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1. Is Nonsymbolic Arithmetic Truly “Arithmetic”? Examining the Computational Capacity of the Approximate Number System in Young Children

   https://doi.org/10.1111/cogs.13299  

   Cheng, Chen; Kibbe, Melissa M. (June 2023, Cognitive Science)

   Abstract Young children with limited knowledge of formal mathematics can intuitively perform basic arithmetic‐like operations over nonsymbolic, approximate representations of quantity. However, the algorithmic rules that guide such nonsymbolic operations are not entirely clear. We asked whether nonsymbolic arithmetic operations have a function‐like structure, like symbolic arithmetic. Children (n =74 4‐ to ‐8‐year‐olds in Experiment 1;n =52 7‐ to 8‐year‐olds in Experiment 2) first solved two nonsymbolic arithmetic problems. We then showed children two unequal sets of objects, and asked children which of the two derived solutions should be added to the smaller of the two sets to make them “about the same.” We hypothesized that, if nonsymbolic arithmetic follows similar function rules to symbolic arithmetic, then children should be able to use the solutions of nonsymbolic computations as inputs into another nonsymbolic problem. Contrary to this hypothesis, we found that children were unable to reliably do so, suggesting that these solutions may not operate as independent representations that can be used inputs into other nonsymbolic computations. These results suggest that nonsymbolic and symbolic arithmetic computations are algorithmically distinct, which may limit the extent to which children can leverage nonsymbolic arithmetic intuitions to acquire formal mathematics knowledge. 

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2. Random Variate Generation with Formal Guarantees

   https://doi.org/10.1145/3729251  

   Saad, Feras A; Lee, Wonyeol (June 2025, Proceedings of the ACM on Programming Languages)

   Generating random variates is a fundamental operation in diverse areas of computer science and is supported in almost all modern programming languages. Traditional software libraries for random variate generation are grounded in the idealized Real-RAM model of computation, where algorithms are assumed to be able to access uniformly distributed real numbers from the unit interval and compute with infinite-precision real arithmetic. These assumptions are unrealistic, as any software implementation of a Real-RAM algorithm on a physical computer can instead access a stream of individual random bits and computes with finite-precision arithmetic. As a result, existing libraries have few theoretical guarantees in practice. For example, the actual distribution of a random variate generator is generally unknown, intractable to quantify, and arbitrarily different from the desired distribution; causing runtime errors, unexpected behavior, and inconsistent APIs. This article introduces a new approach to principled and practical random variate generation with formal guarantees. The key idea is to first specify the desired probability distribution in terms of a finite-precision numerical program that defines its cumulative distribution function (CDF), and then generate exact random variates according to this CDF. We present a universal and fully automated method to synthesize exact random variate generators given any numerical CDF implemented in any binary number format, such as floating-point, fixed-point, and posits. The method is guaranteed to operate with the same precision used to specify the CDF, does not overflow, avoids expensive arbitrary-precision arithmetic, and exposes a consistent API. The method rests on a novel space-time optimal implementation for the class of generators that attain the information-theoretically optimal Knuth and Yao entropy rate, consuming the least possible number of input random bits per output variate. We develop a random variate generation library using our method in C and evaluate it on a diverse set of continuous and discrete distributions, showing competitive runtime with the state-of-the-art GNU Scientific Library while delivering higher accuracy, entropy efficiency, and automation. 

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3. Early numeracy and mathematics development: A longitudinal meta-analysis on the predictive nature of early numeracy.

   https://doi.org/10.1037/edu0000925  

   Liu, Yuting; Peng, Peng; Yan, Xueye (January 2025, Journal of Educational Psychology)

   In this meta-analysis of 54 longitudinal studies with over 58,000 students in kindergarten through 12th grade, we examined the predictive nature of early numeracy measured at or before the first year of formal schooling in relation to later mathematics. Results showed that early numeracy significantly predicted mathematics measured after 6 months or later, r = .49, 95% confidence interval [0.47, 0.52]. After controlling for all moderators in a model, results indicated that (a) different early numeracy including numbering, relations, and arithmetic operations did not differ much in their predictions of different later mathematics; (b) early numeracy as a whole was more predictive of later advanced mathematics skills (word problems) than of later foundational mathematics skills (calculations and fact fluency); (c) early numeracy’s prediction of later mathematics was stronger with longer prediction intervals; and (d) the earlier early numeracy was assessed, the stronger its prediction of later mathematics. Together, these findings suggest that early numeracy may be a unitary construct. Early numeracy does not merely serve as a steppingstone with temporary effects on foundational mathematics; instead, it likely triggers a snowballing effect, cumulatively influencing mathematics development over time. 

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4. Scaled Population Division for Approximate Computing

   https://doi.org/10.1109/ISLPED58423.2023.10244709  

   Bharathi, Kunal; Khatri, Sunil P.; Hu, Jiang (August 2023, Proceedings International Symposium on Low Power Electronics and Design)

   In this paper we present an approximate division scheme for Scaled Population (SP) arithmetic, a technique that improves on the limitations of stochastic computing (SC). SP arithmetic circuits are designed (a) to perform all operations with a constant delay, and (b) they use scaling operations to help reduce errors compared to SC circuits. As part of this work, we also present a method to correlate two SP numbers with a constant delay. We compare our SP divider with SC dividers, as well as fixed-point dividers (in terms of area, power and delay). Our 512-bit SP divider has a delay (power) that is 0.08× (0.06×) that of the equivalent fixed-point binary divider. Compared to a equivalent SC divider, our power-delay-product is 13× better. Index Terms—Approximate Arithmetic, Stochastic Computing, Computer Arithmetic, Approximate Division, Fast Division 

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5. Scaled Population Division for Approximate Computing

   Kunal Bharathi (September 2023, Proceedings International Symposium on Low Power Electronics and Design)

   IEEE (Ed.)

   In this paper we present an approximate division scheme for Scaled Population (SP) arithmetic, a technique that improves on the limitations of stochastic computing (SC). SP arithmetic circuits are designed (a) to perform all operations with a constant delay, and (b) they use scaling operations to help reduce errors compared to SC circuits. As part of this work, we also present a method to correlate two SP numbers with a constant delay. We compare our SP divider with SC dividers, as well as fixed-point dividers (in terms of area, power and delay). Our 512-bit SP divider has a delay (power) that is 0.08× (0.06×) that of the equivalent fixed-point binary divider. Compared to a equivalent SC divider, our power-delay-product is 13× better. Index Terms—Approximate Arithmetic, Stochastic Computing, Computer Arithmetic, Approximate Division, Fast Division 

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