Number sense and arithmetic fluency are fundamental to early mathematical development. However, these capacities generally fail to predict mathematical achievement in older adolescents and adults. We propose that later mathematical development is driven by coming to understand the higher-order principles that bring structure to mathematics. To evaluate this proposal, we tested whether college students ( = 134) apply arithmetic principles – inverse, associativity, and commutativity – to efficiently verify arithmetic sentences mixing multiplication and division operations such as 18 × 7 ÷ 3 = 42. This was the case. People were more accurate and faster when verifying arithmetic sentences that could be simplified by the application of arithmetic principles compared to control problems. People found problems that required the associativity principle to be more difficult (i.e., they made more errors and took longer) than those that required the inverse principle, and problems that additionally required the commutativity principle to be more difficult still. Converging evidence for the use of these principles came from their strategy self-reports. Critically, individual differences in applying these principles predicted mathematical achievement even after controlling for number sense, arithmetic fluency, and verbal achievement. These findings have implications for theories of mathematical development and may point the way to future interventions for increasing the mathematical achievement of younger children.

中文翻译：

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算术原理的应用预测大学生的数学成绩

数感和算术流畅性是早期数学发展的基础。然而，这些能力通常无法预测年龄较大的青少年和成年人的数学成绩。我们认为，后来的数学发展是通过理解为数学带来结构的高阶原理来驱动的。为了评估这个提议，我们测试了大学生（= 134）是否应用算术原理（逆、结合性和交换性）来有效验证混合乘法和除法运算的算术句子，例如 18 × 7 ÷ 3 = 42。情况确实如此。与控制问题相比，人们在验证可以通过应用算术原理简化的算术语句时更加准确和更快。人们发现需要结合性原理的问题比需要逆性原理的问题更困难（即，他们犯的错误更多，花费的时间更长），而另外需要交换性原理的问题则更加困难。使用这些原则的证据来自他们的战略自我报告。至关重要的是，即使在控制了数感、算术流利度和语言成绩之后，应用这些原则的个体差异也可以预测数学成绩。这些发现对数学发展理论具有重要意义，并可能为未来提高幼儿数学成绩的干预措施指明道路。