TAP CHi KHOA HOC BHSP TPHCM Ngd Minh Biec
D^Y HQC KHAI NIEM B^O HAM TRONG MOI QUAN HE L I £ N M 6 N V O I VAT LI
N G O MINH DLCC'
TOM TAT
Dgy hoc lien mon la xu hudng mdi giup hge sinh thdy dugc nhiing ling difng ciia kien thitc todn trong cdc khoa hge khde. Phin ddu bdi bdo se chi ra cdc nghia cua khdi ni?m dgo hdm va tim hieu nhiing ling dung cua chung trong chuang trinh vdl li pho Ihdng. Ket qud cua phdn lich ndy se chi ra nhiing diem cdn tinh den de ddm bdo dugc moi quan he lien mon giUa todn hoc vd vdl li lien quan den khdi niem ndy. Ket qud phdn tich sdch gido khoa todn sau do se cho thdy mSi quan he lien mdn ndy da dugc quan tdm den hay chua.
TH khda: day hge lien mon, d^o ham, toe dg bien thien.
ABSTRACT
Teaching the concept of derivative in Physics interdiscipllnarily Interdisciplinary teaching is a new trend which can help students see the application of mathematical knowledge in other scientific fields. The first part of the article will show the
significance of derivative and figure out their applications in high-school Physics. The results of this analysis identifies issues worth considering to ensure the interdisciplinary relationship between Mathematics and Physics related to this concept The results of the analysis of Math textbooks later reveals if this relationship has been taken into account or
not.
Keywords: interdisciplinary teaching, derivative, rate of change.
Hien nay lifln quan din viec dli mdi phuong phap day hge toan, cd hai xu hudng dang rit dugc quan tam: day hoc lien mon va day hge theo mfl hmh hda. Day hge tiieo hai xu hudng nay la mgt each mang Iji nghTa cho cac kiln thuc toaii hge, gittp hge smh nhan thiy img dyng hiju qua cua toan hge ttong thyc tiln cugc sflng cung nhu ttong cac khoa hpc khac. Va nlu phai tim ra mgt khac niem toan hge nao cd nhilu nhitag img dung da d?ng va hieu qua thi djo ham cd le la mgt ttng ctt vien sang gia. Cung phai ndi them ring, vat li chinh la manh dit mau md nhit dl chttng ta ed thl gieo tting nhung img dyng da dang nay ctta dao ham. Lich stt hinh thanh va tiln ttiln cua khai ni|m djo ham con cho til ttiiy dugc mli quan h? gin bd tuong hfl gitta dao him va vat li. Vat Ii cung cip nhttng bai toan ma viec giai quylt chung la dgng lye thuc diy ra ddi khai niem dao ham (bai toan tim van tflc vat till). Theo chilu ngugc 1^1, dao htoi dem din mflt cong cy todn hoc diy quyln lye dl nghifln cttu va giai quylt nhilu vin dl khac nhau ttong v$t li.
TAP CHi KHOA HOC BHSP TPHCM sS 7(85) nam 2016
Ttt nhttng dilm do chung tfli cho ring, dl viec d^iy bgc khdi mfm djio ham d?t duac nhilu hieu qua hon thi cin phai dat nd ttong moi quan he hen mon vm vat li. Vd nlu nhu thl thi sdch gido khoa (SGK) todn Vi?t Nam hi$n nay khi dua vao khSi mem dao ham cd tinh din mil quan he Uen mdn nay hay ehua? Viec tim hieu eau tta lot cho vin dl tten cung chinh ta myc tieu ma bai bdo cua chung tfli mufln hudng dSn.
D I thuc hien yeu ciu ndy, cdng vi?c ma chung toi djt ra cho minh la tim hieu ttndc tien cdc nghTa cua d?o ham va nhttng ttng dyng ctta chung ttong chuang trinh v^t li phfl fliflng. Kl tilp do la phdn tich SGK todn dl khao sat xem SGK dua vao quy ttinh tilp cfin khdi niflm nay nhu till nao? Cac nghTa ndo cua dao ham cd flie xuat hito, nghTa ndo cin phai xuit hito thi lai da khflng cd ca hfli hinh thanh ttong quan he ca nhan cua hpc sinh. Cdc t l chttc toan hpc ndo lien quan din khdi ni^m djo ham va lito quan dfln viec ttng dyng dao ham ttong cdc bdi todn vjt Ii. Ttt kit qud cua nhflng nghien cttu nay, chttng tfli se bilt dugc till ehl day hpc todn hito nay lien quan din khdi niem dao ham da dam bdo dugc mfli quan h$ lien mon vdi v jt li hay chua.
1. Nghla eua dao hdm va cac ling dijng trong chuxmg trinh vjt Ii pho thSng 1.1. Cac nghla cUa khdi ni^m dgo ham
Trong lich stt todn hge, dao ham ctta ham so tai mgt dilm (neu tfln tai) cd tiie mang nhilu y nghTa khde nhau vi gdn vdi nhflng dac tinng khac nhau:
Nghla hinh hjc: D»o ham t^ mot dilm bing vdi he sfl gdc cua tilp tuyen tji diem ay.
Nghia xip xi: Mgt hdm si /(x) cd djo ham tai x„ thi cd the xip xi nd bang mot hdm sfl tuyin tinh (hdm sfl tilp tuyin) quanh lto cto :«:„ ctta theo cong thttc xap xi:
/ ( X ) » ^ / ( X „ ) + /'(A:,)(X-X„)
Nghla ting quit: Dao ham cua mgt ham sfl dac tnmg cho tic do thay ddi (bien thito) cua ham si theo biln si cua nfl.
1.2. ifng dung tfo hdm trong chuong trinh vat lipho Ihdng
Khdi niem d^o hdm dugc stt dyng ttong chuong ttinh vat li phd thflng vdi hai muc dich chinh:
Dac trung cho tflc do bien thien tttc thdi ctta mflt d^i lugng vat li (chu yeu la bien thien theo thdi gian). Thii|t ngtt "dac trung tic dp biln thien" dugc chung tfli dttng theo nghla: Dao hdm cua ham sfl tai diem Xo phto dnh tucmg quan ve tflc do tien ve 0 cua Ay so vdi Ax khi x->xo. Ndi riflng, dao ham duang tai Xo cho thiy ham se dflng bito ttong mgt lto cdn ctta Xj, f (xo) cdng Idn thi f(x) se ttog cang nhanh khi x tdng mot lugng nhfl so vdi Xo (tuong ty cho dgo hdm dm). Ttt "tdc do" liy y tudng ttt vat Ii de cdp din su ttog (gitoi) nhanh hay chdm cua y theo x
Gidi thich cac xap xi stt dyng ttong vat li: Cac xip xi ttong vat li cd thl dugc gidi thich tiieo cong thttc xip xi affin: / ( X ) » / ( I „ ) + /'(A:„)(X-X„). x i p xi nay hilu theo
42
TAP CHi KHOA HQC BHSP TPHCM Ngd Minh BAc
nghTa hinh hpc la xip xi dudng cong bdng tilp tuyen cua nd quanh lto e|in cua tiep diem.
1.2.1. Ifng difng nghia toe do bien thien trong vgt li Trong SGK Vat Ii Idrp 10 va 11
Trade khi khdi niem dao ham dugc dua vao dgy ehinh thttc d chuong trinh toto cufli nam 11, khdi niem nay dd xuit hito ngam in d khd nhilu tinh hulng khac nhan ttong SGK Vat li ldp 10 vd 11. Ci cdc tinh huflng nay, nd ddc tnmg cho tie dfl bien dfli . AM tiic thdi ctta mot dai lugng u(t) nao do tiieo thdi gian vd dugc xdc djnh bang ti so — khi A/ rat be (tiln din din 0).
Ching ban vdi dinh nghla "vto tic ttie thdi" ttong chuang "Dgng hge chdt diem "
SGK Vat li 10 ban ntog cao (tt.l3 -14 vd tr.22):
"Xet van tdc trung binh cua chdt diim chuyen dgng thdng trong khodng thai gian tit t din t + At. Chgn Cit rdt nho, nhd den rnd-c gdn bing 0... Khi do Vp dgc tnmg cho do nhanh chgm vd chiiu cua chuyin dgng. Ta co thi ditng vector van tdc trung binh khi At rdt nho di dac tnmg cho phuang chiiu, do nhanh chgm cua chuyen dgng vd ggi do Id vector vgn tdc tdc thoi tgi thdi diem t..."
Khai niem tie dfl biln thien cdn xuit hien ttong "dinh lugt Fa-ra-ddy ve cdm ung
^ien fti" SGK Vat Ii 11 Ntog cao (tt.186):
"...do lan cua sudt dien dgng cam itng trong mach kin ti le vdi tSc ttd biin thiln cua tit thdng qua mfch... Niu trong khodng thai gian &t dii nhd, tir thong qua mach biin thien mgt luangts<^ thl—la tic # hien thien cda tie thdng... Cong thuc xdc
tSl
dinh sudt dien dgng cdm dng duoc viet dudi dgng sau: e^
Nhu vay, do nhu ciu cua minh SGK vat Ii da sdm dua vdo khai niem tflc dg bien thien tttc thoi va hilu nd nhu la gidi ban ctta tflc dp biln thien ttnng bmh — khi A(
ddn ve 0. Dilu ndy dtog 16 ra da tjo thdi ca tiiuto Igi cho viec hinh thdnh dugc nghTa tic dfl biln tiiito khi day bgc khai niem d?o ham ttong chuong ttinh mfln toto neu nhu chttng ta tto dyng dugc mfli quan hfl lien mon nay.
Trong SGK V^tli 12
Luc ndy, thl chi vjt li cho ring dgo hdm dd dugc gitog day chinh thttc culi nam Idp 11 d SGK toto, vl v|y khai niem d?o hdm vd cac nghTa cua nd.khflng cin phdi xuat hito mgt cdch "ngdm dn" ntta. Ndi rieng nghTa tie do biln thien ctta d?o hdm dd xuat hi^n mot cdch tudng minh va rd rtog ttong nhilu tinh hulng khac nhau. Chtog toi sB din ra ddy mflt si vi du tieu bilu:
Ddu tien la dinh nghTa khdi niem "gia tic gdc" trong bdi "Chuyin dgng quay ctta v|it rin quanh mot tryc cfl dinh":
TAP CHi KHOA HQC BHSP TPHCM SS 7(85) nim 2M
Gia tdc goc tiic thdi (ggi tdt Id gia tdc goc) cita vgt rdn quay quanh mdt trve-4.
thdi diim I la dgi lugng dgc trung cho sft biin thiln cda toe d$ gdc d thdi diem ddyt,
dugc xdc dinh bing dgo ham cda tdc do goc theo thdi gian. '.
Chttng ta se phto tieh kT hem djnh nghia nay: Mflt mjt gia toe la d?i lugng #|g tnmg cho su biin thien ciia tdc dg gdc, mjt khde nd dugc xdc djnh mgt each rS rtog ^ i^o hdm cua tic dfl gdc theo thdi gian. Djnh nghTa ndy chi ra ring d ^ ham da x i ^ hien d day vdi nghTa ddc tnmg cho toe dfl bien fliito cua mot d?u lugng. Ndi cdch khfa nlu mflt d?i lugng vat li Id ddc tinng cho tic dp biln tiiien ctta mgt dgi lugng khde thi nd cd thl dugc xae dinh bdng dao hdm ctta dgi lugng dd.
d mflt tinh huflng khde, bdi "Dien ttt tindng" ttong phin "Dien trudng bien fliien vd ttt tindng" SGK Vat Ii 12 ban co bto (tt.l09) cd doan:
... Vdy, bilu thttc ctta ddng di$n i se cd dgng: i = C— (21.2)
Bilu thttc (21.2) cho thiy cd su Uen quan mat thilt gitta cudng do ddng dien trong mach vdi toe dg biln thito ciia cu*ng dp dif n tnrdng ttong tu dien."
Trong nhto dinh ndy ro rtog ring SGK Vat li da dua ra cdch hieu— Id tflc dp dt . biln thien eiia cudng dfl dien tnldng E.
Them vdo dd, nhitag tinh hulng dgo ham xuit hito "ngim in" trudc day thl bay gid da dugc chinh xae hoa. Chtog ban vdi khdi niem "sudt dien dgng" ttong bdi "Dinh lugt cdm ttng dito ttt" d ldp 11 dugc xdc dinh la: sudt dien dgng cdm img ti le vdi loc A4>
do bien thien titc thdi cuo tir thong theo thoi gian vd dugc xdc dinh bdng e^ =—— ( vdi At rdt be). Nhung cimg khai niem ndy thi trong bdi "Dai cuong ve dong dito xoay chilu" (tt.63) cua SGK ldp 12 duge ttinh bdy nhu sau:
Vi tir thong d> qua cugn ddy bien thien theo t nen trong cugn ddy xuat hien sudt dien dgng cdm ling dugc tinh theo dinh lugt Fa-ra-ddy:
rfd)
e = NBScosmrot "
dt
Budc chuyen tiep ndy tudng chimg Id dien ra tu nhien, nhung that ra nd chi hgp li neu nhu dgc trung tflc dg bien thien ctta khdi niem dao ham dugc hinh thdnh ttong the che day hge toto md thfli.
1.2.2. Ung dung nghia xdp xi trong vat li
Mot ttng dyng quan ttpng khac ciia dao ham ttong vat li do la dl gidi thich cdc xdp xi. Chtog ta diu bilt ring, vat Ii stt dyng rdt nhilu cdc xdp xi hdm sfl ttong nghien cim cua minh. Khflng ngoai le, chtog tfli phdt hito thdy d chuong ttinh vat li phfl thdng xuit hien nhttng xdp xi hdm mgc dinh duge thtta nhto ma khflng gidi thich.
Trong bai phuong trinh dao dpng ctta con lie dan SGK Vat li 12 ban ntog cao cd neu nhto xet sau: Khi a nhd (a « Irad) cd thi coi gdn dung sina » a.
44
TAP CHi KHOA HQC DHSP TPHCM Ngd Minh Biec
Sdch Bdi tap Vgt li 12 Ntog cao (fr.79) cdn dua ra them cac xdp xi khde: Vdi
£ « 1, cd the dimg nhiing cong thitc gdn dung:
-^»=i+ir; vrr£»i+-
\-s _ 2
Ndi riflng, neu xet tai i„ = 0, thi ham si> y = sinx se cd phuong trinh tilp tuyin Id:
J' = /'(0)(x-0)+/(0)<=>y = cos0.x+0<=>j' = x. Thl nen viec xip xi ham so bdi tilp tuyin cua nd se cho ta xip xi hdm: sinx i» j : , khi x trong lto cto cua 0 (x rdt be).
Le dT nhien viec gidi thich hay chttng minh cdc xdp xi ndi trto Id nhiem vy ctta toto hge (vi neu khflng the thi hoc sinh se dat cdu hfli vi ddu md cd dugc nhitag xip xi ndy). Va bdi vi tiie ehl day hge vat Ii cin stt dung cac xip xi hdm si nen dl hgp Ii hon thi chuong trinh gitog day toto phai gidi quylt nhu cdu ndy mgt cdch fln thoa.
Tdm Iai, ttong the che day hoc vat Ii, khdi niem dgo hdm dugc stt dung vdi nghTa tudng minh la ddc tnmg cho tic do bien thien vd la cflng cy dl gidi thich cac xdp xi. Vi the, viec img dyng cdng cy dgo ham ttong vat li chi thyc sy hgp li vd nfli khdp nlu nhu qud ttinh dgy hoc no ttong chuang trinh todn phfl thflng cd thl Idm xuit hien dugc nhitag dac trung co bto ndy.
Nhimg luto diem tten Id co sd de chung tfli di din kit luto: Viec dgy hoc khdi niem ttgo hdm that sie cdn thiet phai dgt trong moi quan he Hen mdn vdl vgt li.
Ro rtog hem, quan diem lien mfln nay the hito d chfl: Qua trinh dgy hpc khdi niem dgo hdm cto phdi 1dm xudt hito dugc d hpc sinh hai dac trung co bto: ddc tnmg tflc dp bifln thien tttc thdi vd dgc trung xap xi. Theo quan diem ctta chung tdi thi quan ttgng hon Id dac tnmg tflc dfl bien thien. Sy thieu vtog dae trung ndy trong mli quan he ca nhto ctta hge sinh cd thfl ngan cto viec img dung nd mflt each hieu qud ttong day bgc vat li.
2. Khdi niem dao ham trong the che day hoc toan
Trong khudn kho mflt bdi bdo khoa hoc chttng tdi se chi ttinh bay tom luge nhitag ket qua ttt viec phto tich the che day hpc todn lien quan den khdi niem dao ham (Mflt phto tich diy dtt vd chi tiet horn, cdc ban cd the tham khdo trong luto van: Khdi niem dgo hdm trong dgy hge todn vd vgt li d trudng pho thong cua cttng tdc gid [2]).
2.1. vi cdch tiip can khdi niem
Viec dgy hpc khai ni|m dao hdm vto dugc SGK nhin nhto theo quan dilm lien mfln khi dl cgp din vai ttd cua cflng cy nay trong cdc nganh khoa hpc khde dgc biet la vat li. Hem ntta, SGK cdn dua ra hai bdi toto vat li Id bai toto tim vat tflc tttc thdi va tim cuomg dp ddng dito de dto dat tdi khdi niem dao hdm.
Tuy nhien qua ttinh dua ra va gidi quylt cdc bdi toto vat li chi nhim myc tieu lam xuat hien nhu ciu phai tinh gidi han lim " , md khdng gidi thich y nghTa cua
x-x,
gidi han ndy nhu la tflc do bien thien tttc thdi ctta hdm sfl theo biln ctta nd.
TAP CHi KHOA HOC DHSP TPHCM Sd 7(85) nam 2016
2.2 Ve dinh nghla cia dgo ham
SGK dinh nghTa dgo ham Id gidi han (neu ton tgi): iim -^^
Dl thuto loi cho nghTa tflc dp biln thien xudt hito, khai niem sfl gia Ax = x-i„va [,y = / ( , ) _ /(;(__) cdn dugc hilu nhu Id sy biln thien ctta hai dgi lugng x vd y. Theo do ti sfl ^ se cho tiiiy dugc tic dfl biln thito tnmg binh cua y tiieo x. Nhung tiieo nghiSn cttu ctta cdc tac gid trade chdng han, kit qua ctta tac gid Nguyin Thi Cim Trinh ttong Luto van Thac sT: Nghien ciru didactic vi isxtrong Todn hge vd trang Vgt li: "McM ddng vai trd la mgt ki hi^u hinh thiic, vai trd ddi tucmg Idn cong cudeu ma nhgt trong tri thitc duac tiip thu... Ax chi sdng duac khi gdn vdi md hinh Vdl li" [8].
Chtog tfli nhto fliiy ring y niem vl sfl gia nhu la sy gia ttog (bito thien) khdiig thl hien ro rtog ttong cdch hilu cua hge sinh, cung vdi dd quan niem ve tflc dp bien thien cung se khd xuit hien ttong thl ehl dgy hpc toto.
2.3. vi y nghla vgt li cia dgo ham
SGK chi dttng lgi d viec dua vdo ba cflng tiittc tinh vto tflc, gia tflc vd cudng dfl ddng dien bing dao hdm. Viec SGK khdng tiln xa hon dl chi ra dugc y nghTa tflng quat vl tie dp biln fliien in ben dudi cdc ttng dung ndy cfl flil ngto cto viec ung dung dao hdm trong cac tinh huflng khac cua vat li.
2.4. Ve dgc trung xap xi cda ttgo ham
Dac trung xdp xi ctta dgo ham thl hien d cflng thttc xdp xi:
/ ( x ) » / ( x „ ) + /'(x„)(x-x,)C)
Theo dd mot ham si cd dao ham cd thl dugc xip xi bdi hdm sfl tiflp tayen (ham tuyin rinh) d gin lto cto cua tilp dilm vd dao ham chinh Id h^ sd bjc nhat ctta ham tuyin tinh ndy. SGK dua ra xdp xi (*) dudi dgng mflt cflng thttc tinh gin dtog tt-ong bai
"Vi phto" va hinh thdnh xip xi nay khflng di ttt con dudng xap xi hinh hge (xdp xi do thi bing tilp tuyin cua nd quanh lto cto tiep diem) ma di ttt dinh nghTa dao ham theo gidi ban. Han ntta, mot kit qua nghito cim ctta tac gia Btti Thi Thu Hiln [7] cung da ehi ra nghTa xdp xi hdm ctta khdi niem dgo hdm khflng he tfln tgi ttong quan he cd nhan ctta hpc sinh.
2.5. Cdc td chiec todn hge lien quan din khdi niem dgo hdm
Chtog tdi thiy ring ttong thl ehl day hpc toto khflng cd mdt kieu nhiem vu ndo Hfln quan din nghTa tflc dp bien thien ctta khdi niem dgo ham. Chi trtt ra mgt tradng hgp duy nhit ttong SGK Ntog cao 12 phin "Tinh don dieu eua ham sfl" (Bdi 10/tt 9):
Trong bdi toto nay cd yflu ciu rinh "tflc dfl tdng dto sd tiong timg nam", tuy nhien dudng nhu y thttc dugc nghTa tflc dp tang cua dgo hdm chua dugc hinh thtoh
46
TAP CHi KHOA HOC BHSP TPHCM Ngd Minh Birc
ttong quan niem ctta hpc sinh nen ttndc khi dgt cau hfli, SGK cd dua vao mflt thflng bdo: "Dgo hdm cua ham sfl bilu thi tflc dfl tdng dto sfl..."
Cdc phto tich thl ehl SGK toto d tten da chi ra ring n^Ai'ii toe ttp biin thiin va nghla xap xi khd cd kha ning xuat hien trong mii quan hi cd nhan cua cdc em hge sinh (mgt thyc nghiem dugc tdc gid tiln hdnh da kilm chttng nhto dinh nay). SGK khflng nhto manh vd 1dm rd cdc nghTa ndy dto den qud trinh ttng dung dao ham nhu la mot cdng cu nghien cttu va hge tap vgt li ggp nhilu khd khto. Ben canh dd vi^c ltog ttanh cac kieu nhiem vu lien quan den hai ddc trung co bto nay cua dgo ham ctog cho thdy ring, the ehl dgy hoc toto khflng hi mufln tao dilu kito cho cdc nghTa ndy cd the hinh thanh vd tfln tai ttong quan niem cd nhto cua hoc sinh.
3. Ket luan
Mdt mat the ehl day hpc vat U (the ehl Ii) dua ra cdc tinh hulng xuit hito nghTa tie do bien thien tttc thdi cua mflt dai lugng vd nlu tto dung dugc se tgo thuto loi cho viec hinh thdnh nghTa ndy ttong the che dgy toto (the che I2).
Mat khde, I, cung dgt ra mflt nhiem vy quan trgng cho I2 khi ddi hdi I2 phdi lam xudt hito dugc hai dgc trung eg bto ctta dao ham (ddc biet la nghia tflc dp bien thien).
Bdi le chi khi hinh thdnh dugc nhitag nghTa nay, viec ung dung dgo ham trong day hge vat Ii d tradng phd thflng mdi cd the dito ra nfli khdp vd hieu qud.
Ro rtog ring, viec dgy bgc khdi niem dgo ham rdt cdn phdi dgt ttong mfli quan he lien mfln vdi vat li dl hge sinh khdng nhitag cd thl hieu ro bto chit ctta khdi niem md cdn cd thl ttng dyng hieu qua nd ttong cdc ngtoh khoa hpc khde.
Va niu nhu thi th) SGK todn Viet Nam hien nay chira tinh den mdt cdch thda ttang moi quan h^ lien mdn vdl vgt li. He lyy cua thieu sdt nay khflng nhitag ngto eto viec thiu hilu khdi niem dgo ham ctta hoc sinh mgt cdch thyc chit ma cdn tao ra htog rao ngto cto cdc em thdy dugc cdc ling dung da dgng ctta khdi niem quan trgng nay ttong vat li va cd cdc ngdnh khoa hge khde.
TAI LIEU THAM KHAO
1. Le Thi Hodi Chdu (2014), "Mfl hinh hda ttong dgy hpc khdi niem dgo ham". Tap chi Khoa hge, Trudng Dai hpc Su pham TPHCM, (65).
2. Ngfl Minh Dtte (2013), Khdi niem dao hdm trong dgy hge Todn vd Vat 11 d trudng phi thdng, Luto vto Thac sT, Tradng Dgi hge Su phgm TPHCM.
3. Nguyen Thl Khoi (Tflng chu bien), Phgm Quy Tu (Chu bien), Luang Tin Dat, Le Chto Hung, Nguyin Ngoc Hung, Pham Dinh Thilt, Bui Trgng Tuto, Le Trgng Tudng (2014), Vgt li 10 (Ndng cao), Nxb Gido due.
TAP CHi KHOA HOC BHSP TPHCM •S^ 7(85) nam
4 Nguyto Thl Khfli (Ting chu bien), Vu Thanh Khilt (Chu bito), Nguyin Dttc Hift,;, Nguyen Nggc Hmig, Nguyin Dtte Tham, Pham Dinh Thiet, Vu Dmh Tuy, Phjial;
Quy Tu (2014), Vgl li 12 (Ndng cao). Nxb Gido due.
5 Nguyen Thl Khfli (Ting chu bien), Nguyto Phuc Thuin (Chtt bito), Nguyin Ngje!
Hung, Vu Thanh Khilt, Pham Xuto Qui, Pham Djnh Thiet, Nguyin Tran T * ' (2014), Vdl li 11 (Ndng cao), Nxb Gido due,.
6. Trin Van Hao (Tflng chu bien), Vu Tuin (Chu bien), Ddo Ngpc Nam. Le Vto Tit^:
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(Ngiy Tda soan nliin Sugc bai. 22-6-201S: ngiy phin bi?n ainh gii. 14-7-2015:
ngiy chip nhin OSng 22-9-2015)
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