!"#$%&"#$'(%&()'(&"*&+'",'(%&)('-&%.&,(&"()'(
'*$/"'01.'(%&(2'0&230$#'*(
4*56(78*9(:.'"#$*#8(;#<$/(=')'.&,8(
>"$%'% ?%1#'0$@a AB&.2'"8 4$/1el De la Salle” 51&"#'C(?#1'%8.(
:'#1)0'%(%&(:$)8*8DE'C()&0.'*(F(#$&"#$'*(%&()'(&%1#'#$G"C(>"$@&.*$%'%(?*0'0')(%&(51&"#'( :&%&.'#$G"(H&I8.0$@'(%&)(;,1'F(
51&"#'C(?#1'%8.( -8*&'#<$/J<802'$)6#82(
Resumen-- El presente estudio se fijó como meta el demostrar la influencia positiva que tiene el ajedrez sobre varias áreas del conocimiento en especial sobre las matemáticas. Para lo cual se introdujo este juego de manera sistemática y planificada a un grupo de 15 estudiantes del 6to año de Educación Básica de la Unidad Educativa “Hermano Miguel De la Salle” de la ciudad de Cuenca en febrero del 2012 y se prolongó su práctica durante aproximadamente 5 meses, demostrando que el ajedrez mejoró el promedio final en la asignatura de matemáticas de estos niños con respecto a las notas del primer trimestre, aplicando como prueba de hipótesis el estadístico “T de Student” con un nivel de confianza del 95%. Los niños que aprendieron y practicaron el ajedrez mejoraron significativamente sus notas de matemáticas en comparación con las del grupo que no recibieron las clases de ajedrez (20 estudiantes). Por este motivo se propone el “juego ciencia” como herramienta alternativa para mejorar el rendimiento en matemáticas de niños y niñas en escuelas y colegios por su de fácil aplicación y por sus excelentes resultados a corto plazo. Palabras claves: ajedrez, matemáticas, rendimiento escolar, razonamiento lógico, aprendizaje, Ecuador.
Abstract--The present study set a goal to demonstrate the positive influence of chess on several areas of knowledge especially in mathematics.. The game was introduced in a systematic way at the ‘Hermano Miguel De la Salle’ elementary school in Cuenca to a group of 15 6th grade students in February 2012 during approximately a 5 month period. The results show that the game of chess improves the final grades in the subject of Mathematics, applying as hypothesis test the Student's t statistic for 95% confidence intervals. The group of children who learned and practiced the game of chess significantly improved its math scores compared to the group that did not receive the chess classes (20 students). For that reason, the "science game" is proposed to be applied in elementary and high schools as an alternative tool to improve the children’s mathematics
performance for its easy application and its excellent short-term results.
Keywords: chess, math, school performance, logical reasoning, learning, Ecuador.
1. INTRODUCCIÓN
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
?*0'*( F( 80.'*( $"W1$&01%&*( *8"( #8"*0'"0&*( I'.'( I'%.&*( %&( D'2$)$'C( I.8D&*8.&*( F( '108.$%'%&*( &%1#'0$@'*( W1&( L1*#'"( &( $2I)&2&"0'"( %$@&.*'*( 2'"&.'*( I'.'( #8"0.'..&*0'.( )8*( 2')&*( W1&( 'W1&-'"( '( )'( &%1#'#$G"( F( '*E( &)&@'.( &)( .&"%$2$&"08( &*#8)'.C( *8L.&( 08%8( &"( &)( 3.&'( %&( 2'0&230$#'*( %&( )8*( "$+8*( W1&(&*01%$'"(&"(*1*($"*0$01#$8"&*6(
:.&"0&( '( &*0'( I.8L)&230$#'C( &*01%$8*( #$&"0ED$#8*( <'"( %&28*0.'%8(W1&(&)('-&%.&,(#828(<&..'2$&"0'(&%1#'0$@'(0$&"&( &D$#'#$'(I'.'(&)&@'.(&)(.&"%$2$&"08(&*#8)'.C(*8L.&(08%8(&"(&)( #'2I8( %&( )'*( 2'0&230$#'*( I8.( &)( .',8"'2$&"08( )G/$#8( $"0.E"*( %&"0.8( %&)( -1&/8( W1&C( $"#)1*$@&C( *&( 0.'"*D$&.&( '( 80.8*(32L$08*(%&()'(@$%'6((
\81$*&( ]'1%.&'1( &"( &)( '+8( T^^Q( .&')$,G( 1"'( $"@&*0$/'#$G"( &"( 5'"'%3( *8L.&( )'( .&)'#$G"( %&( )'*( 2'0&230$#'*( #8"( &)( '-&%.&,( W1e 'I'.&#& &" las .I$)'#$8"es A5<&** $" ?%1#'0$8"” %el H.6 N8L&.t :&./1*8"( UTV6( ( H$#<'( $"@&*0$/'#$G"( %&"82$"'%'( A?*01%$8( #82I'.'0$@8( *8L.&( &)( 'I.&"%$,'-&( %&( )'*( 2'0&230$#'*( &"( &)( W1$"08( '+8_C( %&( *&I0$&2L.&(%&(T^^R('(-1"$8(%&(T^^QC(#8"()'(I'.0$#$I'#$G"(%&( 0.&*(/.1I8*C(#8"(1"(080')(%&(`ab(&*01%$'"0&*(%&(W1$"08(/.'%8C( &@$%&"#$G( W1&( &)( /.1I8( W1&( .&#$L$G( &)( #1.*8( 0.'%$#$8"')( %&( 2'0&230$#'*( 'I8F'%8( I8.( &)( '-&%.&,( %&*%&( &)( I.$"#$I$8( 01@$&.8"(2&-8.(.&"%$2$&"08(&"()'(.&*8)1#$G"(%&(I.8L)&2'*(F( &"()'(#82I.&"*$G"(*8L.&(&)(/.1I8(#8"0.8)6(
?)( '-&%.&,( D1&( $"0.8%1#$%8( &"( &)( *$*0&2'( &%1#'0$@8( D.'"#8c #'"'%$&"*&('(I.$"#$I$8*(%&(T^S`6((?"(d'"#81@&.C(5'"'%3C(&)( 5&"0.8(%&(?*01%$8*(%&(4'0&230$#'*(F(;-&%.&,C(."8#$&"%8(
!"# $%&&'!"$()*# '*+&'# ,-."&# ",'/&'0# 1# '!# /'2"&&%!!%# /'# !"2# 3"4(!(/"/'2#'*#5"+'56+($"27#3"#$&'"/%#-*"#2'&('#/'#!(4&%2#/'# +&"4",%# 8"&"# "2(2+(&# "# !%2# '2+-/("*+'2# /'# 9"*"/6# '*# :"+'56+($"2;###
<!# ",'/&'07# "!# 2'&# -*# ,-'.%7# =">%&'$'# '!# "8&'*/(0",'# /'# !"2# 5"+'56+($"2#8%&#!"2#3"4(!(/"/'2#(*+'!'$+-"!'2#?-'#/'2"&&%!!"# $%5%# 8%&# ','58!%@# '!# 8'*2"5('*+%# /'/-$+(>%7# '!# &"0%*"5('*+%#!).($%#1#!"2#'2+&"+'.("2#/'#8'*2"5('*+%#ABC;#
D!#3"4!"&#/'#3"4(!(/"/'2#(*+'!'$+-"!'2#?-'#2'#/'2"&&%!!"*#$%*# '!# ",'/&'07# 2'# /'4'# 5'*$(%*"&# +"54(E*# !"# "+'*$()*# 1# $%*$'*+&"$()*7#"*6!(2(2#1#2F*+'2(27#$&'"+(>(/"/#'#(5".(*"$()*# 1#&"0%*"5('*+%#!).($%G5"+'56+($%;##<!#+(8%#/'#&"0%*"5('*+%# ?-'# 2'# '58!'"# '*# '!# ,-'.%# '2# '!# 5(25%# ?-'# '*# !"2# 5"+'56+($"2##AHC;####
<*# <2+"/%2# I*(/%2# 2'# 3"*# &'"!(0"/%# >"&(%2# 8&%1'$+%2# 8"&"# '2+"4!'$'&#!"#'*2'J"*0"#/'!#",'/&'0#'*#!"2#'2$-'!"2#8K4!($"2;##
L(*# '54"&.%7# *%# 2'# $%*%$'*# 3"2+"# '!# 5%5'*+%# +&"4",%2# /'# (*>'2+(."$()*# ?-'#3"1"*#"4%&/"/%#'!# +'5"#/'!# ",'/&'0# 1# 2-# (*=!-'*$("# '*#!"# "2(.*"+-&"# /'# 5"+'56+($"2#'*# *-'2+&%#8"F27# &"0)*#8%&#!"#$-"!#2'#3"#8&%8-'2+%#&'"!(0"&#'!#8&'2'*+'#'2+-/(%# ?-'# 2'# "8!($)# "# BM# '2+-/("*+'2# /'!# NO# .&"/%# /'# </-$"$()*# P62($"# /'# !"# I*(/"/# </-$"+(>"# QR'&5"*%# :(.-'!# S'# !"# L"!!'T7 /'#9-'*$"7#/'#!%2#$-"!'2#UM#*(J%27#$%&&'28%*/('*+'2# "!# .&-8%#'V8'&(5'*+"!7#&'$(4('&%*#$!"2'2#/'# ",'/&'0# /-&"*+'# "8&%V(5"/"5'*+'# M# 5'2'2# W/'# ='4&'&%# "# ,-*(%# /'# XYUXZ7# /'5%2+&"*/%# ?-'# '!# ",'/&'0# 5',%&)# 2(.*(=($"+(>"5'*+'# 2-# 8&%5'/(%#=(*"!#/'#5"+'56+($"2#$%*#&'28'$+%#"#!"2#*%+"2#/'!# 8&(5'&# +&(5'2+&'# '*# $%58"&"$()*# $%*# !"2# *%+"2# /'!# .&-8%# $%*+&%!#?-'#/(25(*-1'&%*;#
2. METODOLOGÍA
["# (*>'2+(."$()*# =-'# /'# +(8%# 'V8'&(5'*+"!7# -+(!(0"*/%# !%2# 8&(*$(8(%2# /'!# 5E+%/%# $('*+F=($%;# # \"&"# '!# '='$+%# 2'# +&"4",)# $%*# /%2# .&-8%2@# -*# .&-8%# $%*+&%!# WXY# "!-5*%2Z# 1# -*%# 'V8'&(5'*+"!# WUM# "!-5*%2Z# '!'.(/%2# "!# "0"&;# # <!# .&-8%# 'V8'&(5'*+"!# &'$(4()# '!# +&"+"5('*+%# 5'/("*+'# !"2# $!"2'2# /'# ",'/&'0#1#'!#%+&%#*%;# # ["#8%4!"$()*#'2+->%# $%*=%&5"/"#8%&# !%2#*(J%2#1#!"2#*(J"2#/'!#N+%#"J%#/'#</-$"$()*#P62ica QDT /e la I*(/"/ </-$"+(>a QR'&5"*% :(.-el De la Salle” /e !a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
\"&"#!"#(*>'2+(."$()*#*%#2'#&'"!(0)#-*"#5-'2+&"#2(*%#?-'#2'# /'$(/()#(*>'2+(."&#"#+%/%#'!#-*(>'&2%#?-'#'2+"4"#$%*2+(+-(/%# 8%&#'!#+%+"!#/'#H^#"!-5*%2#5"+&($-!"/%2;##S'#'2+"#8%4!"$()*# 2'# 2'!'$$(%*)# "!# "0"&# "# BM# "!-5*%2# 5'/("*+'# -*# 8&%.&"5"# (*=%&56+($%# 8"&"# 8"&+($(8"&# /'!# 8&%1'$+%;# # <2+"# 5-'2+&"# &'2-!+)#"/'$-"/"#8"&"#5"*',"&#$%*#8&'$(2()*#$"/"#>"&("4!'#1# 8"&"#$%*+&%!"&#$%&&'$+"5'*+'#$"/"#.&-8%#/'#!"#(*>'2+(."$()*;#
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
<*# !"2# $!"2'2# /'# ",'/&'0# 2'# (*$!-1'&%*# >"&(%2# ','&$($(%2# /'# &"0%*"5('*+%# !).($%7# /'2/'# !%2# 562# '!'5'*+"!'2# $%5%# 8%&# ','58!%#-*"#$"8+-&"#%#-*#5%>(5('*+%#'28'$F=($%# /'#"!.-*"# 8('0"# 3"2+"# 8&%4!'5"2# 562# $%58!',%2# $%5%# '*# '!# $"2%# /'# 4-2$"&#-*#,"?-'#5"+'#'*#>"&("2#,-."/"27#%#/'#','&$($(%2#8"&"# %4+'*'&#>'*+","#/'#8('0"2#%#8"&"#'58"+"&#'!#,-'.%#'*#'!#$"2%# /'#('&/('*/%;###
D/'562# /'# !%2# ','&$($(%2# %# 8&%4!'5"2# 8-&"5'*+'# ",'/&'$F2+($%27# %+&"# 8"&+'# (58%&+"*+'# &'2-!+"# !"# 8&6$+($"# /'# 8"&+(/"27# /%*/'# !%2# *(J%2# 8%/F"*# /'2"&&%!!"&# 2-2# '2+&"+'.("2# /'#,-'.%#'*#$"/"#5%>(5('*+%#8%*('*/%#"2F#/'#5"*(=('2+%#2-# $"8"$(/"/# /'# "*6!(2(2# 1# 2F*+'2(27# 5'5%&("7# $%*$'*+&"$()*# 1# "+'*$()*# ?-'# &'2-!+"# 462($"# 8"&"# !"# =%$"!(0"$()*# '*# %+&"2# 6&'"2# /'!# $%*%$(5('*+%# $%5%# 8%&# ','58!%# '*# !"# &'2%!-$()*# /'#8&%4!'5"2##5"+'56+($%2#AMC;##`+&"#"8&%V(5"$()*#2(5(!"&# 2'#'*$-'*+&"#'*#'!#!(4&%#/'!#'V$"58')*#5-*/("!#/'#",'/&'0# D*"+%!(#a"&8%>#ANC;##
\"&"#%4+'*'&#!"#*%+"#'*#!"#5"+'&("#/'#5"+'56+($"2#+"*+%#'*#'!# 8&(5'&# +&(5'2+&'# $%5%# 2-# 8&%5'/(%# =(*"!7# 2'# "$-/()# "# !%2# &'.(2+&%2# /'# $"!(=($"$(%*'2# /"/%2# 1# $'&+(=($"/%2# 8%&# !"# 2'$&'+"&F"# /'# !"# (*2+(+-$()*# '/-$"+(>"# 1# 2'# &'.(2+&"&%*# '2+%2# /"+%2#'*#-*"#3%,"#/'#$6!$-!%#8"&"#2-#"*6!(2(2#8%2+'&(%&;#
["2# >"&("4!'2# &'='&'*$("!'2# ?-'# 2'# -+(!(0"&%*# '*# '!# '2+-/(%# =-'&%*@# !"# '*2'J"*0"# /'# ",'/&'0# $%5%# >"&("4!'# (*/'8'*/('*+'# 1# !"2# 5"+'56+($"2# $%5%# !"# >"&("4!'# /'8'*/('*+';# ["2# +E$*($"2# 8"&"# !"# &'$%!'$$()*# /'# !"# (*=%&5"$()* =-'&%*#!"#%42'&>"$()*7#!"#'*+&'>(2+"#1#'!#+'2+;###
[%2# /"+%27# -*"# >'0# &'$%!'$+"/%27# =-'&%*# 8&'8"&"/%27# $!"2(=($"/%27#$%/(=($"/%2#8"&"#2-#=6$(!#(/'*+(=($"$()*#1#2'#!'2# "2(.*)#!%2#>"!%&'2#*-5E&($%2#$%&&'28%*/('*+'2;##[-'.%#>(*%# !"# ="2'# /'# "!5"$'*"5('*+%# /'# !%2# /"+%2# '*# -*"# 3%,"# /'# $6!$-!%;# # <!# 8&%$'2"5('*+%# /'# !"# (*=%&5"$()*# 2'# &'"!(0)# 5'/("*+'# '!# 8&%.&"5"# '2+"/F2+($%# L\LL# WStatistical Packageforthe Social Sciences) $%*#!"#'2+"/F2+($"#/'2$&(8+(>"# "# +&">E2# /'# 8&%$'/(5('*+%2# $-"*+(+"+(>%27# 8%&$'*+",'27# 5'/(/"2# /'# +'*/'*$("# $'*+&"!7# /(28'&2()*7# 2(5'+&F"7# '*+&'#
!"#!$%&%'!(%)$"*+,$"-'*%-(.)#)('-*/%0!#%1)+-!%+)%'!("#*$")%+)% 2-03")$-$4%%%
5!$% 6#7.-'!$% 8"-/-9*+!$% )(% )/% )$"8+-!% $)% )$"#8'"8#*#!(% )(% "!#(!% *%líneas% 0*#*% 1!$"#*#% /*% ")(+)('-*% +)% /*$% (!"*$% +)%
1*")17"-'*$%)(%/!$%+!$%6#80!$%)(%)/%0#-1)#%"#-1)$"#)%&%)(%)/% 0#!1)+-!% .-(*/:% &% )/%gráfico de distribución normal% 0*#*% -(+-'*#% /*% 9!(*% +)% #)'2*9!% &% *')0"*'-3(% +)% /*% 2-03")$-$% )(% 8(*%+-$"#-;8'-3(%<%+)%/*$%(!"*$%+)%1*")17"-'*$4%%
=*#*%/*%>)#-.-'*'-3(%+)%/*$%2-03")$-$%$)%8"-/-93%/*%0#8);*%<%+)% ?"8+)("%@8)%#)$8/"*%-+)*/%0*#*%18)$"#*$%0)@8)A*$%'!1!%)(%)/% 0#)$)(")%)$"8+-!%BCDEFG4%%%
3. EVALUACIÓN DE RESULTADOS Y DISCUSIÓN %
H/% )$"8+-!% '!("3% '!(% /*% 0*#"-'-0*'-3(% "!"*/% +)% EF% $8I)"!$% BCDEFG%)("#)%(-A!$%&%(-A*$4%%J)%)$")%8(->)#$!K%LF%)#*(%(-A!$% @8)% '!##)$0!(+)% */% MNO% &% NP% )#*(% (-A*$% @8)% #)0#)$)("*% )/% LQO%'!1!%$)%-(+-'*%)(%/*%$-68-)(")%"*;/*4%
<RS5R%T4%
JT?<UTSVWTXC%JH%5R%=YS5RWTXC%JH5%H?<VJTY%?HZ[C%?H\Y4% %
SEXO NÚMERO EXP. CONTROL
HOMBRE LF% NN% N]%
MUJER NP% ]% ^%
TOTAL EF% NF% LP%
%
Las notas de matemáticas dentro de cada grupo
5!$%#)$8/"*+!$%!;")(-+!$%)(%/*%1*")#-*%+)%1*")17"-'*$%)(%)/% 0#-1)#% "#-1)$"#)% &% )/% 0#!1)+-!% .-(*/% )(% /!$% +!$% 6#80!$% -(>)$"-6*+!$%$)%0#)$)("*(%)(%)/%$-68-)(")%6#7.-'!4%
% _-68#*%N4%%=#!1)+-!%+)%`*")17"-'*$%+)%/!$%+!$%6#80!$%
5*% 1)+-*% +)% /*$% (!"*$% +)% 1*")17"-'*$K% "*("!% )(% )/% 0#-1)#% "#-1)$"#)% '!1!% )(% )/% 0#!1)+-!% .-(*/% +)% /!$% 6#80!$% )a0)#-1)("*/% &% +)% '!("#!/K% (!$% #)./)I*% /!% )$0)#*+!4% % % 5*$%
'*/-.-'*'-!()$% .-(*/)$% +)% 1*")17"-'*$% +)% /!$% (-A!$% @8)% #)'-;-)#!(%/*$%'/*$)$%+)%*I)+#)9%*81)("*#!(%'!(%#)$0)'"!%*/% 0#-1)#%"#-1)$"#)K%(!%*$,%/*$%!;")(-+*$%0!#%/!$%(-A!$%+)/%6#80!% '!("#!/% @8)% +-$1-(8&)#!(4% % H$")% 2)'2!% +*% '8)("*% +)% /*% -(./8)('-*%0!$-"->*%+)/%*I)+#)9%+)1!$"#*+*%)(%>*#-!$%)$"8+-!$% )(%!"#!$%0*,$)$%bFc4%
Análisis estadístico de dispersión y simetría de las notas de matemáticas
5*$%1)+-+*$%+)%+-$0)#$-3(%&%$-1)"#,*%+)/%6#80!%)a0)#-1)("*/% )(% /*$% (!"*$% +)% 1*")17"-'*$% $)% 08)+)(% *0#)'-*#% )(% /*% $-68-)(")%"*;/*d%
%
<RS5R%TT4
`HJTJR?%JH%JT?=HU?TXC%e%?T`H<UfR%JH5%ZUV=Y% H\=HUT`HC<R5%HC%5R?%CY<R?%JH%`R<H`g<TWR?%
%
5*$%1)+-+*$%+)%+-$0)#$-3(%&%$-1)"#,*%+)/%6#80!%+)%'!("#!/%)(% /*$%(!"*$%+)%1*")17"-'*$%$)%08)+)(%*0#)'-*#%)(%/*%$-68-)(")% "*;/*d%
%
<RS5R%TTT4%
`HJTJR?%JH%JT?=HU?TXC%e%?T`H<UfR%JH5%ZUV=Y%JH% WYC<UY5%HC%5R?%CY<R?%JH%`R<H`g<TWR?4%%%
%% %
5*% +-$1-(8'-3(% +)% /*% +)$>-*'-3(% )$"7(+*#% )(% )/% 0#!1)+-!% .-(*/%'!(%#)/*'-3(%*/%0#-1)#%"#-1)$"#)%+)/%6#80!%)a0)#-1)("*/% B+)%N4EEF%*%N4NhEG%(!$%-(+-'*%@8)%/!$%+*"!$%$)%*')#'*#!(%17$% *% $8% 1)+-*:% )(% '*1;-!% )(% )/% 6#80!% '!("#!/% $)% 0#)$)("*% 8(% -('#)1)("!%+)% L4P^]%*% L4NQh4% % H$"*% #)/*'-3(%+)18)$"#*%@8)% )/%0#-1)#%6#80!%!;"8>!%8(%1)I!#%#)(+-1-)("!4%
H/%'!).-'-)(")%+)%>*#-*'-3(%"8>!%8(%'!10!#"*1-)("!%$-1-/*#4%% H(% )/% '*$!% +)/% 6#80!% )a0)#-1)("*/% +-$1-(8&3% +)% M4EP^% *% ^4E^NK% /!% '8*/% -(+-'*% 8(*% 1)(!#% >*#-*;-/-+*+% +)% /!$% +*"!$:% 0!#% )/% '!("#*#-!K% )/% 6#80!% '!("#!/% !;"8>!% 8(% -('#)1)("!% )(% )$")%0*#71)"#!%*/%.-(*/-9*#%/*%-(>)$"-6*'-3(%+)%LKP^]%*%LKNQh4% H$"!% +)18)$"#*% @8)% )/% 6#80!% @8)% #)'-;-3% /*$% '/*$)$% +)% *I)+#)9%!;"8>!%8(%#)(+-1-)("!%17$%30"-1!4%
% !"#$$%
!"#!%% !&#'("
!&#(%%
!(#%% !(#$% !"#%% !"#$% !&#%% !&#$% !)#%%
!*+,-./01,/ 2,3./4-3 5-678 !"#$!%#&'"(
9:;+<:=
>?2><@A>;+ B=
Notas de Matemáticas
No. Media Desviació n Estándar
Coeficient e Variación
Asimetrí a
1 Trimestre NF% NhKL^M% NKEEF% MKEP^% iKEFF%
Promedio Final NF% NhK^PP% NKNhE% ^KE^N% KPL]%
Notas de Matemáticas
No. Media Desviación Estándar
Coeficiente Variación
Asimetría
1 Trimestre LP% NMKFFP% LKP^]% NNKM^N% iKEFh%
Promedio Final
!
!"#! #$%&'()*#! %+,%-#! .+#! &'/0)#! '+! 1#$! +0(#$! 2%+#1'$! ,'! &#('&3(%-#$!,'1!4).50!'65')%&'+(#1!-0+!)'$5'-(0!#1!5)%&')! ()%&'$()'7!5.'$!10$!,#(0$!5)'$'+(#+! .+!1%4')0!$'$40!8#-%#! 1#! ,')'-8#! %+,%-#,0! 50)! '1! 9#10)! 50$%(%90! :;:<=7! '$! ,'-%)7! 50)! '+-%&#!,'!$.!&',%#7!#!,%2')'+-%#!,'1!4).50!-0+()01!>.'!(.90! .+!9#10)!+'4#(%90!,'!?:;@AB7!10!-.#1!%&51%-#!>.'!10$!,#(0$!#1! 2%+#1%C#)! 1#! '65')%'+-%#! '$(3+! &3$! )'5#)(%,0$! 50)! ,'D#/0! ,'! $.!&',%#;!
Prueba de la hipótesis en las notas de matemáticas;
E#)#!-0&5)0D#)!1#!8%5F('$%$!,'!()#D#/0!$'!#51%-F!1#!5).'D#!G! ,'! H(.,'+(! #! 10$! ,#(0$! ,'! 10$! 5)0&',%0$! 2%+#1'$! ,'! &#('&3(%-#$! ,'! 10$! ,0$! 4).50$! -0+! '1! $02(I#)'! '$(#,*$(%-0! HEHH7!0D('+%'+,0!10$!$%4.%'+('$!)'$.1(#,0$J!
GKL"K!MN;!
EOPQLK!G!KE"MRKSK!K!"TH!EOTUQSMTH!SQ!UKGQUVGMRKH!SQ! "TH!STH!WOPETH!
! !
!
De #-.'),0 a la (#Dla ,e ,%$()%D.-%F+ G -0+ XY:7:@ Z[@% ,' +%9'1!,'!-0+2%#+C#\!]!AA!4)#,0$!,'!1%D')(#,!Z41!'+!1#!(#D1#\7! '1!)#+40!,'!#-'5(#-%F+!,'!1#!^0!'$(3!!'+()'!:!]!_7`[Ba!50)!10! (#+(07!#1!8#D')!0D('+%,0!.+!9#10)!,'1!'$(#,*$(%-0!(!,'!<7A[:!$'! )'-8#C#!1#!8%5F('$%$!+.1#!]!$'!#-'5(#!1#!8%5F('$%$!#1(')+#(%9#! >.'!$'!)'5)'$'+(#!'+!'1!$%4.%'+('!4)32%-0J!
!
b%4.)#!<;!!c0+#!,'!)'-8#C0!]!#-'5(#-%F+!,'!1#!8%5F('$%$!+.1#!-0+![@d!,'! -0+2%#+C#!'+!'1!5)0&',%0!,'!&#('&3(%-#$!
!
Q+! ,'2%+%(%9#7! 10$! )'$.1(#,0$! ,'! 1#! 5).'D#! (! ,'! H(.,'+(! ,'&.'$()#+! >.'! 10$! 5)0&',%0$! ,'! &#('&3(%-#$7! (#+(0! ,'1! 4).50!-0+()01!-0&0!,'1!'65')%&'+(#1!$0+!,%2')'+('$7!10!>.'! '>.%9#1'! #! ,'-%)! >.'! 1#$! +0(#$! ,'! &#('&3(%-#$! ,'1! 4).50! '65')%&'+(#1! $0+! $%4+%2%-#(%9#&'+('! &'/0)'$! >.'! 1#$! ,'1! 4).50!-0+()01;!
4. CONCLUSIONES Y TRABAJO FUTURO
S'!10$!)'$.1(#,0$!0D('+%,0$!'+!+.'$()#!%+9'$(%4#-%F+!]!,'!10$! '$(.,%0$!'6()#+/')0$7!$'!,'&.'$()#!>.'!'1!#/',)'C!-0+$(%(.]'! .+#! 8'))#&%'+(#! 5',#4F4%-#! eBf! >.'! 5')&%('! '1'9#)! '1! 5)0&',%0!'+!1#!#$%4+#(.)#!,'! &#('&3(%-#$!]!-.]#!%+-1.$%F+! '+! 10$! -.))*-.1.&! '$-01#)'$! 8#! $%,0! 5)05%-%#,#! 50)! #.(0)%,#,'$! ',.-#(%9#$! -0&0! '1! '6?$'-)'(#)%0! ,'! Q,.-#-%F+! ,'! 10$! Q$(#,0$! P+%,0$! G'))'11! L'117! #$*! -0&0! 50)! 1#! R0&%$%F+! ,'! #/',)'C! '+! 1#! Q,.-#-%F+! ,'! 1#! b',')#-%F+! M+(')+#-%0+#1!,'!K/',)'C!ZbMSQ\!]!)'-%'+('&'+('7!'+!&#)C0! ,'!<:_<7!'+!Q$()#$D.)40?b)#+-%#!50)!'1!E#)1#&'+(0!Q.)05'0! >.'! #5)0DF! $.! %+-0)50)#-%F+! '+! 10$! -.))*-.1.&! ,'! 1#$! '$-.'1#$!,'!(0,0$!10$!5#*$'$!&%'&D)0$;!!
Q+!'1!'$(.,%0!,'1!-#$0!-0+-)'(0!)'#1%C#,0!#!+%g0$!]!+%g#$!,'1! `(0! #g0! ,'! ',.-#-%F+! D3$%-#! ,'! 1#! P+%,#,! Q,.-#(%9#! h^')&#+0 U%4.el De la H#11'i ,e la -%.,#, ,e R.'+-# $e 5.','+!'6()#')!1#$!$%4.%'+('$!-0+-1.$%0+'$J!!!!
_\ Q1!+%9'1!,'!#/',)'C!,'1!4).50!'65')%&'+(#1!2.'!&.]!D#/0! #1!-0&'+C#)!1#!%+9'$(%4#-%F+7!$%+!'&D#)40!#1!2%+#1%C#)!10$! +%g0$! ,'&0$()#)0+! .+! D.'+! -0+0-%&%'+(0! D3$%-0! ,'1! /.'40!]!50-0!-0+0-%&%'+(0!,'!'$()#('4%#!]!(3-(%-#;! !
<\ "0$! 5)0&',%0$! 2%+#1'$! ,'! &#('&3(%-#$! ,'1! 4).50! '65')%&'+(#1! 2.')0+! $%4+%2%-#(%9#&'+('! &'/0)'$! >.'! 10$! 5)0&',%0$!,'1!4).50!-0+()01;!
!
A\ Q1! 5)0&',%0! 2%+#1! ,'! &#('&3(%-#$! ,'1! 4).50! '65')%&'+(#1! &'/0)F! $%4+%2%-#(%9#&'+('! -0+! )'$5'-(0! #1! 5)%&')!()%&'$()'!-0+!.+!+%9'1!,'!-0+2%#+C#!,'1![@d;! !
=\ Q1! 5)0&',%0! 2%+#1! ,'! &#('&3(%-#$! ,'1! 4).50! -0+()01! ,%$&%+.]F!-0+!)'$5'-(0!#1!5)%&')!()%&'$()';!!
!
@\ R0&0! )'$.1(#,0! ,'1! '$(.,%0! $'! ,'&0$()F! >.'! '1! #/',)'C! (%'+'!&#]0)!%+21.'+-%#!'+!'1!4j+')0!&#$-.1%+0!,'+()0!,'1! 4).50!'65')%&'+(#1!>.'!'1!2'&'+%+0;!
!
Q+!'1!3&D%(0!,'!'$(#$!-0+-1.$%0+'$!]!-0+!'1!5)05F$%(0!,'!>.'! 1#! +%g'C! -.'+('! -0+! .+#! 05-%F+! >.'! 1'$! 5')&%(#! &'/0)#)! '1! )'+,%&%'+(0! #-#,j&%-07! $0D)'! (0,0! '+! 1#! &#(')%#! ,'! &#('&3(%-#$7!5)'$'+(#&0$!#14.+#$!)'-0&'+,#-%0+'$J!
_\ M+-1.%)! #1! #/',)'C! '+! 1#$! #-(%9%,#,'$! ',.-#(%9#$! ,'+()0! ,'!1#!M+$(%(.-%F+;!
!" #$%&'$'()*+,&*-,*.&*-,/0(,'1.2%*$2'()0,('/'-*+20,%*+*, &*,'(-'3*(4*,5'&,*6'5+'47,
,
8" 9*%*/2)*+,*,&0-,%+0:'-0+'-,/0(,/&*-'-,;<-2/*-,5',*6'5+'4, %*+*,1.',%.'5*(,2$%*+)2+,*,-.-,*&.$(0-7,
,
=" >*-, /&*-'-, 5', *6'5+'4, )'(5+?*(, .(*, :+'/.'(/2*, $?(2$*, 5', @, /&*-', -'$*(*&, /0(, .(*, 5.+*/2A(, 5', =B, $2(.)0-, /0$0, ;*-', 5.+*()', C, $'-'-, /0$0, '(, (.'-)+*, 'D%'+2'(/2*,0,$<-7,
,
E','-)*,$*('+*,-',%05+<,/0()+*++'-)*+,&*,:*&)*,5',*)'(/2A(,F, /0(/'()+*/2A(,F,5',+*40(*$2'()0,&AG2/0,1.',)2'('(,&0-,(230-, F,(23*-,'(,&*-,'5*5'-,/0$%+'(525*-,'()+',&0-,@B,F,@@,*30-H,F, &'-, *F.5*+?*, *, $'60+*+, '&, +'(52$2'()0, */*5I$2/0H, '-%'/2*&$'()','(,'&,<+'*,5',&*-,$*)'$<)2/*-7
J2(*&$'()', /0$0, )+*;*60, *, :.).+0, -', %05+?*, *$%&2*+, '-)', '-).520, ;*60, 52:'+'()'-, /0(52/20('-, -0/20K'/0(A$2/*-H, (.)+2/20(*&'-H, *$;2'()*&'-H, ')/7, %*+*, 5'$0-)+*+, *-?, 1.', &*, %+</)2/*, 5'&, *6'5+'4, +'1.2'+', .(*, $?(2$*, 2(L'+-2A(, /0$%*+*50,/0(,&0-,+'-.&)*50-,'(,;'(':2/20,5',(.'-)+*,(23'4, F,6.L'().57,
Bibliografía
,
M@N,,O2(2-)'+20,5',P5./*/2A(,5',P/.*50+H,QR'-.&)*50-,%+.';*-,/'(-*&'-, SPR,P/.*50+,!BBTHU,V))%WXX'5./*/20(7G0;7'/XY%K
/0()'()X.%&0*5-X50Y(&0*5-X!B@!XBTX+'-.&)*50Z+.';*-[P\7%5:H,].2)0H, !BBT7,
M!N,,R7,J'+G.-0(H,Q9V'--,2(,'5./*)20(,+'-'*+/V,-.$$*+FHU,5',A review of key chess research studies for the BMCC chess in education "a wise move"H,^'Y,_0+`H,@aaC7,,
M8N,,b7,S*&&<(,F,b7,J'+(<(5'4H,QP(-'3*+,$*)'$<)2/*-,/0(,+'/.+-0-,5', *6'5+'4HU,Tendencias pedagógicas, L0&7,@H,(c,@CH,%7,dTH,!B@B7,, M=N,,b7,O7,e&?*-H,QE'-*++0&&*+,&*,2()'&2G'(/2*,*,)+*LI-,5'&,*6'5+'4HU,O*5+25H,
P52/20('-,Z*&*;+*H,S7f7H,!BBdH,%%7,@gK@a7,
MCN,,R7,h+2(/V'+0H,QV))%WXX`/:'7'.HU,^0L2'$;+',!B@87,MP(,&?('*N7,fL*2&*;&'W, V))%WXX`/:'7'.X-2)'-X5':*.&)X:2&'-Xh+2(/V'+0ij9JP7%5:7,Mk&)2$0,*//'-0W, !=,@!,!B@=N7,
MdN,,f7,j*+%0LH,QP&,*6'5+'4,*%+'(5'+,F,%+0G+'-*+HU,5',20 conversaciones con Anatoli KarpovH,\*+/'&0(*H,9?+/.&0,5',&'/)0+'-H,!BB!H,%%7,!=@K!=T7, MgN,,O7,f$2GAH,QP(/.'()+0,'5./*/2A(,!B@!K!B@8HU,5',Dame la mano- El
ajedrez como herramienta educativaH,\*+/'&0(*H,!B@@7,, ,