Our society has a tendency to judge everything by the bottom line: Is the answer right? And how quickly did you get to the answer? Montessori is going against the grain in its approach to mathematics (and education in general). If you choose Montessori for your child, then you need to be willing to go against the grain yourself and support your child’s Montessori way of learning. –Michael Duffy (Math Works, 2008, p.67)
Why do you see Montessori materials for math in the Upper Elementary classroom? Maria Montessori discussed the importance of developing a "mathematical mind". In essence, this is the idea that children understand the "Why?"; the ability to think, use logic and reason. The materials are the best way for students to engage in the process of math concepts while they build a foundation for abstraction.
In UE, I will sometimes give the children the answer to a math problem. I do this in an effort to take their focus away from getting the correct answer and shift that focus to the process. If we are working in the abstract and an answer is incorrect, I would rather the student find out what went wrong, rather than a simple redo for the answer. This allows for critical thinking.
Montessori materials begin with a concrete representation of math. Over time and self-discovery, children move to the abstract. Montessorian and author Michael Duffy presents the case for Montessori math in his book for parents called "Math Works". He delves into the brain science of it all.
Regarding materials, he says...
They simultaneously engage multiple areas of your child’s brain, including all four lobes of her cerebrum. Her occipital lobe sees the colors and shapes of the materials. Her temporal lobe takes in the auditory cues that accompany their use and triggers memory associations. Her parietal lobe is engaged in the purposeful movement of the materials. And her frontal lobe provides the problem- solving processing to put all the sensory input together and find a solution to the problem. The simultaneous activation of all four lobes creates a multiplicity of neural connections and builds a network in your child’s brain. -- Michael Duffy, (2008, p. 55)
The Montessori math materials will produce a much deeper kind of learning that will stretch the brain to its fullest potential -–Michael Duffy (2008, p. 57)
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