Wednesday, June 24, 2015

Offering Young Children Foundations of Our Number System: The Decimal System and the Absorbent Mind

    We are approaching the four-year-old child to share the heritage of our number system. We cannot begin to teach the components of place value, or the decimal system, to a young child until we are conscious of the components we need to relate. How does it really work? In the Montessori Method, we offer numbers bigger than ten through the Golden Bead Materials, or what w
    e call the layout of the Formation of Numbers. In traditional schools, this is called "base ten" material-- however, this differs substantially in the intangible reverence given for the transference of these objects of human culture and in the sensorial connection to other objects in the environment-- size and dimension references to objects they have already seen. We share these materials with great wonder and awe, inviting the child to participate in one of the greatest discoveries, or constructions, of global civilization. 

    We begin early math only the numbers 1-10. Why? It is the basis for all other numeration. With that basis, it is now time to orient the child toward the Decimal System. As adults, we are not very aware of the ways that the base-ten place value decimal system functions; we do not remember how we learned it as children. It must be brought to our own conscious mind, as a Guide, if it can be part of the child's conscious mind. We cannot proceed to conversations about 11-20 before discussing the decimal system. 

    Our decimal system has a few features to be made conscious:
    • We may only use the digits 1-9.
    • It is based on place value, meaning each space indicates a size of the quantity.
    • It is structured around families of hundreds, tens, and ones-- within the groups of thousands, millions, billions, and so forth, in relation of three-digit families.
      • The three categories repeat in families
      • The spatial pattern for categories, repeating onward, is point, line, square… to cube-the new point
    • Zero is the place holder.

    When we share these concepts to the young child, perhaps even the four year old child, we must make these concepts materialized and available to the senses. None of this will be accessible to the young mind until the child can see the materialized abstractions-- unless we wait until the child is older and able to discuss this for the first time in the plane of abstraction. But this is needless; the child is quite capable of manipulating these ideas and symbols, of understanding this rigid framework of the decimal system, with materials that have made this accessible to the eyes and hands of the child. It is not advanced, truly, to make values up to tens and hundreds of thousands discernible to a child in pre-K, if we begin from the perspective of the senses. It is a minimum, an obligation we have, to induct the child into our culture during these early sensitive periods. 

         In sequence for presenting the decimal system, we follow this pattern: (1) introduce the names, (2) give the layout, (3) form quantities. I can introduce these materials as soon as the child knows 1-9 in quantity, name, and symbol. We give the names in a three-period lesson, and then the layout of the materials, then practice the formation of quantity. It is important to note that the learning does not occur during the lesson but during the practice with the material, where the qualities and rules of the decimal system are absorbed. The goal is to show the child to the quantities and names of the decimal system-- in the introduction. The introduction also shows the child the relationship between one category and the next. In all areas, the child feels the sense of the quantities relative to each other, giving the child the framework for handling large quantities in the framework of the decimal system. The indirect aim is to become conscious of the static laws underlying and governing the organization of quantities in the system-- not more than nine as a digit, only three categories per family, each category is a relationship of ten on successive levels, the levels repeat, and the relationship of one level or family to the next is of one thousand. The child experiences all of these laws in the layout of the decimal system. 

    What aim is fulfilled by the formation of quantities? It consolidates the association of names to the categories. As an aside, we cannot call the categories place value because we are not using numerals in places yet; for now, we are only dealing with categories as quantities with a materialized impression leading to the place value.

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