A. Nogelo, K. Liu, L. Lee, S. Bell, K. Davidson, K. Adams
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As the subject matter of math classes expands to include material on higher dimensions, it is important to develop new ways of seeing and thinking about mathematical concepts. Geometry lends itself well to visual aids. There is a LOT of thinking going on about how to teach math and geometry to children of all ages. Here are some interesting sites that cover this area. Most of these links directly address the issue of teaching geometry in schools. Others are pages of "related" subject matter. Related subject matter includes everything from origami pages to children's art pages, to maritime flag codes and Japanese fish prints. Learning abstract concepts is a pleasure when one can take the new concept and relate it to something in the"real" world.
According to artist and teacher Norman Shapiro, geometry is a tool for performing practical and artistic tasks. He is one of many educators who have made an extensive curriculum available on the web for fellow teachers of art or mathematics. Most of his exercises are simple - all you need is markers, a ruler and some paper.
The pedagogical philosophy behind his classroom plans is as follows:
1.Concrete experiences sequenced in an orderly way develop children's powers of perception and visualization.
2.Words and vocabulary grow out of the hands-on activities. They are given on a need-to-know basis.
3.Art activities provide a priceless, non-threatening way to motivate and activate children to learn. The end product is the one the children can most identify with. Student art gives the teacher an easy-to-read tool to diagnose achievement and levels of learning. Most importantly, art encourages children to learn by finding it out for themselves. When it comes to art, most children will want to "get it right," and will think it is within their power to do so. [excerpted]
These sites are related - they provide opportunity to explore geometric patterns.
N. Shapiro's site - what children can learn about geometry through art.
Teaching math concepts using patterns. Shapes can be put together to create patterns, radiating, tiling patterns. Geometry can be found in objects from maritime flags to hawaiian fish prints and hopi baskets.
One point Mr. Shapiro makes is that exercises for understanding geometry should be geared toward different levels of age and understanding. Art is one method of visualizing and exploring geometry. Geometry is also found in historical architecture, and in may old and new religious traditions. Geometry and religion may be a somewhat more abstract connection than geometry and art, but the connection is just as gratifying to study. Interpreting patterns in geometry was often in religion considered analogous to explaining the complicated patterns of life.
Younger children may find connections between geometry and art gratifying, and older people may enjoy investigating geometry in religion, science and architecture. These sites provide information on geometry used in religion, science, and ancient architecture.
The Seven Wonders of the World.
The history of sciences related to astronomy.
The Many Dimensions - looking at patterns and numbers.
Information on sacred geometry.
As if this wasn't enough information, there is a wealth of other geometry-related sites out there. Here are some collections of the best. The funnest part of looking at these other sites is the fact that the collection of links available reinforces the ideas stated above: that geometry is fun when it is seen in the everyday, and in art, science and history. There are pointers here to everything from the Geometry Center in Minnesota to the directory of Origami home pages.
A basic introduction to geometry of matrices and vectors.
Searchable internet geometry-related pages.
The Mathematics Archive, K-12 internet resource pages. YAY!!
A journal written by highschoolers in beginners, advanced, challenge format.
Mathematics education home page at Vanderbilt University.
The mega-math workbook - students enjoy speculating about what happened to the dinosaurs, pondering the shape of the universe or imagining the Big Bang. We think that they can get as much intriguing puzzlement from wondering about the size of infinity and finding out that there are some very simple-sounding problems that would take huge computers longer than the estimated age of the universe to solve [excerpted]
Download a program that compares Euclidean and non-Euclidean geometries.