predmetni kurikulum - matematika
TRANSCRIPT
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Matematika!"#$%&'()
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Cjelovita kurikularna
reforma
Rani i pred!kolski, osnovno!kolskii srednjo!kolski odgoj i obrazovanje
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!"#$%&' ()*+!$, *#-$, (.+/'$, '0#1*#$' 2#&$'3 /%0'&%3Sonja Bani4, prof. matematike i fizike, Srednja 5kola „Ivan 6vear“, Ivani4 GradSanja Jane5, prof. matematike, Osnovna 5kola „Petar Zrinski“ 7abar, 7abarJosip Kli8inovi4, prof. matematike i informatike, X. gimnazija „Ivan Supek“, ZagrebIvana Lovi4 6tenc, dipl. u8itelj, Osnovna 5kola Antuna Gustava Mato5a, Zagrebdr. sc. Irena Mi5urac, Sveu8ili5te u Splitu, Filozofski fakultet, Odsjek za u8iteljski studijAna Ostoji4, prof. matematike i informatike, Osnovna 5kola „Meje“ Split, SplitGordana Pai4, mag. educ. math., Osnovna 5kola „Dr. Ivan Merz“, ZagrebSnje9ana 6i5i4, prof. matematike, Tehni8ka 5kola Zagreb, ZagrebEva 6palj, mag. math., XV. gimnazija, Zagreb (voditeljica):ur;a Trupini4, nastavnik razredne nastave, Osnovna 5kola „Mato Lovrak“, Nova Gradi5ka!"#$'<, ()*+!$, *#-$, (.+/'$, '0 2,-'$'<, 0# ()*+!$+ ' #-3'$'()*#)'&$+ /%-*=.+Mirjana Konosi4, Agencija za odgoj i obrazovanje (voditeljica):ur;a Kulu5i4, Agencija za odgoj i obrazovanjeAntonela Czwyk Mari4, Agencija za odgoj i obrazovanje!"#$'<# ()*+!$, *#-$, (.+/'$, '0 ,.(/,*)$, *#-$, (.+/'$,Ru9ica Vuk),>$'!.' .%%*-'$#)%* ()*+!$, *#-$, (.+/'$,Jurko Maroevi4, Ministarstvo znanosti, obrazovanja i sporta,.(/,*)$# *#-$# (.+/'$#Boris Joki4 (voditelj)Branislava Baranovi4Suzana HitrecTomislav Re5kovacZrinka Risti4 Dedi4Branka Vuk",.)+*#Igor Medi4Ana Mihaljevi4
!"#$%!"&!$ '()$'(&(* !"+,"-!%." /)01*0,"*",0*",$'"
Prijedlog!"#$%&% '()*.
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Pred Vama se nalazi prijedlog nacionalnog kurikuluma nastavnog predmeta. Nacionalni kuriku-lumi nastavnih predmeta dio su sustava nacionalnih kurikulumskih dokumenata koji je Okvirom nacionalnog kurikuluma (!"#) odre$en kao sustav dokumenata kojima se na nacionalnoj razini iskazuju namjere povezane sa svrhom, ciljevima, o%ekivanjima, ishodima, iskustvima djece i mla-dih osoba, s organizacijom odgojno-obrazovnoga procesa i s vrednovanjem. Sustav nacionalnih kurikulumskih dokumenata prikazan je na Slici A.
Svi nacionalni kurikulumski dokumenti oblikovani su s idejom o djetetu i mladoj osobi kao o sredi&njem sudioniku odgojno-obrazovnoga procesa. Djeci i mladim osobama, roditeljima, od-gojno-obrazovnim radnicima kurikulumski dokumenti jasno ukazuju na odgojno-obrazovna o%e-kivanja i ishode koja postavljamo pred djecu i mlade osobe. Razvojni su i otvoreni dokumenti koje je mogu'e promijeniti kao odgovor na potrebe djece i mladih osoba, odgojno-obrazovnih radnika i ustanova, novih znanstvenih i tehnolo&kih spoznaja i onih proiza&lih iz prakse.
Nacionalnim kurikulumima nastavnih predmeta odre$uju se svrha, ciljevi, struktura, odgoj-no-obrazovni ishodi i razine njihove usvojenosti, u%enje i pou%avanje, povezanost s drugim pred-metima, odgojno-obrazovnim podru%jima i me$upredmetnim temama te vrednovanje usvoje-nosti odgojno-obrazovnih ishoda u predmetu.
Domene/koncepti u organizaciji predmetnog kurikuluma %ine gradivnu strukturu odre$enog predmeta i prote(u se kroz cijeli period pou%avanja predmeta. Unutar svake domene/koncepta odre$eni su odgojno-obrazovni ishodi.
Odgojni-obrazovni ishodi predstavljaju jasne i nedvosmislene iskaze o tome &to o%ekujemo od u%enika u odre$enoj domeni/konceptu predmeta na kraju odre$ene godine u%enja. Odre$eni su kao po(eljna znanja, vje&tine i stavovi koji se napredovanjem u odgojno-obrazovnom sustavu uslo(njavaju. Kroz godine u%enja ishodi %ine zaokru(enu, logi%nu cjelinu u%enja i pou%avanja u odre$enoj predmetnoj domeni/konceptu. Kao cjelina kroz sve godine u%enja i pou%avanja odre-$uju ukupna iskustva u%enja u odre$enom predmetu.
Svaki je ishod oblikovan kao cjelina koja, uz formulaciju ishoda, uklju%uje i razradu ishoda, pre-poruke za njegovo ostvarivanje i opis razina usvojenosti. )itanje ishoda stoga, osim na sâmu formulaciju ishoda, mora biti usmjereno i na ostale njegove komponente.
Razrada ishoda uklju%uje preciznije odre$enje aktivnosti i sadr(aja u okviru pojedinog ishoda ili skupine ishoda.
Za veliku ve'inu ishoda odre$ene su razine njihove usvojenosti. Opisi razina usvojenosti pre-ciznije odre$uju dubinu i &irinu svakog ishoda i opisuju o%ekivana postignu'a u%enika na kraju odre$ene godine u%enja, %ime se olak&ava planiranje i provedba vrednovanja.Osim razrade samih odgojno-obrazovnih ishoda, u ve'ini kurikuluma nastavnih predmeta navo-de se i preporuke za njihovo ostvarivanje.
Od u%enika se o%ekuju ostvarivanje svih odgojno-obrazovnih ishoda.
UPUTE ZA )ITANJE
Slika !. Sustav nacionalnih kurikulumskih dokumenata izra"enih u okviru Cjelovite kurikularne reforme
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Predmetni kurikulumi i Kurikulumi za stjecanje kvalifikacija u redovnom sustavu strukovnog i umjetni9kog obrazovanja
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:
1. c 1. d dječaci 10 8 djevojčice 9 12
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KAMION
AUTOMOBIL
BICIKL
ŽUTI X
CRVENI X
PLAVI X
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∙∙ ∙ ∙
∙ ∙ ∙ ∙ ∙ ∙ ∙∙ ∙ ∙
(𝑥𝜖{3,4,5… } )
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∙
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∙ ∙ ∙ ∙ ∙
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∙ ∙ ∙ ∙ ∙ ∙
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∙
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𝑎𝑏𝑐
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𝑥𝜖𝑄 , 5.2 ≤ 𝑥 ≤ 9.4, 𝑥𝜖𝑄 , 1.2 ≤ 𝑥 < 3.6, 𝑥𝜖𝑄 , 2.3 ≤ 𝑥 ≤
|𝑥| > 3, |𝑦| ≤ −3, |𝑥 + 1| = 6
∙ ∙∙ ∙ ∙
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∙ ∙ ∙
|𝑥| = 5𝑥𝜖{−3,−2,−1… }
∙ ∙ ∙
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356.43 = 3 ∙ 10 + 5 ∙ 10 + 6 ∙ 10 + 4 ∙ 10 + 3 ∙ 10
∙ 10 ∙ 10 ∙ 10 ∙ ∙ 10 ∙ ∙ ∙ ∙ ∙ 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10∙ 10 ∙ ∙ 10 ∙ ∙ ∙
𝑎𝑥 + 𝑏 = 0
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𝑎𝑥 + 𝑏 = 0
𝑎𝑥 = 𝑏
𝑎𝑥 + 𝑏 = 0
∙
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10 10 10 10 10 10 10
10 10 10 10 1010 10
∙ 10 10 10
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∈
,....5,4x
∙ ∙∙ ∙ ∙
∙ ∙ ∙
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∙
Primjer : (2𝑥) , , .
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𝑥 ∙ 7 ∙ 𝑦, 0.5 ∙ (𝑥 ± 𝑦), 6𝑦 − (2 ∙ 𝑥 ± 3 ∙ 𝑦), 7 ∙ (𝑥 ± 𝑦) − (𝑥 − 2 ∙ 𝑦), 0.5 ∙ 𝑥 ∙ , 𝑥 ∙ 𝑦
−3 ∙ 𝑥 ∙ 6 ∙ 𝑥,34∙ 𝑥 ∙
14∙ 𝑥, 6 ∙
−3𝑥7
± 8𝑦 , 0.01 ∙ −2𝑥 ±45𝑦 , 2 ∙ (3𝑥 − 1) − (5𝑥 + 2)
(3𝑥 − 6𝑦)(3𝑥 + 6𝑦), (3𝑥 + 6𝑦)(3𝑥 + 6𝑦), (3𝑥 − 6𝑦)(3𝑥 − 6𝑦)
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𝑎𝑥 + 𝑏 = 0𝑎𝑥 + 𝑏 = 0
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𝑦 = 𝑎𝑥 + 𝑏
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𝜋
𝜋
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7.63 ∙ 10 𝑘𝑚 1.392 ∙ 10 𝑘𝑚
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(𝑎𝑥 − 𝑏𝑦)(𝑎𝑥 + 𝑏𝑦), (𝑎𝑥 + 𝑏𝑦)(𝑎𝑥 + 𝑏𝑦), (𝑎𝑥 − 𝑏𝑦)(𝑎𝑥 − 𝑏𝑦), 𝑎 ∙ (𝑏 ∙ 𝑥 ± 𝑐 ∙ 𝑦) − (𝑑 ∙ 𝑥 ± 𝑒 ∙ 𝑦) 𝑎, 𝑏, 𝑐, 𝑑, 𝑒 𝜖 𝑁, 𝑍, 𝑄 𝑖 𝑅
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𝑎𝑥 = 𝑏
𝑎𝑥 + 𝑏 = 0𝑎𝑥 + 𝑏 = 0
𝑥 = 𝑎
𝑥 = 𝑎 𝑥 = 𝑎 𝑥 = 𝑎𝑥 = 𝑎
𝑥 = 𝑎, 𝑎𝑥 = 𝑏, 𝑎(𝑥 ± 𝑏) = 𝑐
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1. razred trogodišnje srednje škole (70 sati)
)
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𝑎𝑥 + 𝑏 = 0
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𝑎𝑥 + 𝑐 = 0
𝑓(𝑥) = 𝑎𝑥 𝑓(𝑥) = 𝑎𝑥 + 𝑐
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𝑎𝑥 + 𝑐 = 0
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f(x) = axf(x) = ax + c
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+ 16
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𝑁(𝑥) = 10000 ∙ 2
𝐶 = 𝐶 ∙ 1.03
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𝑓(𝑥) = 𝑥 𝑓(𝑥) = 𝑓(𝑥) = √𝑥
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∙
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, ∙ , −
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š − š
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3 ∈ 𝐴 3 ∈ 𝐵 𝐴 ∪ 𝐵 =< −∞, 15 > 𝐴 ∩ 𝐵 = {3} 𝐴\𝐵 = 𝐴
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√−1 = 𝑖 √−16 = 4𝑖.
√3 + 3√2 − 3√3 + 4√2 √3 2 − √3 3 − 2√2
√3 + 3√2 − 3√3 + 4√2 √3(2 − √9) 3 − 2√4√
√, 𝑎, 𝑏, 𝑐 ∈ 𝑁
√ax + b = cx + d
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3x + 4x − 1 = 0
𝑓(𝑥) =
𝑓(𝑥) = √𝑎𝑥 + 𝑏
𝑓(𝑥) =
𝑓(𝑥) = √𝑥. 𝑓(𝑥) =
𝑓(𝑥) = √𝑥
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𝑓(𝑥) = 𝑓(𝑥) = √𝑥𝑦 = 𝑥
𝑓(𝑥) = 𝑎𝑥𝑓(𝑥) = 𝑎𝑥 + 𝑐
𝑓(𝑥) = 𝑎(𝑥 − 𝑥 ) + 𝑦 𝑓(𝑥) = 𝑎𝑥 + 𝑏𝑥 + 𝑐
𝑓(𝑥) = − 𝑥 + 12𝑥 − 180 𝑥 𝑓(𝑥)𝑥
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𝑎𝑥 + 𝑏𝑥 + 𝑐 > 0, 𝑎𝑥 + 𝑏𝑥 + 𝑐 ≥ 0, 𝑎𝑥 + 𝑏𝑥 + 𝑐 < 0, 𝑎𝑥 + 𝑏𝑥 + 𝑐 ≤ 0, 𝑎, 𝑏, 𝑐𝜖𝑄
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𝑓(𝑥) = 𝑎 , 𝑓(𝑥) = 𝑏 ∙ 𝑎 , 𝑓(𝑥) = 𝑙𝑜𝑔 𝑥
𝑦 = 𝑥
𝑁(𝑥) = 10000 ∙ 2
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𝑠𝑖𝑛 𝑥 + 𝑐𝑜𝑠 𝑥 = 1, 𝑡𝑔𝑥 ∙ 𝑐𝑡𝑔𝑥 = 1, = 𝑡𝑔𝑥
𝑐𝑜𝑠𝑥 = , 𝑥 ∈ ⟨ , 2𝜋⟩
𝑠𝑖𝑛 𝑥 + 𝑐𝑜𝑠 𝑥 = 𝑠𝑖𝑛 𝑥 − 𝑐𝑜𝑠 𝑥.
𝑓(𝑥) = 𝑠𝑖𝑛𝑥, 𝑓(𝑥) = 𝑐𝑜𝑠𝑥, 𝑓(𝑥) = 𝑡𝑔𝑥, 𝑓(𝑥) = 𝐴 sin(𝑏𝑥 + 𝑐) + 𝑑, 𝑓(𝑥) = 𝐴 cos(𝑏𝑥 + 𝑐) + 𝑑
𝑓(𝑥) = Asin(𝑏𝑥), 𝑓(𝑥) = Acos (𝑏𝑥)
𝑓(𝑥) = 𝐴𝑠𝑖𝑛(𝑏𝑥 + 𝑐) + 𝑑,𝑓(𝑥) = 𝐴𝑐𝑜𝑠(𝑏𝑥 + 𝑐) + 𝑑
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D(t) = · sin( · (𝑡 − 79) + 12
𝑠𝑖𝑛𝑥 = 𝑎, 𝑐𝑜𝑠𝑥 = 𝑎, 𝑡𝑔𝑥 = 𝑎, 𝐴 sin(𝑏𝑥 + 𝑐) + 𝑑 = 0, 𝐴 cos(𝑏𝑥 + 𝑐) + 𝑑 = 0
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ℎ
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𝑧 = 2
𝑅𝑒(𝑧) = 2, 𝐼𝑚(𝑧) < 3, |𝑧| = 2, |𝑧| ≥ 3
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, , ,
lim → , lim → , lim → 𝑎
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𝑉 = 200(50 − 𝑡) 𝑚
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∙
, ∙ , −
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š − š
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√−1 = 𝑖 √−16 = 4𝑖.√3 + 3√2 − 3√3 + 4√2 √3 2 − √3 3 − 2√2
√3 + 3√16 − 3√31 + 4√2 √3(2 − √9) 3 − 2√4√
√, 𝑎, 𝑏, 𝑐 ∈ 𝑁
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√𝑎𝑥 + 𝑏 = 𝑐𝑥 + 𝑑=
𝑥 − 5𝑥 + 6 = 0𝑥 − 𝑚𝑥 + 6 = 0
4𝑥 − 3𝑥 + 2 = 0 𝑥 + 𝑥
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𝑚(𝑥 + 1) = 𝑥(𝑚 − 1)
𝑓(𝑥) =
𝑓(𝑥) = √𝑥.
𝑓(𝑥) =
𝑓(𝑥) = √𝑥
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𝑓(𝑥) = 𝑓(𝑥) = √𝑥𝑦 = 𝑥
𝑓(𝑥) = 𝑎(𝑥 − 𝑥 ) + 𝑦 𝑓(𝑥) = 𝑎𝑥 + 𝑏𝑥 + 𝑐
𝑓(𝑥) = − 𝑥 + 12𝑥 − 180 𝑥 𝑓(𝑥)𝑥
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𝑓(𝑥) = 𝑎 ,𝑓(𝑥) = 𝑎 + 𝑐, 𝑓(𝑥) = 𝑎 ,𝑓(𝑥) = 𝑏 ∙ 𝑎 (𝑥) = 𝑙𝑜𝑔 𝑥,𝑓(𝑥) = 𝑙𝑜𝑔 𝑥 + 𝑐, 𝑓(𝑥) =𝑙𝑜𝑔 (𝑥 + 𝑐)
𝑦 = 𝑥
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𝑁(𝑥) = 10000 ∙ 2
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𝑠𝑖𝑛 𝑥 + 𝑐𝑜𝑠 𝑥 = 1, 𝑡𝑔𝑥 ∙ 𝑐𝑡𝑔𝑥 = 1, = 𝑡𝑔𝑥
𝑐𝑜𝑠𝑥 = , 𝑥 ∈ ⟨ , 2𝜋⟩
𝑠𝑖𝑛 𝑥 + 𝑐𝑜𝑠 𝑥 = 𝑠𝑖𝑛 𝑥 − 𝑐𝑜𝑠 𝑥.
𝑓(𝑥) = 𝑠𝑖𝑛𝑥, 𝑓(𝑥) = 𝑐𝑜𝑠𝑥, 𝑓(𝑥) = 𝑡𝑔𝑥, 𝑓(𝑥) = 𝐴 sin(𝑏𝑥 + 𝑐) + 𝑑, 𝑓(𝑥) = 𝐴 cos(𝑏𝑥 + 𝑐) + 𝑑
𝑓(𝑥) = Asin(𝑏𝑥), 𝑓(𝑥) = Acos (𝑏𝑥)
𝑓(𝑥) = 𝐴𝑠𝑖𝑛(𝑏𝑥 + 𝑐) + 𝑑, 𝑓(𝑥) = 𝐴𝑐𝑜𝑠(𝑏𝑥 + 𝑐) + 𝑑
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D(t) = · sin( · (𝑡 − 79) + 12,
𝑠𝑖𝑛𝑥 = 𝑎, 𝑐𝑜𝑠𝑥 = 𝑎, 𝑡𝑔𝑥 = 𝑎, 𝑐𝑡𝑔𝑥 = 0 𝐴 sin(𝑏𝑥 + 𝑐) + 𝑑 = 0, 𝐴 cos(𝑏𝑥 + 𝑐) + 𝑑 = 0, 𝐴𝑡𝑔(𝑏𝑥 + 𝑐) + 𝑑 = 0, 𝐴𝑐𝑡𝑔(𝑏𝑥 + 𝑐) + 𝑑 = 0, 𝑠𝑖𝑛𝑥 < 𝑎, 𝑐𝑜𝑠𝑥 ≤ 𝑎, 𝑡𝑔𝑥 > 𝑎, 𝑐𝑡𝑔𝑥 ≥ 0 𝐴 sin(𝑏𝑥 + 𝑐) + 𝑑 ≥ 0, 𝐴 cos(𝑏𝑥 + 𝑐) + 𝑑 < 0, 𝐴𝑡𝑔(𝑏𝑥 + 𝑐) + 𝑑 ≥ 0.
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ℎ
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1 ⋅ 2 + 2 ⋅ 3 + 3 ⋅ 4+. . . +𝑛(𝑛 + 1) = ( )( ) , ∀𝑛𝜖𝑁
+ + +. . . + =( )
, ∀𝑛𝜖𝑁
3|5 + 2 , ∀𝑛𝜖𝑁
𝑧 = 2
𝑅𝑒(𝑧) = 2, 𝐼𝑚(𝑧) < 3, |𝑧| = 2, |𝑧| ≥ 3
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, , ,
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lim → , lim → , lim → 𝑎
𝑉 = 200(50 − 𝑡) 𝑚 .
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∙
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𝑧 𝐵𝑎(𝑂𝐻) + 𝑦𝐻 𝑃𝑂 → 𝑧 𝐵𝑎 (𝑃𝑂 ) + 𝑣𝐻 𝑂
š − š
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√−1 = 𝑖 √−16 = 4𝑖.√
√ √ √
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𝑥 − (𝑏 + 3𝑎)𝑥 + 3𝑎 − 2𝑏 + 3𝑎𝑏 = 0
√𝑎𝑥 + 𝑏 = 𝑐𝑥 + 𝑑
𝑥 − 5𝑥 + 6 = 0𝑥 − 𝑚𝑥 + 6 = 0
4𝑥 − 3𝑥 + 2 = 0 𝑥 + 𝑥 +
𝑚(𝑥 + 1) = 𝑥(𝑚 − 1)
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(𝑓 ∘ 𝑔)(𝑥) (𝑔 ∘ 𝑓)(𝑥) (𝑓 ∘ 𝑔 ∘ 𝑓)(𝑥) 𝑓(𝑥) = 𝑥 − 𝑥 + 3 𝑔(𝑥) = 𝑥 + 2
𝑓(𝑥) 𝑓(𝑥 − 1) = 𝑥 + 3𝑥 − 1
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𝑓(𝑥) =
𝑓(𝑥) = √𝑥.
𝑓(𝑥) =
𝑓(𝑥) = √𝑥
𝑓(𝑥) = 𝑓(𝑥) = √𝑥𝑦 = 𝑥
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𝑓(𝑥) = 𝑎(𝑥 − 𝑥 ) + 𝑦 𝑓(𝑥) = 𝑎𝑥 + 𝑏𝑥 + 𝑐
𝑓(𝑥) = − 𝑥 + 12𝑥 − 180 𝑥 𝑓(𝑥)𝑥
> 0, ≤ 0
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γ
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𝑓(𝑥) = 𝑎 , 𝑓(𝑥) = 𝑎 + 𝑐,
𝑓(𝑥) = 𝑎 , 𝑓(𝑥) = 𝑏 ∙ 𝑎 ,
𝑓(𝑥) = 𝑙𝑜𝑔 𝑥, 𝑓(𝑥) = 𝑙𝑜𝑔 𝑥 + 𝑐,
𝑓(𝑥) = 𝑙𝑜𝑔 (𝑥 + 𝑐)
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𝑁(𝑥) = 10000 ∙ 2
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𝑠𝑖𝑛 𝑥 + 𝑐𝑜𝑠 𝑥 = 1, 𝑡𝑔𝑥 ∙ 𝑐𝑡𝑔𝑥 = 1, = 𝑡𝑔𝑥
𝑐𝑜𝑠𝑥 = − , 𝑥 ∈ ⟨− ,−𝜋⟩
𝑓(𝑥) = 𝑠𝑖𝑛𝑥, 𝑓(𝑥) = 𝑐𝑜𝑠𝑥, 𝑓(𝑥) = 𝑡𝑔𝑥, 𝑓(𝑥) = 𝑐𝑡𝑔𝑥, 𝑓(𝑥) = 𝐴 sin(𝑏𝑥 + 𝑐) + 𝑑, 𝑓(𝑥) = 𝐴 cos(𝑏𝑥 + 𝑐) + 𝑑, 𝑓(𝑥) = 𝐴𝑡𝑔(𝑏𝑥), 𝑓(𝑥) = 𝐴𝑐𝑡𝑔(𝑏𝑥)
𝑓(𝑥) = Asin(𝑏𝑥), 𝑓(𝑥) = Acos (𝑏𝑥)𝑓(𝑥) = 𝐴𝑡𝑔(𝑏𝑥), 𝑓(𝑥) = 𝐴𝑐𝑡𝑔(𝑏𝑥)
𝑓(𝑥) = 𝐴𝑠𝑖𝑛(𝑏𝑥 + 𝑐) + 𝑑, 𝑓(𝑥) = 𝐴𝑐𝑜𝑠(𝑏𝑥 + 𝑐) + 𝑑
D(t) = · sin( · (𝑡 − 79) + 12
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𝑠𝑖𝑛𝑥 = 𝑎, 𝑐𝑜𝑠𝑥 = 𝑎, 𝑡𝑔𝑥 = 𝑎, 𝑐𝑡𝑔𝑥 = 0 𝐴 sin(𝑏𝑥 + 𝑐) + 𝑑 = 0, 𝐴 cos(𝑏𝑥 + 𝑐) + 𝑑 = 0, 𝐴𝑡𝑔(𝑏𝑥 + 𝑐) + 𝑑 = 0, 𝐴𝑐𝑡𝑔(𝑏𝑥 + 𝑐) + 𝑑 = 0, 𝑠𝑖𝑛𝑥 < 𝑎, 𝑐𝑜𝑠𝑥 ≤ 𝑎, 𝑡𝑔𝑥 > 𝑎, 𝑐𝑡𝑔𝑥 ≥ 0 𝐴 sin(𝑏𝑥 + 𝑐) + 𝑑 ≥ 0, 𝐴 cos(𝑏𝑥 + 𝑐) + 𝑑 < 0, 𝐴𝑡𝑔(𝑏𝑥 + 𝑐) + 𝑑 ≥ 0, 𝐴𝑐𝑡𝑔(𝑏𝑥 + 𝑐) + 𝑑 < 0.
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𝑋 ℎ𝑌 𝑏𝑜𝑑
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1 ⋅ 2 + 2 ⋅ 3 + 3 ⋅ 4+. . . +𝑛(𝑛 + 1) = ( )( ) , ∀𝑛𝜖𝑁
+ + +. . . + =( )
, ∀𝑛𝜖𝑁
3|5 + 2 , ∀𝑛𝜖𝑁
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𝑧 = 2
𝑧 = 2 |𝑧 + 2 − 𝑖| < 3
≈
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𝑉 = 200(50 − 𝑡) 𝑚
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∙
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𝑧 𝐵𝑎(𝑂𝐻) + 𝑦𝐻 𝑃𝑂 → 𝑧 𝐵𝑎 (𝑃𝑂 ) + 𝑣𝐻 𝑂
š − š
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𝑣 𝑘𝑚/ℎ 𝑣 𝑘𝑚/ℎ𝑣 < 𝑣 √𝑣 𝑣
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√−1 = 𝑖 √−16 = 4𝑖.
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𝑥 − (𝑏 + 3𝑎)𝑥 + 3𝑎 − 2𝑏 + 3𝑎𝑏 = 0
√𝑎𝑥 + 𝑏 = 𝑐𝑥 + 𝑑
𝑥 − 5𝑥 + 6 = 0
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𝑥 − 𝑚𝑥 + 6 = 0
4𝑥 − 3𝑥 + 2 = 0𝑥 + 𝑥
+
𝑚(𝑥 + 1) = 𝑥(𝑚 − 1)
(𝑓 ∘ 𝑔)(𝑥) (𝑔 ∘ 𝑓)(𝑥) (𝑓 ∘ 𝑔 ∘ 𝑓)(𝑥) 𝑓(𝑥) = 𝑥 − 𝑥 + 3 𝑔(𝑥) = 𝑥 + 2
𝑓(𝑥) 𝑓(𝑥 − 1) = 𝑥 + 3𝑥 − 1
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𝑓(𝑥) =
𝑓(𝑥) = √𝑥.
𝑓(𝑥) =
𝑓(𝑥) = √𝑥
𝑓(𝑥) = 𝑓(𝑥) = √𝑥𝑦 = 𝑥
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𝑓(𝑥) = 𝑎(𝑥 − 𝑥 ) + 𝑦 𝑓(𝑥) = 𝑎𝑥 + 𝑏𝑥 + 𝑐
𝑓(𝑥) = − 𝑥 + 12𝑥 − 180 𝑥 𝑓(𝑥)𝑥
≤ 0 .
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γ
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𝑓(𝑥) = 𝑎 , 𝑓(𝑥) = 𝑎 + 𝑐, 𝑓(𝑥) = 𝑎 , 𝑓(𝑥) = 𝑏 ∙ 𝑎 ,𝑓(𝑥) = 𝑙𝑜𝑔 𝑥, 𝑓(𝑥) = 𝑙𝑜𝑔 𝑥 + 𝑐,𝑓(𝑥) = 𝑙𝑜𝑔 (𝑥 + 𝑐)
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𝑁(𝑥) = 10000 ∙ 2
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𝑠𝑖𝑛 𝑥 + 𝑐𝑜𝑠 𝑥 = 1, 𝑡𝑔𝑥 ∙ 𝑐𝑡𝑔𝑥 = 1, = 𝑡𝑔𝑥
𝑐𝑜𝑠𝑥 = − , 𝑥 ∈ ⟨− ,−𝜋⟩
𝑓(𝑥) = 𝑠𝑖𝑛𝑥, 𝑓(𝑥) = 𝑐𝑜𝑠𝑥, 𝑓(𝑥) = 𝑡𝑔𝑥, 𝑓(𝑥) = 𝑐𝑡𝑔𝑥, 𝑓(𝑥) = 𝐴 sin(𝑏𝑥 + 𝑐) + 𝑑, 𝑓(𝑥) = 𝐴 cos(𝑏𝑥 + 𝑐) + 𝑑, 𝑓(𝑥) = 𝐴𝑡𝑔(𝑏𝑥), 𝑓(𝑥) = 𝐴𝑐𝑡𝑔(𝑏𝑥), 𝑓(𝑥) = 𝐴𝑡𝑔(𝑏𝑥 + 𝑐) + 𝑑, 𝑓(𝑥) = 𝐴𝑐𝑡𝑔(𝑏𝑥 + 𝑐) + 𝑑
𝑓(𝑥) = Asin(𝑏𝑥), 𝑓(𝑥) = Acos (𝑏𝑥)𝑓(𝑥) = 𝐴𝑡𝑔(𝑏𝑥), 𝑓(𝑥) = 𝐴𝑐𝑡𝑔(𝑏𝑥)
𝑓(𝑥) = 𝐴𝑠𝑖𝑛(𝑏𝑥 + 𝑐) + 𝑑, 𝑓(𝑥) = 𝐴𝑐𝑜𝑠(𝑏𝑥 + 𝑐) + 𝑑𝑓(𝑥) = 𝐴𝑡𝑔(𝑏𝑥 + 𝑐) + 𝑑, 𝑓(𝑥) = 𝐴𝑐𝑡𝑔(𝑏𝑥 + 𝑐) + 𝑑
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D(t) = · sin( · (𝑡 − 79) + 12,
𝑠𝑖𝑛𝑥 = 𝑎, 𝑐𝑜𝑠𝑥 = 𝑎, 𝑡𝑔𝑥 = 𝑎, 𝑐𝑡𝑔𝑥 = 0 𝐴 sin(𝑏𝑥 + 𝑐) + 𝑑 = 0, 𝐴 cos(𝑏𝑥 + 𝑐) + 𝑑 = 0, 𝐴𝑡𝑔(𝑏𝑥 + 𝑐) + 𝑑 = 0, 𝐴𝑐𝑡𝑔(𝑏𝑥 + 𝑐) + 𝑑 = 0, 𝑠𝑖𝑛𝑥 < 𝑎, 𝑐𝑜𝑠𝑥 ≤ 𝑎, 𝑡𝑔𝑥 > 𝑎, 𝑐𝑡𝑔𝑥 ≥ 0 𝐴 sin(𝑏𝑥 + 𝑐) + 𝑑 ≥ 0, 𝐴 cos(𝑏𝑥 + 𝑐) + 𝑑 < 0, 𝐴𝑡𝑔(𝑏𝑥 + 𝑐) + 𝑑 ≥ 0, 𝐴𝑐𝑡𝑔(𝑏𝑥 + 𝑐) + 𝑑 < 0.
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ℎ
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+ + +. . . + =( )
, ∀𝑛𝜖𝑁
17|2 + 5 ⋅ 3 , ∀𝑛𝜖𝑁3 > 2 + 3𝑛, ∀𝑛 ≥ 3
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𝑧 = 2
𝑧 = 2 |𝑧 + 2 − 𝑖| < 3
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≈
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𝑉 = 200(50 − 𝑡) 𝑚 .
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𝑓(𝑥) = 𝑎 , 𝑓(𝑥) = 𝑎 + 𝑐, 𝑓(𝑥) =𝑎 , 𝑓(𝑥) = 𝑏 ∙ 𝑎 , 𝑓(𝑥) = 𝑙𝑜𝑔 𝑥,𝑓(𝑥) = 𝑙𝑜𝑔 𝑥 + 𝑐, 𝑓(𝑥) = 𝑙𝑜𝑔 (𝑥 +𝑐)
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𝑁(𝑥) = 10000 ∙ 2
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𝑠𝑖𝑛 𝑥 + 𝑐𝑜𝑠 𝑥 = 1, 𝑡𝑔𝑥 ∙ 𝑐𝑡𝑔𝑥 = 1, = 𝑡𝑔𝑥
𝑐𝑜𝑠𝑥 = − , 𝑥 ∈ ⟨− ,−𝜋⟩
𝑓(𝑥) = 𝑠𝑖𝑛𝑥, 𝑓(𝑥) = 𝑐𝑜𝑠𝑥, 𝑓(𝑥) = 𝑡𝑔𝑥, 𝑓(𝑥) = 𝑐𝑡𝑔𝑥, 𝑓(𝑥) = 𝐴 sin(𝑏𝑥 + 𝑐) + 𝑑, 𝑓(𝑥) = 𝐴 cos(𝑏𝑥 + 𝑐) + 𝑑, 𝑓(𝑥) = 𝐴𝑡𝑔(𝑏𝑥), 𝑓(𝑥) = 𝐴𝑐𝑡𝑔(𝑏𝑥), 𝑓(𝑥) = 𝐴𝑡𝑔(𝑏𝑥 + 𝑐) + 𝑑, 𝑓(𝑥) = 𝐴𝑐𝑡𝑔(𝑏𝑥 + 𝑐) + 𝑑
𝑓(𝑥) = Asin(𝑏𝑥), 𝑓(𝑥) = Acos (𝑏𝑥)𝑓(𝑥) = 𝐴𝑡𝑔(𝑏𝑥),𝑓(𝑥) = 𝐴𝑐𝑡𝑔(𝑏𝑥)
𝑓(𝑥) = 𝐴𝑠𝑖𝑛(𝑏𝑥 + 𝑐) + 𝑑, 𝑓(𝑥) = 𝐴𝑐𝑜𝑠(𝑏𝑥 + 𝑐) + 𝑑 𝑓(𝑥) = 𝐴𝑡𝑔(𝑏𝑥 + 𝑐) + 𝑑, 𝑓(𝑥) = 𝐴𝑐𝑡𝑔(𝑏𝑥 + 𝑐) + 𝑑
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—
D(t) = · sin( · (𝑡 − 79) + 12
𝑠𝑖𝑛𝑥 = 𝑎, 𝑐𝑜𝑠𝑥 = 𝑎, 𝑡𝑔𝑥 = 𝑎, 𝑐𝑡𝑔𝑥 = 0 𝐴 sin(𝑏𝑥 + 𝑐) + 𝑑 = 0, 𝐴 cos(𝑏𝑥 + 𝑐) + 𝑑 = 0, 𝐴𝑡𝑔(𝑏𝑥 + 𝑐) + 𝑑 = 0, 𝐴𝑐𝑡𝑔(𝑏𝑥 + 𝑐) + 𝑑 = 0, 𝑠𝑖𝑛𝑥 < 𝑎, 𝑐𝑜𝑠𝑥 ≤ 𝑎, 𝑡𝑔𝑥 > 𝑎, 𝑐𝑡𝑔𝑥 ≥ 0 𝐴 sin(𝑏𝑥 + 𝑐) + 𝑑 ≥ 0, 𝐴 cos(𝑏𝑥 + 𝑐) + 𝑑 < 0, 𝐴𝑡𝑔(𝑏𝑥 + 𝑐) + 𝑑 ≥ 0, 𝐴𝑐𝑡𝑔(𝑏𝑥 + 𝑐) + 𝑑 < 0.
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—
𝑋 ℎ𝑌 𝑏𝑜𝑑
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—
+ + +. . . + =( )
, ∀𝑛𝜖𝑁
17|2 + 5 ⋅ 3 , ∀𝑛𝜖𝑁3 > 2 + 3𝑛, ∀𝑛 ≥ 3
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—
𝑧 = 2
𝑧 = 2 |𝑧 + 2 − 𝑖| < 3
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≈
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—
𝑉 = 200(50 − 𝑡) 𝑚
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—
о е а а а е а а а е о а а а о е е ј е е о а а еј је ео а е е е е а е а о а а о је е о е е а о а а
е о а е а а а а а е а е е е а а аа а а а а о о е о је а а е а е а е
о о о е а
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dr. sc. Irena Mišurac 26. veljače 2016. Sveučilište u Splitu Filozofski fakultet u Splitu Odsjek za učiteljski studij
Izdvojeno mišljenje o dijelu dokumenta „Kurikulum nastavnoga predmeta Matematika“
Poštovani, kao članica stručne radne skupine, kao metodičarka nastave matematike i kao
profesorica matematike nisam u potpunosti zadovoljna konačnim dokumentom. Moje nezadovoljstvo posebno se odnosi na ishode učenja koji su zamišljeni kao znanja, vještine i stavovi koje očekujemo od učenika na kraju pojedinoga razreda. Želim naglasiti da usprkos ovom izdvojenom mišljenju, dajem punu potporu Cjelovitoj kurikularnoj reformi koju smatram dobro osmišljenim i važnim projektom u osuvremenjivanju odgojno - obrazovnoga sustava te u razvijanju životnih kompetencija naših učenika. Voljela bih da se ovo moje očitovanje protumači kao poticaj za daljnja promišljanja i refleksije o kvaliteti kurikuluma kako bi u konačnici bio što učinkovitiji i kvalitetniji.
Ulazeći u reformu, pošla sam od činjenice da rezultati naših učenika u matematici nisu zadovoljavajući. Rezultati vanjskih vrednovanja (kako nacionalnih tako i međunarodnih – PISA i TIMSS) pokazali su nisku razinu matematičke pismenosti naših učenika, veliki je broj negativnih ocjena upravo iz matematike, brojni učenici prisiljeni su plaćati dodatne instrukcije kako bi savladali programom propisane sadržaje, stav učenika prema matematici je izrazito loš, a naučeno u školi učenici ne uspijevaju primijeniti u svakodnevnim situacijama. Sve navedeno traži od nas da preispitamo uzroke ovih pojava i pokušamo poboljšati uvjete učenja kako bi se rezultati učenika u matematici popravili. Gledajući postojeće nastavne programe matematike uočavamo da je sadržaj jako opširan i zahtjevan, posebno u predmetnoj nastavi u osnovnoj školi te u srednjim školama. Zbog toga se često događa da učitelji „jure“ kroz teme kako bi ih u predviđenoj satnici realizirali, nemajući vremena zaustavljati se na dijelovima koje učenici nisu uspjeli na zadovoljavajućoj razini usvojiti. Uz to, količina sadržaja koja se iz razreda u razred gomila dovodi do toga da učenici sve teže povezuju naučeno, površnije uče i više zaboravljaju, pa na kraju ne steknu niti osnovna znanja i vještine koje su im za život potrebne. Ovo obično prate i loše ocjene te pad samopouzdanja i vjere u sebe, pa se događa da mnogi učenici „bježe“ od matematike čime dugoročno gube i oni i naše društvo u cjelini. Kako je odmah naglašeno da neće biti povećanja broja sati matematike, kao jedan od načina podizanja kvalitete nastave, a time i rezultata učenika, vidjela sam mogućnost da kroz ovu reformu rasteretimo sadržaje koji se u matematici uče. Pri tome ne mislim na neselektivno izbacivanje pojedinih tematskih cjelina, već odabir onih najvažnijih i najreprezentativnijih koncepata koji svakom čovjeku trebaju kako bi bio matematički pismen i kako bi matematičke spoznaje mogao primjenjivati u svom životu.
Tijekom osmišljavanja kurikuluma matematike stalno sam se zalagala da mi matematičari odredimo koji su koncepti najvažniji i treba ih znati svaki učenik, a koji
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koncepti spadaju u matematičku nadgradnju i vezuju se uz određeno strukovno obrazovanje ili predstavljaju dio matematičke teorije. Znanja i vještine koji su temeljni i potrebni svima trebaju se učiti kroz obavezno obrazovanje namijenjeno svima, dok nadgradnju treba prilagoditi potrebama, interesima i mogućnostima učenika te satnici matematike. Nažalost, kao rezultat našega višemjesečnoga rada i dalje ne vidim da su se sadržaji na ovaj način rasteretili, a čini mi se da je u konačnom dokumentu isplanirano još više koncepata nego što ih se učilo do sada. Kroz cijelu je vertikalu dodana nova domena Podaci, statistika i vjerojatnost čija je važnost neupitna, ali koja traži dodatno vrijeme za njeno poučavanje te uključuje još više matematičkih sadržaja. Uza sve sadašnje sadržaje, vraćeni su i neki koncepti koje su ranije reforme izbacile iz programa u cilju rasterećenja (npr. dvojni razlomci ili kompleksni brojevi).
Gledajući ishode po razinama, mišljenja sam da su neki ishodi u razrednoj nastavi presloženi i preambiciozni za učenike nižih razreda, posebno kada se od učenika 1. razreda očekuje da se „služi podacima i prikazuje ih piktogramima i jednostavnim tablicama“. Ishode nižih razreda često opterećuje i nepotrebno komplicirana formulacija kako ishoda tako i razina njihova ostvarivanja. Razrada ishoda koja bi trebala pomoći učiteljima u odabiru sadržaja učenja također je na mnogim mjestima nejasna i nepotpuna. Smatram da u nižim razredima nije bilo neophodno inzistirati na domeni Algebra i funkcije koja niti u NOK-u nije bila predviđena za učenike ove dobi, a što je u konačnici opet rezultiralo ambicioznim ishodima poput onog da učenik 2. razreda „prepoznaje uzorak i kreira niz objašnjavajući pravilnost nizanja“. U preporukama za ostvarivanje ishoda na nekoliko mjesta pojavljuju se naputci koji određuju da se nešto mora raditi na određeni način, a da uz to ne piše zašto se to tako mora raditi. Kao primjer navodim uputu uz zbrajanje s prijelazom desetice u kojoj piše „uvijek je potrebno uz dulji zapis računa zapisati i njegov kraći oblik“ (7+8 =15). Uputa može biti iznesena kao preporuka i to isključivo uz pravovaljanu argumentaciju, a učitelji moraju sami procijeniti hoće li je slijediti ili ne. Sve navedeno otežava pravilno razumijevanje ishoda za razrednu nastavu i u konačnici može za posljedicu imati dodatno opterećivanje mlađih učenika neprimjerenim sadržajima.
Gledajući ishode učenja u predmetnoj nastavi osnovne škole, čini mi se da je količina pojmova koji su predviđeni ovim kurikulumom veća negoli je bila do sada. Kao primjer izrečenoga navesti ću 5. razred u kojem se uz sve sadržaje koje propisuje sadašnji program (osim najvećega zajedničkoga djelitelja i najmanjega zajedničkoga višekratnika koji su prebačeni u 6. razred), predviđa još i učenje skupova i operacija s njima, učenje potencija broja 10, linearna jednadžba, centralna simetrija, grafovi, volumen i preračunavanje mjernih jedinica. Mišljenja sam da je 5. razred posebno osjetljiv budući su učenici u njemu ionako istraumatizirani prelaskom iz razredne u predmetnu nastavu te promjenom učitelja koji ih poučava matematiku. Ova reforma pokušala je ublažiti ovaj prijelaz na način da je 5. razred uključila u 2. ciklus obrazovanja zajedno s 3. i 4. razredom čime se htjelo naglasiti da su učenici ovih razreda po svojim kognitivnim i razvojnim karakteristikama slični te da bi i pristup poučavanju i učenju trebao biti sličan. Smatram da zbog svega navedenoga nije primjereno opteretiti učenike u 5. razredu ovako velikom količinom sadržaja, posebno sadržaja koji su apstraktni za učenike na ovom stupnju kognitivnoga razvoja (poput skupova i
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potencija broja 10!). Gledajući prijedlog kurikuluma mišljenja sam da je količina sadržaja i u 3. ciklusu preopsežna. Smatram da je bilo prostora za rasterećivanje, a osobno sam predlagala da ih rasteretimo koncepata koje učenici osnovne škole najčešće ne mogu u potpunosti razumjeti niti ih onda mogu na pravi način iskoristiti (navest ću primjere sustava linearnih jednadžbi, linearne funkcije, ili sukladnosti i sličnosti trokuta).
Gledajući prijedlog ishoda za srednju školu moram naglasiti da je novina u ovoj reformi bio pristup u kojem su škole grupirane prema broju tjednih sati matematike. To znači da isti program imaju sve škole s 3 sata matematike, neovisno je li riječ o gimnaziji ili strukovnoj školi. U ovu kategoriju škola spada na primjer jezična gimnazija, ali i brojne strukovne škole (tehničar za mehatroniku, odjevni tehničar, pomorski nautičar ili smjerovi iz sektora šumarstvo, prerada i obrada drva i slično). Smatrala sam da u školama koje imaju manje od 4 sata nastave matematike tjedno treba adekvatno „rasteretiti“ sadržaje kako bi se ono što se uči moglo kvalitetno i s razumijevanjem obraditi. Predlagala sam da dio sadržaja stavimo u izborne module kako bi ih pojedine škole (na primjer gimnazije) prema potrebi mogle odabrati. Gledajući ishode koji su predloženi, primjećujem da se sada u svim školama s 3 sata matematike tjedno planiraju raditi sadržaji koji su se do sada u gimnazijskim programima radili. Uz to, ishodi za škole s 3 sata matematike tjedno, gotovo u potpunosti su isti kao i ishodi za učenike iz škole s 4 i 5 sati matematike tjedno! Usprkos činjenici da učenici iz škola s 5 sati matematike tjedno imaju na raspolaganju 66% više sati, učenici koji matematiku imaju 3 sata tjedno moraju ostvariti gotovo sve ishode kao i učenici s 5 sati tjedno (razlika je kroz cijelu srednju školu uglavnom u matricama i determinantama, krivuljama 2. reda, matematičkoj indukciji i Diofantskim jednadžbama). Čak su i razine ishoda koji se od njih očekuju jednake. Ovaj nesrazmjer smatram potpuno neprihvatljivim i to na štetu učenika iz škola s manjim brojem sati. Uzme li se u obzir i činjenica da škole s manjim brojem sati matematike često biraju učenici koji nemaju sklonost prema matematici, apsurdnost ove situacije još više dolazi do izražaja.
Frontalna i predavačka nastava koja je dominirala u tradicionalnoj školi, usprkos svim svojim nedostacima bila je ekonomična i omogućavala je da se u malo vremena svima izloži isti sadržaj. U suvremenoj, budućnosti okrenutoj školi naglašava se samostalno učeničko istraživanje, otkrivanje i promišljanje, samostalno dolaženje do zaključaka, argumentiranje i dokazivanje, a sve to traži vrijeme. Pitam se kako će se ovako veliki broj ishoda sadržajno orijentiranih realizirati u istoj satnici u kojoj ni do sada nismo imali zadovoljavajuće rezultate? Kako će učitelji i nastavnici koji uvedu suvremene načine rada poput suradničkog učenja, problemske nastave ili diskusije stići proći sve sadržaje kako bi učenici stekli kompetencije predviđene postavljenim ishodima učenja?
Uz navedeno, gledajući ishode, vidimo da je orijentacija u cijelom dokumentu ostala tradicionalno usmjerena na nastavne sadržaje, a bitno manje na procese (vještine) koje bi matematikom trebalo razviti ili stavove koje matematikom stječemo. Iako je NOK (2010.) izjednačio važnost (sadržajnih) koncepata i procesa (kompetencija) koji se razvijaju kroz nastavu matematike, ovim dokumentom ta je ravnopravnost procesa i koncepata bitno narušena na štetu procesa. Smatrala sam da u kurikulum treba ubaciti zasebne ishode koji bi tražili da učenik (neovisno o sadržaju učenja) samostalno zaključuje, da
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postavlja pitanja, da povezuje različite matematičke prikaze, da voli matematiku, da ima pozitivan stav o njoj i da ima razvijeno samopouzdanje. U tom slučaju bi i učiteljima bila poslana jasna poruka da se od njih očekuje mnogo više od same realizacije nastavnih sadržaja. Čitajući ishode smatram da je ostala tradicionalna usmjerenost na (preširoke) matematičke sadržaje, a za njihovu realizaciju i dalje će biti jako malo vremena. Naravno da je uz ove nedostatke dokument donio i mnoga dobra rješenja, posebno u pokušaju da spiralno poveže neke matematičke koncepte koji se kroz cikluse nadovezuju, razvijaju i proširuju.
Vjerujem da je nastavom matematike prvenstveno važno učenike naučiti logički misliti i naučiti ih kako voljeti i kako učiti matematiku. Nije potrebno znati baš sve iz matematike da bi učenik bio matematički pismen. Ako zna razmišljati i zna učiti matematiku, ako razumije pojmove koje je upoznao, uvijek može nadograditi svoju kognitivnu mrežu znanja i vještina novim pojmovima ili vještinama ukoliko mu se potreba za tim pojavi. Smatram da je najveća opasnost istraumatizirani učenik koji počinje izbjegavati matematiku i sve aktivnosti vezane uz nju, koji počinje gajiti negativne emocije prema njoj i koji na kraju niti sebi niti društvu ne može pomoći primjenjujući matematičke spoznaje i vještine, a matematika zaista može biti predivan predmet i prezanimljiva igra uma. Željela bih da i naši učenici uživaju u njoj kao što uživamo mi koji je razumijemo i iskreno se nadam da ćemo u dodatnom promišljanju o kurikulumu matematike taj cilj i doseći.
dr. sc. Irena Mišurac
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Sonja Banić, prof. 25. veljače, 2016. SŠ Ivan Švear Ivanić Grad
Izdvojeno mišljenje o dijelu dokumenta „Kurikulum predmeta matematika“
Poštovani, kao članica stručne radne skupine koja je osmišljavala i pisala kurikulum predmeta
Matematika nisam u potpunosti zadovoljna dijelom konačnog dokumenta i želim obrazložiti svoje izdvojeno mišljenje. Dio dokumenta kojim nisam zadovoljna su ishodi učenja u četverogodišnjim programima srednje škole, prvenstveno za plan matematike od 3 sata tjedno (105+105+105+96 sati).
Želim naglasiti da u potpunosti podržavam Cjelovitu kurikularnu reformu, koja je kvalitetno osmišljen i vođen projekt. Smatram da smo u kurikulumu predmeta Matematika napravili značajan iskorak u pravom smjeru. Primjena novog kurikuluma znači kvalitetne promjene i napredak za nastavu matematike te za sustav u cjelini.
Novi kurikulum predviđa suvremeni pristup nastavi, u kojem od učenika očekujemo da samostalno istražuju, otkrivaju pravilnosti, proučavaju i rješavaju probleme te primjenjuju matematiku u situacijama iz životnog i školskog okružja. Također očekujemo da surađuju, rade u timovima, na projektima te prikazuju i obrazlažu svoje radove. Samo suvremena nastava omogućuje da se ostvare ciljevi predmeta matematika, da učenici komuniciraju matematičkim jezikom, matematički rasuđuju, rješavaju problemske situacije te razviju samopouzdanje i pozitivan stav prema matematici. Kao što je objašnjeno u poglavlju 6. kurikuluma predmeta, Učenje i poučavanje, suvremeni pristup zahtijeva da se pojedinoj temi (konceptu) posveti puno više vremena nego što je to bilo do sada, u tradicionalnoj nastavi. Citiram: „Suvremeni pristup nastavi matematike u kojem dominira istraživački pristup, u kojemu se matematika otkriva kroz rješavanje problemskih situacija traži dodatno vrijeme, ali i drugačiji pristup učenju i poučavanju matematike. … Upravo stoga suvremena nastava traži više vremena kako bi učenicima omogućila kreativnost i samostalnost u pristupu i zaključivanju.“ Budući da u postojećem nastavnom planu nema promjene broja sati pojedinog predmeta, više vremena moglo se dobiti jedno smanjenjem broja i opsega matematičkih koncepata.
Plan matematike od 3 sata tjedno (105+105+105+96 sati) odnosi se na mnoge četverogodišnje strukovne škole raznih profila, ali i na jezičnu i klasičnu gimnaziju. Obuhvaća učenike vrlo raznolikog predznanja i matematičkih sposobnosti te različitih očekivanja obzirom na struku i nastavak školovanja. To ga je učinilo vrlo zahtjevnim za stručnu radnu skupinu.
Smatram da u konačnom rješenju za taj plan nismo u dovoljnoj mjeri rasteretili sadržaje i da neće biti moguće provoditi suvremen pristup nastavi, a istovremeno ostvariti sve predviđene ishode u punom opsegu. Ishodi učenja trebali su biti jednostavniji i manje zahtjevni, kako bi bili dostupni učenicima u svim programima koji imaju tu satnicu. Kroz preporučene sadržaje trebala se dati sloboda i odgovornost nastavnicima da rad i ishode učenja prilagode potrebama i sposobnostima svojih učenika. Ovako će nastavnici biti
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primorani juriti kroz sadržaje nauštrb suvremenog pristupa nastavi. U takvoj situaciji neće biti moguće ostvariti ciljeve predmeta matematika. U ostalim nastavnim planovima s više sati tjedno rješenja su bolja, ali se i u njima, smanjenjem sadržaja u ishodima a povećanjem preporučenih sadržaja, mogla dati veća sloboda i odgovornost nastavnicima u određivanju širine i dubine pojedinih ishoda učenja. Smatram da bismo na taj način omogućili veću motivaciju, kvalitetniju nastavu i bolje ostvarenje ciljeva predmeta Matematika.
Sonja Banić, prof.