Competencias Alge

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$#%2+D#(.%+#("#("(.@%4%#+>#7+-,$&$"7$#4"#(.3$)'(47#D+'5#,'+).$-#%+.84"3;#("5#.4%(#%$+>#'$%1.&%#+"#,1,4.%I#7+3"4&48$#,'+7$%%$%#&2(&(E$#,.(7$#D24.$#&2$@#('$#)$4"3#&(132(.3$)'(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


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

5$.#?.3$)'(A#b(.#$%#$.#7(%+#5$#&'()(*+%#7+-+#$.#5$#M('+.@"#V4$'("#%+)'$#.(#4"&$','$&(74="#5$.#%43"+#c#UV4$'(";#KYZ^a[#.+%#$%&154+%#5$#J$%.4$#d++&2#UKYZ]a#@#B('4.@"#B(&N#UKYZPa#%+)'$#$''+'$%#>'$71$"&$%#$"#$.#?.3$)'([#$.#("?.4%4%#7+-,(G'(&48+#5$#.(#.$"31(#-(&$'"(#7+"#$.#.$"31(*$#5$.#?.3$)'(#5$#e("%#X'$15$"&2(.#UKYZ_a[#@#$.#$%&154+#7./"47+#%+)'$#+,$G'(74="#5$#.(#4"7=3"4&(#5$#X4..+@#@#`+*("+#UKYZYa#@#X4..+@;#`+*("+#@#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$#7+-1"47(74="#UX4..+@;#`+*("+#@#\143;#P^^ZaA#!"#$.#,'$%$"&$#('&/71.+#%$#$O,+"$";#$"#1"(#,'4-$'(#%$774="#@#(#-("$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]#UX4..+@#@#`+*("+;#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


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("&$%#01$#.$@$'("#$"#8+N#(.&(#$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$#S@3+&%E@#5$#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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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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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$#.(#$71(74="#$"#&



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R"#1%+#7+-,$&$"&$#4-,45$#01$#$.#1%1('4+#7(43(#$"#74$'G&(%#&$"5$"74(%#7+3"4&48(%;#01$#+)%&'1@$"#.(#,+%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

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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)$)1"&1'Ah#Ug\+'#$%&$#-



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XCJJip;#!A;#ìwFli;#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

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REFERENCIAS BIBLIOGRFICAS


###


VFJBpViSF;#ACA#UKYvjaA#\'+7$%%$%#+>#("(.@%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#UKYZâA#b2$#4"&$','$&(&4+"#+>#&2$#$01(.#%43"k#f@-)+.#>+'#("#$0148(.$"7$#'$.(&4+"#8%A#("#+,$'(&+'#%@-)+.;#$"#V(',.1%;#À#U$5AaA#Proceedings of the Fourth Conference of the Interna-tional Group for the Psychology of Mathematics Education;#,,A#Kx_GKxYA#d$'E$.$@;#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]Â

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#UKYYâA#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

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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

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