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7@-*.g;<&P5.%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%cf
00h#9;+,#"-mf/*!f-&-n/ )lo#k#"1#3;0p
00$Ni,8B+,#"--!m)#!&D#@#"4#@qA$
00ANi,8B+,#"--!m)#!&D#@#"3r#q#s63*'AH
00HNi,8B+,#"--!m)#!&D#@#"4#-CAJ
00JNi,8B+,#"--!m)#!&t#E63h-563#s63*'-!#"?-Ap
00Ni,8B+,#"-9u8I@@&@#E63h#"!*G6fm)#!&
E-*G6#Fg@Ad
c%'%Hb!"L@<0*0*+,!*15U
!"L@<bQON@]7T.7@
-*.g;<&P5.%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%'h
0AS18I(8I+,#"-mfv+,#"3D+,#"w*!f-&-n )lo#
k#"1#3;Ad
0A$?8xE-9li,8Bh#9;+,#"-I#)(8I)!m)#!&D#@
#"4#@q#s63*'HH
0A0?8xE-9li,8Bh#9;+,#"-I#)(8I)!-&-m)#!&D#
@#"3r#q#s63*'0
0AA?8xE-9li,8Bh#9;+,#"-I#)(8Im)#!&D#@
#"4#-C#s63*'d7
0AH?8xE-9li,8B)(8Ih#9;+,#"-I#)!m)#!&t3

#s63*'77
0AJ?8x(8I+,#"p)!fh#9;m)#!&E-*G6#C#y,
mz@@&@#E63h7
c%d%;+,!*15U*TL@<ON].J*
<!UJ[+*R].05%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%&&f
c%#%@*QL.*D*D!.*M].)**0**
ONK%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%&PP
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'%c%(*U*D%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%&Ph
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“Vận dụng tư duy thuật toán vào việc xác định hình để giải các bài toán hình
học không gian ở trường phổ thông trung học”(1996);A!8].
;rGóp phần phát triển tư duy thuật giải cho học sinh trung
học phổ thông thông qua dạy học một số nội dung phương trình" (2007).(
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dựng và vận dụng một số quy trình có tính thuật toán trong dạy học chủ đề
Hình học tọa độ không gian 12 – Nâng cao”
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1.1.2. Khái niệm về thuật toán và khái niệm quy trình có tính thuật toán
1.1.2.1. Khái niệm thuật toán
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1.1.2.2. Khái niệm quy trình có tính thuật toán
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7)@35;m;<*4O].n
m o
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oE-N
P P
v m Em oowr R d I
α
= −
lL@<ON)7[*Q*9B:
REZU*EZ2@<EZ7Wm!7W-"*o%
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*9].L@<ONL.@<U+S,7Q
.*0*%
1.1.3.Tính chất đặc trưng của thuật toán và quy trình có tính thuật toán.
1.1.3.1. Tính chất đặc trưng của thuật toán
/?**DvcwE*0;=7a@].
l/N17i‰•*+].ES,0*@q@%
l/N-4=/0*t!.!"Z+
U*D%
l/N7=76/0*-4L07=?M].*
7a@.%

l/NXb/0*J7Q*0**J*%
1.1.3.2. Tính chất đặc trưng của quy trình có tính thuật toán
/?**Dv&ewE*0CJ*HA(i7a7*Q].L@<O
N
lFO Z+!614?@<UK7i%
l•*+7|b7NbQEO+
.!KEO+S[*7i+!8a+
,U*D.U.@!"Z@)[%
l/@7.!"@)[E!.-*U*D1K0+<7*
74-4L0%
(L@<ONO7a7*QM*"
•*+OQ.!KEO+S[*7i+
!8a+,.7[U.@!"N@)[
&&
O*DL0@*T@)[%/*L@<ON
<N17i-77[7M7]%
Thuật giải
/?**Dv&cwE*0(9C^*O
;)**41=+*@K*T*t7*074
R%F"*+*!"*W*ZL617i|0
L@<*0*%/t*D0L@<*0*KE)*.7*74-**D@U
*T*0*%
r/*0*?8.@U*7[*Q ZZ
S,U*D7[7@iE-4=!.!"Z+
7?*-4L0*47X**mn(Š/o].+*
*@.m‹Š/Š/o0)**0*].+*7Os
NK0L.@].*0*
+ Tính đơn trị: /N7@i].*0*73*2*@|.
@*0*0*7@i%(8..*M„JKU*D
J.@J7"*[<0*J-4L0%

/NKO*N<R].*0*)7O.OQ
@<*.*4iU7U*D*0*.4)*%
+ Tính dừng: /*0*M!.!"ZMU*D
.7 S@.0*7*74-4=E7[-4L0"%
+ Tính đúng đắn: /*0*0*700N7=76R0*
*0*L47=K7T7a@.E7=*D."%/
*0*-~-4L0!.*a-7M7]E2!O@)[%
+ Tính phổ dụng: /*0*0*b7[+*
OJK@=+*ZZ*DbQ-.%()NKE)*.
!@.Z*0*E@W*t7O15UZ@<,7Q*0*
t+*%
&P
+ Tính hiệu quả: ŒM*DL0].*0*N"*%(8.
*0*U*D.E"N)**.E7*0E9U*D7R
7[M].U*9%
Quy tắc tựa thuật giải
/?**Dv&cwE*0(9C^*Or_6U.
*0*7[*Q ZZS,U*D7[?
@<U17i|*47X**].+*
*@.0)**0*].+*7Os%
_6U.*0*5*D+**0*!.
+ Mỗi chỉ dẫn trong quy tắc đó có thể chưa mô tả hành động một cách
xác định;
+ Kết quả thực hiện mỗi chỉ dẫn không đơn trị;
+ Quy tắc không đảm bảo chắc chắn rằng sau một số hữu hạn bước thì
đem lại kết quả là lời giải của lớp bài toán.
aJO!"4@!+**0*!L6U.
*0*Y,@*RL.@ONL@<
7*0*%
(L.*DT*0*@KMY*+*L@<ON

E7*Q-*D+*Z.L@<ON*0*
“thuật giải có tính đơn trị còn quy trình có tính thuật toán thì không có tính
đơn trị”%
&c
,5-5n<d<SB>A9<o\kFY=EmB>@b=RB>=>Fe==C`BWp<W<A@_>`==E<qB=:
MFY=>Fe==C`BWD=:MFYr`B9=sC5
1.2.1. Tư duy thuật toán
1.2.1.1. Khái niệm chung về tư duy
/?**Dv&dwE*0(9HA/!U0
@O.Z!U*D[@Z"**D"*L.
DONL].=%
1.2.1.2. Tư duy thuật toán
/4"].R7[*
%R*Qi-0A*47!.
m/&o/U*DZ.?@<U17iJ[+*

m/Po5NL@<Z.7[U*D?
@<U17i
m/co^*LO.L@<*9@.@!"7"*[@*•
L@<*9@.@+7"*[
m/'o0N1L@<@<*47
m/doG!Z-.JU*D*D
*D"*%
•*-0A@"].%/M
7M*Q*DAUU*DE"M!.Q*D
AU15U%7m/&o74m/do7[*
7
1.2.2. Tư duy sáng tạo
/?**Dv&dwE*0(9HA
&'

/!@.7[*<+*%/*7"*+*
!*@L@<!E@.*+*-K*40**+*
7"*+*1 *E+*7"*+**)**+*7OOQ7"*+*]
L.05<%*+*7O!€O8.1 *B*<-*7O5
7[<*Q@.7)*!"1 *%
/!NU7EB*<!
5*.].NU7%OQO*
!-*!*U-EU<@.R*0
5.*4%
1.2.3. Mối liên hệ giữa quy trình có tính thuật toán với việc phát triển tư
duy thuật toán và tư duy sáng tạo
l_@<ONO"**D*4+*
EO*T7T7aTO!UW*>@*Q
<_@<ONO**Qi<
"].L.7tm/&o74m/do%bQL
@<ONU*D.M*"m/&oU*D.
?@<UE-7M7]17iE.m/Po
]5N?
l_@<ONO"**D*4+*!
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1.3.1. Sự cần thiết của việc dạy học quy trình có tính thuật toán cho học
sinh Trung học phổ thông.
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*D!.
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7[T0].*DU7O.EbQR@q7aN<RE
M=O].L@<U*DE7O!B*D
Q*.!"RA].)*U*D%
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L?U-*Q@.%
1.3.2. Tác dụng của việc dạy học quy trình có tính thuật toán
lFO<Rb*@)

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om
α
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m o
α
lG„bL@<ON*=!*7O7i
+@!8E<3*)**0**E!*!€<7[
0*67Mt75?@<U7Q*0*L4*%
&e
l:L@<ON*=!**4@<)*
*0*].*?OAR*EL.7O**O
7*T-*D@VD-0A!!*%
Ví dụe_@<*4@<aMmGoO5y*41=+*
7)`
∆
<!**4.M<-N
m E oR d A= ∆
%\q@
!*O<aMmGo*41=+*7)` 
∆
 -N
m E oR d A= ∆
%
l/U*D?L@<ON*=!*O*T
@<)**0*-.*E!*O7*T-*D15

UL@<*0*-*U.)**0*J[+*
R05%
Ví dụfmC*e#@.&cdEC*<&P5.o%/<;<
*4O].7*Q
m'p cpPoM −
7)`
& P
 &
P
x t
d y t
z t
= +


= − −


=

Quy trình 1
C+&/<H/
d
u
uur
].7)`
C+PH*4@<
om
α
L.

d
u
uur
H//
C+c/<
om
α
dH =
%H;•O].7)`%
Quy trình 2
C+&I* ;<*4 O]. 7) `  
⇒
H d∈ ⇒
.7;&zm*0!„zo
C+P/<H/
d
u
uur
].7)`N
MH
uuuur
C+cG„b7*T-*D
d
u
uur
%
MH
uuuur
‘$<z
C+'^47*Q;R+*z<7[%

&f
lHbL@<ON]7T.7
-*.g;<&P5.E!*OQ*0*7[*T*
@!*-.E!*<&P5.%/L.7O5
.K[]7T.7-*.g;<
&P5.%
lHbL@<ONKOM@*Q
EL.*D<3*15UL@<ONO
Q@*Q!!*%H*DbL@<ON
K!€D!*O*L?<*E
K7TE5NEX[Y*.2*%/t7O
!**415UL@<7Q*0*L4**D7R
MU7O.E*DO.E*D7*O.%FOM@KM*47Q
7)*@)*7*+*+*7M7]zKAU%
,545)Z=dFeB@>:^B9,
/@&EA7 @q7[EL@<ON
KENK].L@<ONK
%‰7 O*7[.*@3].E"**D*Z.
+*7W)*!27[.*@3].*D15
UL@<ON7"*+**D@*Q].
!*@X