Principles of Montessori materials or equipments
(excerpts from "Montessori – A Modern Approach" by Paula Polk Lillard, pg. 59-64)

The Montessori equipment is widely publicized and its role often misunderstood. Because of their visibility, the Montessori materials tend to be overemphasized in relation to the other elements in the Montessori method. In addition, their purpose is often confused. They are not learning equipment in the conventional sense, because their aim is not the external one of teaching children skills or imparting knowledge through "correct usage." Rather, the aim is an internal one of assisting the child’s self-construction and psychic development. They aid this growth by providing the child with stimuli that capture his attention and initiate a process of concentration.

If the teacher has materials to offer that polarize the child’s attention, he will find it possible to give the child the freedom he needs for this development.

In order to serve their purpose of internal formation, the materials must correspond to the child’s inner needs. This means that any individual material must be presented to the child at the right moment in his development. The sensitive moment for introduction to any individual child must be determined by observation and experimentation. The teacher watches for the quality of concentration in the child and for a spontaneous repetition of his actions with a material.

Because educational materials of the past has been designed for a passive child waiting to receive instructions, Montessori considered her materials a "scientific departure" from the past. Her materials instead are based on

The conception of an active personality – reflex and associative- developing itself by a series of reactions induced by systematic stimuli which have been determined by experiment. This new pedagogy accordingly belongs to the series of modern sciences…. The ‘method’ which informs it – namely, experiment, observation, evidence or proof, the recognition of new phenomena, their reproduction and utilization- undoubtedly place it among the experimental sciences.

This new approach to education, suggested to her by the work of Itard and Seguin, was regarded by Montessori as her "initial contribution to education" and "the key" to the continuation of her work.

In addition to meaningfulness to the child, there are at least five principles involved in the determining of Montessori materials : First, the difficulty or the error that the child is to discover and understand must be isolated in a single piece of materials. This isolation simplifies the child’s task for him and enables him to perceive the problem more readily. A tower of blocks will present to the child only a variation in size from block to block – not a variation in size, color, designs, and noises, such as are often found in block towers in toy stores.

Second, the materials progress from simple to more complex design and usage. A first set of numerical rods to teach seriation vary in length only. After discovering length sensorially through these rods, a second set, colored red and blue, in one meter dimension, can be used to associate numbers and length and to understand simple problems of addition and subtraction.

Third, the materials are designed to prepare the child indirectly for future learning. The development of writing is a good example of this indirect preparation. From the beginning, knobs on materials, by which the child lifts and manipulates them, have acted to coordinate his finger and thumb motor action. Through the making of designs that involves using metal insets to guide his movements, the child has developed the ability to use a pencil. By tracing sandpaper letters with his finger, he has developed a muscle memory of the patterns of forming letters. When the day arrives that the child is motivated to write, he can do so with a minimum of frustration and anxiety.

Fourth, the materials begin as concrete expressions of an idea and gradually become more and more abstract representations. A solid wooden triangle is sensorially explored. Separate pieces of wood representing its base and sides are then presented, and the triangle’s dimensions discovered. Later, flat wooden triangles are fitted into wooden puzzle trays, then on solidly colored paper triangles, then on triangles outlined with a heavy colored line, and finally on the abstraction of thinly outlined triangles. At a certain stage in this progression, the child will have grasped the abstract essence of the concrete materials, and will no longer be dependent upon or show the same interest in them.

When the instruments (materials) have been constructed with great precision, they provoke a spontaneous exercise so coordinated and so harmonious with the facts of internal development, that at a certain point a new psychical picture, a species of higher plane in the complex development, is revealed. The child turns away spontaneously from the material, not with any signs of fatigue, but rather as if impelled by fresh energies, and his mind is capable of abstractions.

Lastly, Montessori materials are designed for auto-education, and the control of error lies in the materials themselves rather than in the teacher. The control of error guides the child in his use of the materials and permits him to recognize his own mistakes.

"Control of error" is any kind of indicator which tells us whether we are going toward our goal, or away from it…. We must provide this as well as instruction and materials on which to work. The power to make progress comes in large measure from having freedom and an assured path along which to go; but to this must also be added some way of knowing if, and when, we have left the path.

This dialogue with the materials puts the child in control of the learning process. In time, he will be able to see it and will correct his own errors.

A block of wood, in which the child places cylinders of varying sizes in corresponding holes, is an example of control of error designed within the materials. If the cylinders are not matched in the correct holes, there will be one cylinder left over. Again, it is not the problem alone that interests the child and aids his progress.

What interests the child is the sensation, not only of placing the objects, but of acquiring a new power of perception, enabling him to recognize the difference of dimension in the cylinders.

It is not necessary to design the control of error into all the materials in such a mechanical way as the cylinder block. As the materials progress in complication, the control of error is shifted to the child himself, who has gradually developed his ability to recognize differences of dimension by sight. Control of error is also introduced at a later stage by providing the child with models with which to compare his work. He can find the answers to a certain set of mathematical problems, for example, on a chart board designed for that purpose and freely available to him.