Although math itself is an abstract process, Montessori believed it can be more easily understood if the child is able to experience that process through the manipulation of concrete materials, or materialized abstractions. The mathematical mind, as defined by Montessori, is an ordered and abstract mental process using the language of signs and symbols. She believed that coming to this frame of mind is a developmental process that occurs at individual rates, and that the children absorb the signs and symbols through accurate observation.
Although math shares the quality of moving from concrete to abstract with other areas of the curriculum, the sequence differs a bit from those in the practical life and sensorial areas. There are a number of different groupings in which lessons can be given simultaneously; however, in math there is one section, the 0-10 activities, in which the child must be completely proficient before the teacher can introduce any other math work. When the child has gained mastery of the 0-10 activities, the teacher has a choice of moving the child into a combination of one, two, or three of the following sections of the math curriculum: the decimal system, linear counting and addition.
Within each of these areas of study, there is a specific sequence that moves from simple to complex. The teacher will decide which of these areas to pursue after careful observation and recordkeeping of the child, and as a result of her interpretation of the child’s interests, needs, and abilities.
Research on child development and learning that has been done in the time period since Montessori created her method simply reinforces her theory that the manipulation of concrete materials makes math all the more meaningful to students. In math, as in other areas of the classroom, the teachers follow the child and implement the curriculum as Montessori envisioned it. As a result, they are able to enjoy the knowledge of how meaningful this type of learning is for the children.