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braicas FilloyRojano

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  • INVESTIGACIN DIDCTICA

    Resumen. !"#$%&$#&'()(*+#,'$%$"&(-+%#.(%#./"$(%#-($%&'(%#01$#2("#314(5+#"1$%&'+#&'()(*+#$"#6.3$)'(#!517(&48(#$"#.+%#9.&4-+%#8$4"&474"7+#(:+%;#2(7$-+%#24"7(,4+'-(;#,'$%$"&(-+%#1"#$*$-,.+#5$#("?.4%4%#5$#.(#7+-,$&$"74(#$"#$.#&$''$"+#5$#.(#'$%+.174="#(.3$)'(47(#5$#,'+).$-(%#5$#$"1"74(5+#8$')(.;#@#'$.(74+"(-+%#1"#7+"*1"&+#5$#'$%1.&(5+%#%+)'$#.+%#,'+7$%+%#7+3"4&48+%#5$#.+%#(.1-"+%#(#.+%#01$#%$#.$%#$%&?#$"%$:("5+#$%&(#'$%+.174="#(.3$)'(47(#5$#,'+).$-(%;#01$#2$-+%#+)&$"45+#(#.+#.('3+#5$#$%&+%#(:+%APalabras clave. B+5$.+%#&$='47+%#.+7(.$%;#%4%&$-(%#-(&$-?&47+%#5$#%43"+%;#-+5$.+%#5$#7+-,$&$"74(;#,'+7$%+%#7+3"4&48+%;#'$%+.174="#(.3$)'(47(#5$#,'+).$-(%;#?.3$)'(#$517(&48(;#.$7&1'(#@#$%7'4&1'(#5$.#?.3$)'(A#

    Local theoretical study of the development of algebraic competenceSummary. C"#&24%#,(,$'#D$#,'$%$"&2$#-(4"#31454"3#45$(%#+>#+1'#D+'E#+"#!517(&4+"(.#F.3$)'(#51'4"3#&2$#.(%&D$"&@G#8$#@$('%;#%&'$%%4"3#&2$#%$-4+&47#,+4"+>#84$D#&2(%2(,$%#4&A#H$#%2+D#(.%+#("#("([email protected]%4%#+>#7+-,$&$"7$#4"#(.3$)'(47#D+'5#,'+).$-#%+.84"3;#("5#.4%(#%$+>#'$%1.&%#+"#,1,4.%I#7+3"4&48$#,'+7$%%$%#&2(&(E$#,.(7$#D24.$#&[email protected]#('$#)$4"3#&(132(.3$)'(47#,'+).$-#%+.84"3;#'$%1.&%#D$#2(8$#3+&&$"#&2'+132+1&2$%$#@$('%AKeywords. J+7(.#&2$+'$&47(.#-+5$.%;#-(&2$-(&47(.#%43"#%@%&$-%;#7+-,$&$"7$#-+5$.%;#7+3"4&48$#,'+7$%%$%;#(.3$)'(47#,'+).$-#%+.84"3;#$517(&4+"(.#(.3$)'(;#(.3$)'(#'$(54"3#("5#D'4&4"3A

    EL ESTUDIO TERICO LOCAL DEL DESARROLLODE COMPETENCIAS ALGEBRAICAS#Filloy, Eugenio1; Puig, Luis2 y Rojano, Teresa1 K#L$,('&(-$"&+#5$#B(&$-?&47(#!517(&48(#M$"&'+#5$#C"8$%&43(74="#@#5$#!%&154+%#F8("N(5+%;#B

  • INVESTIGACIN DIDCTICA

    ###

    ENSEANZA DE LAS CIENCIAS, 2008, 26(3)328

    5$.#?.3$)'(A#b(.#$%#$.#7(%+#5$#&'()(*+%#7+-+#$.#5$#M('[email protected]"#V4$'("#%+)'$#.(#4"&$','$&(74="#5$.#%43"+#c#UV4$'(";#KYZ^a[#.+%#$%&154+%#5$#J$%.4$#d++&2#UKYZ]a#@#B('[email protected]"#B(&N#UKYZPa#%+)'$#$''+'$%#>'$71$"&$%#$"#$.#?.3$)'([#$.#("?.4%4%#7+-,(G'(&48+#5$#.(#.$"31(#-(&$'"(#7+"#$.#.$"31(*$#5$.#?.3$)'(#5$#e("%#X'$15$"&2(.#UKYZ_a[#@#$.#$%&154+#7./"47+#%+)'$#+,$G'(74="#5$#.(#4"7=3"4&(#5$#[email protected]#@#`+*("+#UKYZYa#@#[email protected];#`+*("+#@#f+.('$%# UP^^_a;# $"&'$# +&'+%A#!"# $%&$# 9.&4-+# %$#'$,+'&("#'$%1.&(5+%#01$#,+"$"#5$#-("4#$%&+#01$#-172(%#5$#.(%#,'+51774+"$%#5$#.+%#$%&154("&$%#$"#%4&1(74+"$%#$"#01$# $">'$"&(";# ,+'# ,'4-$'(# 8$N;# .(# '$%+.174="#5$# 74$'&(%#&('$(%#(.3$)'(47(%#"+#,$'&$"$7$"#"$7$%('4(-$"&$#(#(.31"+#5$#.+%#%4%&$-(%#5$#%43"+%#@(#-$"74+"(5+%;#%4"+#01$#&(.$%#,'+51774+"$%#%$#1)47("#$"#"48$.$%#g4"&$'-$54+%h;#$"&'$#.(#('4&-+'-(.;#5$#$"%$:("N(;#5$#,'+7$%+%#7+3"4&48+%#@#5$#7+-1"47(74="#[email protected];#`+*("+#@#\143;#P^^ZaA#!"#$.#,'$%$"&$#('&/71.+#%$#$O,+"$";#$"#1"(#,'4-$'(#%$774="#@#(#-("$G'(#5$#("&$7$5$"&$%#5$#.(#>+'-1.(74="#5$#.(#&$+'/(#5$#.+%#modelos locales;#1"#,('#5$#$*$-,.+%#5$#>$"=-$"+%#+)G%$'8(5+%#$"#$.#$%&154+#Operacin de la incgnita;#,('(#.+%#71(.$%;#$"#$.#&4$-,+#$"#01$#>1$'+"#45$"&4#7(5+%;#"+#%$#7+"&()(#7+"#$O,.47(74+"$%#,.(1%4).$%#5$%5$#$%01$-(%#5$#("?.4%4%#5$#.(%#&$+'/(%#3$"$'(.$%#5+-4"("&$%A#!"#1"(#%$31"5(# %$774=";# %$# 2(7$# '$>$'$"74(# (# &'()(*+%# 01$# $%G&154("#(.#?.3$)'(#7+-+#.$"31(*$A#!"#.(#&$'7$'(#%$774=";#%$#5$%7'4)$"#$.$-$"&+%#5$.#7+-,+"$"&$#5$#7+-,$&$"74(A#!"# .(# 71('&(;# %$# $O,+"$"# 7(%+%# 01$# 4.1%&'("# $.# ("?.4%4%#5$%5$#$.# 7+-,+"$"&$#5$# .+%#,'+7$%+%#7+3"4&48+%#@#01$#(#+'('+"#$"#$.#$%&154+#Operacin de la incgnita#@#$"#$%&154+%#,+%&$'4+'$%;#$"#$.#7+"&$O&+#5$#.(##'$%+.174="#5$#,'+).$-(%#(.3$)'(47+%#8$')(.$%A

    1. ENTRE EL LENGUAJE ALGEBRAICO Y EL LENGUAJE NATURAL: EPISODIOS DEL ES-TUDIO!"#$%&'()*!+$!,&!(*')-*(.&!.#$%&154+#Operacin de la incgnita;#'$(.4N(5+#(#,'4"74G,4+%#5$#.+%#(:+%#KYZ^;#7+-,'$"5$#8('4(%#$&(,(%#+#%1)G$%&1G54+%;#$"#$.#,'4-$'+#5$#.+%#71(.$%#%$#("(.4N("#.(%#'$%,1$%&(%#$%,+"&?"$(%# 5$# $%&154("&$%# 5$# %$71"5('4(# ("&$# &('$(%# 5$#'$%+.174="#5$#$71(74+"$%#.4"$(.$%;#$"#.(%#01$#$%#"$7$%('4+#+,$'('#7+"#.+#5$%7+"+745+#U.(#4"7=3"4&(aA#J+%#'$%1.&(5+%#5$#$%(#$&(,(#%$#'$,+'&('+"#$"#1"(#%$'4$#5$#('&/71.+%#,1).47(G5+%#(#,('&4'#5$#KYZ]#[email protected]#@#`+*("+;#KYZ];#KYZj(;#KYZj);#KYZYaA#R"#%$31"5+#%1)$%&154+;#$.#5$#La lecto-escritura al-gebraica;#%$#..$8=#(#7()+#7+"#$%&154("&$%#5$#%$71"5('4(#(#014$"$%#@(#%$#.$%#2()/(#$"%$:(5+#.(#'$%+.174="#5$#$71(74+G"$%#.4"$(.$%;#,$'+#01$#(9"#"+#&'()(*()("#7+"#$O,'$%4+"$%#()4$'&(%;# $0148(.$"74(# 5$# $O,'$%4+"$%# (.3$)'(47(%# "4# 7+"#%4%&$-(%# 5$# $71(74+"$%A# J(%# >+'-(%# )?%47(%# 5$# "+&(74="#5$.#?.3$)'(#%+"#$%$"74(.-$"&$#.(%#-4%-(%#01$#.(%#5$#.(#('4&G

    -'(%$#ga#(1-$"&(5+#5$#5+%h#("&$%#5$#,'+7$5$'#(#%1#%4-)+.4N(74="#@#$%#$"#$%$#,'+7$%+#5$#(%43"(74="#5$#%43"4G#7(5+%;#$"#5+"5$#%$#+)%$'8(#1"(#4"7+"%4%&$"74(#,+'#,('&$#5$.#$%&154("&$A#L472(#4"7+"%4%&$"74(#%$#5$'48(#5$.#2$72+#5$#01$#$"#.(#>'(%$#+'434"(.#g(h#@#g5+%h#%+"#-$545(%#5$#7("G&45(5$%#5$#.(#-4%-(#-(&$'4(#+#71$%&4=";#@#$.#$%&154("&$#%$#.(%#(%43"(#(#7+%(%#54%&4"&(%#U8$.+745(5#@#5/(%;#$"#1"#7(%+;#@#,$%+#@#&4$-,+;#$"#$.#+&'+a;#$O7$,&+#$"#.(#+'(74="#g

  • INVESTIGACIN DIDCTICA

    329ENSEANZA DE LAS CIENCIAS, 2008, 26(3)

    J(%#'$%,1$%&(%#$%,+"&?"$(%#5$#.+%#$%&154("&$%#(#,'$31"G&(%# 7+-+# .(# ("&$'4+'# '$8$.("#01$# .+%# %43"4#7(5+%# 7+.+G014(.$%#5$#.(%#,(.()'(%#,'$5+-4"("#$"#$%(%#$5(5$%#@#$..+#4"24)$#.(#&'(51774="#(.#%4%&$-(#-(&$-?&47+#5$#%43"+%#5$.#?.3$)'(#UfBf(.a#5$#>'(%$%#>+'-(5(%#,+'#$%(%#,(.()'(%A!"#$.#7(%+#4"8$'%+;#71("5+#%$#.$%#,$5/(#(#.+%#$%&154("&$%#01$#[email protected]$'("#$"#8+N#(.&(#$O,'$%4+"$%#()4$'&(%#&(.$%#7+-+k########################################;#ab;#_ab;#aP;(5$-?%#5$#,'+5174'#.$7&1'(%#&$O&1(.$%#7+-+k

    ga#-?%#b#%+)'$#5+%h;(.31"+%#$%&154("&$%#&$"5/("#(#(%43"('#%43"4#7(5+%#3$+-(&4N(# $.# 7('?7&$'#5$# .$"31(#$%7'4&(#5$.# ?.3$)'(#@#'$71''$#(#.(#.4"3u/%&47(#@#(#.(#%$-4=&47(#,('(#("(.4N('#.(#%4"&(O4%#@#.(#%$-?"&47(#(.3$)'(47(%A#!%&$#(1&+'#2(7$#1%+#5$#.(#3'(-?&47(#3$"$'(&48(#@#&'("%>+'-(74+"(.#,('(#,'+G5174'#$O,'$%4+"$%#(.3$)'(47(%#%4-,.$%#@#&'("%>+'-('.(%;#7+"#)(%$#$"#5$%7'4,74+"$%#5$#%1%#formas super ciales#@#profundas UV4'%2"$';# KYZv;# KYY^;# P^^KaA# \+'# %1# ,('&$;#w$("G\24.4,,$#L'+12('5#5$%(''+..(#1"(#"+74="#5$#%43"4#G7(74=";#7+"#.(#71(.#'$.(74+"(#.+%#(%,$7&+%#5$#referencia, sentido, interpretacin @# connotacin;# 01$# %+"# .(# )(%$#5$# %1# ("?.4%4%# 5$# .+%# %43"4#7(5+%# +# %43"4#7(74+"$%# 5$.#?.3$)'(# $%7'4&(# UL'+12('5;# KYYPaA# \+'# +&'+# .(5+;# J14%#`(5>+'5# &+-(#.(# 45$(#5$#[email protected]+&%[email protected]#5$#01$# .(#7+3"474="#21-("(#$%&?#(&(5(#(.#1%+#5$#%43"+%;#5$#&(.#-("$'(#01$#@("+#%$#7+"%45$'(#7$"&'(.#.+#01$#.+%#%43"+%#'$,'$%$"&("#%4"+#.+#01$#"+%#,$'-4&$"#2(7$'A#!%#-?%;#`(5>+'5#%+%&4$"$#01$#.+%# %43"+%# >+'-("#,('&$#5$# %4%&$-(%#5$# %43"+%#01$# %+"#,('&$#5$#1"(#71.&1'(#@#,+'#.+#&("&+#&'(%74$"5$"#.(%#7+3G"474+"$%#4"548451(.$%#U`(5>+'5;#P^^^aA#L$%5$#$%&(#,$'%G,$7&48(;#$.#(1&+'#("(.4N(#.(#$-$'3$"74(#5$.#,$"%(-4$"&+#(.3$)'(47+#$"#.+%#$%&154("&$%#@#.(#$-$'3$"74(#5$.#%4-)+G.4%-+#(.3$)'(47+#$"#.(#24%&+'4(#U`(5>+'5;#P^^KaAJ+%#&'$%#(7$'7(-4$"&+%#-$"74+"(5+%#%$#"1&'$"#5$#54%74G,.4"(%#7+-+#.(#.4"3u/%&47(;#.(#%$-4=&47(#@#.(#24%&+'4(A#L$#-("$'(# %4-4.(';# $.# (7$'7(-4$"&+# 5$#modelos tericos locales#4"7+',+'(#$.$-$"&+%#5$#.(#%$-4=&47(#@#.(#24%&+G'4(#@#(5+,&(#1"(#,$'%,$7&48(#)(%(5(#$"# .(#,'(3-?&47(;#>(8+'$74$"5+#(%/#$.#$%&154+#5$.#%43"4#7(5+#$"#1%+#-?%#01$# $.# 5$.# %43"4#7(5+# >+'-(.A# L$# $%&$#-+5+;# $.# >+7+#

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  • INVESTIGACIN DIDCTICA

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    ENSEANZA DE LAS CIENCIAS, 2008, 26(3)330

    5$# (&$"74="# %$# 5$%,.(N(# 2(74(# .(# (7&4845(5# 5$# .+%# %1G*$&+%# 7+"# $.# .$"31(*$# 5$.# ?.3$)'(A#J(# 3'(-?&47(# 7+-+#$.#%4%&$-(#>+'-(.#@#.(#,'(3-?&47(#7+-+#$.#7+"*1"&+#5$#,'4"74,4+%# 5$# 1%+# 5$.# .$"31(*$# %+"# 7+"7$)45(%# 7+-+#5+-4"4+%# 7+-,.$-$"&('4+%;# $%,$74(.-$"&$# 71("5+# %$#.+%# '$.(74+"(# 7+"# -+5$.+%# 5$# $"%$:("N(# 5$.# ?.3$)'(A#M+"#$%&$#$"&'$&$*45+#&$='47+;#%$#()+'5("#,'+).$-?&47(%#5$#.(#4"&$'(774="#.$"31(*$#"(&1'(.G?.3$)'(;#>+'-1.("5+#modelos locales#,('(#$.#("?.4%4%#5$#>$"=-$"+%#$%,$7/G#7+%# [email protected];#`+*("+#@#\143;# P^^ZaA#!"# .(%# %4314$"&$%#%$774+"$%;#%$#4.1%&'(#$%&$#&4,+#5$#("?.4%4%#,('(#$.#7(%+#5$#.(#'$%+.174="#5$#,'+).$-(%#(.3$)'(47+%#8$')(.$%;#$">(G&4N("5+#(%,$7&+%#'$.(74+"(5+%#7+"#.+%#,'+7$%+%#7+3"4&4G8+%#01$#&4$"$"#.13('#$"#.(#&'(51774="#5$.#$"1"74(5+#5$#1"#,'+).$-(#8$')(.#(.#.$"31(*$#(.3$)'(47+A

    3. EL PROCESO DE RESOLUCIN ALGEBRAI-CO DE UN PROBLEMA VERBAL: EL COMPO-NENTE DE COMPETENCIA

    3.1. El mtodo cartesiano como modelo de compe-tenciaM+-+#@(#2$-+%#4"547(5+#$"#.(#4"&'+51774=";#1"(#5$#.(%#7('(7&$'/%&47(%# 5$.# $">+01$# 5$# .+%#-+5$.+%# &$='47+%# .+G7(.$%#$%#.(#7+"%45$'(74="#5$#71(&'+#7+-,+"$"&$%#5$#.+%#-+5$.+%A#\$'+;#(5$-?%;#$%&+%#7+-,+"$"&$%#$%&?"#4-)'4G7(5+%;#(%/#01$#.(%#+)%$'8(74+"$%#01$#,'$%$"&(-+%#$"#.(#%$774="#j#%+)'$#.+%#,'+7$%+%#7+3"4&48+%#>+'-("#,('&$#5$.#7+-,+"$"&$#5$#(7&1(74="#5$#.+%#%1*$&+%;#,$'+#$%$#7+-,+G"$"&$#5$#.+%#,'+7$%+%#7+3"4&48+%#%$#-4'(#5$%5$#$.#,1"&+#5$#84%&(#01$#5(#$.#7+-,+"$"&$#5$#7+-,$&$"74(A#!"#$%&(#%$774="#,'$%$"&(-+%#(.31"+%#5$#.+%#$.$-$"&+%#01$#7+"%G&4&[email protected]$"#$.#7+-,+"$"&$#5$#7+-,$&$"74(#5$#.(#'$%+.174="#(.3$)'(47(#5$#,'+).$-(%#5$#$"1"74(5+#8$')(.AJ(# 7+-,$&$"74(# $"# .(# '$%+.174="# (.3$)'(47(# 5$# ,'+).$G-(%# 5$# $%(# >(-4.4(# &4$"$# $.$-$"&+%# 01$# ,'+84$"$"# 5$#8('4+%#?-)4&+%#01$#$%&?"#,'$%$"&$%#$"#$.#,'+7$%+#5$#'$G%+.174="A#!.#7$"&'+#5$.#,'+7$%+#$%#$.#,(%+#5$.#$"1"74(5+#5$.#,'+).$-(;#01$#%$#,'$%$"&(#$%7'4&+#$"#.$"31(*$#"(&1'(.;#(#1"(#$O,'$%4="#5$.#.$"31(*$#5$.#?.3$)'(k#1"(#$71(74="A#\+'#&("&+;#$"#.(#'$%+.174="#(.3$)'(47(#5$#,'+).$-(%#$%&?#4-,.47(5(;#,+'#1"#.(5+;#.(#7+-,$&$"74(#$"#(-)+%#.$"31(G*$%;#@;#,+'#+&'+;#.(#7+-,$&$"74(#$"#$.#,'+7$%+#5$#,(%+#5$#1"#&$O&+#$%7'4&+#$"#$.#.$"31(*$#"(&1'(.#(#1"#&$O&+#$%7'4&+#$"#$.#.$"31(*$#5$.#?.3$)'(A#!%&+#$%#,(&$"&$#%4#%$#$O(-4G"(#$"#.(#24%&+'4(#$.#-+'-(# 7("=G"47(Axa#F,.47(74="#5$#.(#>='-1.(#+#(.3+'4&-+#5$#%+.174="#(#.(#$71(74="#$"#>+'-(#7("="47(Ava#C"&$','$&(74="#5$.#'$%1.&(5+#5$#.(#$71(74="#$"#&

  • INVESTIGACIN DIDCTICA

    331ENSEANZA DE LAS CIENCIAS, 2008, 26(3)

    @;#,+'# &("&+;#%$#2(#5$#(7+-,(:('#5$# .(#7+-,$&$"74(#5$#,'+7$%+#$"#1"#&4,+#5$#("?.4%4%#-$54+%G#"$%[#@#$.#%$O&+;#$"#.(#%+.174="#5$#$71(74+"$%#U+#%4%&$-(%a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

  • INVESTIGACIN DIDCTICA

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    ENSEANZA DE LAS CIENCIAS, 2008, 26(3)332

    b+-$-+%#,+'#$*$-,.+#1"(#$71(74="#7+-+#

    f4#4"&$"&(-+%#'$7+''$'#5$%5$#.(#4"7=3"4&(#$.#7(-4"+#5$.#("?.4%4%;#1&4.4N("5+#.(%#'$.(74+"$%#$"&'$#7("&45(5$%#01$#$%G&?"#$O,'$%(5(%#$"#$%&(#$71(74=";#$'$"74(#>1"5(-$"&(.#$"&'$#$.#("?.4%4%#5$.#$"1"74(G5+#,'+,4+#5$.#-

  • INVESTIGACIN DIDCTICA

    333ENSEANZA DE LAS CIENCIAS, 2008, 26(3)

    "+745(%#L,# UKYZa;#L'# UPKva# @#z-# UKvKa;# .(%# 7("&45(5$%#5$%7+"+745(%#z,;#z';#@#\;#@#.(%#'$.(74+"$%#$"&'$#$%(%#7("G&45(5$%#L,#!#z,#c#\;#L'#!#z'#c#\#@#z'#t#z-#c#z,#U1"#"1$8+#&$O&+a#%$#'$,'$%$"&(#-$54("&$#$.#3'(>+#5$#.(##31'(#_A

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  • INVESTIGACIN DIDCTICA

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    ENSEANZA DE LAS CIENCIAS, 2008, 26(3)334

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  • INVESTIGACIN DIDCTICA

    335ENSEANZA DE LAS CIENCIAS, 2008, 26(3)

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  • INVESTIGACIN DIDCTICA

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    ENSEANZA DE LAS CIENCIAS, 2008, 26(3)336

    5$#1"#fBf#-?%#()%&'(7&+;#&'(5174$"5+;#5$%,1(-4.4(%#5$#,'+).$-(%A#l+# %$# 7+"#$'$#1"# %$"&45+#(.#B(-4.4(%#5$#,'+).$-(%A

    R"#1%+#7+-,$&$"&$#4-,45$#01$#$.#1%1('4+#7(43(#$"#74$'G&(%#&$"5$"74(%#7+3"4&48(%;#01$#+)%&'[email protected]$"#.(#,+%4)4.45(5#5$#2(7$'#1"#1%+#(5$71(5+#5$.#BM#,('(#'$%+.8$'#,'+).$-(%A#!*$-,.+#5$#$%&+#%$'/("k#aa# .(#,'$%$"74(#5$#-$7("4%-+%#(,$.(&48+%#01$#..$8("#(.#5$%$"7(5$"(-4$"&+#5$#,'+7$%+%#$''="$+%#5$#%+.174="#U,+'#$*$-,.+;#%4#$"#.(#'$%+.174="#5$#1"#,'+).$-(#(,('$7$#1"#&4,+#5$#&$O&+#-(&$-?&47+#01$#"+#%$# %()$# 5$%7+54#7('a;#ba# .(# ,'$%$"74(# 5$# +)%&'1774+"$%#,'+8$"4$"&$%#5$#.(#%$-?"&47(#%+)'$#.(#%4"&(O4%#@#847$8$'G%(#U,+'#$*$-,.+;#(.#'$%+.8$'#,'+).$-(%#@#5+&('#5$#%43"4G#7(5+%#(#.+%#%43"+%;#,'$54%,+"4$"5+#$%&+#(.#%1*$&+#(#1"(#)1$"(#1&4.4N(74="#5$#.(#%4"&(O4%a;#ca#.(#,'$%$"74(#5$#-$7(G"4%-+%#4"24)4&+'4+%#U,+'#$*$-,.+;#71("5+#(#1"#,'+).$-(;#5$#1"(#>(-4.4(#5$#,'+).$-(%;#@(#'$%1$.&+;#%$#.$#7(-)4(#$.#8(.+'#5$#(.31"+%#5(&+%aA

    4.4. Sobre el dominio de las tcticas intermedias y las tendencias cognitivas!.#5+-4"4+#5$#.(%#&?7&47(%#4"&$'-$54(%#5$)$#7+.()+'('#$"#$.#5$%(''+..+#5$#&$"5$"74(%#g,+%4&48(%h#7+3"4&48(%#01$#%$#,'$%$"&("#$"#.+%#,'+7$%+%#,('(#(,'$"5$'#7+"7$,&+%#-?%#()%&'(7&+%;#7+-+#%+"k#aa#$.#'$&+'"+#(#%4&1(74+"$%#-?%#7+"G7'$&(%;#71("5+#%$#,'$%$"&(#1"(#%4&1(74="#5$#("?.4%4%;#1"(#,('&$#"$7$%('4(#,('(#(8("N('#$"#.(#7+-,$&$"74(#5$.#BM[#+#ba#.(#,'$%$"74(#5$#1"#,'+7$%+#5$#()'$84(74="#5$#1"#&$OG&+#7+"7'$&+#,('(#,+5$'#,'+5174'#'$3.(%#%4"&?7&47(%#"1$8(%#U,+'#$*$-,.+;#$"#.(#'$%+.174="#5$#,'+).$-(%;#71("5+#%$#$%&?#+,$'("5+#7+"#.+%#8(.+'$%#,('&471.('$%#01$#%$#.$#(%43G"("#(#.(#4"7=3"4&(#5$#1"#,'+).$-(#$"#7(5(#$O,.+'(74="#5$.#BF!f;#@#.1$3+;#,+7+#(#,+7+;#%$#8(#+,$'("5+#$"#$.#&$O&+#()%&'(7&+;#7+"#.(%#'$3.(%#5$.#fBf#-?%#()%&'(7&+;#%4"#2(7$'#'$>$'$"74(#@(#(#.(#%4&1(74="#7+"7'$&(aA

    4.5. Sobre la sintaxis y la memoria\('(#74$'&+%#,'+).$-(%;#$.# $-)('7('%$#$"#1"#,'[email protected]$7&+#5$# '$%+.174="# 8/(# $.# B+'G-(5+%# $%01$-(%# -$"&(.$%# 01$# .$# ,$'-4&$"# '$7+"+7$'#1"#,'+).$-(#4"7.1%+#("&$#.(%#,'4-$'(%#,(.()'(%;#@;#(.#'$G

  • INVESTIGACIN DIDCTICA

    337ENSEANZA DE LAS CIENCIAS, 2008, 26(3)

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f4"#$-)('3+;#(1"01$#$%&+#,1$5$#%17$5$'#$"#-172(%#%4&1(74+"$%#,'+).$-?&47(%;#$"#,+7+#"+%#(@15('/(#,('(#$O,.47('#,'+7$%+%#-?%#7+-,.$*+%#01$# "+%# ,$'-4&("# 5$74'# ,+'# 01$'$"74(#5$.#1%+#5$#1"#-$'$"&$%# (# &4,+%# 5$#,'+).$-(%A#!%&+#$%;#(1"01$#(#1"#%1*$&+#%$#.$#,1$5$#45$"G&4#7('#7+-+#7+-,$&$"&$#$"#.(#'$%+.174="#5$#,'+).$-(%#,+'01$#$-,.$$#$%&'(&+%#5$.#fBf#5$.#BM#,('(#'$%+.8$'G.+%#2(74$"5+;#(,('$"&$-$"&$;#1"#1%+#(1&+-?&47+#5$#$%G01$-(%#-$"&(.$%#01$#2(#(5014'45+;#,'$84(-$"&$;#(7$'7(#5$#.(#'$%+.174="#5$#54%&4"&(%#>(-4.4(%#5$#,'+).$-(%;#%4#

    %$#014$'$#2(7$'#1"(#-$*+'#7('(7&$'4N(74="#5$#.(#7+-,$G&$"74(# >+'-(.# $"# .(# '$%+.174="# 5$# ,'+).$-(%;# %$# 5$)$#7+"%45$'('# $.# ,'+3'$%+#5$#1"#1%1('4+# $"# %1# 7(,(745(5#,('(# '$(.4N('# $.# ("?.4%4%# .=347+Q%$-4=&47+# 5$# .(%# %4&1(G74+"$%#,'+).$-?&47(%A!"# '$%1-$";# 1"# 1%1('4+# 7+-,$&$"&$# 5$# 1"# fBf# -?%#()%&'(7&+#'$(.-$"&$#.+#8(#(#%$'#%4#&(-)4

  • INVESTIGACIN DIDCTICA

    ###

    ENSEANZA DE LAS CIENCIAS, 2008, 26(3)338

    4.11. Sobre el uso de mtodos primitivos y el uso de la memoriaM1("5+#%$#$-,.$("#-/(%# U7=543+%a# ,$'%+"(.$%# ,('(# 4"547('# .(%# (774+"$%# @(#'$(.4N(5(%#@#.(%#(774+"$%#,+'#'$(.4N('#%+)'$#.+%#$.$-$"&+%#5$.#,'+7$%+#5$#'$%+.174="A#!%&+#%134$'$#.(#$O4%&$"74(#5$#1"(#$&(,(#,'$84(#(# .(# $&(,(#+,$'(74+"(.A#!"#$%&(#$&(,(;#%$# ,'$%$"&("# &(-)4A#b2+'"54E$#$(.A;#KYP_a#@#%1%#4-,.47(74+"$%#$"#.(#$"%$:("N(#7+)'("#1"(#"1$8(#,'$%$"74(;#,1$%# .(#"$7$%45(5#5$#(1&+-(&4N(G74="#5$#74$'&(%#+,$'(74+"$%;#01$#,'+84$"$"#5$#.(#5$%7+54G#7(74="#5$#1"(#%4&1(74="#,'+).$-?&47(#7+"7'$&(#U,'+).$G-(%# 5$# $5(5$%;#-$N7.(%;# (.$(74+"$%;#-+"$5(%;# &'()(*+;#$&7Aa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fBf;#@#(%/#%$#'+-,$#7+"#.+%#-$7("4%-+%#("&474,(5+'$%#01$# 4"24)$"#$.#5$%$"7(5$"(-4$"&+#5$# .(%#$%&'(&$34(%#5$#'$%+.174="#"$7$%('4(%A

    NOTAS

    KA#J(#'$3.(#C#547$#&$O&1(.-$"&$k#g\$'#2("7#'(&4+74"("54#-$&2+51-#01('$"5(#%1"&+-(3"4&154"$%#51+)1%#-+54%#54>>$'$"&4)1%#$O,'$%%(;#01+ (5# 54>#71.&(&$-# 54'$7&$# ,$'71''$"5(-# &$'-4"+%# 4"7+3"4&+%# ,'+#7+3"4&4%#%1,,+"4-1%k#4&(#$"4-#&++-,('(&4+"$%#4"&$'#51+#01(.4(#2(G)$)1"&1'Ah#Ug\+'#$%&$#-

  • INVESTIGACIN DIDCTICA

    339ENSEANZA DE LAS CIENCIAS, 2008, 26(3)

    %1,1$%&+#7+"+745+%#,('(#'$7+''$'#54'$7&(-$"&$#.(#54#71.&(5;#@(#01$#(%/#%$#&$"5'?#$.#-4%-+#"9-$'+#5$#7+-,('(74+"$%#$"&'$#5+%#7+%(%#431(.$%ha#UL$%7('&$%;#Kv^K;#,A#xxaA#L$%7('&$%#%1)'(@(#01$#2(@#01$#7+"%&'14'#&("G&(%# $71(74+"$%# 7+-+#7("&45(5$%# %$#2(@("#5$%43"(5+#7+"# .$&'(%# $"#$.#,'4-$'#,(%+#5$.#-A#l(%%(';#P^^KaA_A#!"#`(5>+'5#@#\143#UP^^va#,1$5$#8$'%$#7=-+#1"#(.1-"+#-('7(#$"#1"(#$O,'$%4="#(.3$)'(47(#7+-,1$%&(#5+%#$O,'$%4+"$%#01$#%$#'$#$'$"#(#5+%#7("&45(5$%#5$.#,'+).$-(#$"7$''?"5+.(%#$"&'$#,(''+-#F'4&2-$&47#&+#F.3$)'(A#For the Learning of Mathematics;#YUPa;#,,A#KPGPjA#

    XCJJip;#!A#@#` iwFli;#bA#UKYYKaA#b'("%.(&4"3#>'+-#"(&1'(.#.("G31(3$#&+#&2$#-(&2$-(&47(.#%@%&$-#+>#(.3$)'(47#%43"%#("5#847$G8$'%(k#(#7.4"47#%&[email protected]#D4&2#724.5'$"#4"#&2$#,'$G(.3$)'(47#%&(3$A#Proceedings of the Psychology of Mathematics Education, North American Chapter;#,,A#PYG_jA#S4'34"4(#b$72;#RfFA#

    XCJJip;#!A;#`iwFli;#bA#@#fiJF`!f;#FA#UP^^_aA#bD+#B$G("4"3%# +># &2$# !01(.# f43"# ("5# f$"%$%# +># f1)%&4&1&4+"# ("5#M+-,('4%+"#B$&2+5%;# $"# \(&$-(";# lAFA;# L+32$'&@;# dA# @#4..4+O;#wA#U$5%AaA#Proceedings of the Twenty-seventh Annual Conference for the Psychology of Mathematics Education;#];#,,A#PP_GPPYA#e+"+.1.1;#e(D(44;#RfFk#M`Lz;#M+..$3$#+>#!517(&4+";#R"48$'%4&@#+>#e(D(44A

    XCJJip;#!A;#`iwFli;#bA#@#\RCz;#JA#UP^^ZaA#Educational Al-gebra. A Theoretical and Empirical ApproachA#l1$8(#p+'Ek#f,'4"3$'A

    X`!RL!lbeFJ;# eA# UKYZ_aA#Didactical Phenomenology of Mathematical StructuresA#L+'5'$72&k#`$45$.A

    X`CLBFl;#JABA#UKYY^aA#J+%#3'(>+%#&'4"+-4(.$%#7+-+#-$&(G.$"31(*$# 5$# .+%# ,'+).$-(%A#MatemticasA#Revista del De-partamento de Matemticas de la Universidad de Sonora;#17-18;#,,A#jK~jYA

    REFERENCIAS BIBLIOGRFICAS

  • INVESTIGACIN DIDCTICA

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    ENSEANZA DE LAS CIENCIAS, 2008, 26(3)340

    VFJBpViSF;#ACA#UKYvjaA#\'+7$%%$%#+>#("([email protected]%4%#("5#%@"&2$G%4%#4"#&2$#%+.1&4+"#+>#('4&2-$&47#,'+).$-%;#$"#V4.,(&'47E;#wA;#H4'%N1,;#CA;#d$3.$;#!AzA#@#H4.%+";#wA#U$5%AaA#Soviet Studies in the Psychology of Learning and Teaching Mathematics. Vol. XI. Analysis and Synthesis as Problem Solving Methods;#,,A#KGKvKA#f&(">+'5;#MFk#lMbBA

    VC!`Fl;#MA#UKYZ^aA#b2$#4"&$','$&(&4+"#+>#&2$#$01(.#%43"k#[email protected])+.#>+'#("#$0148(.$"7$#'$.(&4+"#8%A#("#+,$'(&+'#%@-)+.;#$"#V(',.1%;#`A#U$5AaA#Proceedings of the Fourth Conference of the Interna-tional Group for the Psychology of Mathematics Education;#,,A#Kx_GKxYA#d$'[email protected];#M(.4>+'"4(;#RfFk#R"48$'%4&@#+>#M(.4>+'"4(A

    VC!`Fl;#MA#@#XCJJip;#!A#UKYZYaA#!.#(,'$"54N(*$#5$.#?.3$)'(#$%7+.('#5$%5$#1"(#,$'%,$7&48(#,%47+.=347(A#Enseanza de las Ciencias;#v;#,,A#PPYGP]^A

    VC`fel!`;#LA#UKYZvaA#The grammar of symbolic elementary algebraA# R",1).4%2$5# 5+7&+'(.# 54%%$'&(&4+";# R"48$'%4&@# +>#d'4&4%2#M+.1-)4(;#S("7+18$';#M("(5(A

    VC`fel!`;#LA#UKYY^aA#F7014%4&4+"#+>#(.3$)'(47#3'(--(';#$"#d++E$';#zA;#M+));#\A#@#V4$'(";#MA#U$5%Aa#Proceedings of the Fourteenth Annual Conference for the Psycology of Mathe-matics Education;#B+'$.+%;#B#F.3$)'(47#b24"E4"3A#Educational Studies in Mathematics, xxUPa;#,,A#K]jGKx]A

    `iwFli;#bA# UKYY]aA# J(#-(&$-?&47(# $%7+.('# 7+-+# .$"31(*$A#l1$8(%# ,$'%,$7&48(%# 5$# 4"8$%&43(74="# @# $"%$:("N(A#Ense-anza de las Ciencias,#KPUKa;#,,A#]jGjxA

    bei`lLCV!;#!AJA#$(.A#UKYP_aA#The Psychology of AlgebraA#l1$8(#p+'Ek#b2$#B(7-4..("#M+-,("@

  • INVESTIGACIN DIDCTICA

    341ENSEANZA DE LAS CIENCIAS, 2008, 26(3)

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    Local theoretical study of the development of algebraic competenceFilloy, Eugenio1; Puig, Luis2 y Rojano, Teresa1 K#L$,('&(-$"&+#5$#B(&$-?&47(#!517(&48(M$"&'+#5$#C"8$%&43(74="#@#5$#!%&154+%#F8("N(5+%;#B