Newcombe Nora S, Levine Susan C, Mix Kelly S
Department of Psychology, Temple University, Philadelphia, PA, USA.
Department of Psychology, University of Chicago, Chicago, IL, USA.
Wiley Interdiscip Rev Cogn Sci. 2015 Nov-Dec;6(6):491-505. doi: 10.1002/wcs.1369. Epub 2015 Sep 29.
There are many continuous quantitative dimensions in the physical world. Philosophical, psychological, and neural work has focused mostly on space and number. However, there are other important continuous dimensions (e.g., time and mass). Moreover, space can be broken down into more specific dimensions (e.g., length, area, and density) and number can be conceptualized discretely or continuously (i.e., natural vs real numbers). Variation on these quantitative dimensions is typically correlated, e.g., larger objects often weigh more than smaller ones. Number is a distinctive continuous dimension because the natural numbers (i.e., positive integers) are used to quantify collections of discrete objects. This aspect of number is emphasized by teaching of the count word sequence and arithmetic during the early school years. We review research on spatial and numerical estimation, and argue that a generalized magnitude system is the starting point for development in both domains. Development occurs along several lines: (1) changes in capacity, durability, and precision, (2) differentiation of the generalized magnitude system into separable dimensions, (3) formation of a discrete number system, i.e., the positive integers, (4) mapping the positive integers onto the continuous number line, and (5) acquiring abstract knowledge of the relations between pairs of systems. We discuss implications of this approach for teaching various topics in mathematics, including scaling, measurement, proportional reasoning, and fractions.
物理世界中存在许多连续的定量维度。哲学、心理学和神经学方面的研究主要集中在空间和数字上。然而，还存在其他重要的连续维度（例如时间和质量）。此外，空间可以细分为更具体的维度（例如长度、面积和密度），数字可以离散地或连续地概念化（即自然数与实数）。这些定量维度上的变化通常是相关的，例如，较大的物体通常比较小的物体更重。数字是一个独特的连续维度，因为自然数（即正整数）用于量化离散物体的集合。在小学早期，通过数词序列和算术教学强调了数字的这一方面。我们回顾了关于空间和数字估计的研究，并认为广义数量系统是这两个领域发展的起点。发展沿着几条路线进行：（1）能力、持久性和精度的变化，（2）广义数量系统分化为可分离的维度，（3）离散数字系统即正整数的形成，（4）将正整数映射到连续数轴上，以及（5）获得关于成对系统之间关系的抽象知识。我们讨论了这种方法对数学各个主题教学的影响，包括缩放、测量、比例推理和分数。
