Math Manipulatives: From Physical to Virtual

I have been thinking and planning for my upcoming NCTM presentation on taking the hands-on, physical math manipulatives we use to help students learn and model mathematics to the virtual realm in order to provide  unlimited capabilities to explore and understand. It's something that I think is often thought of as an either or - either teachers use hands-on, physical models or they use virtual models, such as The Geometer's Sketchpad, and my thoughts are we should be using both. It's a way to help students see and understand the power of technology by exposing them to the limitations of the physical and then providing them the opportunity to see the capabilities to go beyond with technology.

Some simple examples of what I mean:

1) Geoboards - students begin to explore polygons with geoboards, but are limited by the number of pegs on the geoboard, the number of rubber bands and the physical limitations of both (you can only stretch a rubber band so far before it snaps and hits you in the eye!).  On a virtual geoboard, which is very easy to create in Sketchpad using a grid and snap points, you can use the polygon tool to create infinite polygons of many sides and shapes. You can measure sides, angles, area, perimeter. Students can explore and question and test to their hearts content and no students get injured in the process! There are no physical limitations to hinder discovery and understanding.

2) Algebra Tiles - a great way to introduce students to multiplying and factoring polynomials is with algebra tiles. But again, eventually you are limited by the physical constraints of the tiles - you can only work with fairly simple polynomials due to the number of tiles you have available. Using Sketchpad, you can create infinite examples and explore many possibilities and test conjectures - there are no limits and students still get the hands-on manipulation, but can go even further by then graphing and looking at the equations, so that they see all the connections.  It becomes a multiple representational cornucopia.

As people consider technology and appropriate technology for classroom, look for technology that doesn't just replicate what is done in the class, but rather allows for exploration and discovery beyond what the physical models allow and leads to deeper understanding and a greater appreciation for the beauty and power of mathematics.

To end, just a quick how-to construct perpendiculars using The Geometer's Sketchpad.  Unlimited possibilities.