PERSPECTIVAS TE+RICAS EN EL PROCESO
DE ELABORACI+N DE CONJETURAS E IMPLICACIONES PARA LA PRÁCTICA: TIPOS Y PASOS
!Cañadas, María Consuelo1; Deulofeu, Jordi2; Figueiras, Lourdes2; Reid, David A.3 y Yevdokimov, Oleksiy4
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Theoretical perspectives in the process of producing conjectures and implications for practice: types and steps
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RESULTADOS Tipos de conjeturas
6!2+(5%(!*'!<+)!%$&')5%-+;%.$')!('B'('$;%+*+)!'$!'<!+2+(5+R
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%$*7;;%M$!'92L(%;+!+!2+(5%(!*'!7$!$k9'(.!!!$%5.!*'!;+).)!
*%);('5.)0!bZ! <+! %$*7;;%M$! '92L(%;+! +! 2+(5%(! *'! ;+).)! *%R
$@9%;.)0!cZ!<+!+$+<.-L+0!dZ!<+!+I*7;;%M$!>!eZ!<+!;.$H'57(+!
I+)+*+!'$!<+!2'(;'2;%M$G!V');(%I%9.)!+!;.$5%$7+;%M$!;+*+!
7$.!*'!'<<.)G!
Tipo A: Inducción emp4rica a partir de nDmero ! nito de casos discretos
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5.)0!'$!<.)!;7+<')!)'!27'*+!*'5';5+(!7$!2+5(M$!;.$)5+$5'G!
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Problema 1
1$!7$!;.$5'N5.!*'!7$!H7'-.!9+5'9@5%;.0!7$!2(.B').(!*'!'*7;+;%M$!)'R
;7$*+(%+!2('-7$5M!+!)7)!')57*%+$5')!)%!2.*(L+$!'$;.$5(+(!7$+!('<+;%M$!
'$5('! <.)! $k9'(.)! 2+(')! 9+>.(')! K7'! A! >! <+)! )79+)! *'! *.)! $k9'(.)!
2(%9.)!%92+(')G!6!;.$5%$7+;%M$!2(')'$5+9.)!+<-7$+)!*'!<+)!('<+;%.$')!
K7'!*');7I(%'(.$!<.)!')57*%+$5')d
j!s!=!t!=!! ! "j!s!=!t!"=!s!Y!t!""
^!s!=!t!Y! ! "^!s!Y!t!"=!s!_!t!""
"`!s!=!t!_!s!Y!t!Y!! 4`!s!=!t!"_!s!_!t!"=
"4!s!Y!t!_! ! ! GGGGGGGGG
"A!s!=!t!""!s!_!t!_! =`!s!_!t!4=!s!""!t!"X!s!"=!t!"_
V')27[)!*'!5(+I+H+(!'$!')5'!2(.I<'9+0!7$.!*'!'<<.)!'N;<+9Md!o')5.!2.R
*(L+! ;.$5%$7+(!)%'92('G!U7$;+! )'!+;+I+p!WC9%5D!>! m'$*'().$0!"XYX0!
2G!"4=ZG
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;.$H'57(+!*'!a.<*I+;Dd!5.*.!$k9'(.!2+(!9+>.(!K7'!A!27'R
*'!)'(!');(%5.!;.9.!<+!)79+!*'!*.)!2(%9.)!%92+(')G!1<!2(.R B').(!D+!2<+$5'+*.!'<!2(.I<'9+!;.9.!7$!2(.I<'9+!+I%'(5.!
2+(+!%927<)+(!<+!2+(5%;%2+;%M$!*'!<.)!')57*%+$5')G!6$5'!')5'!
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)'(!;+75'<.).)!;7+$*.!)'!-'$'(+<%/+!+!2+(5%(!*'!&+(%+)!'N2'R (%'$;%+)G!?.$)%*'('9.)!'<!)%-7%'$5'!2(.I<'9+d!
V'5'(9%$+(!;7@$5.)!)'-9'$5.)!*'!<.$-%57*')!*%B'('$5')!27'*'$!5(+/+(R )'!2+(+!;.$';5+(!<.)!;<+&.)!*'!7$!-'.2<+$.!;7+*(+*.!*'!Y!N!YG
m+>!+<79$.)!K7'!'$B.;+$!'<!5(+I+H.!+79'$5+$*.!)7;')%R
&+9'$5'!'<!$k9'(.!*'!)'-9'$5.)!'$!-'.2<+$.)!*'!5+9+R 3.)!"!!!"0!4!!!40!=!!!=0!>!<<'-+$!+!<+!;.$;<7)%M$!*'!K7'!
'<!(')7<5+*.!2+(+!7$!-'.2<+$.!n!!!n!')!4!t!=!t!u!t!n!t!
"G! C%$! '9I+(-.0! '<! (')7<5+*.! 2+(+! '<! ;+).! *'<! -'.2<+$.!
Y!!!Y ')!"X0!>!$.!4`!WU+5%.$+<!?.7$;%<!.B!h'+;D'()!.B!
S+5D'9+5%;)0!4``=ZG!
Tipo F: Inducción emp4rica a partir de casos dinámicos
\+!I+)'!2+(+!')5+I<';'(!7$+!;.$H'57(+!*'!')5'!5%2.!')!7$!
$k9'(.! %$!!$%5.! *'! +;.$5';%9%'$5.)! ;.$5%$7.)0! K7'! ).$!
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+;.$5';%9%'$5.)!*%$@9%;+9'$5'!('<+;%.$+*.)G!
T.(!'H'92<.0!'$!<+!!!-7(+!"!)'!(';.-'!7$+!('2(')'$5+;%M$!
'$!<+!K7'!)'!2%*'!+!<.)!')57*%+$5')!K7'!9+$%27<'$!'<!5(%@$R -7<.!6:?!;.$!7$!2(.-(+9+!*'!-'.9'5(L+!*%$@9%;+!>!K7'!
.I)'(&'$!;M9.!&+(L+!'<!27$5.!v!2+(+!*'5'(9%$+(!;M9.!)'!
D+!;.$)5(7%*.!Wl$%22%$-!>!O'%*0!'$!2('$)+ZG!1<!27$5.!v!
*'<!%$5'(%.(!*'<!5(%@$-7<.!')!<+!%$5'()';;%M$!*'!<+)!9'*%+R
$+)0!2'(.!')5.!$.!')!;.$.;%*.!a prioriG!
#$+!*'!<+)!2(%9'(+)!;.$H'57(+)!K7'!B.(97<+$!<.)!')57*%+$5')!
')!K7'!'<!27$5.!v!$7$;+!27'*'!')5+(!B7'(+!*'<!5(%@$-7<.G Tipo Gd!Analog4a!
C'!27'*'!')5+I<';'(!7$+!;.$H'57(+!2.(!+$+<.-L+!).I('!+<R -k$! D';D.! 2+(5%;7<+(! >+! ;.$.;%*.0! .! %$;<7).! ;.$H'57(+(!
7$+!('-<+!-'$'(+<!).I('!<+!I+)'!*'!.5(+!('-<+!-'$'(+<!>+!
;.$.;%*+G!
Problema 2
V+*.!7$!5(%@$-7<.!6:?!>!7$!27$5.!T!*'$5(.!*'<!5(%@$-7<.0!;.$)5(7>'!
<+)!5(')!<L$'+)!*')*'!;+*+!&[(5%;'!60!:0!?!+<!27$5.!TG!fE7[!27'*')!*';%(!
).I('!<+)!('<+;%.$')!'$5('!<+)!<L$'+)!>!<.)!<+*.)!*'<!5(%@$-7<.g
#$.!*'!<.)!(+/.$+9%'$5.)!K7'!)%-7'$!+<!2<+$5'+9%'$5.!
*'<!2(.I<'9+!')!'<!)%-7%'$5'd!
C[! K7'! )%! *.)! <L$'+)! ;.(5+$! <.)! <+*.)! '$! '<! 27$5.! 9'*%.0! '$5.$;')! <+!
5'(;'(+!<L$'+!5+9I%[$!<.!D+(@0!2.(K7'!<+)!9'*%+$+)!)'!'$;7'$5(+$!'$!7$!
27$5.G!1)5.>!)'-7(.!*'!K7'!)%!*.)!<L$'+)!;.(5+$!<.)!<+*.)!;.$!7$+!(+/M$!
*'!4d"0!<+!5'(;'(+!5+9I%[$!<.!D+(@!Wl$%22%$-!>!O'%*0!4``YZG
67$K7'!<+!;.$H'57(+!)'+!B+<)+!'$!')5'!;+).0!')5'!5%2.!*'!
;.$H'57(+)! (')7<5+$! ('+<9'$5'! 2(.*7;5%&+)! 2+(+! <+! (').R
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$7*.0! '$! <+! <%5'(+57(+! *'! '*7;+;%M$! 9+5'9@5%;+0! <+)!
2('-7$5+)! ).$! 2<+$5'+*+)! ).I('! <+! ;.92+5%I%<%*+*! .! '<!
7).!+2(.2%+*.!*'!;.$;'25.)!)%9%<+(')!*');(%5.)!;.$!<.)!
9%)9.)! $.9I(')0! 2.(! 7$! <+*.! '$! '<! @('+! *'! <+! '*7;+R
;%M$!9+5'9@5%;+0!>!2.(!.5(.!<+*.!'$!<+!%$&')5%-+;%M$!*'!
<.)!9+5'9@5%;.)G!67$K7'!')5+)!*%B'('$;%+)!'N%)5+$0!D+>!
5+9I%[$! )'9'H+$/+)! %92.(5+$5')! K7'! $.)! ;.$*7;'$! +!
%$;<7%(!+K7L!<+!+$+<.-L+!;.9.!7$!5%2.!*'!;.$H'57(+!('<'R
&+$5'!'$!'<!;.$5'N5.!*'!<+!'*7;+;%M$!9+5'9@5%;+G Tipo I: Abducción
1)5'!5%2.!*'!;.$H'57(+!)'!')5+I<';'!).I('!<+!I+)'!*'!7$!
).<.!;+).0!'H'92<.!.!+;.$5';%9%'$5.G!\+!;.$H'57(+!)7(-'!
;.9.!7$+!('-<+!-'$'(+<!K7'!2'(9%5'!'N2<%;+(!7$!+;.$5'R
;%9%'$5.!%$'N2<%;+I<'!*'!.5(+!9+$'(+G!w)5'!')!7$.!*'!<.)!
)%-$%!!;+*.)! K7'! T'%(;'! +5(%I7>'! +! <+! +I*7;;%M$! WO'%*0!
4``=ZG!
T.(!'H'92<.0!+<!%$5'$5+(!*'9.)5(+(!<+!;.$H'57(+!B.(97<+R
*+!2.(!+$+<.-L+!'$!'<!2(.I<'9+!4d!oC%!*.)!<L$'+)!;.(5+$!
<.)!<+*.)!'$!7$+!(+/M$!*'!4d"0!<+!5'(;'(+!5+9I%[$!<.!D+(@p0!
)'! <<'-+! +! 7$+! )%57+;%M$! )'9'H+$5'! +! <+! 9.)5(+*+! '$! <+!
!!-7(+!4G!C'!.I)'(&+!K7'!<+!5'(;'(+!(+/M$!'$!('+<%*+*!')!
;+)%!"dA0!<.!;7+<!('2(')'$5+!7$+!;.%$;%*'$;%+!).(2('$*'$R 5'G!T.(!5+$5.0!)'!2.*(L+!;.$H'57(+(!K7'0!'$!-'$'(+<0!o)%!*.)!
<L$'+)!;.(5+$!<.)!<+*.)!'$!7$+!(+/M$!*'!4d"0!'$5.$;')!<+!
5'(;'(+!<L$'+!;.(5+(@!'<!<+*.!'$!7$+!(+/M$!*'!"dApG!C%!')5'!
(')7<5+*.!B7'(+!;%'(5.0!'<!D';D.!*'!K7'!D+>+!.;7((%*.!'$!
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'$!'<!'H'92<.!+$5'(%.(Z!.!27'*'!)'(!');.-%*+!*'!7$!;.$R H7$5.! *'! ('-<+)! >+! ;.$.;%*+)G! 1$! +9I.)! ;+).)0! +<-7$+!
&'(%!!;+;%M$!*'!<+!('-<+!');.-%*+!*'I'!('+<%/+()'!2+(+!K7'!
)'+!;.$)%*'(+*+!&@<%*+G!T.(!'H'92<.0!;.$)%*'(+$*.!'<!)%R -7%'$5'!2(.I<'9+0!<.)!')57*%+$5')!2.*(L+$!;.$H'57(+(!K7'!
<+!B7$;%M$!<')!(')7<5+!B+9%<%+(G Problema 3
1$;7'$5(+!'<!2+5(M$!K7'!)%-7'!<+!)%-7%'$5'!)7;')%M$!*'!$k9'(.)!$+57(+R
<')!>!'N2([)+<+!;.9.!7$+!B7$;%M$d!=0!_0!"=0!4"u
C%!<.)!')57*%+$5')!D+$!5(+I+H+*.!;.$!2(.-(')%.$')!+(%5R 9[5%;+)0!27'*'$!2'$)+(!'$!+2<%;+(!<+!BM(97<+d!an J aA K Wn R!"Z!d0!)%'$*.!an!'<!5[(9%$.!'$[)%9.!*'!<+!)7;')%M$!
>!d!<+!*%B'('$;%+!'$5('!*.)!$k9'(.)!;.$)';75%&.)G!C%$!
'9I+(-.0!(@2%*+9'$5'!'$;7'$5(+$!<+!*%!!;7<5+*!*'!*';%R
*%(!7$!&+<.(!2+(+!d W?+3+*+)0!4``4Z0!2.(K7'!<+!)7;')%M$!
*+*+!')!;7+*(@5%;+G!#$+!;.$H'57(+!B+<)+!*'!')5'!5%2.!<')!
+>7*+!+!*');7I(%(!K7'!D+>!7$+!('<+;%M$!B7$;%.$+<!'$5('!
;+*+! 5[(9%$.! >! )7! 2.)%;%M$! *'$5(.! *'! <+! )7;')%M$0! <.!
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-'$'(+<!*'<!2+5(M$!>!<+!;.92('$)%M$!9%)9+!*'<!;.$;'2R 5.!*'!)7;')%M$G
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Tipo L: Con.eturas basadas en la percepción
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2'(;'257+<!*'!)7!2<+$5'+9%'$5.G!\+!I+)'!*'!')5'!5%2.!*'!
;.$H'57(+)!')!7$+!;7%*+*.)+!('2(')'$5+;%M$!*'<!;.$5'$%R
*.!*'<!2(.I<'9+!;.9.!7$+!%9+-'$!9'$5+<G!U.!D+>!7$+!
+5'$;%M$!%$9'*%+5+!+!<+)!('<+;%.$')!K7'!'N%)5'$!'$5('!<.)!
'<'9'$5.)!*'<!2(.I<'9+!2.(K7'!'<!B.;.!%$%;%+<!')5@!'$!<+!
;('+;%M$!*'!7$+!$7'&+!('2(')'$5+;%M$!*'<!2(.I<'9+G!#$+!
&'/!K7'!')5+!('2(')'$5+;%M$!'N%)5'0!')!+!9'$7*.!I+)5+$R 5'!+*';7+*+!2+(+!('2(.*7;%(!<+)!('<+;%.$')!'$5('!<.)!'<'R 9'$5.)! 9+5'9@5%;.)! K7'! +2+(';'$G! V'! a7/9@$! W"XXjZ!
2(.2.$'!'<!)%-7%'$5'!2(.I<'9+!K7'!5.9+9.)!+K7L!;.9.!
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Problema 4: El problema de Schroeder-Bernstein
C'+$!6!>!:!*.)!;.$H7$5.)G!C72.$-+9.)!K7'!'N%)5'!7$+!B7$;%M$!f 7$.!+!
7$.!*'!6!D+)5+!:0!>!.5(+!B7$;%M$!g!7$.!+!7$.!*'!:!D+)5+!6G!V'9.)5(+(!
K7'!'$5.$;')!'N%)5'!7$+!I%>';;%M$!h *'!6!D+)5+!:G
T.*'9.)!D+;'(!<+!)%-7%'$5'!('2(')'$5+;%M$!*'<!2(.I<'9+d!
A!>!B!).$!('2(')'$5+*.)!2.(!*.)!<L$'+)0!>!f!>!g!2.(!*.)!
"!';D+)! *');'$*'$5')! W]%-7(+! =ZG! \+)!"!';D+)! ;(';%'$5')!
('2(')'$5+$!<+)!B7$;%.$')!%$&'()+)!f-A!>!g-AG
6D.(+!2.*'9.)!;.$)%*'(+(!<+)!;+*'$+)!*'!"!';D+)!+);'$R
*'$5')! K7'! ('2(')'$5+$! ')5+)! B7$;%.$')0! >! ;<+)%!!;+(<+)!
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e!?<+)'!"d!<+!;+*'$+!5'(9%$+!).I('!6G e!?<+)'!4d!<+!;+*'$+!5'(9%$+!).I('!:G
e!?<+)'!=d!<+!;+*'$+!$.!5%'$'!$%$-k$!!!$+<0!2.(K7'!5%'$'!
7$!$k9'(.!%$!!$%5.!*'!2+).)0!.!)M<.!7$.G
1)5+!2+(5%;%M$!5+9I%[$!2'(9%5'!D+;'(!7$+!2+(5%;%M$!*'!<.)!
'<'9'$5.)!*'!6!'$!5(')!;.$H7$5.)!*%)H7$5.)0!>!+)L!5'$'(!7$+!
%$57%;%M$!*'!K7'!<+!I%>';;%M$ h!')d!C%!x '$!6!')5@!'$!7$+!;+R
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*'!;<+)'!40!'$5.$;')!hPx) J g-APx)0!>!)%!N!')5@!'$!7$+!;+*'$+!
*'! ;<+)'! =0! '$5.$;')!hPx) J fPx)G!6D.(+! '<! 2(.I<'9+! ')!
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\+!('2(')'$5+;%M$!D';D+!'$!')5'!'H'92<.!*'!<.)!;.$H7$R 5.)!6!>!:0!>!<+!('2(')'$5+;%M$!*'!"!';D+)!*');'$*'$5')!.!
+);'$*'$5')!$.!')!<.!9%)9.!K7'!<.)!;.$;'25.)!9+5'9@5%R
;.)!K7'!('2(')'$5+$0!2'(.!).$!)7!!;%'$5'9'$5'!;<+(+)!2+(+!
;.$)5(7%(!<+!B7$;%M$!h!K7'!).<7;%.$+!'<!2(.I<'9+G
Tipos de conjeturas y pasos
T+(+!'<!+$@<%)%)!*'!<+)!;.$H'57(+)!')!k5%<!*%&%*%(!'<!2(.;'R ).!'$!*%B'('$5')!2+).)G!1)5.)!2+).)!D+$!)%*.!*')+((.<<+R
*.)!;.$!*'5+<<'0!2+(+!<+)!;.$H'57(+)!*'!5%2.!"0!2.(!O'%*!
W4``"Z0!2.(!?+3+*+)!W4``4Z!>!?+3+*+)!>!?+)5(.!W4``_ZG!
1$!')5+!)';;%M$!$.)!('B'(%('9.)!+!<.)!2+).)!2+(+!<+)!;.$R H'57(+)! *'! 5%2.! "! >! )7-'(%('9.)! <.)! 2+).)! ('<+;%.$+*.)!
2+(+!<.)!.5(.)!5%2.)!*'!;.$H'57(+)!%*'$5%!!;+*.)G Pasos en con.eturas de tipo A
O'%*!W4``"Z!;.$)%*'(+!<.)!)%-7%'$5')!2+).)!2+(+!<+!%$*7;R
;%M$!+!2+(5%(!*'!7$!$k9'(.!!!$%5.!*'!;+).)!2+(5%;7<+(')d e!bI)'(&+(!7$!2+5(M$G
e!?.$H'57(+(!W;.$!*7*+Z!K7'!'<!2+5(M$!)'!+2<%;+!-'$'(+<R 9'$5'G
e!T.$'(!+!2(7'I+!<+!;.$H'57(+G e!a'$'(+<%/+(!<+!;.$H'57(+G
?+3+*+)!W4``4Z!>!?+3+*+)!>!?+)5(.!W4``_Z!.B(';'$!)%'5'!
2+).)! 9@)! *'5+<<+*.)! 2+(+! *');(%I%(! ')5'! 5%2.! *'! ;.$H'R 57(+)d!
e!bI)'(&+;%M$!*'!;+).)G e!b(-+$%/+;%M$!*'!;+).)G
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*+!).$!'N2'(%'$;%+)!;.$!;+).)!2+(5%;7<+(')!*'<!2(.I<'9+!
2<+$5'+*.G!T+(+!'<!2(.I<'9+!"!2(')'$5+*.0!'<!2(%9'(!2+).!
*'I'(L+!')5+(!<%9%5+*.!+!<+!.I)'(&+;%M$!*'!7$.!.!*.)!;+R ).)d!^!s!=!t!Yc!"^!s!_!t!""G
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<.)!')57*%+$5')!'$79'('$!)7)!)79+)!'$!.(*'$!;(';%'$5'!
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*'!9+$'(+!$+57(+<!K7'!'<!2+5(M$!2.*(L+!+2<%;+()'!+!<.)!)%R -7%'$5')!;+).)!*');.$.;%*.)G!1)5.!')!*%B'('$5'!+!;.$H'57R (+(!K7'!<+!('-<+!2.*(L+!+2<%;+()'!2+(+!5.*.)!<.)!;+).)G!1$!'<!
2(.I<'9+!"0!'<!2+5(M$!.I)'(&+*.!')!o5.*.)!estos!$k9'(.)!
2+(')!27'*'$!)'(!'N2(')+*.)!;.9.!<+!)79+!*'!*.)!$k9'R (.)!2(%9.)pG!#$+!2('*%;;%M$!)'(L+!o=4!5+9I%[$!27'*'!)'(!
'N2(')+*.!;.9.!<+!)79+!*'!*.)!$k9'(.)!2(%9.)pG La formulación de una con.etura!;.$)%)5'!'$!D+;'(!7$+!
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;%'$*.!o5+<!&'/!5.*.)!<.)!$k9'(.)!2+(')!27'*'$!)'(!'NR 2(')+*.)!;.9.!<+!)79+!*'!*.)!$k9'(.)!2(%9.)p0!)'!')5@!
B.(97<+$*.!7$+!;.$H'57(+!2+(+!'<!'H'92<.!a.<*I+;DG Validar la con.etura0! ;.9.! )'3+<+! O'%*! W4``"Z0! ')! 2.R
$'(!+!2(7'I+G!1)5.!<<'&+!+!&'(%!!;+(!7$+!2('*%;;%M$!2+(+!
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GeneraliTación de la con.etura %92<%;+ 7$!;+9I%.0!K7'!
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K7%/@)!;.$!<+!%$5'$;%M$!*'!;.$&'$;'(!+!.5(+!2'().$+!*'!K7'!
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<.-(+*.!H7)5%!!;+(!<+!;.$H'57(+!*'!a.<*I+;DG
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*%.!*'!<+!;.$5%$7%*+*!*'!;+).)G
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27'*'$!)'(!2<+$5'+*.)!'$!;.$5'N5.)!97>!*%B'('$5')d Problema 5a: El problema del apretón de manos
n!2'().$+)!)'!('k$'$!'$!7$+!;'<'I(+;%M$!>!;+*+!7$.!')5(';D+!<+!9+$.!
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*%+$5')!)%!')!2<+$5'+*.!'$!<.)!)%-7%'$5')!5[(9%$.)G Problema 5b: El problema de diagonales
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27'*'!*+(!<7-+(!+!*.)!5%2.)!*'!;.$H'57(+)!*%B'('$5')d Problema 6
6<-7$.)!2(.I<'9+)!$+57(+<')!27'*'$!)'(!'N2(')+*.)!;.9.!<+!)79+!*'!
*.)!.!9@)!$k9'(.)!;.$)';75%&.)G!T.(!'H'92<.!_!s!=!t!Ac!"`!s!"!t!4!
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Problemas y conjeturas: implicaciones en el ámbi- to educativo
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