UAH DUNG CAP PHAM TRU "GAI RIEHG" UA "CAI CHUHG' VAO HOAT DONG KHAM PHA CAC KIEN THlfC HINH HOC
O GS. TS. O A O T A M * - NGUYEN C O N G C H U A N '
1. J. Bruner ddnh nhieu cdng sue vdn dyng li thuye't cOa J. Piaget de xdy dyng md hinh dqy hgc dyo vdo sy hqc tdp khdm phd cuo hge sinh (HS). Bruner de xudt md hinh dqy hoc dugc dgc trung bdi bdn ye'u to chO yeu: hdnh ddng tim tdi, khdm phd eua hge vien; cdu true tdi uu eua nhdn fhuc; cdu true eua ehuang trinh dgy hgc vd bdn chdt cua sy thudng phgt. Trong dd cdu true tdi uu cua nhdn thuc td yeu td' then ehdt (1).
Theo Bruner «hgc vien Id ngudi fy lyc, tich eye hanh dgng tim tdi, khdm phd ddi fugng hgc tap dehinh fhdnh eha minh ede nguyen fde, cdc y fudng ca bdn fu ede finh hudng hge tap ey the...". Trong hoc tdp khdm phd, hge vien di qua ba gioi doqn, ba hinh thuc hdnh ddng hqc tdp: Ddu fien edn phdi
^ao tdc hdnh ddng fren cdc tdi lieu dd cd (hdnh dgng phdn tich), sau dd hdnh ddng ten cdc hinh anh ve ehung (hdnh ddng md hinh hod) vd cudi cung rut raduge cdc khdi niem, cdc quy tdc chung fu nhung md hinh dd (hdnh dgng ki hieu hod) (1).
J. Bruner dd chi ra bdn li do cho viec stJ dyng phuong phdp ndy: Thuc ddy fu duy; Phdt trien dgng lyc ben frong han Id tdc ddng ben ngodi;
Hgc cdch khdm phd; Phdt frien fri nhd.
Cdc thdnh to ndng lye khdm phd gdm: Ndng lyc md hinh hod ede Idp ddi tugng, hien fugng Todn hgc fheo mdt sdquan he vd finh chdt ehung cua ehung; Ndng lyc ehuyen di chuc ndng hdnh dgng nhd chuyen ddi ede ddi fugng eua hogt dgng; Ndng lye the hien cdc quan diem bien chung cua fu duy Todn hgc frong viec phdt hien khdm phd kien thue mdi (2).
Bdi viet ndy de cdp vd'n de vdn dyng cap phqm tru «cdi rieng, cdi chung" nhdm trang bj eho HS nhCrng tri thuc cdn thiet vdo hoqt dqng khdm phd cdc kien thuc hinh hge.
2. Cdi chung v d c d i rieng: Cdi rieng Id phqm frij dung de chi mdt sy vdt, mdt hien tugng nhd't djnh. Cdi chung Id phgm tru dugc dOng de ehi nhi/ng mdt, nhCrng thuge tinh tqp Iqi trong nhieu sy vqt, nhieu hien tugng. Gida cdi rieng, cdi chung
Tap Clii Giao due s6 2 2 0 (ki 2.8/2009)
cd mdi tien he bien ehung vdi nhau: cdi chung chi tdn tgi trong cdi rieng, bieu hien thdng qua cdi rieng; cdi rieng ehi tdn tqi trong mdi lien he vdi cdi chung, boo hdm cdi chung; cdi rieng Id cdi todn bd, phong phu hon cdi chung, cdi chung Id cdi bd phdn nhung sdu sde hon cdi rieng (3).
Khai thdc moi lien he giCro cdi rieng, cdi chung dugc nhieu tdc gid nghien euu vdn dyng vdo dqy hge khdm phd. «Khdm phd, phdt frien tu mdt bdi todn fhdnh nhieu bdi todn mdi fheo quan diem mdt cdi rieng ndm frong nhieu cdi ehung khde nhau" vd «Tim tdi cdc kie'n thuc mdi, bdi todn mdi fu nhieu trudng hgp rieng theo tu fudng nhieu cdi rieng duge bao trum bdi mgt cdi chung cdi tdng qudt" (2). «Mgt cdi rieng ed the Id frudng hgp dde bief cua nhieu cdi ehung khde nhau" vd
«Mgt cdi ehung, dem ddc bief hod tung bd phdn khde nhau, bdng nhdng cdch khde nhau se cho nhieu cdi rieng khde nhau"(4).
Quo viec tie'p can phuong phdp dqy hge khdm phd, mdi lien he bien chung cOo cap phqm trO
«Cdi rieng vd cdi chung" vd tdng hgp y kien cOa cdc tdc gid, chOng tdi xin neu ra mgt so djnh hudng nhdm khdm phd phdt hien cdc kien tht/c thdng quo dqy hqc chO de hinh hqc.
Djnh hudng 1. Dy dodn, phdt hien, dinh hudng Idi gidi cdc bdi todn hinh hgc theo fu tudng di tu trudng hgp rieng den trudng hgp chung, lay frudng hgp rieng soi sdng cho trudng hgp chung vd van dyng trudng hgp rieng ggi y phuang phdp degidi quyet trudng hgp chung. luc Id to di tu sy khdo sdt mdt tdp hgp dd'i tugng song mdt tdp nhd hon - hay chi mdt dd'i tugng - chua trong mdt tdp ddu.
Vi dy 1. Cho hai hinh binh hdnh ABCD vd ABEF ndm trong hai mat phdng khde nhau. Tren cdc cqnh AD, BE Idn tugt lay hoi diem M, N sao 'TnrdngDai hoc Vinii
" Cao hoc 15 Toan, Tnrdng Dai hoc Vinh
, AM BN ' . . , . , ^ , 1 ctio TT^^Tfj: • Chung minh rdng M N ludn song song vdi mgt mat phdng cd djnh.
Oe d y dodn dugc mdt phdng ed djnh, edn ehung minh vd phuang phdp gidi, to td ehue eho HS cdc hoqt dgng sou;
- Cho M hxing A, keo theo N trung B, suy ra M N chinh td AB;
- Cho M trung D, keo theo N trung E, suy ra M N ehinh td DE.
N h u v d y , m d t phdng cd djnh edn tim phdi song song vdi DE vd AB.
Quan sdt hinh ve vd dya vdo djnh nghTa vd tinh ehdt cuo dudng thdng song song vdi mdt phdng eho phep to dy dodn mdt phdng ed djnh cdn chO'ng minh td (DCEF).
To se chung minh M N song song vdi (DCEF) bdng viec chi ro: M N song song vdi mgt dudng thdng ndo dd ndm trong mdt phdng (DCEF) hogc ehung minh M N ndm trong mdt mdt phdng ndo d d song song vdi mdt phdng (DCEF),...
BN . . . MD 1^' P P ^° '®" ^^"'"S tdi viee su' dyng djnh ti Tolet de chO'ng minh.
Tren AF to tdy mgt diem P sao cho AP AM _ BN PE~ MD~ NE'
Xet bd phdn phdng (ADF). Tu he thue, suy ra M P / / D F ( 1 )
Xet bg phdn phdng (ABEF). Tu he thue PA MA
- = - , s u y r a N P / / E F ( 2 )
Tu (1) vd (2) suy ro (MNP) / / (DCEF). Vdy M N / / (DCEF).
Dinh hudng 2. De tim tdi Idi gidi bdi todn hinh hgc ndo dd ta ed the di gidi bdi todn tdng qudt bao trum nd vd khi bdi todn ndy dugc gidi thi bdi todn dd cho chi Id trudng hgp rieng cua bdi todn ndy md thdi.
Dd td viec chuyen tu viec khdo sdt mgt dd'i tugng ndo dd song khdo sdt mdt nhdm ddi tugng ndo dd ehua dd'i tugng ndy, tu viec khdo sdt mgt nhdm hep dd'i tugng song viee khdo sdt mdt nhdm ddi tugng quan trgng hon vd boo hdm nhdm thO- nhd't. Ddi khi bdi todn tdng qudt tqi de gidi hon, dieu dd tudng ehung nhu mdu thudn (5).
Vi dy 2. Trd tqi vi dy 1.
To nghT den bdi todn tdng qudt hon sou ddy: Cho hai dudng thdng AB vd DE ed djnh vd eheo nhau. Ta tdy hoi diem M, N Idri luqrt tren A D , BE sao eho AM _BN
MD~ NE'
ChO'ng minh rdng dudng thdng M N tudn song song vdi mdt mdt phdng c d djnh.
Bdi todn ndy ed the gidi nhu sou:
- Qua hai dudng thdng eheo nhau tdn tqi duy nhdt mdt cap mdt phdng song song vdi nhau ehua DE, AB. Do AB, CO cd' djnh neii cdp mdt phdng ndy ed djnh.
- Qua M ta dyng mdt mat phdng song song vdi hai mdt phdng ndi tren. Gid sO' mdt phdng ndy edt BE tqi N ' . Theo djnh li Tolet trong khdng gian to ed AM BN'
, ket hgp vdi gid thiet bdi
-r, , , „. AM l u q i o triiet -—-^
MD N'E'
todn cho to N s N ' . Tu dd thu dugc dieu phdi cht/ng minh vd bdi todn xetdvidy 1 dugc gidi quy^.
Dinh hudng 3. Khdm phd, phdt trien td mdt bdi todn thdnh nhieu bdi todn mdi theo quan diem mdt cdi rieng nam trong nhieu cdi chung khde nhau vd tu cdi chung dd dem ddc biet hod theo tung bd phdn khde nhau ta thu dugc nhieu bdi todn mdi vd eudi eOng to thu dugc mdt chudi cdc bdi todn. Cd the minh hog dieu ndi tren qua vi dy sou ddy:
Vi dy 3. Cho A S A B . Ggi M td trung diem cOo cgnh A B , mdt dudng thdng d bdt ki edt cdc cqnh SA, S M , SB tan lugt tqi A ' , M ' , B'. Cht/ng mi
nh
rang: SA SBSA' SB' SM ' SM'
Bdi todn ndy ed rdt nhieu tdi gidi: ed the ke them cdc dudng thdng phy quo A vd B song song vdi d rdi su' dyng djnh li To tet, ed the su dyng phuang phdp dien tieh de chO'ng minh, cd the st>
dyng phuang phdp vecto,...
Neu ta xem ke't qud bdi todn tren Id mdt cdi chung. G V ed the td chO'C eho HS tim ro cdc bdi todn mdi thdng qua cdc hoqt ddng sou:
- Xet trudng hgp tom gidc SAB deu cqnh a.
Ta thu dugc bdi todn sou:
Bdi todn 1. Cho torn gidc SAB deu cqnh o. M td trung diem cOa AB, mdt dudng thdng d bd't ki
Tap eiii Giao due s6 2 2 0 (ki 2 - 8/2009)
cdt ede cgnh SA, SM, SB Idn lugt tgi A ' , M ' , B'.
SM' Chunq minh rdnq: — - H —
^ ^ SA' SB'
- Xet trudng hgp dudng thdng d di quo diem A, to di de'n bdi todn:
Bdi todn 2. Cho tom gidc SAB. Gqi M Id trung diem cuo AB, mdt dudng thdng d bd't ki di qua A cdt cdc cqnh SM, SB tdn tugt tqi M ' , B'. Chung . > 2SM SB ,, ., minh rang: "^T7r~T7T co gio trj khdng ddi.
- Xet trudng hgp dudng thdng d di quo trgng tdm tom gidc, to cd the ddt bdi todn:
Bdi todn 3. Cho tam gidc ABC. Mdt dudng thdng d hd\ ki edt cdc cgnh SA, SB Idn tugt tgi A ' ,
I SA SB .^
a) Chung minh rdng dudng thdng d ludn di quo mdt diem ed djnh.
b) Hdy tim vj tri cOa dudng thdng d soo eho dien tich tom gidc SA'B' nhd nhd't,...
Neu to xem bdi todn tren Id cdi rieng. N h d di chuyen lien tudng trung diem cuo dudng thdng AB td trgng tdm cuo he hoi diem sang trgng tdm tam gidc. Nhd lien tudng dogn thdng, tom gidc, KJ dien chOng deu Id cdc don hinh dde biet trong khdng gian mdt chieu, hai ehieu, bo ehieu. To di den bdi todn:
Bdi todn 4. Cho hinh chdp SABC. Ggi G td trgng tdm tam gidc ABC, mdt mdt phdng (a) edt cdc cqnh SA, SB, SC, SG Idn tugt tqi A ' , B', C , SA_ SB_ SC . SG SA' SB'
Bdi todn ndy cd rd't nhieu tdi gidi: to ed the su dyng phuong phdp tdch bd phdn phdng ro khdi hinh hgc khdng gian de gidi, phuong phdp vecto, st/dyng the tieh,...
Ta tie'p tyc xem bdi todn ndy Id cdi ehung. GV to chuc cho HS phdt hien bdi todn mdi thdng quo cdc hoqt ddng:
- Xet tu dien SABC deu to thu dugc bdi todn:
6d/ todn 5. Cho h> dien SABC deu cgnh a. Ggi G td trgng tdm tom gidc ABC, mdt mdt phdng (a) cdt cdc cqnh SA, SB, SC, SG Idn lugt tgi A ' , B', C ,
G
/ /-L - . 1 1 1 1 1 v 6 . Chung mmh rang:G'. Chunq minh rdnq: T ! " : + ^7r-:-t--^^ = 3 „ ^ a a c^' CD' sc SG
_L J_ J_
SA' SB' SC 3SG' - Xet mdt phdng (a) di quo dinh A cOo tu dien to cd bdi todn:
Bdi todn d. Cho hinh chdp SABC. Gqi G td
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trgng tdm tom gidc ABC, mdt mdt phdng (a) quo A edt ede cqnh SB, SC, SG Idn lugt tgi B', C , G'.
/~i ' . I i -, SG SB SC
Ctiung mmh rang: ^ ^ " ^ " ^ cd gid trj
khdng ddi,...
Djnh hudng 4. Phdt hien nhdng sai ldm trong Idi gidi hoqc trong qud trinh dy dodn de xudt gid thuyet vd kiem nghiem gid thuyef cua HS bang cdch lay xet mdt sd trudng hqp rieng soi sdng.
Theo (5) «Ta ed the thu thdch nhiJng gid thuye't bdng cdch dua ung dyng ehung vdo mdt so trudng hgp dgc biet: neu ung dyng duo de'n mdt ke't qud soi thi chdc chdn gid thuye't tuong ung Id soi.
Neu cdc thu thdch deu duo de'n ke't qud dung thi chua chdc gid thuye't tuong ung dd dung, nhung tdng hn rdng nd dOng tdng ten...".
Theo (7) «Cho mgt menh de... ta ed the xet bdt cu trudng hgp rieng ndo md nhd dd ta cd the nghiem Iqi menh de mdt cdch de ddng. Neu trudng hgp ndy khdng duge nghiem dung, thi nd bj pho djnh ngay tue khde, vd edng viec cuo to the td xong. Neu ngugc Iqi menh de dugc xdc nhdn, thi sy khdo sdt cuo to khdng phdi Id vd ich, vd nd ggi y cho ehung ta ve phuong hudng de tie'n hdnh viee nghien euu".
Djnh hudng 5. Ngodi ra, GV cdn quan tdm td ehuc cho HS cdc hoqt ddng nhdm tim tdi cdc kien thuc rrfdi, bdi todn mdi tu nhieu trudng hgp rieng theo lu tudng nhieu cdi rieng dugc bao trum bdi mdt cdi chung cdi tdng qudt.
3. Trong qud trinh vdn dyng phuang phdp dqy hqc khdm phd vdo dqy hge hinh hgc d trudng phd thdng, neu GV biet tgo ra ede tinh hudng phu hgp vdi trinh do nhdn thuc cuo HS de tren co sd kien thue dd ed HS khdo sdt tim tdi phdt hien kien thuc mdi thi viee hge tdp khdm phd se dem Iqi ket qud tdt hon so vdi nhieu hinh thuc hoc tdp khdc.U Tai lieu tham khao
(1) Phan Trong Ngo. Day hoc va phuong phap day hoc trong nha truong. NXB Dgi hgc suphgm, H 2005.
(2) Dao Tam - L.e Hien Duong. Tiep can cac phuong phap khong truyen thong trong day hoc Toan o truong Dai hoc va truong pho thong. NXB Dgi hge suphgm, H 2008.
(3) Giao trinh triet hoc. NXB Li ludn chinh tri, H 2006.
(4) NguySn Canh Toan. Phuong phap luan duy vat bien chung vdi viec hoc, day, nghien cihi Toan hoc, tap 1. NXB Dgi hgc qude gia Hd Ngi, 1997.
(5) G. Polia. Giai bai toan nhu the nao? NXB Gido cfuc, H I 997.
