The numbers 1, 2, \dots, 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, \dots, 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.