Early childhood mathematics, in Montessori education, serves as the conceptual bridge between the concrete world of practical life and sensorial thinking, which the child has begun to leave behind, and the abstract world of complex thought the child is trying very hard to enter. In practical life, the child has learned to follow a series of ordered steps in a number of activities from hand washing and table scrubbing to tying a bow and pouring equal servings of liquid into multiple cups. These ordered steps are not unlike the steps required in following an arithmetic operation. In the Sensorial area the child has grown accustomed to dealing with groups of ten, providing a connection to the decimal system while advancing their understanding of spatial characteristics and dimension. This practice, with ordered steps, leads to concrete understanding of the world that mathematics is designed to measure.
The Montessori math materials, like all other Montessori materials, possess a number of characteristics which attract the child and provide the circumstances appropriate to their development. First, they are visually appealing and able to be manipulated, which invites the child to touch them. Second, they possess inherent controls of error, allowing the child to make and learn from their mistakes without the addition of any possibility of embarrassment or shame they might experience in being corrected by a teacher. Lastly, they allow the child to bridge the gap between difficult concepts. From having worked with groups of ten in previous works, the children are able to then count to larger numbers by counting in tens, hundreds, or even thousands. From this, they can begin to count by twos, threes, fours and so forth, laying the groundwork for multiplication.
In this way, with the properly ordered use of tangible, clearly colored, and thoughtfully designed materials, the child is gifted with a deep-seated inherent understanding of the decimal system, fractions, logarithmic scales, the concept of zero, place values, odd and even numbers, and so on. With material after material, the child is given the power not only to hold in their minds these difficult concepts, but also to recall and understand what they mean in the real world in a way that few adults ever become capable of.