqng todn hqe truu tuqng cd y nghla hon, thuc ddy ddng co vd ndng cao ket qud hqc top cuo HS (3). Cdc md hinh todn tfch eye duqc thiet ke bdng nhi/ng phuong tien edng nghe, nhu mdy tinh bd tui vd mdy tinh dien h/. Id nhCrng edng cy thiet yeu de dqy, hqe vd Idm todn. Ode biet, nhirng md hinh todn tich eye duqc thiet ke bdng phdn mem dqng tren mdy tfnh cung cdp nhirng hinh dnh trye quan ve cdc y h/dng todn hqc, thuc ddy viee sdp xep vd phdn tich cdc dCf lieu vd tinh todn mot edeh ed hieu qud vd chinh xdc. Chung ed the ho trq nhi/ng khdo sdt todn cua HS trong mqi fTnh vyc todn hqe, bao gdm hinh hqc, dqi so, gidi tich, thdng
Tap chi Giao due so 2 1 9 (ki i - 8/2009)
ke, do dqe vd sd. Vdi nhCrng cdng cy cdng nghe phu hqp, HS cd the top trung vdo viee dua ra quyet dinh, phdn dnh, suy ludn vd gidi quyet cdc vdn de todn hqe (5; h-.l 3).
MHA ho trq HS kien tqo tri thire todn the hien trong cdc yeu to sau: - Hd h^q HS quan sdt, khdm phd, hinh thdnh khdi niem todn hqc; - Tdng eudng khd ndng phdng dodn; - Phdt frien ndng lye ty hqc, khd ndng Idm viec theo nhdm vd Phdt trien tu duy logic, h/ duy sdng tqo vd tu duy phe phdn cho HS. Hon nua, quo thye nghiem tde gid dd dua ra bdng cdc tieu ehi de ddnh gid tinh hieu qud cuo md hinh ddng the hien trong bdng sou:
7"/eu chi True quan
Tuong tac giCia GV, HS va mo hinh HS thao tac duOc
GiCr t)lt bien cac tinh ctiat toan hoc Ho trq tim tdi kham pha Md rpng duqc
Than thien Cong
Diem toi da 1,0 1,5 2,0 2,0 1,5 1,0 1,0 10,0 Mdt md hinh duqc xem Id t ^ neu ed tdng diem tu 7 - 10, Id dqt neu ed tdng diem h> 5 - dudi 7. Nhirng md hinh cd tdng diem dudi 5 thi edn cd nhung chinh sua de ddp ung cdc tieu chf dd phdn h'ch d tren. • (1) Soner D u r m u s , Erol Karakirik. "Virtual manipulatives in mathematics education: a theoretical framework". ISSN: 1303-6521, Vol 5, Article 12. 2006.
hup://www.tojet.net/articles/5112.doc.
(2) Frank Swetz and J.S.Hartzler. Matliematical Modeling in the Secondary Scliool Curriculum.
NCTM.USA. 1991.
(3) Nguyen Dinh Ho^ng Nhan. Tde ddng ciia md hinh do duac tpa dp hda trong ho tra HS kien tao tri thuc todn. Luan van Thac si, dai hoc Hue, 2008.
(4) Tran Dung. Sit dung md hinh hod todn hoc trong chucmg trinh todn phd thong de ndng cao khd ndng gidi quyet vdn de cho ngudi hoc. Luan v3n thac si, dai hoc Hu^, 2007.
(5) Tran Vui. "Tich hap cdng nghe thdng tin vdi nghien euu bdi hoc de giiip gido vien todn tu ndng cao ndng luc vd hodn thien nghiep vu su pham". Sd GD-DT Thira Thien - Hue, 2008.
Tai lieu tham khao
1. Doan Quynh (tdng chu bien). Hinh hoc 12 (nSng cao). NXB Gido due, H.2008.
2. Sophie va Pierre Rene de Cotreva (Montreal, Quebec, Canada). Cabrilog - Innovative Math Tools - User Manual.
