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Holt S, Fan JE, Barner D. Creating ad hoc graphical representations of number. Cognition 2024; 242:105665. [PMID: 37992512 DOI: 10.1016/j.cognition.2023.105665] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/04/2023] [Revised: 11/09/2023] [Accepted: 11/11/2023] [Indexed: 11/24/2023]
Abstract
The ability to communicate about exact number is critical to many modern human practices spanning science, industry, and politics. Although some early numeral systems used 1-to-1 correspondence (e.g., 'IIII' to represent 4), most systems provide compact representations via more arbitrary conventions (e.g., '7' and 'VII'). When people are unable to rely on conventional numerals, however, what strategies do they initially use to communicate number? Across three experiments, participants used pictures to communicate about visual arrays of objects containing 1-16 items, either by producing freehand drawings or combining sets of visual tokens. We analyzed how the pictures they produced varied as a function of communicative need (Experiment 1), spatial regularities in the arrays (Experiment 2), and visual properties of tokens (Experiment 3). In Experiment 1, we found that participants often expressed number in the form of 1-to-1 representations, but sometimes also exploited the configuration of sets. In Experiment 2, this strategy of using configural cues was exaggerated when sets were especially large, and when the cues were predictably correlated with number. Finally, in Experiment 3, participants readily adopted salient numerical features of objects (e.g., four-leaf clover) and generally combined them in a cumulative-additive manner. Taken together, these findings corroborate historical evidence that humans exploit correlates of number in the external environment - such as shape, configural cues, or 1-to-1 correspondence - as the basis for innovating more abstract number representations.
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Affiliation(s)
- Sebastian Holt
- Department of Psychology, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA.
| | - Judith E Fan
- Department of Psychology, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA; Department of Psychology, Stanford University, 450 Jane Stanford Way, Stanford, CA 94305, USA
| | - David Barner
- Department of Psychology, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA; Department of Linguistics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA
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Piantadosi ST. The algorithmic origins of counting. Child Dev 2023; 94:1472-1490. [PMID: 37984061 DOI: 10.1111/cdev.14031] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/20/2022] [Revised: 09/16/2023] [Accepted: 09/19/2023] [Indexed: 11/22/2023]
Abstract
The study of how children learn numbers has yielded one of the most productive research programs in cognitive development, spanning empirical and computational methods, as well as nativist and empiricist philosophies. This paper provides a tutorial on how to think computationally about learning models in a domain like number, where learners take finite data and go far beyond what they directly observe or perceive. To illustrate, this paper then outlines a model which acquires a counting procedure using observations of sets and words, extending the proposal of Piantadosi et al. (2012). This new version of the model responds to several critiques of the original work and outlines an approach which is likely appropriate for acquiring further aspects of mathematics.
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O’Shaughnessy DM, Cruz Cordero T, Mollica F, Boni I, Jara-Ettinger J, Gibson E, Piantadosi ST. Diverse mathematical knowledge among indigenous Amazonians. Proc Natl Acad Sci U S A 2023; 120:e2215999120. [PMID: 37603761 PMCID: PMC10469040 DOI: 10.1073/pnas.2215999120] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/21/2022] [Accepted: 05/12/2023] [Indexed: 08/23/2023] Open
Abstract
We investigate number and arithmetic learning among a Bolivian indigenous people, the Tsimane', for whom formal schooling is comparatively recent in history and variable in both extent and consistency. We first present a large-scale meta-analysis on child number development involving over 800 Tsimane' children. The results emphasize the impact of formal schooling: Children are only found to be full counters when they have attended school, suggesting the importance of cultural support for early mathematics. We then test especially remote Tsimane' communities and document the development of specialized arithmetical knowledge in the absence of direct formal education. Specifically, we describe individuals who succeed on arithmetic problems involving the number five-which has a distinct role in the local economy-even though they do not succeed on some lower numbers. Some of these participants can perform multiplication with fives at greater accuracy than addition by one. These results highlight the importance of cultural factors in early mathematics and suggest that psychological theories of number where quantities are derived from lower numbers via repeated addition (e.g., a successor function) are unlikely to explain the diversity of human mathematical ability.
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Affiliation(s)
| | | | - Francis Mollica
- School of Informatics, University of Edinburgh, EdinburghEH8 9AB, United Kingdom
| | - Isabelle Boni
- Department of Psychology, University of California, Berkeley, CA94720-1650
| | | | - Edward Gibson
- Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, Cambridge, MA02139-4307
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Mistry PK, Strock A, Liu R, Young G, Menon V. Learning-induced reorganization of number neurons and emergence of numerical representations in a biologically inspired neural network. Nat Commun 2023; 14:3843. [PMID: 37386013 PMCID: PMC10310708 DOI: 10.1038/s41467-023-39548-5] [Citation(s) in RCA: 4] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/17/2022] [Accepted: 06/16/2023] [Indexed: 07/01/2023] Open
Abstract
Number sense, the ability to decipher quantity, forms the foundation for mathematical cognition. How number sense emerges with learning is, however, not known. Here we use a biologically-inspired neural architecture comprising cortical layers V1, V2, V3, and intraparietal sulcus (IPS) to investigate how neural representations change with numerosity training. Learning dramatically reorganized neuronal tuning properties at both the single unit and population levels, resulting in the emergence of sharply-tuned representations of numerosity in the IPS layer. Ablation analysis revealed that spontaneous number neurons observed prior to learning were not critical to formation of number representations post-learning. Crucially, multidimensional scaling of population responses revealed the emergence of absolute and relative magnitude representations of quantity, including mid-point anchoring. These learnt representations may underlie changes from logarithmic to cyclic and linear mental number lines that are characteristic of number sense development in humans. Our findings elucidate mechanisms by which learning builds novel representations supporting number sense.
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Affiliation(s)
- Percy K Mistry
- Department of Psychiatry & Behavioral Sciences, Stanford University School of Medicine, Stanford, CA, 94304, USA.
| | - Anthony Strock
- Department of Psychiatry & Behavioral Sciences, Stanford University School of Medicine, Stanford, CA, 94304, USA
| | - Ruizhe Liu
- Department of Psychiatry & Behavioral Sciences, Stanford University School of Medicine, Stanford, CA, 94304, USA
| | - Griffin Young
- Department of Psychiatry & Behavioral Sciences, Stanford University School of Medicine, Stanford, CA, 94304, USA
| | - Vinod Menon
- Department of Psychiatry & Behavioral Sciences, Stanford University School of Medicine, Stanford, CA, 94304, USA.
- Department of Neurology & Neurological Sciences, Stanford University School of Medicine, Stanford, CA, 94304, USA.
- Wu Tsai Stanford Neuroscience Institute, Stanford University School of Medicine, Stanford, CA, 94304, USA.
- Graduate School of Education, Stanford University, Stanford, CA, 94304, USA.
- Stanford Institute for Human-Centered AI, Stanford University, Stanford, CA, 94304, USA.
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Pitt B, Ferrigno S, Cantlon JF, Casasanto D, Gibson E, Piantadosi ST. Spatial concepts of number, size, and time in an indigenous culture. SCIENCE ADVANCES 2021; 7:eabg4141. [PMID: 34380617 PMCID: PMC8357228 DOI: 10.1126/sciadv.abg4141] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/04/2021] [Accepted: 06/28/2021] [Indexed: 05/27/2023]
Abstract
In industrialized groups, adults implicitly map numbers, time, and size onto space according to cultural practices like reading and counting (e.g., from left to right). Here, we tested the mental mappings of the Tsimane', an indigenous population with few such cultural practices. Tsimane' adults spatially arranged number, size, and time stimuli according to their relative magnitudes but showed no directional bias for any domain on any spatial axis; different mappings went in different directions, even in the same participant. These findings challenge claims that people have an innate left-to-right mapping of numbers and that these mappings arise from a domain-general magnitude system. Rather, the direction-specific mappings found in industrialized cultures may originate from direction-agnostic mappings that reflect the correlational structure of the natural world.
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Affiliation(s)
- Benjamin Pitt
- Department of Psychology, UC Berkeley, Berkeley, CA, USA.
| | - Stephen Ferrigno
- Department of Psychology, Harvard University, Cambridge, MA, USA
| | - Jessica F Cantlon
- Department of Psychology, Carnegie Mellon University, Pittsburgh, PA, USA
| | - Daniel Casasanto
- Department of Human Development, Cornell University, Ithaca, NY, USA
| | - Edward Gibson
- Department of Brain and Cognitive Sciences, MIT, Cambridge, MA, USA
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