Our understanding of proportions can be both symbolic, as when doing calculations in school mathematics, or intuitive, as when folding a bed sheet in half. While an understanding of symbolic proportions is crucial for school mathematics, the cognitive foundations of this ability remain unclear. Here we implemented a computerized training game to test a causal link from intuitive (nonsymbolic) to symbolic proportional reasoning and other math abilities in 4th grade children. An experimental group was trained in nonsymbolic proportional reasoning (PR) with continuous extents, and an active control group was trained on a remarkably similar nonsymbolic magnitude comparison. We found that the experimental group improved at nonsymbolic PR across training sessions, showed near transfer to a paper-and-pencil nonsymbolic PR test, transfer to symbolic proportions, and far transfer to geometry. The active control group showed only a predicted far transfer to geometry. In a second experiment, these results were replicated with an independent cohort of children. Overall this study extends previous correlational evidence, suggesting a functional link between nonsymbolic PR on one hand and symbolic PR and geometry on the other.

中文翻译：

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训练儿童的非符号比例推理及其对他们符号数学能力的影响。

我们对比例的理解既可以是象征性的（如在学校数学中进行计算时），也可以是直观的（如将床单对半折叠时）。尽管对符号比例的理解对于学校数学至关重要，但这种能力的认知基础仍然不清楚。在这里，我们实施了一个计算机化的训练游戏，以测试4年级儿童从直觉（非符号）到符号比例推理以及其他数学能力的因果关系。实验组接受了连续范围的非符号比例推理（PR）训练，而活动对照组接受了非常相似的非符号幅度比较培训。我们发现实验组在整个训练过程中的非符号PR测试方面有所改善，显示几乎转移到纸笔式非符号PR测试中，转换为符号比例，而远转换为几何。主动对照组仅显示出预测的远距离转移。在第二个实验中，将这些结果复制到一个独立的儿童队列中。总体而言，这项研究扩展了先前的相关证据，表明一方面非符号PR与另一方面符号PR和几何之间存在功能联系。