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1. Schneider, R.M., & Barner, D. (2020). Children use one-to-one correspondence to establish equality after learning to count. 42nd Annual Meeting of the Cognitive Science Society.

   Schneider, R.M. (January 2020, 42nd Annual Meeting of the Cognitive Science Society.)

   null (Ed.)

   Humans make frequent and powerful use of external symbols to express number exactly, leading some to question whether exact number concepts are only available through the acquisition of symbolic number systems. Although prior work has addressed this longstanding debate on the relationship between language and thought in innumerate populations and seminumerate children, it has frequently produced conflicting results, leaving the origin of exact number concepts unclear. Here, we return to this question by replicating methods previously used to assess exact number knowledge in innumerate groups, such as the Piraha, with a large sample of semi- numerate US toddlers. We replicate previous findings from both innumerate cultures and developmental studies showing that numeracy is linked to the concept of exact number. However, we also find evidence that this knowledge is surprisingly fragile even amongst numerate children, suggesting that numeracy alone does not guarantee a full understanding of exactness. 

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2. Schneider, R.M, Feiman, R., Mendes, M., & Barner, D. (2021). Pragmatic impacts on children's understanding of exact equality. 43rd Annual Meeting of the Cognitive Science Society.

   Schneider, R.M (January 2021, 43rd Annual Meeting of the Cognitive Science Society)

   null (Ed.)

   The distinctly human ability to both represent number exactly and develop symbolic number systems has raised the question of whether such number concepts are culturally constructed through symbolic systems. Although previous work with innumerate and semi-numerate groups has provided some evidence that understanding exact equality is related to numeracy, it is possible that previous failures were driven by pragmatic factors, rather than the absence of conceptual knowledge. Here, we test whether such factors affect performance on a test of exact equality in 3- to 5-year-old children by modifying previous methods to draw children’s attention to number. We find no effect of highlighting exact equality, either through framing the task as a “Number” game or as a “Sharing” game. Instead, we replicate previous findings showing a link between numeracy and an understanding of exact equality, strengthening the proposal that exact number concepts are facilitated by the acquisition of symbolic number systems. 

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3. Measuring Emerging Number Knowledge in Toddlers

   https://doi.org/10.3389/fpsyg.2021.703598  

   Silver, Alex M.; Elliott, Leanne; Braham, Emily J.; Bachman, Heather J.; Votruba-Drzal, Elizabeth; Tamis-LeMonda, Catherine S.; Cabrera, Natasha; Libertus, Melissa E. (July 2021, Frontiers in Psychology)

   null (Ed.)

   Recent evidence suggests that infants and toddlers may recognize counting as numerically relevant long before they are able to count or understand the cardinal meaning of number words. The Give-N task, which asks children to produce sets of objects in different quantities, is commonly used to test children’s cardinal number knowledge and understanding of exact number words but does not capture children’s preliminary understanding of number words and is difficult to administer remotely. Here, we asked whether toddlers correctly map number words to the referred quantities in a two-alternative forced choice Point-to-X task (e.g., “Which has three?”). Two- to three-year-old toddlers ( N = 100) completed a Give-N task and a Point-to-X task through in-person testing or online via videoconferencing software. Across number-word trials in Point-to-X, toddlers pointed to the correct image more often than predicted by chance, indicating that they had some understanding of the prompted number word that allowed them to rule out incorrect responses, despite limited understanding of exact cardinal values. No differences in Point-to-X performance were seen for children tested in-person versus remotely. Children with better understanding of exact number words as indicated on the Give-N task also answered more trials correctly in Point-to-X. Critically, in-depth analyses of Point-to-X performance for children who were identified as 1- or 2-knowers on Give-N showed that 1-knowers do not show a preliminary understanding of numbers above their knower-level, whereas 2-knowers do. As researchers move to administering assessments remotely, the Point-to-X task promises to be an easy-to-administer alternative to Give-N for measuring children’s emerging number knowledge and capturing nuances in children’s number-word knowledge that Give-N may miss. 

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4. The development of early numeracy in deaf and hard of hearing children acquiring spoken language

   https://doi.org/10.1111/cdev.13793  

   Shusterman, Anna; Peretz-Lange, Rebecca; Berkowitz, Talia; Carrigan, Emily (August 2022, Child Development)

   Abstract Most deaf and hard-of-hearing (DHH) children are born to hearing parents and steered toward spoken rather than signed language, introducing a delay in language access. This study investigated the effects of this delay on number acquisition. DHH children (N = 44, meanage = 58 months, 21F, >50% White) and typically-hearing (TH) children (N = 79, meanage = 49 months, 51F, >50% White) were assessed on number and language in 2011–13. DHH children showed similar trajectories to TH children but delayed timing; a binary logistic regression showed that the odds of being a cardinal-principle (CP) knower were 17 times higher for TH children than DHH children, controlling for age (d = .69). Language fully mediated the association between deaf/hearing group and number knowledge, suggesting that language access sets the pace for number acquisition. 

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5. The logic of cardinality comparison without the axiom of choice

   https://doi.org/10.1016/j.apal.2024.103549  

   Harrison-Trainor, Matthew; Kulshreshtha, Dhruv (April 2025, Annals of Pure and Applied Logic)

   We work in the setting of Zermelo-Fraenkel set theory without assuming the Axiom of Choice. We consider sets with the Boolean operations together with the additional structure of comparing cardinality (in the Cantorian sense of injections). What principles does one need to add to the laws of Boolean algebra to reason not only about intersection, union, and complementation of sets, but also about the relative size of sets? We give a complete axiomatization. A particularly interesting case is when one restricts to the Dedekind-finite sets. In this case, one needs exactly the same principles as for reasoning about imprecise probability comparisons, the central principle being Generalized Finite Cancellation (which includes, as a special case, division-by-$$m$$). In the general case, the central principle is a restricted version of Generalized Finite Cancellation within Archimedean classes which we call Covered Generalized Finite Cancellation. 

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