Cards and Counters: This exercise matches numeral cards with the corresponding quantities, confirming the child’s knowledge of the numbers 1 through 10 in their correct sequence. It also gives a visual representation of odd and even numbers and indirectly prepares the child to learn divisibility, multiples and submultiples. Decimal system These exercises familiarize the child with the different categories in the decimal system, the difference between them and their numerical representations. It introduces the child to complex numbers and the role of zero as a place holder. Golden Beads – Introduction to the Decimal System: A set of materials representing the quantities of unit, ten, hundred, thousand, with the actual difference in volume between the categories. Association of Beads and Cards: Gives the child the symbol for the quantity represented by the beads, introduces the color coding (green, blue, red) of the categories, reinforces the fact that there are no numerals beyond 9, and introduces zero as a symbol which gives value. Formation of Complex Numbers: This exercise presents the combining of categories to form complex numbers and gather the corresponding quantities. The child learns to read large numbers and prepares for understanding the hierarchy of numbers. Changing Exercises: The child gathers large quantities of golden bead materials and counts categories to exchange ten of the lower category for one of the higher category. Arithmetic Operations of the Decimal System (collective exercises): This subsection of the Decimal System gives an impression of addition, subtraction, multiplication and division to the child, as well as how they relate to each other. The exploration of categories continues through exchanging, and the child becomes more comfortable with large numbers. • • Addition: The children combine quantities to gain impressions about the nature of addition, static and dynamic. Multiplication: The children combine identical quantities to gain impressions about the nature of multiplication, static and dynamic. They also observe the relationship between addition and multiplication as a series of additions. Subtraction: The children take smaller quantities from larger quantities to gain an impression of subtraction as the reverse process of addition. The children experience exchanging among categories within the context of subtraction. Short Division: The children gain impressions of division as a process of sharing out, and experience exchanging among categories within the context of division. Long Division with Bows: This exercise helps highlight that the answer in division is what one person gets after the sharing out process. • • • The Stamp Game: The children work with an individual exercise that represents the decimal quantities more abstractly on stamps. This allows the child to practice all four operations and exchanging without the collaboration of classmates. The Dot Game: The child gains further understanding of addition in the decimal system, practices working with large numbers, and realizes that there are never greater quantities in any category than nine. This work focuses the child’s attention on exchanging and is the first abstract exercise in the decimal system work. Story Problems: The child either reads or works with a friend who reads simple word problems, then works out the mathematical equations. Teens and Tens: This subsection of the Decimal System gives the language and corresponding symbols and quantities for the teens and tens (twenty, thirty, etc.), and their respective relationships. 11-19 Beads Only: The child works with the golden ten bars in combination with the colored bead bars to form quantities for eleven through nineteen. This work runs parallel to the work with complex numbers. Teen Board: The child connects the teen quantities with the symbols. Ten Boards and Beads: The child works with beads and symbols to associate the names of the numbers from twenty to ninety-nine with the corresponding quantities. Linear Counting (100 and 1000 chains): The child works with long chains of connected ten bars to count from one number to the next in a linear fashion. With this exercise, the child becomes familiar with the sequence of numbers, counting each individual bead. This material provides a sensorial impression of the difference between two numbers when laid side by side, and provides an introduction to squaring and cubing. Skip Counting (1 – 9 chains): The child works with chains of connected bead bars to experience skip counting of multiples and to prepare for squaring and cubing. This work indirectly prepares the child to memorize multiplication tables. 38