Symmetry and WIMS – pagina 1
• Groups through images • Rosettes with flowers • Breaking symmetry • Groups through images • Rosettes with flowers • Breaking symmetry
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
WIMS
Symmetry and WIMS – pagina 7
concepts concepts
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
WIMS
Tools
´ • Symetrie • Beautiful images • Deep mathematical
Symmetry
Symmetry
Tools
Isometries
´ • Symetrie • Beautiful images • Deep mathematical
´ Symetrie
Symmetry and WIMS – pagina 8
Symmetry and WIMS – pagina 6
In both cases we need to show them “something new” (possibly something likable).
• I hate mathematics, I never understood mathematics, I do not want to have anything to do with mathematics • I already know everything I need to know
Prospective teachers
Isometries and teacher training
Symmetry and WIMS – pagina 4
(H. S. M. Coxeter, Introduction to geometry, John Wiley & Sons Inc., second edition edition, 1969, p. ix)
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
WIMS
Tools
Symmetry
Geometries? • Isometries and teacher training
• Teaching geometry • Why isometries? • Geometry or
Isometries
Symmetry and WIMS – pagina 2
The scope of geometry was spectacularly broadened by Klein in his Erlanger Programm (Erlangen program) of 1872, which stressed the fact that, besides plane and solid Euclidean geometry, there are many other geometries equally worthy of attention.
Why isometries?
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
WIMS
Tools
Symmetry
Geometries? • Isometries and teacher training
• Teaching geometry • Why isometries? • Geometry or
Isometries
Symmetry
Symmetry and WIMS – pagina 5
Isometries
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
WIMS
Tools
Symmetry
Isometries
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
(ibid., p. 67) fet
[. . . ] Euclidean geometry is by no means the only possible geometry: other kinds are just as logical, almost as useful, and in some respect simpler. According to the famous Enlargen program (Klein’s inaugural address at the University of Erlangen in 1872), the criterion that distinguishes one geometry from another is the group of transformations under which the proposition remain true.
Geometry or Geometries?
Symmetry and WIMS – pagina 3
• Primary: University teacher training degree “Scienze della formazione primaria” (5 years) • Secondary: University degree (3 year + 2 year) and “Tirocinio formativo attivo” (1 year)
Teacher training
• Primaria: grade 1 (6 years) to grade 5. • Secondaria di primo grado: grades 6, 7 and 8. • Secondaria di secondo grado: grades 9 to 13.
Cycles in Italy
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
WIMS
Tools
Symmetry
Geometries? • Isometries and teacher training
• Teaching geometry • Why isometries? • Geometry or
Isometries
Teaching geometry
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
12 June 2014
Marina Cazzola Dipartimento di Matematica e Applicazioni Universita` di Milano-Bicocca
Isometries, symmetry, teacher training and WIMS
Geometries? • Isometries and teacher training
• Teaching geometry • Why isometries? • Geometry or
Isometries
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
Analogy
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
Beautiful images
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
Beautiful images
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
WIMS
Tools
• Groups through images • Rosettes with flowers • Breaking symmetry
concepts
´ • Symetrie • Beautiful images • Deep mathematical
Symmetry
Isometries
Beautiful images
Symmetry and WIMS – pagina 15
Symmetry and WIMS – pagina 13
Symmetry and WIMS – pagina 11
Symmetry and WIMS – pagina 9
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
Difference
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
Wallpaper patterns
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
Beautiful images
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
Beautiful images
Symmetry and WIMS – pagina 16
Symmetry and WIMS – pagina 14
Symmetry and WIMS – pagina 12
Symmetry and WIMS – pagina 10
Seoul
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
Breaking symmetry
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
C5
Shaping an idea
Symmetry and WIMS – pagina 23
Symmetry and WIMS – pagina 21
Trento
Milano
t
Symmetry and WIMS – pagina 18
four reflections (σr , σt , σ s e σq ) with respect to the dashed lines
D4
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
Breaking symmetry
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
Falso gelsomino Trachelospermum jasminoides 5. (C5)
Rosettes with flowers
Symmetry and WIMS – pagina 24
Symmetry and WIMS – pagina 22
Verbena Verbena officinalis ∗5. (D5)
Symmetry and WIMS – pagina 20
Given a symmetry group you can build images with that symmetry
Groups and images
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
WIMS
WIMS
Symmetry and WIMS – pagina 19
Tools
Tools
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
• Groups through images • Rosettes with flowers • Breaking symmetry
The composition of two reflection with intersecting axes is a rotation
concepts
´ • Symetrie • Beautiful images • Deep mathematical
Symmetry
Isometries
• Groups through images • Rosettes with flowers • Breaking symmetry
Given a symmetry group you can build images with that symmetry
concepts
´ • Symetrie • Beautiful images • Deep mathematical
Symmetry
Isometries
Groups and images
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
Symmetry and WIMS – pagina 17
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
(Something is missing)
s
r
WIMS
q
Given a figure, you can find its symmetry group
WIMS
• Groups through images • Rosettes with flowers • Breaking symmetry
concepts
´ • Symetrie • Beautiful images • Deep mathematical
Symmetry
Isometries
Tools
◦ the tool to describe “symmetry” of a figure is its symmetry group i.e. the set of all isometries of the plane that leave the figure unchanged
• Groups
Groups through images
Tools
• Groups through images • Rosettes with flowers • Breaking symmetry
concepts
´ • Symetrie • Beautiful images • Deep mathematical
Symmetry
Isometries
Deep mathematical concepts
Tools
Symmetry and WIMS – pagina 25
Tools
• matematita • Publishing • Il ritmo delle forme • Publishing • Images for mathematics • Interactive • Rosettes • Wallpaper patterns • Kaleido • Simetria WIMS
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
• matematita • Publishing • Il ritmo delle forme • Publishing • Images for mathematics • Interactive • Rosettes • Wallpaper patterns • Kaleido • Simetria WIMS
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
WIMS
WIMS
Symmetry and WIMS – pagina 32
http://www.xlatangente.it/
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
• matematita • Publishing • Il ritmo delle forme • Publishing • Images for mathematics • Interactive • Rosettes • Wallpaper patterns • Kaleido • Simetria
• matematita • Publishing • Il ritmo delle forme • Publishing • Images for mathematics • Interactive • Rosettes • Wallpaper patterns • Kaleido • Simetria
Symmetry and WIMS – pagina 31
Tools
Tools
http://www.quadernoaquadretti.it/
Symmetry
Symmetry
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
Isometries
Isometries
Symmetry and WIMS – pagina 30
Symmetry and WIMS – pagina 28
A magazine for secondary school students
Symmetry
Tools
Publishing: books
Isometries
Il ritmo delle forme
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
Symmetry
Symmetry and WIMS – pagina 29
Symmetry and WIMS – pagina 26
• training courses for pre-service and in-service teachers; • problem-based mathematical laboratories in school (both in primary school and at a higher level); • interactive exhibitions; • web-based mathematical game contests; • iconographic references on mathematical topics.
http://www.matematita.it/
matematita: products&offers
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
• originates from the experience of promoting mathematics by four Italian universities: Milano, Milano-Bicocca, Pisa and Trento • focus on informal learning as one of the main prerequisites to any subsequent more formal learning • aims to identify the right form of contents and methods for this type of communication
http://www.matematita.it/
Interuniversity Research Center for the Communication and Informal Learning of Mathematics
Isometries
Publishing: books&CDrom
Symmetry and WIMS – pagina 27
doing mathematics with the pencil
Also “mate” = maths “matita” = pencil
The word “matematita” resembles the word “matematica” which means “mathematics”.
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
WIMS
• matematita • Publishing • Il ritmo delle forme • Publishing • Images for mathematics • Interactive • Rosettes • Wallpaper patterns • Kaleido • Simetria
Tools
Symmetry
Isometries
matematita
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
WIMS
• matematita • Publishing • Il ritmo delle forme • Publishing • Images for mathematics • Interactive • Rosettes • Wallpaper patterns • Kaleido • Simetria
Tools
Symmetry
Isometries
matematita
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
Wallpaper patterns
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
Interactive
Symmetry and WIMS – pagina 39
Symmetry and WIMS – pagina 37
Symmetry and WIMS – pagina 35
Transparent mathematics: minimal surfaces and soap bubbles
matetrentino, mathematical explorations of Trento and its surroundings University of Milano-Bicocca – Dip. di Matematica e Applicazioni
matemilano, mathematical explorations of the city
Symmetry and WIMS – pagina 33
Symmetry, playing with mirrors
Interactive exhibitions
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
Symmetry and WIMS – pagina 34
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
Wallpaper patterns
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
Rosettes
Symmetry and WIMS – pagina 40
Symmetry and WIMS – pagina 38
Symmetry and WIMS – pagina 36
1 “interactivity and freedom: not really the kind of exhibition I am familiar with, where you are only allowed to watch, sometimes to listen, but never to touch”
◦ propose problems that can be experienced at different levels, from different viewpoints
• multilevel
◦ “interattivita` e liberta` di sperimentazione della mostra (non certo le mostre cui sono abituata io in cui si deve solo guardare, qualche volta ascoltare, ma mai toccare!)”1
• interactive
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
WIMS
• matematita • Publishing • Il ritmo delle forme • Publishing • Images for mathematics • Interactive • Rosettes • Wallpaper patterns • Kaleido • Simetria
Tools
Symmetry
Isometries
The exhibits
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
The website (∼ 10 000 images, constantly evolving) is designed to be user-friendly while still ensuring a high level of scientific correctness alongside top quality relevant images. Each image has a presentation with a full (mathematical) description, possibly connecting with other images.
• Use images to communicate mathematical ideas; • make matematita’s collection of images and animations available by creating an online website.
http://www.matematita.it/materiale/
Images for mathematics
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
WIMS
WIMS
WIMS
WIMS
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
WIMS
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
Symmetry and WIMS – pagina 45
• matematita • Publishing • Il ritmo delle forme • Publishing • Images for mathematics • Interactive • Rosettes • Wallpaper patterns • Kaleido • Simetria
• matematita • Publishing • Il ritmo delle forme • Publishing • Images for mathematics • Interactive • Rosettes • Wallpaper patterns • Kaleido • Simetria
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
Tools
Tools
Symmetry and WIMS – pagina 44
Symmetry and WIMS – pagina 42
◦ recognize symmetry ◦ build symmetric figures
• symmetry
◦ converse problem
Symmetry and WIMS – pagina 46
• what is an isometry? how do you deal with isometries? • isometries and figures: apply an isometry to an image
University of Milano-Bicocca – Dip. di Matematica e Applicazioni
• Issues
WIMS
Tools
Symmetry
Isometries
Symmetry
Symmetry
Issues
Isometries
Isometries
Symmetry and WIMS – pagina 43
• matematita • Publishing • Il ritmo delle forme • Publishing • Images for mathematics • Interactive • Rosettes • Wallpaper patterns • Kaleido • Simetria
Simetria
Tools
• matematita • Publishing • Il ritmo delle forme • Publishing • Images for mathematics • Interactive • Rosettes • Wallpaper patterns • Kaleido • Simetria
Kaleido
Symmetry
Tools
Symmetry and WIMS – pagina 41
Isometries
Symmetry
Kaleido
Isometries
Kaleido
