If you think of the window of development between birth and age six as marked by the construction of labels for files in a cognitive file cabinet, the period between six and twelve is when children fill up those files. Children now are interested in learning to master the procedures of mathematical operations, to strengthen their understanding of the concepts on which those operations are based and to gain fluency and speed in recalling facts for use. Classrooms offer opportunities for children to engage in large group and socially-driven math experiences, taking advantage of children's natural propensity to be with their friends at this age. In Lower Elementary, children may explore a wide range of math lessons, building on their understanding of basic operations from their earlier environment and expanding that understanding to include larger numbers or more complex relationships. In Upper Elementary, learners move away from the concrete manipulatives through which they first learned mathematical concepts to abstract applications.
Throughout this Plane of Development, Montessori classrooms offer children experiences with math concepts that combine their fluency, number sense, critical thinking and problem solving skills with a joyful approach to the content, pushing back on the idea that some people "just aren't good at math." Quite the opposite - we believe that the mathematical mind is a universal tendency of human development, and we seek to preserve the child's enthusiastic approach to it by matching our expectations to their development.
Look for foundational lessons in basic numeracy, including expansions on the Bead Cabinet from early childhood, skip counting, place value into the millions or higher, and fraction materials. Notice more advanced applications of those operations, including lessons on the distributive, associative and commutative properties, multiples, factors, multi digit multipliers and divisors. Notice the geometry materials, through which children learn to define and describe geometric shapes and concepts, to compare between those shapes and to develop reliable indicators of congruence and equivalence.
Simultaneously, you'll see children practicing the speed with which they can recall math facts. But remember: rote memorization is not the goal. Rather, we want children to be fluent in math facts because that fluency allows them to explore math relationships with more ease. So, while you'll see children practicing their times tables and completing finger charts, they'll do so because it makes it much more engaging to work with the Checkerboard multiplication material or the Pegboard for factoring when they have already memorized those facts.
And, of course, you'll see all of these experiences demonstrated in applied, practical ways, driven by the real needs of the classroom. Children who are interested in building more hiking paths on their campus will need to understand geometry and measurement in order to design those paths. Children who want to create new spaces for learning in their classroom will need to demonstrate spatial reasoning and physics. Children who want to finance a long-sought class trip need basic numeracy and budgeting skills. The elementary classrooms move between the real-life, socially-conscious experiences that motivate children at this age and the skills they need to make them happen.