The Mini-Computer
One device used throughout the program was a mini-computer. This was a 2 by 2 grid of squares, the squares represented the numbers 1, 2, 4, and 8. Checkers could be placed on the grid to represent different numbers in a similar fashion to the way the binary numeral system is used to represent numbers in a computer.
The mini-computer is laid out as follows: a white square in the lower right corner with a value of 1, a red square in the lower left with a value of 2, a purple square in the upper right with a value of 4, and a brown square in the upper left with a value of 8. Each mini-computer is designed to represent a single decimal digit, and multiple mini-computers can be used together to represent multiple-digit numbers. Each successive board's values are increased by a power of ten. For example, a second mini-computer's squares will represent 10, 20, 40, and 80; a third, 100, 200, 400, and 800, and so on.
Students are instructed to represent values on the mini-computers by adding checkers to the proper squares. To do this only requires a memorization of representations for the digits zero through nine, although non-standard representations are possible since squares can hold more than one checker. Each checker is worth the value of the square it is in, and the sum of the checkers on the board(s) determine the overall value represented. Most checkers used by students are a solid color- any color is fine. The only exception is checkers marked with a caret (^), which are negative.
An example of representing a number: 9067 requires four boards. The leftmost board has two checkers in the 8 and 1 squares (8000 + 1000). The second board has none, as the value has zero hundreds. The third board has checkers in the 4 and 2 squares (40 + 20), and the rightmost board has checkers in the 4, 2, and 1 squares (4 + 2 + 1). Together, these 7 values (8000 + 1000 + 40 + 20 + 4 + 2 + 1) total up to 9067.
This would be considered a standard way to represent the number as it involves the fewest checkers possible without involving negatives. It would be simpler to replace the last board with a positive checker in the 8 and a negative checker in the 1, but this is not taught as the standard.
Arithmetic can be performed on the mini-computer by combining two numbers' representations into a single board and performing simplification techniques. One such technique is to replace checkers from the 8 and 2 squares of one board with a checker on the 1 square of the adjacent board to the left. Another technique is to replace a pair of checkers in the same square with one checker in the next higher square, such as two 4's with an 8.
Read more about this topic: Comprehensive School Mathematics Program