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    CONTENT

    INTRODUCTION

    PART 1 :

    THEORY OF MATHEMATICS LEARNING-ZOLTAN PAUL DIENES

    THE FOUR PRINCIPLE

    THE SIX STAGES

    PART 2 :

    LESSON PLAN APPLYING DIENES SIX STAGES OF

    MATHEMATICS LEARNING

    BEFORE THE LESSON

    DURING THE LESSON AFTER THE LESSON

    PART 3 :

    FINDINGS AND IMPLICATIONS OF THE DIENES THEORY

    REFERENCES

    APPENDIX

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    INTRODUCTION

    Matheat!"# !# a$ a%#t&a"t!'$ !$ e(!#te$"e) *hethe& !t +a# ,!#"'e&e, %. '&

    ha# %ee$ "&eate, %. a$/!$, !# #t!00 a #%e"t ' ,e%ate#) Yet '$e th!$4 !# "e&ta!$)

    Matheat!"# !# !$,!#5e$#a%0e !$ 5&e,!"t!$46 ,e#"&!%!$46 a$, $,eta$,!$4 ee$t# !$

    the +'&0, a&'$, #) It ha# a0#' 5ae, the +a. '& the 5&'0!e&at!'$ ' $e&'#

    te"h$'0'4!"a0 a$, #"!e$t!!" %&ea/th&'4h# a$, a,a$"ee$t#)

    The $ae ' Z'0ta$ Pa0 D!e$e# #ta$,# +!th the 0e4e$,a&. !4&e# +h'#e

    +'&/# 0eae a$ !$,e0!%0e !5&e##!'$ a$, "'$t&!%t!'$ !$ the !e0, ' atheat!"#

    e,"at!'$) H!# $ae !# a##'"!ate, +!th a$!50at!e ate&!a0# !$ tea"h!$4

    atheat!"# #"h a# the 0'4!" %0'"/# a$, the 0t! 7 %a#e %0'"/# #e, '& the

    tea"h!$4 ' 50a"e a0e) He a0#' the'&!8e, that 4ae#6 #t'&!e#6 a$, ,a$"e

    9e)4)6D!e$e#6 1;36 1 the "'$#t&"t!!t.

    5&!$"!50e> the atheat!"a0 a&!a%!0!t. 5&!$"!50e> a$, the 5e&"e5ta0 a&!a%!0!t.?

    /$'+0e,4e e%',!e$t 5&!$"!50e) A0'$4 +!th that he a0#' #44e#te, @ #ta4e# '

    atheat!"# 0ea&$!$4 that he05 %!0, #t&'$4 "'$"e5ta0 !$ #t,e$t# $,eta$,!$4)

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    The "h&'$'0'4!"a0 ' "'4$!t!e ,ee0'5e$t '& !$,!!,a0 "a$$'t a%!,e the

    a"t' ' a4e) The "h!0, a$, the a,0t 5ha#e ' th!$/!$4 !# $'t the #ae) The a%!0!t.

    ' th!$/!$4 4&e+ 5 +!th a4e) H'+ee&6 the "'4$!t!e ,ee0'5e$t a0#' &e0e"t# '$

    the #&&'$,!$4 a$, the #'"!a0 !$te&a"t!'$) The&e'&e ea"h 5e'$ a"h!ee ,!e&e$t

    5ha#e ' "'4$!t!e ,ee0'5e$t) The&e'&e6 a$ a55&'a"h 0!/e D!e$e# The'&. '

    atheat!"a0 0ea&$!$4 4!e# the "ha$"e '& a "h!0, t' a(!!8e the!& "'4$!t!e #/!00#)

    Th!# 5&'e"t %e4a$ +!th the 4&'5 e%e& t&. t' /$'+ a$, $,eta$, the

    !e+# ' the the'&!#t6 Z'0ta$ Pa0 D!e$e# a$, h!# "'$t&!%t!'$# t'+a&,# the !e0, '

    atheat!"# a$, e,"at!'$#) Ate& that +e a550. '& $,eta$,!$4 ' the the'&. '$

    a 0e##'$ that +e 50a$$e, #!$4 the #!( #ta4e# ' D!e$e# the'&. ' atheat!"#

    0ea&$!$4 a$, h!# '& 5&!$"!50e#) The 't"'e ' the $,eta$,!$4 ' the the'&.6 the

    !50ee$tat!'$ ' the the'&. !$ "0a## a$, the !50!"at!'$ ' the the'&. a&e #ha&e, %.

    5&e#e$tat!'$ a$, ,'"e$tat!'$ ' th!# 5&'e"t !$ PLG @1 "0a##)

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    THEORY OF MATHEMATICS LEARNING-ZOLTAN

    PAUL DIENES

    THE FOUR PRINCIPLE

    THE SIX STAGES

    PART

    1

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    LESSON PLAN APPLYING DIENES SIX STAGES OF

    MATHEMATICS

    LEARNING AND THE FOUR PRINCIPLES

    BEFORE THE LESSON

    DURING THE LESSON

    AFTER THE LESSON

    PART

    2

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    BEFORE THE LESSON

    STEP 1 : The !t #te5 +a# t' ,ete&!$e, +h!"h t.5e ' #t,e$t# t' %e "h'#e !$

    the 5&'e"t) It +a# e!the& 5&!a&. '& #e"'$,a&. #"h''0 #t,e$t#) Ate&

    4' th&'4h the D!e$e# The'&.6 !t a55ea that D!e$e# The'&. '&e

    a550!"a%0e '$ "h!0,&e$ a$, %e"a#e the 4&'5 +a$t t' ,e#!4$ 4ae#

    a"t!!t!e# that 4et the .'$4 #t,e$t# !$te&e#te, !$) The&e'&e the

    5&!a&. #"h''0 #t,e$t# ' #ta$,a&, #!( !$ Pe$a$4 +e&e "h'#e$)

    STEP 2 : The #e"'$, #te5 +a# t' !,e$t!. the '#t #!ta%0e t'5!" t' a550. the

    D!e$e# The'&. !$ Matheat!"# Lea&$!$4 ) Th&ee t'5!"# +e&e #44e#te,6

    +h!"h +e&e &a"t!'$6 a"t'&!8at!'$ a$, !$te4e& $%e&) Ate& #'e

    ,!#"##!'$#6 the 4&'5 a4&ee, 5'$ #e0e"t!$4 the t'5!" ' !$te4e&

    $%e& %. the #%t'5!" ' 0t!50!"at!'$ ' 5'#!t!e a$, $e4at!e

    $%e) The ,e"!#!'$ +a# %a#e, '$ the #t,e$t# 5&e&e!#!te

    /$'+0e,4e a$, the a$!50at!e# '& "'$"&ete ate&!a0# that +e "a$

    5&'!,e !$ the 0e##'$)

    STEP 3 : The th!&, #te5 +a# t' 5&e5a&e the a"t!!t!e#6 the #0!,e#6 the

    a$!50at!e# a$, the 0e##'$ 50a$ a550.!$4 the #!( #ta4e# ' the

    D!e$e# The'&.) D!e&e$t t.5e# ' a"t!!t!e# +a# 50a$$e, t' !50. the

    D!e$e# 5&!$"!50e ' atheat!"a0 a&!a%!0!t. a$, the #e ' the #0!,e#

    'the& tha$ +&!tte$ '$ +h!te %'a&, a$, ha$,# '$ a"t!!t!e# ' the

    a$!50at!e# +a# t' !50. the D!e$e# 5&!$"!50e ' 5e&"e5ta0

    a&!a%!0!t.)

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    STEP : The '&th #te5 +a# t' 5&e5a&e the "0a##&'' '& the 0e##'$) The 0a%

    +a# "h'#e$ %e"a#e the #5a"e +a# a50e e$'4h '& the #t,e$t# t'

    +a0/ a%'t a$, 4et !$t' 4ae# a"t!!t!e#) The a&&a$4ee$t ' ta%0e a$,

    "ha! '& 4&'5 a"t!!t!e# +e&e ,'$e %e'&e the #t,e$t# "ae !$t' the

    "0a##) The 0a% a0#' "'$#!,e&e, t' %e a "'$,"!e 50a"e t' tea"h a$, t'

    !0 the 0e##'$)

    STEP : The&e +a# $' #t&ate4!" 5'!$t t' 50a"e the "ae&a %e"a#e the

    #t,e$t# a"t!!t!e# +!00 %e '$ 4'!$4 th&'4h't the 0e##'$6 the&e'&e the

    4&'5 a4&ee '& a e%e& t' %e the "'$,"t!$4 tea"he& a$, the 'the&

    t+' t' !0 the 0e##'$ !$ 't!'$ 9 'e, a&'$, the "0a## =) A e%e&

    '"#e, '$ the #t,e$t# a$, a$'the& '$e '"#e, '$ the tea"he&) The

    !0!$4 +a# #!$4 a ta%0et "ae&a a$, a #a&t5h'$e "ae&a)

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    The F0'+ Cha&t Beforethe Le##'$:

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    U$,eta$, the the'&. Ch''#e #t,e$t#

    STEP 1

    Dete&!$e the t'5!" Dete&!$e the #%t'5!"

    STEP 2

    P&e5a&e 0e##'$ 50a$ '& 2 5e&!',# De#!4$ a"t!!t!e# P&e5a&e a$!50at!e#

    STEP 3

    Dete&!$e the e$e A&&a$4ee$t ' ta%0e# a$, "ha!

    STEP

    De0e4at!'$ ' ta#/# 9 t' tea"h ? t' !0 = P&e5a&e LCD 5&'e"t'&6 "ae&a#

    9 ta%0et #a&t5h'$e =STEP

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    DURING THE LESSON

    STEP 1: Tea"he& a&'#e the #t,e$t# !$te&e#t '$ 5'#!t!e a$, $e4at!e $%e

    %. !$t&',"!$4 the 5'#!t!e a$, $e4at!e ' the #&&'$,!$4 '%e"t a$,

    $at&e)

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    Suggested by the professor that we shoud g!"e e#a$pes that

    reay reated to the $ea%!%g of pos!t!"e a%d %egat!"e %u$bers

    su&h as the fu&tuat!o%s of the te$perature' It !s so the

    stude%ts w! %ot ha"e $!s&o%&ept!o%'

    Late& '$6 tea"he& te#te, the #t,e$t# 5&e&e!#!te /$'+0e,4e '$ the

    a,,!t!'$ ' 5'#!t!e a$, $e4at!e $%e)

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    STEP 2: A550.!$4 #ta4e 16 the &ee 50a. ' D!e$e# the'&. ' atheat!"#

    0ea&$!$4) At th!# #ta4e the ,.$a!" 5&!$"!50e +a# !50!e, %. 0ett!$4 the

    #t,e$t# 50a. +!th the "a&,#)

    St,e$t# +e&e 4!e$ a #et ' 50a.e& "a&,# "'$#!#t ' &e, a$, %0a"/)

    Let the #t,e$t# #e the "a&,# t' !$, the &e0at!'$ that "a$ a,e 2

    %0a"/ "a&,# a$, 3 %0a"/ "a&,# %e"'e @ %0a"/ "a&,#)

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    STEP 3: A550.!$4 #ta4e 26 50a.!$4 %. the &0e# ' D!e$e# the'&. ' atheat!"#

    0ea&$!$4) Tea"he& !$t&',"e, the 4ae# a$, the &0e# t' %e 50a. %. the

    #t,e$t#)

    The &0e# a&e : 1) I %0a"/ "a&,# eet %0a"/ "a&,# the &e0at!'$ a,e

    #t a0#' %e !$ %0a"/)

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    2) I &e, "a&,# eet &e, "a&,# the &e0at!'$ a,e

    #t a0#' %e !$ %0a"/

    3) I &e, "a&,# eet %0a"/ "a&,# '& !"e ea the

    &e0at!'$ a,e #t %e !$ &e,)

    Tea"he& !$te&e$t!'$ +a# $e"e##a&. t' 4!,e a$, a/e #&e the

    #t,e$t# 4et the #t&"t&e ' the a"t!!t.)

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    STEP : A550.!$4 #ta4e 36 the "'5a&!#'$ #ta4e '& #ea&"h '& "'$a0!t!e# '

    D!e$e# the'&. ' atheat!"# 0ea&$!$4)

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    Tea"he& 4ae a e+ e#t!'$# a$, 0et the #t,e$t# !4&e, 't the

    a$#+e& #!$4 the "a&,#) It +a# #' the #t,e$t# a%0e t' &e"'4$!#e the

    5atte&$ ? #t&"t&e ' the a"t!!t!e#)

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    STEP : A550.!$4 #ta4e 6 the &e5&e#e$tat!'$ #ta4e ' D!e$e# the'&. '

    atheat!"# 0ea&$!$4)

    St,e$t# !a4!$e, that the %0a"/ "a&,# a# the &e5&e#e$tat!'$ '5'#!t!e $%e a$, the &e, "a&,# a# the &e5&e#e$tat!'$ ' $e4at!e

    $%e)

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    STEP @: A550.!$4 #ta4e 6 the #.%'0!8at!'$ #ta4e ' D!e$e# the'&. '

    atheat!"# 0ea&$!$4)

    At th!# #ta4e #t,e$t# %e4a$ t' "ha$4e the &e5&e#e$tat!'$ "'0'& ' the"a&,# !$t' atheat!"a0 '&)

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    STEP ;: E$&!"he$t ' a"t!!t!e# !$ '& ' 4ae# t' #t&e$4th 5 the #t,e$t#

    "'$"e5ta0 $,eta$,!$4)

    Gae# $ae : I$te4e& M0t!50!"at!'$ N%e& Batt0e

    P0a.e : G&'5# ' t+' t' !e

    Mate&!a0# : De"/ ' "a&,#6 A"e +'&th 116 a"/ +'&th 126 ee$ +'&th 136!$4 +'&th 16 #"&at"h 5a5e&

    S/!00 : N%e& &e"'4$!t!'$6 5'#!t!e !$te4e $e4at!e !$te4e a$,0t!50!"at!'$

    H'+ t' P0a. : B0a"/ "a&,# a&e 5'#!t!e $%e> &e, "a&,# a&e $e4at!e

    $%e) P0a.e #50!t a ,e"/ ' "a&,# t' "a&,# ea"h) Ea"h50a.e& ha# t' "'%!$e t+' ' the!& %e#t "a&,# t' 4et the%!44e#t $%e&) Ree%e& that t+' $e4at!e$%e a/e a 5'#!t!e 5&',"t a$, -2 !# 4&eate& tha$ -;)

    E(a50e ' &e#0t# :

    P0a.e& 1: 5&',"t !# -2 90''#e= P0a.e& 2: 5&',"t !# 2< 9+!$=

    The h!4he#t 5&',"t +!$# a00 "a&,# !$ the 50a.!$4 "'$te&)

    I the "a&,# 5&',"t# hae the #ae a0e6 the "a&,# a&e 50a"e, !$ a "e$te& 5!0e)

    The $e(t ha$, !# 50a.e, $'&a00. a$, the +!$$e& ' the $e(t 0t!50!"at!'$

    $%e& %att0e ta/e# the "e$te& 5!0e a# +e00)

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    STEP

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    R(NC(NG(N )E*BEL(+(R(N

    ,EL(S - @ SIDDI

    SESI - 2J1

    SUB+E,- MATEMATI

    BIL )EL(+(R - 2J *INGGU -

    *(S(- 1 AM 2J MINIT TE*)(T :

    GURU - AZMER ASRAF BIN MOHD HUSSAIN

    TO)I, - NOMBOR BULAT

    SUB.TO)I, - PENDARABAN MELIBATAN NOMBOR POSITIF DAN

    NEGATIF

    H(SIL

    )E*BEL(+(R(N-

    Pa,a a/h!& #e#! 5e%e0aa&a$6 5e0aa& a5 $t/:

    1= e$4e$a05a#t! !$te4e& 5'#!t! ,a$ $e4at!)

    2= e$.e0e#a!/a$ 5e$a%aha$ ,a$ 5e$'0a/a$ e0!%at/a$$'%'& 5'#!t! ,a$ $e4at!)

    3= e$.e0e#a!/a$ 5e$,a&a%a$ e0!%at/a$ $'%'& 5'#!t! ,a$$e4at!)

    ,E)ERLU(N

    )ENG(+(R(NB(H(N ,U(NTITI

    B(H(N )ENG(+(R(N P'+e& P'!$t S0!,e1

    B(H(N (UDIO /ISU(L P&'e/t'& 1

    (L(T(N

    a, Pe&a!$a$

    Pe$ Ma&/e&

    Pa5a$ Pt!h

    1

    2

    1

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    )ERING,(T0

    *(S((,TI/ITI GURU (,TI/ITI *URID C(T(T(N

    )ENGEN(L(N

    9 !$#=

    Set I$,/#!

    Ra#!'$a0 :

    Me&a$4#a$4

    5e$4etaha$

    5e0aa& te$ta$4

    #!at %e&0a+a$a$

    #e#at

    '%e/?#!at?5e&/ataa$ ,a$

    $'%'&)

    Me$.a5a &!,

    Me$a$,a/a$ /eha,!&a$

    &!,

    Me$$//a$ "!&!-"!&! #!at

    .a$4 %e&0a+a$a$ ,a0a/eh!,5a$)

    Me$4/ehe$,a/! &!,

    $t/ %eaa-#aa

    e%e&!/a$ #!at%e&0a+a$a$ %a4! #e#at'%e/)

    Re#5'$

    M&!, ,!5!0!h #e"a&a

    &a+a/ $t/ e%e&!a+a5a$ /e5a,a #!at

    %e&0a+a$a$ .a$4,!%!$"a$4/a$)

    eah!&a$Be&!/!&:

    e%e&/e#a$a$

    e$4e0a&/a$

    !,ea

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    )ERING,(T0

    *(S((,TI/ITI GURU (,TI/ITI *URID C(T(T(N

    )E*B(NGUN(N

    )er!%g1at 2-

    )er$a!%a%

    Bebas

    91 !$!t=

    Ra#!'$a0:

    Me&a$4#a$4 !,ea

    /&eat! &!,

    ,e$4a$ %e&a!$

    #e"a&a %e%a#

    ta$5a at&a$

    5e&a!$a$)

    Me$e&a$4/a$ te$ta$4

    '%e/t! 5e$4aa&a$ ha&! !$!)

    Me$4a#!$4/a$ &!,

    e$4!/t /50a$)

    Set!a5 /50a$

    e5e&'0eh #at #et /a,5e&a!$a$ .a$4 te&,!&!,a&!5a,a /a, %e&+a&$ae&ah ,a$ h!ta)

    Me%e&!/a$ a/t!!t!

    5e&a!$a$ /a, .a$45e&taa !a!t e$$//a$h%$4a$ a$ta&a 3 /a,

    h!ta ,a$ 2 /a, h!ta %a4!e$4ha#!0/a$ @ /a, h!ta)9ta$5a at&a$?a&aha$=

    Me%at "atata$)

    Me%e&! 5e&hat!a$

    #ea#a 4&e$a0a$/a$ #e#!5e%e0aa&a$)

    Pe0aa& e$.e0e#a!/a$

    a/t!!t! 5e&taa #e"a&a%e&/50a$)

    Pe0aa& e$44$a/a$

    /a, 5e&a!$a$ %a4!e+a/!0! h%$4a$

    '5e&a#! .a$4e0!%at/a$ ,a$'%'&:

    P'+e& P'!$t

    S0!,e

    N!0a! M'&a0 :

    '$#e$t&a#!

    ,a$5e$40!%ata$

    &!,)

    Me&a$4#a$4

    #!/a5 !$4!$

    tah &!,)

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    )ERING,(T0

    *(S((,TI/ITI GURU (,TI/ITI *URID C(T(T(N

    )er!%g1at 3-

    )er$a!%a%Berstru1tur

    91J !$!t=

    Ra#!'$a0:

    M&!,

    e$,a5at/a$

    /'$#e5

    5e&a!$a$ /a,

    e0a0!

    5e$4a0aa$

    %e&a!$ /a, ,a$

    t$/ aa& 4&)

    Me%e&!/a$ #e,!/!t t$/

    aa& /e5a,a &!, $t/e$.e0e#a!/a$ 5e&a!$a$/a,)

    Me$$//a$ /'$#e5 La&

    /a,93= ,a&a% Ba&!# /a, 92=a/a$ e$4ha#!0/a$ @)

    Me$,a5at/a$ #at"'&a/?#t&/t&5e&a!$a$ /a, .a$4e0!%at/a$ h%$4a$,a $'%'&)

    Me$,e$4a& /e5a,a

    %a$ta$?&/a$ .a$4,!%e&!/a$ 4&)

    N!0a! '&a0 :

    Pe&hat!a$ ,a$

    '/#)

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    )ERING,(T0

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    1J)91 X 3= 9; X < = K

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    Tha$/# a$, &e+a&, the #t,e$t# +h'+!00!$4 t' 5a&t!"!5ate !$ the 5&'e"t

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    IMPLICATION OF THE THEORY

    THE STRENGTH

    THE WEAKNESS

    PART

    3

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    Co%&us!o% Of The D!e%es Theory !% Ge%era

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