University of Birmingham AISB/IACAP World Congress 2012 - Alan Turing 2012

2-6 July 2012

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Mathematical Practice and Cognition II

Location: CPD6

Session A, Monday, 2 July, 10:30-13:00

10:30-13:00 Rafael Núñez
The study of mathematical practice and cognition as an empirical endeavor: The case of the cognitive science of the number line
(Invited Talk)
11:30-11:40 Comfort break (no refreshments served)
11:40-12:20 Matthew Inglis and Lara Alcock
Watching Experts and Novices Read Proofs
12:20-13:00 Andrew Aberdein
The Parallel Structure of Mathematical Reasoning

Session B, Monday, 2 July, 14:30-17:00

14:30-15:10 Alan Smaill
Mathematical Notation and Analogy
15:10-15:50 Alison Pease and Ursula Martin
Seventy Four Minutes of Mathematics: An analysis of the third Mini-Polymath project
15:50-16:00 Comfort break (no refreshments served)
16:00-17:00 Liesbeth De Mol
Taking the machine seriously. A study of 'mechanized mathematics'
(Invited Talk)

Session C, Tuesday, 3 July, 8:30-11:30

8:40-9:20 Sandra Visokolskis
Discovery in Mathematics as an Experiential Practice of Privation
9:20-10:00 Manfred Kerber, Christoph Lange and Colin Rowat
Formal Representation and Proof for Cooperative Games
10:00-10:30 Coffee/Tea Break
10:30-11:10 Brendan Larvor
What Philosophy of Mathematical Practice Can Teach Argumentation Theory about Diagrams and Pictures
11:10-11:20 Gabriella Daróczy
Dialogue with a Mathematical Assistant Cognitive Aspects of Designing Rule Based Dialogue Guidance (poster)

Session D, Tuesday, 3 July 14:00-16:30

14:00-15:00 José Ferreirós
Convergence, not Reduction: On cognitive science and studies of mathematical practice (Invited Talk)
15:00-16:00 Panel discussion

Further details of invited talk:

Rafael Núñez: The study of mathematical practice and cognition as an empirical endeavor: The case of the cognitive science of the number line

Abstract: The investigation of mathematical practice and cognition can be informed by multiple sources such as history, philosophy, and descriptive studies. In this talk I'll argue that the field also needs multidisciplinary empirical (experimental) hypothesis-testing methods. In order to illustrate my point I'll focus on a few empirical questions involving the nature of the number line - fundamental to modern mathematics - and its relation to cognition, cultural practices, and biological constraints. What are the cognitive origins of the number line? Are the intuitions underlying the number line "hard-wired"? Is the number line a cultural construct? Contemporary research in the psychology and neuroscience of number cognition has largely assumed that the representation of number is inherently spatial and that the number-to-space mapping is a universal intuition rooted directly in brain evolution. I'll review material from the history of mathematics as well as empirical results from two of our recent experimental studies to defend a radically different picture: the representation of number is not inherently spatial and the intuition of mapping numbers to space is not universal. In one study we show that there are non-spatial representations of numbers that co-exist with spatial ones, as indexed by instrumental manual actions, such as squeezing and bell-hitting, and non-instrumental actions, such as vocalizing. Moreover, the results suggest that the number-to-line mapping - a spatial mapping - is not a product of the human biological endowment but that it has been culturally privileged and enhanced via specific practices. The other study, which we carried out in the remote mountains of Papua New Guinea, shows experimentally that individuals from a culture that has a precise counting system (and lexicon) for numbers greater than twenty - but no measurement practices - lack the intuition of a number-to-line mapping, suggesting that this intuition is not universally spontaneous, and therefore, unlikely to be rooted directly in brain evolution. The number-to-line mapping appears to be learned through - and continually reinforced by - specific cultural practices, such as measurement tools, writing systems, and elementary mathematics education. It is over the course of exposure to these cultural practices that well-known brain areas such as the parietal lobes are recruited to support number representation and processing. Implications for the study of mathematical practice and cognition will be discussed.