Department of Educational Psychology, University of Minnesota, 165 Education Sciences Building, 56 East River Rd., Minneapolis, MN 55455, USA.
Cognition. 2011 Dec;121(3):363-85. doi: 10.1016/j.cognition.2011.08.005. Epub 2011 Sep 21.
Mathematics has a level of structure that transcends untutored intuition. What is the cognitive representation of abstract mathematical concepts that makes them meaningful? We consider this question in the context of the integers, which extend the natural numbers with zero and negative numbers. Participants made greater and lesser judgments of pairs of integers. Experiment 1 demonstrated an inverse distance effect: When comparing numbers across the zero boundary, people are faster when the numbers are near together (e.g., -1 vs. 2) than when they are far apart (e.g., -1 vs. 7). This result conflicts with a straightforward symbolic or analog magnitude representation of integers. We therefore propose an analog-x hypothesis: Mastering a new symbol system restructures the existing magnitude representation to encode its unique properties. We instantiate analog-x in a reflection model: The mental negative number line is a reflection of the positive number line. Experiment 2 replicated the inverse distance effect and corroborated the model. Experiment 3 confirmed a developmental prediction: Children, who have yet to restructure their magnitude representation to include negative magnitudes, use rules to compare negative numbers. Taken together, the experiments suggest an abstract-to-concrete shift: Symbolic manipulation can transform an existing magnitude representation so that it incorporates additional perceptual-motor structure, in this case symmetry about a boundary. We conclude with a second symbolic-magnitude model that instantiates analog-x using a feature-based representation, and that begins to explain the restructuring process.
数学具有超越未经训练的直觉的结构层次。是什么样的认知表现形式使抽象的数学概念变得有意义？我们在整数的背景下考虑这个问题，整数是在自然数的基础上加上零和负数得到的。参与者对整数对进行了更大或更小的判断。实验 1 证明了逆距离效应：当在零边界上比较数字时，当数字彼此靠近（例如，-1 与 2）时，人们的反应速度会更快，而当它们相距较远（例如，-1 与 7）时，反应速度会更慢。这一结果与整数的直接符号或模拟大小表示法相冲突。因此，我们提出了模拟-x 假设：掌握新的符号系统会重新构建现有的大小表示法，以编码其独特的属性。我们在反射模型中实例化了模拟-x：心理上的负数轴是正数轴的反射。实验 2 复制了逆距离效应并证实了该模型。实验 3 证实了一个发展预测：尚未重新构建其大小表示法以包括负大小的儿童会使用规则来比较负数。综合来看，这些实验表明了一种从抽象到具体的转变：符号操作可以改变现有的大小表示法，使其包含额外的感知运动结构，在这种情况下是关于边界的对称。我们以第二个符号-大小模型结束，该模型使用基于特征的表示实例化了模拟-x，并开始解释重新构建过程。
