scholarly journals Counting and the ontogenetic origins of exact equality

2021 ◽  
Author(s):  
Rose M. Schneider ◽  
Erik Brockbank ◽  
Roman Feiman ◽  
David Barner
Keyword(s):  
Matching Task ◽  
Important Data ◽  
Number Concepts ◽  
Counting System ◽  
Number Words ◽  
Exact Number ◽  
Number Systems ◽  
Us Children ◽  
Exact Equality ◽  
Symbolic Number

Humans are unique in their capacity to both represent number exactly and to express these representations symbolically. This correlation has prompted debate regarding whether symbolic number systems are necessary to represent exact number. Previous work addressing this question in innumerate adults and semi-numerate children has been limited by conflicting results and differing methodologies, and has not yielded a clear answer. We address this debate by adapting methods used with innumerate populations (a “set-matching” task) for 3- to 5-year-old US children at varying stages of symbolic number acquisition. In five studies we find that children’s ability to match sets exactly is related not simply to knowing the meanings of a few number words, but also to understanding how counting is used to generate sets (i.e., the cardinal principle). However, while children were more likely to match sets after acquiring the cardinal principle, they nevertheless demonstrated failures, compatible with the hypothesis that the ability to reason about exact equality emerges sometime later. These findings provide important data on the origin of exact number concepts, and point to knowledge of a counting system, rather than number language in general, as a key ingredient in the ability to reason about large exact number.

scholarly journals Exact number concepts are limited to the verbal count list

2021 ◽  
Author(s):  
Benjamin Pitt ◽  
Edward Gibson ◽  
Steven T. Piantadosi
Keyword(s):  
Data Analysis ◽  
Numerical Approximation ◽  
Matching Task ◽  
Analysis Model ◽  
Single Population ◽  
Number Concepts ◽  
Number Words ◽  
Exact Number ◽  
Number Representations

Previous studies suggest that mentally representing exact numbers larger than four depends on a verbal count routine (e.g. “one, two, three...”). However, these findings are controversial, as they rely on comparisons across radically different languages and cultures. We tested the role of language in number concepts within a single population – the Tsimane’ of Bolivia – where knowledge of number words varies across individual adults. We used a novel data analysis model to quantify the point at which participants switched from exact to approximate number representations during a simple numerical matching task. The results show that these behavioral switchpoints were bounded by participants’ verbal count ranges; their representations of exact cardinalities were limited to the number words they could recite. Beyond that range, they resorted to numerical approximation. These findings resolve competing accounts of previous findings and provide unambiguous evidence that large exact number concepts are enabled by language.

Sounding the depths at the confluence of numerosity and language

2015 ◽  
Vol 1 (1) ◽  
Author(s):  
Caleb Everett
Keyword(s):  
Recent Work ◽  
Numerical Cognition ◽  
Number Words ◽  
Number Systems ◽  
The Relationship

AbstractResearchers in a variety of disciplines are currently exploring how number words and other culturally variant symbols for quantities enable and enhance numerical cognition. In this article I survey the ways in which fieldwork among Amazonian languages is helping to elucidate the relationship between numerical cognition and language. I highlight several noteworthy findings in recent work on this topic, address their implications, and also consider the potential of future fieldwork on languages with typologically remarkable number systems.

Children's Conceptual Structures for Multidigit Numbers and Methods of Multidigit Addition and Subtraction

1997 ◽  
Vol 28 (2) ◽  
pp. 130-162 ◽  
Author(s):  
Karen C. Fuson ◽  
Diana Wearne ◽  
James C. Hiebert ◽  
Hanlie G. Murray ◽  
Pieter G. Human ◽  
...  
Keyword(s):  
Problem Solving ◽  
Teaching And Learning ◽  
Number Concepts ◽  
Number Words ◽  
Conceptual Structures ◽  
Common Framework ◽  
Approach To Teaching ◽  
Addition And Subtraction

Researchers from 4 projects with a problem-solving approach to teaching and learning multidigit number concepts and operations describe (a) a common framework of conceptual structures children construct for multidigit numbers and (b) categories of methods children devise for multidigit addition and subtraction. For each of the quantitative conceptual structures for 2-digit numbers, a somewhat different triad of relations is established between the number words, written 2-digit marks, and quantities. The conceptions are unitary, decade and ones, sequence-tens and ones, separate-tens and ones, and integrated sequence-separate conceptions. Conceptual supports used within each of the 4 projects are described and linked to multidigit addition and subtraction methods used by project children. Typical errors that may arise with each method are identified. We identify as crucial across all projects sustained opportunities for children to (a) construct triad conceptual structures that relate ten-structured quantities to number words and written 2-digit numerals and (b) use these triads in solving multidigit addition and subtraction situations.

scholarly journals How Counting Leads to Children’s First Representations of Exact, Large Numbers

2014 ◽  
Author(s):  
Barbara W. Sarnecka ◽  
Meghan C. Goldman ◽  
Emily B. Slusser
Keyword(s):  
Young Children ◽  
Successor Function ◽  
Number Words ◽  
Large Numbers ◽  
Individual Number ◽  
The Individual ◽  
First Time ◽  
Exact Equality

Young children initially learn to ‘count’ without understanding either what counting means, or what numerical quantities the individual number words pick out. Over a period of many months, children assign progressively more sophisticated meanings to the number words, linking them to discrete objects, to quantification, to numerosity, and so on. Eventually, children come to understand the logic of counting. Along with this knowledge comes an implicit understanding of the successor function, as well as of the principle of equinumerosity, or exact equality between sets. Thus, when children arrive at a mature understanding of counting, they have (for the first time in their lives) a way of mentally representing exact, large numbers.

scholarly journals A new look at old numbers, and what it reveals about numeration

2021 ◽  
Author(s):  
Karenleigh A. Overmann
Keyword(s):  
Context Dependence ◽  
Mental Calculation ◽  
Number Words ◽  
Number Systems ◽  
Counting Methods ◽  
And Behaviors ◽  
New Look

In this study, the archaic counting systems of Mesopotamia as understood through the Neolithic tokens, numerical impressions, and proto-cuneiform notations were compared to the traditional number-words and counting methods of Polynesia as understood through contemporary and historical descriptions of vocabulary and behaviors. The comparison and associated analyses capitalized on the ability to understand well-known characteristics of Uruk-period numbers like object-specific counting, polyvalence, and context-dependence through historical observations of Polynesian counting methods and numerical language, evidence unavailable for ancient numbers. Similarities between the two number systems were then used to argue that archaic Mesopotamian numbers, like those of Polynesia, were highly elaborated and would have served as cognitively efficient tools for mental calculation. Their differences also show the importance of material technologies like tokens, impressions, and notations to developing mathematics.

scholarly journals Cognitive mediators of US—China differences in early symbolic arithmetic

PLoS ONE ◽  
2021 ◽  
Vol 16 (8) ◽  
pp. e0255283
Author(s):  
John E. Opfer ◽  
Dan Kim ◽  
Lisa K. Fazio ◽  
Xinlin Zhou ◽  
Robert S. Siegler
Keyword(s):  
Standardized Tests ◽  
Number Line ◽  
Chinese Children ◽  
Numerical Magnitude ◽  
Magnitude Comparison ◽  
Cognitive Mediators ◽  
Number Line Estimation ◽  
Mathematics Knowledge ◽  
Us Children ◽  
Symbolic Number

Chinese children routinely outperform American peers in standardized tests of mathematics knowledge. To examine mediators of this effect, 95 Chinese and US 5-year-olds completed a test of overall symbolic arithmetic, an IQ subtest, and three tests each of symbolic and non-symbolic numerical magnitude knowledge (magnitude comparison, approximate addition, and number-line estimation). Overall Chinese children performed better in symbolic arithmetic than US children, and all measures of IQ and number knowledge predicted overall symbolic arithmetic. Chinese children were more accurate than US peers in symbolic numerical magnitude comparison, symbolic approximate addition, and both symbolic and non-symbolic number-line estimation; Chinese and U.S. children did not differ in IQ and non-symbolic magnitude comparison and approximate addition. A substantial amount of the nationality difference in overall symbolic arithmetic was mediated by performance on the symbolic and number-line tests.

Mental Numerosity Line in the Human’s Approximate Number System

2016 ◽  
Vol 63 (3) ◽  
pp. 169-179 ◽  
Author(s):  
Xinlin Zhou ◽  
Chaoran Shen ◽  
Leinian Li ◽  
Dawei Li ◽  
Jiaxin Cui
Keyword(s):  
Number System ◽  
Spatial Layout ◽  
Matching Task ◽  
Approximate Number System ◽  
Current Experiment ◽  
Number Systems ◽  
Subitizing Range ◽  
Arabic Digits

Abstract. Previous studies have demonstrated existence of a mental line for symbolic numbers (e.g., Arabic digits). For nonsymbolic number systems, however, it remains unresolved whether a spontaneous spatial layout of numerosity exists. The current experiment investigated whether SNARC-like (Spatial-Numerical Association of Response Codes) effects exist in approximate processing of numerosity, as well as of size and density. Participants were asked to judge whether two serially presented stimuli (i.e., dot arrays, pentagons) were the same regarding numbers of dots, sizes of the pentagon, or densities of dots. Importantly, two confounds that were overlooked by most previous studies were controlled in this study: no ordered numerosity was presented, and only numerosity in the approximate number system (beyond the subitizing range) was used. The results demonstrated that there was a SNARC-like effect only in the numerosity-matching task. The results suggest that numerosity could be spontaneously aligned to a left-to-right oriented mental line according to magnitude information in human’s approximate number system.

scholarly journals Symbolic Number Skills Predict Growth in Nonsymbolic Number Skills in Kindergarteners

2017 ◽  
Author(s):  
Ian M. Lyons ◽  
Stephanie Bugden ◽  
Samuel Zheng ◽  
Stefanie De Jesus ◽  
Daniel Ansari
Keyword(s):  
Formal Education ◽  
Number System ◽  
Kindergarten Children ◽  
Formal Schooling ◽  
Exact Number ◽  
Repeated Practice ◽  
Two Systems ◽  
Comprehensive Test ◽  
Symbolic Number ◽  
Number Skills

There is currently considerable discussion about the relative influences of evolutionary and cultural factors in the development of early numerical skills. In particular, there has been substantial debate and study of the relationship between approximate, nonverbal (approximate magnitude system, AMS) and exact, symbolic (symbolic number system, SNS) representations of number. Here we examined several hypotheses concerning whether, in the earliest stages of formal education, AMS abilities predict growth in SNS abilities, or the other way around. In addition to tasks involving symbolic (Arabic numerals) and non-symbolic (dot arrays) number comparisons, we also tested children’s ability to translate between the two systems (i.e., mixed-format comparison). Our data included a sample of 539 Kindergarten children (mean=5.17yrs, SD=0.29yrs), with AMS, SNS and mixed comparison skills assessed at the beginning and end of the academic year. In this way, we provide, to the best of our knowledge, the most comprehensive test to date of the direction of influence between the AMS and SNS in early formal schooling. Results were more consistent with the view that SNS abilities at the beginning of Kindergarten lay the foundation for improvement in both AMS abilities and the ability to translate between the two systems. Importantly, we found no evidence to support the reverse. We conclude that, once one acquires a very basic grasp of exact number symbols, it is this understanding of exact number (and perhaps repeated practice therewith) that facilitates growth in the AMS. Though the precise mechanism remains to be understood, these data challenge the widely held view that the AMS scaffolds the acquisition of the SNS.

scholarly journals The history of number words in the world's languages—what have we learnt so far?

2021 ◽  
Vol 376 (1824) ◽  
Author(s):  
Andreea S. Calude
Keyword(s):  
Human Evolution ◽  
Experimental Studies ◽  
Consistent Pattern ◽  
Theme Issue ◽  
Number Words ◽  
The Past ◽  
Language History ◽  
Number Systems ◽  
The World ◽  
History Of

For over 100 years, researchers from various disciplines have been enthralled and occupied by the study of number words. This article discusses implications for the study of deep history and human evolution that arise from this body of work. Phylogenetic modelling shows that low-limit number words are preserved across thousands of years, a pattern consistently observed in several language families. Cross-linguistic frequencies of use and experimental studies also point to widespread homogeneity in the use of number words. Yet linguistic typology and field documentation reports caution against positing a privileged linguistic category for number words, showing a wealth of variation in how number words are encoded across the world. In contrast with low-limit numbers, the higher numbers are characterized by a rapid and morphologically consistent pattern of expansion, and behave like grammatical phrasal units, following language-internal rules. Taken together, the evidence suggests that numbers are at the cross-roads of language history. For languages that do have productive and consistent number systems, numerals one to five are among the most reliable available linguistic fossils of deep history, defying change yet still bearing the marks of the past, while higher numbers emerge as innovative tools looking to the future, derived using language-internal patterns and created to meet the needs of modern speakers. This article is part of the theme issue ‘Reconstructing prehistoric languages’.

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