The earliest material we give the children consist of circles cut from 1-whole through 10-tenths. They can use these primary as well as elementary for sensorial purposes (experiencing equivalencies) and design work purposes (tracing and creating figures).
In primary, they can begin simple operations - no changing of the denominator or going over 1-whole. They can experience all 4 basic operations in this manner.
In elementary, they continue (sometimes begin) this work with operations and go much further with it, dividing and multiplying fractions by fractions, converting into decimal fractions, and going far beyond simple operations through the use of a plastic material of multiples of each fraction circles (lots of unit-wholes, halves, thirds, etc. through lots and lots of tenths). There is so much they can do with this portion alone, without even going into different denominators.
Why did the Montessoris only go to 10 with this material? Probably because it is the basis of our number system, and because it directly leads into the use of decimal fractions with the yellow board and all those little colored cubes.The fact is, there is even more fraction material in the traditional albums:
- Metal squares cut into further squares and triangles, with total pieces in some squares totalling more than 10 (here the children sensorially experience division of more than 10) - while not specifically a fraction work (it is used in geometry), the material still provides a fraction experience.
- Fraction skittles for use in dividing by fractions
- Clear plastic overlays to help the children abstract beyond denominations of 10; you provide 2 rectangles with lines cutting into 3 and into 4, along with blank transparencies for them to make their own divisions for their own purposes.
- And as previously mentioned, the decimal fraction material with the yellow board and cubes to represent each decimal place - along with the decimal checkerboard for continuing one's work from the multiplication checkerboard.
Use caution when utilizing other fraction materials. Consider if it is truly useful or a distraction from the foundation they are building now (going ahead too soon; not fast enough; getting off the path of delving deep); be sure it does not cause them to become too dependent upon the materials such that they later cannot get away from them.
Many children develop an intense interest in fractions at age 5 or 6; they work with it a LOT (hence the fractions are found in the primary class), and then they step away for a time; and when they come back, they are ready for more. They were building a foundation that, if not filled with distractions, is truly full and strong.
Trust the materials described in the albums - go deep with them. There is a LOT that can be culled from just the basic materials. Then, if a child has found a true deep interest that even the depths of the material can't provide, then look at your situation, clarify the needs and fulfill them with appropriate further basic material. Anything beyond basic: let the child create - it's his interest, let him own it.