el lenguaje en el aprendizaje de las matemticas la evaluacin

7/25/2019 El Lenguaje en El Aprendizaje de Las Matem__ticas La Evaluaci__n

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El lenguaje en el aprendizaje

de las matemáticas

La evaluación formativa en la práctica

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El lenguaje en el aprendizaje

de las matemáticas

La evaluación formativa en la práctica

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#? *$ ' $ %$:Language for Learning Mathematics

2006 C$ LEEOriginal edition copyright 2006 Open University Press UK Limited A *+5 5'

EDICIONE" MO!A#A, ". L. (2010)M?$ L&$, 12. 28004 - M$''.'$$.5 - $$@'$$.5

D&+5 5$'5I"BN: 978-84-7112-604-7D5 L*$: M-36.837-2009

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Agradecimien!o ................................................................

C)mo e dearrolla e!e libro ...........................................

CAPK"#LO PIMEO: Incremen!ar el dic"ro, incre-men!ar el aprendi%aje ..................................................Di$c&#$ ) E'al&acin !a#a el a!#endi*aje , 18.El len- g&aje ma%em+%ic: Una ba##e#a a $&!e#a#, 19.Inc#e- men%a# el di$c&#$ ma%em+%ica, inc#emen%a# el a!#endi- *aje, 19.Benefici$ de la im!licacin de l$ al&mn$

en el di$c&#$ ma%em+%ic, 21.E>+ ))+'+/), 21.+, 22.I<;)' ' '; ++ )+ *+ '+*/@'+, 22.Cm&nidade$ f&e#a del a&la, 24.E$%ablece# cne(ine$ en ma%em+%ica$, 26."+*+ ; ;++ ++ */);, 27.In'e$%iga- cin-Accin, 28.0E +) *+ /<+/')/, 31.E+;'* *+ )/) *+ I<+/')/-A))/, 32.

CAPK"#LO II: Leng"aje ma!em&!ico: Q"' e $ q"' no e ...El #egi$%# ma%em+%ic, 35.Ca#ac%e#$%ica$ im!#%an - %e$ del #egi$%# ma%em+%ic, 38.$)'(;'/ ++)/-), 39.!/'>/ )+', 41.E ; *+ ' +'

7+++)/' (/(//)'

CAP*TULO VI

Con!enido

11

13

17

33

E*/)/+ M'', !. L.

'' '// ; ///)'*, 42.E *+ *+ '''(' + + +/ '+/), 43.Dific&l%ade$ "&e $&#gen al &%ili*a# el leng&aje ma%em+%ic en la cla- $e, 44.E +;'+ '+/) ) ;' +;*'+;', 44.A+*+ ' +>+'+J ) ; '-+/), 44.H')+ (/+J, 45. Cncl&$in, 47.

CAPK"#LO III: Empe%ar a hablar en la clae de ma!em&-!ica .............................................................................O#gani*a# la cla$e, 49.I);/ ' * + + */);:

F+' ' ')/;* '*+);'*', 55.N'*/+ +<'' '', 56.N );+ '*' *' ;' +;+'+;/<)'*', 58.#' ')/;* /);/<', 59.El %iem! e$ &n fac%# im!#%an%e en el a!#endi*aje, 61.C#ea# &n cn%e(% de leng&aje ma%em+%ic, 63.Plan%ea- mien%$ !#+c%ic$ !a#a &$a# en la cla$e, 68.O+)+;' +>/)')/ )'' *+ ;' /*+' '+/)', 68.C+/ ' +*'))/ '+/)' *+ , 70.!'(+; ''(' ;' +<' ' ' )/)', 73.P+*/ ' '; ;+ + /<;)+ + + +> 76.P+*/;+ '; /<++ (+ '' ))+'+/), 79.Cncl&$in, 81.

CAPK"#LO I$: E#al"aci)n para el Aprendi%aje ..............In%#d&ccin, 83.Obje%i'$ de a!#endi*aje ) c#i%e#i$ de e'al&acin, 85.O(+/< *+ '+*/@'+, 85.C/+/ *+ +<';')/ '/<', 87.#//@' (+/<

*+ '+*/@'+ )/+/ *+ +<';')/, 89.M ++- *+ * *+ ;' )/+/ *+ +<';')/, 91.P#eg&n%a$ ) #e$!&e$%a$, 93.F'<+)+ + '(/++'*+);'*, 96.F' )/)' *+ )+' +;' ')/</*'*+ +/;+)+*', 100."'(' + ;- +*;)/*, 102.Re%#alimen%acin , 104.Q;+ ;' +'/+')/ ++)/<'?, 105.Cm!ae- #$ ) a&%e'al&acin , 112.0D//);'*+ + ' '/-

)')/ *+ ' ';+<';')/ ' +<';')/ ++)'+, 114.F' )/)' *+ ++)+ ' ';-+<';')/ ' +<';')/ ++ )'+, 116.Cncl&$in, 123.

49

83

8 C+/*

E*/)/+ M'', !. L.

CAPK"#LO $: A#an%ar en la com"nicaci)n ma!em&!icacon "na finalidad ...........................................................In'l&c#a# a l$ al&mn$ en el !#ce$ de a!#endi*a- je, 125."'('' + ; +*;)/*, 127.L' ++)-)/ + /'+, 132.I/)' ' '; + +)+ *+ ++'@', 135.El #e% cm &n fac%# im!#%an%e en el a!#endi*aje del al&mn, 137.Alfabe%i - *acin ) ma%em+%ica$, 139.E)/(/ ' ';+) ; /, 140.P+', '(', +)/(/, ++ <-<+ ' +*')', 141.C;* +)/(/ );* , 144.

Cncl&$in, 146.

CAPK"#LO $I: La f"en!e de idea: Prof"ndi%ar en la!eor(a ..............................................................................Una !e#$!ec%i'a gene#al de la %e#a "&e 'inc&la el len- g&aje cn el a!#endi*aje, 148.0Leng&aje ma%em+%ic ) %e#a$ de a!#endi*aje, 151.E'al&acin !a#a el A!#endi*aje, 156.La cla$e de ma%em+%ica$ cm di$c&#$ de la cm&nidad, 161.E */); + ;' +'/+' *+'+*/@'+, 161.#' );/*'* *+ */); '+-/), 162. E */); ); +*'/), 164.El !en$amien% ) el di$c&#$ e$%+n e$%#echamen%e ligad$ al !#ce$ de a!#endi*aje, 165.0Cambi$ en el !a!el del !#fe$# ) del al&mn en &na cm&nidad de di$c&#$ ma%em+%ic, 166.0E '+ *+ +, 167.E '+*+ ';, 170.Cambia# la !#+c%ica, 173.C'(/'' )/)' +*/'+ ' /<+/')/-'))/, 174.Cn-

cl&$in, 177.

CAPK"#LO $II: Ahondar en la pr&c!ica ..............................Una cm&nidad de di$c&#$ ma%em+%ic, 179.Ca#ac%e- #$%ica$ cla'e$ de &na cm&nidad de di$c&#$, 185.Cambia# la !#+c%ica, 187.P)/)' +/@'*', 187.C'-(/ +/(+, 189./El final el !#inci!i?, 191.

BIBLIOGAFKA ......................................................................KNDICE DE A#"OE! & MA"EIA! ....................................

125

147

179

195203

E*/)/+ M'', !. L.

9C+/*

$' ''*+@) + '(' *+ ;) ++ )+-' *+ %'=/)/+ ;+ ' )'('* )/ '' +<+-

' + + )/)' ' /*+' ;+ )/++ ++ /(. G'-)/' <;+ /+ (;+' *//)/ '' (',';+' *+('/ ;+<' /*+'. G')/' <;+ '.

A / * ' ')/' ' B('', );' )/'@' '(/*;' + ';*' ' *+'' / ))//+ , ;;+, ' / '/*.

E*/)/+ M'', !. L.

CAP*TULO VI

Agradecimien!o



E+e #!b)& c&%+!e%e !e+e ca'+#& &)ga%!0ad& de +a#$&d& (e #& c&%ce'+& e e#ab&)e% de % ca'+#& a# !-g!e%+e. S!% e$ba)g&, cada 'e)&%a (e))2 a) e# #!b)& 'a)ad!fe)e%+e ')&'!+& /, '&) c&%!g!e%+e, %& ca'+#& 'e-de% )e#+a)#e de $2 !%+e) (e &+)&. E# #!b)& !ge +)e#%ea a)g$e%+a#e d!fe)e%+e: %a 'e)'ec+!a ge%e)a# de##e%ga"e / de# a')e%d!0a"e de #a $a+e$2+!ca, #a bae +e)!-ca de #& c&%ce'+&, / #a !$'#e$e%+ac!% ')2c+!ca de d!c&

c&%ce'+&.L& Ca'+#& P)!$e)& / VII &f)ece% %a 'e)'ec+!a ge-

%e)a# de #a !dea ')!%c!'a#e. E# Ca'+#& P)!$e)& ace %a!%+)&dcc!% a# ca$'& de a'#!cac!% de# #!b)& a c&$& a #&c&%ce'+& (e e a% dea))&##ad& a +)a de %a !%e+!-gac!% a'#!cada. E# Ca'+#& VII c&%c#/e c&% % )e$e% de#& &b"e+!& de #& c&%ce'+& / de !g%!f!cad&, / #a !deac#ae 'a)a c&%eg!) ca$b!& c)c!a#e d)a%+e #a ')2c+!ca')&fe!&%a#.

L& Ca'+#& II / VI e'#&)a% #a bae +e)!ca. E# Ca'+-#& II !%e+!ga / e'#!ca e% ( c&%!+e e# #e%ga"e $a+e$2-

8%Ed!c!&%e M&)a+a, S. L.

13Refe)e%c!a b!b#!&g)2f!ca

CAPÍTULO VI

Cómo se desarrolla este libro



+!c& / e'&%e #& (e debe% a')e%de) #& a#$%&!* a# +!#!0a)

e# #e%ga"e $a+e$2+!c& 'a)a e')ea) !dea, / +!#!0a) /c&%+)&#a) #& c&%ce'+& $a+e$2+!c&. E# Ca'+#& VI ea$!%ae% de+a##e e# & de# #e%ga"e e% e# c&%+e+& #!+e)a)!& / e% e#a')e%d!0a"e de #a $a+e$2+!ca. E'#&)a +a%+& #a bae de #a')!%c!'a#e +c%!ca de #e%ga"e / c&$%!cac!% 'a)a a'&/a)e# a')e%d!0a"e / ')&$&e) #a Ea#ac!% 'a)a e# A')e%d!0a- "e!**, c&$& e# efec+& e'ecf!c& (e e+a ')&'e+a 'ede+e%e) &b)e e# a')e%d!0a"e de $a+e$2+!ca de #& a#$%&.

Ta$b!% e'#&)a #& a'ec+& +e)!c& de# ca$b!& / c$& #a!%e+!gac!% a'#!cada c&%+)!b/e a dea))&##a) % c&%&c!-$!e%+& ')&fe!&%a#.

L& Ca'+#& III, IV / V c&%+!+/e% e# %7c#e& de# #!b)&, /dec)!be% e# $&d& de !$'#e$e%+a) #& c&%ce'+& e% #a c#a-e de $a+e$2+!ca / c$& e# e%f&(e &b)e e# #e%ga"e ee%#a0a c&% #a !dea &b)e e# dea))&##& de #a Ea#ac!%'a)a e# A')e%d!0a"e. E# ca'+#& III a%a#!0a #& $2 e#e$e%+a#

'a)a c&$e%0a) a c&%ce%+)a)e e% e# #e%ga"e d)a%+e #ac#ae. E# Ca'+#& IV +)a+a de #a Ea#ac!% 'a)a e# A')e%d!-0a"e / c$& e 'ede +!#!0a) ef!ca0$e%+e '&+e%c!a# 'a)a$e"&)a) e# a')e%d!0a"e e% %a c#ae e% #a (e e a')e%de/ e ab#a &b)e $a+e$2+!ca. E# Ca'+#& V c&%+!%7a c&%e# e%f&(e ')2c+!c&, / e'#&)a &+)a !dea 'a)a !%&#c)a) a#& a#$%& e% #a c&%e)ac!% / e% e# ')&ce& de a')e%-d!0a"e.

8%Ed!c!&%e M&)a+a, S. L.

14 E# #e%ga"e e% e# a')e%d!0a"e de #a $a+e$2+!ca

**%S!e$')e deea$& e!+a) e# e!$& e)ba#, 'e)& +a$b!% (e)e$&a#e"a)%& de #a )e!+e)ac!% (e '&%e ##e%a) +&d& e# #!b)& de )efe)e%c!a aa$b& e&. A'e, a ece e !%c#/e% e')e!&%e c&$& 9%!& / %!a,9a#$%& / a#$%a / &+)a ece e +!#!0a e# $ac#!%& e% ge%e)a# & a#g7%ge%)!c& c&$& ')&fe&)ad& / a#$%ad&. (N. del E.)

**%E% !%g# e% e+& 7#+!$& a& #a e')e!% c&% #a (e de%&$!%a) #&(e +)ad!c!&%a#$e%+e e%!$& de%&$!%a%d& “evaluación formativa”, e e+2b+!+/e%d& '&) 9Ae$e%+ f&) Lea)%!%g (Evaluación para el Aprendizaje).

E% e+a &b)a e$'#ea)e$& a 'a)+!) de a&)a e+a 7#+!$a e')e!%. N& &b+a%-+e, e% e# b++#& de# #!b)& $a%+e%e$& #a de%&$!%ac!% $2 c&%&c!da, 9ea-#ac!% f&)$a+!a, 'a)a %& c&%f%d!) a #a 'e)&%a !%+e)eada e% e+a +e$2-+!ca, 'e)& %& a# +a%+& de e+a $&d!f!cac!% +e)$!%&#g!ca. (N. del R.)



8%Ed!c!&%e M&)a+a, S. L.

15C$& e dea))&##a e+e #!b)&

Ca'+#& P)!$e)&

I%c)e$e%+a) e# d!c)&I%c)e$e%+a) e# a')e%d!-

0a"e

Ca'+#& II

Le%ga"e $a+e$2+!c&Q e / ( %& e

Ca'+#& III

C&$e%0a) a ab#a)de $a+e$2+!ca

e% #a c#ae

Ca'+#& IV

Ea#ac!% 'a)ae# A')e%d!0a"e

Ca'+#& V

Aa%0a) e% #a c&$%!-cac!% $a+e$2+!ca c&%

%a f!%a#!dad

Ca'+#& VI

La fe%+e de# c&%&c!-$!e%+&-

P)&f%d!0a) e% #a +e&)a

Ca'+#& VII

A&%da) $2 e%#a ')2c+!ca

Perspectivageneral

Origen Teórico ImplementaciónPráctica



De f&)$a &ca!&%a# !%+)&d0c& c!+a e% #e+)a c)!a.

+a ')&cede% de #a c#ae e% #a (e e !d& ')&fe&)a &be)ad&)a, & de #& ')&fe&)e c&% (!e%e e +)aba"ad&.Ta$b!% a/ c&$e%+a)!& (e c&%+!e%e% b)ee dec)!'c!&-%e de de+e)$!%ad& a#$%& / a#g%a ')eba (e c&%f!)-$a% (e e# !%c)e$e%+& de# d!c)& #e a/d a 'e)fecc!&%a) c&%&c!$!e%+& $a+e$2+!c&.

8%Ed!c!&%e M&)a+a, S. L.

16 E# #e%ga"e e% e# a')e%d!0a"e de #a $a+e$2+!ca

E< + << < < == < = <<. E< =

= = +, = - <. $ +, = < = << = =< < = << @ / @= . C= -< = = >=> =, / + @ < = +-< @ , = << / =

= + < > . & > = = <=+ = , E>= A* ->< = < < .

&< + = = < +. C / < = = =<= = . " << /- = < < + =< <<, <+ + =

E M<, $. L.

17#/ ++/

CAP(TULO PRIMERO

Incremenar el disc!rso,incremenar el aprendi%aje

*' < 14. (N. del E.)

/, =< < @ <<, @ < -

>+= <<. $ +, + / @ <= = = < / < @, <<, @ / . E < -= <+ = = < >= < ; = < @ =< @</ = < << <.E / @ < = < =

= = < = > << <= . L = -< = <<, @ =< - =< @ < <<; ->< = << <<.

Discurso y Evaluaci"n para el Aprendiaje

I< = << =< = E>= A . & - /<> =< E>= AK>= / <=K < <<, < < =+- (BLACk @ ., 2002, 2003). $ < >+=- << =, < < = +-

/, @ > E>= A, =, < >= =. " <<, < + + /> /-< = >+= << < =-, / < E>= A, @ = = > <. A=< <=< =< -= <<, << = / = < << @, =<, + < <<. C= <

E M<, $. L.

18 E = <<

<,

>< = >= =<> @ </.

El lenguaje matem!tico: Una barrera a superar

&< = << = = + = + =-

< @ > / < <<. L = < = <<; / = = @=. " = =, < << + == <. L = + = = >+=<, @ / = /< << @ = +

< <<. $> = = =< = <<, < = < = < < = = = < =. $ +, = = <<, >< -< = < @ . L /- < + = = << @= = <

= @ = < =< = /< = =.

Incrementar el discurso matem!tico, incrementar el aprendiaje

L = = = <<- = = >< = < E>= A. & > = <= = + = + @ -

E M<, $. L.

19I< =, <

< => =<. $

< << < < @ < < <=. "=- < + <<, + => / =<, < - => = >= @ = <<>. L = = + ->< / + << @, <<,

= < /< => <. D = = = = , <+ / = = = < <<.

C= = = , << - = / = @ = < =<. L = = + = / <<, = = =

< = , @ = + = < <-< =< @ = = . D < , / + = = < +, => =, @ /= = <. C= = =< = << = , / @ < = > =, = = <

= <. L =- = /= =< = @ <, < @ > -+ =, , <<=@ = < = = . $>< < << / @ << < =< < <= <.

I< = = /- < < < F.C= =< < < << = < << = . C-

E M<, $. L.

20 E = <<

, , < <=, = =

/= =< = <, =< @ . A = = = <= <> = <= = = =- < /= =, <. &<= = /= =@ , = < @ < = + <<. " <<,

< <= = > <. C= >= = @ +<, < / <-. C /= / = = << / = <=< =. $ = < < = < / @ = = ++, = /<. C /= =

< =< < <<. L << / < = =, = -= =< = = = < < @ = > <. L + / << @= = = -> < /=<. C= < <> -@ = =< + @ / < << <

< < = .

Beneficios de la implicaci"n de los alumnos en el discurso matem!tico

E#poner concepos maem&icos

I>< = < = = < << < < = < @ < -

E M<, $. L.

21I< =, <

@ . E = @=

= <+, @ < =< @ - < = =. A -< << @ = = > + <<.

E< = << => = / + @ / = -< =. $ < F -< =, < < / =

. "< < = /= = = , = @ / + < <. E / <+ < < / @, <<, < @ = =. E / = < / <-> = < < - <= =; ,

< = >= =<>.

Reo

C= = < < = = <-< = <+ / @ << @ < / = . $+ =

+ < @ < =< = + = =- = < . " =< < = <<, > / <+= + > +, = . A = = <<, /+ < = = > / -> + @ = = = < - /<.

E M<, $. L.

22 E = <<

In"ol!crar a los al!mnos en el proceso

de aprendi%ajeE = < = < >=

. < = /< E>= A. C= < >= + @ /, @ < @ <. $+, < >= -

<<, + = +< @ < ; ,+ =< = <<.

"< -< = = = + =< @ < > / = = @ = @ = + = @ = . E = -

= < < = = @ + / /- . C= = < -< +<, /=@ = = << < = /< , < =< =<.

L = <+ >= = / , @ <

<> < + <+ = @ = . L = < = <= = == > < <=+. L =<+ /< F. C= @= =<=-< < <<, < / <= < = = <.

L = - / = <<=@ = < = -= = = <; / < =

E M<, $. L.

23I< =, <

= = = = = =<. "= <

= <= <-<> < = @ < + .

Comunidades fuera del aula

E +> = = = <-

= /< + = = < =. L/< = /< K , = = < / @ + =< K =<, < <<> =<< = << / -< = <. < = >= = , @ = < < @ @ F < . " = @

= = /<>. $ @=- < = + = << @ /< = = @ /> < @ / < = < =-<= =. $ +, /< = /= =< < = = << =<. C = << = = @ =</ = , @ -

/ << = = < <=.

E < -< = << = + @ . L = = =< = / < > == + << @ = =< < +. A= < < < <<> =@ =. E< +, = - = =< ><, < =< < = <. C < = = < -

E M<, $. L.

24 E = <<

/< << <

<= =<. C /= = < = < @ >/ <> +=< = . E< = << =. % < +< < = +< = = < = = =< =>.

L E>= A <+=@ +-

= < +<= <<. E =, <+ +<> / = @ =. L =<>= @ +> @= </ < + @ < = /> ==< < =< .L <=< = =< < < < +=< = < +, = =

= <+ /< = F. E< -<< @= = = = + =< = = <+ @ + = < += .

M= = < = <- = > / @ <, @ - = = < +. $ << = < < (DECk, 2000) @ ><

< . L = = <= > < < << = + /, < < <>, < / = F. L -<< = / + = < = = < < << /< /=< / ( <- < ). E< = = <=> <=<. $ = = << << < @= /, <F = <

E M<, $. L.

25I< =, <

<<. E < =< =

< < < > @ <<, << @ = <. A / =< = / = = > = <. E << F =< >< >; < = = >,@= @ @.

Establecer conexiones en matem!ticas

I< = < < @= = <+ < -/< <<. E< =< F = <. L / + <<=: /, < "<-

, ++, "@< "< G/, @ =><. L = =< < =@ /< , > = = /< @= >= @ - = < < . L << <<=@= < <<; < << K+, <, <<, <<K/ < = < = <<=@ = < >=<>,

= @ = . L =- <+=@ < = /-, @ = += < . = < < K, += <-, <<K, << = =, = , < < "<, == < <+ <. L - << = = < = + = < @ . % = +, , < @ =</ < << =@

E M<, $. L.

26 E = <<

<. D + = <-

= << < >=- @ < <<. L = << = / , , @ >, @ < = =<<, =@ < >+, < /.

Tender !n p!ene enre los disc!rsosL = < , =< <-

< = = / @ < <-<; < +<< = >+= @ /- <<. L = =< <<=<, < / << <=< = / << = =

/ <, < < @ / =< = <<. "=< = = = >+= @ = = < << =, <+ - = < =< < <<-. E , <+ = >= < -= << @ = / .

E << = < <-< < @ / = = = = , < < < /= @ <- < <<. A = =</ < << @ = == F = <<. $+, << = =< /=< = @ / @ @ -<.

A =< = << = ,= = < = = = =

E M<, $. L.

27I< =, <

< . <F = <

= @ =< = = << <= = <<, @ = . +<<, @ < @= @ < = . $ =F = @ @ =<-. E< = /< <= => @ : +, = /, < @ = /< +

= < = <<, @= @ < = = + < = < <-<.

Investigaci"n-Acci"n

L < < + << = / /< + = = =< = << = = /<. $ +, = = < < =< /= @ = / = < / = = = -< = < + = < = =

@ / /=<=.

C= < < < +, @ / << = <<=-< F = = + =-. L / <+ >< (LEE, 2004)= @ < < +, < =+ = < < /. E = + < = /+ , @ + > < . C

E M<, $. L.

28 E = <<

+> = <, = =

/=< < = ,<+ < = >=+ @ <+ E>= A. & > = = + =< + @ = @ <-, + < < =; @ << > = @=, . L

= =F /=+, < @ < = /- < = <=> =< <+.

&< < < C<= 'I <- = < = < <<.

" / ><- el estudio de

una situación social para tratar de mejorar la calidad de la acción en la misma. $= +<> < < => /< = < <= < @ > < < = << =+ </ >, = =< @=- <= << @ <.

(ELLI!%, 1991, . 69, <. <. . 89, => .)

M < = << @<< @ + = + = < = = <. E -< = < =< < =< = -< = << < = = . M =<+J =< = ? H <-< @ = > /. L = <+ /= < = = . " - <=> < @ , << = @, /= <=+ = /=+ @ = .

E M<, $. L.

29I< =, <

A, @ @ , @= = -

= < << @ +> = + = + < /.C< < ><

>< < = < = = + = <> <=. E , /= <= =<- < = + < <, = < @ > =<. L -

< @ > < >-<- <=> < =<.

1.= /= > <- = <+ <=,

2. <<= + < >= + < < = +->+,

3.= <= < -<< = / .

A < + @ /-= <. &< < + @ + = <+ =< = << @ +. E

/ =< / < /= >< @< @ <= =<.E<+ @ > < = >/= + < . M = = < (G#IffI%H$, 190, . 43)/+ = > / < =<+, <+ + <= @ /= /- =<. #< = =</ >< @ = = / +.

E M<, $. L.

30 E = <<

El pro$eco de in"esigaci'n

E < = = + < @ < ; +, + = =-= < = =< = + + < >. E = = + +, = 9* (< 13 @ 14F), = += > <, = = = + += <<. $+

= <++ < ++ = /= @ < =. %-+ >< = < = = =< << / - =< = .

# < = >+ ><-. M /=< / < = >+ = . E+

< + = =< =- = ><= < < . E< =<+ / , = >+ + =< = @ = + +<> < =. %+ =<+ = = / @ <+ < = > <. A / F

<>< / = =, = =<+ + = =< >< + / =+ <<= .

E / < < < F; @=@ <. $ +, > =< @ < << = < <-+. M = = ><- < , =< /, > @ / => =< .

E M<, $. L.

31I< =, <

*E=>< EF 3. E$!. (N. del T.)

El res!lado de los ciclos de In"esigaci'n-Acci'n

E =< /= = < = = <+ = <<< @ + = . D- = < < = @= = + = -< <<. %+ =+ = = - + = + = -

, @ << @ + =/ <+ = @ < = = < .

L ><- = < < / /. L >< (MA$!, 2002) = < = = @ / + /= = =-, F = / @ =

+ < . L ><- = / = - @ < < =< @ + = <<=@ = += < < << - = <. L ><- = = = < - = .

E M<, $. L.

32 E = <<

Una par!e del aprendi%aje para aprender a hablar como "n

ma!em&!ico, e er capa% de "!ili%ar el leng"aje !an!o para

recordar $ con!rolar la im&gene ma!em&!ica peronale,como de !ranmi!irla a lo dem&.

($i!!, 1995, <F. 40.)

E ?@ <@A <AG A ? A @F@ D A ? A? ?@< < < ?A? <@? @F@?.CA?@ ?@ @F@ B D J G?@

<A ? B@? A A ?@G <- @F@?, B?@ AF? ? ? <@??? A A < <? ?A? ?@F@? D < @ F ? F? ?A? @F@.

B?@J ? ?A? @F@ D @A- ? @@? @F@? AD A A A ? <? ? <@? @F@?, ? DA ? A?, ?@@AD A ?A <- ($i!!, 1987; ab#&dE, 1990; E&"Ck, 1992). E ? A <@ ?A? @F@ A A?@?

ME? !@, '. .

33&? F?

CAP-TULO II

Leng%aje ma$em($ico:Q%) e# ' !%) no e#

A? <@ ? <? A ?@ B.

' ? A? @ A BA? <@ ?-A? @F@ @@ @ A A, < ? B?? A @F@D ? <?? DA? <?.

E A ? <? ? @F@? ?? @@, A A < <@ A? <??, ?< ? F@? L?@? ? A. E A ? F? ? A?

PA@? @@? B? < D @ FQ,A @F@ F? <?B D B P F A @FA ? A @A A@< < AQ. aAA ? ?? <? ? <-@, ?A ? ? B@ F? @-F@, D A ? <? A ?@ ?, <? A@. E?@ A < @J @ ? ?<-@? L?@?, <@A? D ?? <

A A (ab#&dE, 1990). $ @@, ? A@? A?A A A ? <? ? @-F@?, @ B? <@AJ ? ? @F@? A.

E A ?B ?@ B ? @-F@? @BG? ? @@? @F@? D, A,< A ? <? ?A? <?? A ?<. " ? ? <?J @F@

@A D <A A PQ @F@?<A <A ?@F ?<A?@? A? ?@ ?-@@, <? D A@@ A ?@@F@. ? A? <@< ? @F@?A @@ AB? ? ?A < ? J, ?@@-? D <@? A D ?. ? A?PQ @F@? A ? ?A <<@ @F@ D A A@ A A A <A <? ?A? ? @F@? D <?A? AB? <?. " ? ? A@ ?@

ME? !@, '. .

34 E A < ? @F@?

@F@ B < ?@, < ? ? <-

@@ A ? A? ? ?A < A ?J, <-?J D ?A? <??. d @@ <?-? F <?J A @?A? ?A? A? A ?, @ ?? @B A ?A? ?? A. No !iene en- !ido hacer poe'a o li!era!"ra en Ma!em&!ica (E&"Ck, 1992,<F. 222). CA A ? A@ A <<? ? <@? @F@? ? @ A

@, ? A? <A < A@ ?@ A< <? @ ?A? <?@?.

El registro matemático

E @F@? ?@ A A <@A, A ?< < <? ? A ? ?@ @-

F@ ($i!!, 1987, <F? 75-110). E ?@ @F@ ? @ A@ ??, A BA ?<-, <?J ? @G?, ?@A@A? F@?, <? A ?A@ ? <-?J A ? B@@ @F@?. E ?@ B-A @ A <A @ A A ?A?? <? <@? D <?? @F@?. ?@ (Haida, 1975) ? ?J A A@ <?

? A ? ? ? @? ??, <-? AB? ?? < <? ?@@? <-@?. E? A A@ A < <? -<@? A? A @?@ <?@ D< A @. E ?@ @F@ ? A - A@ B? D J? L?@? AD ?A ? ? @BG? A? ?? D AA J ? ? ?A? @F@. E?@ -?@ ? ?@F@; BA D ?@@@, <@ < AB? ? A ? @ ?A? @F@, ? @F@? ?@? D

ME? !@, '. .

35A @F@: %AG ? D AG ?

@ ?, D <@ < ?A J A

@A A < ? D BA.EB@@, ? ?@? BA @??<?: < <, d @ <? ?<?A ? < < @ <@ ?A?, @? PBQ P<@ Q, D ? A K?, D @? ?<? ($i!!, 1987; E&"Ck,1992, Haida D !a&i", 1993, !#&ga", 1998). ? A?@ A ?A A ?@ ?< < <@

A @ A< A@A; K? <, < -<, A @ ?A << G, F@ D ? <-?J. ? A? ? A@ A D A <?J A@ ? @F@? A ,?, A ? ?@. ' , <A ? ?@? <A ? ? D ? < A? ?@ < < ? <@? @F@?. ? ?@? @G ? <A

A@ ? @@? ?@@A?, A ?I A? <?? < A? @?A<? D < A? @? (gE&gE", 1995). E -@, ? ? A? A ?@ @ ? < <@ A? A P?Q. d AB,?@ ? A ?<@ A ? A? <@ ?AB ?. CA ? <?? ? @ ? ?@ A < A@ ?@, ?@? K@? ?

?@ <@ A A BA? A ? A A? ? <@? @F@?.

!A? ?<@? A ? <@? <@ ?@ @F@ ? ?A@ B?? D ?. AJ @F@ B- ? A@, B, A, <-@ @@. !#&ga" (1999, <F?. 58-59), <A? A ?@F@? F? ?<@? ?< A@ ?@B ? A@? A ?. ' ,@G BJ A D ? ?@ B- F? @F@ A @? ? <?J, ?@

ME? !@, '. .

36 E A < ? @F@?

<A@ A ? A? @ A <A@AJ F? -

B gC'E

* <@ ?@ B <<? ?A? @?, A ? <? ? ??<@? A B.

E ?@ @F@ B @ <? ?A-<A?, ?J A ?. " ?@ <??@? A@? AJ. E @ $@F? ?@ ?A@: PE A @FA @FA, A <@A? ? A ?A ? A?

? ? @@?Q. " ?@ <? A@? ?@?. ' , @ $@F? @G ? <?@ ?A@ : PE A @FA @FA,? ?A? ? A? @A ? ? F? -@?, ?A@ ? A A @A <@-A?Q. E?@? <? @ A ? A F? -? < ? A? @F@?, < @G A @B @@ A, A ? ? -

?? ?@ B. ? <?? A <? A ?@ B ? <@@, < < A ?<? A?@ A ?@ @ @? ?A@? < A J. E < @, ? <? PFAQ D P@A Q ? A@?, D A ? ?A< A AP@FA @FAQ @ FA @ D, ?@ -@@, K ? <@A? A -

? A ?< B ?A @A.E ?@ B @F@? @G ? @-

<, ? <?. E ?@ <? ? A -BJ <@ A? ?@? G? D ?< @@? @F@?. E A? B <?B D ?A<?J <? <?? ? @?@? ?A? @F@ D ?@ @AD @ PB A@-@ D ?@@Q (!#&ga", 1995, <F. 14) A ? @ K

ME? !@, '. .

37A @F@: %AG ? D AG ?

*General Cer!ifica!e of Sec"ndar$ Ed"ca!ion, ?

A? 16 I? i@ D g?. (N. del E.)

? @@? @F@?. E ?@ @F@ @G ?

<@: P? A?A? <?? A @F-@ A?@ A <<?J ?@@ @ ?A -A F? ? D ? <?Q (E&"Ck, 1992, <F.222). E ?? @F@ @ ?A< @@ ? <?? D < ?A@ ? F? ?<?.? ?: PE K A? A ?, A? <A # , ? ?< D A K <-?? A ? ($ )R, A # > $ ?< -

? D <@ < ? <? @F@?, @? ? <? <J . ? ??@ J A ? @F@@ <@@D @ ?J D A ? AB @@ A ? ?AJ. G ? @ A ?A? @-F@ ? @ A? <?? D ? @, AA A?B, ?A? ?@ < A @@?A@@ ? F? <@ D ? <?. @-

@? @F@? A A A @ ?@?@? ?? <??. iA? @ F? < A- ? @F@? ? ig"al <A @?? A@?. $A ? A AJ, ( N 1)2 = 2 + 3 A @ ( + 1)2 = 2 + 2 + 1 A A f (2) = 5 f ? A AJ .E?@ @? <?, D A < <@? ?A@-@ @?. E @@ ?@AJ @F@

@ A ? ?.

Características importantes del registro matemático

" A? ?@ @F@ ?@B ? @F@?. E ?@ B@F@ < A? ?@ @F@, ?, ? ? <? A@, ? , D ? <@? @F@? ? A? ?@ @F-

ME? !@, '. .

38 E A < ? @F@?

@ B. ? A? <A D A@

?@ @F@ < <? ?A? <<? ? D -<@? @F@?, <? A @ B ?@G A@ ?@ @F@ B. ? ? ?@@F@ B, <?@ < -@ <?J ? ? @F@? < <-@ ? A?, <A ?@? ? ?@ <? <? ?A? ? ? ?@. $ @@, ? <@@A A <? @ @@ A AA ?A ?@

A < ? @@? @F@? A ? A-? A@, @ < AG A?. E ?A A - ?? < <? ?A? ? D <-?J, A@ BA A D ?A? <<? ? <?J. E @B ?A? ? ? @ L D A? ?A BA? A ?@ <?, @<, A-@, A ? B@ @F@.

E? <@@ A, , A@ ? A@ J @?@ ?-, D ? A @ A ?@ F? @ < D@ @. ? A <J @-F@? D < ?@ J ? ?? A ?< < ?A <?J <?J. ?A? A < <? ?A? ? A@F@ AF @ @ <?J, < <D

<A ? F? <@@? <? ?A? -<@? @F@? D ? L, ? ,?F F? <@@? ? A@ ?@ @F@.

Vocab%la"io e#ec*fico

E ?@ @F@ @ A BA ?<(#E&b&" D "iCH#'#", 1976; 'Ha&d D &#HE&, 1984;Haida D !a&i", 1993; a$'#", 1997). @ <? A? B @? @?:

ME? !@, '. .

39A @F@: %AG ? D AG ?

1."<? A @ ? ? -

A K A @F@, <? A ?A@ < A ? @F@? A @@,2."<? A @ A ? ?J A-

@F@: <@A?, ?J??, @, F-,

3."<? A @ @? ?? @@ A @F@ A @A:, <, , BA, B, @.

E?@ ? <? A ?A BA ?@ @F@ DA ? A? < @F@?. E < ? A AA ? <-? A@? A ? @F@? <A ??? ? <? A@? ?@A? , ?? A <? <? ? A @ A ? @@ @F@?. E?@ ?

A @F@?, @@ A < A ? <@ A ? B?@ A .$ <, ? A < <@ K B? A@ A A ?? < @?<@ @@ <?? ?@ K@ <@, ? A A ? @ A, A? A@? <?? -F ?A <. E A < @F@ ?@ ? -? A ? @@ ?AJ. ? <?

PA Q ? @A < ?@ A ?@ ? ?, D @. ' , A@ ? ? @F@? ? A ? ?@@? D -@@A?, D < DA ? A?. ?AA@ ? A A? <? ? < -@? ?? A? @F@?, < <, PA-Q (A G@ D A@< A K < ??), P?Q ( ? A @FA D ? A -@) D P@@Q (A @ @G@ A , AB < A @ @ AB ?A< A <A@ < A).

ME? !@, '. .

40 E A < ? @F@?

Sin$a&i# comleja

C A ?@? ?@ B <?-@ A@? ? A? A ?@A @F@?. ?<?? B? @F@? O ?A< <? D A? ?A@ B <?-BO B@ A ?@? ? B? <." ?< ? ? A G?@ ? < A <- @F@; ? , ? A@ <? A ?

A@ B <?B

* B <?, < ?@? ? B@.

a @AJ @? A <F A @@ B a:

E A@ K? @? ? < A? <-<J?@?, < ?A ?@? @A ?.CAA K A <A ? <? a/b -

D <A @ AA B K @ <-@ b = 0 ? nmero racional. ? ? K? @?? ? @? <A ? <?? a /1. ' @? A?@ A@ K? @? < A @? ? K? ?, @? A A@ K? ? %.

('adE& D H#&"i"g, 1987, <F. 1.)

E?@ @ <B @AJ D, <J, @ A ?@ A <@ ? <? A? A A@. E< ? <? P?@?QPA?@Q B? ?@? < A @ ? ?A ?. E? B@ A ? @A A -J <@ @F@ K? ? ?A@ B <?B, BB AB ?@ A?B

ME? !@, '. .

41A @F@: %AG ? D AG ?

*aAA ?<I ? A@ <?B ? <? < (P<A ? <?Q) ? F? A@ A? <?B ,

? , P? <A <?Q. G? F? $i!!, El leng"aje ma!em&!ico en

el a"la, !@, 2002, <F. 111. (N. del E.)

?<AG? J. ' A B <?B <

A J ? ?A@@ <?, AA ?-@? < ? F? ? A@ A ?@ @-B. $ <, P? A K A-A K A <? <? a/b ; @?A a D b ? K? @? D b <A ? AAK <@ 0Q. " ?@@, ?@F < ?@ ?A-@ <@, A ? ? @@ A@ B <?B< @A ? @F@? <? <

?@?.E ?? ? A B <?B < AJ @F@ <A B @B @A<? ??. ? P <@ AJ ? B @?Q ? A ?A@P@BQ, AA ? @A P@AQ <? A? P?@?Q. @?, P@AQ P?@?Q ?@A A < A @ ? ?@ A

@@. ' , ?@ <A ?A@ A?, DA ? < < A @ ? BA B-?J. E A ? A? ? ? @F@?, ?@ ? ? A?, ? A ? <?- 3 = 15 ⇒ = 5, ?A B?J @@?. @ B <?B @F@? A ?A? ?@F ? D @B ? A?A?<@? , ? @ A < A ?-

@ A ? <? D A? ? <@? @F-@?.

El %#o de la me$(fo"a a"a $"an#mi$i"%n #ignificado

@F < @? ? B? ?A?@F@: ? A? @? ?, A A-J ? "na m&q"ina D A AJ ? A eq"idad . E?@?@F? ? K@? ? <@? @F@-

ME? !@, '. .

42 E A < ? @F@?

?, < ?A <? ?< ? . E @,

? A @F ? < - A ? ?@AD D ? A@ ?@ @F@. aA A A A, A? A?B @F- <A ?A@ A ?. A A AJe "na eq"idad A AD ?@ A ? AD ?K? @B? ? <? A ? ?A-. ' A ? @F D ? <-@? ?AD@? ? A?, @F

J < ? ?< @F. EA? <??@@ A @F A A ? (<<, A AJ ? A FA A e como A FA) @ A <@ < A?JD @ A ? A? @@ ? ?@ @F@.

' ? A? ?@F ?@A? ?@ -

@F@ ? <??, @@ < ?@ @ A A D << ?, ? <A ? A , A D A A A ?A@ @ AB.

($i!!, 1987, <F. 109*.)

El ode" de la# alab"a# en el "egi#$"o ma$em($ico

< <?J @ ?@ @-F@ @ < B A < ?A <@ @F@. E BA @? AD <@@ < ? <? ?-@ @F@, A ? A <@ A @ , A@, @? D ? ? @F@-? < ?A <?. ' ? < ? A? A

< ?A? <<? ? < ? ? @F@?

ME? !@, '. .

43A @F@: %AG ? D AG ?

*$F. 160 @AJ ?@. (N. del E.)

A , <A < ?<@ -

?@ @F@ D @ F? < < <@ ?@@?@.

Dificultades que surgen al utilizar el lenguaje matemático en la clase

El leng%aje ma$em($ico como %na #eg%nda

leng%a

$A?@ A ?< A ? <? ?@F@? ? ? ?@@ A A @ A? ? @?@? A A @A, ? ? ?I @F@? ? AA ?A A @ <A DA ?

A? ?A< A? ? ? <A < A@ ?@ @F@. a< A @ A A ? A? ?<?, F@ D ?@? A, ? @ A J A@A A A@ A <<? @? ? D <@?. E? ? < @F@ < < <? ? ? D <-@? A ?A ?<, D < A ?

A? < ? A@A.

A"ende" a +e&"e#a"#e, como %n ma$em($ico

E <<J?@ A@ AA A @A ?A ?B < <? A A@ ?. ? @F@-? @ A@@ JA?. E A ?A@ < <? <@? @F@?, ? @J D J @ ?@? <@?. ? ? -A? @A? <? @F@? A@ <?

ME? !@, '. .

44 E A < ? @F@?

?@ A, < @G A? ? <?J A

? ? @F@? @ A. E?@ A PA A A @ ? ?A @BG? < <? ?@ Q (E&"Ck,1992, <F. 219) $ @@, < <?? A @F@? ? A? @G ? <? <? A @F@. E? <? A< A @F@? @ @? ? <@? @F@? ($i!!, 1987; gE&gE", 1995).

J ? @A @@ A @F@ @< A ? ?I <?@ @F@. a A? ?@ ?@ @F@ < ?A << <-<J?@, ? A ? A? <?? @F@?: P? I? ?@ < <? - @F@? D A? A @F@ < , D A ?A? <<? <?? @F@?,? @<@ A @F@ @?Q ($i!!,

1995, <F. 179).

+Hace"lo bien,

!A? <?? < A D A ?<A?@ A- @ < < @F@?, D <-@ ? @ < D ? <? A ?

A? ?AB . E B@ <?A ? @F@? @ ?@ @@A ? A?A ? A? ?A ??. ' ?J D A?<A?@ @, @? @? ? F? ?@? D, < @@, B@A? A ?<A?@AB ?@@AD ? ?. aA? D A-? A A @<J ? A D A ?<A?@ @ BJ AD A @ ?B A <. E?@ - <? <A ?A< A ?@FA <@@ A? A? ? ?A? ? ? -

ME? !@, '. .

45A @F@: %AG ? D AG ?

@? ? ?B A <. $ A ? ?

P@Q < < ?? < ?@K<? ? ? AB.a A? A? ? <A< P Q D A

?@F ?A? < . A <B?@ <A < <, < ? A ?< < B? ?@K<? A ?@F ?A-? J P Q. ? <?? @ A - A ?A, < DA? ?A< A@ -

BJ A D A ? P@Q << ?A A <, D ?A @ A A??@ K A @ A ?A<< < <B. ? A? B? <@ << A @ @ A A <AI ?A? G@?.

' <A B ? A? A < ? @?, ?< D A ?@ A

?A? ? ? B? D ?<@?, D A ? ?A?? < ?A <?J, < ?@ ?A-F F<@. ?BJ D A?? A@BA-J <A DA? ? A @? ??? D @? A < A ?A@ B. E? @? A ? A? ? B A ?@F P Q.

E?@ @? @? A A A? D A-

? <? ?A @ P Q. !A? ?@FB? A @ A B <@ ?-@? D ? <A< A ?@ B ? ; ?@ ? AP@ @ <Q (dECk, 2000). ? ?@A-@? <? ? < <? A @ A B < A ? <?? <A@A A <? ?< J <A ?A <, A A? ? @? BAJ < ?@? A< A ? A<? ?K ?A? <?,A <? AD K AJ ?A ??A? & . <@@ F? @B

ME? !@, '. .

46 E A < ? @F@?

? ?A A @ A <A ?A <

< A? ? <@? @F@? ?A << ?AD DA A? @ A@ ?A? A@?< <. E?@ ? @ @ <-. ' ? A? A@ A@ K @,?@ ? PAQ <A ? @? A <A <- < D ? PQ <A @ ?A? <?. E? <@@ <A@ B?@ < <?B A @ A A@ @-

B @ < < @F@? D A ? @B @ A < <? ?A? -? ?A?.

Conclusión

E A ? @F@? ? <@A D A@ A

BA D ? <?J ?<? ??@ @F@. aA? ? @?@? ?-@ @F@ ? <@@ A?? < ? A-? A ?@F < @F@?; < <, <-? A @ ? F? ??, A? @F-, A?J A <?J @F@ ? F? ? <?. E << <? ? @ ?A? @F@? D ?A? A A@

? A? A@. d ?@ , <A-@? @ ? ?A?? < A ? A? ? <-? A@ A @F@ < , B?-@ D A ?A? ?.

<@ A? D A? ? ? , ? A@ A ?@ @F@ -B <?J A ? <?, @<, A@ D ?. " ? ? < A A@- ?@ ? ?@ D A? @F@? . ' , ? ? ? A ? F? <-@@? A@ D ?@ ?@ B,

ME? !@, '. .

47A @F@: %AG ? D AG ?

< A D < < @ <@ A ?-

A? @F@ F? <, D A A ? ?? ? @F@? ? ? <? A@ ?@ B.

E? <@@ A ? A? A@ A @-F@ < ?A << @B. 'F <? <??A? @? D ?A? A?, @ A ?@F <-<? < < F?. G <F @? ?<@? @F@? ?@A? AB? D ?A?

? @F@?. ? ?@A@? A ? <? A@- A @F@ < <? ?A? ?, ?<? A? @ ? D ?A <?. d?@ , ? ?@F ?<A?@? D <@ ? ? <? D <?? D, < K@,@ A <F @F@? A .

ME? !@, '. .

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CAP,TULO III

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55E @ I@

Nadie le%an#a la mano

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57E @ I@

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e. (de A'!#e C%#').

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!# e% #!#e'e (de Ge&&a N#'a).

El tiempo es un factor importante

en el aprendizaje "<@ <@ < @ @@ <<<@ @

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65E @ I@

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67E @ I@

Planteamientos prácticos para usar en la clase

Of!ece! $na e&plicaci+n cla!a de $na ideama#em)#ica

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1, 1, 2, 3, 5, 8, 13, 21...

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Introducción

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Objetivos de aprendizaje y criterios de evaluación

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KE M*<*, %. L.

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134 E =* ** * *<<*

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KE M*<*, %. L.

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* /.

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136 E =* ** * *<<*

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= * <*+** .

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Alfabetización y matemáticas

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Pe"&a%, haba%, e&c%ibi%, ee% + )#)e% a %edac'a%

E <* =* =< * **+* N+O / * = < *< *< *<<-, * <, * <* <* *= < <* ** ** *<. E+ *+ * * *<<* =*< *. " * *= = =< *< * * = * * *<* +/ = @*. L* * *-

*, >* <*+ * /=-. A, */<* * =*, =< < =* * *<=** = =* << ** > < +*. L* <=* *= < *< = = >* <* ** + * = * < = >* /*-*.

% * <=* =<* +/= * @ *, <*+ < * * *- /=<*. = * *= = - *+ =* = =* = < =,

KE M*<*, %. L.

141A>** * =* *<<* =* /**

= * = *< *< =

< =* @** <*< >*=*. @<*-* = * <= + + = * *<<* ** *<** /. M= / -* < *< <* / = < = +/- * *. % +*, = * = *= =* <=* =* * /* *-<<*, *== =* * = @=,+* =</= = *, @* /=*

/* *<<*. E + * * = * +* *<< */ /* -<* = < ** = <* < =* * =* =<*. E@*, + =-</* =* *< <*< * *<<*; <-* *< *<< = * =* *. C=* * *= = +* <**, >=* * <*< *<

*<<.E+ <*<, *<<* / @ =. " <*<, + * =<*/* =* @* **, *= ++ *=* <* = >* * +. E *- * *+* = < = + < <* * * = * **+* + = * <** ( , < <*+ < @<).

L *= <*+** < /* >=* ***+* = +* = = ; *= *, = * =* + =*=* = =* /*<* < < **. E /= * * * =<* + * = ** = =*= <* = = =*> *=*. % < *= = =<*+*.

'* *<>* <* <*+ * < *= *= ** * = *** /= =<* * =* >< = + =, * -

KE M*<*, %. L.

142 E =* ** * *<<*

= + * * * <* =*

=* = < *<< ** > = +*. L*** /= @ < @** >* *. '<* ** > ** <** * * = -*< +=* ** *** ** -<* * = < = < = >*. E /+>* *< *** /= *>=* @< * =* = <* = @** < <*

= . L *= =* = *** = < ** ***, I *<*<* /, = * *>=* < = * + = = * =<-* *<. A <=* =* ***** *+* =* @* <* ** = * * ***, *= -<* * * = *** /= ** *=

.% <=*< * *+* =* /* -*< = * *<<* <*< =:

Pie"&e" + = <<* =* =/-< <.

Habe" * <* *, /+< =* =* * @* = =-

*.E&c%iba" =* ** * <* =

*.Lea" ** ** <* * ** >

=* +.Redac'e" de "(e)# e b#%%ad#% +** * <*-

<* + * /* * <=*, = *- * * = <*+* *= = = *-* =* **.

KE M*<*, %. L.

143A>** * =* *<<* =* /**

C(-"d# e&c%ibi% + c(-"d# "#

E =* *<<* *=+ ** * +*<*<; *+ * * = **, = =<** <*+**- =* *. *+ + * . M /@ * >*, *= *+* * /* = * * * =<* *=* * *= * < /*-

< +* +*<, * =<* * <.% +*, * < = + <*

* <*<:

=* *= <* * = < *-< =* * = * * * * ** =<** < <,

=* = < /@ **** = *= =* *<* = * - ** = <,

=* *<* =< + = < **= =>* <* *,

=* <<* > = +* = * <* = < *= * <*,

=* = @= * / =*+, = * , = ** * ** */* *=**< * - =<,

=* <* <*/* * * = * *** *<* <* * * *< + ** = <.

E < = *= <*+:

KE M*<*, %. L.

144 E =* ** * *<<*

=* + *; + * * =

< <* * * = =*,=* + *+* *** = * =* <* *< +,

=* => =* * = >* /* <* , =, *=<* *=<* + =* ** * +* /=* *,

=* <*+** =< = ** + =*

@* <*<; *= <* *+* * * = *= + =* @* ><;* ** < = < /= = **, =* <<==* *< *< *<<.

"= * * * * <=* ** N* *< * @* *O. C=* *=

*, *+* */* = *, <* **- ** + /* < <* /=-* *+* = < *, @*< =< *<<. %= <=* < >< = < I< = <* ** ** -< * *<<*. L *= >> * -=<* = * < < = ** =<* *+ ** =* = +*, =*

, <* "<*, =* *<* =/= = +< * <* = +* *=*. D <* **, = <=* = /=-: * */* ** = *, * +* =<* =* < **<.

% / *=* * = *= * * * /** * * * *+* = * *<-<*, * <=* +* <** * *.L *= = * <*+** *<<*= * < ** = =* ,=* *+ <*+** =, * *, *+*,

KE M*<*, %. L.

145A>** * =* *<<* =* /**

/@*, */*, @* =</* = *<,

* * * =* * < *<<.&*+** = = * * < = ** =* /=* * * = -< * * *<<* <* >=*< .

Conclusión

E < *<= <* ** * * *= **. C=*< = *+ + < *<<-, =< **+* @ *<<*, =*, * >=* <* * >** ** * = *** ** *. i= *< *= *=* = * *

=* <** / * >* = <. &*< / <=*< <* * * = *= + ==> * * *. E / + * * =* **< * <* *, * *<* = * = <, */= *<>* <*+* * *<<=*=** ** = * *= = *+, - ** = * *<* = * *<> ** *= * *+* <* *

** *>** = **.

146 E =* ** * *<<*

KE M*<*, %. L.

Introduccin

E => +H>? B+<G + += >J<+ += += ? =+<<+ +H>?= +><<=. EB++<G < ?G < + =I++ <H+ ++< + +<+? ?F= =<H+ = <=?>+= =<+= = +=. M=><+<G @H? B=>> >< -?+ +<+ +>F>+= + ?= +><+>?<+ =.

L+ ><H+ BH+ ? + < <=J

<H J => + H+ << < + =+?=, =><<> + = ><= <=<= +<+ ?=+< +< += += +>F>+=. C=< ? + += >J-<+ = ? +>< <>+> +<+ <++< ? + =?=>+-+; <=<, =?= <+= F=+=, > ? +=+<+= += +<+ ? + = ? =J + =?<-. EB<+< + ><H+ ? = B +?H ? >+< ++, <@< <+= +<+ +< += += <F>+= + += >@+< +<+ <+<. $ +<, + =< ?>< +<+ =< ><+, < ? F= ++>, => = +H>?, B J => = =.

NE= M<+>+, $. L.

147#<+= <F+=

CAP&TULO VI

La fuente de ideas:Profundi"ar en la teor$a

E << ?+< ? ?+ <=>@+ <+ + >-

<H+ ? >+ ?+ +<+, B< <?+ = @H?= =H= >< ?+ +>-F> +<+. D >G< P?+ =-?<= (+>F>)Q B < ?G = ?+ ><H+K> +<+ ?+ += + ? ?= ?+ = + <<+->+ +<+ <<+. "<= +<+ B++< ?+> >J< + E@+?+J +<+ A<+ <@= <@> @+<+= ><H+= +.

Una perspectiva general de la teoría que vincula el lenguaje con el aprendizaje

D= ?> @=>+, += =?+= >?+<= += ? <?=> ==> ? + += =@<> ?+ P?+ =?<=Q, =>

G=> = <<=+< +<+ += +>F>+=.E =?<= > @<== =+= ?>+=; +?H ?= +<+ < = <== + ><+@G= = ?+= = <?-= =>?+>= = ?+ >< =H (@< < CADE, 1986 "iMM, 1996+). "< >+>, ?+ += +>F>+= =?<= ? ?< ?= +=>=,>+= ? ><+ <+ += <=?=>+= =<->+= <=< = +?=. E =?<= ?< >

F> + ?+J += +=++= ?+?>+ + +=, ? = P?+ >+ = ?=+ +<+=+<<+< + @+ >>?+ =+ ?+ ?+Q(ME#CE#, 1995, F. 79).

%+G ?> >G< P?+ =?<=Q <-? +< @+< += @<=+= +>+= ? ><=+< ? <=?>+ @>+= +<+ <<= +<-+ +>F>. E >G< + = ?>+ ?<+> +?-= +I= EE.&&. (@< < E#%$Ch %!MA,1995; $iL'E# $Mi%h, 1996; $hE#i, 2002). E +<>?+<, =?=+ +<+ =<< ?+ += +>F>+= ? = =-

NE= M<+>+, $. L.

148 E ?+ +<+ += +>F>+=

@?@ =K < C= +-

+ "<=<= M+>F>+= (Na$ional Con#eil of Tea- che" of Ma$hema$ic#, C%M, 1991) + P++<, +< ++< +>F>+=Q. E <>, >+= ?+-= = =<+ ? = +?= +=> B? =?=+=, <=+ + = =+>= B<=+= <><= <= + +=. L= <=<= ><@ <+ ? +>+ >+= F= ><+ +=. E=?<= +=H B?=> + += = P@<= +?+>Q

(C%M, 2000).M <+ ? +< ?= ?+ +<-+ += +>F>+= > ? =><+ . E <<>+ Na$ional C%""ic%l%m ("+ E=>?= ++)+<+ += M+>F>+= (DE$, 1989) BH+ ? = +?=++<+ =? ><++, <=<+ <+ <?>+=+>F>+=. E=> =>++ +=+ < C<>(C!CkC#!f%, 1982), ? H+ =><+>+= ? <++

+ ?+J +>F>+ + +=, < PM+>-F>+= +<+ += >< 5 16 +I=Q ? +<+: PE =?=> F= +, += +>F>+= <@ ? +<+<++<, ?+< ++< <+J. E ?++>F> ==> +<++= =H= @-= ><+= +=+=Q (DE$/!, 1988, F. 3). E?+ =<@ ?+>+> +<+ +<< +>F>-+=; ?+> < ?= + = +?= ?+ +>-

F>, < <F =><+< =?= >= +>F->=. E= <, =<F @+?+= +>F>= F=>>=. L+ =I++ += +>F>+= + <=? ?+, +@++ = +>+=+ < ?+, = <=?>+= + ? = @+K+ +> ?+(D&#ki $hi#E, 1991, F. 1).

M!#gA (1995) =><J ? ? = +?-= <<=>+ =? +>@+ +>F>+ <=+ =>B>= ? <?, = <?+ +<+ >< GB> += +>-F>+=. $? ++ +<+ =<< =? <++> =+><= ? =<@+ += +>@+= +>F>+=

NE= M<+>+, $. L.

149L+ ?> +=: "<?+< + ><H+

? => @+ +>F>, ? =< ? +>< +-

@ +<+ =? GB> Ke S$age 3 (? B+ ? = <+-+ + = +?= 14 +I= i+><<+) <+>F= <>+> Gene"al Ce"$ifica$e of Seconda" Ed%- ca$ion, gC$E (B+ <++ + = +?= 16 +I= i+><<+ g+=). $ = +?= = ++= =-<< ? => <=+ @+ =< <-= ? + <++ + +<+< =?= >+<+= +>F>+=,=? ><++ <+> =<F @+?+ +<+ F=

=>@+ ? = +?= ? =<+ = ==<== +>F>= < = =><+ ? => -@+. "< >+>, + +++ ?+< <+J ?+ +<+ <=+ =><?>?<++ + ?=+< => -= <=+ ? = @+> +>F>, =<>+> +<+ GB> = +?=.

L+ ><H+ ? = +?= =<F @+?+= <= +>F>= ?+> < ?= ++ ?+

+>F> >+G = <+ += ><H+= gE#gE(1995), ? +<+ ? = <= =>?+>= ?=+ = =?<= ? ?>+ = +>F>= ?+> F=P+>F>=Q = +, = <, ?+> F= <++-= > @<><= +? ? ? <=@< <-+= +> ?= += +>F>+=. EB=> ><+=><H+= ? =++ + <>++ +<< + B<=+<-= ? +>F> > +>+< + >+

? +>F> (@< < h!LLAD =., 1998,LA'E EgE#, 1991, EgE#, 1999). C?+> +< K-< +?= =+ ++= P++< +>F>=Q,< <F +>+< + >+ , =?-+, ><F +< +, + ?+ ? ><=, +<+< +>F>+=. "< >+>, ?+ += @>++= ?+ ?>+ +<+ B<=+< += += +>F>+= = + @<><= +? ? = ++ P+<Q +>F->+=. E= @H? + ><+@G= ?+ = +?= + +=<+<= ++= ?=+< +< = >=+>F>=.

NE= M<+>+, $. L.

150 E ?+ +<+ += +>F>+=

&+ += = ?+ ?+ ? > ?+ ?>?<+ +<-

>?+< <++ < <=< = +?=, >+G < >+ ? = +<. A<< +>F>+= ? <= +?<< ?>?<+ +>F>+=. $K ?> @=->+ P>HQ P><++Q, += +>F>+= <+ ?+>+ > + ?>+, < <++=>>? ?+ ><+J ? =>+>> = P<<?+ = <=<+Q (E'A$ %$A%$A#!i, 1994, F. 169) <?+ ?+ <=+= ? =+ =+<<+ ? ==-

>+ > ? = += +>F>+=. L= +?-= = ?>?<+ + ?+ +>F>+ ?+ +-<+ <++> ?@+= <+= <++< =?B<+ <B+< =< + <++J, =><+>-+= >= ? + + =+<<+. E=> <H+ -==>< ?+ K=?+ +><= <+, ?>> <++< <++< G>=, =?<< -B= ==>+ =+<<+< +<?>= J= +<+

?=+<, <+< +<>< =?= <=?>+=. E=>+= <F>+=,?=> ? > =? < ?+, = ? -+< <F>+= =?<= +>F>.

Lenguaje matemático y teorías de aprendizaje

E +> =< += ><H+= +<+ -

+< =><?>@= <++, >+ B+ '!gLA$E#fELD (1984), ? = +=J ><++ "iAgE%. E=><?>@= ?=><+ J = +?= ?><+ ?=> + ? = ?<< + P=><?<Q ? <++>@+ +<+ = ==. D==>+ ><+=-=J +<+ O=K ?+, > =< ><+=> + +? = ?+ ?> B><+O , =? ?+<, +>+ ? ++ @? + <+< =? <> ('! gLA$E#fELD, 1987). "+<+ ? >+ ?+< +<+, ++ +? + <<++< =>-> =? < ==>+ >@ +>?+ +<+ ++< +

NE= M<+>+, $. L.

151L+ ?> +=: "<?+< + ><H+

?@+ B<+. $ + ?@+ B<+ = ?><+

+=+ +++ +< +>?+ =>?+>, + ++->+J + +=+J ? >< ?+< +? +?< > (LE#MA, 2000).

$ +<, += ?@+= B<+= @@+= < =+?= + += = += =< ++=, + = +<>, < ?+ H> +< =+ ?>?<+, < ?+, + ?+, <=< =>F =?>> +<= += ><H+= =><?>@=>+=.

"?=> ? = <=<= = +?= ? =><?<=?= <= =+= +<+ += ++<+= +>>= >B> ?+ ><+J ?<=, = <>+>@> < ?G + ?+J + ? = ><<?, < ?G = <=<= +?= <?+ = >-. E= <= >< ?>+ => <+ =><?>-@= +<+ <+< ?+ ?+ ?+J.

(C!bb, 1988, F. 92.)

L+= ><H+= =?>?<+= =>K+ ?+ K +<+, + ? ?+ = <++< + ><+J =+. 'g!%$k (1962), E#%$Ch

(1985), b#&E# (1996) !!D, C!bb ACkEL (1995) ++<?>+ =< J = I= +< + =>?+<= + =+ + ><+@G= + ><+==J =?= +I<=

F= >>= ?=+ <<+>+= =H=, ?= = H=>=, ? <+ +<> ? =+.E= <, = I= =+<<+ P?= >+= F= -@++=Q ('g!%$k, 1981, F. 162) + ><+@G= ><+-= =+= ? > ?+< <+ <+>, < B?=@+, ?+ @<++. "+<+ = =-?->?<+=>+=, +<+ = <<=>+ ? @?? =>F ++ @ F= ++>+ +<+ ?+< =? ?+< ?+ ?>?<+, =< P?>?<+Q + ><+@G= <F>+= =+-=. "< >+>, ?+ += = <>= <><? ++ @? =+<< =? < +<+ @-

NE= M<+>+, $. L.

152 E ?+ +<+ += +>F>+=

?+ + +<= =+<< > ?+

><++ ?+. D +?< += ><H+= =-?>?<+=, += ?= =+<< = +?= <-< > ?+< >< <=+= ?+ +><H+ -><=J+ =J F= >+< ?+ +><H+ ><+-=J+, ?+ ?J >+ +=++ ('g!%$k,1981). E= <, >+= ?= ?<< +> ? +> =+ ?+ = ><++ ><+= <-=+=.

E= =+< ? > >< <+= F= @++==+ B><, = <, +<+ ><= <+ ? = +<+ +<+ ?=. C?+?< ?J >+ F= @++ +><+@=+ -=+<+> < ?+ += B><+ =? =+<< <? -+> = ?+ ?J =+... C?+?< ?J >+F= @++ + = B><+ <? + = =+ +K> +>= =< ?+ ?J >+ <+> ><+.

('g!%$k, 1981, F. 162.)

"< >+>, + ?+J =+>=+><+ ?+ += =@>+ +<+ +<< , ?=> ? += ?>+= ?+- +>F>+= ? ><+?<= ?+ +<<<+ +<++ ?+J, = =+ ? <=< <?+=>= @>=. L+= ><H+= =?>?<+= <@ H>? +<+ =<+< <+= <+< + ++

= +?= +<+ ?=+< ?+ += +== +>F->+=.

&+ ?+J +@<+ <?@ ?+ ?+ +<= ? =+<<+ ?+ @ ? > K (!!D ACkEL, 1990; ME#CE#, 2000; DAiEL$,2001). E=>+< ?+ ?+ @+ = >= += ?+= ? < += ><H+= <F>+= =+=, +<>?+<, += += LA'E EgE# (LA'E EgE#,191, EgE#, 1999). E +< >?+ LA'E EgE#

><H+ <F>+ =+ =?<J =? B<+ +<-=. !=<@+< J = =>?+>= ++ + <

NE= M<+>+, $. L.

153L+ ?> +=: "<?+< + ><H+

+= <F>+= + ?+ +> <= P?+

+<>+J <G<+ H>+Q. %+= += = ><+=-> F> ?+ +=. & +? +< +>F>-+= F> + ?+ + += ++<+ =< <=< +>F>+=, =?<+ +<+ =< ?<=+ += +>F>+=, +?? ?+?<+ =>+== =+ <=?>+ + +<+ ?= ?+>=. $+<, ?+= = <>= + + ><H+ <F>+=+ = <<+= +<+ =+<<+< ?+ ?+

+<+ ?+ += +>F>+=. PL+ +++ +?<< > = ?+ ?J <+ ?G=>+=, J =>+=, =< ?G ?G +>?+=, >+= += =><+=, =, <+= =+= < ><+= =>= +=>= ><<++= =<Q (ADLE#, 2000, F. 35). C++< + +++ +?-<< > ? @? + ?+ ><+J = <?<== ?+ ?+ =? ><+=+<-

+ +<+ = +<>+>=. L= >= +<= >? =< @== = + ++< + @=+ >- >+. E F >< +?=, < , = ?<?<= + ?+ + += = ? +< +-<+ <>+< + +++ +?=J >."+<+ <>< + +++ +?=J > +>F>+=, = +?= << + < += <+= F + +=, ? >= ?+<F ?>+=. E=>

= @F <=> + ?= = <?<== + +=.EB=> ?+ ><+J + >< + @=+ + @-=+ = <?<== <?<= +<+ ?+>>@ ? >B> +<+. "+<+ <>+< += + < +<+ +>F>+=, +=+>@+= =>+< <+++= +<+ ? <->+ ? = >= +<= O< ?= ?+ +>F> <<+>+= H=+=O = ++@== +<+ +< @= =+ +>F> +-<+ = +<>+>= + <F>+ +<+ +>-F>+=.

NE= M<+>+, $. L.

154 E ?+ +<+ += +>F>+=

L+= ><H+= LA'E EgE# >+G <@+ F= +<-

+ + <>++ F ? >B> +<-+. D=< = +<+= <>= ++<: ++<P= ><Q P++< =<Q. L+ +<>+J + ?+ <F>+ + =< ++ ++< ? +<>-+> <. E => >B>, +<< +>-F>+= ><+I+ +<< + ++< ? +>F>,+ ? += ><H+= ><= +?><= O<, += ME#CE# (1995), "iMM (1991) LAb!#DE

(190)O + ><H+ + =><?J =+ + >-+, gE#gE. LA'E =>++ < =? ><H+ P++< =><Q (++< +<> +<) P++< =<Q (<-B+< =<) > ? ? = ++ <<=>= @+== = ?+ > ?+< + +++ +?=J >. E =>?+> +< +++< +<>+ +> + ?+ + @?-<+<= += <F>+= + =+. "< >+>, = =>

?> @=>+, =?<= + += = ?+ <<+>+ +<+, < =+> ?+ @?<+ + +< ?+ <+ +>@+; = <= +?= =>+ ++<>+> P>< Q P=<Q + <F>+ +>F>+=.E= <>+> ? =+ >@+= >+> +<+ ++< >< += +>@+= +>F>+= =< += =+= +?+<= + >< <>.

L+ ><H+ <F>+ =+ >+G <@+ ? = +?-

= + + + += +<> @<=+= ?+= += ? + <+ =?= >+=, += ?+= <=-><?>?<+< + + ? +<>+ + ?+ + +=. E= =>+ <=><?>?<+J + >+ <= +<+. E + ><H+ + <F>+ =+ (E-gE#, 1999), +< + < + ?+ = > @<><= +. L+ ><H+ <F>+ =+ ><+>+ > +<<, ? J >-+ =>F @?++= ><+= = <+ +<> + ?-+. L= <+= ><+==J >= =B+, +<>, < =>+= ><H+=, + ? >

NE= M<+>+, $. L.

155L+ ?> +=: "<?+< + ><H+

=>F ><+ + ?+ + ? = ><+=.

E= H ?=+< += +>F>+= + ?+ += +==>H+=, ?=> ? =+< +>F>+= =>F + + <-+< +<> ?+ ?+ ? =>?+ +>F>+=.

PL+= >+= = =><? >B> ?++>@+, = +?= =+<<+ ?+ >+, ? = ?+<+ B+<= + ?+ + ? >+ +<>Q(h!LLAD =., 1998, F. 270). E=I+< + = +?= ++>+< + >+ +? ? = ++ ?=+< +=

+= +>F>+= +<+ <=@< <+=, = ? >@<>+> +<+ ?+ += +>F>+=. gE#gE (1995)=<+ ? P<Q =J ? B=>< ?+ = <++<> ?+ ?+, ? +<+ < < P=G +>-F>+=Q, ? +>+< ?+ =>?<+ ? =>G <-++ ?> @=>+ + +?><+ ?=>+ < +?+ =+ +>F>=. PL+ =>?<+ J+ =< ? >+, = =+< J <?< ? ?+

? =>G <++ =>+>?=Q (gE#gE, 1995,F. 31). E= <, ? @? +>+ + >+ ?+>F>, +? ? P=+ +>F>+=Q, ?+ +<- + ++< ? +>F>.

Evaluacin para el Aprendizaje

L+ += >J<+ + E@+?+J +<+ A<+ +@+?+J <+>@+ =>F B<=++ < ><= +?><=, +<>?+< bLACk iLLiAM (1998+) bLACk =.(2003). $ +<, +?H ?< B+< J +?+=+<+= <+< + ++ = +?= +<+ ?=+< ?+ ? <> =+ +<+ ?= ->@ + E@+?+J +<+ A<+.

& > +@ + @+?+J <+>@+ = > <><+>+J ? = ?+ +<+><H=>+=+ ?+?< ==>+ ? <+ <?< ?+, + =+ += +== +<+ ?+?<

NE= M<+>+, $. L.

156 E ?+ +<+ += +>F>+=

><+. &+ +<+><H=>+ @>+ + @+?+J <+>@+ =

? + <+J = ?>++; = G=>+ ++ @ >, ++ <=J = +?=,>= + <><+>+J = <+>@+. "< >+>, == +?= > ?+ ? = B<=+ +<+J, + <><+>+J. $ >+ +<> <-+ <?+< =?<= ? = =+<<+ = -+<> += ?==, = <<+ = +==?>+= +<+ + <><+>+J >= = +?-

= ? +@++< =? +<+.!><+ += >J<+ + @+?+J <+>@+ < $ADLE# (1989), ? =I+J ? >< + +>@+ + @+?+J <+>@+ <= + =?+ = +-=. L+ <<+ = + <J =>?+> ?++?+ >< >@ =+ +>?+ =>+ -> / <=J / ++ +?. L+=?+ = ? + +J = @++ + + o" el a"endi

+<+ +< =+ +?+ +?<< >@ =+. E=>+ ? << =>?+> > ? >+< + + @+ + +?+ =?G= =< ++ >+<+= +=F= =>+ @+. E +? = ? +<->+> +>@ => <= + <++< = @H?= -= >< + @+ + +?+, + =J >< +=<>= += ? <H+ +< + +?+, + =?-> +>@+ <+>@+. E=>= +<?>= + +< ? +

+?>@+?+J = ?+ +<+><H=>+ =+ + <F>+ + @+?+J <+>@+.

L= <== <><+>+J, ?=>+> +?>@+?+J > ?+< >@+ = +<>< =+ ?+ +=. L= +?= <<+<+ +< += +?+= =? ?+ ?< J + +<. $ + <=J ?+ ? ?>+ <+<?>+<+ + += = < = <+<+> >+ +<> <= +<+J =, <F =-+<<+< @ ++ ? B + @+?+J <+>@+.L+ @+?+J >< ?+= B ?+ ++ < +<>

NE= M<+>+, $. L.

157L+ ?> +=: "<?+< + ><H+

= =>?+>= +<+ ?=+< ?+ +>F>. % ?

++<, >< @+?+< ><++ ><= +?+ = <><= =>+=. L+ @+?+J >< +I<== ?+ <<+>+ ? K> +<+ ++< + E@+?+J +<+ A<+ <? =+<<+ + ++ = +?=+<+ +?> @+?+<=; = <, >< + @+ =>+- +>?+ =? >, =><+= >>, >+< += +<+ +< + +?+ >< =? =>+ +>?+ >@ =+.

L+ @=>+J <+ =< + @+?+J <+>-@+ (bLACk =., 2002, 2003) =><J ? B=> @H?==><= >< + +<+ =+<<+< + <F>+ +@+?+J <+>@+ > ? > <=< =? <. L= += <?= >+> += <= = <=<= =? < + <+J + =?= =>?-+>=, + <+ ? <H+ ?+ ?+ <F->+ +=, + ? <+ =+<= +>= +<+>+<

? +H+ =>+ ?+= <F>+= @+?+J <-+>@+. L+= +<+><H=>+= <+= =>= +=<+ += =?>=:

"+<+ +?>+< + <><+>+J >< = ? <-H+ += =I++= <=<, +H+ ? =+<<-+< ?@+= <+= +H+. E +<>?+<, = =-+<<+< <>= ? <>+< @

=?<= + += +<+ ? ?<+ F= <+J= + @ >>, =><+ / -> ? = +?= >?@<+ = >, =< J +@++< +<+. L= <=<= = +?= P+ >H+ ? +@+<Q; >++ + <+J +<+ << + +J +<++.

L= <=<= ++< =?= J>== =< = +->+>= ? ?<+ <@+< ? +<+>@. E <>, +< + << ? =+?= H+ =>+< +>@+> += <= +<+.

NE= M<+>+, $. L.

158 E ?+ +<+ += +>F>+=

"+<+ ?=+< += @+?+= += >+> < <=<

< +? +<+ <+>@+, +H+ ?+< + <+J +?<+ +<+ ++>+< +==?>= +>@+= +<+, +<+ ? +? H+ =>+< + >< <> ><=< => <=.

L+ @+?+J +>+ >+> + + >@+J + ++?>=>+ = +?=; = ?+ +<> =+>@+ =+<< + >+ ? =>?+>. L+ +?>-

@+?+J ?+ ? + =+>@ +?> + >@+J + +?>=>+.

&+ ?=J <>+> + ><+>?<+ =< E@+?+-J +<+ A<+ = ? = += ?<+<= =+ <+ +>+ =+. L= += <?= + <F>+ <=?>+ King# Med&a O'fo"d#hi"e A##e#men$ P"ojec$ i * (kM!fA") >?@< ?+< ><+= ? <H- =+>@, <=+ @=>+<= ><=<=<= <? (bLACk =., 2003).

E <<= + @+?+J <+>@+ = ?+ >+<+=. + <=?>+= +>= +I+= + + <F>+B=>> + <=+ ?+ =+J <F+. "< ><+<, =J = <H+ +=?<+< ?+ =+J =?=>+-

+, = <=< =?< =? < G> +<+ <<+<+= = += =>++= +><<+ =?=+?>+= ><++ +?+. E=> =J ? ?<<< ? <+>@+> > + ><+@G= <<++= ==>= =+<< + <=+. E=> <=>+ <>++ +=+ ? +?H = ><+=>, <H+ =< ? +?>>+, + ? += <+= ?<+<+= =J+= +=I++ +<+ =J ? >< ?+< =>.

(bLACk iLLiAM, 1998, F. 15.)

NE= M<+>+, $. L.

159L+ ?> +=: "<?+< + ><H+

*"<> E@+?+J f<+>@+ k= M+ !B<=<. (N. del T.)

"E##E!&D B<=+ ? ?> F= <>+> =< +

@+?+J <+>@+, +=H +<+ += <=?=>+= = +?= + = += <?= < +=I++.

?<== +?= +=<+ + +<< FB=, = ? = >>+ +<+<, +=+< ?<=, H+ +I = ?= =+=><=, >< > +<+ ><+=+>@+= ++<> += >+<+= +=... L+ @+?+J <-

+>@+ <=? <+ @+<+ ? + => ?-> ?< ++ F= ><++ =+<, ?+ +>>? F==<+ ++ +<+... %= = <=<= ? ?<< <F>+ + @+?+J <+>@+ deben "ehace" lo#

con$"a$o# de en#e+ana a"a modifica" lo# h*bi$o# ad!%i"ido#

o" #%# al%mno#. MF= +K, +?= = I= +=->= + ?= =I+ =>F +<=+= + >+ ? +? + +@<=.

("E##E!&D, 1991, F. 92*.)

L= +?= <H+ =><+< <==>+ + >+< +<> ?+?< +, =+> < = +< H =?<, = >+G < =?<+. !><+?=>J <H+ =< ? = +?= <++= <<+ <><+>+J <+>@+ ?+ =I+ ?+ ?H+ K>(%&$%ALL gi""$, 1996). L+= <+= = +?=

=< =? +<+ + ++ =? <+ +++<+ =< +<= <J=<= (DECk, 2000) > ?+< ?+ =? +++ +<+ @?<+<= <= +<+. "< >+>, =+<< ?= + ++ = +?= +<+ ? > +<> =-?<= ?+ += +>F>+=, = ?+ <<+>+@+=+ +<+ ++>+<= + ?=+< + @+?+J ? <+>@ +?>+< =? +<+.

NE= M<+>+, $. L.

160 E ?+ +<+ += +>F>+=

*L+ ><+ ?<=@+ = +>@+ + +?><+. (N. del T.)

La clase de matemáticas como discurso

de la comunidad

El discurso es una herramienta de aprendi"aje

E =?<= = < ?+ <>+> <<+->+ +<+. PE +> <?+< += +<+ +<><<+J +<?>= +<+ @< + ><= = ?+ +<-> =+>@+ +<+. C?+ = ><++ +=

+= = =>+= + <H>+= <+++= + ? G=>+== <++= <++=Q (C%M, 2000, F. 348)."< + = +?= ? +<>? =?= += ?<+ + +>@-+ >+>@+ =>+ +<+ <+ + +<+ =?==+>=. L+= @=>+= + =><+ ?< + = +?= ? +<>+ =?= =+>= +>-F>= += = > <=?>+ ? <> +<+ +>F> (#&$$ELL C!#i, 1991,

!#D, C!bb ACkEL, 1995). E =?<= = <>+> <-? + +< += +>F>+= > =?=J ++=, ?<+ + = =+>= += >F>+= +>>= +@<><= +>J. E + + ? =+?= <?+ =?= <+= += +<+ ? ><= += ?-+ +=+<, + ? =?= =+>= = +=> =+ >+= +<+ = ==.

E ?+ = ?+ <<+>+ + ><+@G= + ?+ =

+?= =+<<+ =? > +>F>+=, =>> @?+ ><= =+= ('g!%$k,1978; E#%$Ch %!MA, 1995; b#&E#, 1996). C=<+< +=><+= + += ? =?<= >+ <+< =+<< =+ >, P=@H+?=><+ +>J = ? +>+ =+> =>?+> + ? +>+ =?= ><+= =+=Q(MCAi#, 2001, F. 198). E =?<= + += +>-F>+= <H+ <+< P? =?< <+ < +<-< ? > ? <> +>F> ? = +@?> <F>Q (MCAi#, 2001, F. 199). AF= =

NE= M<+>+, $. L.

161L+ ?> +=: "<?+< + ><H+

= +?= + ++< =< = <>= ?

= ?=+, < , ?>+< =?= <=?=>+=, ? ? +<+ += +>F>+= ? =>F ++= ==<>= (LAM"E#%, 1988, ALkE#DiE, 1997); PA<-= + +<>+< =J += +>@+=, = =-> ? += Q (bA#E$, 1992, F. 128).

L= +?= +< +>F>+= ><+= P+<>+ + =>>?J ><+>@+ += =>?+= += ?+<Q (C!bb =., 1992, F. 119). A +<>+<

F + += (E#%$Ch %!MA, 1995), = +?==<>+ ?+?< ><+J >< =? <+ +>@++>F>+ + = F=. L+ ?J F =?<= = ? += +<+ =+<. %+> ?>< = >= +>+ ?+ =>?<+ +>@+ > ?= + ?=>+< ++< >+> += B<== <-=< + =?= +I<=, P<<+<+= =?=<+= B<== B><+= ><+=Q (E#%$Ch %!MA,

1995, F. 171).

Una comunidad de discurso matem#tico

L= +?= +< ++ ?+ = <=-<= = <> <+< +<> + ?+ =?<=+>F> + +=, + ?>+< ?+ +<+ =+<<-+< B<=+< >=, + =><?>?<+< >B> =+ + += +<+ ? = +?= ?= ?+@<+ =<> <= +<+ +>F->+=.

$iL'E# $Mi%h + < P?+ =?<-= +>F>Q +<+ =<< ?+ += + ?:

E + <=< = @<=+ +<+ +< <?=>+= >+<+= +>F>+= <>+= @+=+=; =>+<+ +>@+ >>?+ + += + ><+< =?<=, +?+<+ = +?= + >< += += +>F>+=, @<+< =?

NE= M<+>+, $. L.

162 E ?+ +<+ += +>F>+=

< >>. $ =<+ ? = +?= = ? += +>F>+= ><+= +<>+ +>@+> P=-?<= + ?+Q.

($iL'E# $Mi%h, 1996, F. 20.)

"+<+ ? = +?= = @?< + @=>+J =?<= =< += >= +>F>= = +< @+<+= =. L+ <+ = ? +> ++ <=>. P$+@ ? +> + += =+

=?< +<+ =+< ++<, = +?= =><+<F <==->+ + << =?= += J>==, + ?=>+< +<+-= ? = ?=+= + +<>< =?= ><<>+=+><+>@+=Q ($iL'E# $Mi%h, 1996, F. 22). E= <>+>? =?<= =>+ ?+ P=?<+Q @++ F= +F P<> J =>Q P+< + <=?=>+Q.

&+ += +>F>+= = ?+ ?+ + ?= +?<+ << +<+ =>+ +><+. E +<-

+ + =I++ +>F>+= + =?+ = ?+<F>+ =+ ? =H+. E= ?+ ?+ =?<-= +>F>, < < ? F= +<+, ??> =?<==, + ? B=> ?= =?<== ? + += =>++ ><+= <F>+= =+==H+=. E ?+ ? = ?>+ + += +>-F>+= +<> =>>? <=>< +>F> ("iMM,1987), < >+G < =?<== ><+= ?-

+=, >+= += ?+= =+= = +?=.A +<< +>F>+=, B=> ?>= ? <?+ = =?<= ?>+ <+ <?>+<+ + += +=>+ =?<= ?+ ?+ +>F>+ F= ++. E=>=?>= = <>+>= <F>=. $ ?> >< =?<= ? = +?= ?=+ =? < +>=+ ? = = B + += +>F>+= = >+H <?+<, =<F ++= >+< +<> =-?<= + +=.

NE= M<+>+, $. L.

163L+ ?> +=: "<?+< + ><H+

El discurso culto ! pedag%gico

E di#c%"#o edaggico = =?<= ?>+ +<+=I+< +<< + +=, + <+ +<>?+< ++< +?+. E ?+ += +>F>+= + <+= Ph+ <- K< 5Q @+ ?+ =?< H= ++<> =<< =+<. %= = +?= = + +?++<+ => =?<=. E di#c%"#o c%l$o += +=+== +>F>+= = ? =<+< =?<-

= += ?+= +>F>+= F= ++=. E-><+I+ ?@+= <+= ?=+< ?+ ? <> + =+?= +<>+< = =?<== ?+= F=B>=+= ?=><+ =+ =<++= P?>+=Q. E ++= +>F>+= = +?= +?< ?>?<+ ?++ ? <=>< +>F> <J=> ?+< += +>F>+=.

E + <=< = ++>+< + = +?= +<+ +<->+< => =?<= ?>, = <+= += ++J ? ? <+< +<> + +>?+ =I++ +>F>+= = ? +><?<, + = +<>, + ?<++= +< => +?++>.

E >@ ?+>+ + ?+J = =?<? = =>?+>= > +<> = ><+= @-+= =?<= +J, +?? = = B+ + +<- =? ?+J. E= =?< ? =+<< ?@+= <-+= ?=+< ?+ +<+ =+< ?+<=, P<+= ++<+=Q, ?+ = <><H+ =< <= +>@= +=?+= F= B>=+= =?<= ?>.

(ME#CE#, 1995, F. 80.)

A ++< =?= += +>F>+=, = +?= > +<>?+ =++< ?=?+<= =?<= ?>. E=?<= P+>F>= ?>=Q ? =< F>><+=> + + +=, =+@ ? <=< ?+ -?+ = +<= <<+ =?<= PB<>Q

NE= M<+>+, $. L.

164 E ?+ +<+ += +>F>+=

=?<=. PE + <=< = ><+?< ? = +

? =?<= ++G <J=> +?+< + +<-+< F, +>+< <?>+=, =?<< B= + @+ <+, B<+< +<?>= < <?+=Q (ADLE#,1998, F. 174).

L+ +<>+J = +?= =?<= +>F>- ?>+ <=>< +>F> <+ <+= <>==+>@= +<+ ? <=<. EB=> + =+ =-><?< ?+ +H+ ++ + =>= <+= +<>-

+J =?<= +=, + <=>< +>F> + +==>?+= =+=, ><+<= <+= <>< ?>= +<> = = <+= >+>= ++-J +<+ +?= =>?+>=.

L= +?= + += =<F <= ><+= ?-+= ++<> + += +>F>+=: + =?+, ->< ++<, += = +, >G><+. M>@+< =?<= + += +>F>+= = ? >@ ? -

< < + = +?= +<+ +< >+G + =>+=?+= + ?+ ? + + ?+ +>F>+=F= B>=+. L+ +< + += +>F>+= = ><++- +< >< = H>= <>?+= ><+= ?+-= +<+ ? +> +<+ +>F>+=.C?+ = +?= = ++= =+< ?+<= =>+= ?@+= <+= ?>+< ?+ ? - <=>< +>F>, >= = ? < ?

+ ? =?<= ?>. E=>+ = ><+ <+ =<+<? +? + +< +>F>+=.

El pensamiento y el discurso están estrechamente ligados al proceso de aprendizaje

$K $fA#D (2000, 2001) + ?+J = <H+=<+< ?+ = +?+ +<+ =+>, = P<F->+> ?@+> + =+> =HQ. L+ ?+-

NE= M<+>+, $. L.

165L+ ?> +=: "<?+< + ><H+

J + +<>+J =>F @?++=; + +<>+J

?+ ?+ B ?+J ><= +<>+>=.A<< = << F= <>+> ?+> + -=+<< <+= += ? ? =?> +<>+ +>@+-= ?+= =>++=, => +=, ?+ += +>F>+=: P"< =?>, += <F>+= = <H+ @< ?+ <+J =?<= +>F> + ? ??>+ ? <=?>+ = ? > @F +=H ?+ +<>+J +Q (LE#MA, 2001, F. 100).

E = +<>+< += <F>+= =?<=@+= = ? = <+= >@= +<+ +<< ($fA#D, 2001, F. 49).E =+>, =?<= +<+ =>F @?+= ?+ + <<=>+J +<+, ?++<>+J += <F>+= =?<= ?+ ?+? =H =+ <?+ <+= >+-=?<=@+= ? ?H+ ?<= <+ + ?+J. "+<> + <-=< = +< += <+= >+-=?<=@+= < >+> +->+< + +<>+J =?= =>?+>= + ?+. L=+?= +< F= ><+= = =?<+ < ?-+<=; ?+< = =+< =?<= = ? ?<> <?-=< <= +<+.

Cambios en el papel del profesor y del alumno

en una comunidad de discurso matemático

& = <=?>+= >+< ?= ?++>F> +<+ = ? ++ @ = F= =<=<= ? @?<+ + =?= +?= + @<=+J;+=H = < + + < + =?= =>?+>= G=>= + =? <=<. L+ <+J ++ <++>. L+== = @<> <=+= ?= + ?+< =+< >< F= =< +>F>+=.

L+ <+J = <=<= =?= +== + ? =?+><+. M+> ? >< =>< =< ? =

NE= M<+>+, $. L.

166 E ?+ +<+ += +>F>+=

+ += +?+=, >>+ +@<?+< +@+< ?

<H+ +@<< F= +<+, < =<+ ? =+?= >+ ?+ J ? F= = @?<+ =?<= + +=, F= = +=+ = ?+ <=+ K+<+ +<< F= =< +>F>+=. C++ +? =@<> ? < @+<+ ?+ ?+ +<+ ? ++ +<> =?= += =< +>F->+=. L= =>?+>= + + +>+< +<> + <=-

=++ + =>>?J + ?+ => <-> ? <=< <=+ < + += =+= =?+<+.

El papel del profesor

E + <=< =?<= ?+ += +>-

F>+= = +?>+< + = +?= +<+ +< ?+? = ?>+ +<+ B<=+< = >= +>F>=."+<+ ? +?= +?+= ?+ +< +< +=+= +>F>+=, > ? =< ++= +<>?+< =-?>< =< =>= >=. "< >+>, <+ +<> < ? <=< +>F>+= = =I+< + = +?= +=<+= ? =?<= +>F> ($fA#D, 2000). E + > = ? B+< + =?> +<+:

L+ +< +<> > + <=< = ? =-<+< ? <<=>+> + =><+ ?>?<+ += +>-F>+= >+ >+< +<> =?<= + +=: =J ?H+ ?+ <= = =@H, = >+G +<>+>, + =?<< == =?=, =><+>+=,>=, >.

('A !E#$, 2001, F. 74.)

"< >+>, = <=<= > @<== <= ?++= +>F>+= + ? = ?>+ =?<=; +-

NE= M<+>+, $. L.

167L+ ?> +=: "<?+< + ><H+

+< <?>+= +>@+= ? +> >>

= +?=, =?+<, @+?+< ><, >. $ +<-, ? = <= F= <>+>= = +<>+> ? =?<= ? = +<+ = = =><?+ = +?= +<+ ?=+< <+= B<=J ? = <-+= += +>F>+=, +=H +<+ >+< +<> =?<= +<+. E <=< <F ?<+< <-+= ?=+< ?+ +>F> +<+ ? = +?==+ J = ++, + ?+ ? =>+< +>@+=

? = B+ ?= += ++<+= B<== +>F->+=. $? + = <>< ? = +?= =+ ? ++< + ?=+< + +< += += +>F>+=, ?+=? ?+ = == <+ +<>?+< ++< -+= +=.

E < <=< ++ ? F= =?> +<>?+ ?=+ =?<= +>F> ? ?-+< + =?= +?= += +>F>+=. L= >= >+

+<> =?<= , < >+>, = ++= @+?+< +<=J ++ @? >+< += F= <-=+=. L+ E@+?+J +<+ A<+ = + >>+-> ?+< H+ + H+ + +=. L= <=<=+ + < + =?= +?= + + ?++ +< ?>=, =>F <+<+= +<+ +-+<=, + ?+ ? +<+ =+>=+< =?= =+= <-<+= <+=, = @?@ F= <>@= +<+ =?= +?= +@<> ? =>F =I+ <, < ? + ?- = =+>=+ F= =? ><++.

h @?> + =<?>+< + =I++... + <>+-+ + >= <= +<+ <<<. E=>++ =+?>+< <+>@+ +<+ ++< + += H+ =?>...M ><++ + <=J += += =? -<>+>. A ? =?<H+ ? =? <=J +++I++ ? < <>+>.

(g.)

NE= M<+>+, $. L.

168 E ?+ +<+ += +>F>+=

"=+< =?<= += ?+= =?<=

@?@ + >B>?++< + <=<. $? ><++ +<-> =?<= ? = +?= + = => J-= <@ <+= +<+ ? >< =?<= ?> +?+ +>F>+ F= ++.

$ <> ? = <=<= +? + =?= =>?+>= +=+<<+< <+= ++<, =<< =+< ? = <>++< @+= >>?+= F= +=, + >< =< >-

= < ><= <= ?+= F= B>=+= =?<= +J.(ME#CE#, 1995, F. 83.)

C>+< + += ?+ ?+ =?<=+>F>, =+ ? <=< > + <>+ ?>= ? <+< +<> +=+< += <+= ?-+J ? = ?>+. L+ <==++ <+ =<

<=< +<+ +<+>+< ? + ?+ +<> ?+=< >@= K, ? = ? <+J <><+>+J =? <F>+ ? >= +<> +<=J <=>< +>F> ? = + += =? ?+J. L= +?= + + + += ? =-?<= =+<<+ =? < >< =+, = <, =? < ?+ +>?<+. ? +<>+< ? =-?<= ?> +>F>+= = +?+, <F>+ >@+J.

E=> <?< ? += +>@+= + += = =>>?+ +<+ ? <>+ + ++ < +<>+< =-?<= P+ <<= += + ?+ <+ +<>+ F= += +>F>+=, ?+ ==, + +=<>= ++= +>F>+= ? = + =+<<++=>+ +<+Q ("iMM, 1995, F. 11).

A>+< + <>< +>+< ?+ ?-+ =?<= +>F> B <?= += +<F>+ ?= <=<=, >< = ? ?<+ ==?>=:

NE= M<+>+, $. L.

169L+ ?> +=: "<?+< + ><H+

=>++<= + ?+ + =>?=; ?-

< + =>?= = ? = >+ > =-> ? =< <?+,++< =?= <+= =< += +++= +<-

+ = +?= = + +@ + +<<=@ (DECk, 2000); = <=<= -<< ? =?= +?= = ++= <+< + +?+ + +?+,

=>?+< F= =?<= =+> =?=

+?= , < >+>, ++< = = ==,=?< + >< ? <@+> + <=< + =I++ +<+, <>< ? =+?= +=?+ <> <+ <==++ => <=.

El papel del alumno

E= <>+> ? = +?= +@<>+ ? =? + +++ + += +>F>+=. + = = B =<<><= +=@= <+J, = +<>+>= +>@= <= +<+. L= +?= >+< +<-> =?<= + << =?= += =. %+G> ? +=?< + <==++ =? < +<+- , +=H ? =+ ++ +<+ ?G

J +<.$<F F= F +<+ ?= +?= ? +<+ ><= +<>-

+< +> =?<=. "+<+ +?= = F= H++< + +=. =>+>, = = @>+ + B<=+< =?=+= +>F>+= +<+ +?+< + +<+<, B<+< =-+< += +=, ?+ +=H =< @+?++= < <-=< < = ==. L+ >J <= = +?+< += +?= + +<<, +<+ ? ?+ =-><= + @<=+J. A?= =>+<F F= +- ? ><=, ?F ?+ +<+ +< =<H+ <=? B<= =?= += +>F>+= << <?=

NE= M<+>+, $. L.

170 E ?+ +<+ += +>F>+=

<?=. "<= ? + ? +I< =-

?G= + >+ + += ? +?+< + ?= =<+ ?++< = ?+ >+<+ H. $ +<, + +<H+ =+?= +<>+<F =?<= = =><= <=+== +> + += = +?+, = <, = >+=+= <=?=>+= = @+<++= < =? ><?J + ? =-?<= <++ = +<>+>= ? @<= H<= ++.

L= +?= =<+ ? = H +<>?+< =?= +=

+>F>+=. $ +<, ?+ = <?>+, + +-<H+ < ? = <>+> B<=+< =?= += + ? =+ + +<+< =?= =+>=. "+<+ = +?=, +<B+= = ? + <=+ =< ++=+<+ =+< J ?>+< <<>+> ?+ +<+ ?><+= <=+= ?+ ><=. E= =@ ? = <-=<= +=?+ ? => = H, < ? + >@+J =<F F= F. E + + ? = +?=

+?< =><+ ?= ?+ +<+ B+< =?=+=, >+G +@<> ? = = F= F +<< +>-F>+=.

AF= B< ? =+ = +?= ?= F=+ + +=, >+G = = <?< ? =+ =?= <B. E=> = = = ? +<-, + ? <?+ = +?= =>F +=>?<+-= + ? = = @> + <B+< =< += =.

%+= += += ? = + =?< => < +?>+ +? <=< +?, B? ?< + >+<+ +<+, ?+ + =? @ <?< ? = +?= -=. $ ? <=< > ?>+ >+< +?+= += +=, <H+ <= + ? = +?= =>F+=>?<+= + ?=+< =? ?@ < += <B= ?> +.

C?+ = +?= ++ =+ =< =? +<-+ +>F>, =>F <+<+= +<+ <==++<= . "<F B<=+<= ?+ +<+ +> => @?<+<= ? +<+ F= <?

NE= M<+>+, $. L.

171L+ ?> +=: "<?+< + ><H+

+. %+G =>+<F ++>+= +<+ < + =? <-

=< = > ?>+=, =>+ +<+< + +<+ >>+< >< + J >< . $ = +?=?= > ? +<<, < >+> = =<+ ? <-B =< B<= J ? <<=+<.

E ?+ ?+ =?<= +>F>, = +?=:

+<>+ <+ =<+ <= +<+, =?= >@= -

=?<=,>+ +<> ? +<+ +<>, >= =+?= =>F @?<+= ? =?<= +<-+ =?= +I<=,

+?< +< ++ = ?+<= -+=,

+< + ><+++< +?+ + <+< ?+ +>>? +? += ?+= B<=++= = < <+ F>-

+ +<> =?<= +<+.

&+ ?+ =?<= +>F> = ? >< +<+ +<>. A=H = J B<=J ? <=<:

%= >H+= ? =>+< >= <=>, =>+< <-+<+= +<+ =?+< <+ F>+, +=H >< @<>< ? >@ K ? <+ +@++< +<-

+... = =>?+>= = =>H+ =?<= >+> +<+ +< <=-?=>+= ?@++= +<+ B<=+< <> =? +>+ >>... + += ++ =< ? F>+> ?=J = F= <+>= =<@@H+ <=<++, +<+ =< ? ? <++= + = +<?+>= =>?+= ?=?=>+>=, ++ =>?+> H+ =>< ? <<=+-+ + J.

(#<>.)

E + = +?= ?+ ?+ =?<=+>F> = +<>+<. E <=< = <><= >+<+<> <? =+ ? +<+ <= + +<>+-

NE= M<+>+, $. L.

172 E ?+ +<+ += +>F>+=

J. A + ? = +?= ><+=<+ =?= += ?

?+ ? ?+ =< +<>, ><+=<+ += ?= ><+=><+= += ><+= F= <+++= =>+=, ?+ <+ +<> <= +<+.C+<>< += += >+G <> ? =>G ==+<+ =< +<++= ><+= => <<++= ?+= ? =+ F= K>= +<+ <=@< <+= +>-F>=; + +<><, = <>+ += ?+= = <-+ = <== =+> >=.

Cambiar la práctica

"+<+ ?< => +H>? =< + += >J<+ =+= =<>= <=> <, ?< B++< >-<H+= ? < + = <=<= ?<+> + ><+=J =? <F>+. M?+= += += =<>+= ?< =+<<-

++= +> + ><+@G= + @=>+J +>@+, ? =?+ +<+ <=+ ++< + <F>+. L+= += ?-< @+++= + ><+@G= ><++ ><= <=<=, ?= =<@< + += +<+ =+<<+< =? <> ?+ += +>F>+= P<+ <+>@+Q."< >+>, >= = += = = <=?>+ <=+= ? <?<++ <+< < =? <-=J.

L= += >?= < =? ?+ ==+H+ += <+= = <=<= =< ? =>->? ?+ ?+ <F>+ ?>+ + <>?+, > + +<+ B<+< < = == = +=? = @>+= P<+ ? + <F>+Q(g#iffi%h$, 1990, F. 43). "+< ? ?+= += <->= +>@+=, < , Na$ional C%""ic%l%m ("+ E=>?= ++) Na$ional N%me"ac S$"a$eg (E=><+>-+ ++ +<+ += = >+= F? +<>-G>) + <+ ?+ + ? + >+ +<F>+ <=+ + <BJ (DEE, 1938; $Ch,

NE= M<+>+, $. L.

173L+ ?> +=: "<?+< + ><H+

1991). $ > <=+ = @+<= =

<=<= = + ++< =+<<+<, =<F <?+@<> + +?+ B=>> >< =? <+ <F>+ +>?+ = @+<= ? =?=>>+ =? + <=+= + =I++ (hi%EhEAD, 1989; A%ki, 1992)). L= <-=<= =>+ ? = += = =+<+= +-<+ ? = += >+ ? => a"a ello# mi#mo#

(f&LLA, 1991, F. 112); =J >= = += <-== = +=+<F +<> = <= =+-

>= <+= <=< =< ? = ?+ ?+<F>+. &+ ?+ @?= =+>==+<= =<H+ ?+ <>?+ +?++. A ><+@G= ++<>+J + ?+, = <=+= + +=<+<= P>= ? + =+<<+ < =? <=JQ (h!LLAD =., 1998, EgE#, 1999), <+ <?+< ? =+ +>+< =>+ >+."+<>+< + ?+ <+ +<> ? + = <? = @+<= ? > <-=+= + =I++ (ELLi!%, 1987; hA#g#EA'E$, 1998), < >+>, >+G <+ +<> <= + = >@= ? = ? + + +J =>?+=><++=.

Cambiar la pr#ctica mediantela inestigaci%n-acci%n

L+ @=>+J +J <@ ?+ <<+>+ +<+ ?= <=+= <@= < =? <=J. L=>= ? < >+ @=>+J +J = ?-><+ ?+ =J <@++ ?+ = ><+>+ @=>-+<, +++< +>= <+< ><H+=. L= @=>+<= +=I++ ? ><+< =? + @=>+J +<+ +<+ (+ <+ = =J=, ?=? ><+< += ==) <? + @@

NE= M<+>+, $. L.

174 E ?+ +<+ += +>F>+=

><+++ G. L= <?= +?<= + ><+@G= =? B-

<+ <@ ?+ >@+J +<+ B<+< ? -> ? <@+> + =>?+J = ><== <=> + =>?= =><<=. E ?+< =+< =>=P<?=Q, = @=>+<= = > =<+J;= ?=+, +<+ <@=+ ?<+> + @=>+J (AL-%#iCh%E# "!$Ch, 1989, F. 26).

M= P<?=Q ?+ ><G => + <+ ? =+?= +<H+ +<< +>F>+= >+ ,

+=, <+H+ >H+. %+> + <+H+ +>H+ +<>H+ +?+= += P+<<<+=Q +<-+ ? +<H+ ? =>++ +>?+> +><?+=+ += +>F>+=; < , ?+ >G, ?++ <+= >= <+= +<H>+= +<+<=@< <+=. E ++= +=+>?<+= +<H+ F= F<+<+< + = +?= +<+ =?<+< =>+= +<<<+=. A=<@+< += +== <+H+ +<J ? = =>?-

+>= ?=++ ? F= ?+ +<+ B<=+< =?=+=. #BG =< += = >H+ +<J ? +=>+ >= = +?= =>+ += +-H+ > +< >< =< + <J =? +<-+. %+= <?= +>= <@+H+ H = +H+>, ? ?+ J + ><H+ ? B<-=++ + =+ = +?= ?=+< ?+ +<++?<< >< =< =?= += +>F>+=, + =>++ <-

+<++ += B<+= ? <>< <=>+= ><H+= <F>+. L+ B<+ ? + <+-+< +> ? =>++ : ? H+ ><+++< <++ B<=J =+ ><H+ += <++= <F>+= ++=.

"? =< H +<+ = <=+= ? ?>+ +@=>+J-+J =>++<= + <+J ? =->>? =? <+ <F>+. EB=> = ++= <-=<= ? <++ + @=>+J-+J. &+ = -> <B@ ? +>+ <?>+= +<+ =? < +<+<+< + B<+ =?= +?=. E + =?+ +-

NE= M<+>+, $. L.

175L+ ?> +=: "<?+< + ><H+

+ = @=>+<= +>+ <?>+= <=> + ?F

<H+ =< =+ (H>) ><F= + =J +>?+< ?+ +<+ ><+ J = ? B<."+<+ <@< + =>++, = <=+= ?>>+< =+<+< = = <=, + += >J<+ = ? B-< + <+J <+J = ? ++< @=>+<. "++< += +== ? >+ =<+-J < >J< = +< <=< (? <-=< P<B@Q <+<+ +<+ >+<= > +<+ <B-

+< =< =?= +>= < =< += =><<=+=++= =+= <B=, < =< <=<),=<+ + +J +<+ =+= -= ?G +< =?G=. L+= >+= + +<= =>+ >+> <=< @=>+<, + ?=> = =<<H+ =< <? <=<= >+G @=>+<. $ +<, <=<-@=>+-< > += = ++ <B+< =< += >-

= += + +J. M><+= = <+ =+>=, <=< > >< =< + =>?+J =?= <<-+= <+. $ +<, + >< @, @=>+<-=+< <=><+ >=, + <?>+=, <@+ =>?+-= +< =< ? = +> +< ?+ +=."?=> ? + <+J <+J = +++ = +>= +++=, + =+ @=>+< > <-<+, < + += +F== <+>+>-=+<

>>+ <<, + + +J + ? + +< ? = <@+. PD =>+ +<+ (= <, =+F-= = + =I++ +=>+ + @=>+J) = +?<?+ >@+ <> ? <> ? @=>+<+=> =? <+J = +>=, ? + +-J + = < = ? = <+ +<> P@>Q ? =>F = @=>+Q (hA%Ch $hi&, 1998, F. 308).

L+ @=>+J-+J = ? <++< + + <+;= ? <= =<, =< > ?+ = +=? +<> ? <? @?= =+>= =+-

NE= M<+>+, $. L.

176 E ?+ +<+ += +>F>+=

>=. E < + ?+ <+ +<> => >, < =

? +>< <>+> + @=>+J, +<> <? +@=>+J = P?+ ++J ==>F>+ ?++Q($%Eh!&$E, 1980), <? <, ?+?<+ ? =++ <+ ? >, + >+ +<+ <=< +<+ ><=>+= += += +<+= ? = ? <>. L+@=>+J-+J = < ?+ <=+ <+ =+<<+< < + =I++, + =?+=?++= < <=<= +<+ @+< + + >+= <>=

+<= > +<+ +< +<> =? <<++ D=+<< "<=+ C>?+. A?= <=<=>?+<F =<F= +<+ ++< + @=>+J =+<< =? <F>+ =? + +<>++ ++, +??+<+ ? => =+ ? = <= +<= > <?<-== ? F= ==>F>.

Conclusin

E=> +H>? + >++ + + ><H+ ? + ><F= = += < + =I++ B++= K> +H>?. L+ ><H+ = <>+> +<+ ++< +<F>+ <=+; = <=<= =< = +<<=+<F?+ + += < >+> =>+<F =>+< @-= ? ><F <=?>+ ? < ><

+<+ +<+ =?= +?=. L+ ><+>?<+ +<+; =+-> ?+ =>F =><+> @?+=. L=+?= +< + ?=+< + +< += += +>F>+=+> ?+, + B<=J + + =+>+ =< = ><+=><, + =>+< + =< +<. $+<, +< ?+ +>F> > >< ><=; ><+= = +?= ++ +>F>=+>+ + >+ = +>F>=. E= <, ?+ =+?= +< + B<=+< =?= += +>F>+= +=+ +<+ ? = +>F>= P?>=Q, = + + @< P?>=Q, = ++= P+<Q +>F>+=.

NE= M<+>+, $. L.

177L+ ?> +=: "<?+< + ><H+

C?+ = +?= =>F <+<+= +<+ +<>?+< =?

+<+ +>F>, >+> =>?+> <=<? +< ?= >@ + E@+?+J +<+ A<-+ ><+= +@++ +<+ + +<+ F=> =. L+ += = @<> ?+ ?+ =?<= +>F> ? >@ ==> +<-< +>F>+=; = <, = +?= =?> >+> +=+= ? =>F +< + ++ ? +=+<. E <=< +? +>+ <= =H=

?+ ?+ =?<= +>F>; <=< >+ = ? =>+ +>+ +<+. E +?- +?< ? + +>@ <= +<+ ++=?< <+ +<> =+> + +<> <+, -><+ + <==++ =? < +<+.

C?+ = <=<= = @ ? = += =? <F>+ = =+<=, ?=+< +<+= >-@+= >?+< =>+< == += >J<=. L+ @=-

>+J-+J <> ? = >, <@= <+- = +=, ? =>>?+ ?+ +<> ><++ >?++ + <F>+ <=+.

178 E ?+ +<+ += +>F>+=

NE= M<+>+, $. L.

L < < + < < = <-

<. E< + = = , @ < ? = < F. %+ =< = + @ = <<. E < I< F-<= ? = < + = = << @ ? @ F <.

Una comunidad de discurso matemático

& = <<=@ = = =<< = = = @ = = << < @ +. L >< = = =<< <+ @ << < = +, = =, = @ =. L < = > @ << + @

KE M<, $. L.

179# ++

CAP$TULO VII

Ahondar en la pr#cica

= <> = ? =

@ = =F = ? - . E<= << = = = <<, = +<< @ <-<, @ = < <. E = <=< = = < =+< = =< < <<. " <<, = <+ = = < < = = = -

< @ + = = <<.C= = <= = < @ - = < = @ <<. L = < = = < << ? = << = <>. E < <-< < +N < <<, = <-

< = <. C= ?= = = = <<, <-+=@ @ < = < = =. E +, @ < < = = = = @ <<.

E?< = = = <<.L << : = <=

=, = = == = = < <<, @ = =<N = = = = = = =<= <<, , < <<. E?< < =, = < > <<, = = < << = <. E < << > , < @ =<; = > > *

KE M<, $. L.

180 E = <<

*' < . 41. (N. del E.)

@ . E< <F< = F

=. E < << < >, F = < << << <<. >< = = =< < > << ? = << @ +> = < = < = + = =< . +<<, = = < >< <

= < > <<=:

J< N << ? = @ , @= < = +,

J = < < < = = = << ; < = <> = =< + = < << > = - ,

J = + < < <<>, @ = = ? @ = > @= >= = - <.

L = = <= = = = <<<. L = << = = => , = -=<= = = <=< => = =-<= <; E <?< , @ <-< + < < << << < ( <-F + <<) < =->N ('A !E#$, 2001, . 72).

KE M<, $. L.

181A <

C F << = <-

< = => =<= = < @ F = =. L =<= =+@ < - < = al mi%m! &iem"! = - < <F + <<.#+ = << = = = = = = + =<?< (&$i$ki, 1996). "< < => =<= -

= = < = = +, = < = = . " @= = < =<= = <<, < = = << = < = < < =. L = >= = <> + @ =+ =?<<> + => -

< = +< >< + =N.La% ide&idade% %e f!$ma * $ef!$ma a &$a% de la "a$-

&ici"aci0 e 'a c!m'idad. L < => -< = < = < <=< << << << = = >= = = + = @ <<. L <

< <<= = => @ - < < = = -< < N (LE#MA, 2001, . 88). L -<<= %@ + <. " = =< <> @ > +<<, = <> @ <<>. E = <+ < =, + @ += < @ < < . C< <+ = < = < = = <-<.

KE M<, $. L.

182 E = <<

D= < > C< <= <+- ; ? = < < -< >< = @ < ? = =. C = F < ?< = <+ @ << <+ = = < +. D < >= + = <<= <<.

C< = = << - . E = + = = @ ? <= - <+. A< =<: H = C-

KE M<, $. L.

183A <

Chanelle

A @<, C< = <>-< +<N, <=+ > = +N. L=<+ =@ F <+, @ = <F< + = << = <-<. + + = <+, @ F < +I =N = + <?< I =N = +F + . M <-

F: N! haba hablad! de 1me$!% c'ad$ad!% e "a$- &ic'la$, "e$! m'ch!% i/!% hicie$! la &$a%ici0 de%de 4 c!m! 3d!% al c'ad$ad! a 4 c!m! 3' 1me$! al c'ad$ad!. N! &!d!%: Nia * Cha&elle e"$e%a$! %'% d'da%.

" , C< < @ = = > =. %++ = = = = ><+

+ =< < < = << <+:32C0m! "!d$a% ecaja$ la% "el!&a% de g!lf a#', Cha&e-

lle? "$eg'&, al %e/ala$ 'a "i$mide de ba%e c'ad$ada.32D!% de lad! a lad! * '! ecima? "$eg'&0 acila&e,3E&!ce%, 2' c'ad$ad! %e$a la mej!$ f!$ma? N!, %e$a ' $ec&g'l!, l! cambia$.

< @ = <= @ ? +

= = @ = <+N.A F >< =+ = .

C< <>, <= <<= <> @ < @-< >=< @ =< < <+ <>. " = <=< = = F < <+= <>

= +< .La "a$&ici"aci0 de ' al'm! e el di%c'$%! "e$mi&e #'e !&$!% &ambi "a$&ici"e. A= = < <> = = =. E +<> = < = > , I =.L < +F >< <, <+= => = =-

< < = =< = </?> = ?N (LE#MA,2001, . 89).

Alg'!% al'm!% e%& me!% "$e"a$ad!% #'e !&$!% "a$a f!$ma$ "a$&e de la c!m'idad. A= = =- < = = =>

KE M<, $. L.

184 E = <<

Collee

C< +F < = < =

=. "F = = = = , @ < - <>+ < . C< = < , = , =< = < = -? < < @ =< =.

= < . ?=

=, < < < <+ . L <<= + = = << ? < < < . L = < <> < =? = = < -< =< = =. C =, = <F< = = =< =

= = +<> = < = > ?, - <N (LE#MA, 2001, . 100). !< ? = = < =, + = = < = = = = F (ACk @G#A'E$, 2001, . 263-265).

Características claves de una comunidad de discurso

L = <<=@ = =

< =< <F<:T!d!% l!% miemb$!% de la c!m'idad &iee la !"!$&'i-

dad de c!&$ib'i$ al di%c'$%!. & = =

KE M<, $. L.

185A <

Sha!n

$= >F F = <. E = N @ <=+ = < -, = < = <+=. $=F < < =- =. <F = @ < =, = < @ = =

= <.

= + <= =

@ < = + =+<. L = @ << < =< < + =. "<<, > > = <+= =.

La c!m'idad elab!$a * c!m"a$&e defiici!e%, * de- %a$$!lla c!!cimie&!. E = = <= + + @ =<-. E< <, =

+ @ <<< = < + @ - = =. E <+ << @ < = = =<.

La c!m'idad g!+a de ' ambie&e de c!fia+a * $e%- "e&!. % = = = + . & < << =

= = + = < @ = <. C= = < < <> = , @= = +< . E = = < < = = , <+ = + = < = . E= < +, = > < <

= < <> @ < = + <= = < .

La c!m'idad e%& $elaci!ada c! la% ide&idade% de %'% miemb$!%. M< = < > < < < = =-, + = + . L += = ? = <-< = = = - << @ = .

KE M<, $. L.

186 E = <<

Cambiar la práctica

L + < = = = ?< < - = = =, < , @ < <, <> @ @ >= = + =< <-. E =, < < <N,

= + @ ? -< <F @ . D-= ? = @ = = + =<.

Pr#cica eori"ada

"< < + = @ + < =< = = <-F = + = < @ <=- <F. % + -+ + = = +=NN, @ << + = -. L < +< ? + <-. L <F >< < =-

> = << << (A#Gi$ @ $CH, 1974,%!#ff, 1999). M= >< =< = < @ = (AL%#iCH%E# @ ., 1993). $+, = = < <F = =F + @ < < < <F @ : <F = = + < @ < @ ? <F. E < < = <F < < - = < , @ << <-

KE M<, $. L.

187A <

+ = <. L =< <

< < < <F @ < < <N.L F< <F @ < = < <-

= = < ?> F<- ($CH, 1983) @ < "$ai% .

E ?, < @ ( <F @ <)< << . D+ < =<=< <<=<> = < = = <= <= < @ < >> = =< < <= . < -<... E ? = =F < =- < + = = , I < , <= =<< @<.

(CA## @ kEMMi$, 1986, . 34.)

L < < = = < < < @ , = = <=@ <.L < + =-< ; < = < -, @ = <= <F < < < + , = - +, < > -

. L + = < < = = =< + < <F.L + = = <> = < .

i <F ? < + = -+ + <= = , ? = = = >= <-+ @ + +< = < <<-. < >< <, ==

KE M<, $. L.

188 E = <<

< + < <= @ <.

E< < < <, = < @ =<=< <<=<->, <+ << @ ?+, <.L < < + < , - @ <, =< < = <F = @ +F < <, =-= = ? <. L = < < <F, @ = <F =

< < <.

Cambio sosenible

L >< = = - = + +, < = + =, + < =< <

<. E + < < = =< + < <> < <. != = - =< =<:

1.P$'eba% fiable%. E < <> =< + =+ + = + < = -

< =. L =< + = <+ = @ + = <= = = = < = =.

2.Idea% "$c&ica%. & < << + << < = = <. i><+<, - < =< > = @ + =, ?< < =< <

KE M<, $. L.

189A <

= < @ -

@, <<, = ." = , << <F, @-F <, + =- @ ? + <> = <+ @ = +< = <> = = >= -< N @ > <=<= > +. L= >< = = = < = i<<=<

E= $= > <>, @ @<=<= = =- + = <.

4.Reflei!e% ace$ca de la acci0. E @ - = == < = >- << ? < + ? = = < => <@ <>. C< @ -

=< < > = < -@ = ? + .

5.Tiem"!. M < > = <. E @< kM!fA" (BLACk @., 2003), =< = = < < +F = < = +F + >< @ << , > = < <- < => = <. E 2000 = = @< = +> +

KE M<, $. L.

190 E = <<

< , = =+ <<

= < = = <+ <. $ +, = <, = <N =, @ = > + < .

E =@ F < < < + @ + + = . $> = + < F-

< < <, ? > +<= . " ==<, = F =< <-, @ < ? @ + =-. AI F, = = @ = = = @ < , +F > = <F < << +.

6.Fleibilidad. G < < < = <N. $ +, > = -+ = @ ?+ < => =<, == + = = < @ <. L =< F <= < = < ?<= +< = > <<

< = => <. L -< = = = << + @= < =, > @ < = ?<.

¿El final o el principio?

L << ; = -+ =< < < < = > = = = N <-

KE M<, $. L.

191A <

<. E< + << < =

<=< = > = = = N<<. G < < + -+ <<. L = = = ++ << = < <<,+ = = < <<.E>I = = = =? + = < <@ = < >= = =<=. L

= + + << < @ = =@ +. C ? @ >F= < < <<.

H+ + << <+ < =<< =. C= = @ +- = << @ ?, > <<, =

= = @ < << <> < =< @ > +.%+ <=< << = -< + + << + =: <, @ = < <+> = = = +<> . L = + #'e @ = > = + c0m!

. L <=-< <<.

E< + + = = +<> @= + << = = + @ <<.# == + = > , @ < = = + @ > + +. E <<= , + < <, ? @ > <. + + , @

KE M<, $. L.

192 E = <<

<, @ =

= , = <> <= =< <<N, = @ = = =-< = @ < < = .

L < + = =< < = > <, = = @ - <+ =< < = <

= = << = . L < <= = < = = <=, @ = < , <+ > + =. L CF<= ii @ 'i @ - < CF<= iii, i' @ ' I = >, + - @ . E ?< <; < <

<< = <=+ = < ; , <- = <, @ =-, @ ? = < @ = -+ = .

E = = + @ , ,= = =, = < = = = @ <<, @

= = <<. " <<, <+ = < @ ? = << , @ < + = << < =< +- = <. D=+ = = N -<<.

KE M<, $. L.

193A <

AAIA (2003). S$+%-$$3. %. AAIA, A==< =@ A-D< @=D< @=C A<, [email protected].

AdE, J. (1998). Q < : @=<C< D <+<@ = =@H<= = < < @<< = =-= R. E< S(343$# C(3( # 3'$ L$( %

M3'$3(", = >=@ A. +A'$#. $=@: <D@ =$=@ d>@< = EC=< 'C.

P (2000). Q'= >@ =@ < @ C-=<R. N#(" S34#($ ( M3'$3(" E#4"3( 8(3).

AIchE, h. %$'ch, %. (1989). Qd= SG@=C< =@T

>>@= =@ C< >@ =@ @ @@?R C- !(#$ J4+ % E#4"3( 19(1): >. 21-31.P; %$'ch, %. '$"Eh, B. (1993). T$"'$ I$3(3$ 3'$( W*:

(3#4"3( 3 3'$ $3'# % "3( $$"'. =<@: =C.AGI', c. 'ch#, d. (1974). T'$8 ( P"3("$. '< F@<=:

J=-B.AI#, J. ". (1992). Q< @@: < R. T$"'( #

T$"'$ E#4"3( 8(4): >. 381-390.BA#E', d. (1992). Q @= = < @<<R. E< T'(*( V("$,

= >=@ . #$"A#. =<@: h=@ & '=C=<.BAc, %. +IIA", d. (1998). QA< < @== @-

<<R. A$$3 ( E#4"3( 5(1): >. 7-74.

O<E=< "=@, '. .

195@< =@

CAPÍTULO VI

Bibliografía

http://www.aaia.org.uk./

http://www.aaia.org.uk./

BAc, %. +IIA", d. (1998). I(#$ 3'$ B+"* B7. =<@: <c=.

P; hAI'$#, c.; EE, c.; "A'hA, B. +IIA", d. (2002). W- *( I(#$ 3'$ B+"* B7 ? A$$3 % L$( ( 3'$ C+. =<@: #E.

P; hAI'$#, c.; EE, c.; "A'hA, B. +IIA", d. (2003). A$- $3 % L$( ? P433( (3 (3 P"3("$. BC<: $><<D@ %@.

B#E, J. (1996). T'$ C4+34$ % E#4"3(. c@, "A: h@-D@ <D@ %@. (@. .: L $#4""(=, /4$3 #$ +

"4+34. "@. *=@, 1997.)BE, . (1988). QE<<< < C<@<< <@< =D-=<: = -<D=D< < =-<D=D< DC=<=< <@ < >@=@<R. B(3(' J4+ % E#4"3(+ P8"'+8 58: >. 1-14.

cA, +. E""I', '. (1986). B$"( C(3("+. E#4"3(, K- +$#$ # A"3( R$$"'. : F@ %@. (@. .:T$; ";3(" #$ + $$<9. L ($3("(=-""(= $ + %"(= #$+ /%$#. B@=<. "@L<H =, 1988.)

cAdE#, c. (1986). Qc@== =C@R. E< H#!* % R$- $"' T$"'(, = >=@ ". +I$c. #CD =@:"<. (@. .: QE C@= CR. E< ". c. +I-$c (c=>.). L ($3("(= #$ + $$<9. V+. III. P- %$$ 8 A+4. B@=<. %N-"<@= ECN< c<, 1990, >. 627-709.)

cAE, '. (2005). F3($ A$$3 ( 3'$ S$"#8 C+- . =<@: h=@ A@<=.

c$BB, %. (1988). Q <=< < =@ = @<< < <-@C=<R. E#4"3(+ P8'+(3 23(2): >. 87-103.

P; AcE, E. +$$d, . (1992). QI<@=< < @<< < @== C=<R. E#4"3(+ S34#($ ( M- 3'$3(" 23(1): >. 99-122.

c$cc$F, +. h. (1982). M3'$3(" C43. =<@: h" '-=<@ $. (@. .: L 3$3( ; "4$3. I%$ C"*"%3. "@. 'C@N< G<@ %@=<<=

%@==@=, "<@= ECN< c<, 1985.)dA#IE', h. (2001). V83*8 # P$#8. =<@: =CF@. (@. .: V83*8 8 + /$#;. B@=<. %-N, 2003.)

O<E=< "=@, '. .

196 E <C < >@<H

dE'/+$ (1988). M3'$3(" % A$ 5 3 16. =<@: h"'$.dE+E, J. (1938). L(": T'$ T'$8 % I04(8. #CD =@: h<@

h=. (@. .: L=(": 3$; #$ + ($3("(=. "K=.F=<= cCC@ E=<N, 1950.)

dE' (1989). M3'$3(" ( 3'$ N3(+ C4("4+4. =<@.dI#, . 'hIE, B. (1991). L4$ ( M3'$3("+ E#4"3(

R$$"' # P"3("$. BC<: $>< <D@ %@.d+Ec, c. (2000). S$+% T'$($: 3'$( +$ ( 3(3(, /$+(38 #

#$$+/$3. <=<, #c: %== %@, =@ & F@<.EI$, J. (1987). QEC=< =@, >@ >==> <

=< @@R. B(3(' J4+ % E#4"3(+ S34#($ 35(2):>. 149-169. (@. .: Q=@L CD, ==L >@ <DN<-N<R < J. EI$. L I$3("(=-A""(= $ $#4""(=. "@. "=@, 2005, 5 ., >. 105-123.)

P (1991). A"3( R$$"' % E#4"3(+ C'$. "=< <:$>< <D@ %@. (@. .: E+ "!( $#4"3( #$#$ + ($3("(=-""(=. "@. "=@, 2000, 3 .)

E*#c, G. (1992). Q" =@<<CR. d=C-<= L= < %"E, dhA" #h.

E*A#', J. 'A'A$#I, A. (1994). Q<C < CD < @==R. C4+34+ P$/$"3($ 3'$ M3'$- 3(" C+, = >=@ '. E"A#. #@<: C@.

FA#, ". G. (1991). T'$ N$ M$( % E#4"3(+ C'$.=<@: c EC=< . (@. .: L 4$ ((- %("# #$+ "!( $ + $#4""(=. B@=<. $@=, 2002.)

GEGE#, . J. (1995). Q'= =<@C=< < C=< >@=-R. E< C34"3(( ( E#4"3(, = >=@ . %. 'EFFE

J. GAE. h=D, : @< E@C A=.GIFFIh', ". (1990). QA=< @@: @ @== >@ =@

<< ==?R E< S3%% D$$+/$3 ( S"'+: "3( $$"' //"', = >=@ %. $"A. cD=<: "C-<-C "@.

hAIdA, ". A. . (1975). Q'= > = ==<CR. E<I3$"3( !$3$$ L(4(3(" # M3'$3("+ E#4"3(.c=><C: #E'c$.

P "AI#, J. . (1993). W(3( S"($"$: L(3$"8 # D("4($ P$. =<@: F@ %@.hAdcA'E, . (1993). Qd= <= @ < =C?R

M3'$3(" ( S"'+ 22 ("=).

O<E=< "=@, '. .

197B=@L

hAGEA*E', A. (1998). I3$3(+ H#!* % E#4"3(+ C'$. d=@@, #@<: C@ A %C@.

hAch, G. 'hI, c. (1998). Q%@=<@ @@ < =<-@C=< = <= < C=<R. E< M3'$- 3(" R$$"' D(: $"' % (#$3(38, = >=@ A.'IE%I#'A J. I%AIc: C@ A %C@.

h$A#d, d.; 'I##E, d.; AchIc$E, +. cAI#, c. (1998). I#$3(38 # A$"8 ( C4+34+ W#. c@, "A: h@D@ <-D@ %@.

JA+$'I, B. (1994). I$3(3( M3'$3(" T$"'(. =<@:

F@ %@.GE, A. #. dE#I'I, A. (1996). Q = <@-D<=< =< >@=@<: =@ @D, -<,< >@<@ <@D<=< =@R. P8"'+("+ B4++$3( 119: >. 254-284.

AB$dE, c. (1990). Q<C < R. E< M3'$3(" # C(3(, = >=@ %. #E'hE J. I%AIc. c@-: c@ <D@ %@.

A"%E, ". (1988). T'$ T$"'$@ R+$ ( R$($3( 3'$ M$- ( % M3'$3("+ K( (3'$ C+. "<: I<C =@ @ =< <, "< ' <-D@.

A*E, J. +E#GE, E. (1991). S(343$# L$(: +$(3(3$ /$(- /'$+ /3("(/3(. c@: c@ <D@ %@.

EE, c. '. (2004). T'$ R+$ % L4$ ( 3'$ L$( % M3'$- 3(". d%, EC=< 'C, $=@.

E"A#, '. (2000). Q'= >@= = ==-CC@ @@ <

< < @<<R. N#(" S34#($ ( M3'$- 3(" E#4"3( 8(3): >. 55-71.

P (2001). QcCC@, C@D >==: ==CC@ >>@=- = C< < < @<< = R.E#4"3(+ S34#($ ( M3'$3(" 46: >. 87-113.

"A'$#, J. (2002). R$$"'( 84 P"3("$, 3'$ D("(/+($ % N3("(. =<@: =C F@.

"c#AI, . (2001). Q+=@< < @: H<

>=< = @< @= C<T @==C=<R. E#4"3(+ S34#($ ( M3'$3(" 2000(42):>. 197-209.

"EcE, #. (1995). T'$ G4(#$# C34"3( % K+$#$. cD-

O<E=< "=@, '. .

198 E <C < >@<H

=<: "C<C "@ . (@. .: L "34""(= 4(- # #$ ""(($3: $+ '!+ #$ /%$$ 8 +4. B@-=<. %N, 1997.)

"EcE, #. (2000). W# & M(#. =<@: =C. (@..: P+! 8 $3$: "= 4 $+ +$4)$ / /$- )43. B@=<. %N, 2001.)

"$GA#, c. (1995). QA< < = =C@ = @< @>=@= <D =@ < Gc'E R. ==@, I<-C = EC=<, <D@ = =<=<, =<@.

P (1998). W(3( M3'$3("++8. =<@: F@ %@.

P (1999). Qc=<D<=< =@ <? >@=< < - @<R. d=C<= L= < B@ '= =@ -@ <= @<< ", c=D<@.

#AI$#A c$#cI $F EAchE' $F "AhE"AIc' (1991). P%$(+ S3## % T$"'( M3'$3(". =<, *A: #c".

P (2000). P("(/+$ # S3## % S"'+ M3'$3(". -=<, *A: #c".

$EB#, ". . #Ich$'$#, A. . (1976). Q <C =(c'E) R. M3'$3(" ( S"'+ 5(5): >. 18-20.

%EE#$d, %. (1991). Q=@ >@ >>@= = =@DDC=<R. I< A$$3 % P4/(+@ A"'($$$3: M3(3( # S"'+ S4""$, = >=@ %. +E'$#. A@:' < <@.

%I"", d. (1987). S/$*( M3'$3("++8. =<@: =C &< %C. (@. .: E+ +$4)$ 3$3(" $ $+ 4+."@. "=@-".E.c., 2003, 3 .)

P (1991). Qc=C<< R. E< L4$ ( M-

3'$3("+ E#4"3(, = >=@ . dI# B. 'hIE. BC-<: $>< <D@ %@.

P (1995). S8!+ # M$( ( S"'+ M3'$3(". =<@:=C.

P (1996). QdD@ =C<=<R. E< C4("3( ( M3'$3(" K-12 # !$8#, = >=@ %. c. EI$ . -=<, *A: #c".

P (1996). Q%<@ @ = %"E c=<@<R. d=C<= L-

= < 20 c=<@< = I<@<=< G@=C> =@ %== = " EC=<, < *<, E>M.%$AI, d. JA+$'I, B. (2002). Q< => < -

< D=><: C< < @ == =@

O<E=< "=@, '. .

199B=@L

@=< < <R. J4+ % M3'$3(" T$"'$ E#4- "3( 5: >. 351-380.

''E, '. J. c$+I#, . B. (1991). Q< : S=<=T < S< =TR. d=C<= L= < 13 A<<C "-< = #=@ A@< c>@ = I<@<=< G@=C>=@ %== = " EC=<, BC@.

'AdE, A. J. h$#I#G, d. +. '. (1987). U#$3#( P4$ M3'$3(". $=@: $=@ <D@ %@.

'AdE, . (1989). QF=@D < < < = <-@C=< R. I34"3(+ S"($"$ 18: >. 119-144.

'ch#, d. A. (1983). T'$ R$%+$"3($ P"3(3($. A@=: AD-C@. (@. .: E+ /%$(+ $%+$7(: "= /($ + /%$(+$ "4# "3>. B@=<. %N, 1998.)

P (1991). E#4"3( 3'$ R$%+$"3($ P"3(3($. A@=: AD-C@. (@. .: L %"(= #$ /%$(+$ $%+$7(.H"( 4 4$ #($< #$ + $$<9 8 $+ /$#(9)$ $ + /%$($. B@=<. %N-".E.c., 1992.)

'FAd, A. (2000). Q'@< ()=C@ < >=@ <@=C@: C< = < = <D @< = =R. J4+ % R$$"' ( M3'$3("+ E#4"3( 31(3): >. 296-327.

P (2001). Q@ =@ = =C@ < @: ==-< << =C<< = @< =@ =C - @<<R. E#4"3(+ S34#($ ( M3'$3(" 46:>. 13-57.

'hEI#, ". G. (2002). QA << : D=>< =C@=C< < @==R. J4+ % M3'$-

3(" T$"'$ E#4"3( 5(3): >. 205-233.'hAd, h. $hE, A. (1984). C'(+#$ R$#( M3'$3(".

=<@: J=< "C@@ .'I*E, E. A. '"Ih, ". '. (1996). QBC< =C@ =C<

< @==R. E< C4("3( ( M3'$3(" K-12 # B$8#, = >=@ %. c. EI$. =<, *A: #c".

'E#h$'E, . (1980). Q' C@R, '=< F@@ <D@,*<=CD@.

A%'$#, F. (1997). Q+ =C@ <C!R M3'$- 3(" ( S"'+ 26(1).$FF, B. (1999). Q <= < <: = >= <

@ C=<R. E< T"(3 K+$#$ ( P%$(+ P"3(-

O<E=< "=@, '. .

200 E <C < >@<H

"$, = >=@ . J. 'E#BEG J. A. h$*Ah. ":@< E@C.

#'A, %. GI%%', c. (1996). Q@ = =C< -@< < =@D <: >==R. B(3(' E#4"3(+ R$$"' J4+ 22(4): >. 389-404.

'I'I#, . (1996). Q" <CR. E< C4("- 3( ( M3'$3(" K-12 # !$8#, = >=@ %. c. EI$.=<, *A: #c".

*A# $E', B. (2001). QEC=< =@ = <=< < CC@R. E#4"3(+ S34#($ ( M3'$3(" 46: >. 59-85.

*$# GA'EFEd, E. (1984). QA< <@=C=< = @ =<@C-DR. E< T'$ I$3$# R$+(38, = >=@ %. +AA+Ic. =<-@: +. +. #=@=< & c=.

P (1987). Q@<< =<@CD DR. E< P!+$ % R$/$$33( ( 3'$ T$"'( # L$( % M3'$3(",= >=@ c. JA#*IE. h, #J: @< E@C.

*G$', . '. (1962). T'4'3 # L4$. c@, "A:"I %@ # =@. (@. .: P$($3 8 +$4)$.BC<= A@. %K, 1977; B@=<. %N, 1995,3 .)

P (1978). M(# ( S"($38. c@, "A: h@D@ <D@%@. (@. .: E+ #$++ #$ + /"$ /("+=(" 4/$($. B@=<. c@L, 1995, 3 .)

P (1981). Q < = @ < C<=<R. E< T'$ C- "$/3 % A"3((38 ( S($3 P8"'+8, = >=@ J. +. +E'ch.A@=<, #: ". E. '@>. (@. .: Q K< C<-=< >LC C>@=@R < . *G$' (1995). O!

$"(#. *=. 3. "@. *=@, >. 139-168.)+AEdI#E, *. (1997). Q<< C < C =<-

=< =@R. E< S(343$# C(3(: S"(+ S$(3(" # P8"'+("+ P$/$"3($, = >=@ . d. J. +hI'$#.h, #J: @< E@C A=.

+E#GE, E. (1999). C4(3($ % P"3("$. c@: c@- <D@ %@. (@. .: C4(##$ #$ /"3(": /$#(9)$, ((%("# $ (#$3(##. B@=<. %N, 2001.)

+E'ch, J. *. (1985). V83*8 # 3'$ S"(+ F3( % M(#.c@, "A: h@D@ <D@ %@. (@. .:V83*8 8 + %"(= "(+ #$ + $3$. B@=<. %N,1995, 2 .)

O<E=< "=@, '. .

201B=@L

+E'ch, J. *. $"A, c. (1995). Qd=C@ < @<< < @==: ==CC@ >>@=R. E< C34"3(( ( E#4"3(, = >=@ . %. 'EFFE J. GAE. h=D, :@< E@C A=.

+hIEhEAd, J. (1989). Qc@< D< =@ @= C=< = <, Sh= = I >@=D >@?TR C!(#$ J4+ % E#4"3( 19(1): >. 41-52.

+$$d, . AcE, E. . (1990). Q D=>< = ==@D=C < @=C> <@=<R. E< T%( "'(+- #$@ M3'$3("+ E#4"3(: I3$3(+ P$/$"3($, -

= >=@ . %. 'EFFE . +$$d'. h, #J: @< [email protected]; c$BB, %. AcE, E. (1995). Q=< =< @<< < -

< < <@ ==R. E< C34"3(( ( E#4"3(, = >=@ . %. 'EFFE J. GAE. h=D, :@< E@C A=.

Ac, *. GA*E', B. (2001). Q"< <<@=C =C: S$< =C < = , <C @ >@ @R. E#4"3(+ S34#($ ( M3'$3(" 46, >.229-271.

O<E=< "=@, '. .

202 E <C < >@<H

A+;;< +:, 49, 55, 59, 81, 96, 116, 146, 171, 172.AdLE!, J., 154, 165.AL#!ich#E! @ o"ch, 175.

A<: + ::, 71, 136.A*;, 96.I @, 27, 52, 55, 78, 79, 93, 96, 115, 126, 140, 170-173.A. c=:+, 51, 52, 54, 67, 99, 112, 184.I d:+<:, 20, 168, 172.I +;=, 98, 112, 158.I. E=<+ , 17-19, 22, 25, 29, 52, 63, 72, 82, 83-123, 148, 156-160, 168, 178.I. o*;=:, 25, 85, 86, 89, 92, 106.I. +: , 23, 25, 63, 125, 126, 133, 135, 158, 166, 172, 178.I. # +; , 25, 47.A, 139.A:*, <, 78, 84, 116, 128.

A#kin, J. M., 174.A<. A+;;< , 27, 53, 63, 96.A<;+++, 105.A<;+, 113, 115.A<;+;+, 112.A<;++, 22, 123.A<;=<. A<;=<+, 25, 46, 51, 57, 106, 112-116, 120, 123, 157.

BA!nE", d., 162.B: , 19, 26, 32, 33, 42, 44, 45, 49, 75, 81, 115, 118, 140, 153, 175, 181.BLAck, ., 18, 156, 158, 159, 190.

B!$nE!, J., 152, 161.B$#LE!, !., 105.

c, 25, 29, 32, 46, 71, 78, 82, 83, 111, 114, 118, 119, 143.c* ::;*, 147, 159, 188, 189, 191, 192.

HE+: M;, ". L.

203!+: **C+:

CAPÍTULO VI

Índice de Autores y Materias

c*:, 28, 67, 84, 156, 158, 166, 169, 182, 183.I. c;<: <:, 147, 158, 187, 192.

I. C+;+, 141, 147, 160, 173-178, 187-191.c, +, 108.c;<, 73.c+ <:+ ++;, 154, 155.I (/:=), 25, 47, 170.c;, 182, 183.cLA!kE, "., 88.c+, 155.c;;, 184.c:, 50, 72, 78, 104, 110, 112, 113, 172, 190.I :<:;, 61, 117.c;+, 54, 96.

c: :;<K, 61.c<++, 36, 122, 139, 143, 148, 152, 153, 166, 169, 185.c<, 22, 36, 98, 148-156, 161-173, 178, 179-186, 193.c+:, ++:, 34, 35, 37, 47, 82, 145, 150.c?:. h+ +?:, 26, 52, 76, 86, 95, 151, 165, 192.c, :<, 22, 25, 39, 62, 101, 115, 116, 160, 172.c<, , 52, 97, 111, 168, 171.c:;<+;=: +, 151.c;. L+<: , 186.c=:+, 51, 67, 99, 117, 183.c;:. c: :+, 118.

I =<+, 46, 85-93, 104-108, 114, 115, 135, 143.c<;<, +<;<+, 44, 151, 152, 181, 182.

d*;, 23, 51-57, 94-99, 126, 154, 165.d*;, 17, 23, 25, 161.I, +, 17, 20, 25, 127.I :+:, 17, 23, 161.d+, ?:, 70, 72, 78, 112, 177.d: +;=, 139.dE" Depa"amen" fo Ed#ca"ion and Skill! (d;; E<++ @ #D++:), 149.dC, 32, 56, 60, 83.d. i=:;+, 31.

d:+<:, 17-32, 33-38, 47, 50, 127, 135, 146, 148-151, 155-173, 178, 179-186, 193.I. c: + +< , 17-29, 148-1156, 161-173, 176, 178, 179-186, 193.I. c ; , 51.I. c< , 23, 126, 148, 163, 169-173, 178, 185, 193.I +<;, 164, 169.I ;C;+, 17, 19, 21, 28, 33, 35, 37, 42, 80, 85, 162, 165, 167, 193.I ;<, 79, 81, 180.I +, 164, 169.I :+, 75.d, , 18, 33, 45, 94, 125, 144, 150, 159, 170, 175-178.d<:, 58, 83, 95, 100, 102, 119, 141, 143, 172, 186.

d$khin, k. @ "hi!E, B., 149.d&Eck, c., 25, 46, 160, 170.

E++, 24, 26, 60, 84, 132.ELLio#, i., 29, 174.

HE+: M;, ". L.

204 + A<;: @ M;:

E, 109.E, 49-82, 128.

I *, 49-82, 128.E< ; :+<, 24.E:+*, 18, 38, 66, 69, 70, 75, 110, 138, 139-146, 164, 169.E:+<+, 20, 22, 50, 60, 96, 99, 117, 123, 132, 168, 172, 186.E:+, <, 65, 106, 112, 157, 174.E:; +=+ ;C;+, 34-38, 47, 76, 78, 150, 181.E:;; : =; K, 56, 98, 99.E=<+, 72, 112-120, 156, 158, 160, 192.I. c;: , 72, 114, 115, 117, 121.I ; +:, 72, 112-120.I ;=, 18, 22, 156, 158, 160.I , 17-18, 23, 25, 29, 50, 52, 63, 72, 82, 83-123, 148, 156-160, 168, 178.

E%An" @ #"A#"A!oni, 151.E?:, 17, 29-30, 52, 123, 134, 171, 193.I. i: ;C;+:, 50, 141.

F<, <, 45, 62, 164.F;=. E=<+, 160.F$LLAn, M. G., 174.

Gc"E (Geneal Ce"ifica"e of Seconda' Ed#ca"ion), 37, 150.G, 59.GE!GEn, k. J., 36, 45, 150, 155.

G!iFFi#h", M., 30, 173.G<:, 52, 53, 55, 77-80, 102-104, 128-138.I. #* , 127.

h*, 17, 33, 50, 76, 81, 111, 142-146, 150, 155, 177.h+ +?:, 57.h+ *K, 25, 45-46.hA#h, G. @ "hi$, c., 176.h;: , 32, 47, 81, 100, 113, 119, 148, 153-156, 160, 161, 180.hoLLAnd, d. @ +:., 150, 156, 174.

i;, 23, 38, 135, 150, 155, 159, 160, 174, 177, 180, 182-185, 186.

i *, 87.i;=. i;=+, 50, 52, 54.i=:;+, 29, 30, 60, 77, 78, 109, 130, 174, 175, 184, 190.I-++, 28, 29, 32, 174, 178.I-++. d, 32.i=<+ : <:, 126, 146.

JA&o!"ki, B., 137, 139.J<:;+, 24, 140, 142.

King! Med%a', O&fod!hie Foma"i$e A!!e!men" Pojec" (kMoFA), 159, 190.

kL$GE!, A. @ dEni"i, A., 109.

LABo!dE, c., 33, 155.LAME!#, M., 162.LA%E, J., 150, 153, 155.

HE+: M;, ". L.

205+ A<;: @ M;:

L< ;C;+, 17-28, 33-48, 40-54, 62, 63-68, 76, 78, 81, 83, 84, 95, 129, 140,147-151, 154, 166, 168, 177.

I @ , 147.II ;C;+. Va!e: L< ;C;+.LE!MAn, "., 152, 166, 182, 184, 185.L*: ;?;, 34, 41, 77, 103, 120.L>@, 80, 129, 135.L+<: +;, 186.

MA"on, J., 32.M; :, 121.McnAi!, !., 161.ME!cE!, n., 148, 153, 155, 164, 169.M;-++, 113.

M;C, 42, 47.M ;:: , 151.Mo!GAn, c., 36, 37, 149.M;=+, 109, 123, 159, 175, 187, 190.

n =; , 56.n*:, 20, 43, 67, 79, 81, 100, 126, 162, 180, 187.I. :, 43.n: ;-:+<:=:, 166.

o*:=, *:=+, 51, 54, 57, 84, 102, 149.

o, 49, 63, 81, 149.I +:, 49, 63, 81.

, <, 125, 128, 133, 170, 172, 178, 179, 193.II :, 20, 47, 125, 126, 146, 158, 162, 164, 165, 166-170, 172, 178, 179, 193.:; <+, 100.:, 52-57, 94-100, 123, 140-146, 161-166, 177, 193.I. E = ;, 128.I. E:, 97.E!!Eno$d, ., 160.:=+, 26, 126.iAGE#, 151.

iMM, 36, 43, 45, 148, 155, 163, 169. ;:, 46, 111, 112, 122, 137, 168.o#A!i, d., 139.C+;+ ;, 187, 188, 190.C+;+: :+<:=:, 151, 166.<;. <;:, +<:;:, 61, 95, 97, 98, 123, 157, 162, 172.<;:, 51, 93-104, 108, 144, 165, 168.I ; +:, 61, 98, 101.I <+:, 54, 100.:, 63, 85, 87, 93, 113, 114, 138, 172, 184.<;, 27, 47, 163.

<: *, 56, 116.

!-;+<+, 30.! :, 20, 43, 81, 94, 101, 103, 181.!<+, *: <;:, 36, 37, 39, 47.

HE+: M;, ". L.

206 + A<;: @ M;:

!?, ?, 22, 24, 31, 34, 60, 61, 63, 75, 116, 128, 151, 161, 171, 173, 176, 190.!:;, 27, 35-43, 47, 52, 69, 165, 169, 180.

I ;C;+, 27, 35-44, 47, 69, 74, 79, 82, 136, 163, 165, 180, 192.!;, 50.!+:, 44, 166, 179.!;, 105, 109.!::*, 24, 60, 123, 127, 128, 133, 137, 146, 167, 169, 171, 178, 186, 193.!:<:;: <=+:, 76, 96, 172.!:<;: , 87.!;;+, 109-116, 123, 128, 132, 143, 158, 169.I. A ;:, 46, 111, 112, 122, 132.I. c;:, 104, 107, 109, 110, 115.I +;=., 104, 106, 110.!:, 45, 96, 177, 192.

"AdLE!, !., 41, 157."chn d A 173

207+ A<;: @ M;:

el lenguaje en el aprendizaje de las matem__ticas la evaluaci__n

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el lenguaje en el aprendizaje de las matemticas la evaluacin