Without paper or pencil (nor screen)
While prohibiting the traditional tools of writing, mental calculation can be enriched with objects that make it possible to track the progress of an operation. For example, rather than considering a number from its decimal representation, some calculators rather visualize it by imagining it written on an abacus. By juggling internally with the beads of the soroban rather than with the usual digits, or by cleverly using the support of the fingers, new horizons open up for developing our calculation abilities in new directions.
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The abacus mindset: soroban and mental arithmetic | Tangente
The enduring use of the abacus has made Asia the continent where mental arithmetic is most highly developed. Competitions are held there, and the leading experts regularly set new records.
Finger multiplication: tables at your fingertips | Tangente
Even before they had words for numbers, human societies developed counting systems of varying sophistication to suit their activities. To count beyond a handful of units, they used the tools they always had to hand: their own bodies. The result was a range of ingenious methods for performing certain operations.
At Ratchinski's public school
For fifteen years, Sergei Ratchinski developed his own teaching method, based on creativity.
Napier's bones: multiply without tables | Tangente
With the advent of calculators and, later, smartphones, people have become less and less likely to work out a multiplication, division or square root by hand. Yet for centuries this was a problem, inspiring a host of ingenious ways to streamline calculations. One such invention was Napier's bones, which effectively automated multiplication.
Conquering carries: Genaille–Lucas rulers | Tangente
Napier's rods had simplified multiplication, but handling carries remained an unsolved problem. The obstacle they posed remained insurmountable for three and a half centuries, until Genaille and Lucas finally found a solution.
Math games: 2048 and more | Tangente
All right… a little screen time after all—not to do the arithmetic for us, but to practise mental arithmetic while having fun. The games featured on this page can be downloaded to a smartphone.