Posts Tagged: arithmetic
mental and environmental arithmetic
Paper: Decade effects in mental addition
| 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 |
Classification of addition types for adding a single-digit addend a (with a∈{1…9}) to an augend Au (u denoting the augend’s unit). (See Table 1 for details.)
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.
Four different addition types (A) and their hypothetical frequency (B, assuming uniform distribution of addends 1–9 and full decomposition of super-complements). (See Figure 4 in Appendix A1 for details.)
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
Related: Addition as interactive problem solving | Thinking by doing? | Immediate interactive behavior (IIB) | Arabic vs. Roman arithmetic | Taxonomy of actions | The cognitive basis of arithmetic | Interactive coin addition | The functional task environment
Resources: open access article | PDF download | Google Scholar
Paper: Interactive coin addition
| ‘Can you do Addition?’ the White Queen asked. ‘What’s one and one and one and one and one and one and one and one and one and one?’ ‘I don’t know,’ said Alice. ‘I lost count.’ |
| Lewis Carroll, Through the Looking-Glass, Chapter IX. |
Hansjörg Neth, Stephen J. Payne
Interactive coin addition: How hands can help us think
Abstract: Does using our hands help us to add the value of a set of coins?
Chapter: The cognitive basis of arithmetic
| The truths about numbers are in us; but still we learn them. |
| G.W. Leibniz (1765), Nouveaux essais sur l’entendement humain, p. 85 |
Helen De Cruz, Hansjörg Neth, Dirk Schlimm
The cognitive basis of arithmetic
Overview: Arithmetic is the theory of the natural numbers and one of the oldest areas of mathematics. Since almost all other mathematical theories make use of numbers in some way or other, arithmetic is also one of the most fundamental theories of mathematics. But numbers are not just abstract entities that are subject to mathematical ruminations — they are represented, used, embodied, and manipulated in order to achieve many different goals, e.g., to count or denote the size of a collection of objects, to trade goods, to balance bank accounts, or to play the lottery. Consequently, numbers are both abstract and intimately connected to language and to our interactions with the world. In the present paper we provide an overview of research that has addressed the question of how animals and humans learn, represent, and process numbers.
Paper: Arithmetic with Arabic vs. Roman numerals
| … how information is represented can greatly affect how easy it is to do different things with it. (…) it is easy to add, to subtract, and even to multiply if the Arabic or binary representations are used, but it is not at all easy to do these things — especially multiplication — with Roman numerals. This is a key reason why the Roman culture failed to develop mathematics in the way the earlier Arabic cultures had. |
| D Marr (1982): Vision, p. 21 |
Dirk Schlimm, Hansjörg Neth
Modeling ancient and modern arithmetic practices: Addition and multiplication with Arabic and Roman numerals
Abstract: To analyze the task of mental arithmetic with external representations in different number systems we model algorithms for addition and multiplication with Arabic and Roman numerals. This demonstrates that Roman numerals are not only informationally equivalent to Arabic ones but also computationally similar — a claim that is widely disputed. An analysis of our models’ elementary processing steps reveals intricate trade-offs between problem representation, algorithm, and interactive resources. Our simulations allow for a more nuanced view of the received wisdom on Roman numerals. While symbolic computation with Roman numerals requires fewer internal resources than with Arabic ones, the large number of needed symbols inflates the number of external processing steps.