User:Phlsph7/Arithmetic - Tools and methods

By tools and methods

Calculations in mental arithmetic are done exclusively in the mind without relying on external aids.

Another distinction focuses on the tools and methods used to perform arithmetic operations and covers many approaches besides the regular use of pen and paper. For mental arithmetic, the calculations are performed exclusively in the mind without external tools like pen and paper or electronic calculators. Instead, mental arithmetic makes use of visualization, memorization, and certain calculation techniques to solve arithmetic problems. One such technique is the compensation method. It consists in altering the numbers to make the calculation easier and then adjusting the result afterward. For example, instead of calculating $85-47$, one calculates $85-50$ which is easier because it uses a round number. In the next step, one adds $3$ to the result to compensate for the earlier adjustment. Mental arithmetic is often taught in primary education to train the numerical abilities of the students.

The human body can also be employed as an arithmetic tool. The use of hands in finger counting is often introduced to young children to teach them numbers and simple calculations. In its most basic form, the number of extended fingers corresponds to the represented quantity. This system is limited to small numbers while more advanced systems employ different approaches to represent larger quantities as well. The human voice is used as an arithmetic aid in verbal counting.

Tally marks are a simple system based on external tools other than the body. It relies on strokes drawn on a surface or notches in a wooden stick to keep track of quantities. Some forms of tally marks arrange the strokes in groups of five to make them easier to read. The abacus is a more advanced tool to represent numbers and perform calculations. An abacus usually consists of a series of rods, each holding several beads. Each bead represents a quantity, which is counted if the bead is moved from one end of a rod to the other. Calculations happen by manipulating the positions of beads until the final bead pattern reveals the result.

Mechanical calculators automate this process. They present the user with some form of input device to enter numbers by turning dials or pressing keys. They include an internal mechanism usually consisting of gears, levers, and wheels to perform calculations and display the results. For electronic calculators and computers, this procedure is further refined by replacing the mechanical components with electronic circuits like processors that combine and transform electric signals to perform calculations.

Sources

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• Sternberg, Robert J.; Ben-Zeev, Talia (12 October 2012). The Nature of Mathematical Thinking. Routledge. ISBN 978-1-136-48750-7.
• Reynolds, Barbara E. (12 March 2008). "Abacus". In Selin, Helaine (ed.). Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. Springer Science & Business Media. ISBN 978-1-4020-4559-2.
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• Dowker, Ann (27 March 2019). Individual Differences in Arithmetic: Implications for Psychology, Neuroscience and Education. Routledge. ISBN 978-1-317-62743-2.