EL ESTUDIO TEÓRICO LOCAL DEL DESARROLLO DE COMPETENCIAS ALGEBRAICAS

Resumen. !"#$%&$#&'()(*+#,'$%$"&(-+%#.(%#./"$(%#-($%&'(%#01$#2("#314(5+#"1$%&'+#&'()(*+#$"#6.3$)'(#!517(&48(#$"#.+%#9.&4-+%#8$4"&474"7+#(:+%;#

2(7$-+%#24"7(,4<#$"#.(#,$'%,$7&48(#%$-4=&47(#01$#.+#4">+'-(;#,'$%$"&(-+%#1"#$*$-,.+#5$#("?.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$#2$-+%#+)&$"45+#(#.+#.('3+#5$#$%&+%#(:+%A

Palabras 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 competence

Summary. 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+&&$"#&2'+132+1&#&2$%$#@$('%A Keywords. 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 TEÓRICO LOCAL DEL DESARROLLO DE 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<O47+#

P#L$,('&(-$"&+#5$#L45?7&47(#5$#.(%#B(&$-?&47(%#Q#L$,('&(-$"&+#5$#B(&$-?&47(#!517(&48(

R"48$'%45(5#5$#S(.$"74(#Q#M$"&'+#5$#C"8$%&43(74="#@#5$#!%&154+%#F8("N(5+%;#B<O47+

%--$$>T(+.A7+- .14%A,143T18A$%

&'+*("+T74"8$%&(8A-O

!.#4"&$'<%#,+'#7+"+7$'#$"#,'+>1"545(5#$.#,$"%(-4$"&+#

5$#.+%#$%&154("&$%#01$#%$#4"474("#$"#$.#(,'$"54N(*$#5$.#

?.3$)'(#2(#..$8(5+#(#-172+%#4"8$%&43(5+'$%#(#("(.4N('#

.(%# 4"&$''$.(74+"$%# 5$.# .$"31(*$# (.3$)'(47+# 7+"# $.# .$"G 31(*$#"(&1'(.#@#7+"#$.#5$#.(#('4&-<&47(A#J+#("&$'4+';#$"#

'(N="#5$#01$#,+'#.+#3$"$'(.;#$"#&<'-4"+%#71''471.('$%;#

.(#4"&'+51774="#(.#?.3$)'(#&4$"$#7+-+#("&$7$5$"&$%#-?%#

,'=O4-+%#(#$%+%#5+%#.$"31(*$%;#@#&(-)4<"#,+'01$;#5$%5$#

.+%# ,'4-$'+%# $%&154+%;# %$# 24N+# $845$"&$# .(# $"+'-$# 4"G

#1$"74(#01$#.+%#-4%-+%#&4$"$"#$"#.(#7+"%&'1774=";#,+'#

,('&$#5$#.+%#%1*$&+%;#5$#.(#%4"&(O4%#(.3$)'(47(#@#$"#$.#1%+#

01$# <%&+%# 2(7$"# 5$# 5472(# %4"&(O4%# $"# .(# '$%+.174="# 5$#

,'+).$-(%A

!.#(7$'7(-4$"&+#&$='47+#5$#.+%#modelos locales#(.#$%&154+#

5$#.(#(5014%474="#5$.#.$"31(*$#(.3$)'(47+#4"7+',+'(#$.#$.$G -$"&+# %$-4=&47+# 5$# .+%#sistemas matemáticos de signos#

UV4$'("# W# X4..+@;# KYZY[# \143;# KYY][# \143;# P^^_[# X4..+@;#

`+*("+#W#\143;#P^^Za;#.+#71(.#,$'-4&$#$.#("?.4%4%#5$#.(%#

4"&$''$.(74+"$%#("&$%#-$"74+"(5(%#5$%5$#1"(#,$'%,$7&48(#

01$# 7+"74)$# .(# ('4&-<&47(;# $.# ?.3$)'(# @# .(# .$"31(# -(&$'G

"(#-4%-(#7+-+#%4%&$-(%#5$#%43"+%A#R"#)1$"#"9-$'+#5$#

4"8$%&43(74+"$%# '$(.4N(5(%# (# ,('&4'# 5$# .+%# (:+%# KYZ^# 5("#

71$"&(# 5$# .(%# -("$'(%# $"# 01$# $.# (''(43+# (.# ,$"%(-4$"&+#

"1-<'47+#@#(#.+%#%43"4#7(5+%#7+.+014(.$%#5$#.(%#,(.()'(%#

,$'-$(#.(#4"&$','$&(74="#@#1%+#5$#.(%#.$&'(%#@#5$#.(%#$O,'$G

%4+"$%#(.3$)'(47(%#$"#.(%#$&(,(%#4"474(.$%#5$.#(,'$"54N(*$#

###

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&-<&47(#+#$.#.$"31(*$#"(&1'(.#@#$.#?.3$)'(A#J(#,$'%,$7&4G 8(#%$-4=&47(#(5+,&(5(#$"#.(#&$+'/(#5$#.+%#modelos locales#

$"3.+)(#$"#$.#("?.4%4%#5$#'$.(74+"$%#$"&'$#%4%&$-(%#5$#%43G

"+%#(#.(%#,'+51774+"$%#,'+,4(%#5$#.+%#%1*$&+%#@#<%(#$%#1"(#

7('(7&$'/%&47(#$%$"74(.#5$#5472+#(7$'7(-4$"&+#&$='47+A i&'(# 7('(7&$'/%&47(# 5$# .(# &$+'/(# 5$#modelos locales $%# .(#

,'+,1$%&(# 5$# ("(.4N(';# 5$# -("$'(# .+7(.4N(5(# @# $%,$7/#7(;#

54%&4"&+%#(%,$7&+%#+#7+-,+"$"&$%#5$#.+%#>$"=-$"+%#5$#.(#

-(&$-?&47(#$517(&48(;#7+-+#%+"#.+%#7+-,+"$"&$%#5$#7+-G ,$&$"74(#>+'-(.;#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+#Operación de la incógnita;#,('(#

.+%#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+#Operación de la incógnita# @# $"#

$%&154+%#,+%&$'4+'$%;#$"#$.#7+"&$O&+#5$#.(##'$%+.174="#5$#

,'+).$-(%#(.3$)'(47+%#8$')(.$%A

1. ENTRE EL LENGUAJE ALGEBRAICO Y EL LENGUAJE NATURAL: EPISODIOS DEL ES- TUDIO!"#$%&'()*!+$!,&!(*')-*(.&

!.#$%&154+#Operación de la incógnita;#'$(.4N(5+#(#,'4"74G ,4+%#5$#.+%#(:+%#KYZ^;#7+-,'$"5$#8('4(%#$&(,(%#+#%1)G$%&1G 54+%;#$"#$.#,'4-$'+#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(G 5+%#(#,('&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

-<&47(;#$%#5$74';#"9-$'+%;#%43"+%#5$#+,$'(74=";#%43"+#5$#

431(.5(5#@#.$&'(%A#f4"#$-)('3+;#.+%#%43"4#7(5+%#@#-+5+%#5$#

+,$'(74="#5$#&(.$%#"+&(74+"$%#54#$'$"#5$#1"#7(-,+#(#+&'+A#

!"#$%&(#$&(,(;#%$#,15+#+)%$'8('#01$#.+%#%1*$&+%#&$"/("#(9"#

3'("#54#71.&(5#$"#.(#.$7&1'(#@#$%7'4&1'(#5$#.(%#$O,'$%4+"$%#

(.3$)'(47(%;#.(#71(.#%$#-("4>$%&()(#7+-+#1"(#&$"%4="#$"&'$#

$.#1%+#5$#.+%#%4%&$-(%#5$#%43"+%#5$#.(#('4&-<&47(#UfBf('a

@#5$.#.$"31(*$#"(&1'(.#@#.(#"$7$%45(5#5$#5('.$#%43"4#7(5+

(# .(%# +'(74+"$%# @# %43"+%# (.3$)'(47+%# UX4..+@# W# `+*("+;#

KYYKaA#F##"#5$#4"8$%&43('#.+%#,'+7$%+%#5$#&'(51774="#5$.#

?.3$)'(#(.#.$"31(*$#"(&1'(.#@#847$8$'%(;#$"#%4&1(74="#5$#$"G

&'$84%&(#4"548451(.;#%$#.$%#,$5/(#(#.+%#$%&154("&$%#01$#1%('("#

%/-)+.+%#(.3$)'(47+%#,('(#$O,'$%('#>'(%$%#7+-+#ga#(1-$"G

&(5(#5$#5+%hA#F#7+"&4"1(74="#%$#,'$%$"&(#1"#>'(3-$"&+#5$.#

54?.+3+#$"&'$#1"#$%&154("&$#Ufa#@#$.#$"&'$#84%&(5+'#UCa;#(#,'+G ,=%4&+#5$#$%&(#&('$(k

fk#l+#$"&4$"5+#$%+A

Ck#b$#5("#1"(#+'(74="#@#&$#,45$"#01$#.(#$%7'4)(%#1%("5+#.$&'(%#@#%43"+%

5$#+,$'(74="A

fk#!%#5$74';#%4-)+.4N("5+#(.3+m

Ck#f/;#,$'+#n01<#014$'$%#5$74'#7+"#%4-)+.4N("5+o fk#m

Ck#L(-$#1"(#+'(74="#$"#$%,(:+.#$"#.(#71(.#1%$%#g(1-$"&(5+hA fk#e(%#(1-$"&(5+#.(#8$.+745(5A

Ck#p#$%+#n01<#%43"4#7(o

fk#q1$#.(#,$'%+"(#$%&?#(2+'(;#)1$"+;#@$"5+#-?%#'?,45+A Ck#np#+&'(#+'(74="#$"#$%,(:+.#01$#4"7.1@(#&(-)4<"#g5+%ho

fk#d1$"+m;#@+#7+"&4"1()(#(1-$"&("5+#-4#8$.+745(5#51'("&$#5+%#5/(%A rms

fk#!"#.+%#9.&4-+%#5+%#5/(%;#<.#(1-$"&=#5$#,$%+A rms

fk#f1#,$%+#(1-$"&=#$"#5+%#E4.+%A

!"#$%&$#7(%+;#$.#$%&154("&$#2(7$#1%+#5$#%43"4#7(5+%#&+-(G 5+%#5$.#.$"31(*$#7+.+014(.#,('(#'$%,+"5$'#(#.(#,'$31"&(#5$.#

$"&'$84%&(5+'A#!%#7.('+#01$#%$#'$014$'$#(%43"('.$#%43"4#7(G 5+#(#.(#>'(%$#ga#(1-$"&(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;#$O7$,&+#$"#.(#+'(74="#g<.#(1G -$"&=#%1#,$%+#$"#5+%#E4.+%h;#$"#.(#71(.#%$#'$#$'$#(#1"#,$%+#

4"474(.#Uaa#@#.+%#dos#E4.+%#01$#(1-$"&=A#!"#+&'+#7(%+;#1"#

$%&154("&$#'$%,+"5$#(#.(#-4%-(#,'$31"&(#2(74$"5+#$.#54)1G

*+#01$#(,('$7$#$"#.(##31'(#KA

F01/;#.(#,(.()'(#g(1-$"&(5+h#7.('(-$"&$#"+#%$#(%+74(#(#

.(#(774="#5$#%1-('#@;#$"#7(-)4+;#7+''$%,+"5$#(#.(#(774="#

5$#(3'("5('#+#$O,("54';#(774="#01$#(>$7&(#(#.(#3'(>/(#5$#

.(#.$&'(#g(hA

X431'(#K

J(%#'$%,1$%&(%#$%,+"&?"$(%#5$#.+%#$%&154("&$%#(#,'$31"G

&(%# 7+-+# .(# ("&$'4+'# '$8$.("# 01$# .+%# %43"4#7(5+%# 7+.+G 014(.$%#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;#a^P; (5$-?%#5$#,'+5174'#.$7&1'(%#&$O&1(.$%#7+-+k

ga#-?%#b#%+)'$#5+%h;

(.31"+%#$%&154("&$%#&$"5/("#(#(%43"('#%43"4#7(5+%#3$+-<G

&'47+%#U(.$*(5+%#5$.#?.3$)'(a#(#$%&(%#$O,'$%4+"$%A#!.#%4314$"G

&$#>'(3-$"&+#5$#$"&'$84%&(#4.1%&'(#.+#("&$'4+'A

Ck#J$$#$"#8+N#(.&(#.(#%4314$"&$#$O,'$%4="#r(,1"&("5+#(#.(#,'4-$'(#5$#.(%

$O,'$%4+"$%#$".4%&(5(%#(''4)(sA fk#!.#.(5+#-?%#("72+#%+)'$#5+%A

Ck#p#(2+'(;#%4#5$*(%#5$#,$"%('.(#7+-+#1"(#>='-1.(;#n7=-+#.(#.$$'/(%;#$"#01<

%4&1(74+"$%#&$#.(#2(%#&+,(5+o

fk#d1$"+;#$%#,('(#$"7+"&'('#1"#'$%1.&(5+A Ck#nM+-+#71?.o

fk#d1$"+m

L$%,1<%#5$#1"(#.('3(#,(1%(;#$.#$"&'$84%&(5+'#4"&$'84$"$#

@#2(7$#.(#%4314$"&$#,'$31"&(;#(2+'(#'$#'4<"5+%$#(#.(#$OG ,'$%4="#a t#bA

fk#M+-+#"9-$'+%m;#,+'#$*$-,.+;#j^#t#P^A

Ck#F%/#01$#%4#$%&+#r%$:(.("5+#a t#bs#$%&?#(2/#$"#$.#,4N(''=";#%43"4#7(#

j^#t#P^A

fk#l+;#.+#01$#014$'+#5$74'#$%;#,+5'/(#%43"4#7('#&(-)4<"#(.3+#-?%A Ck#nM+-+#01<o

fk#i&'(#4"7=3"4&(A Ck#nq1<#4"7=3"4&(o

fk#\+'#$*$-,.+;#a 431(.#(;#"+m;#)1$"+;#%4#a $%#431(.#(#P^;#n(#01<#$%

431(.#bo Ck#p#(2/m

fk#J(#4"7=3"4&(#$%#b.

J+#01$#+)%$'8(-+%#$"#$%&$#54?.+3+#$%#01$#$.#$%&154("&$#

&4$"5$#(#4"&$','$&('#.(%#$O,'$%4+"$%#()4$'&(%#7+-+#>='-1G .(%#3$+-<&'47(%#U,+'#$*$-,.+;#g$.#.(5+#-?%#("72+#%+)'$#

5+%ha# +# )4$"# &4$"5$# (# g7$''('.(%h;# @(# %$(# )1%7("5+# 1"#

'$%1.&(5+# ,+'# -$54+# 5$# (%43"('.$%# 8(.+'$%# "1-<'47+%#

,('&471.('$%#(#.(%#.$&'(%;#+#)4$"#7+"84'&4$"5+#1"(#5$#.(%#

.$&'(%# $"# 1"# 5(&+# @# .(# +&'(# $"# 1"(# 4"7=3"4&(A# !%&+# 9.&4G -+#$%#7('(7&$'/%&47+#5$#$O,'$%4+"$%#g7$''(5(%h#7+-+#.(%#

$71(74+"$%A

!"#(-)+%#7(%+%;#.(%#.$&'(%#@#.+%#%43"+%#5$#+,$'(74="#%+"#

.$/5+%#7+"#%43"4#7(5+%#,'+,4+%#5$#.(#('4&-<&47(#5$#.(#,'4G -('4(;#7+-+#%+"#.+%#5$#.(%#>='-1.(%#3$+-<&'47(%#+#.+%#5$#

.(%#$71(74+"$%;#(#,$%('#5$#01$#$.#%43"+#5$#431(.5(5#$%&?#(1G

%$"&$A#!%#5$74';#$.#%1*$&+#g7+-,.$&(h#.(#$O,'$%4="#()4$'&(;#(#

#"#5$#,+5$'#.$$'.(#$"#7+"&$O&+%#01$#.$#%+"#>(-4.4('$%A

!.#$*$-,.+#("&$'4+'#4"547(#$.#&4,+#5$#("&$7$5$"&$%#%$-?"G

&47+%#5$.#f4%&$-(#B(&$-?&47+#5$#f43"+%#5$#.+%#$%&154("G

&$%#5$#$"&'$#KP#@#K]#(:+%#5$#$5(5#@#01$#%+"#.(#)(%$#%+)'$#

.(#01$#5$%(''+..('?"#%1%#7+-,$&$"74(%#(.3$)'(47(%A#M1("G 5+#<%&(%#$%&<"#7+"%+.45(5(%;#.(%#$O,'$%4+"$%#()4$'&(%#U%4"#

%43"+# 5$# 431(.5(5a# 5$"+&('?"# "1$8+%# +)*$&+%# -(&$-?G

&47+%;# .+%# 71(.$%# %$# 1)47("# $"# 1"# "48$.# -?%# ()%&'(7&+# $#

4"8+.17'("#7+"7$,&+%#-?%#3$"$'(.$%#7+-+#$.#5$#"9-$'+#

3$"$'(.#Ua#@#b $"#.(#$O,'$%4="#

###########a#@#$.#5$#+,$'(74="#

%1%,$"545(#U.(#%1-(#$"#a t#baA#\$'+#-4$"&'(%#$%&+#9.&4-+#

%17$5$;#.+%#$%&154("&$%#&'("%4&('?"#5$#.(#$%,+"&("$45(5#5$#

%1%# .$7&1'(%# @# ,'+51774+"$%# $%7'4&(%# 5$# $O,'$%4+"$%# (.G 3$)'(47(%#2(74(#1"(#.$7&+G$%7'4&1'(#(.3$)'(47(#01$#+,$'(#

%$39"#'$3.(%#U%+74(.-$"&$a#$%&().$745(%A#!"#$%$#7(-4"+;#

.+%#%1*$&+%#2()'?"#5$#'$-+"&('#1"(#%$'4$#5$#54#71.&(5$%#

@(#45$"&4#7(5(%;#'$.(74+"(5(%#7+";#,+'#$*$-,.+;#.(#,1$%&(#

$"#$71(74="#5$.#$"1"74(5+#5$#1"#,'+).$-(;#.(#71(.#,'$G

%1,+"$# ,'+7$%+%# 5$# &'(51774="# 5$.# .$"31(*$# "(&1'(.# (.#

?.3$)'(#@#$%#.(#)(%$#5$.#-<&+5+#7('&$%4("+#5$#'$%+.174="#

5$#,'+).$-(%A

2. INTERACCIÓN ÁLGEBRA-LENGUAJE NA- TURAL: ACERCAMIENTOS TEÓRICOS

\+'# .(# 4-,+'&("74(# 5$# 7+"+7$'# (# >+"5+# 7=-+# %$# '$.(G 74+"("#$"&'$#%/#.(#.$"31(#-(&$'"(#@#$.#?.3$)'(#%4-)=.4G 7(;# $"# $.# (,'$"54N(*$# @# .(# $"%$:("N(# 5$# $%&(# 9.&4-(;# %$#

2(# 7+"%45$'(5+# 7+"8$"4$"&$# $%&154('.(# U$.# ?.3$)'(a# $"

%1# 7+"5474="# 5$# .$"31(*$A# R"(# %$'4$# 5$# 4"8$%&43(74+G

"$%# %$# 2("# 5$547(5+# (# $..+# 5$%5$# 54%&4"&(%# ,$'%,$7&48(%#

U8<(%$# `+*("+;# KYY]aA#F%/;# $.# &'()(*+# 5$# L(845# V4'%2G

"$'# $">(&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$#.(#3'(-?&47(#3$"$'(&48(#@#&'("%>+'-(74+"(.#,('(#,'+G 5174'#$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#G 7(74=";#7+"#.(#71(.#'$.(74+"(#.+%#(%,$7&+%#5$#referencia, sentido, interpretación @#connotación;# 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^^KaA

J+%#&'$%#(7$'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 teóricos 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+#

a#t#b P

a#t#b P

###

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+%# UX4..+@;# `+*("+# @# \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$%+%#7+3"4&4G 8+%#01$#&4$"$"#.13('#$"#.(#&'(51774="#5$.#$"1"74(5+#5$#

1"#,'+).$-(#8$')(.#(.#.$"31(*$#(.3$)'(47+A

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

3.1. El método cartesiano como modelo de compe- tencia

M+-+#@(#2$-+%#4"547(5+#$"#.(#4"&'+51774=";#1"(#5$#.(%#

7('(7&$'/%&47(%# 5$.# $">+01$# 5$# .+%# -+5$.+%# &$='47+%# .+G 7(.$%#$%#.(#7+"%45$'(74="#5$#71(&'+#7+-,+"$"&$%#5$#.+%#

-+5$.+%A#\$'+;#(5$-?%;#$%&+%#7+-,+"$"&$%#$%&?"#4-)'4G 7(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&1@$"#$.#7+-,+"$"&$#5$#7+-,$&$"74(#5$#.(#'$%+.174="#

(.3$)'(47(#5$#,'+).$-(%#5$#$"1"74(5+#8$')(.A

J(# 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

"(#$"#.(#24%&+'4(#$.#-<&+5+#5$#'$%+.174="#7+"#7+"&$"45+#

2$1'/%&47+#01$#$%#,$'&4"$"&$#("(.4N('#$"#$%&$#7(%+;#01$#$%#

$.#-<&+5+#7('&$%4("+#UBMa;#@(#$"#.(#,'+,4(#>+'-1.(74="#

5$#L$%7('&$%;#$"#%1%#Regulae ad directionem ingenii#@#$"#

%1#Géometrie;#+#$"#&$O&+%#("&$'4+'$%;#7+-+#2$-+%#2$72+#

$"# \143# UP^^_)a;# \143# @# `+*("+# UP^^]a# @# \143# UP^^xaA#

!"# $>$7&+;# $.# -<&+5+# ,1$5$# ,'$%$"&('%$# >+'-(.-$"&$;#

5$%3.+%(5+#$"#1"(#%$'4$#5$#,(%+%;#01$#$.#'$%+.1&+'#45$(.#

'$7+''$#.4"$(.-$"&$k

Ka#R"(#.$7&1'(#("(./&47(#5$.#$"1"74(5+#5$.#,'+).$-(#01$#

.+#'$517$#(#1"(#.4%&(#5$#7("&45(5$%#@#5$#'$.(74+"$%#$"&'$#

7("&45(5$%A

Pa#!.$774="#5$#1"(#7("&45(5#01$#%$#8(#(#'$,'$%$"&('#7+"#

1"(#.$&'(#U+#5$#1"(%#71("&(%#7("&45(5$%#01$#%$#8("#(#'$G ,'$%$"&('#7+"#.$&'(%#54%&4"&(%aA

_a# `$,'$%$"&(74="# 5$# +&'(%# 7("&45(5$%# -$54("&$# $O,'$G

%4+"$%#(.3$)'(47(%#01$#5$%7'4)$"#.(#'$.(74="#U('4&-<&47(a#

01$# $%(%# 7("&45(5$%# &4$"$"# 7+"# +&'(%# 01$# @(# 2("# %45+#

,'$84(-$"&$#'$,'$%$"&(5(%#,+'#1"(#.$&'(#+#1"(#$O,'$%4="#

(.3$)'(47(A

]a#!%&().$74-4$"&+#5$#1"(#$71(74="#U+#&("&(%#7+-+#.$&'(%#

54%&4"&(%#%$#2(@(#5$74545+#4"&'+5174'#$"#$.#%$31"5+#,(%+a;#

431(.("5+#5+%#$O,'$%4+"$%;#5$#.(%#01$#%$#2("#$%7'4&+#$"#$.#

&$'7$'#,(%+;#01$#'$,'$%$"&$"#.(#-4%-(#7("&45(5A

ja# b'("%>+'-(74="# 5$# .(# $71(74="# $"# 1"(# >+'-(# 7("=G

"47(A

xa#F,.47(74="#5$#.(#>='-1.(#+#(.3+'4&-+#5$#%+.174="#(#.(#

$71(74="#$"#>+'-(#7("="47(A

va#C"&$','$&(74="#5$.#'$%1.&(5+#5$#.(#$71(74="#$"#&<'-4G

"+%#5$.#,'+).$-(A

!.# ,'4-$'# ,(%+;# .(# .$7&1'(# ("(./&47(;# '$,+%(# %+)'$# .(#

7+-,$&$"74(#$"#$.#.$"31(*$#"(&1'(.;#,$'+;#$"#,('&471.(';#

%+)'$#.(#7+-,$&$"74(#$"#$%$#&4,+#$%,$74(.#5$#&$O&+%#-(G

&$-?&47+%#01$#%+"#.+%#,'+).$-(%#('4&-<&47+G(.3$)'(47+%#

5$#$"1"74(5+#8$')(.#U7>A#\143#@#M$'5?";#KYZZa;#$"#.+%#

01$# "+# 2(@# $.4,%4%;# @# 71@(# .$7&1'(# 2(# 5$# 43"+'('# 71(.G 014$'# %43"4#7(5+# 01$# "+# &$"3(# 01$# 8$'# 7+"# .(%# 7("&4G 5(5$%# @# .(%# '$.(74+"$%# 5$# .(%# 01$# 2().(# .(# 24%&+'4(# 5$.#

,'+).$-(A#!"#$%$#,'4-$'#,(%+;#$.#&$O&+#$"#.$"31(*$#"(G

&1'(.#%$#,'$,('(#,('(#%1#&'(51774="#(.#%4%&$-(#5$#%43"+%#

5$.#?.3$)'(;#01$#%=.+#2().(#5$#7("&45(5$%#@#'$.(74+"$%A#

J(#.$7&1'(#("(./&47(#5$%7+-,+"$#$.#&$O&+#5$.#,'+).$-(#

$O,.+'("5+#%1#7(-,+#%$-?"&47+;#7+"84'&4<"5+.+#$"#1"#

"1$8+#&$O&+#$"#$.#01$#%=.+#$%&?"#.(%#7("&45(5$%#@#'$.(G 74+"$%# 01$# ,$'-4&$"# '$%+.8$'# $.# ,'+).$-(;# 7+"# .+# 01$#

$.# "1$8+# &$O&+# '$517$# @# (-,./(# $.# $"1"74(5+;# @(# 01$#

%=.+#&4$"$#7("&45(5$%#@#'$.(74+"$%;#,$'+#$%&(%#7("&45(5$%

@#'$.(74+"$%#,1$5$"#$%&('#-$"74+"(5(%#$O,./74&(-$"&$#

$"#$.#$"1"74(5+;#+#,1$5$#01$#2(@(#01$#$"7+"&'('.(%#$"#

%1#7(-,+#%$-?"&47+A

!"# .+%# ,(%+%# %$31"5+;# &$'7$'+# @# 71('&+;# $%&?"# ,'$%$"G

&$%#&("&+#.(%#7+-,$&$"74(%#5$#,'+7$%+#,'+,4(%#5$.#BM#

7+-+# .(%# 7+-,$&$"74(%# $"# $.# %4%&$-(# 5$# %43"+%# 5$.#

?.3$)'(# 01$# &4$"$"# 01$# 8$'# 7+"# .(# 5$%43"(74="# )41"/G 8+7(#$"&'$#%43"+%#$.$-$"&(.$%#U.$&'(%a#@#7("&45(5$%;#$.#

%43"4#7(5+#5$#.(%#$O,'$%4+"$%#(.3$)'(47(%#7+-,1$%&(%

@# %1%# '$3.(%# 5$# %4"&(O4%# @# .(# >+'-(# 7+-+# <%&(%# %$# '$G

#$'$"#(#7("&45(5$%#,+'#$.#4"&$'-$54+#5$#.(%#'$.(74+"$%#

$"&'$#7("&45(5$%;#@#.+#01$#L$%7('&$%#..(-()(#g$O,'$%('#

1"(# 7("&45(5# 5$# 5+%# -("$'(%# 54>$'$"&$%h^K# UL$%7('&$%;#

Kv^K;#,A#xxa;#01$#$%#.+#01$#5(#%$"&45+#(#.(#7+"%&'1774="#

5$# .(# $71(74=";# @# 7+"%&4&1@$# $.# %43"4#7(5+# (.3$)'(47+#

5$.#%43"+#431(.#$"#1"(#$71(74="A

J+%#,(%+%#014"&+#@#%$O&+#7+"&4$"$"#5+%#7+-,$&$"74(%#$%G

&'47&(-$"&$# (.3$)'(47(%k# $.# 014"&+;# $"# $.# 7?.71.+# .4&$'(.;#

01$;#$"#$%&$#7(%+;#7+"%4%&$#$"#.(#&'("%>+'-(74="#5$#$O,'$G

%4+"$%#(.3$)'(47(%#,('(#..$8('.(%#(#1"(#>+'-(#7("="47(;#

@;#,+'#&("&+;#%$#2(#5$#(7+-,(:('#5$#.(#7+-,$&$"74(#5$#

,'+7$%+#$"#1"#&4,+#5$#("?.4%4%#-$54+%G#"$%[#@#$.#%$O&+;#

$"#.(#%+.174="#5$#$71(74+"$%#U+#%4%&$-(%a#-$54("&$#(.3+G '4&-+%#+#>='-1.(%A

X4"(.-$"&$;# $.# 9.&4-+# ,(%+;# $"# 01$# %$# '$3'$%(# (.# ,'+G ).$-(#8$')(.;#$O43$#.(#7+-,$&$"74(#$"#$.#7+"&$"45+#5$.#

,'+).$-(#01$#,$'-4&$#$8(.1('#.(#(5$71(74="#5$.#'$%1.G

&(5+A

F2+'(#)4$";#$%&(#5$%7'4,74="#5$.#BM#7+-+#1"(#%$'4$#5$#

,(%+%#45$(.$%#$%#1"(#5$%7'4,74="#-(7'+%7=,47(;#01$;#,+'#

&("&+;#"+#(3+&(#$.#7+"*1"&+#5$#.(%#7+-,$&$"74(%#"$7$%(G '4(%#,('(#.(#'$%+.174="#(.3$)'(47(#5$#,'+).$-(%;#(1"01$#

7+"&$"3(# %1%# '(%3+%# -?%# 4-,+'&("&$%A# !.# '$%+.1&+'# &$"G 5'?#01$#.454('#(5$-?%#7+"#.(%#,$71.4('45(5$%#,'+,4(%#5$.#

,'+).$-(#01$#%$#&'(&$#5$#'$%+.8$'A#!%#5$74';#.+%#,'+).$G -(%#,1$5$"#&$"$'#$%&'17&1'(%#54%&4"&(%#01$#2(3("#,'$74%+#

5$%$"7(5$"('# 7+-,$&$"74(%# 54%&4"&(%;# $"# 54%&4"&+# 3'(5+#

+#5$#>+'-(#54%&4"&(A#!.#$%&154+#,1$%#5$#.(#$%&'17&1'(#5$#

.+%#,'+).$-(%#('4&-<&47+G(.3$)'(47+%#5$#$"1"74(5+#8$'G )(.;#5$%5$#$.#,1"&+#5$#84%&(#5$#%1#'$%+.174="#,+'#$.#BM;#

7+"&'4)1@$#(#(-,.4('#.+%#$.$-$"&+%#5$#.(#7+-,$&$"74(#(.G 3$)'(47(A#R"#$%)+N+#5$#$%&$#$%&154+#$%#.+#01$#$O,+"$-+%#

(#7+"&4"1(74="A

3.2. Sobre estructura de los problemas aritmético- algebraicos de enunciado verbal

l1$%&'+# $">+01$# %+)'$# .(# 5$%7'4,74="# 5$# .(# $%&'17&1'(#

5$#$%&(#7.(%$#5$#,'+).$-(%#&4$"$#%1#+'43$"#$"#$.#$%&154+#

5$# .+%# ,'+).$-(%# ('4&-<&47+%# 5$# $"1"74(5+# 8$')(.# 5$#

8('4(%#$&(,(%;#'$(.4N(5+#$"#\143#@#M$'5?"#UKYZZaA#F2/#

%$#$.()+'=#1"(#8$'%4="#5$.#-<&+5+#5$#("?.4%4%#@#%/"&$%4%;#

(5(,&(5(# (# .(# '$%+.174="# 5$# $%(# >(-4.4(# 5$# ,'+).$-(%#

('4&-<&47+%;#01$#%$#1%=#7+-+#2$''(-4$"&(#-$&+5+.=347(#

,('(#5$%7'4)4'#.(#$%&'17&1'(#5$#1"(#%+.174="#('4&-<&47(#

5$# 1"# ,'+).$-(# 71(.014$'(# 5$# $%(# >(-4.4(A#F# ,('&4'# 5$#

$..+;# $"# \143# @# M$'5?"# UKYY^a# %$# 7+-,('=# $.# ,'+7$%+#

5$# '$%+.174="# 5$# ,'+).$-(%# ('4&-<&47+G(.3$)'(47+%# 5$#

$"1"74(5+#8$')(.#-+5$.(5+#,+'#1"(#8$'%4="#5$.#-<&+5+#

5$#("?.4%4%#@#%/"&$%4%;#(5(,&(5(#(#.+%#,'+).$-(%#('4&-<G

&47+%;#7+"#$.#-+5$.(5+#,+'#$.#-<&+5+#7('&$%4("+A#!"#$.#

-+5$.(5+#,+'#$%(#8$'%4="#5$.#-<&+5+#5$#("?.4%4%#@#%/"G

&$%4%;# $.# ("?.4%4%# %$# '$(.4N(# (# ,('&4'# 5$# .(# 4"7=3"4&(# 5$.#

,'+).$-(#@#%$#7+"7.1@$#71("5+#@(#"+#%$#..$3(#(#+&'(%#

7("&45(5$%# 5$%7+"+745(%# U4"7=3"4&(%# (1O4.4('$%a# %4"+#

(# 7("&45(5$%# 7+"+745(%# U5(&+%# 5$.# ,'+).$-(a;# $%&+# $%;#

71("5+#.(#4"7=3"4&(#%$#2(#'$51745+#(#5(&+%A#!.#,'+517&+#

5$.#("?.4%4%#$%#$"&+"7$%#1"#7+"*1"&+#5$#'$.(74+"$%#$"G

&'$#.(%#7("&45(5$%#5$.#,'+).$-(#$".(N(5(%#5$#&(.#-("$'(#

01$#,1$5$"#'$,'$%$"&('%$#$"#>+'-(#5$#1"#54(3'(-(#7+"#

>+'-(# 5$# ?')+.# 01$# 7+"517$# 5$%5$# .(# 4"7=3"4&(# (# .+%#

5(&+%#5$.#,'+).$-(A#J(#%/"&$%4%#7+"%4%&$#$"#$.#'$7+''45+#

5$#$%$#54(3'(-(#$"#%$"&45+#4"8$'%+;#5$%5$#.+%#5(&+%#(#

.(#4"7=3"4&(;#'$(.4N("5+#.(%#+,$'(74+"$%#('4&-<&47(%#7+G ''$%,+"54$"&$%#+;#%4#%$#014$'$;#$%7'4)4$"5+#.(#$O,'$%4="#

('4&-<&47(# 01$# '$%1$.8$# $.# ,'+).$-(A# !%&(# 8$'%4="# 5$.#

-<&+5+#5$#("?.4%4%#@#%/"&$%4%;#,+'#&("&+;#71("5+#%$#1%(#

,('(#'$%+.8$'#$%$#&4,+#5$#,'+).$-(%#@#7+"517$#(#%1#%+G .174=";#.+#2(7$#$"#$.#fBf('#5$#.(#('4&-<&47(A#C.1%&'(-+%#

.+#5472+#7+"#$.#$"1"74(5+#5$#1"#,'+).$-(;#.(#'$,'$%$"G

&(74="#5$#%1#("?.4%4%#$"#1"#54(3'(-(#@#.(#$O,'$%4="#('4&G -<&47(#01$#'$%1.&(#5$#.(#%/"&$%4%A

!.#,'+).$-(k

$/!0123/456!74!/6!84/6!0616!81694:

M1(&'+#,4$N(%#5$#&$.(#5$#j^#-#7(5(#1"(#%$#8("#(#1&4.4N('#

,('(#2(7$'#P^#&'(*$%#01$#"$7$%4&("#_#-$&'+%#5$#&$.(#7(5(#

1"+A#!.#'$%&+#5$#.(#&$.(#%$#1&4.4N('?#,('(#2(7$'#()'43+%A#f4#

,('(#2(7$'#7(5(#()'43+#%$#"$7$%4&(#]#-;#n71?"&+%#()'43+%#

,1$5$"#2(7$'%$o

!.#54(3'(-(#5$#%+.174="k

J(#$O,'$%4="#('4&-<&47(#(#.(#01$#%$#&'(517$#$.#,'+).$-(#

-$54("&$#$%$#("?.4%4%#@#%/"&$%4%k

U]#!#j^#"#P^#!#_a###]

\'+).$-(%#7+-+#<%&$#,1$5$"#'$%+.8$'%$#&(-)4<"#7+"#$.#

1%+#5$.#fBf#5$.#?.3$)'(;#&'(5174$"5+#$.#$"1"74(5+#(#1"(#

$71(74="#@#.1$3+#'$%+.84$"5+#<%&(A#F%/;#%4#%$#5$745$#$"#$.#

,(%+#%$31"5+#5$.#BM#5$%43"('#7+"#.(#.$&'(#x#(#.(#7("&45(5#

g"9-$'+# 5$# ()'43+%h# @;# $"# $.# ,(%+# 71('&+;# $O,'$%('# 5$#

5+%#-("$'(%#54>$'$"&$%#.(#7("&45(5#g&$.(#&+&(.h;#.(#$71(G 74="#01$#'$%1.&(#%$'?#]x#t#P^y_#c#]yj^;#71@(#%+.174="#$%#

,'$74%(-$"&$#.(#$O,'$%4="#('4&-<&47(#5(5(#,+'#.(#%/"&$%4%#

("&$'4+'A#J+#01$#2(7$#01$#.(#%+.174="#U4"547(5(a#5$#$%&(#

$71(74="# 7+4"745(# 7+"# $%(# $O,'$%4="# ('4&-<&47(# $%# 01$#

.(#7(5$"(#5$#+,$'(74+"$%#4"547(5(%#$"#.(#$71(74="#,1$G 5$# 4"8$'&4'%$# (>$7&("5+# .(%# +,$'(74+"$%# 4"8$'%(%# %=.+# (#

7("&45(5$%#7+"+745(%A#L472+#5$#+&'+#-+5+;#$.#7+"*1"&+#

5$# 7("&45(5$%# @# '$.(74+"$%# $O,'$%(5(%# $"# $%(# $71(74="#

,1$5$#'$7+''$'%$#@$"5+#5$%5$#.(#4"7=3"4&(#(#5(&+%#7+-+#

%$#2(7$#$"#$.#("?.4%4%#@#%/"&$%4%A#i)%<'8$%$#01$#$%&(#$71(G 74="#$%#5$#.(%#01$#..(-(-+%#$"#X4..+@#@#`+*("+#UKYZj)a#

g$71(74+"$%#('4&-<&47(%hA

l+# $%# 54>/74.# ,$'7(&('%$# 5$# 01$# .(%# $71(74+"$%# 01$# ..(G -(-+%#g(.3$)'(47(%h;#$%#5$74';#(01$..(%#$"#.(%#01$#.(#4"G 7=3"4&(#(,('$7$#$"#(-)+%#-4$-)'+%#5$#.(#$71(74=";#"+#

,1$5$"#4"8$'&4'%$#5$#.(#-4%-(#-("$'(;#@(#01$#$%#,'$74%+#

+,$'('#7+"#.(#4"7=3"4&(#,('(#'$%+.8$'.(%A#F%/#01$#$.#7+"G

*1"&+#5$#7("&45(5$%#@#'$.(74+"$%#$O,'$%(5(%#$"#$%(#$71(G 74="#"+#,1$5$#'$7+''$'%$#5$%5$#.(#4"7=3"4&(#(#.+%#5(&+%#

7+-+#%$#2(7$#$"#$.#("?.4%4%#@#%/"&$%4%A

"9-$'+#5$#()'43+%

]#-

_#- j^#- P^#&'(*$%

]#,4$N(%

&$.(#&+&(.

&$.(#()'43+%

O O

G t X431'(#K

&$.(#&'(*$%

###

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(%#$"#$%&(#$71(74=";#<%&$#"+#7+"%431$#'$5174'#

.(#4"7=3"4&(#(#5(&+%;#%4"+#01$#%$#81$.8$#(#.(#4"7=3"4&(#(.#

1%('#.(#'$.(74="#01$#7+''$%,+"5$#(#%1#%$31"5(#+71''$"G 74(A# J+# -+%&'('$-+%# '$71''4$"5+# (# 1"# ,'+).$-(# 8$')(.#

$"#1"(#5$#71@(%#%+.174+"$%#(,('$7$#$%(#$71(74="A#F%/#,+G 5'$-+%#"+-)'('#.(%#7("&45(5$%#@#'$.(74+"$%#$O,'$%(5(%#

$"#.(#$71(74="#,+'#%1%#%43"4#7(5+%#$"#$.#7+"&$O&+#5$#.(#

24%&+'4(#01$#"(''(#$.#$"1"74(5+#5$.#,'+).$-(A

!.#,'+).$-(#,'+84$"$#5$#V(.-@E+8(#UKYvja#@#@(#2(#%45+#

1%(5+#$"#\143#@#M$'5?"#UKYZZa#@#\143#UKYYxaA#J+#..(-(G '$-+%#g$.#,'+).$-(#5$.#2$"+hA

$/!0123/456!74/!;4<2

!"#1"(#3'("*(#7+.$7&48(#%$#,'$84=#01$#1"(#74$'&(#7("&45(5#

5$#2$"+#(.-(7$"(5+#,('(#$.#7+"%1-+#5$.#3("(5+#51'('/(#

KYZ#5/(%;#,$'+#$.#2$"+#51'=#PKv#5/(%#@(#01$#$'(#5$#.(#-$*+'#

7(.45(5#@#$.#3("(5+#7+"%1-4=#KvK#E3#-$"+%#,+'#5/(#5$#.+#

01$#%$#2()/(#,'$84%&+#01$#3(%&('/(A#nM1?"&+#2$"+#%$#2()/(#

(.-(7$"(5+#$"#.(#3'("*(o#UV(.-@E+8(;#KYvj;#,?3A#Y^a f4#$%&$#,'+).$-(#%$#&'(517$#(#.(#$71(74=";#

;#$%#,+'01$;#5$#.(#24%&+'4(#01$#"(''(#

$.# $"1"74(5+;# %$# 2("# $O&'(/5+# .(%# 7("&45(5$%# 7+"+745(%#

g5/(%#,'$84%&+%h;#L^,#UKYZa;#5/(%#'$(.$%h;#L_'#UPKva#@#g2$"+#

(2+''(5+#54('4+h;#z-#UKvKa;#.(%#7("&45(5$%#5$%7+"+745(%#

g3(%&+#54('4+#,'$84%&+h;#z_,;#g3(%&+#54('4+#'$(.h;#z';#@#g2$"+#

(.-(7$"(5+h;# \;# @# .(%# '$.(74+"$%# $"&'$# $%(%# 7("&45(5$%#

L^,#!#z,#c#\;#L'#!#z'#c#\#@#z'#t#z-#c#z,A

F2+'(#)4$";#$.#1%+#5$#$%(%#7("&45(5$%#@#'$.(74+"$%#$"#$.#

("?.4%4%#5$#.(#4"7=3"4&(#7+"517$#(#1"+#5$#.+%#5+%#54(3'(G -(%# %4314$"&$%;# 01$# "+# ,1$5$"# (7()('# $"# 5(&+%# ,+'01$#

5$#"1$8+#(,('$7$#.(#4"7=3"4&(;#7+"#.+#01$#$.#("?.4%4%#"+#

,1$5$#7+"7.14'%$#5$#>+'-(#01$#.(#%+.174="#5$.#,'+).$-(#

%$(#1"(#$O,'$%4="#('4&-<&47(#+)&$"45(#,+'#.(#%/"&$%4%A

J+%#54(3'(-(%#5$#.(##31'(#P#%$#7+''$%,+"5$"#7+"#.+%#5+%#

4"&$"&+%# ,+%4).$%# 5$# 5$%,$*('# 7(5(# 1"(# 5$# .(%# +71''$"G

74(%# 5$# .(#x# 4"84'&4$"5+# .(%# +,$'(74+"$%# 4"547(5(%# $"# .(#

$71(74=";#4"&$"&+%#01$#7+"517$"#(##

,$'+# "+# (# .(#x# 431(.(5(# (# 1"(# $O,'$%4="# ('4&-<&47(A# !"#

$>$7&+;#$"#$.#7(%+#5$#$71(74+"$%#7+-+#<%&(;#"+#)(%&(#7+"#

4"8$'&4'#.(%#+,$'(74+"$%#4"547(5(%#,('(#5$%,$*('#.(#4"7=3G

"4&(;#%4"+#01$#$%#,'$74%+#(5$-?%#+,$'('#7+"#.(#4"7=3"4&(A f4#$.#,'+).$-(#5$.#2$"+;#$"#8$N#5$#'$%+.8$'%$#7+"#$.#BM#

U@;#,+'#&("&+;#$.#fBf(.#5$.#?.3$)'(a;#%$#21)4$'(#4"&$"&(5+#

'$%+.8$'#7+"#$.#-<&+5+#5$#("?.4%4%#@#%/"&$%4%#U@#$.#fBf^('#5$#

.(#('4&-<&47(a#@#$"#$.#("?.4%4%#%$#21)4$'("#$%&().$745+#,'$G 74%(-$"&$#.(%#'$.(74+"$%#L,#!#z,#c#\;#L'#!#z'#c#\#@#z'#t#z-# c#z,;#$"&+"7$%#$.#,'+).$-(#"+#21)4$'(#,+545+#'$%+.8$'%$#

,+'01$#$%$#("?.4%4%#"+#,$'-4&$#'$5174'#.(#4"7=3"4&(#(#5(&+%A#

J+%#54(3'(-(%#5$#.(##31'(#P#.+#-1$%&'("#7+"#7.('45(5A J(#54>$'$"74(#>1"5(-$"&(.#$"&'$#$.#("?.4%4%#5$.#$"1"74(G 5+#,'+,4+#5$.#-<&+5+#5$#("?.4%4%#@#%/"&$%4%#@#$.#5$.#BM#

'$%45$#$"#01$#$.#$%)+N+#.=347+G%$-4=&47+#01$#1"+#'$(.4N(#

71("5+# 1%(# $.# BM# ,'$8<# $.# 1%+# 5$.# fBf(.# 5$.# ?.3$)'(A#

!%&+#7+"..$8(#"+#%=.+#$.#1%+#5$#.$&'(%#,('(#5$%43"('#.(%#

7("&45(5$%#01$#%$#5$&$'-4"("#$"#$.#("?.4%4%;#%4"+#.+%#"1$G 8+%#%43"4#7(5+%#5$#.(%#+,$'(74+"$%#@#.(%#'$.(74+"$%#('4&G -<&47(%#@;#$"#,('&471.(';#5$.#%43"+#431(.#01$#%+"#,'+,4+%#

5$#$%$#fBf(.A#`$%1.&(#$"&+"7$%#01$#$%$#("?.4%4%#%$#2(7$#

7+"%45$'("5+#5$#.(#-4%-(#-("$'(#.(%#7("&45(5$%#7+"+G 745(%#@#.(%#5$%7+"+745(%A#\+'#$.#7+"&'('4+;#$.#("?.4%4%#5$.#

$"1"74(5+#,'+,4+#5$.#-<&+5+#5$#("?.4%4%#@#%/"&$%4%#%$#5$G

%(''+..(#%4&1?"5+%$#$"#.(#4"7=3"4&(#5$.#,'+).$-(#@#7+"%4G 5$'("5+#7+"#01<#5(&+%#2()'/(#01$#+,$'('#,('(#+)&$"$'.(;#

@#$"#$.#$%)+N+#.=347+G%$-4=&47+#"+#%$#7+"&$-,.(#.(#,+%4G )4.45(5#5$#+,$'('#-?%#01$#7+"#7("&45(5$%#7+"+745(%A J+%#54(3'(-(%#("&$'4+'$%;#01$#'$#$*("#1"(#%+.174="#-+G 5$.(5(#,+'#$.#-<&+5+#5$#("?.4%4%#@#%/"&$%4%;#"+#%+"#(5$G 71(5+%#,('(#5('#71$"&(#5$#.(#.$7&1'(#("(./&47(#,'+,4(#5$.#

BMA#\+'#$..+;#,('(#5$%7'4)4'#.(#$%&'17&1'(#5$#.+%#,'+).$G -(%#5$%5$#$.#,1"&+#5$#84%&(#5$#%1#'$%+.174="#(.3$)'(47(#

8/(#$.#BM;#2$-+%#$.()+'(5+#+&'+#&4,+#5$#54(3'(-(#7+"#

>+'-(#5$#3'(>+;#(5(,&(5+#5$##X'45-("#UKYY^aA

#!%+%#3'(>+%#'$,'$%$"&("#.(#.$7&1'(#("(./&47(#5$.#$"1"74(G 5+# 5$# 1"# ,'+).$-(# 8$')(.# ('4&-<&47+G(.3$)'(47+# ,'+,4(#

5$.#BM#,+'01$#%1%#8<'&47$%#'$,'$%$"&("#7("&45(5$%#@#%1%#

('4%&(%;#'$.(74+"$%#$"&'$#7("&45(5$%;#5$#-+5+#01$#$.#3'(>+#

-1$%&'(#.(#'$5#5$#'$.(74+"$%#$"&'$#7("&45(5$%#01$#%$#2(#

5$&$'-4"(5+#$"#$%(#.$7&1'(#("(./&47(A#F5$-?%;#$"#$.#3'(>+#

%$#54%&4"31$"#.(%#7("&45(5$%#7+"+745(%#5$#.(%#5$%7+"+74G 5(%#'$,'$%$"&("5+#-$54("&$#7/'71.+%#"$3'+%#.+%#8<'&47$%#

7+''$%,+"54$"&$%#(#.(%#7("&45(5$%#7+"+745(%#U.+%#..(-(G '$-+%#g8<'&47$%#+%71'+%ha#@#-$54("&$#71(5'(5+%#7+"#$.#

4"&$'4+'# 8(7/+# .+%# 8<'&47$%# 7+''$%,+"54$"&$%# (# .(%# 7("&4G 5(5$%#5$%7+"+745(%#U.+%#..(-('$-+%#g8<'&47$%#7.('+%haA#

M+-+#.(%#71(&'+#+,$'(74+"$%#('4&-<&47(%#)?%47(%#%+"#)4G

"('4(%;#.(%#'$.(74+"$%#7+''$%,+"54$"&$%#%+"#&$'"('4(%;#5$#

-("$'(#01$#$"#.+%#,'+).$-(%#('4&-<&47+G(.3$)'(47+%#-?%#

7+''4$"&$%#.(%#('4%&(%#&4$"$"#&'$%#8<'&47$%^PA

F%/;#.(#.$7&1'(#("(./&47(#5$.#,'+).$-(#5$.#2$"+#U&+-(5+#

7+-+# $%,(74+# &$O&1(.a# 01$# ,'+517$# .(%# 7("&45(5$%# 7+G x

PKv $#KvK#% x KYZ

x

PKv $#KvK#% x KYZ

x

PKv $#KvK#! KYZ#% x##@##

U a U

a

X431'(#P

z,

z'

\ L'

z-

L,

\

z'

z,

\

\

L,

z-

L'

# #

! !

$ "

"+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

J(#.$7&1'(#("(./&47(#5$.#,'+).$-(#5$#.(#&$.(#,('(#&'(*$%#$"#

.(#01$#%$#5$&$'-4"("#.(%#7("&45(5$%#@#'$.(74+"$%#01$#("G

&$'4+'-$"&$# 2$-+%# '$,'$%$"&(5+# -$54("&$# 1"# 54(3'(-(#

,1$5$#'$,'$%$"&('%$#&(-)4<"#-$54("&$#$.#3'(>+#5$#.(##G 31'(#]A#!"#$..(;#.(%#7("&45(5$%#7+"+745(%#g%+"#"9-$'+#5$#

,4$N(%#5$#&$.(h;#l,#U]a;#g"9-$'+#5$#&'(*$%h;#l&#UP^a;#g&$.(#

,+'#()'43+h#z(#U]#-a;#g&$.(#,+'#,4$N(h;#z,#Uj^#-a#@#g&$.(#,+'#

&'(*$h#z&#U_#-a[#.(%#7("&45(5$%#5$%7+"+745(%#g"9-$'+#5$#

()'43+%h;#l(;#g&$.(#5$#.+%#()'43+%h;#b(;#g&$.(#5$#.(%#,4$N(%#

+#&$.(#&+&(.h;#b,;#@#g&$.(#5$#.+%#&'(*$%h;#b&;#@#.(%#'$.(74+"$%#

l4#!#z4#c#b4;#4#!#{(;#,;#&|;#b,#c#b(#t#b&A

!"#$%&+%#3'(>+%#(5$-?%#$%#,(&$"&$#,+'#01<#1"(#'$5#5$#'$.(74+G

"$%#,$'-4&$#01$#%$#+)&$"3(#.(#%+.174="#5$.#,'+).$-(#-$54("&$#

$.#1%+#5$.#fBf_('#5$#.(#('4&-<&47(#@#.(#+&'(#"+A#!"#$>$7&+;#,('(#

01$#"+#2(@(#01$#+,$'('#7+"#7("&45(5$%#5$%7+"+745(%#$%#,'$G 74%+#01$#$"#1"(#('4%&(#&$'"('4(#5+%#5$#.+%#8<'&47$%#%$("#+%71'+%#

U%$("#7("&45(5$%#7+"+745(%a;#$"&+"7$%#$.#8<'&47$#7.('+#%$#,1$G 5$#7+"8$'&4'#$"#+%71'+#U.(#7("&45(5#5$%7+"+745(#7(.71.('%$#(#

,('&4'#5$#7("&45(5$%#7+"+745(%a#'$(.4N("5+#.(#+,$'(74="#('4&G -<&47(#7+''$%,+"54$"&$A#J(#4"7=3"4&(#5$.#,'+).$-(#%$#,1$5$#

+)&$"$'#(#,('&4'#5$#.+%#5(&+%#%4$-,'$#01$#2(@(#1"(#-("$'(#5$#

4'#+%71'$74$"5+#.+%#8<'&47$%#7.('+%#%17$%48(-$"&$#2(%&(#..$3('#

(#<.A#!"#$.#3'(>+#7+''$%,+"54$"&$#(#.(#.$7&1'(#("(./&47(#5$.#,'+G ).$-(#5$#.(#&$.(#,('(#&'(*$%#$%&$#7(-4"+#$O4%&$;#@#7+4"745$#7+"#

$.#7(-4"+#01$#5$%7'4)$#$.#54(3'(-(#5$#("?.4%4%#@#%/"&$%4%#5$#.(#

#31'(#KA#!"#$.#5$#.(#.$7&1'(#5$.#,'+).$-(#5$.#2$"+;#$.#7(-4"+#

"+#,1$5$#$O4%&4'#,+'01$#"+#2(@#"4"31"(#('4%&(#01$#&$"3(#5+%#

8<'&47$%#+%71'+%A#`$%1.&(#7+2$'$"&$;#,1$%;#7+"#.(#&$'-4"+.+3/(#

("&$'4+'-$"&$#4"&'+51745(;#..(-('#g('4&-<&47+%h#(#.+%#3'(>+%#

01$#7+-,('&$"#7+"#$.#5$#.(##31'(#]#.(#,'+,4$5(5#01$#2$-+%#

$O,.47(5+#@#g(.3$)'(47+%h#(#.+%#01$#"+#.(#&4$"$"#U7+-+#$.#5$#

.(##31'(#_aA

J+%# 3'(>+%# '$,'$%$"&("# ,1$%# .(# .$7&1'(# ("(./&47(# 5$# .+%#

$"1"74(5+%# 5$# .+%# ,'+).$-(%# 71("5+# %$# '$%1$.8$"# -$G 54("&$#$.#1%+#5$.#BM;#,$'+#$%&(#.$7&1'(#("(./&47(#$%#%=.+#$.#

,'4-$'#,(%+#5$.#-<&+5+A#\('(#+)&$"$'#.(#$71(74="#

]x#t#P^y_#c#]yj^#+#.(#$71(74="#########################################;#

2(@#01$#'$7+''$'#+&'+%#&'$%#,(%+%;#.+%#,(%+%#%$31"5+;#&$'G 7$'+#@#71('&+#5$.#BMA

!"#.(%##31'(%#j;#x#@#v#-+%&'(-+%#7=-+#$%+%#,(%+%#&(-G )4<"#,1$5$"#'$,'$%$"&('%$#$"#.+%#3'(>+%A

f$31"5+#,(%+

b$'7$'#,(%+

M1('&+#,(%+

X431'(#j

X431'(#x

X431'(#v

x

PKv $#KvK#% x KYZ

X431'(#_

Dp

Dr

$p

$m

$r

P

Na

Nt

$a Ta

$t

Np $p

Tt

Tp X431'(#]

###

i)%<'8$%$# 7=-+;# %4# $.# 3'(>+# $%# ('4&-<&47+;# .(# $71(74="#

01$#%$#+)&4$"$#$%#('4&-<&47(#@;#%4#$.#3'(>+#$%#(.3$)'(47+;#

(.3$)'(47(A#b(-)4<"#2$-+%#-+%&'(5+#01$#.+%#54(3'(-(%#

5$#("?.4%4%#@#%/"&$%4%#7+''$%,+"54$"&$%#,'+517$"#1"(#%+G .174="#('4&-<&47(#+#"+#.(#,'+517$"A#F2+'(#)4$";#$%&+#"+#

014$'$# 5$74'# 01$# .+%# ,'+).$-(%# 7+''$%,+"54$"&$%# ,+5(G -+%#7(.4#7('.+%#5$#g('4&-<&47+%h#U$.#5$#.(#&$.(#,('(#&'(G

*$%a#@#5$#g(.3$)'(47+%h#U$.#5$.#2$"+aA#!"#$>$7&+;#.+#01$#$"#

$.#7(%+#5$.#,'+).$-(#5$.#2$"+#"+#$%#('4&-<&47+#$%#1"#,'+G 7$%+#5$#'$%+.174="#7+"51745+#,+'#$.#-<&+5+#5$#("?.4%4%#@#

%/"&$%4%#U$.#01$#'$,'$%$"&(#$.#54(3'(-(#5$#("?.4%4%#@#%/"&$G

%4%a#@#1"(#.$7&1'(#("(./&47(#01$#7+"%&4&1@$#$.#,'4-$'#,(%+#

5$.# BM# U.(# 01$# '$,'$%$"&(# $.# 3'(>+a[# ,$'+# ,1$5$# 7()$'#

01$#$O4%&(#+&'+#,'+7$%+#5$#'$%+.174="#1#+&'(#.$7&1'(#("(./G

&47(#01$#5$&$'-4"$#+&'(#'$5#5$#'$.(74+"$%#$"&'$#7("&45(5$%#

01$#%/#%$(#('4&-<&47(A#p#"+#%=.+#7()$#01$#%$(#(%/;#%4"+#01$#

(%/#$%#$.#7(%+k#%4#.(#.$7&1'(#("(./&47(#5$&$'-4"(;#(5$-?%#

5$#.(%#7("&45(5$%#("&$'4+'$%;#.(%#7("&45(5$%#5$%7+"+745(%#

g5/(%#5$#-?%h;#L^-;#g3(%&+#$"#.+%#5/(%#5$#-?%h;#zL-;#@#

g(2+''+#&+&(.h;#F_&;#@#.(%#'$.(74+"$%#"1$8(%#L,#!#z-#c#F&;#

L-#!#z'#c#zL-;#L,#L-#c#L'#@#zL-#c#F&;#$.#3'(>+#7+''$%G ,+"54$"&$#$%#('4&-<&47+;#7+-+#,1$5$#8$'%$#$"#.(##31'(#Z;#

@#.(#4"7=3"4&(#%$#5$&$'-4"(#7+"#.(#$O,'$%4="#('4&-<&47(;#

############################################! KYZ#01$#%$#+)&4$"$#(.#'$7+''$'#%17$%4G 8(-$"&$#.(%#('4%&(%#UL,;#L';#L-a;#UL,;#z-;#F&a;#UF&;#zL-a#

UL^-;#z^';#z^L-a;#Uz^,;#z^';#z^-a#@#UL^,;#z^,;#\a#7(.71.("5+#$"#7(5(#

,(%+#1"(#7("&45(5#5$%7+"+745(#(#,('&4'#5$#5+%#7+"+745(%A

J+#01$#,1$5$#7(.4#7('%$#5$#('4&-<&47+#+#(.3$)'(47+#$%;#

,+'# &("&+;# $.# ,'+7$%+# 5$# '$%+.174="# U'$,'$%$"&(5+# ,+'#

$.# 54(3'(-(# 5$# ("?.4%4%# @# %/"&$%4%a;# .(# .$7&1'(# ("(./&47(#

U'$,'$%$"&(5(#,+'#$.#3'(>+a#+#.(#$71(74="#01$#&'(517$#$.#

$"1"74(5+;#,$'+#"+#$.#,'+).$-(A

\$'+# (5$-?%# 5$# ,$'-4&4'# 54%71&4'# 5$# $%&(# -("$'(# $.# 7(G '?7&$'#('4&-<&47+#+#(.3$)'(47+#5$#.+%#,'+).$-(%;#$.#3'(>+#

'$,'$%$"&(# 1"(# .$7&1'(# ("(./&47(# ,+%4).$# 5$.# ,'+).$-(;#

01$;#(.#7+"%4%&4'#$"#1"(#'$5#5$#7("&45(5$%#@#'$.(74+"$%;#

'$,'$%$"&(#1"(#$%&'17&1'(#5$.#,'+).$-(A#L$%5$#$%$#,1"&+#

5$#84%&(;#.(%#54%&4"&(%#$%&'17&1'(%#01$#4.1%&'("#.+%#3'(>+%#

'$#$*("# .(# 7+-,.$*45(5# 5$# .(# 7+-,$&$"74(# 01$# 2(@# 01$#

,+"$'#$"#*1$3+#,('(#'$%+.8$'#.+%#,'+).$-(%#-$54("&$#$.#

-<&+5+#7('&$%4("+A#M$'5?"#UP^^Za#2(#1&4.4N(5+#1"(#8$'G

%4="#.43$'(-$"&$#54%&4"&(#5$#$%&+%#3'(>+%#,('(#$%&154('#.+%#

,'+).$-(%#5$#$%&(#>(-4.4(#('4&-<&47+G(.3$)'(47(;#$%&().$G 7$'#4%+-+'#%-+%#$"&'$#$..+%;#5$%7'4)4'#.(#7+-,.$*45(5#5$#

.(#$%&'17&1'(#5$#.(#'$5#5$#'$.(74+"$%#@#'$.(74+"('#&+5+#$..+#

7+"#.(%#54#71.&(5$%#01$#.+%#,'+).$-(%#&4$"$"#71("5+#.+%#

'$%1$.8$"#.+%#(.1-"+%#@#.+%#$''+'$%#01$#7+-$&$"A

3.3. Sobre el proceso de traducción y el trabajo en el nivel de la expresión

!%#2()4&1(.#'$>$'4'%$#(.#-<&+5+#7('&$%4("+#7+-+#$.#,'+7$%+#

5$#,+"$'#1"#,'+).$-(#$"#$71(74+"$%#+#7+-+#1"#,'+7$%+#5$#

&'(51774="#5$.#$"1"74(5+#5$.#,'+).$-(#$"#.$"31(*$#"(&1'(.#

(.#.$"31(*$#5$.#?.3$)'(#$"#01$#$%&?#.(#$71(74="A#p(#2$-+%#

84%&+#01$#$.#BM#"+#%$#.4-4&(#(#$%(#&'(51774=";#%4"+#01$#

&(-)4<"#7+"&4$"$;#,+'#1"#.(5+;#.(#.$7&1'(#("(./&47(;#01$#,'$G ,('(#$.#&$O&+#5$.#,'+).$-(#$.()+'("5+#+&'+#&$O&+#01$#$"#

74$'&(#-("$'(#$%&?#,'$,('(5+#,('(#%$'#&'(51745+#(.#.$"31(*$#

5$.# ?.3$)'(;# @;# ,+'# +&'+;# 1"# &'()(*+# $"# $.# "48$.# 5$# .(# $OG ,'$%4="#$"#$.#.$"31(*$#5$.#(.3$)'(#01$#&'("%>+'-(#$.#&$O&+#

&'(51745+#U.(#$71(74="a#(#+&'+#&$O&+#01$#%$#%()$#'$%+.8$'#U.(#

$71(74="#$"#>+'-(#7("="47(a[#7+"#$.#(:(545+#5$#.(#81$.&(#

(.# -1"5+# 5$.# $"1"74(5+# 5$.# ,'+).$-(;# 7+"# $.# '$%1.&(5+#

+)&$"45+#$"#$.#-1"5+#5$.#.$"31(*$#(.3$)'(47+A

!"#.(#'$%+.174="#(.3$)'(47(#5$#,'+).$-(%#5$#$"1"74(5+#

8$')(.# 2(@# 4-,.47(5+%;# ,+'# &("&+;# 5+%# &4,+%# 5$# ,'+517G 74="#5$#&$O&+%#(.3$)'(47+%A#!.#,'4-$'+#$%#$.#01$#,+5$-+%#

%$314'#..(-("5+#g&'(51774="h#(#7+"5474="#5$#01$#&$"3(G -+%#,'$%$"&$#01$#$.#,'+7$%+#01$#(01/#%$#'$(.4N(#$%#$.#,(%+#

$"&'$#5+%#.$"31(*$%#01$#%+"#54%&4"&+%#"+#%=.+#,+'#.(#"(&1G '(.$N(#5$#%1%#%43"+%;#%4"+#,+'01$#$.#.$"31(*$#5$.#?.3$)'(#

%=.+#&4$"$#7+-+#7+"&$"45+#$.#-1"5+#5$#.(%#7("&45(5$%#@#

.(%# '$.(74+"$%# ('4&-<&47(%# $"&'$# $..(%A# L$%5$# $%&$# ,1"&+#

5$#84%&(;#.(#.$7&1'(#("(./&47(#>+'-(#,('&$#5$.#,'+7$%+#5$#

&'(51774="#7'$("5+#1"#&$O&+#4"&$'-$54+#$"#.$"31(*$#"(G

&1'(.;#71@+#7+"&$"45+#$%#U7(%4a#%=.+#1"(#'$5#5$#7("&45(5$%#

@#'$.(74+"$%A#gM(%4h#,+'01$#.(#$O,'$%4="#5$#.+%#"+-)'$%#

5$#.(%#7("&45(5$%#$"#.$"31(*$#"(&1'(.#7+"..$8(#+&'+%#%43G

"4#7(5+%#01$#.(%#,1'(%#7("&45(5$%A

J(#&'(51774="#%$#'$(.4N(#(%43"("5+#"+-)'$%#(#.(%#7("&45(G 5$%#$"#$.#.$"31(*$#5$.#?.3$)'(#-$54("&$#%43"+%#4"548451(G .$%#U.$&'(%a;#71@(#'$.(74="#7+"#.(#7("&45(5#(#.(#01$#"+-)'(#

%$#$%&().$7$#5$#>+'-(#(')4&'('4(#$"#$.#%$31"5+#,(%+#5$.#

-<&+5+;#+#-$54("&$#$O,'$%4+"$%#7+-,1$%&(%#01$#&'(51G 7$"#.(%#'$.(74+"$%#$"&'$#7("&45(5$%#(#+,$'(74+"$%#$O,'$G

%(5(%# $"# $.# .$"31(*$# 5$.# ?.3$)'(;# $"# $.# &$'7$'# ,(%+# 5$.#

BMA#J(%#$O,'$%4+"$%#(.3$)'(47(%#%4-,.$%#@#7+-,1$%&(%#

01$#'$%1.&("#5$#.(#&'(51774="#%$#'$#$'$"#,1$%#(#.(%#7("&4G 5(5$%#7+"7'$&(%#5$.#,'+).$-(;#7+-+#7+"%$71$"74(#5$#%$'#

$.#'$%1.&(5+#5$#$%(#&'(51774=";#"+'-(.-$"&$#(#&'(8<%#5$#

1"#&$O&+#4"&$'-$54+A#f4"#$-)('3+;#,('(#01$#,1$5(#5$%$"G 7(5$"('%$#.(#&'("%>+'-(74="#5$#.(#$71(74="#$"#1"(#>+'G -(#7("="47(;#$%#,'$74%+#01$#.(%#$O,'$%4+"$%#(.3$)'(47(%#

7+-,1$%&(%#5$*$"#5$#'$>$'4'%$#(#$%(%#7("&45(5$%#7+"7'$G

&(%A#B4$"&'(%#$%&<"#,$3(5(%#(#$..(%^_;#.(%#$O,'$%4+"$%#@#.(%#

$71(74+"$%#01$#%$#,'+51N7("#(,.47("5+#'$3.(%#5$#&'("%G KYZ#! KvK

PKv#" KYZ $#KvK#

U a

$Dm

Dp

Dr

Dm

$p

$m

$r

At

P

, X431'(#Z

>+'-(74="#"+#,('$7$'?"#'$>$'4'%$#(#"(5(A#J(%#$O,'$%4+"$%#

(.3$)'(47(%# ,'+517&+# 5$.# ,'+7$%+# 5$# &'(51774="# 2("# 5$#

5$%,'$"5$'%$#5$#.(%#7("&45(5$%#@#'$.(74+"$%#7+"7'$&(%#5$.#

-1"5+#5$.#,'+).$-(;#"+#,('(#7+"8$'&4'%$#$"#,1'+%#%43"+%#

01$# %$# -("$*("# %4"&?7&47(-$"&$# %4314$"5+# '$3.(%;# %4"+#

,('(#'$>$'4'%$#(#.(%#7("&45(5$%#@#'$.(74+"$%#()%&'(7&(%#5$.#

-1"5+#5$.#?.3$)'(A

!%&+#$%#(%/#,+'01$#1"(#$O,'$%4="#(.3$)'(47(;#4"7.1%+#%4#

&'(517$# $.# $"1"74(5+# 5$# 1"# ,'+).$-(# 8$')(.;# "+# &4$"$#

7+-+#7+"&$"45+#$.#$%&(5+#5$#7+%(%#5$.#-1"5+#(.#01$#%$#

'$#$'$#$.#$"1"74(5+#5$.#,'+).$-(#8$')(.k#<%$#$%#$.#7+"G

&$"45+#5$#.(#$O,'$%4="#01$#$%#$.#&$O&+#5$.#$"1"74(5+#5$.#

,'+).$-(# 8$')(.A# J(# $O,'$%4="# (.3$)'(47(# &4$"$# 7+-+#

7+"&$"45+# $.# $%&(5+# 5$# 7+%(%# 5$.# -1"5+# 5$.# ?.3$)'(# (.#

01$#%$#'$#$'$;#$%&+#$%;#1"#7+"*1"&+#5$#7("&45(5$%#()%&'(7G

&(%#@#1"#7+"*1"&+#5$#'$.(74+"$%#U('4&-<&47+G(.3$)'(47(%a#

$"&'$#$..(%A

F5$-?%;#$"&'$#.(#$O,'$%4="#(.3$)'(47(#@#%1#7+"&$"45+#2(@#

7+''$%,+"5$"74(# )41"/8+7(A# !%+# 2(7$# 01$# %$(# 1"# 47+"+;#

$"#&<'-4"+%#5$#\$4'7$#U\$4'7$;#MA\A#PPvY;#,AKjZa;#@#01$;#

,+'# &("&+;# g.(%# +,$'(74+"$%# 01$# %$# ..$8("# (# 7()+# $"# .(#

$O,'$%4="#-+54#7("#$.#7+"&$"45+[#@#%4#$%&(%#+,$'(74+"$%#

%$# ..$8("# (# 7()+# 7+">+'-$# (# 74$'&(%# '$3.(%;# $.# '$%1.&(5+#

,'+,+'74+"(# "1$8(%# 4">+'-(74+"$%# %+)'$# $.# 7+"&$"45+h#

U!7+;#KYY^;#,A#P]a;#$"#,('&471.(';#$"#$.#7(%+#01$#"+%#+71,(;#

"1$8(%#'$.(74+"$%#$"&'$#.(%#7("&45(5$%#01$#$8$"&1(.-$"&$#

5('?"#1"#8(.+'#"1-<'47+#(#.(#4"7=3"4&(#5$#.(#$71(74="#U+#

.(%#4"7=3"4&(%#5$#.(%#$71(74+"$%aA#\+'#$.#7+"&'('4+;#"+#2(@#

7+''$%,+"5$"74(# )41"/8+7(# $"&'$# .(# $O,'$%4="# $"# $.# .$"G 31(*$# "(&1'(.# @# %1# 7+"&$"45+;# %4$"5+# .(%# $O,'$%4+"$%# 5$.#

.$"31(*$#"(&1'(.#%/-)+.+%;#$"#&<'-4"+%#5$#\$4'7$A#F-)(%#

$O,'$%4+"$%#@#(-)+%#7+"&$"45+%#$%&?"#'$.(74+"(5+%#,+'#$.#

,'+7$%+#5$#&'(51774=";#@;#7+-+#2$-+%#5472+;#(#-$"15+#(#

&'(8<%#5$#&$O&+%#4"&$'-$54+%A

!.#%$31"5+#&4,+#5$#,'+51774="#5$#&$O&+%#(.3$)'(47+%#$%#

$.# 01$# %$# '$(.4N(# $"&'$# 1"# &$O&+# (.3$)'(47+# @# +&'+# &$O&+#

(.3$)'(47+# -$54("&$# &'("%>+'-(74+"$%# (.3$)'(47(%;# 01$#

%$#'$(.4N("#$"#$.#"48$.#5$#.(#$O,'$%4="A#!"#$.#7(%+#5$#.(#

'$%+.174="#(.3$)'(47(#5$#,'+).$-(%;#$%&(#,'+51774="#5$#

"1$8+%#&$O&+%#(.3$)'(47+%#$%#5$#.+#01$#%$#+71,(#$.#014"&+#

,(%+#5$.#BM#@#&4$"$#7+-+#+)*$&48+#,+"$'#.(#$71(74="#$"#

>+'-(#7("="47(A

\('(# $..+# 2(7$# >(.&(# 01$# .(%# $O,'$%4+"$%# (.3$)'(47(%# @(#

"+#%$#'$#$'("#(#.(%#7("&45(5$%#,('&471.('$%#5$.#$"1"74(5+#

5$.# ,'+).$-(;# %4"+# 01$# (5014$'("# "1$8+# %43"4#7(5+# $"#

$.#-1"5+#5$.#?.3$)'(#5$#7("&45(5$%#@#'$.(74+"$%A#!"#$%$#

-1"5+;#.(%#'$3.(%#5$#&'("%>+'-(74="#5$#.(%#$O,'$%4+"$%#

@#$71(74+"$%#(5014$'$"#%1#%43"4#7(5+#7+-+#'$%1.&(5+#5$#

&$+'$-(%#01$#$%&().$7$"#01$;#%4#74$'&(%#'$.(74+"$%#$"&'$#

7("&45(5$%# $%&?"# 5(5(%;# +&'(%# '$.(74+"$%# &(-)4<"# $%&?"#

5(5(%A

\$'+#(5$-?%;#$"#$.#7+"&$O&+#5$#.(#'$%+.174="#(.3$)'(47(#

5$#,'+).$-(%;#.(%#&'("%>+'-(74+"$%#%$#5+&("#5$#%$"&45+#

7+-+#7+"%$71$"74(#5$#.(#7+"%&(&(74="#5$#01$#71(.014$'#

$71(74="# ,1$5$# &'("%>+'-('%$# $"# 1"# 7+"*1"&+# '$51745+#

5$# $71(74+"$%# 01$# %$# %()$# '$%+.8$'# U.(%# >+'-(%# 7("=G

"47(%a;# @# .1$3+# $%# ,+%4).$# 8+.8$'# (# .+%# %43"4#7(5+%# 5$.#

,'4-$'#-+5+#5$#,'+51774="#5$#&$O&+%#(.3$)'(47+%#U.(#&'(G

51774="a#,('(#'$71,$'('#.+%#5$.#-1"5+#5$.#$"1"74(5+#5$.#

,'+).$-(#U$"#$.#,(%+#%<,&4-+aA

J(%#54#71.&(5$%#5$#.+%#(.1-"+%#&4$"$"#01$#8$'#&(-)4<"#

7+"#$.#,(%+#$"&'$#.+%#5+%#&4,+%#5$#,'+51774="#5$#&$O&+%#

(.3$)'(47+%#@#.+%#7(-)4+%#5$#%43"4#7(5+#"$7$%('4+%#,('(#

$..+A# !"# .(# %$774="# %4314$"&$# $O(-4"(-+%# 1"# 7+"*1"&+#

5$#2$72+%#01$#2$-+%#$"7+"&'(5+#$"#"1$%&'(%#4"8$%&43(G 74+"$%;#5$#71@(#)(%$#$-,/'47(#@#>1"5(-$"&(74="#&$='47(#

2$-+%#5(5+#71$"&(#$"#X4..+@;#`+*("+#@#\143#UP^^Za

4. EL PROCESO DE RESOLUCIÓN ALGEBRAI- CO DE UN PROBLEMA VERBAL: EL COMPO- NENTE DE LOS PROCESOS COGNITIVOS 4.1. Sobre la cognición

J+%#,'+7$%+%#7+3"+%74&48+%#01$#%$#,+"$"#$"#(774="#,('(#

..$8('#(#7()+#.(%#>+'-(%#5$.#,$"%(-4$"&+#-(&$-?&47+#@#

%1#7+-1"47(74="#U7+"#7=543+%#%+74(.-$"&$#$%&().$745+%a#

8("#(#"("5+#.+%#$.$-$"&+%#7+-,.$*+%#7+-+#.+%#01$#%$#

1&4.4N("k#aa#$"#.(#,$'7$,74="#U,+'#$*$-,.+;#$"#$.#7(%+#5$.#

-("$*+#5$#.(%#>+'-(%#3$+-<&'47(%#@#%1%#&'("%>+'-(74+G

"$%a;#ba#$"#$.#54'$774+"(-4$"&+#5$#.(#(&$"74="#@#%1%#'$G .(74+"$%#7+"#.+%#,'+7$%+%#5$#7+-,'$"%4=";#ca#$"#$.#1%+#

7(5(#8$N#-?%#4"&$"%48+#5$#.(#-$-+'4(;#da#$"#$.#5$%$"7(G 5$"(-4$"&+# 5$# ,'+7$%+%# 5$# ("?.4%4%# @# %/"&$%4%# 7(5(# 8$N#

-?%#$"&'$.(N(5+%#7+"#$.#1%+#5$#.(#.=347(;#ea#$"#.(%#7+"G 7$,74+"$%# 2$1'/%&47(%# 1&4.4N(5(%# $"# .(# '$%+.174="# 5$# .(%#

%4&1(74+"$%#,'+).<-47(%;#fa#$"#$.#(,'$"54N(*$;#-1@#.43(5+#

(#.+%#,'+7$%+%#5$#3$"$'(.4N(74="#@#()%&'(774=";#@#01$#'$G 014$'$#5$#1%+%#"+8$5+%+%#5$#.+%#fBf#5$#.(#-(&$-?&47(#

$%7+.(';#$&7A

4.2. Sobre la resolución de problemas

\('(#$*$-,.4#7(';#7+"%45$'$-+%;#$"&'$#-172+%;#&'$%#-<G

&+5+%#7.?%47+%#,('(#'$%+.8$'#,'+).$-(%A

BCFfk# M+"74)4$"5+# .+%# $"1"74(5+%# 5$# .+%# ,'+).$-(%#

7+-+# 5$%7'4,74+"$%# 5$# g%4&1(74+"$%# '$(.$%h# +# g$%&(5+%#

,+%4).$%#5$.#-1"5+h#@#&'("%>+'-("5+#&(.$%#&$O&+%#(#&'(G 8<%#5$#+'(74+"$%#("(./&47(%;#$%&+#$%;#1&4.4N("5+#g2$72+%h#

8?.45+%# $"# g&+5+# -1"5+# ,+%4).$hk# 4">$'$"74(%# .=347(%#

01$#(7&9("#7+-+#5$%7'4,74+"$%#5$#.(%#&'("%>+'-(74+"$%#

5$# .(%# g%4&1(74+"$%# ,+%4).$%h# 2(%&(# ..$3('# (# 1"(# 01$# %$#

'$7+"+7$#7+-+#.(#%+.174="#5$.#,'+).$-(A#F#$%&$#-<&+5+#

.+# ..(-('$-+%# B<&+5+# 5$# C">$'$"74(%#F"(./&47(%# f17$G

%48(%#UBCFfaA#}%&$#$%#$.#B<&+5+#F"(./&47+#M.?%47+#,('(#

'$%+.8$'#,'+).$-(%A

BF!fk#F# &'(8<%# 5$# $%&$# -<&+5+# 5$# '$%+.174="# %$# 1%("#

$O,.+'(74+"$%#7+"#5(&+%#,('&471.('$%#,('(#5$%$"7(5$"('#

$.#("?.4%4%#5$.#,'+).$-(#@#7+"#$..+#%1#%+.174="[#(#<.#"+%#

'$>$'4'$-+%# 7+-+# B<&+5+# F"(./&47+# 5$# !O,.+'(74+"$%#

f17$%48(%#UBF!faA

BMk#F#&'(8<%#5$#.(#'$,'$%$"&(74="#5$#(.31"+%#5$#.+%#$.$G -$"&+%#5$%7+"+745+%#5$.#&$O&+#,+'#-$54+#5$#$O,'$%4+"$%#