Post on 15-May-2018
maTematikis swavlebis
konstruqtivistuli meTodi
konstruqtivizmi da maTematikis swavlebis
Tanamedrove meTodebi
konstruqtivizmis (constructus – laT. agebuli, aSenebuli) – ZiriTadi arsi
mdgomareobs imaSi, rom moswavleebma codnis konstruireba moaxerxon
ZiriTadad sakuTar gamocdilebaze dayrdnobiT da ara mxolod
maswavleblis mier miwodebuli mza informaciis safuZvelze.
tradiciuli swavlebisas axal sakiTxs maswavlebeli xsnis, konstruqtivis-
tuli midgomisas ki codna ar miewodeba mza saxiT, aseTi swavlebis mTavari
xazia moswavlis gamocdilebis mizanmimarTuli gamdidreba da yoveli axali
codnis bunebriviad daSeneba. am gamdidrebul gamocdilebaze. swavlebis
gamowveva dRes maswavleblisgan moiTxovs, rom man Seqmnas iseTi saswavlo
viTareba, romelic ,,CaiTrevs” da aiyoliebs moswavles. swavleba da
maswavlebeli moswavles unda sTavazobdes sakuTari mosazrebebis
Camoyalibebisa da gamocdilebis axsnis, movlenaTa Sefasebis, sxvebTan
komunikaciis, maTematikuri modelebis gamoyenebis sayrdensa da safuZvels.
maTematikis konstruqtivistuli swavlebis safuZvlebia: swavleba codnis konstruirebis gziT; swavleba amocanebis amoxsnis saSualebiT; gakveTilebis dagegmva; moswavleebis winsvlis zusti Sefaseba.
codnis konstruirebis procesisaTvis aucilebelia: arsebuli winare codna; axali codnis asagebi masala; aqtiuri azrovneba.
maswavlebelma unda Seuqmnas moswavleebs situaciebi, romlebic
ubiZgebs maT aucilebeli saazrovno konstruqciebis agebisaken.
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keTebiT swavla konstruqtivizmis praqtikuli ganxorcielebaa. es aris
aqtiuri swavlis nairsaxeoba, romelSic mTavari saswavlo saSualebaa
moswavlis saqmianoba, moswavle swavlobs muSaobis procesSi. swavleba
SeiZleba mimdinareobdes maswavleblis xelmZRvanelobiT, ufrosi megobris
rCevebis daxmarebiT, an TviTganaTlebis pirobebSi.
ZiebiT-aRmoCeniTi / sokrateseuli / evristikuli (,,𝒆𝒆𝒖𝒖𝒓𝒓𝒊𝒊𝒌𝒌𝒂𝒂” – berZ. vpoulob) meTodis safuZvelia _ aqtiuri swavlebis nairsaxeoba, romelic emyareba
moswavlis mier saswavlo sakiTxis arsis aRmoCenas. konstruqtivizmi da
evristika imdenad axlosaa erTmaneTTan, rom maT xSirad sinonimebadac ki
xmaroben. es, cxadia, arazustia: pirvelSi mTavaria ageba, meoreSi _ aRmoCena.
magram, es ori meTodi, rogorc wesi, paralelurad gamoiyeneba. SevafasoT konstruqtivizmi da evristika. ra aqvT maT saerTo da riT
gansxvavdebian:
konstruqtivizmsa da evristikas maT dadebiT mxareebTan erTad siZneleebic
axlavs.
dadebiTi mxareebi: iZleva aqtiur, gaazrebul, Rrma codnas;
Zlier aviTarebs moswavlis zogad gonebriv unar-Cvevebs, damoukidebeli
muSaobis unar-Cvevebs, gergilianobas, SemoqmedebiTobas.
sakmaod saxalisod mimdinareobs
siZneleebi:
saWiroebs SemoqmedebiTobaze agebul da Tan saaTiviT zustad gaTvlil
saswavlo programasa da saxelmZRvanelos;
saWiroebs maswavleblis saTanado momzadebas.
konstruqtivizmisTvis Zalian mniSvnelovania xSiri ukukavSiri, anu
maswavlebeli xSir-xSirad da yuradRebiT unda iRebdes moswavleTa
gamoxmaurebas, unda amowmebdes:
ramdenad gaazrebulad mimdinareobs davalebis keTebis procesi;
calkeuli moswavleebisTvis xom ar darCa gaugebari axali sakiTxebi;
programis gavlis tempi xom araa zedmetad Cqari (an neli);
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naswavli sakiTxebi xom ar rCeba mxolod faqtobrivi codnis, infor-
mirebulobis doneze, xom iqceva aqtiur codnad;
naswavli sakiTxebi aris Tu ara mravalmxrivad gaSlili;
sakmarisad viTardeba Tu ara saTanado maTematikuri da gamoyenebiTi
unar-Cvevebi;
sakmarisad viTardeba Tu ara moswavleTa SemoqmedebiTi unari, intuicia,
mixvedriloba.
yovelive amis misaRwevad saWiroa gaRrmavebuli, mravalmxrivi, auCqarebeli
swavleba, ZiriTadad orientireba naswavlis xarisxze _ da ara moculobaze.
ricxvebis gaazrebiT swavla
V-VI klasebSi
gamWoli kompetenciebi V-VI klasebSi
maTematikaSi dawyebiTi ganaTlebis yovel stadiaze saswavlo programis
safuZvels warmoadgens qvemoT mocemuli gamWoli kompetenciebi da TiToeul
maTganTan dakavSirebuli ZiriTadi cnebebi (ix. suraTi 1):
Tvla (mxolod dawyebiTi
safexuris qveda klasebSi);
ricxvebze moqmedebebis
gaazreba;
raodenoba;
kavSirebi;
warmodgenis formebi;
proporciuli azrovneba
(mxolod dawyebiTi
safexuris mexuTe-meeqvse
klasebSi).
es kompetenciebi aris
urTierTdamokidebuli,
Tanabarzomierad mniS-
vnelovani da, garkveulwilad,
erTmaneTsac gadafaravs.
magaliTad, Sinaarsiani, kargad gaazrebuli daTvla gulisxmobs imis codnas,
rom am procesSi konkretuli raodenobebi Sesabamisi ricxvebiT gamoixateba.
am codnis dakavSireba aTobiTi sistemis principebTan moswavleebs ricxvebis
gaazrebisTvis karg safuZvels sTavazobs.
suraTi 1. gamWoli kompetenciebi V-VI klasebSi
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samive kompetencia – Tvla, raodenoba da kavSirebi – ariTmetikuli
moqmedebebis gaazrebaze axdens gavlenas, rac, Tavis mxriv, proporciul
azrovnebas exeba. es oTxi didi idea Tavis TavSi moicavs maTematikaSi
gamoyenebul warmodgenis formebs, kerZod cifrebs, algoriTmebs da
sxvadasxva saxis simboloebs.
moqmedebebis gaazreba
moswavleebs ricxvebze moqmedebebis arsSi wvdoma da procedurebis codna
moeTxovebaT.
qvemoT mocemulia ariTmetikuli moqmedebebis swavlebis zogadi strategiebi.
mniSvnelovania, maswavleblebma:
yuradReba gaamaxvilon gamoTvla-gamoangariSebaze orientirebul,
problemebis gadaWraze dafuZnebul savarjiSoebze:
mozardebi iseTi situaciuri savarjiSoebiT moamaragon, sadac problemis
gadaWrisas ricxvebze moqmedebebis (magaliTad, Sekrebis) ganxorcieleba
mravali sxvadasxva gziT SeiZleba, es moswavleebs Tavdajerebulobas
SesZens da maT codnasac ganamtkicebs;
waaxalison moswavleebis mier damxmare masalebisa da TvalsaCinoebis
saSualebebis gamoyeneba problemebis modelirebisas, SesaZlebloba
miscen moswavleebs SeimuSaon problemebis gadaWris sakuTari, maT mier
Camoyalibebuli strategiebi; daxmareba gauwion muSaobis procesis ukeT
gaazrebasa da azrebis gadmocemaSi Ria kiTxvebiT, aseve gamokiTxvis
saSualebiT;
waaxalison moswavleebis mier argumentaciis strategiebis mosafiqre-
blad; Zalian mniSvnelovania, waaxalison moswavleebi gauziaron
sakutari mosazrebebi maswavlebelsa da Tanaklaselebs;
ganmaviTarebeli ukukavSiris miwodebis mizniT, droulad Seafason
calkeuli moswavlis mier miRebuli codna da mis mier gamoTqmuli
azrebi, raTa man sakuTari mcdari warmodgenebisa da dasaxvewi ideebis
Sesaxeb zustad gaigos da Secdomebi gamoasworos.
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raodenoba
ras niSnavs “raodenoba”?
mexuTe-meeqvseklaselebisTvis ricxvebis gaazreba mTeli ricxvebiT,
aTwiladebiT, wiladebiTa da procentebiT gamosaxul raodenobebzea
damokidebuli. isini raodenobas iazreben ara marto modelebiTa da
vizualuri damxmare masalebis saSualebiT, aramed ricxvebis ufro
abstraqtulad warmodgeniTac. am dros ricxviTi RerZebis gamoyeneba
gansakuTrebiT sasargeblo gamocdilebas iZleva. marTalia, aseTi RerZi
calkeul ricxvebs Sesabamis raodenobebTan ar akavSirebs, magram friad
sasargeblo instrumentia, raTa bavSvma dainaxos kavSiri sxvadasxva
raodenobebs Soris.
raodenoba moqmedebebTan, cifrebis poziciasTan da wiladebTan kavSiris WrilSi
mexuTe-meeqvse klasebis pedagogebs Soris zogjer SeiniSneba tendencia,
aqcents gamoTvlis procedurebis swavlebaze axdendnen, raodenobis cnebis
gaazrebis nacvlad. arada, bavSvebisTvis uaRresad didi mniSvneloba aqvs
ricxvebisa da raodenobis cnebebis jerovnad gaazrebas Sinaarsian
konteqstze dayrdnobiT.
am dros Semdegi savarjiSoebis gamoyenebaa mizanSewonili:
didi ricxvebis mniSvnelobebis Sedareba Sesabamis konteqstze
dayrdnobiT (magaliTad, imis gasagebad, ramdenad didia 1 miliardi 1
milionze, isini drois konteqstSi SeadareT erTmaneTs – 1 milioni wami
11 dReze cota metia, 1 miliardi wami ki daaxloebiT 32 weliwads
utoldeba!). SeecadeT, Sinaarsiani da, amave dros, saintereso konteqstze
dafuZnebuli amocanebi SearCioT;
nacnob konteqstze dafuZnebuli amocanebis amoxsna (magaliTad, ,,Tu baf-
Tis gasakeTeblad 1-metriani lentis 3/8 gWirdeba, sul am lentisgan
ramdeni bafTa gamova?).
raodenoba Sefasebis, fardobiTi sidideebis, proporciulobisa da ra-cionaluri ricxvebis WrilSi
mexuTe-meeqvse klasebSi SefasebiTi unarebis ganviTarebis saqmeSi kritikuli
mniSvneloba aqvs raodenobis SegrZnebas.
bavSvebma wiladebis odenobebis warmosaxvis unaric unda ganiviTaron da
Semdeg am unarze dayrdnobiT sxvadasxva raodenobebis gansazRvraSi
imecadinon.
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iseTi aqtivobebi, sadac mozardebi wiladebis, aTwiladebisa da procentebis
modelirebas damxmare masalebiTa (magaliTad, seqtorebiani wreebi,
wiladebis RerZebi, kedeli-sveti-kubis modelebi) da sqemebiT (magaliTad,
ricxviTi RerZebi, 10×10-ze cxaurebi) axdenen, maT ZiriTadi wiladi
odenobebis – magaliTad, 1/4, 0,75 da 50% _ vizualizaciis SesaZleblobas
aZlevs. bavSvebi am YTvalsaCinoebas sxva wiladi odenobebis
gansazRvrisTvisac iyeneben (magaliTad, qilaSi darCenilia saRebavis
TiTqmis¼1/4; Jurnalis gverdebis 50%-ze meti sareklamo gancxadebebs ukavia).
raodenoba ricxvebze moqmedebebis efeqtebis gaazrebis WrilSi
dawyebiT klasebSi bavSvebi ecnobian Sekreba-gamoklebis efeqtebs: Sekrebisas
raodenoba matulobs, gamoklebisas ki mcirdeba. mexuTe-meeqvse klasebSi maT
evalebaT ukve gamravleba-gayofis efeqtebis ufro Rrmad gaazreba _ imis
gageba, Tu ratom xdeba, rom:
1-ze did ricxvze gamravlebiT raodenoba izrdeba (magaliTad, 4×1,5=6);
1-ze patara ricxvze gamravlebiT raodenoba mcirdeba (magaliTad, 4×0,5=2);
1-ze did ricxvze gayofisas raodenoba mcirdeba (magaliTad, 3,5:7=0,5);
1-ze patara ricxvze gayofisas raodenoba izrdeba (magaliTad 4 : 0,5= 8).
moswavleebma moqmedebebis Sesrulebis efeqtebze konkretul ricxvebze ope-
rirebis saSualebiTac unda imecadinon – magaliTad, ricxvis 10-ze, 100-ze da
1000-ze gamravleba-gayofiT, an 0,1-ze, 0,01-ze da 0,001-ze gamravlebiT.
wiladebis cnebis jerovani gaazreba mozardebs momdevno klasebSi ufro
rTuli moqmedebebis SesrulebaSic daexmareba da ricxvebis gaazrebis
deficitiT gamowveuli Secdomebisganac daicavs (magaliTad, wiladebis
Sekrebisas mricxvelisa da mniSvnelis mimatebis procesSi – roca 1/2+1/3
operacias axorcieleben, 2/5-is nacvlad, swori pasuxi, 5/6, rom miiRon).
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kavSirebi
kavSirebi ricxvebis Sedareba-dalagebisas
mexuTe-meeqvse klasebSi aTobiT sistemaSi arsebuli kavSirebis codna kri-
tikulad mniSvnelovania mravalniSna mTeli ricxvebisa da aTwiladebis
gasaazreblad, aseve gamoTvlis Sinaarsiani strategiebis gansaviTareblad.
am asakobriv jgufebSi aTobiTi sistemis kargad mcodne moswavles Semdegi
cnebebis aTviseba SeuZlia:
Cveni ricxviTi sistema aTobiT kavSirebzea dafuZnebuli (magaliTad, 10
erTeuli qmnis 1 aTeuls, 10 aTeuli – 1 aseuls, 10 aseuli – 1 aTaseuls);
aTwiladebis sistema aTobiTi sistemis erTgvar gafarToebas warmoadgens
(magaliTad, 10 meaTedi qmnis 1-s, 10 measedi – 1 meaTeds, 10 meaTasedi – 1
measeds);
cifris mier gamoxatul sidides ricxvSi misi pozicia gansazRvravs
(magaliTad, 5 347-Si 3-iani gamosaxavs 3 aseuls);
ricxvis CanawerSi nulianis gamoCena Sesabamisi saTanrigo raodenobis
ararsebobas aRniSnavs (magaliTad, 6,05-Si meaTedebis TanrigSi arc erTi
erTeuli ar aris);
ricxvi misi gamomsaxveli cifrebis poziciuri mniSvnelobis safuZvelze
iSleba (magaliTad, 27,82 = 20 + 7 + 0,8 + 0,02);
ricxvSi SesaZlebelia raodenobebis gadajgufeba. magaliTad, 417
SeiZleba ganvixiloT rogorc:
4 aseuli, 1 aTeuli da 7 erTeuli;
3 aseuli, 11 aTeuli da 7 erTeuli;
3 aseuli, 10 aTeuli, 17 erTeuli da a.S.
marTalia, kedeli-sveti-kubis modelebi, aTwiladebis RerZebi da 10×10-ze
cxaurebi mozardebs ricxvebis CanawerebSi cifris poziciis mniSvnelobis
gagebaSi exmareba, magram es damxmare masalebi ver uzrunvelyofen poziciis
cnebis avtomaturad gaazrebas. poziciuri sistemis gasagebad moswavleebs
sakuTari codnis konstruirebis SesaZlebloba unda mivceT. amaSi maT
daexmareba iseTi aqtivobebi, rogoricaa Sesabamisi modelebis xSiri
gamoyeneba da amocanebis amoxsnisas ideebis damxmare TvalsaCinoebis
meSveobiT warmodgena.
wiladebis Sedareba-dalagebis nayofieri strategiaa ricxvebis Sedareba
niSnul ricxvebTan. magaliTad, 3/8, 5/6, da 7/8-is dalagebisas zrdadobis
mixedviT moswavlem SeiZleba gaiTvaliswinos iseT niSnulebTan siaxlove,
rogoricaa 0, 1/2, da 1:
• 3/8 naklebia 1/2-ze (4/8);
• 5/6 1/6-iT naklebia 6/6-ze, Sesabamisad 1-Tan axlosaa;
• 7/8 1/8-iT naklebia 8/8-ze da aseve axlosaa 1-Tan.
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moswavle SeiZleba mixvdes, rom am jgufSi 3/8 erTaderTi ricxvia, romelic
1/2-ze naklebia da, Sesabamisad, aris yvelaze patara ricxvi. 5/6-isa da 7/8-is
Sedarebisas ki amCnevs, rom radgan meeqvsedi mervedze ufro didi nawilia,
amitom 5/6 naklebia 7/8-ze.
wiladebs, aTwiladebsa da procentebs Soris arsebuli kavSirebi
mexuTe-meeqvse klasebSi bavSvebi swavloben, rom wiladebi, aTwiladebi da
procentebi urTierTkavSirSia. isini aTwiladebs aTobiT sistemaSi
wiladebis gamoxatulebad aRiqvamen (anu, aTwiladebi igive wiladebia, sadac
mniSvnelia 10, 100, 1000 da a.S.).
wiladebis RerZi gviCvenebs, ra nawilia gamuqebuli.
moswavleebi imasac swavloben, rom procenti igive wiladia, romelic mTelis
100 tol nawilad dayofas efuZneba. Tumca bevri maTgani am kavSirs ver
xedavs, procenti da wiladi erTmaneTisagan damoukidebeli cnebebi hgonia.
dawyebiT klasebSi wiladebs, aTwiladebsa da procentebs Soris kavSirebis
saCveneblad aqcenti ufro modelebis gamoyenebaze unda gakeTdes da ara
gadayvanis wesebis Seswavlaze (ix. suraTi 13).
modelebi, magaliTad 10×10-ze
cxaurebi moswavleebs wi-
ladebs, aTwiladebsa da pro-
centebs Soris kavSiris
danaxvaSi exmareba.
10×10-ze cxaurze TvalsaCinod
vxedavT mTelis ra nawilia
gamuqebuli (ix. suraTi 12).
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9
warmodgenis formebi
eWvgareSea, rom yvela asakobrivi jgufis moswavleebisTvis erT-erTi
yvelaze mniSvnelovani maTematikuri ideaa ricxvebis Caweris poziciuri
sistema da Cvens mier ricxvebis warmodgenis saSualebebze am sistemis
zegavlena. swored poziciuri sistemaa is instrumenti, romelic Zalian didi
da Zalian patara ricxvebis gamoxatvis SesaZleblobas gvaZlevs.
marTalia, Sefardebis, anu fardobis cneba wiladebTan uSualo kavSirSia,
magram mas Tavisi sakuTari warmodgenis forma gaaCnia. misi simbolo (:) ori
raodenobis urTierTkavSirs aRniSnavs. miuxedavad imisa, rom fardoba ,,sami
oTxze” (3:4) SeiZleba gamoixatos wiladebis (3/4), aTwiladebis (0,75),
procentebis (75%) simboloebiT da aseve sityvebiTac (sami meoTxedi),
fardobis Sinaarsis interpretaciisTvis yvela es warmodgena erTnairad
sasargeblo da mosaxerxebeli ar aris. fardoba SeiZleba nawilis mTelTan
Sedarebas gamoxatavdes – vTqvaT, klasSi biWebis raodenobis FSefardeba
mTeli klasis moswavleebis raodenobasTan (magaliTad, klasSi aris 25
moswavle da maTgan 15 biWia. biWebis mTel klasTan Sefardeba iqneba 15 : 25).
is aseve SeiZleba aRniSnavdes mTelis erTi nawilis danarCen nawilTan
Sedarebas _ vTqvaT, klasSi biWebisa da gogonebis Sefardebas (magaliTad,
Tu klasSi 15 biWia da 10 gogona, maSin maTi Sefardeba, Sesabamisad, iqneba
15:10). toli fardobebis amocnobis unari (magaliTad, 4-dan 3 lurji burTula
imave Sefardebas qmnis, rogorc 12-dan 9 lurji burTula) proporciulobis
gaazrebis safuZvelia.
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proporciuli azrovneba
obieqtebis erTmaneTTan Sedareba bavSvebs samyaros organizebis, misi
gaazrebisa da gadmocemis SesaZleblobas aZlevs. patarebi Tavidan Tvi-sobrivi SedarebebiT iwyeben (magaliTad ufro didi, ufro patara).
gamocdilebasTan erTad, aditiur Sedarebebsac swavloben (magaliTad 2-iT
meti, vidre..., 50-iT naklebi, vidre...), bolos ki multiplikaciur Sedarebebs euflebian (magaliTad 3-jer ufro grZeli.. raRacis naxevari raodenobis...…)
daakvirdiT am suraTs:
ori sididis urTierTTanafardobis, anu maT Soris arsebuli multip-
likaciuri kavSiris gageba, proporciulobis gaazrebis safuZvelia. zemoT
moyvanil magaliTSi imis codna, rom kaci bavSvze 2-jer ufro maRalia,
migviTiTebs, rom maTi simaRleebis fardobaa 200:100, anu 2:1 (orive am
fardobaSi pirveli sidide meores 2-jer aRmateba).
proporciulobis gaazreba bavSvis maTematikuri unarebis ganviTarebaSi mni-
Svnelovani nabijia. es aris or urTierTkavSirs Soris arsebuli mimarTebis
gagebis unari – anu ara mxolod erTi fardobis farglebSi arsebuli
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kavSirebis aRqma, aramed or an met fardobas Soris arsebuli kavSirebis
gageba.
or fardobas Soris arsebuli multiplikaciuri kavSirebis sailustraciod
Semdeg magaliTs gaecaniT:
,,oTxma gogonam 12 patara pica Tanabrad gaiyo, 3-ma biWma ki 9 patara pica
aseve Tanabrad gainawila. vis ufro meti pica Sexvdeba?”gogonebisa da
picebis fardoba aris 4:12. biWebisa da picebis fardoba aris 3:9. orive
SemTxvevaSi picebis raodenoba bavSvebis raodenobas 3-jer aRemateba.
gamomdinare iqidan, rom orive SemTxvevaSi fardoba tolia, biWebsac da
gogonebsac pica Tanabrad SexvdebaT.
proporciuli azrovneba rom daiwyon, mniSvnelovania, rom dawyebiTi
klasebis zeda safexurebze moswavleebs Sesabamisi savarjiSoebiT
mecadineobis saSualeba mieceT. aseTi saxis mecadineobebi bavSvebs maRal
klasebSi iseTi sakiTxebis ufro formaluri swavlisTvis umzadebs niadags,
rogoricaa proporciebi, procentebi, igiveobebi da algebra.
proporciuli kavSirebis gadmocema wiladebis, Sefardebebisa da pro-centebis meSveobiT
proporciuli kavSirebis gamosaxva SesaZlebelia wiladebiT, SefardebebiTa
da procentebiT. magaliTad:
ojaxebis daaxloebiT 1/3-s saxlSi ZaRli hyavs;
gundis gamarjvebebisa da damarcxebebis Sefardeba 2:3-zea;
bazrobaze nivTi mis adrindel fasze 10%-iT iafi Rirs.
bavSvebi swavloben, rom proporciulobis gamosaxatad gamoyenebuli
wiladebi, Sefardebebi da procentebi diskretul raodenobas ar
warmoadgenen; isini gadmoscemen iseT cnebebs, rogoricaa ,,raRac raodenoba
yveladan”, ,,raRac raodenoba - sxva raodenobasTan mimarTebaSi”, ,,raRac
raodenoba erTTan mimarTebaSi”.
gamomdinare aqedan, zemoT moyvanil magaliTSi, saubari ar aris zustad 3
ojaxze, an 2 da 3 TamaSze, romelic gundma moigo an waago, arc konkretulad
10-larian fasdaklebaze – aq igulisxmeba 1 ojaxi yoveli 3-dan; 2 mogeba
yovel 3 wagebaze da 10-lariani fasdakleba TiToeul 100 larze. aseTi
codniT SeiaraRebul mozards proporciuli gansjiT zusti raodenobis
gansazRvra SeuZlia. ZaRlebTan da ojaxebTan dakavSirebul magaliTSi, 60
ojaxidan 20-s eyoleba es Sinauri cxoveli; Tu gundis wageba-mogebis
Sefardeba aris 2 : 3, maSin 15 naTamaSebi matCidan mas 6 mogebuli eqneba, 9 ki
12
- wagebuli; 10%-iani fasdakleba ki imas gulisxmobs, rom 25 larad
Sefasebuli nivTi 2,5 lariT ufro iafad gaiyideba.
proporciuli azrovnebis ganviTareba araformaluri aqtivobebis saSualebiT
dawyebiTi safexuris zeda klasebis moswavleebma mravali iseTi amocana unda
amoxsnan, romelic proporciul azrovnebas moiTxovs. kerZod, am asakobriv
jgufSi proporciulobis swavlebisaTvis Semdegi saxis amocanebis
gamoyenebaa mizanSewonili:
ucnobi sididis gamoTvla SefardebaSi. magaliTad: ani da irakli
velosipediT erTi da imave siCqariT moZraoben. anis 24 km-is gasavlelad 1.5
saaTi sWirdeba. ra dro dasWirdeba iraklis 20 km-is gasavlelad?
ori Sefardebis erTmaneTTan Sedareba. magaliTad: anim velosipediT
moZraobisas 12 km 45 wuTSi dafara, iraklim ki 8 km 40 wuTSi gaiara. romeli
ufro swrafad moZraobda?
raodenobrivi Sedarebebi. magaliTad: anim da iraklim erTi da igive
manZili dafares, risTvisac anis 35 wuTi dasWirda, iraklis ki - 30. romeli
ufro swrafad moZraobda?
sazomi erTeulis sididis gamoTvla. magaliTad, anim velosipediT 18
km 1.5 saaTSi gaiara. ramden kilometrs gadioda is saaTSi?
sazomi erTeulebis gadayvana. magaliTad: iraklim 15 wuTSi 3550 metri
gaiara. ramden kilometri/saaTSi siCqariT moZraobda is?
wiladi raodenobis gamoTvla. magaliTad: anim unda gaiaros 24 km. man
ukve dafara 16 km. gasavleli manZilis ra nawili dafara anim?
Tanabari fardobebis gamoTvlis meqanikuri procedurebis swavleba
(magaliTad, jvaredini gamravleba ucnobi sididis dasadgenad)
proporciulobis gaazrebis ganviTarebaSi SesamCnev Sedegs ar iZleva.
proporciulobis cnebiTi gaazreba maSin xdeba, roca bavSvebi mecadineobisas
maTTvis gasageb strategiebs praqtikulad iyeneben. rogorc qvemoT aris
naCvenebi, proporciulobis swavlis sawyis etapebze isini konkretul
damxmare masalebsa da sqemebs mimarTaven.
13
aRsaniSnavia, rom moswavleebi proporciulobis cnebas im maTematikur
procesebs ukavSireben xolme, romlebic ukve aTvisebuli aqvT. magaliTad,
moswavle xvdeba, rom toli Sefardebebis gansazRvra iseve xdeba, rogorc
toli wiladebisa _ 3:5 fardoba 6:10-is tolia, iseve rogorc 3/5=6/10.
gaxsovdeT: tol Sefardebebsa da tol wiladebs Soris principuli gansxvavebaa!
toli wiladebi erTsa da imave raodenobas gamoxataven (maaliTad, Zafis
3/5 igive zomisaa, rac igive Zafis 6/10).
amave dros, toli Sefardebebi SeiZleba sxvadasxva raodenobas
gamoxatavdnen (magaliTad, 3 vaSlisa da 5 qliavis Semcveli jami da jami,
sadac 6 vaSlia da 10 qliavi; arada, orive jamSi vaSlebisa da qliavebis
fardoba tolia).
moswavleebma maswavleblebis meTvalyureobiT proporciuli msjelobis
meTodebi unda gamoiyenon. magaliTad, toli Sefardebebis dasafiqsireblad
fardobebis cxrilis Sedgena sakmaod efeqturi instrumentia.
14
ricxvebis konstruqtivistuli swavleba V-VI klasebSi
(kerZo sakiTxebi)
gayofis maTematikuri arsi da gayofis swavlebis meTodologia
gayofis zogadi SemTxvevaa naSTiani gayofa da misi kerZo saxeobaa unaSTo
gayofa.
gayofa _ esaa gamravlebis sapirispiro moqmedeba. misi arsia ucnobi Tanamam-
ravlis gamoTvla namravlisa da meore Tanamamravlis mixedviT. igi hgavs
gamoklebas, romelic Sekrebis sapirispiro moqmedebaa. Tumca, erTi arsebiTi
gansxvaveba da sirTulea: gayofisas _ SeiZleba, gasayofi gamyofze zustad
ar gaiyos da mogvrCes naSTi. Tanac es sakiTxi gayofis swavlebis
dasawyisSive unda gairkves, radgan Tuki moswavlis praqtikaSi, gayofis
swavlebis sawyis etapze, unaSTo gayofis magaliTebi damkvidrda, mas Se-
eqmneba mcdari Sexeduleba: gayofa, iseve rogorc gamokleba, yovelTvis
unaSTod sruldeba. amitom SemdgomSi mas gauWirdeba naSTiani gayofis
gaazreba.
kidev erTi sirTulec gvxvdeba. gamokleba mxolod erTaderTi saxisaa,
gayofa ki _ orisa:
(1) naSTiani gayofa (da misi kerZo saxeoba _ roca naSTi nulis tolia);
meoregvarad, esaa SemcvelobiTi gayofa.
gamravleba = tol SesakrebTa Sekreba
gayofa = gamravlebis sapirispiro moqmedeba
amitom
gayofa = ricxvis daSla tol SesakrebTa jamad
maSasadame:
ganayofis moZebna = ramdenjer unda SevkriboT gamyofi, raTa miviRoT
gasayofi.
nimuSi: 21 : 4 = ? ramdenjer unda SevkriboT 4, raTa miviRoT 21? 4+4+4+4+4 =
20, xolo 21-Si meti oTxeuli aRar moTavsdeba. amitom 21 : 4 = 5 (naSTi 1).
anu raki 21 = 5 × 4 + 1, amitom ganayofia 5 da naSTia 1.
arsebiTad: viRebT oTxeulebs da vkrebT, sanam 21-mde ar mivaRwevT.
aseve, rom yofiliyo 20 : 4, maSin miviRebdiT 5 (naSTi 0), anu, rac igivea, 20 :
4 = 5.
sayofacxovrebo viTareba: 21 lariT (an 20 lariT) ramdeni cali 4-lariani
wignis yidva SeiZleba?
15
(2) wiladuri gayofa anu Tanabari dayofa.
21 = 4 • 𝒙𝒙, 𝒙𝒙 = 21 : 4 = 214 = 5 𝟏𝟏
𝟒𝟒.
arsebiTad: vanawilebT pirobiTad oTx ujraSi, rigrigobiT vamatebT TiTo-
TiTos, sanam ar amoiwureba 21 an 4-ze naklebi ricxvi ar dagvrCeba.
sayofacxovrebo viTareba: 21 lari rogor gavunawiloT Tanabrad 4 bavSvs?
gayofis namdvili pirveladi arsi naSTiani gayofaa da ara Tanabari gayofa.
marTlac: Tanabrad rogor gavyoT, magaliTad, 21 oTxze? unda gamoviyenoT
naSTiani gayofa, es gza bunebrivi da TvalsaCinoa: pirdapir daviwyoT
gamokleba (oTxeulebisa) da gamovakloT manam, sanam 4-ze nakleb ricxvs ar
miviRebT. amitom, jer kargad unda vaswavloT gayofa (SemcvelobiTi) da
mxolod amis Semdeg da amis safuZvelze daviwyoT gayofis gamoyeneba
Tanabrad dayofisTvis, anu wiladebisTvisac.
esaa mTeli saskolo maTematikis erT-erTi umniSvnelovanesi sakiTxi:
tradiciulad, gayofa iswavleba an II gziT, an I da II erTmaneTSia aRreuli.
amgvari mcdari swavlebis gamo moswavle verc naSTs swavlobs kargad da
verc wiladebs.
logikurad, wiladebs winaswar sWirdeba naSTiani gayofa, naSTian gayofas
ki wiladebi _ ara. piriqiT: wiladebis naadrevi Semotana angrevs naSTian
gayofas!
marTlac, Tanabari dayofisas anu wiladis Semdeg naSTian gayofas TiTqos
azri ekargeba _ raRaSi gvWirdeba naSTi, rodesac pirdapirac SeiZleba
gavyoT? mag.: 90 : 11 = 90/11.
amgvarad, gayofa martivad, mxolod mTeli ricxvis miRebiT ar mTavrdeba.
magaliTad, rodesac mTel ricxvebs 6-ze vyofT, unaSTod es mxolod eqvsidan
erT SemTxvevaSi gamova. konteqstis ararsebobis pirobebSi, naSTs SeiZleba
ornairad movepyraT: morCenil odenobad anu naSTad davtovoT an wiladebad
davyoT.
16
realur konteqstSi naSTs pasuxSi zogjer sami sxvadasxva gziT epyrobian:
naSTs anuleben da rCeba mxolod mTeli ricxvi – arasruli ganayofi;
naSTi “gvaiZulebs” pasuxad momdevno (ufro maRali) mTeli ricxvi
davweroT;
xdeba pasuxis damrgvaleba uaxloes mTel ricxvamde (mrgvaldeba metobiT
an naklebobiT - damrgvalebis wesis gamoyenebiT).
1. gaqvT 30 kanfeti, rac 7 megobars Tanabrad unda gaunawiloT. ramden cal
kanfets miiRebs TiToeuli maTgani?
pasuxi: 4 cal kanfets.
2. TiToeuli saTli 8 litr siTxes itevs. Tu kasrSi 46 litri siTxe asxia,
ramden saTls aavsebs igi?
pasuxi: 5 da 6/8 saTli (gadayvanili wiladSi).
3. Tokis sigrZe 25 metria. misgan ramdeni 7-metriani Toki gamova?
pasuxi: 3 Toki (naSTi ganulebulia).
4. borani 8 manqanas itevs. ramdenjer unda gadakveTos mdinare boranma im-
isTvis, rom mdinareze 25 manqana rom gadaitanos?
pasuxi: 4-jer (momdevno mTeli ricxvi).
17
5. eqvsma bavSvma 50 saReWi rezini unda gainawilos. daaxloebiT ramdeni
cali Sexvdeba TiToeul maTgans?
pasuxi: daaxloebiT 8 cali TiToeul bavSvze (damrgvalebuli uaxloes mTel ricxvamde).
Secdoma iqneba, Tu dawyebiT klasebSi moswavleebs davavaldebulebT,
magaliTad, naSTSi morCenili 3 yovelTvis `naSTi 3~-is saxiT daweron an
mxolod unaSTo amocanebi amoxsnan. umjobesia, naSTis Semcveli amocanebi
SevTavazoT da davakvirdeT im gzebs, riTac patarebi am problemas arsebuli
konteqstis farglebSi gaumklavdebian. naSTze raime mkacri wesebis SemoReba
saR azrs moklebulia. im bavSvebs, romlebic wiladebze saubars iwyeben,
iseTi amocanebi SesTavazeT, sadac mTeli SeiZleba wilad nawilebad
daiSalos (magaliTad, pica, kanfetebi, rZis jamebi da a.S.). zogierTi
moswavle problemas ise gadaWris, rogorc es zemoT sqemaze aris mocemuli.
tradiciuli swavlebis Sedegia tipuri Secdoma naSTiani gayofisas.
102000 : 50000 = 102 : 50 = 2 (naSTi 2)
an Semdgomi SekveciT:
102000 : 50000 = 102 : 50 = 51 : 25 = 2 (naSTi 1).
es Secdoma gamowveulia wiladuri anu Tanabari gayofis mTavari wesiT _
SekveciT. es wesi ki naSTiani gayofisTvis mcdaria. marTlac:
raki 50000 + 50000 = 100000, amitom 102000 : 50000 = 2 (naSTi 2000). maSasadame, roca gasayofi bolovdeba nulianebiT, maSin:
(1) unda daviwyoT qveSmiweriT an zepirad gayofa Cveulebrivad;
(2) Tuki gasayofi nulianebamde unaSTod gaiyo, maSin gayofas SevwyvetT da
ganayofs mivuwerT gasayofis nulianebs, magaliTad:
Tuki unaSTod ar gaiyo, maSin gayofas Cveulebrivad ganvagrZobT bolomde,
vidre ar miviRebT Cveulebrivad naSTs. naSTiani gayofisas nulebis Camocileba-damaxsovreba ar SeiZleba!
2719 00: 9 = ? gasayofs Camoaciles nulianebi da miiRes:
18
2719 : 9 = 302. Semdeg ki ganayofs miuweres is Camocilebuli nulianebi:
2719 00 : 9 = 302 00 (naSTi ?) _ magram aq ganayofic ki ukve mcdaria,
radgan sinamdvileSi 271900 : 9 = 30211 (naSTi 1).
naSTiani gayofa TvalsaCinoa ricxvTa sxivzec. qvemoT naCvenebia, 170 rogor
gavyoT 30-ze. naxazidan Cans, rom 170-Si eteva 5 cali ocdaaTeuli da kidev
rCeba naSTi 20. ese igi: 170 : 30 = 5 (naSTi 20).
gayofis arsis safuZvelze, gayofisas SeiZleba gamoviyenoT gamravleba da
Sekreba. amiT xSirad advilad sruldeba gayofa.
gayofis advili gza:
I. 1500 : 250 = ? 4 × 250 = 1000; 2 × 250 = 500; 4 + 2 = 6;
maSasadame, 1500 : 250 = 6.
II. 90 : 11 = ? 8 × 11 = 88; 88 + 2 = 90;
maSasadame, 90:11 = 8 (n. 2).
mxolod naSTian gayofas emyareba qveSmiweriT gayofis algoriTmi.
19506 : 4 = ? gayofa qveSmiweriT Sesruldeba dafasTan dawvrilebiTi
xmamaRali msjelobiT _ srulad iwereba qveSmiweriT gayofa. Tan TiToeul
nabijs axlavs axsna-ganmartebebi, raTa wesi gaSinaarsdes da gaazrebul
iqnes.
19506 = 19 × 1000 + 5 × 100 + 6. jer gavyoT aTaseulebi. ramdenia gasayofSi
aTaseuli? {cxrameti}. 19 gayofili 4-ze tolia 4-isa. ramdeni aTaseuli iqneba
ganayofSi? ga-nayofis aTaseulebis TanrigSi vwerT 4-s. naSTi ramdeni rCeba?
amisaTvis 19-s unda gamovakloT 4 × 4 = 16. rCeba 3 aTaseuli _ es jer ar
gagviyvia {Sesamowmeblad SevadaroT naSTi 3 gamyofs 4 _ marTlac naklebia}.
magram 3 aTasi 4-ze rom gavyoT, ganayofi ukve aTasze naklebi iqneba. amitom
gadavdivarT aseulebze.
gvaqvs 3 aTasi da kidev 5 aseuli, sul _ 35 aseuli.
19
rogorc vxedavT, naSTiani gayofis gareSe SeuZlebelia qveSmiweriT gayofis
algoriTmis dasabuTeba da gaazreba. gauazreblad qveSmiweriT gayofis al-
goriTmis swavla zedmeti xdeba.
naSTiani gayofa dawyebiTi klasebis mTeli ariTmetikis gvirgvinia, igi
aerTianebs Tvlas, Sekreba-gamoklebasa da gamravlebasac.
da bolos, naSTi, oRond gayofis gareSe, gvWirdeba I-II klasebSive _ uamisod
moswavle ver gaiazrebs orniSna ricxvis agebulebas, Tvlis safuZvels da,
miT umetes, oceulebiT Tvlas.
20
wiladebis swavlebis meTodologia
yvela ariTmetikuli moqmedeba, sabolood, unda daviyvanoT sxvadasxva
simravlis wevrebis raodenobis garkvevaze, anu Tvlaze. swored Tvla aris
yvelaze pirveladi, bunebrivi, konkretuli ariTmetikuli moqmedeba.
moswavlisTvis gaazrebadia is, rac Tvlas daukavSirdeba. am meTodikur
arguments zedmiwevniT eTanxmeba logikur-maTematikuric: maTematikaSi
mecnierulad gamarTulia mxolod is cneba, romelic raime saxis simra-
vleebze daiyvaneba.
ariTmetikis mecnieruli dafuZnebisa da ariTmetikis gaazrebuli swavlebis
saukeTeso Tanmimdevroba mocemulia qvemore sqemaze (masSi yvelgan
igulisxmeba moqmedeba saTanado farglebSi _ oRond wina farglebis sakmao
ganmtkicebis Semdeg). es sqema zedmiwevniT aRwers ariTmetikis agebis
aramarto meTodikur, aramed mecnierul-maTematikur Tanmimdevrobasac _ es
ori Tanmimdevroba erTmaneTs emTxveva!
sqemaze TvalsaCinoa moqmedebaTa ierarqia: romelia maRali rigis moqmedeba
da romeli _ dablisa. magram saqme isaa, rom yoveli moqmedeba mWidrod unda
daemyaros mis wina, ufro dabali rigis moqmedebas, raTa safuZveli ar
gamoecalos!
sqemaze sxva ramec TvalsaCinoa: zepiri (gonebaSi) moqmedebis pirveladoba
weriTTan mimarTebaSi. weriTi moqmedeba winaswar kargad ganmtkicebul imave
zepir moqmedebas unda daemyaros, Tanac wamyvani mainc zepiri moqmedeba unda
darCes. magaliTad, gaumarTlebelia qveSmiweriT Sekreba-gamokleba manam,
sanam moswavle ar gaiazrebs aramarto Sekreba-gamoklebis cnebebs, aramed
zepiri Sekreba-gamoklebis xerxebsac. amas ki I-II klasebi sWirdeba. ese igi
21
qveSmiweriT Sekreba-gamokleba III klasamde araa saWiro. Sesabamisad,
qveSmiweriT gamravleba-gayofis dawyeba umjobesia IV klasidan, radgan III
klasSi moswavlem jer qveSmiweriT Sekreba-gamokleba unda gaiTavisos. da
aseve: aviTvisoT jer zepiri moqmedebebi wiladebze da aTwiladebze da
mxolod amis Semdeg _ weriT moqmedebaTa wesebi. isev da isev:
jer TviT cnebebis Rrma gaazreba da mxolod amis Semdeg _ moqmedebebi!
am gegmis darRveva, romelime isris anu mWidro kavSiris gawyveta
ariTmetikul moqmedebaTa codnas gaazrebulobas daukargavs da
sqolastikur manipulaciebSi gadaiyvans maT. xolo am gegmis dacva
moswavlis mier moqmedebaTa wesebis gaazrebul dauflebas ganapirobebs (es
ki udidesi maTematikuri miRwevaa); migvaRwevinebs saskolo maTematikis
erTerT umTavres zogad miznamde _ ricxvis gumanis ganviTarebamde.
gegmidan Cans wiladis cnebis swavlebis mTavari Tanmimdevroba:
jer bolomde da Rrmad unda iyos damTavrebuli naSTiani gayofa; da, rac
mTavaria, jer safuZvlianad unda SevaswavloT wiladi (Tavisi gamyof-
jeradebiT, farTobiT wreebiTa da monacemebis damuSavebiT, rasac mTeli V
klasi sWirdeba), da mxolod amis Semdeg mivawodoT aTwiladebi.
marTlac, aTwiladi, arsebiTad, igive wiladia (maTematikurad _
racionaluri ricxvi), oRond sxvagvarad Cawerili. Tuki moswavles bolomde
ar aqvs gaazrebuli wiladis cneba, wiladebis Sedareba da maTze
ariTmetikuli moqmedebebi da ukve aTwiladebs vaswavliT, es niSnavs Semdegs:
CvenTvis mTavaria moqmedebaTa Sesruleba (rac aTwiladebze ufro advilia)
da ara - cnebis gaazreba da azrovneba. anu, CvenTvis mTavaria, moswavlem
kalkulatoriviT imuSaos uSecdomod _ da ara is, rom wiladis arsi esmodes.
marTlac, rogor SeiZleba kacma gaiazros aTwiladebze _ anu sxvagvarad
Caweril da kerZo saxis wiladebze moqmedebebi _ Tuki jer wiladebze ar
gauazrebia? Tuki moswavles azriT ar esmis, romelia meti _ 3/7 Tu 5/9, maSin
ra dros aTwiladebze moqmedebebSi meqanikuri gawafvaa? da, sazogadod, ra
saWiroa, rom kalkulatoris saqmes esoden didi yuradReba davuTmoT?
moswavlem jer wiladi da maTze moqmedebebi unda gaiazros kargad da mere
vaswavloT aTwiladi.
maTematikurad wiladi (vigulisxmoT dadebiTi) igivea, rac racionaluri
ricxvi _ ori naturaluri ricxvis Sefardeba. yovel aseT ricxvs Seesabameba
erTaderTi wertili ricxvTa sxivze. magram am sakmaod abstraqtul cnebamde
meTodikurad sakmaod grZeli gzis gavlaa saWiro, raTa wiladis cneba
gaSinaarsianebuli iyos (ix. sqema).
am gzis Tanmimdevrobis darRveva an aCqareba dauSvebelia, radgan es
safuZvels gamoaclis maTematikis erT-erT umTavres cnebas. moswavlem Rrmad
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unda gaiTavisos wiladis cnebis esa Tu is done, mere ki gadavides momdevno
doneze da gaizros, rom es yovelive _ arsebiTad erTidaigivea.
mTliani mTelis nawili, wriuli diagrama →→ grovis nawili, saTanado raodenoba → erTi memniSvneledi aRebuli mricxveljer, anu nawilis namravli naturalur ricxvze →→ wiladi, rogorc ori naturaluri ricxvis Sefardeba, ukve nawilebisgan ganyenebiT →→ erTze meti wiladebi, mTeli ricxvis toli wiladebi →→ nebismieri ori naturaluri ricxvis naSTiani ganayofis Cawera Sereuli wiladis saxiT → → wiladebi ricxvTa sxivze →→
Sefardeba da proporcia, skalis masStabi ...
amocanebi mTelisa da misi nawilebis gamoyenebiT
aq mocemuli savarjiSo moswavleebs wiladebis gaazrebaSi, aseve wiladSi
mricxvelisa da mniSvnelis gagebaSi daexmareba. mTelisa da misi Semadgeneli
nawilebis gamosaxatad modelebia gamoyenebuli, maT Soris kavSirebis
gadmosacemad ki- wiladebis werilobiTi da verbaluri formebi. am samidan
– mTeli, misi nawili da wiladi ricxvi – orze dayrdnobiT mozardebma,
modelirebis saSualebiT, mesame unda ipovon.
gamomdinare iqidan, rom mTeli gansxvavebul sidideebs warmoadgens, yvela
saxis modelis gamoyeneba SeiZleba. Tumca ,,namcxvris” wriuli modeli aq ar
gamodgeba, radgan mTeli am SemTxvevaSi yovelTvis erTi da igivea,
Sesabamisad – wiladebic mxolod martivia (,,martivSi” igulisxmeba iseTi
wiladebi, romelTa mricxveli aris 1. magaliTad, 1/3, 1/8 da a.S.). qvemoT
TiToeuli tipis savarjiSo sqematuradaa mocemuli. aq naxavT, rogorc
marTkuTxa, aseve grZiv (kiuzineris Cxirebi da wiladebis lentebi) da groviT
modelebs.
23
24
25
26
swavlebis zogadi principebis Sejameba
swavlebis bevri principi, romelic dawyebiTi klasebis nebismier saganSia
gamosadegi, maTematikuri ideebis swavlebisTvisac Sesaferisia. am
principebis praqtikaSi danergva yvela pedagogisTvis umniSvnelovanesia.
zogierT maTgans qvemoT gTavazobT:
moswavleebis mier sakuTari azrebis gamoTqma yvela asakobriv jgufSi
saWiroa. maT unda isaubron maTematikaze maTematikuri terminologiiT,
unda esaubron erTmaneTs da maswavlebels;
cnebebis warmodgena samanipulacio masalebiT, suraTebiT, sqemebiT an
simboloebiT xels uwyobs gageba-gaazrebas da aviTarebs maT sityvierad
Camoyalibebis unarebs.
amocanebis amoxsna maTematikis swavlebis ZiriTadi sayrdeni principi
unda iyos;
mowafeebs intensiuri wvrTna esaWiroebaT mravali sxvadasxva resursisa
da saswavlo strategiis monacvleobiT (magaliTad, ricxviTi RerZebi,
asricxviani cxrilebi, kedeli-sveti-kubis modelebi, konstruqtoris
kubebi, aTujrediani CarCo-cxrilebi, kalkulatorebi, maTematikuri
TamaSebi, fizikuri moZraobebi da a.S);
rac ufro rTulia cneba, miT ufro didia saazrovno unarebis gaa-
qtiurebis saWiroeba. saazrovno unarebis ganviTareba maqsimalurad unda
wavaxalisoT, magaliTad iseTi unarebis, rogoricaa kanonzomierebebis
SemCneva da raodenobrivi Sefasebebi.
maTematikaSi gawafuli moswavleebis maxasiaTeblebi SeuZliaT sakuTari azrebis verbalizacia da sxvis mier gamoTqmuli
mosazrebebis gageba;
ganviTarebuli aqvT ricxvebis gaazrebis unari;
muSaoben nayofierad da problemis gadaWrisas codnis konstruirebas axdenen;
Tanmimdevrulad cdiloben maTematikis arsSi wvdomas;
cnebebs Soris pouloben kavSirebs da xedaven maTematikur kanonzomierebebs;
maTematikuri amocanebis gagebisa da amoxsnis saqmeSi Seupovrobas iCenen;
iviTareben siRrmiseuli da moqnili maTematikuri azrovnebis unarebs.
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