The Role of Conceptual Entities and Their Symbols in Building Advanced Mathematical Concepts
The Role of Conceptual Entities and Their Symbols in Building Advanced Mathematical Concepts
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概念实体及其符号在构建高级数学概念中的作用
DOI：
10.1007/0-306-47203-1_6
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发表时间：
2002
影响因子：
2.8
通讯作者：
J. Kaput
中科院分区：
文献类型：
--
作者：
G. Harel;J. Kaput
数学思维是利用心理对象进行的。例如，假设有人问向量空间 V 及其双对偶 V** 是否同构。在一个层面上，人们询问的是“对象”V和V**，并且，为了开始描述同构，人们可以继续描述两个空间中各自向量之间的对应关系，这再次在心理上被视为对象，尽管它们可能是例如n元组或矩阵。类似地，人们可能需要定义两个功能空间之间的映射，其中映射的域和范围的元素必须在认知上被视为对象，而不是映射本身，映射本身可以被视为具有输入和输出的过程。在另一种情况下，人们可能需要将 MacLane (1971) 意义上的通用构造重新解释为伴随函子对，其中具有特定属性的唯一映射的存在实际上定义了函子之间的自然变换，因此映射必须扮演自然变换作用的对象的角色。这些经验在各个级别的数学中都很常见，但它们在高级数学思维中也广泛存在。本章的目的是开始讨论它们以及它们在帮助我们构建更加复杂的数学概念方面的作用。概念实体形成的思想是皮亚杰（Piaget，1977）在区分形式和内容时提出的。最近，一些研究人员已经认识到它在数学学习中的价值。它被称为封装(Ayers, Davis, Dubinsky & Lewin, 1988)、具体化(Sfard, 1989)、整合操作(Steffe & Cobb, 1988)，例如，这个过程是反思性抽象的一个实例(Beth & Piaget, 1966)，其中“物理或心理行为在更高的思维层面上被重建和重组，因此被理解为 认识者”（第 247 页）。 Greeno (1983) 将概念实体定义为认知对象，心理系统具有可以将该对象作为参数、输入的程序。他将认知对象与依附于对象或作用于对象的属性、操作和关系区分开来。此外，他认为要成为对象，它们必须在个人的心理表征中永久可用（第 277 页）。将功能构建为概念实体是实体化过程的一个例子（Thompson，1985a；Harel，1985；Ayers 等，1988）。理解函数概念的一个层次是将函数视为将域中的元素与范围中的元素关联起来的过程。这种理解水平可能足以处理某些情况，例如逐点解释函数图或求解 f (x)= b 形式的方程中的 x，但不足以有意义地处理涉及函数上的某些运算符（例如积分和微分运算符）的情况，正如我们将在本章后面看到的那样。对于后一种情况，函数的三个组成部分——规则、域和范围——必须封装到一个概念实体中，以便
Mathematical thinking is carried out using mental objects. For example, suppose one asks if a vector space Vand its double dual V** are isomorphic. At one level, one is asking about the “objects” Vand V** and, to begin describing an isomorphism, one may go on to describe a correspondence between respective vectors in the two spaces, which again, are treated mentally as objects, although they might be n-tuples or matrices, for example. Similarly, one may need to define a mapping between two function spaces, where the elements of the domain and range of the mapping must be treated cognitively as objects, as opposed to the mapping itself, which may be treated as a process, with inputs and outputs. In yet another instance, one may need to reinterpret a universal construction in the sense of MacLane (1971) as an adjoint functor pair, where the existence of a unique mapping with a certain property in fact defines a natural transformation between functors–so the mapping must play the role of an object on which the natural transformation acts. Such experiences are quite common in mathematics at all levels, but they feature widely throughout advanced mathematical thinking. The aim of this chapter is to begin to discuss them and their roles in helping us to build ever more complex mathematical concepts. The idea of conceptual entities formation was suggested by Piaget (1977) in his distinction between form and content. Recently, several researchers have recognized its value in the learning of mathematics. It has been called encapsulation (Ayers, Davis, Dubinsky & Lewin, 1988), reification (Sfard, 1989), integration operation (Steffe & Cobb, 1988), for example, this process is an instance of reflective abstraction (Beth & Piaget, 1966), in which “a physical or mental action is reconstructed and reorganized on a higher plane of thought and so comes to be understood be the knower”(p. 247). Greeno (1983) defines a conceptual entity as a cognitive object for which the mental system has procedures that can take that object as an argument, as an input. He distinguishes cognitive objects from attributes, operations and relations, which attach to or act on objects. Further, he suggests that to qualify as objects, they must be permanently available in the individual’s mental representation (p. 277).The construction of function as a conceptual entity is an example of the entitication process (Thompson, 1985a; Harel, 1985; Ayers et al, 1988). One level of understanding the concept of function is to think of a function as a process associating elements in a domain with elements in a range. This level of understanding may be sufficient to deal with certain situations, such as interpreting graphs of functions point-wise or solving for x in an equation of the form f (x)= b, but it would notbe sufficient to deal meaningfully with situations which involve certain operators on functions, such as the integral and differential operators, as we will see later in this chapter. For the latter situations, the three components of function–the rule, the domain, and the range–must be encapsulated into a single conceptual entity so