Subitizing: The role of figural patterns in the development of numerical concepts.
Subitizing: The role of figural patterns in the development of numerical concepts.
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子化：图形模式在数字概念发展中的作用。
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发表时间：
1982
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通讯作者：
E. Glasersfeld
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作者：
E. Glasersfeld
每当我们听到一个数字，我们知道，正如欧几里得已经知道的，它指的是一个概念单位，它本身是由单位组成的。在正常的日常情况下，组件单元没有问题。我们知道它们是什么，因为演讲者告诉我们，或者我们可以看到它们在我们面前，或者我们可以从某些特定的实验情境中回忆起它们。也就是说，在一般情况下，数字词是用来指称实际的或表象的（想象的）知觉项目的。然而，在数学中，情况并非如此，即使像在本文中那样，我们只考虑整数，而不考虑数学家们称之为“数”的所有更奇特的项目。因此，当孩子们学习算术时，他们被期望从我们通常使用数字词的感性情境中“抽象”出数字词的意义。例如，他们被要求弄清楚7 + 5是什么，尽管一开始他们可能会得到有形珠子、跳棋或饼干等形式的拐杖，但他们最终应该在没有任何感性或具象材料帮助的情况下解决这类问题。即使许多孩子都实现了这一期望，但有些孩子实现得很慢，有些孩子根本没有实现，这一事实提出了一个问题，即人们如何才能更准确地规定必须做什么。我们研究小组早些时候发表的两篇论文探讨了这个问题的某些方面。第一篇文章对计数类型进行了分析，阐明了计数抽象单位项目的能力的发展，这些项目可能来自于感觉运动材料，但不再依赖于感觉运动材料（Steffe, Richards, & von Glasersfeld, 1979）；第二篇文章提出了抽象概念结构的理论模型，称为“单位”和“数”（von Glasersfeld, 1981 a）。在接下来的几页中，我将重点关注“subsubtizing”现象，这一现象早已为人所知，但通常被视为一种充其量的奇怪现象
Whenever we hear a number word, we know, as Euclid already knew, that it refers to a conceptual unit that is itself composed of units. Under normal, everyday circumstances there is no problem about the component units. We know what they are, because we are told by the speaker, or we can see them in front of us, or we can recall them from some specific experiental situation. That is to say, under ordinary circumstances number words are used with reference to actual or represented (imagined) perceptual items. In mathematics, however, that is not the case, even if, as in this paper, we consider nothing but whole numbers and leave aside all the fancier items which mathematicians have come to call “numbers”. Thus, when children are taught arithmetic they are expected to “abstract” the meaning of number words from the perceptual situations in which we ordinarily use them. They are asked to figure out what, for instance, 7 + 5 is, and – though at first they may he given a crutch in the form of tangible beads, checkers, or cookies – they are supposed to solve that kind of problem eventually without the help of any perceptual or representational material. Even if that expectation is fulfilled by many children, the fact that some fulfill it slowly and others not at all, raises the question of how one could specify more precisely what it is that has to be done. Two earlier papers from our research group dealt with some aspects of that question. The first presented an analysis of counting types that explicated the development of the ability to count abstract unit items that may have been derived from, but are no longer dependent on, sensorimotor material (Steffe, Richards, & von Glasersfeld, 1979), the second a theoretical model for the abstract conceptual structures called “unit” and “number” (von Glasersfeld, 1981 a). In the pages that follow I shall focus on the phenomenon of “subitizing” which has been known a long time but was usually treated as an oddity that is at best