Posts Tagged: embodiment
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: A taxonomy of (practical vs. theoretical) actions
| The solution to a problem changes the problem. |
| Peer’s Law |
Hansjörg Neth, Thomas Müller
Thinking by doing and doing by thinking: A taxonomy of actions
Abstract: Taking a lead from existing typologies of actions in the philosophical and cognitive science literatures, we present a novel taxonomy of actions.
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.