This article describes UMA (Unified Model of Arithmetic), a theory of children's arithmetic implemented as a computational model. UMA builds on FARRA (Fraction Arithmetic Reflects Rules and Associations; Braithwaite et al., 2017), a model of children's fraction arithmetic. Whereas FARRA-like all previous models of arithmetic-focused on arithmetic with only one type of number, UMA simulates arithmetic with whole numbers, fractions, and decimals. The model was trained on arithmetic problems from the first to sixth grade volumes of a math textbook series; its performance on tests administered at the end of each grade was compared to the performance of children in prior empirical research. In whole number arithmetic (Study 1), fraction arithmetic (Study 2), and decimal arithmetic (Study 3), UMA displayed types of errors, effects of problem features on error rates, and individual differences in strategy use that resembled those documented in the previous studies of children. Further, UMA generated correlations between individual differences in basic and advanced arithmetic skills similar to those observed in longitudinal studies of arithmetic development (Study 4). The results support UMA's main theoretical assumptions regarding arithmetic development: (a) most errors reflect small deviations from standard procedures via two mechanisms, overgeneralization and omission; (b) between-problem variations in error rates reflect effects of intrinsic difficulty and differential amounts of practice; and (c) individual differences in strategy use reflect underlying variation in parameters governing learning and decision making. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

中文翻译：

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包含整数、分数和小数的统一算术模型。

本文介绍了 UMA（统一算术模型），这是一种以计算模型形式实现的儿童算术理论。UMA 建立在 FARRA（分数算术反映规则和关联；Braithwaite 等人，2017）的基础上，FARRA 是一种儿童分数算术模型。与之前所有算术模型一样，FARRA 只关注一种类型的算术，而 UMA 则模拟整数、分数和小数的算术。该模型接受了数学教科书系列一年级到六年级的算术问题的训练；将其在每个年级结束时进行的测试中的表现与之前的实证研究中儿童的表现进行比较。在整数算术（研究 1）、分数算术（研究 2）和小数算术（研究 3）中，UMA 显示了错误类型、问题特征对错误率的影响以及策略使用中的个体差异，类似于以前对儿童的研究。此外，UMA 生成了基础和高级算术技能个体差异之间的相关性，类似于算术发展纵向研究中观察到的情况（研究 4）。结果支持 UMA 关于算术开发的主要理论假设：（a）大多数错误反映了通过两种机制（过度概括和遗漏）与标准程序的微小偏差；(b) 问题间错误率的变化反映了内在难度和练习量差异的影响；(c) 策略使用的个体差异反映了管理学习和决策的参数的潜在差异。（PsycInfo 数据库记录 (c) 2023 APA，保留所有权利）。