A question that I get asked a lot :
'When will the children learn to do sums mentally without using the montessori beads or materials?'
The montessori mathematics materials are brilliant, they introduce the mathematical concepts in a very concrete way. As a child progresses through the materials, they become more abstract but in tiny weeny steps and many things such as colour coding help the child to feel confident when working with large numbers or hard sums. The child gradually progresses from doing addition with the large number rods [holding, counting, seeing quantity] to written addition mentally [sitting at a table with a pencil and paper].
Montessori mathematics materials allow for an enormous amount of repetition. For addition of one digit numbers we have over 12 activities for the children to do! With each activity they are working hands on and using they're senses. By time [and there isn't a specific time or activity when this will happen] the child will begin to mentally recall some number bonds or know how to work things out in their heads. The ability to do addition mentally can happen anytime from 3 years up and Montessori allows each child to take their own individual time to make this leap.
Below are a few pics of Little-N's work. He's been working with the golden bead material a lot and is now able to do addition with 4 digit numbers [with out changing] mentally.
He first showed me he could do this when he told a 5 yr old, 'Did you know that 200 plus 300 is 500?'. The 5 yr old was shocked and came to ask me if Little-N was right, we checked the answer with the golden beads and he was correct. The 5 yr old then set Little-N up with other sums until finally he asked Little-N to work out the 4 digit addition sums mentally while the boy worked out the answer with the golden beads.
I find with most children its a matter of practice and more practice until a switch is flicked within them and they're able to do it all mentally.