I posted a few weeks ago about the decanomial squares activity. I explained that I made one from foam, here are the pictures. In the top picture i have all the true squares stacked in order. Each square is the same measurement as one side of the pink tower. The colours are that of the short bead stair. This activity is an example of how mathematical ideas are absorbed by the children from very early on. Having already been familiar with the pink tower and short bead stair, the child who works will this will already have a sense of the size and order of these squares. From this activity the children will absorb the squareness of numbers from 1 -10, and if you have a very mathematical child they will be able to tell you that the squares are as follow 1x1 2x2 3x3 .....8x8 9x9 which is 1,4,9... 64,81.
Here I have the squares stacked by matching one corner. A montessori child will automatically be able to tell you that that looks just like the Ariel view of pink tower when it is built by matching one corner. Maria Montessori speaks about the direct and indirect aims of the sensorial material used in the early childhood classroom. Many of the indirect aims are absorption of mathematical concepts through concrete materials. I can fully see and understand the later benefit of these sensorial materials because as a child i worked with them and they gave me a different insight into the world of numeracy.
Here is a picture of the decanomial squares when layout in the sequence. Each piece represents a specific equation or area. There are several different ways of showing a child this layout, each presentation will provide the child with a specific understanding of the squareness of numbers. I see it beneficial to show the children all the different ways of laying out this activity.
Believe it or not but this activity as simple and colourful as it may look, can be used to find out the square root of large numbers such as 3136. If you have mathematical children who have worked through the montessori materials well, they should be able to work out the square root of 3136 simply by using this materials [Something even PHD students may not be able to do!].