T e a c h in g g e o me tr y

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UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniSymmetryandWIMS–pagina1

Is o me tri e s , s y mme tr y , te a c h e r tra in in g a n d W IM S Ma ri n a C a zz o la D ip a rt ime n to d i Ma te ma ti ca e A p p lica zi o n i U n iv e rsi t`a d i Mi la n o -B ico cca 1 2 J u n e 2 0 1 4

Is o m e tri e s

Isometries •Teachinggeometry •Whyisometries? •Geometryor Geometries? •Isometriesandteacher training Symmetry Tools WIMS UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniSymmetryandWIMS–pagina2

T e a c h in g g e o me tr y

C ycl e s in It a ly • P ri ma ri a : g ra d e 1 (6 y e a rs) to g ra d e 5 . • S e co n d a ri a d i p ri mo g ra d o : g ra d e s 6 , 7 a n d 8 . • S e co n d a ri a d i se co n d o g ra d o : g ra d e s 9 to 1 3 . T e a ch e r tr a in in g • P ri ma ry: U n iv e rsi ty te a ch e r tr a in in g d e g re e “S ci e n z e d e lla fo rma zi o n e p ri ma ri a ” (5 y e a rs) • S e co n d a ry: U n iv e rsi ty d e g re e (3 y e a r + 2 y e a r) a n d “T iro ci n io fo rma ti v o a tt iv o ” (1 y e a r) W h y is o me tri e s ?

T h e sco p e o f g e o me tr y w a s sp e ct a cu la rl y b ro a d e n e d b y K le in in h is E rl a n g e r P ro g ra mm (E rl a n g e n p ro g ra m) o f 1 8 7 2 , w h ich st re sse d th e fa ct th a t, b e si d e s p la n e a n d so lid E u cl id e a n g e o me tr y, th e re a re ma n y o th e r g e o me tr ie s e q u a lly w o rt h y o f a tt e n ti o n . (H . S . M. C o x e te r, In tro d u ct io n to g e o me tr y , Jo h n W ile y & S o n s In c. , se co n d e d it io n e d it io n , 1 9 6 9 , p . ix) G e o me tr y o r G e o me tri e s ?

[. .. ] E u cl id e a n g e o me tr y is b y n o me a n s th e o n ly p o ssi b le g e o me tr y: o th e r ki n d s a re ju st a s lo g ica l, a lmo st a s u se fu l, a n d in so me re sp e ct si mp le r. A cco rd in g to th e fa mo u s E n la rg e n p ro g ra m (K le in ’s in a u g u ra l a d d re ss a t th e U n iv e rsi ty o f E rl a n g e n in 1 8 7 2 ), th e cr it e ri o n th a t d ist in g u ish e s o n e g e o me tr y fro m a n o th e r is th e g ro u p o f tr a n sf o rma ti o n s u n d e r w h ich th e p ro p o si ti o n re ma in tr u e . (i b id ., p . 6 7 ) fe t

Is o me tri e s a n d te a c h e r tra in in g

P ro sp e ct iv e te a ch e rs • I h a te ma th e ma ti cs , I n e v e r u n d e rst o o d ma th e ma ti cs , I d o n o t w a n t to h a v e a n yt h in g to d o w it h ma th e ma ti cs • I a lre a d y kn o w e v e ryt h in g I n e e d to kn o w In b o th ca se s w e n e e d to sh o w th e m “so me th in g n e w ” (p o ssi b ly so me th in g lika b le ). Sy m m e tr y

Isometries Symmetry •Sym´etrie •Beautifulimages •Deepmathematical concepts •Groupsthroughimages •Rosetteswithflowers •Breakingsymmetry Tools WIMS UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniSymmetryandWIMS–pagina7

S y m ´e tri e

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B e a u ti fu l ima g e s

W a ll p a p e r p a tt e rn s

A n a lo g y

D if fe re n c e

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D e e p ma th e ma ti c a l c o n c e p ts

• Gro u p s ◦ th e to o l to d e scr ib e “symme tr y” o f a fig u re is it s symme tr y g ro u p i. e . th e se t o f a ll iso me tr ie s o f th e p la n e th a t le a v e th e fig u re u n ch a n g e d

G ro u p s th ro u g h ima g e s

Gi v e n a fig u re , y o u ca n fin d it s symme tr y g ro u p r t q s

fo u r re fle ct io n s ( σ r , σ t , σ s e σ q ) w it h re sp e ct to th e d a sh e d lin e s (S o me th in g is mi ssi n g ) G ro u p s a n d ima g e s

Gi v e n a symme tr y g ro u p y o u ca n b u ild ima g e s w it h th a t symme tr y T h e co mp o si ti o n o f tw o re fle ct io n w it h in te rse ct in g a x e s is a ro ta ti o n

G ro u p s a n d ima g e s

Gi v e n a symme tr y g ro u p y o u ca n b u ild ima g e s w it h th a t symme tr y D 4 S h a p in g a n id e a

Mi la n o T re n to S e o u l C 5

R o s e tt e s w it h fl o w e rs

F a lso g e lso mi n o T ra ch e lo sp e rm u m ja smi n o id e s 5 . ( C 5 ) V e rb e n a V e rb e n a o ffici n a lis ∗ 5 . ( D 5 ) B re a k in g s y mme tr y

B re a k in g s y mme tr y

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T o o ls

Isometries Symmetry Tools •matematita •Publishing •Ilritmodelleforme •Publishing •Imagesformathematics •Interactive •Rosettes •Wallpaperpatterns •Kaleido •Simetria WIMS UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniSymmetryandWIMS–pagina25

ma te ma ti ta

In te ru n iv e rsi ty R e se a rch C e n te r fo r th e C o mm u n ica ti o n a n d In fo rma l L e a rn in g o f Ma th e ma ti cs h t t p : / / w w w . m a t e m a t i t a . i t / • o ri g in a te s fro m th e e xp e ri e n ce o f p ro mo ti n g ma th e ma ti cs b y fo u r It a lia n u n iv e rsi ti e s: Mi la n o , Mi la n o -B ico cca , P isa a n d T re n to • fo cu s o n in fo rma l le a rn in g a s o n e o f th e ma in p re re q u isi te s to a n y su b se q u e n t mo re fo rma l le a rn in g • a ims to id e n ti fy th e ri g h t fo rm o f co n te n ts a n d me th o d s fo r th is typ e o f co mm u n ica ti o n ma te ma ti ta

T h e w o rd “ m a te ma ti ta ” re se mb le s th e w o rd “m a te m a ti c a ” w h ich me a n s “ma th e ma ti cs”. A lso “ m a te ” = ma th s “ ma ti ta ” = p e n ci l d o in g ma th e ma ti cs w it h th e p e n ci l

ma te ma ti ta : p ro d u c ts & o ff e rs

h t t p : / / w w w . m a t e m a t i t a . i t / • tr a in in g co u rse s fo r p re -se rvi ce a n d in -se rvi ce te a ch e rs; • p ro b le m-b a se d ma th e ma ti ca l la b o ra to ri e s in sch o o l (b o th in p ri ma ry sch o o l a n d a t a h ig h e r le v e l); • in te ra ct iv e e xh ib it io n s; • w e b -b a se d ma th e ma ti ca l g a me co n te st s; • ico n o g ra p h ic re fe re n ce s o n ma th e ma ti ca l to p ics . P u b li s h in g : b o o k s & C D ro m

Il ri tmo d e ll e fo rme

P u b li s h in g : b o o k s

h t t p : / / w w w . q u a d e r n o a q u a d r e t t i . i t /

A ma g a zi n e fo r s e c o n d a ry s c h o o l s tu d e n ts

h t t p : / / w w w . x l a t a n g e n t e . i t /

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Ima g e s fo r ma th e ma ti c s

h t t p : / / w w w . m a t e m a t i t a . i t / m a t e r i a l e / • U se ima g e s to co mm u n ica te ma th e ma ti ca l id e a s; • ma k e m a te ma ti ta ’s co lle ct io n o f ima g e s a n d a n ima ti o n s a v a ila b le b y cre a ti n g a n o n lin e w e b si te . T h e w e b si te ( ∼ 1 0 0 0 0 ima g e s , co n st a n tl y e v o lvi n g ) is d e si g n e d to b e u se r-f ri e n d ly w h ile st ill e n su ri n g a h ig h le v e l o f sci e n ti fic co rre ct n e ss a lo n g si d e to p q u a lit y re le v a n t ima g e s . E a ch ima g e h a s a p re se n ta ti o n w it h a fu ll (ma th e ma ti ca l) d e scr ip ti o n , p o ssi b ly co n n e ct in g w it h o th e r ima g e s . In te ra c ti v e e x h ib it io n s

S ymme tr y, p la yi n g w it h mi rro rs

m a te mi la n o , ma th e ma ti ca l e xp lo ra ti o n s o f th e ci ty m a te tre n ti n o , ma th e ma ti ca l e xp lo ra ti o n s o f T re n to a n d it s su rro u n d in g s T ra n sp a re n t ma th e ma ti cs: mi n ima l su rf a ce s a n d so a p b u b b le s T h e e x h ib it s

• in te ra ct iv e ◦ “i n te ra tt ivi t`a e lib e rt `a d i sp e ri me n ta zi o n e d e lla mo st ra (n o n ce rt o le mo st re cu i so n o a b it u a ta io in cu i si d e v e so lo g u a rd a re , q u a lch e v o lt a a sco lt a re , ma ma i to cca re !) ” 1 • m u lt ile v e l ◦ p ro p o se p ro b le ms th a t ca n b e e xp e ri e n ce d a t d if fe re n t le v e ls , fro m d if fe re n t vi e w p o in ts

1

“i n te ra ct ivi ty a n d fre e d o m: n o t re a lly th e ki n d o f e xh ib it io n I a m fa mi lia r w it h , w h e re y o u a re o n ly a llo w e d to w a tch , so me ti me s to list e n , b u t n e v e r to to u ch ” In te ra c ti v e

R o s e tt e s

W a ll p a p e r p a tt e rn s

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K a le id o

S ime tri a

W IM S

Is s u e s

Isometries Symmetry Tools WIMS •Issues UniversityofMilano-Bicocca–Dip.diMatematicaeApplicazioniSymmetryandWIMS–pagina46

• w h a t is a n iso me tr y? h o w d o y o u d e a l w it h iso me tr ie s? • iso me tr ie s a n d fig u re s: a p p ly a n iso me tr y to a n ima g e ◦ co n v e rse p ro b le m • symme tr y ◦ re co g n iz e symme tr y ◦ b u ild symme tr ic fig u re s