Tribute to Invariance

Addition and Multiplication Tables in Various Bases

Many children grow superstitious, and think that you cannot carry except in tens; or that it is wrong to carry in anything but tens. The use of algebra is to free them from bondage to all this superstitious nonsense, and help them to see that the numbers would come just as right if we carried in eights or twelves or twenties. It is a little difficult to do this at first, because we are not accustomed to it; but algebra helps to get over our stiffness and set habits and to do numeration on any basis that suits the matter we are dealing with.

Mary Everest Boole
Philosophy And Fun Of Algebra,
London: C. W. Daniel, LTD, 1909

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 children have to memorize (almost) twice as much as their Chinese counterparts, an opportunity is missed to add a little of abstract thinking to rote memorization.

Commutativity of addition and multiplication appears as 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 advantage over looking at it a line at a time. Even more regularity can be observed when several tables (one per a different base system) are studied together.

I am not suggesting to teach preschoolers counting 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.

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