Central Perspective from the 3D reality to the 2D plane Documents Authors Anna StenkvistAnna Stenkvist, Tensta Gymnasium, Stockholm, Sweden To describe and analyze the central perspective as a model for perception To be able to draw a picture according to the central perspective and document the process To understand how the mathematical idea history and the cultural history lead to the visual concept of the central perspective Objectives Windows Movie Maker, Windows Media player, MS PowerPoint, MS Word Software Description The project is about the central perspective, how the three dimensional reality is projected at the two dimensional picture plane. We have created a room were we have constructed a coordinate system in the middle of the room with strings stretched over a wooden frame. The project is part of a collaboration among all the mathematic teachers at Tensta Gymnasium. Learning Areas Mathematics, Cultural History, Fine Art, Physics Levels 15 – 18 years old (gymnasium)central perspective, vanishing point, horizon line, picture plane, projection, orthogonal, parallel, distance, infinity, optic, understanding of space Keywords Project Overview Woodcut from the first book about the central perspective. It has been used as a school- book in geometry. Albrecht Dürer (1471 – 1528)

Central Perspective from the 3D reality to the 2D plane Teacher Planning and Management 1.A room is equipped with a stilleben and a frame with strings streched over it to represent a coordinate system and a picture plane. (Fine Art: 1-3 lessons) 2.The historical background is presented: the connection between the understanding of space through history and the mathematical idea history. The students learn that, according to the central perspective, all lines orthogonal to the picture plane will join at the vannishing point. (Mathematics, Cultural history: 1-3 lessons) 3.A discussion about the development of the central perspective. (Mathematics, Cultural History, Fine Art: 1-2 lessons) 4.The students draw according to the central perspective and learn about related geometrical concepts such as orthogonal, parallel etc. They also learn how to use the camera and the related software to make the film. (Mathematics, Fine Art: 1-2 lessons) 5.The students work two by two. One draw according to the central perspective and the other one document the process. Then they change. Finally they cut the film together. (Mathematics, Fine Art: 4 lessons) 6.The video is analyzed. Are there deficiencies compared to what we see? (Mathematics, Cutural History, Fine Art, Physics: minutes per group)

Central Perspective from the 3D reality to the 2D plane For preparation I would like to recommend the book Perspective as Symbolic Form by Erwin Panofsky, a german art and idea historian. His theories have inspired to this project. He describes how there is an resiprocal action between the history of art and the mathematical idea history in creating the human conception of the universe. The lexicon Sigma, part one, present an excellent overview over the mathematical history. For a short definition of the central perspective klick here.here If you are looking for renaissance paintings you can try the on-line museum Art Renewal Center or Art Cyklopedia.Art Renewal Center Art Cyklopedia Teaching Resources The painting of Jan Van Eyck (1390 – 1441) illustrates the central perspective. In the mirror behind the couple you can see the artist. All the lines orthogonal to the picture plane meet in artist’s eye.

Central Perspective from the 3D reality to the 2D plane Assessment and Standards Assessment Rubrics: Approved To be able to explain the concepts: vanishing point, parallel etc. To be able to use the central perspective and explain how it works as a model for perception in terms of the related concepts above Excellent Be able to explain and analyze the shortcomings of the central perspective as a model for perception Mapping the Standards: In Sweden we have National Standards for all the courses at the Gymnasium. In addition to that Tensta Gymnasium have there own local standards, that are not of course contradicting the national standards, but adds a similar curriculum as they follow at Ross School. All our teaching is based on teams consisting of teachers with the same subjects or of teachers in different subjects. The cultural history is used as a core in all our teaching. We have formulated the enduring understanding and essential questions that they should be able to answer for each subject and project.National Standards In mathematics we have a working plan for integration according to the embedded VCT at this page. Documents

Central Perspective from the 3D reality to the 2D plane Teacher and School Information My name is Anna Stenkvist and I work at Tensta Gymnasium as a teacher in mathematics, philosophy and computer science. Apart from university studies in my subjects, I also have an education in fine art. Therefor it is natural for me to involve art and visualization in my teaching. I can tell that the students like it a lot. Moreover, it offers an other learning style for the students. This way of teaching suits me too. I like to learn new things for myself and to use a lot of different aproaches to involve the students.Tensta Gymnasium Since one year I am dean in mathematics at our school. It means that I am responsible for the development of mathematics in the pedagogical project, the Ross Project, a collaboration between Tensta Gymnasium and Ross School at Long Island. At Tensta Gymnasium the mathematical teachers have devided the year in four periods of integratrion that correspond to the mathematical main topics: geometry, algebra/arethmetics, functions and statistics/probalbility. It enables us to work together and share resources such as different kinds of lecturers for the teachers and project rooms for the students.Ross School I have participated in the Swedish Mathematics Biennal twice, 2006 och 2008, one time with a lecture and the other with an exhibition. Both were about integration beteween mathematics and other school subjects. In 2006 I worked in a project were we developped a web-based learningtool in mathematics for imigrant students. Now I am a participant of a project about Internet and the future in Sweden, at the Royal Swedish Academy of Engineering Sciences. My contribution to the project is user knowledge from the teaching perspective the Royal Swedish Academy of Engineering Sciences Anna Stenkvist