A glance at typical geometry text might lead to the conclusion that geometry is far removed from daily life. Nothing could be further from the truth. The world that surrounds us is composed of shapes, lines, curves, and angles. There is geometry in nature - In the angles and planes of a mineral crystal, and in the symmetry of a butterfly's wings.
An inborn, intuitive understanding of fundamental geometric concepts exist in all human beings. Geometry also is to be found everywhere in the world created by humanity. Bridges, iron girders and the design of a chair all have their roots in geometry. This is something that children begin to absorb early in their lives. Impressions, which may not yet have names to them, are gathered by the child in mind and stored for later use. If correct names are used the child is able to add this vocabulary to the growing store of impressions that is being used.
Whatever the case, the child encounters geometry everywhere. It is a part of the world in which children live, and this must be reflected in the environment that is prepared for children.
In the Montessori classroom, geometry is part of a three-pronged approach to mathematics. When combined with arithmetic, algebra, and geometry it offers a stable and mutual reinforcing approach to mathematics. The binomial square is a good example of this. It explores an algebraic expression (a + b)^2 = a^2 + 2ab + b^2, and this expression appears in a geometric form (squares and rectangles) Arithmetic is incorporated as numerical and hierarchical values are assigned to pronumerals. The algebraic expression is easy to internalize through the geometric pattern that it produces.
The elementary child has an interest in analysis and the finding of relationships. Constructive triangles, for example, were used in the Children's House for work in construction. Now in the elementary, concepts such as equivalence and the Pythagorean theorem are explored using this material. The Geometry materials and presentations of the Montessori elementary classroom invite intellectual activity, which is attractive to elementary children.
The absorbent mind of the first plane disappears as the child enters the second plane. As the child encounters new terms, another avenue for internalization is required. The need for second plane children to know "How?" and "Why?" provides a new vehicle. Etymology additionally provides a fascinating new way to approach the learning of such nomenclature, for example, why is this shape called a pentagon? (Its name comes from the Greek word for five: pente.) The child's fascination with reasons makes the memorization of terms less onerous than it would otherwise have been!
The idea of Cosmic Education is thus realized as a study of geometry results in a survey of the history of our language. History is suddenly related to geometry, and the lives and work of various innovators in the field of geometry - Thales, Pythagoras, and Euclid for example - offer new fields for exploration for the child. It is in this way that the child is brought into contact with humanity, as Dr. Montessori desired.
The materials allow the children to explore a variety of the basic ideas of geometry. The creative aspect of their developing minds is also nurtured as unique answers to the challenges provided by the materials are found. Many of the geometry activities are open-ended. There is no one correct way, nor one correct route to an answer. Many equivalent shapes may be created for one base figure, for example, with the geometry insets or constructive triangle. Constructions using only a straight edge and compass provide many challenges that draw on the creative talents of each child.