| The most important part of the Analytical Engine was undoubtedly the mechanical method of carrying the tens. (…) The difficulty did not consist so much in the more or less complexity of the contrivance as in the reduction of the time required to effect the carriage. (…) nothing but teaching the Engine to foresee and then to act upon that foresight could ever lead me to the object I desired... |
| Charles S. Babbage (1864), Passages from the Life of a Philosopher, p. 114 |
Hansjörg Neth, Stephen J. Payne
Decade effects in mental addition
We examine representational effects of Western numerals on mental arithmetic. An analysis of mental addition tasks using a base-10 place-value notation yields a taxonomy of addition types that is anchored in the notion of complements (i.e. additions with round sums). Two experimental studies use a paradigm of serial addition that presents lists of numbers to adult participants, who mentally represent all intermediate steps. In study 1, participants add sequences of single-digit addends in a self-paced fashion. Study 2 extends this paradigm by simultaneously presenting two addends, thus allowing for a modicum of strategic choice. Both studies vary the number of complements within the lists and measure addition accuracy and latency. Beyond decade and carry effects, our results show that lists containing or enabling complements are easier to add. Addition latencies jointly depend on addition type and problem size. When adders have some discretion about the order of choosing addends, they adaptively exploit the difficulty of addition types by tailoring their sequences to decade boundaries. One motivation for seeking complements lies in enabling subsequent post-complements. Reflecting on the dynamic interplay between numeric representations, strategic choices and cognitive adaptations, we discuss implications for psychological explanations, technology and design.
This article is part of the theme issue A solid base for scaling up: the structure of numeration systems.
Keywords: mental arithmetic, addition strategies, base notation, representational effects.
Reference: Neth, H., & Payne, S. J. (2025). Decade effects in mental addition. Philosophical Transactions of the Royal Society B, 380, 20240220. https://doi.org/10.1098/rstb.2024.0220
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