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.

1

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);

2

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

3

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.

4

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.

5

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).

6

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.

7

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).

8

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.

10

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

11

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).

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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.

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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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