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Sketchpad i promjene u poučavanju matematike

Napisao: Viktorija Sukser, 3.F, Peta gimnazija, Zagreb, 02/10/2007, kategorija: SketchPad

Poučavanje matematike se promijenilo

Napisao:Steven Chanan, SAD;
Prevela s engleskog: Viktorija Sukser, 3.F, Peta gimnazija, Zagreb.

Načina na koji poučavamo matematiku – posebice geometriju – promijenio se, zahvaljujući nekolicini važnih napredaka proteklih godina. Alternative strogo deduktivnom pristupu dostupne su nakon više od stoljeća neuspjeha da se dosegne većina učenika. (Nacionalna procjena obrazovnog napretka – The National Assessment of Educational Progress – utvrdila je 1982. godine da je dokazivanje najneomiljenija matematička tema sedamnaestogodišnjaka, i manje od 50% njih je tu temu ocijenilo kao važnu.) Prvo su 1985. godine Judah Schwartz i Michal Yerushalmy iz Centra za razvitak obrazovanja – Educational Development Center – razvili važan dio softwarea za učenje koji je omogućio učiteljima i učenicima da koriste računala kao alate za poučavanje i učenje, a ne samo za vježbanje. Geometrijske pretpostavke (Geometric Supposers) za kompjutore Apple II poticale su učenike da osmisle vlastitu matematiku, olakšavajući stvaranje jednostavnih geometrijskih likova i pravljenje pretpostavki o njihovim svojstvima. Učenje geometrije moglo bi postati serija otvorenih istraživanja odnosa između geometrijskih likova – proces otkrivanja koje motivira (pokreće) dokaz, radije nego „prožvakavanje“ dokaza teorema koje učenici uzimaju zdravo za gotovo ili ne razumiju.

Standardi za nastavu matematike

Godine 1989. Nacionalni savjet učitelja matematike (National Council of Teachers of Mathematics – NCTM) objavilo je „Nastavni plan i standarde procjene za školsku matematiku“ (Curriculum and Evaluation Standards for School Matheatics), poznato i kao Standardi, koji su zahtijevali značajne promjene u načinu na koji je matematika poučavana. U poučavanju geometrije, Standardi su tražili smanjenje naglaska na prezentaciji geometrije kao cjelovitog deduktivnog sustava i dokazima od dva retka. Diljem nastavnog plana, Standardi su tražili povećani naglasak na otvorenom istraživanju i postavljanju pretpostavki, i veću pažnju za teme u geometriji transformacije. U svojoj potražnji za promjenama, Standardi su prepoznali kakav utjecaj imaju alati tehnologije na način poučavanja matematike, oslobađajući učenike od dugotrajnih, svjetovnih zadaća i dajući im sredstva da vide i istraže zanimljive veze.

Otkrivanje geometrije

Objavljivanjem prvog izdanja „Otkrivanja geometrije: Induktivni pristup“ (Discovering Geometry: An Inductive Approach) Michaela Serrae 1989. godine, Key Curriculum Press pridružio se snagama promjene. „Otkrivanje geometrije“, udžbenik geometrije za srednju školu, ima uglavnom isti pristup za koji su se kreatori Geometrijskih pretpostavaka zalagali: učenici bi trebali stvoriti vlastite geometrijske konstrukcije i sami formulirati matematiku koja bi opisala odnose koje otkriju. Uz „Otkrivanje geometrije“, učenici, radeći u kooperativnim grupama, rade istraživanja koristeći alate geometrije kako bi otkrili svojstva. Učenici traže uzorke pravilnosti i koriste induktivno rasuđivanje za pretpostavljanje. Od njih se ne očekuje da dokažu svoja otkrića sve dok ne savladaju geometrijske pojmove i mogu biti svjesni značenja dokaza. Sada u svom drugom izdanju, „Otkrivanje geometrije“ omogućava učenicima da iskoriste prednosti šireg raspona alata, uključujući i radne listiće i Sketchpad (The Geometer's Sketchpad).

Razine geometrijskog razmišljanja

Pristup je u skladu s istraživanjem koje su proveli nizozemski pedagozi matematike Pierre van Hiele i Dina van Hiele-Geldof. Promatranjima u razredu, van Hieleovi su spoznali da učenici prolaze kroz niz razina geometrijskog razmišljanja: vizualizacija, analiza, neformalna dedukcija, formalna dedukcija i strogost. Standardni geometrijski tekstovi očekuju da učenici od početka primjenjuju formalnu dedukciju. Malo je toga napravljeno da se učenicima omogući vizualiziranje ili da ih se potakne na pravljenje pretpostavki. Glavni je cilj Pretpostavki, „Otkrivanja geometrije“, i, sada, Sketchpada, provođenje učenika kroz prve tri razine, potičući postupak otkrivanja koji pobliže odražava kako se matematika obično pronalazi: matematičar prvo zamisli i analizira problem, praveći pretpostavke prije pokušaja dokazivanja.

Sketchpad je temelj današnjeg edukacijskog softwarea

Sketchpad je utemeljio je današnju generaciju obrazovnog softwarea koji je ubrzao promjenu započetu Geometrijskim pretpostavkama, i koji je nadalje potican izdanjima poput „Otkrivanja geometrije“ i Standarda NCTM-a. Sketchpadova jedinstvena Dinamična geometrija omogućuje učenicima da istraže odnose interaktivno tako da mogu vidjeti promjenu u matematičkim dijagramima dok njima rukuju. S ovim probojem, usporedo s cjelovitošću njegove konstrukcije, transformacije, analitičkih i algebarskih mogućnosti – kao i neograničen opseg koji nude njegovi uobičajeni alati – Sketchpad proširuje domet onoga što se može napraviti s matematičkim softwareom do dosad neviđene širine. U petnaestak godina njegova postojanja, učitelji su prenijeli Sketchpad izvan nastave geometrije u algebru, račun (calculus), trigonometriju i srednjoškolsku nastavu; a daljnji razvoj softwarea poboljšao ga je za te šire primjene. Paradigma Dinamične geometrije koju je uveo Sketchpad, bila je tako dobro prihvaćena – od strane matematičkih i obrazovnih istraživača, učitelja širom nastavnog plana i milijuna učenika – da izdanje Standarda iz 2000. godine sada to naziva Dinamičnom geometrijom. Konkurentski razvoj Macintosha, Windowsa, Jave i priručnih verzija Sketchpada u mnogo raznih jezika osigurava da najmoćniji i suvremeni alat za geometriju bude uvijek dostupan širokom krugu školskog računalnog okruženja diljem svijeta.

Kako je nastao Sketchpad?

Geometer's Sketchpad se razvio kao dio Projekta vizualne geometrije (Visual Geometry Project), kojeg je financirala Nacionalna znanstvena zaklada (The National Science Foundation) pod upravom dr. Eugenea Klotza sa sveučilišta Swarthmore (Swarthmore College) i dr. Doris Schattschneider sa sveučilišta Moravia (Moravia College) iz Pennsylvanije. Kao dodatak Sketchpadu, Projekt vizualne geometrije (VGP) je razvio Zvjezdane oktangule i Platonova tijela: videi, radne bilježnice i radni materijali koje je također objavio Key Curriculum Press. Sketchpadov tvorac i programer, Nicholas Jackiw, pridružio se VPG-u u ljeto 1987. godine. Godinu nakon toga započeo je ozbiljan rad na programu. Sketchpad je za Macintosh bio razvijen u otvorenom, akademskom okruženju, gdje su mnogi učitelji i drugi korisnici eksperimentirali sa ranijim verzijama programa i (do)davali podatke za njegovo oblikovanje. Nicholas je došao raditi za Key Curriculum Press 1990. godine kako bi napravio „beta“ inačicu softwarea isprobanog na nastavi. Jezgra od 30 škola ubrzo je porasla u grupu veću od 50 kako se vijest širila i sve više ljudi čulo za Sketchpad ili vidjelo njegove demonstracije na konferencijama. Otvorenost s kojom je Sketchpad bio razvijen potaknula je nevjerojatan val povratnih informacija i entuzijazma za taj program. Do vremena njegova objavljivanja u proljeće 1991. godine, već su ga koristile stotine učitelja, učenika i ostalih zaljubljenika u geometriju, i već je bio najčešća tema razgovora i najočekivaniji matematički software u skorašnjem vremenu.

Najvrjedniji software za učenike i nastavnike

U prvoj godini Sketchpada, Key Curriculum Press je započeo proučavati kako se program učinkovito koristi u školama. Djelomice financirano bespovratnim sredstvima za mala poduzeća, od strane Nacionalne znanstvene zaklade, to se istraživanje odražava u ovim bilješkama za poučavanje, u materijalima za nastavne planove, i u novim verzijama Skatchpada. Druga verzija programa (Version 2), objavljena u travnju 1992. godine, predstavila je poboljšane mogućnosti transformacije i prezentacije, i donijela alate za grafičko istraživanje rekursija i iteracije u ruke korisnika Sketchpada. Treća je verzija (Version 3) za Macintosh i Windowse, velika nadogradnja objavljena u travnju 1995. godine, proširila analitičke i grafičke mogućnosti programa. Do 1999. godine, nacionalni pregled učitelja „Poučavanje, učenje i računalstvo“ (Teaching, Learning and Computing) – koji je provelo sveučilište Californije, Irvine (University of California, Irvine) – pokazalo je da su tamošnji učitelji matematike naveliko ocijenili Sketchpad „najvrjednijim softwareom za učenike“. Četvrta verzija softwarea (Version 4), predstavljena u jesen 2001. godine, dramatično je proširila korisnost primjene programa u nastavi algebre, predračuna (pre-calculus) i računa (calculus), istovremeno još više olakšavajući njegovu upotrebu u nižim razredima, i povećavajući razvoj ovlaštenih alata u nastavnom planu softwarea. Istraživanja u razredu nastavljaju stvarati temelje za daljnji razvitak softwarea i pratećih materijala.

Uporaba Sketchpada u učionici

Geometar's Sketchpad je bio prvotno stvoren za upotrebu u nastavi geometrije u srednjim školama. Testiranja su ipak pokazala da njegova lakoća korištenja omogućuje i mlađim učenicima u osnovnoj školi da se uspješno služe Sketchpadom, a da ga je moć njegovih karakteristika učinila privlačnim predavačima matematike na sveučilištu i nastavnicima u osnovnoj i srednjoj školi. Predavače na učilištima posebno privlače Sketchpadove moćne mogućnosti transformacije i uobičajeni alati koji omogućuju učenicima da istražuju neeuklidsku geometriju. Čak je i umjetnike i profesionalne mehaničke crtače oduševila Sketchpadova moć i elegancija. To je potvrda mnogostranosti softwarea u kojemu isti alat mogu koristiti šestogodišnjaci i sveučilišni profesori za istraživanje novih matematičkih pojmova. (Svakako valja pogledati primjere koji dolaze instalirani uz Sketchpad za dodatne alate koji pomažu u prilagođavanju programa za potrebe vašeg razreda. Naći ćete alate za konstruiranje pravilnih mnogokuta, određivanje matematičkih simbola, istraživanje neeuklidske geometrije, slaganje i kombiniranje funkcija, i još mnogo toga.) U ovom odjeljku, koncentrirat ćemo se na načine na koje se Sketchpad može koristiti u nastavi geometrije u srednim školama.

Kao učitelj geometrije u školi, htjet ćete voditi svoje učenike ka otkrivanju specifičnog svojstva ili malog skupa svojstava, ili ćete željeti postaviti otvoreno pitanje ili problem i tražiti od učenika da pokušaju otkriti o tome najviše što mogu. Ili ćete, pak, htjeti pripremiti za učenike interaktivnu demonstraciju koja oblikuje određeni pojam. U svakom slučaju, željet ćete da učenici surađuju i raspravljaju međusobno o svojim otkrićima. Sketchpadova karakteristika bilježenja potiče učenike da izraze matematičke ideje. Koji god da način korištenja Sketchpada uzmete, on može poslužiti kao odskočna daska za rasprave i razgovore. Sketchpada u razredu koristi za: usmjereno istraživanje, otvoreno istraživanje i prezentaciju. Vrlo lijepe primjere možete naći u „Istraživanja geometrije sa Sketchpadom“ (Exploring Geometry with The Geometer's Sketchpad), kojeg je 1999. godine objavio Key Curriculum Press i u nizu drugih knjiga.

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