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Space and Perspective is meant to be used as a resource with learners geared towards learning the complexity of our world. The site features multiple cases (postings) that allow us to investigate space, perspective, dimension, and mathematics as it exists in complex, real-world instances. The geometric investigations being developed on this site are guided by Cognitive Flexibility Theory (see Sprio et al, 1988), Realistic Mathematics Education (see Freudenthal), and an emergent, interactionist perspective of learning environments (see Cobb and Yackel, 1996).

Investigations comprise 15 main themes, but are approached through activities helping learners problematize space, capture and represent space, and even representing impossible and invisible spaces. During the actual implementation, this list may grow or change, depending on the class conversations, feedback from students, and as new ideas arise. Learners will be asked to add to the cases on the class website. Below is a summary of some broad categories of investigations.

Photography. An investigation into photography has multiple goals: Investigating varying points of view of same object, investigating shadows and lighting, more recent imaging technologies (x-ray and CAT scans), relations to the telescope and microscope, and comparing how different lenses affect perception and perspective.

Time. The sequence starts with a reflection comparing a past and present experience of the same thing. This investigation asks students to reflect on their POV as a young child and their current perspective, starting a conversation about what changes and what stays the same.

Visual art and creating perspective. Artwork sweeps across cultures, time periods, and offers multiple perspectives to explore. A trip to an art museum or “virtual” museum would be an appropriate activity for this stage. Starting with paintings from different cultures and eras (or movements), students will be asked to locate the point of view or multiple points of view available to the viewer. Artwork that seems especially helpful for this investigation includes cubist paintings, folk and aboriginal art, Renaissance paintings, and modern/post-modern paintings. Subsequently to exploring paintings, a similar investigation can unfold with photography (multiple exposure, etc.) and sculpture (esp. lighting). Exploring the perspective that art portrays (and assumes of the viewer) paves the way for an investigation into creating perspective for others. Anytime something is created by a human (or animal, tree, etc.) a perspective is automatically and perhaps unintentionally created for others’ point of view.

Gestalt/Optical illusions. This investigation is focused more on foreground/background and proximity. The challenge here is to investigate illusions for the multiple perspectives they create and possibly promoting student created illusions.

Motion. Motion changes the “natural” perspective because the environment around changes from the normal standing, walking, and even running position. Yet, we have become accustomed to multiple forms of high-tech travel. What about also traveling underwater and in the air? This neatly directs attention towards investigating moving pictures (video).

Video. The first part is an investigation into the perspective given/taken with video. This part is not as focused on content, but more on film techniques for creating/changing perspective. During the second part, videos are used that have perceptual content, such as Flatland: The Movie (Travis & Johnson, 2007) and Powers of Ten (Eames & Eames, 1977).

Sound. I am not certain about this investigation yet, but a simple exploration into sound waves being represented visually is a way to make the invisible, visible. This may be a good exercise to relate to the telescope, x-ray, etc.

Games. Board games are usually played on a flat surface. This investigation changes the traditional flat game into a torus, which would change the strategy of the game. Students will be asked to find other games and write about the changes from flat to torus. Next, video games are connected to this idea, such as Asteroids, where the character can move off the right side of the screen and come back through the left side. Portal may also work as an example.

Architecture. Architects design spaces that affect our perspective. In order to make their designs come to life, they must transfer from 3D to 2D and back to 3D. Architects usually start by decomposing space into smaller spaces. This investigation explores the perceptual effects of architecture on our relation to space.

Origami/Maps. Origami and maps consist of a transfer between 3D to 2D to 3D. This is similar to the work of architects. In this investigation, students first explore paper folding with origami. Pursuant to this activity, students investigate map making and the various projections implied in the 2D rendering. This draws heavily on coordinate geometry and visualization.

Place. Place has a profound impact on our perspective of the world. Growing up in a desert versus growing up under a canopy of trees can affect what is normal in viewing the sky, the ground, etc. In addition to place, there are cultural practices such as reading direction that can affect what is perceived as normal.

Patterns. There are myriad possibilities when exploring patterns, but it would appear that kaleidoscopes, fractals, and the Fibonacci ratio would relate most to investigations of space.

Imagination. We are able to imagine things that don’t actually exist or exist differently than we are familiar with. This investigation seems like a great way to wrap up multiple perspectives.

Curricular Goals and Description

  • This project seeks to implement a series of perspective building investigations exploring the concept of geometric space in real world contexts. 13 potential investigations have been identified.
  • Cognitive Flexibility Theory (CFT) guides the design of investigations in order to promote a flexible and transferable space concept.
  • Realistic Mathematics Education (RME – out of the Netherlands) also informs design decisions, especially the guiding principle that mathematics instruction should start by exploring the real world instances of a concept. This starting point then leads to a mathematization process (moving towards more abstract notations) in a more natural, student-centered orientation.

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