# Tribute to Invariance

## Addition and Multiplication Tables in Various Bases

In China, children are made to memorize only half of the addition and multiplication tables. The other half is obtained via the commutative law. In the United States, the tables are taught a line at a time. Thus not only do our children have to memorize (almost) twice as much as their Chinese counterparts, we miss the opportunity to add a little abstract thinking to rote memorization.

The commutativity of addition and multiplication appears as a symmetry of the tables with respect to the main diagonal. One may discern other features of the tables that have algebraic interpretation. So looking at the whole table has definite advantages over looking at it a line at a time. Even more regularity can be observed when several tables (one for each different base system) are studied together.

I am not suggesting that we should teach preschoolers to count in various bases. However, even older children enjoy non-routine tasks. Turning something as dull as a table of addition or multiplication into a research tool will definitely enliven class instruction.

Are you game?