Schematic Learning of the Addition and Multiplication Tables---sticks as Concrete Manipulatives

Milosav_M. Marjanoviĉ

The numbers 1, 2, ..., 20 are represented in the form of arrangements of horizontal and vertical lines and, when materialized, these lines are replaced by flat, longitudinal, rectangular sticks each having two sides dyed in two different colors. Sharp individuality of these arrangements is excellent for quick recognition of the numbers they represent. The way of arranging emphasizes the relation of the numbers 1, 2, ..., 10 to five and ten and this ``ten fingers'' model is basic, both conceptually and operationally, for our approach to schematic learning of the arithmetic tables. In case of addition and subtraction, the chosen structures of the arrangements reflect clearly ``crossings the five and ten lines'', serving efficently as illustrations (and explanations) of these methods. The suggested designs of pictured products $m\times n$ are easily seen as $m$ groups of $n$ sticks and, in the same time, as groups of tens and ones. Wall maps of these designs might be used in the class, letting the pupil have them to fall back on and so helping him/her form gradually a store of mental images related to the multiplication table. The use of space holders is also suggested to help the child compose the symbolic codes which immediately follow manipulative activities. Thus, a one-to-one correspondence between manipulative, reflective and symbolic operations is established, what also makes them connected in a child's mind.