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Let the child be presented with the attractive problem of arranging the pieces of the Large Number Rods in such a way as to have all as long as the longest. He first arranges the rods in their right order; he then takes the last rod (1) and lays it next to the 9. Similarly, he takes the last rod but one (2) and lays it next to the 8, and so on up to the 5.
This very simple game represents the addition of numbers within the ten: 9 + 1, 8 + 2, 7 + 3, 6 + 4. Then, when he puts the rods back in their places, he must first take away the 4 and put it back under the 5, and then take away in their turn the 3, the 2, the 1. By this action he has put the rods back again in their right gradation, but he has also performed a series of arithmetical subtractions, 10 – 4, 10 – 3, 10 – 2, 10 – 1.
The teaching of the actual figures marks an advance from the rods to the process of counting with separate units. Because the child has completed the learning of absolute numbers and visualising these with the Large Number Rods, he will now have the foundation to learn separate units in the right context. When the figures are known, they will serve the very purpose in the abstract which the rods serve in the concrete; that is, they will stand for the uniting into one whole of a certain number of separate units.
The synthetic function of language and the wide field of work which it opens out for the intelligence is demonstrated, we might say, by the function of the figure, which now can be substituted for the concrete rods.
The use of the actual rods only would limit arithmetic to the small operations within the ten or numbers a little higher, and, in the construction of the mind, these operations would advance very little farther than the limits of the first simple and elementary education of the senses.
In the sandpaper numbers material there is a box containing numbers cut out in sandpaper. The child is guided to touch the figures in the direction in which they are written, and to name them at the same time.
He is then shown how to place each figure upon the corresponding Large Number Rod. When all the figures have been learned in this way, one of the first exercises will be to place the number cards upon the rods arranged in gradation. So arranged, they form a succession of steps on which it is a pleasure to place the cards, and the children remain for a long time repeating this intelligent game.
After this exercise comes what we may call the “emancipation” of the child. He carried his own figures with him, and now using them he will know how to group units together.
For this purpose we have in the spindle box material, which is a series of wooden pegs, but in addition to these we give the child activity “extensions” in the form of all sorts of small objects––sticks, tiny cubes, counters, etc.
The exercise will consist in placing opposite a figure the number of objects that it indicates. The child for this purpose can use the Montessori Spindle Box which which is divided into compartments, above each of which is printed a figure and the child places in the compartment the corresponding number of spindles.
An example of the activity extension (as mentioned above) is to lay all the figures on the table and place below them the corresponding number of cubes, counters, etc.
This is only the first step, and it would be impossible here to speak of the succeeding lessons in zero, in tens and in other arithmetical processes––for the development of which other lessons activities should be used. The Spindle Box material itself, however, does give some initial idea. In the box containing the pegs there is one compartment over which the 0 is printed. Inside this compartment “nothing must be put,” and then we begin with one.
Zero is nothing, but it is placed next to one to enable us to count when we pass beyond 9––thus, 10. There are many bridging exercises using the golden beads and the number cards that will assist the child to understand the transition from units to tens to hundreds to thousands, and it is not uncommon for a Montessori child to be able to comprehend and work in thousands before the age of six.
The children show much enthusiasm when learning these exercises, which demand from them two sets of activities, and give them clarity of purpose in their work.
As a foundation for writing and arithmetic the child has learnt (in a cognitive fashion), “concrete” experiences through the sensorial and practical life lesson activities. All these early sensorial and practical life activities which have brought order into the child’s mind, would be wasted were they not firmly established by means of written (abstract) language and of (abstract) figures, and this is described as the process of moving from “concrete” to “abstract”. Once this process is established via the Montessori Method, the child is open to an unlimited field for future education. What we have done, therefore, is to introduce the child to “abstract” concepts through the a process of “concrete” cognition. By contrast, in traditional schools we try to teach abstract concepts not yet grounded by relevant concrete experiences, and these “concrete” experiences and the progression of then into the “abstract” are an absolutely vital foundation to enable a child to progress to abstract learning in the first instance. Similar to the process of building a house, by applying the Montessori Method in Maths, we establish a firm foundation first, to prevent the house from being structurally weak over time, even if the house appears all in order from the outside.<
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