I have to say that I am a BIG fan of Geoboards. It is great fun watching kids do stuff on Geoboards to get a tactile feel for what area and perimeter actually mean inside formulas. I've only been using Geoboards for two years, so I still have a lot to learn, but things I think it can immediately address:
* Many kids don't really understand what "unit" means and why units are different from points. For example, hand a Geoboard to any kid who is already familiar with the definition of area and how to find rectangular area, and ask them to build you a rectangle with a certain "easy" area, say 24 squared units. I guarantee you that MORE than half of the class will immediately build you rectangles with the wrong area, because they are counting pegs (or "points") instead of spaces or "units" along the edges of the rectangle.
How do you address / fix this misconception with 7th-graders? That's the fun part. You can address it by walking over to the kids who have their hands raised (they have to get it "checked off" before they draw it down on their paper), and you count the squared units inside with them. You touch each squared unit inside the rectangle -- fortunately there are only a small number of them -- and you count slowly until the kids can see / are convinced beyond a doubt that their originally constructed area is incorrect. And then you just walk away and let them figure out what they did wrong. Within a minute or two, they'll get it on their own by trying new things with the rubber band -- and they'll be very excited about their own discovery that units, not points, are what matter when we measure objects!!
* On a Geoboard, their mistakes are indeed malleable and a "normal" part of the learning process. Kids don't feel discouraged OR frustrated, which they might if they had to erase and re-draw on paper. You are able to completely normalize mistakes by simply making the process of correcting the mistake fun.
* Geoboard allows kids to experiment. Can they build multiple, different-looking rectangles with the same area? Do those rectangles have the same perimeters? Kids can reach out and touch the perimeter units as they count. This is good for their connection between what's visual and real, and what is written on paper. For me, I try to let kids use a continuum of representation -- tactile, to dots on paper that look a lot like what they use on the Geoboard, to then "abstract" images with only labeled side lengths that are not drawn to scale. The continuum is what will help the developmentally more "concrete" thinkers understand abstract formulas.
* Using a Geoboard, kids can build shapes within other shapes. A classic connection is building triangles within a rectangle. Can they build different ones? Can they convince themselves why the different triangles that can "maximally" fill a rectangle will always have the same area? If I then change the triangle to be something smaller, can they fix the rectangle as to help them find the new triangular area? Those are things that they can explore with their hands and talk through with their partners and with me, and eventually they can deduce the area formula of a triangle. The kids can also build overlapping parallelograms and rectangles, for example, to discover the connection between their area formulas. (Naturally, some of this can be done with cutting and pasting paper, too. I did this with my 9th-graders and they all remembered the various 2-D formulas very well on the test AND could easily apply them. But, Geoboard is nice because within minutes the kids can build multiple examples in order to draw a more confident conclusion.)
* And then, can they extend these discovered concepts to abstract diagrams? And when they do, does it fill them with excitement to be able to make that little connection between a newer, more abstract representation and the concrete one with which they are now familiar with? Geoboard is not the end-all, but it certainly paves way to future learning.
I loooove Geoboards for introducing geometry, because I feel that it fits so well into the adolescent need to explore, make mistakes, and transition from the concrete gradually to the abstract! Why give the kids formulas, when they can learn to develop them on their own?
Addendum March 16, 2012: The shearing activity of triangles was also quite lovely! The kids gave me the "Duh! It's so obvious!" response when I asked them to compare areas of triangles before and after being sheared. (I had them cut the triangles into strips, shear them, and then glue the sheared results back down into their notebooks.) Many of the kids are now able to successfully figure out triangular areas without being told a formula, and are also able to re-draw what the sheared triangles had looked like BEFORE they were sheared, in order to better visualize their area calculations. Lovely! The next follow-up geometry activity will again use the Geoboard. Some of my faster students have already begun working on it. It'll guide them through building squared areas such as 25 and 16, in order to learn to recognize which numbers are not perfect squares, and to be able to still estimate the sizes of those decimal square roots. (This is, of course, in preparation for solving Pythagorean Theorem equations a bit further down the road.) Love teaching Geometry to my 7th-graders!!
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