Ratio-based perceptual foundations for rational numbers, and perhaps whole numbers, too?

2021 ◽  
Vol 44 ◽  
Author(s):  
Edward M. Hubbard ◽  
Percival G. Matthews
Keyword(s):  
Numerical Cognition ◽  
Processing System ◽  
Number System ◽  
Rational Numbers ◽  
Approximate Number System ◽  
Access Points ◽  
Whole Numbers ◽  
Two Systems ◽  
Systems View

Abstract Clarke and Beck suggest that the ratio processing system (RPS) may be a component of the approximate number system (ANS), which they suggest represents rational numbers. We argue that available evidence is inconsistent with their account and advocate for a two-systems view. This implies that there may be many access points for numerical cognition – and that privileging the ANS may be a mistake.

Non-symbolic and symbolic number and the approximate number system

2021 ◽  
Vol 44 ◽  
Author(s):  
David Maximiliano Gómez
Keyword(s):  
Numerical Cognition ◽  
Number System ◽  
Rational Numbers ◽  
Approximate Number System ◽  
Symbolic Number

Abstract The distinction between non-symbolic and symbolic number is poorly addressed by the authors despite being relevant in numerical cognition, and even more important in light of the proposal that the approximate number system (ANS) represents rational numbers. Although evidence on non-symbolic number and ratios fits with ANS representations, the case for symbolic number and rational numbers is still open.

The number sense does not represent numbers, but cardinality comparisons

2021 ◽  
Vol 44 ◽  
Author(s):  
José Luis Bermúdez
Keyword(s):  
Experimental Evidence ◽  
Number Sense ◽  
Number System ◽  
Rational Numbers ◽  
Approximate Number System

Abstract Against Clarke and Beck's proposal that the approximate number system (ANS) represents natural and rational numbers, I suggest that the experimental evidence is better accommodated by the (much weaker) thesis that the ANS represents cardinality comparisons. Cardinality comparisons do not stand in arithmetical relations and being able to apply them does not involve basic arithmetical concepts and operations.

scholarly journals Numerical cognition needs more and better distinctions, not fewer

2021 ◽  
Vol 44 ◽  
Author(s):  
Hilary Barth ◽  
Anna Shusterman
Keyword(s):  
Numerical Cognition ◽  
Number System ◽  
Approximate Number System ◽  
The Way

Abstract We agree that the approximate number system (ANS) truly represents number. We endorse the authors' conclusions on the arguments from confounds, congruency, and imprecision, although we disagree with many claims along the way. Here, we discuss some complications with the meanings that undergird theories in numerical cognition, and with the language we use to communicate those theories.

A rational explanation for links between the ANS and math

2021 ◽  
Vol 44 ◽  
Author(s):  
Melissa E. Libertus ◽  
Shirley Duong ◽  
Danielle Fox ◽  
Leanne Elliott ◽  
Rebecca McGregor ◽  
...  
Keyword(s):  
Error Detection ◽  
Number System ◽  
Rational Numbers ◽  
Approximate Number System ◽  
Math Skills ◽  
Detection Mechanism ◽  
Affective Factors ◽  
Rational Explanation

Abstract The proposal by Clarke and Beck offers a new explanation for the association between the approximate number system (ANS) and math. Previous explanations have largely relied on developmental arguments, an underspecified notion of the ANS as an “error detection mechanism,” or affective factors. The proposal that the ANS represents rational numbers suggests that it may directly support a broader range of math skills.

The perception of quantity ain't number: Missing the primacy of symbolic reference

2021 ◽  
Vol 44 ◽  
Author(s):  
Rafael E. Núñez ◽  
Francesco d'Errico ◽  
Russell D. Gray ◽  
Andrea Bender
Keyword(s):  
Essential Role ◽  
Number System ◽  
Rational Numbers ◽  
Theoretical Construct ◽  
Approximate Number System

Abstract Clarke and Beck's defense of the theoretical construct “approximate number system” (ANS) is flawed in serious ways – from biological misconceptions to mathematical naïveté. The authors misunderstand behavioral/psychological technical concepts, such as numerosity and quantical cognition, which they disdain as “exotic.” Additionally, their characterization of rational numbers is blind to the essential role of symbolic reference in the emergence of number.

Toward an integrative approach to numerical cognition

2017 ◽  
Vol 40 ◽  
Author(s):  
Tali Leibovich ◽  
Naama Katzin ◽  
Moti Salti ◽  
Avishai Henik
Keyword(s):  
Collaborative Research ◽  
Animal Studies ◽  
Numerical Cognition ◽  
Number System ◽  
Open Science ◽  
Integrative Approach ◽  
Approximate Number System ◽  
Suggested Model ◽  
Future Developments

AbstractIn response to the commentaries, we have refined our suggested model and discussed ways in which the model could be further expanded. In this context, we have elaborated on the role of specific continuous magnitudes. We have also found it important to devote a section to evidence considered the “smoking gun” of the approximate number system theory, including cross-modal studies, animal studies, and so forth. Lastly, we suggested some ways in which the scientific community can promote more transparent and collaborative research by using an open science approach, sharing both raw data and stimuli. We thank the contributors for their enlightening comments and look forward to future developments in the field.

Sizes, ratios, approximations: On what and how the ANS represents

2021 ◽  
Vol 44 ◽  
Author(s):  
Brian Ball
Keyword(s):  
Number System ◽  
Rational Numbers ◽  
Approximate Number System ◽  
Positive Integers

Abstract Clarke and Beck propose that the approximate number system (ANS) represents rational numbers. The evidence cited supports only the view that it represents ratios (and positive integers). Rational numbers are extensive magnitudes (i.e., sizes), whereas ratios are intensities. It is also argued that WHAT a system represents and HOW it does so are not as independent of one another as the authors assume.

Not so rational: A more natural way to understand the ANS

2021 ◽  
Vol 44 ◽  
Author(s):  
Eli Hecht ◽  
Tracey Mills ◽  
Steven Shin ◽  
Jonathan Phillips
Keyword(s):  
Number System ◽  
Rational Numbers ◽  
Approximate Number System ◽  
Natural Numbers ◽  
Wide Range ◽  
Natural Way

Abstract In contrast to Clarke and Beck's claim that that the approximate number system (ANS) represents rational numbers, we argue for a more modest alternative: The ANS represents natural numbers, and there are separate, non-numeric processes that can be used to represent ratios across a wide range of domains, including natural numbers.

scholarly journals Measuring the Approximate Number System

2011 ◽  
Vol 64 (11) ◽  
pp. 2099-2109 ◽  
Author(s):  
Camilla Gilmore ◽  
Nina Attridge ◽  
Matthew Inglis
Keyword(s):  
Numerical Cognition ◽  
Number System ◽  
Single System ◽  
Approximate Number System ◽  
Methodological Issues ◽  
Educational Benefits ◽  
Adult Participants ◽  
Typical Performance ◽  
The Relationship

Recent theories in numerical cognition propose the existence of an approximate number system (ANS) that supports the representation and processing of quantity information without symbols. It has been claimed that this system is present in infants, children, and adults, that it supports learning of symbolic mathematics, and that correctly harnessing the system during tuition will lead to educational benefits. Various experimental tasks have been used to investigate individuals' ANSs, and it has been assumed that these tasks measure the same system. We tested the relationship across six measures of the ANS. Surprisingly, despite typical performance on each task, adult participants' performances across the tasks were not correlated, and estimates of the acuity of individuals' ANSs from different tasks were unrelated. These results highlight methodological issues with tasks typically used to measure the ANS and call into question claims that individuals use a single system to complete all these tasks.

Probing the Neural Correlates of Number Processing

2016 ◽  
Vol 23 (3) ◽  
pp. 264-274 ◽  
Author(s):  
André Knops
Keyword(s):  
Computational Models ◽  
Numerical Cognition ◽  
Experimental Studies ◽  
Number System ◽  
Neural Correlates ◽  
Approximate Number System ◽  
Numerical Information ◽  
The Past ◽  
Code Model ◽  
Human Neuroimaging

The cognitive and neural mechanisms that enable humans to encode and manipulate numerical information have been subject to an increasing number of experimental studies over the past 25 years or so. Here, I highlight recent findings about how numerical information is neurally coded, focusing on the theoretical implications derived from the most influential theoretical framework in numerical cognition—the Triple Code Model. At the core of this model is the assumption that bilateral parietal cortex hosts an approximate number system that codes for the cardinal value of perceived numerals. I will review studies that ask whether or not the numerical coding within this system is invariant to varying input notation, format, or modality, and whether or not the observed parietal activity is number-specific over and above the parietal involvement in response-related processes. Extant computational models of numerosity (the number of objects in a set) perception are summarized and related to empirical data from human neuroimaging and monkey neurophysiology.

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