Long before a child touches the math materials in a Montessori classroom, that child has been exposed to logic and mathematical concepts. Practical life and sensorial exercises are the concrete foundations for higher mathematical learning. Practical life exercises provide a foundation of concentration, coordination, attention to detail, independence, organization, and problem solving techniques. Sensorial materials provide a foundation of math concepts. The materials fix concepts in the mind and senses so that the concepts are understood and recalled by name.
Students in our Preprimary program begin their exploration of math concepts with Montessori math materials and other teacher-created activities. The youngest children use the materials to explore relatives sizes and quantities and to order and compare objects along various dimensions. As they build familiarity with the materials, students begin to create sets, compare and order quantities, discover patterns and solve problems. Working in the math area, students build their early number sense, counting skills and ability to manipulate numbers. Students use the materials for increasingly sophisticated tasks as they move through their first two years in preprimary and may begin to represent some of their concrete work on paper by the time they are five years old.
Kindergarten students address problem situations that build and extend their number sense and mathematical thinking. Dividing snack foods into fair portions or figuring out how many two-person seats we need on the bus are typical projects that invite student exploration. Materials such as the arithmetic rack, with its groups of five colored beads, support students’ ability to compose and decompose numbers (i.e. 1 + 5 = 6, 2 + 4 = 6, 3 + 3 = 6) and to build the concept of equivalence. They learn to use 5s to make 10 and extend this learning to other combinations that make ten. They begin to explore doubles and near-doubles and collect data that allows them to see and explore patterns in numbers.
By the end of Kindergarten, many students can count beyond 100, match quantities with numerals to 100, and read and write numerals from 0 to 100. They can count by 2s and recognize and generate other simple number patterns as well. They add and subtract with concrete materials and are solidifying the concept of equivalence.
Mathematical knowledge comes to children first in a concrete manner through experience with the materials and then the necessary language is supplied. When children use the math materials in the classroom, there is always a concrete component to the work as well as a representational component. Eventually, mathematical work in the classroom moves children toward abstract thinking.