Mathematics is a special language that has evolved over a considerable span of time as man has made discoveries about measurable relationships that he has experienced in his environment. These relationships have come to be expressed in very abstract terms but initially they were in all probability as concrete and as simplistic as Montessori would come to make them for the young child. Montessori's end goal was to find a way to allow the child to solve problems and make discoveries. She was able to find a way to turn an abstract concept into a concrete form that could be manipulated to achieve a single goal as the child worked with the material. She said, "Articles of mathematical precision do not occur in the little child's ordinary environment. Nature provides him with trees, flowers and animals but not with these. Hence the child's mathematical tendencies may suffer from lack of opportunity with detriment to his later progress. Therefore, we think of our sensorial material as a system of materialized abstractions, or of basic mathematics". (Montessori, The Absorbent Mind, pg. 231)
Mathematics begins in the sensorial area of the classroom as the materials there train the child to observe, compare, classify, make judgements, reason logically and make decisions. This is seen especially as the child moves into variations and extensions. The child has made knowledge his own. He is ready to begin to apply new ideas of his own with already tried and true realities. He might stack the knob-less cylinders upright in a row aside each other. From this he has made a visual discovery. They each have the same number of cylinders. The child may then attempt a horizontal extension using the brown stairs, the pink tower, one knob-less cylinder box and one knobbed cylinder box. Again, he has absorbed through physical movement of the objects the physical attributes of these objects. Indirectly, he has been prepared to move into Mathematics reinforcing Montessori's own words, "We should really find the way to teach the child how before making him execute a task". "Pupils could then come to the real work able to perform it without ever having directly set their hands to it before". (Kramer, Maria Montessori, pg. 90)
Mathematics is presented in such an arrangement that the following three elements are always to be followed. There is always an isolation of the concept being taught. Children learn that concept by arranging materials in such a way as to reveal their function and essential idea. There is a hierarchical pattern of progression in learning. Montessori said, "It is the stimuli for things, not the reasons which attract his attention; it is, then the time to direct sense stimuli methodically, in order that sensations should evolve rationally". (Montessori, The Discovery of the Child, pg. 198)
The red and blue rods are the child's initial introduction to the concept of number. However, a number has three dimensions. It can show quantity. It can be a symbol for quantity and it can be an association between the two because they are related. Remembering the three elements from above and that a number has three dimensions one more factor enters in. Quantity is presented first, symbol second, and the association between quantity and symbol third. Sequin's "Three Period Lesson", also shows the progression which must be achieved within any level of conceptual learning before moving on to a new and higher level of understanding. The first period of the lesson would be the classic presentation by the teacher to the child. The second period would be independent practice by the child. One needs to keep in mind that the materials Montessori designed have a built in element that reveals the essential purpose and idea to be established in the child's mind. That makes the process of doing the work more important than the outcome. As the child works through the second period the essential idea becomes increasingly more established in his mind. The third period of Sequin's lesson reveals internal knowledge on the part of the child that allows him to progress on to the next level of knowledge.
Games of gradation, numerical order and counting begin initially with the red and blue rods as the child sequences them and matches numerals to corresponding rods. In the spindle boxes the child continues to sequence but with an added element, the loose quantity is now added by the child to the stationary numeral. The child must bring his previous knowledge of quantity to use in a new fashion to achieve a new discovery - counting sequentially from one to nine. The colored bead bars follow the spindle boxes with graduation, numerical order and counting all being emphasized and with the indirect purpose of paving the way for either linear counting or movement into the decimal system. Cards and counters provide an association of quantity with symbol. This work also provides a test of sorts to allow the child to show his knowledge of numbers and the correct sequencing of them along with the ability to demonstrate knowledge of counting out a loose quantity and placing it in correct relationship to its symbol.
The child is now ready to use concepts he has already internalized to make a new concept his own. He is ready to move into the decimal system. As with the red and blue rods, quantity is introduced first and the child is shown through the precise arrangement of materials their essential function and idea. In this instance it is place value. A hierarchical pattern of progression initially unknown to the child will gradually emerge. Units are arranged to the far right with the sequence ending on the left. Symbol is introduced next and then the association is made between the two. The progression follows with the nine tray, which shows both the rigidity of the decimal system and the provisions which can be made to increase quantities limitlessly. Games are then played to reinforce association between quantity and symbol and to prepare the child to move into operations such as addition, multiplication, subtraction and division. Linear counting can be introduced after numeration as a parallel activity alongside of the decimal system. Again Montessori in both the layout of the work and the materials she used, was able to find a way to present to the child the essential idea she wanted to be conveyed. In the snake game, the child brings with him previous knowledge of both the ten bars and the colored bead stairs and begins to use them for a new purpose. The precise layout of the colored bead bars reveals multiple ways to create the sum of ten. Counting them and removing them allows for the process of exchange. Again the child through interaction with the materials arrives at a new level of understanding. The continued process of counting and replacing at its end reveals the essential idea to be gained. The child is ready to move into Teen counting with the Teen board. Quantity is introduced first, then the symbol and finally the association between quantity and symbol.
Because the child has done so many prior works that have increasingly internalized within him a sense for order and arrangement in materials he is ready to use the Ten boards and create numbers from twenty-one to ninety-nine. Quantity is always shown first, symbol next and then the association between the two. Montessori's principle of indirect preparation has been preparing the child for movement into multiplication. The hundred bead chain again makes use of previous knowledge and materials familiar to the child to present new information. The hundred board follows with just symbols being used. The short chains follow using the color code established early on in the colored bead bars. They provide for more practice in linear counting, skip counting and prepare the child for multiplication. The long bead chains follow. Again they use previous established and known material to the child with an added dimension. The chains are cubed. The number of beads on the chain corresponds to the cube of the number. This work allows the child continued practice in linear counting, skip counting and preparation for multiplication.
What has made for such tremendous strides in a young child's mathematical mind? Montessori has prepared a math environment that provides for all levels of the child's development. The materials themselves become more abstract as the child moves sequentially through them. The materials, as they develop sequentially, allow the child more freedom to operate on an increasingly more abstract level that was grounded in a concrete reality initially. This is seen for example as the child moves from quantity (the concrete) to symbol (the abstract) and onto the association between the two. This progression is also seen in the stamp game for each of the operations. The child also is able to act with increasing independence. Montessori has provided for a pervasive sense of external order to prevail in all of her materials as they are being used. The external order of the materials contributes towards the child's ability to see and feel the essential purpose behind the work. This is seen for example in the 45 lay out. This has as one of her aims to bring a sense of internal order to the child's mind. She has seized the moment of time in the young child's life when they are in a sensitive period for learning, through both movement and their senses. She has created materials that in their simplistic beauty call out to the child to be used.
BIBLIOGRAPHY
Kramer, Rita, Maria Montessori: 1976, Addison-Wesley Publishing Company
Montessori, Maria, The Absorbent Mind: 1992, Kalakshetra Publications, 10th ed.
Montessori, Maria, The Discovery of the Child: 1993, Kalakshetra Publications
