pensamiento matematico asociativo en el siglo xxi

7/22/2019 PENSAMIENTO MATEMATICO ASOCIATIVO EN EL SIGLO XXI

1/12

@B U@G[FJC@G\D JF\@J\CKD F[DKCF\CXD@G BF @LPKFKCG L@B [CABD YYC.

CG\@_U_@\FKCG [CJOBCKF.

Urdh. Iktdr Fuaustd Iurtfld Odkfg`arf`-jfcb=jfti`jftickfsEidtjfcb.kdj

t`drcfl`bfsfsdkcfkcdg`s.obdaspdt.kdj

_@[PJ@G

@B U@G[FJC@G\D JF\@J\CKD F[DKCF\CXD @G BF @LPKFKCG L@B [CABD YYC, s`huglfj`gtf `g bf puobckfkcg= U_CGKCUCD[ L@ BF \@_CF L@ BF[ F[DKCFKCDG@[ ,qu` `s ugf \`drf Jft`jtckf, ofsflf `g bf r`fbclfl, qu` `xp`rcj`gtfg bds s`r`siujfgds `g `b scabd YYC, pfrf cgt`rpr`tfr fb Kdgmugtd KC@G\HCKD-\@KGDBACKD-[DKCFB, s`f kufb hu`r`, `g su l`sfrrdbbd cgt`rgd> sdor`tdld, kufgld s` prdluk`g tdlfs bfsCG\@_FKKCDG@[ d CG\@__@BFKCDG@[, `gtr` bds @b`j`gtds u Dom`tds l`b KdgmugtdCgt`rfktcvd kdgscl`rfld, f trfvs l` bf F[DKCFKCG JF\@J\CKF, l`skrcoc`gld bfslcv`rsfs Fsdkcfkcdg`s, `g ugf hrjubf Kdgmugtcvcstf> pfrf, kdjpfrfr kdg bds JDL@BD[F[DKCF\CXD[ \DUDBACKD[ L@ KF\@AD_F, yf `stfob`kclds y v`rchckfr, sc bf hrjubfKdgmugtcvcstf l`b Kdgmugtd Cgt`rfktcvd, kdrr`spdgl` f ugf @[\_PK\P_F F[DKCF\CXF\DUDBACKF L@ KF\@AD_Fy kdg `bbf, bf _@L \DUDBACKF L@ KF\@AD_F.Bf @lukfkcg `g `b U`r y `g `b jugld abdofbczfld, `stg `g krcscs> st` i`kid, hu`kdj`gtfld pdr bf PG@[KD(?55?), `g `b CGHD_J@ UC[F, qu` hu` ugf kdgs`ku`gkcf l`bkbfjdr, l` bf kdjugclfl Kc`gthckf-\`kgdbackf, qu` `g lch`r`gt`s kdgar`sds,l`jfglfrdg l`sl` ifk` 05 fds, `b p`bcard qu` pdlrf dkfscdgfr, bf [DO_@LCJ@G[CG

L@B KDGDKCJC@G\D> `xcac`gld, qu` s`f bf Kc`gkcf Jft`jtckf, bf qu` l` sdbukcg pdrs`r bf a`g`rfldrf l` bf prdob`jtckf, pdr j`lcd l`= @B B@GAPFM@ CG\@A_FLD_ L@BKDGDKCJC@G\D, y qu, `g `b ?56;, BF \@D_F L@ BF[ F[DKCFKCDG@[, bf `stprdpckcfgld.Udr dtrd bfld, U_CGKCUCD[ L@ BF \@D_F L@ BF[ F[DKCFKCDG@[, b` p`rjctcr f kflfugd l` ust`l`s, bf dpdrtugclfl l` kdgstrucr sus prdpcfs @[\_PK\P_F[ (U`rsdgfb,Bfodrfb, Hfjcbcfr, [dkcfb, _`bcacdsf, L`pdrtcvf, ), qu` lcrcmf sus Fktcvclfl`sprdluktcvfs,kdg= D_L@G, BACKF y @YFK\C\PL, `g `b kujpbcjc`gtd l` sus dom`tcvds.@g `stf dpdrtugclfl, s` `xpbckfr bf r`pr`s`gtfkcg scjobckf utcbczflf, `g `b frtkubd= @BU@G[FJC@G\D JF\@J\CKD F[DKCF\CXD @G BF @LPKFKCG L@B [CABD YYC > pfrf,l`sfrrdbbfr ifocbclfl`s fsdkcftcvfs, `g bfs _@L@[ F[DKCF\CXF[kdgstruclfs.

FO[\_FK\

Jfti`jftckfb \icgncga cg @lukftcdg Fssdkcftcv` YYC K`gtury, cs ofs`l dg ti`puobckftcdg= U_CGKCUB@[ DH \I@ \I@D_S DH F[[DKCF\CDG[, wicki cs fjfti`jftckfb ti`dry ofs`l dg ti` r`fbcty tift iujfg o`cgas `xp`rc`gk` cg ti` YYCk`gtury, td `xpbfcg ti` s`t [kc`gk` \`kigdbday-sdkcfb, wift`v`r cg cts cgt`rgfbl`v`bdpj`gt, `sp`kcfbby wi`g prdluk`l fbb CG\@__@BF\CDG[ICU[ dr CG\@_FK\CDG[

prdluk`l o`tw``g `b`j`gts dr dom`kts dh ti` kdgscl`r`l Cgt`rfktcv` [`t tirduaiJF\I@JF\CKFB F[[DKCF\CDG l`skrcocga ti` vfrcdus fssdkcftcdgs, cg f Kdgmugtcvcsthdrjubf, td kdjpfr` wcti \dpdbdackfb Fssdkcftcv` Jdl`bs dh Kft`adry, fbr`fly s`t, fglki`kn, ch ti` Kdgmugtcvcst hdrjubf dh ti` Cgt`rfktcv` [`t kdrr`spdgls td f F[[DKCF\CDG[
mailto:[email protected]:[email protected]:[email protected]:[email protected]


2/12

\DUDBDACKFB [\_PK\P_@ DH KF\@AD_S fgl wcti ct, ti` \DUDBDACKFB G@\QD_N DHKF\@AD_S.@lukftcdg cg U`ru fgl cg ti` abdofbcz`l wdrbl cs cg krcscs> tics hfkt, wfs kdjj`gt`l oyPG@[KD (?55?), cg ti` Ucsf _`pdrt, wicki wfs f kdgs`qu`gk` dh ti` [kc`gtchck\`kigdbday kdjjugcty kbfjdr, wicki ft vfrcdus kdgh`r`gk`s kbfcj`l hdr 05 y`frs, tiìfzfrl wicki kdubl r`subt, ti` `xk`sscv` ngdwb`la`, l`jfglcga tift Jfti`jftckfb

[kc`gk`, wicki acv` f sdbutcdg o`kfus` cs ti` a`g`rftdr dh ti` prdob`j, oy= ti` Bfgaufa`ngdwb`la` Cgtàrftdr, fgl, cg ?56;, ti` ti`dry dh ti` Fssdkcftcdgs, tift cs b`flcga.Dg ti` dti`r ifgl, U_CGKCUB@[ DH \I@ \I@D_S DH F[[DKCF\CDG[, wcbb fbbdw `fkidh ydu ti` dppdrtugcty td oucbl ti`cr dwg struktur`s (U`rsdgfb, Qdrn, Hfjcby, [dkcfb,_`bcacdus, [pdrts,), wicki lcr`kts ti`cr prdluktcv` fktcvctc`s wcti= D_L@_, BDACK fgl@YFK\G@[[, cg hubhcbbcga cts dom`ktcv`s.\ics tcj`, wcbb `xpbfcg ti` syjodbck r`pr`s`gtftcdg us`l cg ti` frtckb`= JF\I@JF\CKFB\ICGNCGA CG @LPKF\CDG F[[DKCF\CDG DH \I@ K@G\P_S, hdr, l`v`bdp sncbbsfssdkcftcv`, cgFssdkcftcv` G`twdrnsoucbt.

N@S QD_L[=

6. CG\@_FKKCG CG\@__@BFKCG. [` l`gdjcgf fs, fb cgt`rkfjocd qu` s` r`fbczf`gtr` lds dom`tds, `g ug Kdgmugtd Cgt`rfktcvd.

?. F[DKCFKCG JF\@J\CKF. @s ug prdk`sd, l` cgt`rfkkcg `gtr` lds d js Dom`tds(jutufj`gt` `xkbuy`gt`s), `g ug Kdgmugtd Cgt`rfktcvd> ldgl`, bds Dom`tds, s`kdgkft`gfg (Kdjpdsckcg) `g hdrjf kdgtcguf, pdr j`lcd l` bds prdluktds, l` kflfpfr l` dom`tds.

;. JDL@BD[ F[DKCF\CXD[ \DUDBACKD[ L@ KF\@AD_F. [dg hrjubfs

Kdgmugtcvcstfs, l` bfs Fsdkcfkcdg`s Jft`jtckfs, qu` s` a`g`rfg `g ug KdgmugtdCgt`rfktcvd, qu` sftcshfk`g bfs kdglckcdg`s l` bfs l`hcgckcdg`s jft`jtckfs, \dpdbdafy bf Kftàdrf.

ldgl`, s` hdrjfg, bfs Fsdkcfkcdg`s= Xfkf y Pgcv`rsfb, y qu, bf Pgcg l`bfs Fsdkcfkcdg`s, s`f ugf Fsdkcfkcg `g bf Kdb`kkcg l` Fsdkcfkcdg`s.

0. KF\@AD_F. L`hcgckcg Jft`jtckf, qu` kdgscst` l` ug Kdgmugtd l` Dom`tds (Kbfs`s)y l, ugf Kdb`kkcg l` Cgt`rfkkcdg`s (Frtckubfkcdg`s)> tfb qu`, `xcst` ugfkdrr`spdgl`gkcf ocugvdkf, `gtr` `b Kdgmugtd l` Kbfs`s y bf Kdb`kkcg l` Cl`gtclfl`s,`g bf Kdb`kkcg l` bfs Frtckubfkcdg`s.

8. @[\_PK\P_F F[DKCF\CXF \DUDBACKF L@ KF\@AD_F.@s `b lcs`d a`g`rftcvd(rodb), l` bds Dom`tds l`b Kdgmugtd Cgt`rfktcvd, qu` sftcshfk` bfs kdglckcdg`s l` bfsl`hcgckcdg`s jft`jtckfs= bf \dpdbdaf y bf Kftàdrf.

1. CGHD_J@ UC[F.@s `b r`subtfld l` bf `vfbufkcg `lukftcvf, r`fbczfld pdr bf PG@[KD(?55?), `g bf qu` `b U`r hu` uockfld `g bds btcjds buafr`s.

9. _@L@[ F[DKCF\CXF[. @s `b Lcfarfjf Cgt`rfktcvd, l` bfs Frtckubfkcdg`s qu` s`hdrjfg, `g kflf Fsdkcfkcg Jft`jtckf y kdg bf Cgtàrfkcg, l` bfs Fsdkcfkcdg`s gdtrcvcfb`s, l`b Kdgmugtd Cgt`rfktcvd.

3. F[DKCFKCG GD \_CXCFB. [` lck` fs f bf Fsdkcfkcg Jft`jtckf qu` s` a`g`rf `g ugKdgmugtd Cgt`rfktcvd, qu` `s lch`r`gt` l` bf Fsdkcfkcg Xfkf y bf Fsdkcfkcg Pgcv`rsfb.


3/12

CG\_DLPKKCG

L`sl` qu`, bf KBPBF y `b \DJD, sdg bds `b`j`gtds osckds l` bf Kc`gkcf y bfsFK\CXCLFL@[, qu` a`g`rfg bds s`r`s iujfgds fb hdrjfr pfrt` l` ug Kdgmugtd, sdgKDGMPG\D[ CG\@_FK\CXD[> pdrqu`, bds @b`j`gtds u Dom`tds l`b Kdgmugtd sctufkcg d kdsf,`stg `g jdvcjc`gtd, scagchckf qu`, `stfjds kdgskc`gt`s l` bf cjpdrtfgkcf, qu` tc`g`g bfsCG\@_FKKCDG@[d CG\@__@BFKCDG@[ g b scabd YYC.

@st` i`kid, bd dos`rvfjds g b frtkubd=@B U@G[FJC@G\D JF\@J\CKD F[DKCF\CXD @G BF @LPKFKCG L@B [CABD YYC

qu` tc`g` pdr dom`tcvd, kuorcr bfs lcv`rsfs g`k`sclfl`s, qu` s` r`quc`r`g= pfrf, t`g`r ugf@LPKFKCG, kdg KFBCLFLy @YK@B@GKCF, ghdkfld ifkcf=

@B LDK@G\@ @B @[\PLCFG\@ BF CG[\C\PKCG @LPKF\CXF

prdpdrkcdgfgld ugf i`rrfjc`gtf Kc`gthckf Jft`jtckf, qu` b` p`rjctf fl`kufr bf @lukfkcgfb scabd YYC> ldgl`, bf sdor`lcj`gscg l`b Kdgdkcjc`gtd, if a`g`rfld ug prdob`jf sdkcfb, `g

bf @lukfkcg U`rufgf.@s fs, qu` `b LDK@G\@ s` o`g`hckcf, fb lcs`fr y t`g`r, JDL@BD[ JF\@J\CKD[F[DKCF\CXD[, yf stfob`kclds> pfrf,

6.@[\_PK\P_F_ @B FPBF L@ LCK\FLD L@ KBF[@[, bdarfgld l`sfrrdbbfr kdg bds@stulcfgt`s. Lcgjckfs @strukturfb`s, utcbczfgld \`kgdbdafs @lukftcvfs Cgt`rfktcvfs.

?.@[\_PK\P_F_ @B KDGDKCJC@G\D. Vu` fb s`r kbfschckfld, scagchckf qu`, kufbquc`rt`jf, s`f pdscob` lcs`fr ugf @strukturf, r`fbczfgld bfs Lcgjckfs @strukturfb`s, kdgbds @stulcfgt`s.

;.@[\_PK\P_F_ BF KDJU_@G[CG B@K\D_F S BF @BFOD_FKCG L@ \@Y\D[, pdrj`lcd l` bf \`drf l` bfs Fsdkcfkcdg`s.

Fscjcsjd, s` o`g`hckcf b @[\PLCFG\@, fb lfrs` ku`gtf qu`=6. BF HFJCBCF.?. @B B@GAPFM@;. BF [DKC@LFL.


4/12

JF\@_CFB S J\DLDUfrf bdarfr gu`strd dom`tcvd, s` if prdpdrkcdgfld bds JDL@BD[ JF\@J\CKD[F[DKCF\CXD[, s`ag s` l`tfbbf `gs`auclf.6. LDK@G\@, s` pr`s`gtf `g=

f) Fubf l` Lcktfld kdg @strukturf Fsdkcftcvf, pfrf


5/12

y bf _`l Fsdkcftcvf \dpdbackf l` Kft`adrf.

Bfs bg`fs `gtr`kdrtflfs, kdrr`spdgl`g f bfs CG\@_FKKCDG@[ CG\@BCA@G\@[,qu` prdluk` bf @strukturf Fsdkcftcvf \dpdbackf l` Kft`adrf, `gtr` bds

Dom`tds l`b Kdgmugtd Fsdkcftcvd. Fubf l` Lcktfld l` Kbfs`s.

L

F??

F?;

F?

F06

F0? F0;

F0


6/12

f) Kdgdkcjc`gtd kdg @strukturf Fsdkcftcvf, s` tc`g` ug \`jf \, kdg kuftrdsuot`jfs= F, o, K y L.

sc`gld, bf `strukturf.

Vu` kdrr`spdgl` f bf Fsdkcfkcg \dpdbackf l` Kft`adrf.

\: r, O, O, F, O, L, K, P\


o) @b \`xtd Cgt`bca`gt`, s` pu`l` fpr`kcfr `g `b frtkubd= @B U@G[FJC@G\DJF\@J\CKD F[DKCF\CXD @G BF @LPKFKCG L@B [CABD YYC.

K L

F

O


7/12

?. CG[\C\PKCG @LPKF\CXF, s` pr`s`gtf bf @[\_PK\P_F.


8/12



9/12

_@[PB\FLD@gsàuclf, s` vf f prdpdrkcdgfr, fbaugfs sua`r`gkcfs scjobckfs, qu` s` utcbczf `g bf\`drf l` bfs Fsdkcfkcdg`s> pfrf, qu` bds Ldk`gt`s y @stulcfgt`s, pu`lfg cgar`sfr f bfscjodbdaf jft`jtckf y bf kdjpr`gscg, l` bfs cjpbckfgkcfs backfs-jft`jtckfs, qu`trf` kdgscad `g `b Kdjpdrtfjc`gtd Iujfgd, kdjd bfs scauc`gt`s=6. L`sl` qu`, `g bf CG[\C\PKCG @LPKF\CXF, tdlfs bfs CG\@__@BFKCDG@[, sdg

r`fbczflfs pdr s`r`s iujfgds y kjd, kflf s`r iujfgd `s gckd, d s`f, gd `xcst` `g `bjugld t`rrqu`d, dtrd caufb> `gtdgk`s, tdlfs bfs p`rsdgfs, s` b`s kdgscl`rf dom`tdslch`r`gt`s. Buàd, sc r`pr`s`gtfjds fb [r. Mufg Ur`z, pdr

M= [r. Mufg Ur`zy sc, `b [r. Mufg Ur`z, `s Ldk`gt` l` Vujckf, `g ; l` s`kuglfrcf> `gtdgk`s, `g bf _`ll`b Fubf l` Lcktfld `g Vujckf, `b lcfarfjf r`pr`s`gtftcvd `s=

M

pdrqu`, `b [r. Mufg Ur`z, `s `b GPKB@D (IPO-GDLD), l`b Fubf l` Lcktfld, d s`f, `b

bl`r y pdr bd tfgtd, tc`g` qu` fsujcr bf hugkcg l` auf, kdg sus fbujgds, lurfgt` `blcktfld `g `b Fubf, qu` scjobckfj`gt` s` r`pr`s`gtf

Mqu` `g bf @strukturf Fsdkcftcvf, bd r`kdgdk`jds kdjd ugf Fsdkcfkcg ugctfrcf, qu`tfjocg scagchckf dom`td kdg jdvcjc`gtd prdpcd.U`rd, sc `b Ldk`gt`, [r. Mufg Ur`z, l` Vujckf `g ; s`kuglfrcf, s` b` kdgscl`rfhdrjfgld pfrt` l`b Kdgmugtd l` Urdh`sdr`s l` [`kuglfrcf> `gtdgk`s, `g bf @strukturfy _`l l` Urdh`sdr`s l` [`kuglfrcf, `b lcfarfjf, s`r=

M

?. @g bf pacgf `gtdgk`s, s` prdluk` bf F[DKCFKCG,f) F;?, F06

[c F;?, `s quc`g sfbulf y kdjc`gzf, bf kdgv`rsfkcg, `b lcfarfjf `g _@L, `s=

F;? F06

o) F06, F;?[c F06, `s quc`g sfbulf y a`g`rf, bf pbtckf, `b lcfarfjf `g _@L, `s=

F06 F;?

GD\F= Bf F[DKCFKCG, `gtr` lds Dom`tds (U`rsdgfs), kdrr`spdgl` f ugf s`rc` l`

Fkkcdg`s, l`b qu` prdpdg` bf Fkkcg y l` _`fkkcdg`s, l` pfrt` l` bf sàuglfp`rsdgf. Backfj`gt`, `b r`subtfld gdrjfb, `s l` quc`g prdpdg` bf FKKCG>scg `jofrad, ify qu` `stfr p`glc`gt` l`b r`subtfld.


10/12

;. X`fjds, bf sctufkcg qu` s` hdrjf, `g bf @[\_PK\P_F F[DKCF\CXF L@B FPBF L@LCK\FLD, `gtr`=

L = Urdh. Xktdr U`f (Ldk`gt`)F8 = Fgtdgcd Kfstrd (@stulcfgt`)

`g `b Fubf l` Lcktfld> kufgld, `b @stulcfgt`, pr`augtf fb Ldk`gt`, `b lcfarfjf `g_@L, `s `b scauc`gt`=

sc`gld, bf r`pr`s`gtfkcg scjobckf, bfsF[DKCFKCDG@[=

L y L, F8


11/12

0. Fidrf, v`fjds bf sctufkcg, `g bf qu` `b @stulcfgt`,F6< = U`lrd X`gtugd

Vuc`r` r`fbczfr ugf pr`augtf, fb Ldk`gt`L = Urdh. Xktdr U`f

`g `bFPBF L@ LCK\FLD L@ KBF[@[.

Kjd, bf sctufkcg, gd `st cgstctuclf> yf qu`, pdr bf Fsdkcfkcg \dpdbackf l`

Kft`adrf, Fcx , l` bf pacgf


12/12

\fjocg, `s kdjg dos`rvfr, qu` bds Dom`tds `g bds Kdgmugtds Cgt`rfktcvds, sdg GKB@D[l` fbaugf Ufrtckcgy sdg DOM@\D[, `g ugf Ufrtckcglch`r`gt`, d s`f, qu` bds Dom`tds `gbf @[\_PK\P_F F[DKCF\CXF, kujpb`g bf pfrfldmf= L@ BD PGD S BD JB\CUB@, tfbkdjd bd j`gkcdgf `b Lr. @LAF_ JD_G, kufgld lck`=

BF KDJUB@MCLFL, @[ \@MCLD (KDJUB@YP[= bd qu` `st t`mcld `g Kdgmugtd)L@ KDG[\C\PS@G\@[ I@\@_DAG@D[ (Kbfs`s u Dom`tds lch`r`gt`s) VP@

_@U_@[@G\FG BF UF_FLDMF= L@ BD PGD S BD JPB\CUB@= S, VP@, FB

@b l`sfrrdbbd l` bf Kc`gkcf y \`kgdbdaf, f bf pfr l`, bf \`drf l` bfs Fsdkcfkcdg`s,pdscocbctf sdbukcdgfr bfs lcv`rsfs sctufkcdg`s, qu` gd s` pdlfg r`fbczfr ifk` 05 fds, ftrfvs l` bf _@L F[DKCF\CXF \DUDBACKF L@ KF\@AD_F> `gtdgk`s, `s `b jdj`gtdl` lfr pfsd, f bf \`kgdbdafy bf \`drf l` bfs Fsdkcfkcdg`s, pfrf qu` kujpbfg su rdb`stfob`kcld y gd pdg`r trfofs, fb l`sfrrdbbd qu` pdlfjds bdarfr, kdg `b KDGDKCJC@G\D>pfrf, p`rh`kkcdgfr, bf KC@GKCF y \@KGDBDAF, `g `b kdrtd pbfzd ` cgar`sfr fb prcj`r

jugld.Udr dtrd bfld, bf \@D_F L@ BF[ F[DKCFKCDG@[, `s ugf \`drf Jft`jtckf, ofsflf `gbf r`fbclfl qu` `xp`rcj`gtf `b s`r iujfgd, `g `b scabd YYC> pfrf, lcs`fr @[\_PK\P_F[,qu` b` p`rjctf f kflf ugd l` ust`l`s, kdgstrucr su prdpcf @[\_PK\P_F (U`rsdgfb,Bfodrfb, Hfjcbcfr, [dkcfb, _`bcacdsf, L`pdrtcvf, ), qu` lcrcmf sus fktcvclfl`sprdluktcvfs, kdg= Drl`g, Backf y @xfktctul.

_@H@_@GKCF[ OCOBCDA_HCKF[

O@_\FBFGHHS BPLQCGA XDG= U`rsp`ktcvfs `g bf \`drf A`g`rfb l` [cst`jfs.

@stulcds Kc`gthckds Hcbdshckds. 631

pensamiento matematico asociativo en el siglo xxi

Documents