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1. Testing a Unified Model of Arithmetic

   Braithwaite, David W.; Siegler, Robert S. (January 2022, Proceedings of the Annual Conference of the Cognitive Science Society)

   We describe UMA (Unified Model of Arithmetic), a theory of children’s arithmetic implemented as a computational model. UMA extends a theory of fraction arithmetic (Braithwaite et al., 2017) to include arithmetic with whole numbers and decimals. We evaluated UMA in the domain of decimal arithmetic by training the model on problems from a math textbook series, then testing it on decimal arithmetic problems that were solved by 6th and 8th graders in a previous study. UMA’s test performance closely matched that of children, supporting three assumptions of the theory: (1) most errors reflect small deviations from standard procedures, (2) between-problem variations in error rates reflect the distribution of input that learners receive, and (3) individual differences in strategy use reflect underlying variation in learning parameters. 

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2. Affordances of Fractions and Decimals for Arithmetic

   Braithwaite, David W.; Liu, Qiushan (January 2022, Journal of experimental psychology)

   Rational numbers are represented by multiple notations: fractions, decimals, and percentages. Whereas previous studies have investigated affordances of these notations for representing different types of information (DeWolf, Bassok, & Holyoak, 2015; Tian, Braithwaite, & Siegler, 2020), the present study investigated their affordances for solving different types of arithmetic problems. We hypothesized that decimals afford addition better than fractions do and that fractions afford multiplication better than decimals do. This hypothesis was tested in two experiments with university students (Ns = 77 and 80). When solving fraction and decimal arithmetic problems, participants converted addition problems from fraction to decimal form more than vice versa, and converted multiplication problems from decimal to fraction form more than vice versa, thus revealing preferences favoring decimals for addition and fractions for multiplication. Accuracies paralleled these revealed preferences: Addition accuracy was higher with decimals than fractions, whereas multiplication accuracy was higher with fractions than decimals. Variations in notation preferences as a function of the types of operands involved (e.g., equal versus unequal denominator fractions) were more consistent with an explanation based on adaptive strategy choice (Siegler, 1996) than with one based on semantic interpretations associated with each notation. 

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3. UltraPrecise: A GPU-Based Framework for Arbitrary-Precision Arithmetic in Database Systems

   https://doi.org/10.1109/ICDE60146.2024.00294  

   Li, Xin; Xiao, Mengbai; Yu, Dongxiao; Lee, Rubao; Zhang, Xiaodong (May 2024, IEEE)

   Fixed-point decimal operations in databases with arbitrary-precision arithmetic refer to the ability to store and operate decimal fraction numbers with an arbitrary length of digits. This type of operation has become a requirement for many applications, including scientific databases, financial data processing, geometric data processing, and cryptography. However, the state-of-the-art fixed-point decimal technology either provides high performance for low-precision operations or supports arbitrary-precision arithmetic operations at low performance. In this paper, we present a design and implementation of a framework called UltraPrecise which supports arbitraryprecision arithmetic for databases on GPU, aiming to gain high performance for arbitrary-precision arithmetic operations. We build our framework based on the just-in-time compilation technique and optimize its performance via data representation design, PTX acceleration, and expression scheduling. UltraPrecise achieves comparable performance to other high-performance databases for low-precision arithmetic operations. For highprecision, we show that UltraPrecise consistently outperforms existing databases by two orders of magnitude, including workloads of RSA encryption and trigonometric function approximation. 

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4. Conceptual Knowledge, Procedural Knowledge, and Metacognition in Routine and Nonroutine Problem Solving

   https://doi.org/10.1111/cogs.13048  

   Braithwaite, David W.; Sprague, Lauren (October 2021, Cognitive Science)

   Abstract When, how, and why students use conceptual knowledge during math problem solving is not well understood. We propose that when solving routine problems, students are more likely to recruit conceptual knowledge if their procedural knowledge is weak than if it is strong, and that in this context, metacognitive processes, specifically feelings of doubt, mediate interactions between procedural and conceptual knowledge. To test these hypotheses, in two studies (Ns = 64 and 138), university students solved fraction and decimal arithmetic problems while thinking aloud; verbal protocols and written work were coded for overt uses of conceptual knowledge and displays of doubt. Consistent with the hypotheses, use of conceptual knowledge during calculation was not significantly positively associated with accuracy, but was positively associated with displays of doubt, which were negatively associated with accuracy. In Study 1, participants also explained solutions to rational arithmetic problems; using conceptual knowledge in this context was positively correlated with calculation accuracy, but only among participants who did not use conceptual knowledge during calculation, suggesting that the correlation did not reflect “online” effects of using conceptual knowledge. In Study 2, participants also completed a nonroutine problem‐solving task; displays of doubt on this task were positively associated with accuracy, suggesting that metacognitive processes play different roles when solving routine and nonroutine problems. We discuss implications of the results regarding interactions between procedural knowledge, conceptual knowledge, and metacognitive processes in math problem solving. 

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5. A Decidable Logic for Tree Data-Structures with Measurements

   https://doi.org/10.1007/978-3-030-11245-5_15  

   Qiu, Xiaokang; Wang, Yanjun (January 2019, 20th International Conference on Verification, Model Checking, and Abstract Interpretation)

   We present DRYADdec, a decidable logic that allows reasoning about tree data-structures with measurements. This logic supports user-defined recursive measure functions based on Max or Sum, and recursive predicates based on these measure functions, such as AVL trees or red-black trees. We prove that the logic’s satisfiability is decidable. The crux of the decidability proof is a small model property which allows us to reduce the satisfiability of DRYADdec to quantifier-free linear arithmetic theory which can be solved efficiently using SMT solvers. We also show that DRYADdec can encode a variety of verification and synthesis problems, including natural proof verification conditions for functional correctness of recursive tree-manipulating programs, legality conditions for fusing tree traversals, synthesis conditions for conditional linear-integer arithmetic functions. We developed the decision procedure and successfully solved 220+ DRYADdec formulae raised from these application scenarios, including verifying functional correctness of programs manipulating AVL trees, red-black trees and treaps, checking the fusibility of height-based mutually recursive tree traversals, and counterexample-guided synthesis from linear integer arithmetic specifications. To our knowledge, DRYADdec is the first decidable logic that can solve such a wide variety of problems requiring flexible combination of measure-related, data-related and shape-related properties for trees. 

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