注：本文为 “数学思维过程对象封装” 相关译文。
英文引文，机翻未校。
略作重排，如有内容异常，请看原文。

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What is the object of the encapsulation of a process?

过程封装的对象是什么？

David Tall#, Michael Thomas+, Gary Davis*, Eddie Gray, Adrian Simpson
戴维·托尔、迈克尔·托马斯+、加里·戴维斯*、埃迪·格雷、阿德里安·辛普森

University of Warwick, UK, +University of Auckland, New Zealand, *University of Southampton, UK
英国华威大学、+新西兰奥克兰大学、*英国南安普顿大学

Dedicated to the memory of Robert B. Davis †

谨以此文纪念罗伯特·B·戴维斯（Robert B. Davis）†
 
The reader will notice that the work of Bob Davis features prominently in this article, as it must in any discussion of the encapsulation of a process. On completion of this paper we learned of Bob’s death. His passing is a great loss to us all. He was one of the most intellectually vigorous mathematics educators anywhere. His constant striving for new theoretical positions was a positive stimulus to the community, and a necessary call to develop greater depth and maturity in our thought. Vale, Bob!
读者会发现，鲍勃·戴维斯（Bob Davis）的研究在本文中占据重要地位——事实上，任何关于过程封装的讨论都离不开他的成果。本文完稿时，我们获悉鲍勃已逝世。他的离世对我们所有人都是巨大的损失。他是全球范围内最具学术活力的数学教育家之一。他不断探索新理论视角的精神，为学界提供了积极的推动力，也促使我们不断深化思想、提升思维成熟度。致敬，鲍勃！
 
To appear in Journal of Mathematical Behavior
将发表于《数学行为期刊》（Journal of Mathematical Behavior）

Several theories have been proposed to describe the transition from process to object in mathematical thinking. Yet, what is the nature of this “object” produced by the “encapsulation” of a process? Here we outline the development of some of the theories (including Piaget, Dienes, Davis, Greeno, Dubinsky, Sfard, Gray & Tall) and consider the nature of the mental objects (apparently) produced through encapsulation, and their role in the wider development of mathematical thinking. Does the same developmental route occur in geometry as in arithmetic and algebra? Is the same development used in axiomatic mathematics? What is the role played by imagery?
已有多种理论被提出，用于描述数学思维中从过程到对象的转变。然而，由过程“封装”（encapsulation）产生的这一“对象”，其本质究竟是什么？本文将概述部分理论（包括皮亚杰、迪内斯、戴维斯、格里诺、杜宾斯基、斯法德、格雷与托尔的理论）的发展，并探讨通过封装（表面上）产生的心理对象的本质，以及它们在数学思维更广泛发展中的作用。几何学的发展路径是否与算术和代数相同？公理数学是否采用相同的发展模式？表象（imagery）在此过程中扮演何种角色？

Theories of encapsulation/reification

封装/物化理论

In this article we address the question: what are mathematical objects and how are they constructed by the individual? This does not have a universal answer. Rather we must ask: what is a mathematical object for a given person in a given mathematical context? We contend that as mathematical maturity develops so does the number of available mathematical objects, each intimately linked with (and related to) other mathematical objects. It is the purpose of this article to consider how these objects are constructed.
本文旨在探讨以下问题：什么是数学对象？个体如何构建数学对象？这个问题没有普适答案。相反，我们必须追问：在特定的数学情境下，对特定个体而言，数学对象是什么？我们认为，随着数学成熟度的提升，个体可运用的数学对象数量也会增加，且每个数学对象都与其他数学对象紧密关联（并相互依存）。本文的目的便是探讨这些数学对象的构建方式。

In recent years there has been great interest in the encapsulation (or reification) of a process into a mental object as a fundamental method of constructing mathematical objects. Piaget (1985, p. 49) focused on the idea of how “actions and operations become thematized objects of thought or assimilation”. Dienes (1960), following Piaget, used a grammatical metaphor to formulate how a predicate (or action) becomes the subject of a further predicate, which may in turn become the subject of another. Davis (1984) formulated the same basic idea a quarter of a century later:
近年来，将过程封装（或物化，reification）为心理对象，作为构建数学对象的基本方法，受到了广泛关注。皮亚杰（Piaget, 1985, p. 49）重点关注“动作与运算如何成为主题化的思维对象或同化对象”这一观点。迪内斯（Dienes, 1960）继承皮亚杰的思想，运用语法隐喻阐释了谓语（或动作）如何成为后续谓语的主语，而这一主语又可能成为另一个谓语的主语。25年后，戴维斯（Davis, 1984）提出了相同的核心观点：

When a procedure is first being learned, one experiences it almost one step at time; the overall patterns and continuity and flow of the entire activity are not perceived. But as the procedure is practiced, the procedure itself becomes an entity – it becomes a thing. It, itself, is an input or object of scrutiny. All of the full range of perception, analysis, pattern recognition and other information processing capabilities that can be used on any input data can be brought to bear on this particular procedure. Its similarities to some other procedure can be noted, and also its key points of difference. The procedure, formerly only a thing to be done – a verb – has now become an object of scrutiny and analysis; it is now, in this sense, a noun.
当个体初次学习某一程序时，通常是一步一步进行的；无法感知整个活动的整体模式、连续性与流程。但随着对程序的反复练习，程序本身会成为一个实体——即一个“事物”。它自身可作为输入项或被审视的对象。所有可用于处理输入数据的感知、分析、模式识别及其他信息处理能力，都可应用于这一特定程序。我们能发现它与其他程序的相似之处，也能识别其核心差异。此前，该程序只是一个“待执行的动作”（即动词），如今却成为被审视和分析的对象（即名词）。

(pp. 29–30.)

He formulated the notion of visually moderated sequence where each step is written down and prompts the next until the problem is solved. This becomes an integrated sequence when it is conceived as a whole and may be organized into sub-procedures. He also used the term “procedure” as a specific algorithm for implementing a “process” in an informationprocessing sense (Davis, 1983, p. 257).
他（戴维斯）提出了“视觉调控序列”（visually moderated sequence）的概念：该序列中，每一步都被记录下来并触发下一步，直至问题解决。当个体将其视为一个整体并可能将其拆分为子程序时，它便成为“整合序列”（integrated sequence）。从信息处理的角度出发，他还将“程序”（procedure）定义为执行“过程”（process）的特定算法（Davis, 1983, p. 257）。

At this time information processing was focusing on the way in which a procedure that can be used as an input to another procedure could be conceived as a “conceptual entity” (Greeno, 1983).
同期，信息处理领域的研究重点在于：当一个程序可作为另一个程序的输入时，该程序如何被视为“概念实体”（conceptual entity）（Greeno, 1983）。

The notion of the transformation of a process into an object took new impetus in the work of Dubinsky (1986, 1991) and Sfard (1988, 1989, 1991). Sfard hypothesized two approaches to concept development, one operational focusing on processes, the other structural, focusing on objects.
在杜宾斯基（Dubinsky, 1986, 1991）和斯法德（Sfard, 1988, 1989, 1991）的研究中，“过程向对象转化”的概念获得了新的推动力。斯法德提出，概念发展存在两种路径：一种是“操作型”（operational），聚焦于过程；另一种是“结构型”（structural），聚焦于对象。

A constant three-step pattern can be identified in the successive transitions from operational to structural conceptions: first there must be a process performed on the already familiar objects, then the idea of turning this process into a more compact, self-contained whole should emerge, and finally an ability to view this new entity as a permanent object in its own right must be acquired. These three components of concept development will be called interiorization, condensation, and reification, respectively.
从操作型概念向结构型概念的逐步转化中，存在一个固定的三步模式：首先，必须对已熟悉的对象执行某一过程；其次，需形成将该过程转化为更简洁、自包含整体的认知；最后，需具备将这一新实体视为独立永久对象的能力。概念发展的这三个阶段分别被称为“内化”（interiorization）、“浓缩”（condensation）和“物化”（reification）。
 
Condensation means a rather technical change of approach, which expresses itself in an ability to deal with a given process in terms of input/output without necessarily considering its component steps.
“浓缩”指一种具有技术性的方法转变，具体表现为：个体能够从输入/输出（input/output）的角度处理特定过程，而无需关注其组成步骤。
 
Reification is the next step: in the mind of the learner, it converts the already condensed process into an object-like entity. … The fact that a process has been interiorized and condensed into a compact, self-sustained entity, does not mean, by itself, that a person has acquired the ability to think about it in a structural way. Without reification, her or his approach will remain purely operational.
“物化”是下一步：在学习者的思维中，它将已浓缩的过程转化为类对象实体。……即便一个过程已被内化并浓缩为简洁、自维持的实体，这一事实本身并不意味着个体已具备从结构角度思考该过程的能力。若缺乏物化，个体的思维方式仍将停留在纯操作层面。

(Sfard, 1992, pp. 64–65)

Dubinsky (1986, 1991) and his colleagues (Cottrill et al, 1996) formulate the encapsulation of process into object as three stages of a four-part theory with the acronym APOS. A (step-by-step) action becomes conceptualized as a total process, is encapsulated as a mental object, to later become part of a mental schema.
杜宾斯基（Dubinsky, 1986, 1991）及其同事（Cottrill et al, 1996）将“过程封装为对象”定义为四阶段理论（缩写为 APOS）中的三个阶段。其中，（逐步的）“动作”（Action）被概念化为“过程”（Process），进而被封装为“对象”（Object），最终成为“图式”（Schema）的一部分。

The notions of action and process are characterized in a manner reminiscent of the notions of visually moderated sequence and integrated sequence of Davis:
“动作”与“过程”的定义，与戴维斯提出的“视觉调控序列”和“整合序列”具有相似性：

An action is any physical or mental transformation of objects to obtain other objects. It occurs as a reaction to stimuli which the individual perceives as external. It may be a single step response, such as a physical reflex, or an act of recalling some fact from memory. It may also be a multi-step response, by then it has the characteristic that at each step, the next step is triggered by what has come before, rather than by the individual’s conscious control of the transformation. … When the individual reflects upon an action, he or she may begin to establish conscious control over it. We would then say that the action is interiorized, and it becomes a process.(Cottrill, et al, 1996, p. 171 (our italics))
“动作”是指为获得其他对象而对现有对象进行的物理或心理转化。它是个体对外部刺激的反应，可能是单步反应（如生理反射），也可能是一种从记忆中提取事实的行为。它也可能是多步反应，其特点是：每一步的触发依赖于前一步的结果，而非个体对转化过程的有意识控制。……当个体对“动作”进行反思时，可能会开始有意识地控制它。此时，我们认为“动作”被内化，进而成为“过程”。

The action becomes a process when the individual can “describe or reflect upon all of the steps in the transformation without necessarily performing them.” A process becomes an object when “the individual becomes aware of the totality of the process, realizes that transformations can act on it, and is able to construct such transformations.”
当个体能够“描述或反思转化的所有步骤，而无需实际执行这些步骤”时，“动作”便成为“过程”。当“个体意识到过程的整体性，认识到可以对其施加转化，并能够构建此类转化”时，“过程”便成为“对象”。

The final part of the APOS structure occurs when “actions, processes and objects … are organized into structures, which we refer to as schemas.” When this has been achieved, it is also proposed that:
APOS 理论的最后一个阶段是：“动作、过程与对象……被组织成结构化的整体，即我们所说的图式（schema）”。该理论进一步提出，当这一阶段达成时：

“… an individual can reflect on a schema and act upon it. This results in the schema becoming a new object. Thus we now see that there are at least two ways of constructing objects — from processes and from schemas.”(Cottrill, et al, 1996, p. 172 (our italics))
“……个体能够反思图式并对其施加动作，这使得图式成为新的对象。由此可见，构建对象至少存在两种方式——从过程中构建，以及从图式中构建。”

By considering the developments of concepts in simple arithmetic of whole numbers, Gray & Tall (1994) reviewed how a lengthy procedure such as “count-all” (count one set, count another, put the sets together and count all) becomes compressed into a shorter procedure “count-on”—with variants such as “count-both”, “count-on-from-larger”). Other techniques are also developed, such as remembering “known facts” and “deriving facts” from a combination of number facts and counting. Following Davis (1983) they used the term “procedure” as a specific algorithm for implementing a process and highlighted how a number of different procedures were being used to carry out essentially the same process in increasingly sophisticated ways. They noted that a symbol such as $4+2$ occupies a pivotal role, as the process of addition (by a variety of procedures) and also as the concept of sum. Soon the cognitive structure grows to encompass the fact that $4+2$, $2+4$, $3+3$, $2$ times $3$, are all essentially the same mental object. They therefore proposed the following definitions:
通过研究整数初等算术中的概念发展，格雷与托尔（Gray & Tall, 1994）指出，“逐数计数”（count-all，即先数第一组物体、再数第二组物体、合并两组后总数数）等复杂程序，会逐渐简化为“接着数”（count-on）程序，还会衍生出“双向数”（count-both）、“从大数开始数”（count-on-from-larger）等变体。此外，个体还会发展出其他策略，如记忆“已知事实”（known facts），以及通过数字事实与计数结合“推导事实”（deriving facts）。他们继承戴维斯（Davis, 1983）的观点，将“程序”定义为执行过程的特定算法，并强调：多种不同的程序正以日益复杂的方式，用于执行本质相同的过程。他们发现，$4+2$ 这类符号具有核心作用——既代表加法过程（通过多种程序实现），也代表和的概念。随后，个体的认知结构会进一步发展，认识到 $4+2$、$2+4$、$3+3$、$2$ 乘 $3$ 本质上是同一个心理对象。据此，他们提出以下定义：

An elementary procept is the amalgam of three components: a process which produces a mathematical object, and a symbol which is used to represent either process or object.
基础概念过程体（elementary procept）是三个要素的融合：产生数学对象的过程、数学对象本身，以及用于同时表示过程或对象的符号。
 
A procept consists of a collection of elementary procepts which have the same object.
(Gray & Tall, 1994)
概念过程体（procept）是一组具有相同对象的基础概念过程体的集合。

The name “procept” arose because of the symbol’s dual role as process and concept. The notion of procept is present throughout a large portion of mathematics. Tall & Thomas (1991) had already noted that, for many children, an expression such as $2+3x$ may be conceived as a process which cannot be carried out until the value of $x$ is known – a reinterpretation of the notion of “lack of closure” discussed by earlier
“procept”一词的命名，源于符号兼具“过程”（process）与“概念”（concept）的双重角色。概念过程体的概念广泛存在于数学的多个领域。托尔与托马斯（Tall & Thomas, 1991）已发现，对许多儿童而言，$2+3x$ 这类表达式会被视为“未完成过程”——只有知道 $x$ 的值才能执行，这是对早期学者所讨论的“封闭性缺失”（lack of closure）概念的重新解读。

authors. This “lack of closure” is none other than the focus on the procedure of evaluation rather than on an algebraic expression as a manipulable procept. Gray and Tall (1994) also noted the peculiar case of the limit concept where the (potentially infinite) process of computing a limit may not have a finite algorithm at all. Thus a procept may exist which has both a process (tending to a limit) and a concept (of limit), yet there is no procedure to compute the desired result.
此处的“封闭性缺失”，本质上是个体仅关注表达式的“求值程序”，而非将代数表达式视为可操作的概念过程体。格雷与托尔（Gray & Tall, 1994）还指出了极限概念的特殊情况：计算极限的（潜在无限的）过程可能根本不存在有限算法。因此，存在这样一种概念过程体：它既包含“趋近极限”的过程，也包含“极限”的概念，但不存在计算预期结果的程序。

Table 1 shows a summary of the discussion so far. It does not intend any direct correspondence between the stages of the theories, simply that each passes through a development of growing sophistication from some kind of procedure/process usually performed step-by-step and ending with an object/concept that can be manipulated as an entity in its own right. The intermediate stages specified in each line are not intended to correspond directly. For instance, Greeno’s “input to another procedure” is essentially the same as conceiving a programming procedure as an entity, whereas Dubinsky characterizes the individual’s ability to take control of a repeatable action, and Sfard’s focus is on the ability to think of the process in terms of input/output without needing to consider the intermediate steps.
表 1 总结了前文的讨论内容。该表并非旨在表明各理论阶段之间存在直接对应关系，而仅想说明：每种理论都描述了从“逐步执行的程序/过程”到“可作为独立实体操作的对象/概念”的逐步深化过程。每行中指定的中间阶段也不具有直接对应性。例如，格里诺提出的“作为另一程序的输入”，本质上等同于将编程程序视为实体；而杜宾斯基强调的是个体对可重复动作的控制能力；斯法德则聚焦于“从输入/输出角度思考过程，无需关注中间步骤”的能力。

Theorist (Decade) 理论提出者（年代）	Process Stage 过程阶段	Intermediate Stage 中间阶段	Object Stage 对象阶段
Piaget (50s) 皮亚杰（20 世纪 50 年代）	action (s), operation (s) … 动作（们）、运算（们）……	… ……	thematized object of thought. 主题化的思维对象
Dienes (60s) 迪内斯（20 世纪 60 年代）	predicate … 谓语……	… ……	subject. 主语
Davis (80s) 戴维斯（20 世纪 80 年代）	visually moderated sequence … each step prompts the next 视觉调控序列…… 每一步触发下一步	integrated sequence … seen as a whole, and can be broken into sub-sequences 整合序列…… 被视为整体，可拆分为子序列	a thing, an entity, a noun. 事物、实体、名词
Greeno (80s) 格里诺（20 世纪 80 年代）	procedure … 程序……	input to another procedure … 作为另一程序的输入……	conceptual entity. 概念实体
Dubinsky (80s) 杜宾斯基（20 世纪 80 年代）	action … each step triggers the next 动作…… 每一步触发下一步	interiorized process … with conscious control 内化的过程…… 具有有意识控制	encapsulated object. 封装对象
Sfard (80s) 斯法德（20 世纪 80 年代）	interiorized process … process performed 内化的过程…… 已执行的过程	condensed process … self-contained 浓缩的过程…… 自包含的整体	reified object. 物化对象
Gray & Tall (90s) 格雷与托尔（20 世纪 90 年代）	procedure … specific algorithm 程序…… 特定算法	process … conceived as a whole, irrespective of algorithm 过程…… 被视为整体，与算法无关	procept. symbol evoking process or concept 概念过程体（procept）：可唤起过程或概念的符号

Table 1: The transition between process and object
表 1：过程与对象之间的转化

The wider literature of the various authors suggests further similarities and differences between their ideas. For instance, there seems to be broad agreement that a function as a process is determined as a whole by inputoutput, regardless of the internal procedure of computation. Thus the functions $f(x)=2x+6$ and $g(x)=2(x+3)$ are one and the same as processes—even though the arithmetic procedures to compute them have a different sequence of operations. The intermediate stage(s) intimate how (one or more) specific procedures become seen as a single process without needing to carry out the intermediate steps.
更多学者的文献表明，他们的观点既存在相似性，也存在差异。例如，学界普遍认同：作为过程的函数，其整体由输入-输出（input-output）关系决定，与内部计算程序无关。因此，函数 $f(x)=2x+6$ 与 $g(x)=2(x+3)$ 作为过程是完全相同的——尽管计算它们的算术程序具有不同的操作顺序。中间阶段揭示了（一个或多个）特定程序如何在无需执行中间步骤的情况下，被视为单一过程。

What seems more problematic is to explain precisely what is meant by the “object” hypothesized as being constructed by encapsulation.
更具争议性的问题在于：如何准确解释“封装构建的对象”这一假设概念的具体含义。

What is the “object” of encapsulation?

封装的“对象”是什么？

Dörfler questioned the nature of the object formed by encapsulation:
德弗勒（Dörfler）对封装形成的对象本质提出了质疑：

… my subjective introspection never permitted me to find or trace something like a mental object for, say, the number 5. What invariably comes to my mind are certain patterns of dots or other units, a pentagon, the symbol 5 or V, relations like $5+5=10$, $5\ast 5=25$, sentences like five is prime, five is odd, 5/30, etc., etc. But nowhere in my thinking I ever could find something object-like that behaved like the number 5 as a mathematical object does. But nevertheless I deem myself able to talk about the number “five” without having distinctly available for my thinking a mental object which I could designate as the mental object “5”.(Dörfler, 1993, pp. 146–147.)
……例如，通过主观内省，我始终无法找到或追踪到与数字 5 对应的心理对象。我脑海中总会浮现出特定的点或其他单元的排列模式、五边形、符号 5 或 V、关系式（如 $5+5=10$、$5\ast 5=25$）、表述（如“5 是质数”“5 是奇数”“5/30”）等。但在我的思维中，始终找不到一个“类对象实体”，其行为能与作为数学对象的数字 5 一致。尽管如此，我仍认为自己能够谈论数字“5”，即便我的思维中并不存在一个可明确指定为“5 的心理对象”的实体。

Support for this view comes from interviews on a video produced by Gray & Tall (1993) in which individuals were asked the following two questions:
格雷与托尔（Gray & Tall, 1993）制作的一段视频中的访谈为这一观点提供了支持，视频中受访者被问及以下两个问题：

What does the word “triangle” mean to you?
“三角形”（triangle）这个词对你来说意味着什么？
 
What does the word “five” mean to you?
“5”（five）这个词对你来说意味着什么？

The first was invariably met with a description or definition of a three-sided figure as if the individual had a clear mental picture of what was being described, sometimes adding other properties, such as “the angles add up to 180”. The second invariably caused the difficulties described by Dörfler, with some individuals describing “five objects”, or a property such as “it’s one bigger than four”, yet not being able to describe what the term “five” meant of itself. However, all were secure when asked to operate with the number “five”, for instance, that “five plus five” evoked the response “ten”.
对于第一个问题，受访者总会描述或定义“三边图形”，仿佛他们对所描述的内容有清晰的心理图像，有时还会补充其他属性（如“内角和为 180 度”）。对于第二个问题，受访者则总会遇到德弗勒所描述的困境：部分人会描述“五个物体”或“比四多一”这类属性，但无法解释“5”本身的含义。然而，当被要求对数字“5”进行运算时（如“5 加 5 等于多少”），所有人都能给出准确答案（如“10”）。

We hypothesize that the distinction between the notion “triangle” (which most would consider as an “object”) and “five”, which Dörfler suggests is not, is the difference between what we would term a “perceived object” and a “conceived object”. The first occurs based on perceptual information—seeing a triangle, physically cutting out a triangle, touching it, feeling the corners, counting the edges. The focus is therefore on specific physical manifestations of the notion of a triangle. The second occurs when there is reflection on perceptions and actions, with the focus shifted from the specific physical manifestations to the actions/processes performed upon them.
我们假设，“三角形”（多数人认为是“对象”）与德弗勒认为“非对象”的“5”之间的区别，本质上是“感知对象”（perceived object）与“构想对象”（conceived object）的区别。“感知对象”源于感知信息——如看到三角形、剪出三角形实物、触摸三角形、感知其角的数量、数其边的数量，其核心是三角形概念的特定物理表现形式。“构想对象”则源于对感知与动作的反思，核心从特定物理表现形式转向对这些表现形式所执行的动作/过程。

Piaget, as usual, made pertinent comments in this direction, long before any of the rest of us. He distinguished between empirical abstraction deriving knowledge from the properties inherent in real-world
与往常一样，皮亚杰早在我们之前就对此发表了中肯的观点。他区分了两种抽象：一种是“经验抽象”（empirical abstraction），即从现实世界对象固有的属性中获取知识；

objects and pseudo-empirical abstraction deriving knowledge from the processes which the individual performs on the objects. Empirical abstraction constructs a perceived object, or percept.
另一种是“准经验抽象”（pseudo-empirical abstraction），即从个体对对象执行的过程中获取知识。其中，经验抽象构建的是“感知对象”（perceived object），或称“知觉体”（percept）。

Pseudo-empirical abstraction constructs a conceived object which in many settings takes the form of a procept. For instance, a focus on the process of counting may progress to the idea that different ways of counting the same set give the same result, leading to the concept of number. Even though there may be no ‘mental object’ corresponding precisely to the number “5” as there is with a perceived object, there is a huge cognitive structure built up allowing the individual to use the symbol 5 as if it refers to an object. The number “5” has a concept image, in the sense of Tall & Vinner (1981), consisting of “all the mental pictures and associated properties and processes” related to the concept in the mind of the individual. The symbol “5” or the word “five” can be written, spoken, heard, seen and read. It can be manipulated mentally as if it were an object. This is a common mode of mental activity not just in mathematics, but throughout our whole experience. Lakoff and Johnson (1980) express this succinctly as follows:
准经验抽象构建的是“构想对象”（conceived object），且在多数情况下，这种对象以“概念过程体”（procept）的形式存在。例如，个体聚焦于“计数过程”，逐渐认识到“同一集合的不同计数方式会得到相同结果”，进而形成“数字概念”。尽管可能不存在与“感知对象”（如三角形）对应的、精确的“5 的心理对象”，但个体的认知结构会逐步发展，使得其能够将符号 5 当作“指向某一对象的符号”来使用。按照托尔与文纳（Tall & Vinner, 1981）的定义，数字“5”具有“概念意象”（concept image），即个体思维中与该概念相关的“所有心理图像及相关属性与过程”。符号“5”或单词“five”可被书写、言说、聆听、看见与阅读，且能被个体在心理层面当作对象进行操作。这种心理活动模式不仅存在于数学中，也贯穿于我们的整个生活经验。拉科夫与约翰逊（Lakoff and Johnson, 1980）对此有简洁的表述：

Our experiences with physical objects (especially our own bodies) provide the basis for an extraordinarily wide variety of ontological metaphors, that is, ways of viewing events, activities, emotions, ideas, etc., as entities and substances.(p.25)
我们对物理对象（尤其是自身身体）的经验，为各类“本体隐喻”（ontological metaphors）提供了基础——所谓本体隐喻，即把事件、活动、情感、思想等视为实体与物质的方式。

It is essential in empirical abstraction to be aware of the genuine distinction between objects existing in the physical world and objects constructed in our minds. A stone in the real world can be thrown and a person can be hit with it. We cannot hit someone with a stone in our mind. Of course, the word “stone” is a result of a categorization and so, in some sense, a mature observer can take the “object” to be an instance of the constructed category of stones in our head. However, the reference, for the knower, is to a concrete object in the world: one that is heavy and hurts if one is hit with it. In considering the cognitive development of the individual it is not how we see things as mature reflective adults, but how a child senses them during the construction process.
在经验抽象中，必须明确区分“物理世界中存在的对象”与“我们思维中构建的对象”。现实世界中的石头可以被投掷，也能砸到人；但我们无法用思维中的石头砸到人。当然，“石头”一词本身是分类的结果——从某种意义上说，成熟的观察者会将“石头对象”视为思维中“石头类别”的实例。但对认知者而言，该词指向的是现实世界中的具体对象：一个沉重且砸到人会造成疼痛的实体。在研究个体认知发展时，关键不在于成熟反思型成年人如何看待事物，而在于儿童在构建认知的过程中如何感知事物。

For a child, pseudo-empirical abstraction involves an intimate connection between the objects operated upon and the concepts abstracted through operating upon them:
对儿童而言，准经验抽象涉及“被操作的对象”与“通过操作抽象出的概念”之间的紧密关联：

With regard to icons, Piaget’s distinction between the “figurative” and the “operative” would seem to be of some importance. Number is not a perceptual but a conceptual construct; thus it is operative and not figurative. Yet, perceptual arrangements can be used to “represent” a number figuratively. Three scratches on a prehistoric figurine, for instance, can be interpreted as a record of three events. In that sense they may be said to be “iconic” but their iconicity is indirect.(Von Glasersfeld, 1987, p. 233).
关于符号（icons），皮亚杰对“形象的”（figurative）与“操作的”（operative）的区分似乎具有重要意义。数字并非感知构造，而是概念构造，因此属于“操作的”而非“形象的”。然而，感知排列可用于“形象地表示”数字。例如，史前雕像上的三道刻痕可被解读为“三次事件的记录”——从这个意义上说，这些刻痕具有“符号性”（iconic），但其符号性是间接的。

Empirical evidence collected by Gray and Pitta (1997) suggests that those who move to focus on the conceptual relationships rather than the perceptual arrangements are more likely to develop a flexible and powerful view of symbols and hence to have longer-term success. We therefore suggest that the total cognitive structure of the concept image of number, with its power to manipulate the symbols and to think of their properties, gives number its most powerful status as an object. What matters more is not what it is, but what we can do with it.
格雷与皮塔（Gray and Pitta, 1997）收集的经验证据表明：那些从“关注感知排列”转向“关注概念关系”的个体，更有可能形成对符号的灵活且强效的认知，进而在长期学习中取得更好的成效。因此，我们认为：数字的“概念意象”所具备的完整认知结构——包括对符号的操作能力及对符号属性的思考能力——赋予了数字作为“对象”的最强效地位。重要的不是数字“是什么”，而是我们能用数字“做什么”。

Mental conceptions of objects

对象的心理构想

The apparent “non-existence” of an “object” corresponding to a number is not as strange as at first seems. Consider, for example, the notion of “animal”, which includes cats, dogs and gorillas. When we mentally picture an “animal”, what do we “see”? The name “animal” is a signifier that can be used to signify any of a wide number of particular instances but we appear to fail to have a single mental object which is “an animal.” Nevertheless, to paraphrase Dörfler, although we may fail to have a unique mental object for “an animal” we deem ourselves able to talk about it.
数字对应的“对象”看似“不存在”，但这并不像初看时那样奇怪。以“动物”（animal）概念为例，它涵盖猫、狗、大猩猩等。当我们在脑海中“想象动物”时，我们“看到”的是什么？“动物”是一个能指代大量具体实例的符号，但我们似乎无法形成一个单一的“动物心理对象”。尽管如此，套用德弗勒的表述：即便我们没有“动物”的独特心理对象，我们仍能谈论“动物”。

This is a well-observed phenomenon in the verbal categorization of objects. Rosch (1978) notes that certain “basic” categories are more easily recognized and are often the first level comprehended by children. A basic category such as “dog” usually has a generic mental image that individuals can imagine in a representative way. Higher-order categorizations such as “animal” become so general as to fail to have a single mental image, although the basic categorizations within the category such as “dog”, “kangaroo”, “chimpanzee” are each capable of having a prototypical mental image. Such higher order categories may be supported not only by a variety of images, none of which is broadly characteristic of the whole category, but also by properties such as “being alive”, “having four legs”, “having fur” which may be typical of many members of the category, but not necessarily of all of them.
这是对象语言分类中一个已被充分观察的现象。罗施（Rosch, 1978）指出，某些“基础类别”（basic categories）更易被识别，且通常是儿童最先理解的类别层级。例如，“狗”（dog）这类基础类别，通常具有“代表性心理图像”——个体能想象出一个典型的狗的形象。而“动物”（animal）这类高阶类别则过于宽泛，无法形成单一的心理图像；尽管其包含的基础类别（如“狗”“袋鼠”“黑猩猩”）各自都能形成典型心理图像。高阶类别不仅依赖于“无广泛代表性的多种图像”支撑，还依赖于“活着”“四条腿”“有皮毛”等属性——这些属性对多数类别成员成立，但并非对所有成员都成立。

The same phenomenon occurs for many hierarchies encountered in mathematics. We may have a generic image of the graph of a straight line (often with positive gradient and positive intercept!), but the mental image of a general polynomial is likely to involve a curve with several maxima and minima rather than, say, a cubic with only a single point of inflection. Young children may have a single mental prototype for a square, often misleadingly with horizontal and vertical sides, which prevents the recognition of a square when it is turned through an angle (say as a “diamond”). It is part of the development of mathematical knowledge to be able to broaden our categories of visual imagery to allow hierarchies of concepts, such as squares being rectangles which are in turn
数学中的许多概念层级也存在相同现象。我们可能对“直线图形”有典型心理图像（通常是正斜率、正截距的直线），但对“一般多项式图形”的心理图像，更可能是“具有多个极大值和极小值的曲线”，而非“仅含一个拐点的三次曲线”。幼儿对“正方形”的心理原型通常是“水平垂直边的正方形”，这种认知会导致他们无法识别“旋转后的正方形”（如被称为“菱形”的图形）。数学知识发展的一部分，就是拓宽视觉意象的类别范围，以容纳概念层级——例如，正方形是矩形的特例，而矩形又是

parallelograms, within the category of quadrilaterals, within polygons. This development is supported by a variety of experiences. For instance, a square may be categorized as a rectangle using the definition of having all angles right angles and each pair of opposite sides equal, or it may simply be added to the category of rectangles because it has been deemed to be so.
平行四边形的特例，平行四边形又属于四边形，四边形则属于多边形。这种认知发展需要多种经验支撑。例如，根据“所有角为直角且对边相等”的定义，正方形可被归类为矩形；或者，正方形也可能因“被认定为矩形”而直接归入矩形类别。

In this way we see that mental objects occurring in mathematics have aspects which are found in other areas of experience.
由此可见，数学中的心理对象具有与其他经验领域相似的特征。

Language markers for objects

对象的语言标志

The discourse used to describe everyday objects, or mental images of such objects, is description. Whereas narrative discourse is used to describe a procedure or a succession of events, descriptive discourse usually features the simultaneity of various aspects of a to-be-described object. The possibility is, therefore, that we might ascertain whether an individual has constructed a mental object in relation to a concept by the way that individual talks or writes about the concept. In other words, seeing the concept as an object is likely to lead to descriptive rather than narrative discourse. As Denis (1996) says:
描述日常对象或其心理图像的话语类型是“描述性话语”（descriptive discourse）。“叙事性话语”（narrative discourse）用于描述程序或一系列事件，而“描述性话语”的核心特征是“同时呈现被描述对象的多个方面”。因此，我们可通过个体“谈论或书写某一概念的方式”，判断其是否已构建该概念的心理对象。换言之，若个体将概念视为对象，其话语更可能是描述性的，而非叙事性的。正如丹尼斯（Denis, 1996）所言：

“ … descriptions are not disorganized lists of elements. Descriptions are constrained by the structural organization of described objects.”(p. 169)
“……描述并非元素的无序列表，而是受被描述对象的结构组织所约束。”

If someone conceived of “5” as a mental object what sort of language might they use to describe it? How might this language itself give us pointers to their seeing “5” as an object? Typically objects are described by their properties, their relationships with other objects, and the ways in which they can be used. For example, someone might describe a kitchen pot as generally round, made of metal, with a handle, and used for holding water in which to boil food. Similarly, if we say that 5 is a prime number, the third prime, and the second odd prime; it is the first number n for which there is a non-solvable polynomial of degree n, then we are describing properties of 5, and its relation to other things. Whether these other things—polynomials or other numbers—are themselves conceived as objects at the time is not the point at issue. It is the use of language in a way that intimates properties, relationships, usage of a concept which indicates that the individual is, in fact, conceiving “5” as an object.
若个体将“5”视为心理对象，他们会用何种语言描述“5”？这种语言又如何体现他们将“5”视为对象？通常，描述对象需提及“对象的属性”“与其他对象的关系”及“对象的用途”。例如，描述“厨房锅具”时，会说“通常为圆形、金属材质、带手柄，用于装水烧水”。同理，若我们说“5 是质数、是第三个质数、是第二个奇质数；5 是首个满足‘存在 n 次不可解多项式’的 n”，我们便是在描述 5 的属性及其与其他事物的关系。此时，“多项式”或“其他数字”是否被视为对象并不重要——关键在于，这种“提及概念属性、关系与用途”的语言使用方式，表明个体确实将“5”视为对象。

The role of awareness in the construction of objects

意识在对象构建中的作用

We have seen that when a student first carries out a mathematical procedure (as a visually moderated sequence in the sense of Davis or action in the sense of Dubinsky), then they are generally unaware of the nature of the whole procedure. What they are aware of is the current step being carried out and then the final result. Steffe (in a personal communication) has remarked on the development of a symbol for a
我们已发现：当学生初次执行数学程序时（无论是戴维斯所说的“视觉调控序列”，还是杜宾斯基所说的“动作”），他们通常无法意识到整个程序的本质——仅能意识到“当前执行的步骤”与“最终结果”。斯特菲（Steffe，私人交流）曾就“程序符号的发展”发表过观点：

procedure. This development utilizes imagery of fingers or other collections of objects in an important way as figurative material acted upon mentally and then re-interpreted (or re-presented) to see it also in terms of the results of the operations. This is essentially a change of focus of attention in the same way that Sfard speaks of focusing on structural or operational facets and Gray and Tall speak of the ambiguous use of symbolism to dually represent process or concept. The dual use of figurative imagery as operations or results is embodied in the following comment by Steffe about number units. The discussion can also be interpreted for many, if not all, mathematical procedures:
程序符号的发展，在很大程度上依赖于“手指或其他物体集合的意象”——这些意象作为“心理操作的形象材料”，经重新解读（或重新呈现）后，也能被视为“操作结果”。这本质上是“注意力焦点的转移”，与斯法德所说的“聚焦于结构层面或操作层面”、格雷与托尔所说的“符号模糊性——同时代表过程或概念”具有一致性。斯特菲在谈论“数字单位”时，曾提及“形象意象同时作为操作与结果”的现象，这一现象也适用于多数（若非全部）数学程序：

“I was thinking of, for example, the way in which a child might use the records of past experience that are recorded in the unit items of a sequence to regenerate something of that past experience in a current context. The figurative material that is regenerated may act as symbols of the operations of uniting or of the results of the operations in that the operations may not need to be carried out to assemble an experiential unit item. The figurative material stands in for the operations or their results. In this I assume that the records are interiorized records – that is, records of operating with re-presented figurative material.
“例如，我认为儿童可能会利用‘序列单位项中记录的过往经验’，在当前情境中重现部分过往经验。重现的形象材料可作为‘合并操作’的符号，也可作为‘操作结果’的符号——因为无需实际执行操作，就能整合出‘经验单位项’。形象材料可替代操作或操作结果。在此，我假设这些记录是‘内化记录’——即对‘重新呈现的形象材料进行操作’的记录。
 
My hypothesis is that these operations will continue to be outside the awareness of the operating child until a stand-in is established in which the operations are embedded. … awareness to me is a function of the operations of which one is capable. But those operations must become objects of awareness just as the results of operating. To become aware of the operations involves the operations becoming embedded in figurative material on which the operations operate. To the extent that this figurative material can be re-generated the operations become embedded in it.
我的假设是：在建立‘嵌入操作的替代物’之前，执行操作的儿童始终无法意识到这些操作……在我看来，意识是‘个体所能执行的操作’的函数。但这些操作必须与‘操作结果’一样，成为意识的对象。要意识到操作，需将操作嵌入‘操作所作用的形象材料’中；形象材料的可重现程度，决定了操作的嵌入程度。

The figurative material is also operated on again. In this way the operations are enlarged and modified.”
形象材料还会被再次操作，操作也因此得以扩展与调整。”

(Steffe, private communication 私人交流)

This point of view—that operations are outside conscious awareness until a mental “stand-in”, or symbol, is developed upon which the operations can act mentally—is of critical importance in our perception of the development of mathematical objects. For it is these symbols (in the sense of Steffe) that lead to objects: not only is the scope of the operations extended, as Steffe says, but what it is that the operations operate on becomes conceived as a mental object. One might argue that symbols, in Steffe’s sense, do not lead to objects: if they did, so the argument might go, our job of teaching would be very much easier for we could just give students appropriate symbols when we wanted them to form an object. However, this argument does not take into account that—by the very nature of symbols—we cannot simply “give” a student symbols. Their symbols have to be constructed through a subtle process of reflective abstraction.
这一观点——“在形成‘心理替代物’（即符号）之前，操作处于意识之外；只有形成符号后，操作才能在心理层面作用于符号”——对我们理解“数学对象的发展”至关重要。因为正是这些符号（斯特菲所定义的符号）促成了对象的形成：正如斯特菲所说，不仅操作的范围会扩展，操作所作用的实体也会被构想为“心理对象”。有人可能会反驳：斯特菲所说的符号无法促成对象形成——若能，教学工作会变得简单：只需在希望学生形成对象时，给予他们相应符号即可。但这种反驳忽略了符号的本质属性——我们无法简单地“给予”学生符号，符号必须通过“反思抽象”（reflective abstraction）这一微妙过程由学生自行构建。

Consider, for example the procedure of finding the square root of a number, such as 2, by repeated approximation. Here a student might be told that $\sqrt{2}$ lies between 1 and 2 because $1^2$ is smaller than 2 and $2^2$ is greater than 2. Forming the average of 1 and 2 gives 3/2 which is greater than $\sqrt{2}$ because $(3/2{)}^2$ is greater than 2. The average of 1 and 3/2 is 5/4 which, again is smaller than $\sqrt{2}$. This procedure, repeated often enough gives a decent, but slow, approximation to $\sqrt{2}$. What is uppermost in a student’s mind after carrying out such a procedure? We hypothesize it relates to the answer to the question “What is the value of $\sqrt{2}$?” which may be calculated to some degree of accuracy using the procedure. The number $\sqrt{2}$ has the property that its square is (exactly) 2 and the procedure gives successively better approximations. What the student may not generally be aware of in the early stages—in the sense of being able to articulate it—is the nature of the overall procedure by which this approximation was found, despite the fact that the student can work step-by-step through the procedure to get an answer.
以“通过反复逼近求某数（如 2）的平方根”这一程序为例。学生可能会学到：$\sqrt{2}$ 介于 1 和 2 之间，因为 $1^2<2$ 且 $2^2>2$；取 1 和 2 的平均值 3/2，其值大于 $\sqrt{2}$，因为 $(3/2{)}^2>2$；再取 1 和 3/2 的平均值 5/4，其值仍小于 $\sqrt{2}$。反复执行这一程序，可得到 $\sqrt{2}$ 的近似值（精度尚可，但速度较慢）。学生执行完这一程序后，脑海中最核心的想法是什么？我们假设是“$\sqrt{2}$ 的值是多少”——通过该程序可计算出一定精度的 $\sqrt{2}$ 值。$\sqrt{2}$ 的属性是“其平方等于 2”，而该程序能给出精度逐步提升的近似值。尽管学生能逐步执行程序得到结果，但在初期阶段，他们通常无法清晰表述“该近似程序的整体本质”——这是他们意识不到的内容。

A student’s reflection on their actions—in the absence of actual calculations—may bring back to mind the episode of operating to calculate a good approximation to the square root of two. What might prompt such reflection? One thing could be a student’s wondering whether they could calculate an approximation to $\sqrt{3}$ in a similar way, or being asked to formulate such an approximation. Without actually going through the procedure, this recalling of the operations in a similar context allows the possibility—but not the necessity—of the student operating on mental imagery in the form of figurative material associated with the original calculations. The scope of the operations has become enlarged because, although students could, in principle, carry out the actual calculations to approximate $\sqrt{3}$, they may not, preferring instead to reflect on the figurative material in their mind. The original operations now become objects of awareness, and the sign “$\sqrt{n}$” has the possibility, through repeated re-presentation, to become a symbol: the symbol of the operations embedded in the figurative material. This, we hypothesize, is what transforms $\sqrt{n}$ into a mathematical object. Prior to the re-presented figurative material and the act of operating on this material, $\sqrt{n}$ may be a procedure for which the only articulable awareness the student had was the result of concrete operations.
学生在“不进行实际计算”的情况下反思自身动作，可能会回忆起“计算 $\sqrt{2}$ 近似值”的操作过程。是什么引发这种反思？可能是学生好奇“能否用类似方法计算 $\sqrt{3}$ 的近似值”，也可能是被要求“推导 $\sqrt{3}$ 的近似方法”。在不实际执行程序的情况下，这种“在相似情境中回忆操作”的行为，使学生有可能（但非必然）“对与原始计算相关的形象材料进行心理操作”。操作范围由此扩展：尽管学生原则上可通过实际计算近似 $\sqrt{3}$，但他们可能选择“反思脑海中的形象材料”而非实际计算。此时，原始操作成为意识的对象；符号“$\sqrt{n}$”通过反复重新呈现，有可能成为“嵌入形象材料中的操作的符号”。我们假设，正是这一过程将 $\sqrt{n}$ 转化为数学对象。在“重新呈现形象材料”及“对材料进行操作”之前，$\sqrt{n}$ 对学生而言只是“程序”——他们仅能清晰意识到“具体操作的结果”，而非程序的本质。

The scope of the process-object construction

过程-对象构建的范围

Once the possibility is conceded that the process-product construction can be conceived as an “object”, the floodgates open. By “acting upon” such an object, the action-process-object construction can be used again and again.
一旦承认“过程-产物构建可被构想为对象”，其应用范围便会大幅扩展。通过对“对象”施加“动作”，“动作-过程-对象”的构建模式可被反复运用。

…… the whole of mathematics may therefore be thought of in terms of the construction of structures, … mathematical entities move from one level to another; an operation on such ‘entities’ becomes in its turn an object of the theory, and this process is repeated until we reach structures that are alternately structuring or being structured by ‘stronger’ structures.(Piaget, 1972, p. 70)
……因此，整个数学可被视为“结构的构建过程”……数学实体从一个层级过渡到另一个层级；对这些“实体”的操作，反过来成为理论的对象；这一过程不断重复，直至形成“交替构建更强结构或被更强结构构建”的结构。

The various proponents of process-object construction do not all claim the same scope for the application of their theories. For Dubinsky—utilizing Piaget’s notion of “permanent object”—a process is any cognitive process. The notion of “permanent object” arises through “encapsulating the process of performing transformations in space which do not destroy the physical object” (Dubinsky et al, 1988, p. 45). A “perceived object” in our sense is, for Dubinsky, formed by process-object encapsulation. He also sees the theory of encapsulation applying equally well to the logical construction of formal concepts in advanced mathematics. He also acknowledged that there may be many cognitive processes involved in the construction of a mental object, used in an increasingly coherent manner, leading naturally to his later assertion that objects can also be formed by encapsulating schemas. Dubinsky therefore offers a single, unified theory of encapsulating cognitive processes as cognitive objects after the manner of Piaget.
不同的“过程-对象构建”理论提出者，对其理论的应用范围主张不同。杜宾斯基借鉴皮亚杰的“永久对象”（permanent object）概念，将“过程”定义为“任何认知过程”。“永久对象”的概念源于“对‘空间中不破坏物理对象的转化过程’的封装”（Dubinsky et al, 1988, p. 45）。在杜宾斯基看来，我们所说的“感知对象”，正是通过“过程-对象封装”形成的。他还认为，封装理论同样适用于“高等数学中形式概念的逻辑构建”。此外，他承认“心理对象的构建可能涉及多种认知过程”，且这些过程的连贯性会逐步提升——这也促使他后来提出“对象也可通过封装图式形成”的观点。因此，杜宾斯基借鉴皮亚杰的思想，提出了“将认知过程封装为认知对象”的统一理论。

The scope of procept theory is narrower. This does not mean that it is weaker, since it is designed to give greater insight into the profoundly powerful use of symbols in mathematics to switch effortlessly between concepts to think about and mathematical processes to solve problems. The definition is formulated so that it can refer not only to the construction of procepts, but also to the use of procepts which enables a biological brain with a limited focus of attention to switch between thinking of the problem symbolically and then to perform a mathematical solution process.
概念过程体（procept）理论的应用范围较窄，但这并不意味着其效力较弱——该理论旨在深入阐释“数学中符号的强效作用”：符号能让个体在“思考概念”与“执行解题过程”之间灵活切换。概念过程体的定义不仅涵盖“概念过程体的构建”，还包括“概念过程体的使用”——这使得“注意力范围有限的生物大脑”能够在“符号化思考问题”与“执行数学解题过程”之间切换。

The notion of procept was never intended to have the same broad scope as the theories of Sfard or Dubinsky. Neither the child’s notion of permanent object nor the students’ notion of axiomatic system are procepts because neither has a symbol capable of evoking either process or concept. Nor is the notion of procept defined to be explicitly tied to the situation in which the mental object represented by the symbol is necessarily construction by “encapsulation” from the corresponding mathematical process, even though this is the way in which many procepts are constructed.
概念过程体的概念，从未被设计为具有与斯法德或杜宾斯基理论同等宽泛的范围。儿童的“永久对象”概念与学生的“公理系统”概念都不属于“概念过程体”——因为两者都没有“能同时唤起过程或概念的符号”。此外，概念过程体的定义也未明确要求“符号所代表的心理对象，必须通过对相应数学过程的‘封装’构建而成”——尽管多数概念过程体确实通过这种方式构建。

As has often been emphasized by Dubinsky, there may be many processes involved in encapsulating a mental object not all of which are usually classified as being mathematical. For instance, the Solution Sketcher software (Tall, 1991) allows the user to enter a first order differential equation; internally the program works out the slope of the solution curve through any selected point and draws an appropriate short line segment with this slope. The student can move the pointer around the screen and, without performing any mathematical calculation, observe the direction of a segment of a solution curve change direction. Encouraging the student to build up a solution curve by sticking such segments together end-to-end can give a visual and kinesthetic meaning to a solution of a differential equation, including the insight that there is a unique solution through each point where the equation defines a slope. This suggests that it could be inappropriate to always insist that a mathematical process must be presented first so that it may then be encapsulated as a mathematical concept. Other cognitive processes can be used to build up useful parts of the concept image as foundations for later formal or symbolic development.
正如杜宾斯基常强调的，“心理对象的封装可能涉及多种过程”，且并非所有过程都被归为“数学过程”。例如，“解题草图软件”（Solution Sketcher, Tall, 1991）允许用户输入一阶微分方程；程序会自动计算“任意选定点处解曲线的斜率”，并绘制出对应斜率的短线段。学生可在屏幕上移动指针，无需进行任何数学计算，就能观察到“解曲线段的方向变化”。鼓励学生通过“将这些线段首尾相连”来构建解曲线，能让学生从“视觉与动觉层面”理解微分方程的解，包括领悟到“在方程定义斜率的每一点处，都存在唯一解”。这表明，“必须先呈现数学过程，再将其封装为数学概念”的观点并非绝对成立。其他认知过程（如视觉、动觉认知）也可用于构建“概念意象的有用部分”，为后续的形式化或符号化发展奠定基础。

Sfard’s (1991) theory formulates two ways of constructing mathematical conceptions through the complementary notions of operational and structural activities as “two sides of the same coin”. It is the first of these which involves process-object construction. She makes the important observation that in any given mathematical context it is usually possible to see both operational and structural elements. For instance, in the transition from arithmetic to algebra, Sfard (1995, p.21) considers that, given “the boys outnumber the girls by four”, the number of boys can be formulated in an operational manner as “add four to the number of girls"or structurally as $x=y+4$
斯法德（Sfard, 1991）的理论提出，“操作型活动”与“结构型活动”是“同一硬币的两面”，二者互补，共同构成数学概念构建的两种方式——其中，操作型活动涉及“过程-对象构建”。她还提出一个重要观点：在任何特定数学情境中，通常都能同时观察到“操作型元素”与“结构型元素”。例如，在从算术到代数的过渡中，斯法德（Sfard, 1995, p.21）认为：若“男孩人数比女孩多 4 人”，则“男孩人数”可通过操作型方式表述为“女孩人数加 4”，或通过结构型方式表述为 $x=y+4$（其中 $x$ 代表男孩人数，$y$ 代表女孩人数）。

Kieran (1992) also makes a distinction between the (numerical) evaluation of an expression such as $2x+3$ for a numerical value of $x$ as “operational” and the manipulation of such expressions as “structural”. However, this does not attend to the full subtlety of Sfard’s distinction between structural and operational. For Sfard, the algebraic expression itself has a structure but, once the student conceives the algebraic expressions as manipulable objects, then their manipulation is categorized as “operational”. The latter interpretation is consonant with the original ideas of Piaget where “operations” at one level (in this case algebraic formulae as generalized arithmetic operations) become “objects of thought” at a higher level (algebraic expressions) which can themselves be manipulated.
基兰（Kieran, 1992）也对“操作型”与“结构型”进行了区分：将“对 $x$ 取数值后代入 $2x+3$ 这类表达式进行计算”归为“操作型”，将“对这类表达式进行变形”归为“结构型”。然而，这种区分并未完全体现斯法德理论中“结构型”与“操作型”的微妙差异。在斯法德看来，代数表达式本身具有结构；但一旦学生将代数表达式视为“可操作对象”，对表达式的变形就应归为“操作型”。这一解读与皮亚杰的核心思想一致：某一层级的“操作”（如作为“广义算术操作”的代数公式），会成为更高层级（如代数表达式）的“思维对象”，且这些“思维对象”本身也可被操作。

Structural aspects of mathematics

数学的结构层面

The term “structural” has more than one meaning in the literature. The French texts published under the corporate name of Bourbaki are often referred to as having a structural approach in the sense that they describe the formal “structures” of mathematical axiomatic systems. The set-theoretic “new math” of the sixties was often referred to as a “structural approach”. Sfard uses this interpretation when she describes two distinct aspects of proof by induction:
文献中，“结构的”（structural）一词具有多重含义。以“布尔巴基”（Bourbaki）为集体署名出版的法国数学著作，常被认为采用“结构主义方法”——其核心是描述“数学公理系统的形式结构”。20 世纪 60 年代基于集合论的“新数学”（new math），也常被称为“结构主义方法”。斯法德在描述数学归纳法的两个不同层面时，便采用了这一解读：

• Operational: for a property $P(n)$, prove $P(1)$ and that $P(k)\Rightarrow P(k+1)$ for all $k$, to establish the truth of $P(n)$ for all natural numbers $n$,
操作型：对于性质 $P(n)$，证明 $P(1)$ 成立，且对所有 $k$ 有 $P(k)\Rightarrow P(k+1)$，进而证明 $P(n)$ 对所有自然数 $n$ 成立；
 
• Structural: given a set $S\subseteq N$ prove $1\in S$, and $k\in S\Rightarrow k+1\in S$ to establish $S=N$.
结构型：给定集合 $S\subseteq N$，证明 $1\in S$，且对所有 $k$ 有 $k\in S\Rightarrow k+1\in S$，进而证明 $S=N$。

Sfard (1989, p. 151)

Her description of a function as a “set of ordered pairs” as structural (Sfard, 1991, p. 5), also agrees with the formal Bourbaki approach. However, in referring to structural conceptions as being “supported by visual imagery” while an operational conception is “supported by verbal representations” (ibid., p. 33), she is using a meaning different from that of Bourbaki. In this meaning she refers to the visual structure of a mental object by analogy with the structure of a building. For instance, she considers the visual imagery of a graph of a function to be structural (ibid., pp. 5, 6). She also provides perceptive empirical evidence in which mathematicians use visual and spatial metaphors to provide them with “intimate familiarity” with structure that gives “direct insight into the properties of mathematical objects,” (Sfard, 1994). Such insight may be fallible and is certainly not structural in the Bourbaki sense.
她将“函数定义为有序对集合”归为“结构型”（Sfard, 1991, p. 5），这也与布尔巴基的形式化方法一致。然而，当她提出“结构型概念依赖视觉意象支撑，而操作型概念依赖语言表征支撑”时（ibid., p. 33），所使用的“结构型”含义与布尔巴基不同。在此含义中，她将“心理对象的视觉结构”类比为“建筑结构”——例如，她认为“函数图像的视觉意象”属于“结构型”（ibid., pp. 5, 6）。此外，她还提供了富有洞察力的经验证据：数学家会使用“视觉与空间隐喻”来获得对“结构的深入熟悉”，进而“直接洞察数学对象的属性”（Sfard, 1994）。这种洞察可能存在误差，且显然不属于布尔巴基意义上的“结构型”。

Skemp (1979) formulated a theory that sheds light on these differences in meaning. First he proposed a varifocal theory in which a concept, seen in close detail, could be considered as a schema of related ideas, and a schema, seen as a totality could be considered a concept. This idea (first enunciated in a 1972 Warwick Education Seminar by an American graduate student Robert Zimmer) reveals a remarkable duality between the notions of concepts and schemas a decade before Dubinsky proposed that schemas could be encapsulated as concepts. However, the varifocal theory does not specify how concepts are constructed from schemas, or vice versa. For the construction (of concepts) Skemp proposed a broader theory taking into account the context in which the mental structures are performed. This consists of three modes of building and testing concepts, one in terms of interaction with concrete objects, a second through interaction with other individuals, and a third through internal consistency of the mathematics within the mind of the individual. Each of these can be seen as a method of constructing and refining particular mental concepts, and each can have both operational and structural aspects in the sense of Sfard. However, the third mode has a clear link with the working of professional mathematicians mentioned above (Sfard, 1994). In a conversation one of us had with Skemp in 1985, he hypothesized that Archimedes “Method” conceiving “surfaces made up of lines” was a strategy for building formulae for area whereas the formal method of “proof by exhaustion” was for testing the formulae and for release in formal publications. In the same way many mathematicians use intuitive conceptions as private constructs before releasing formally presented theories for public scrutiny.
斯肯普（Skemp, 1979）提出了一套理论，为理解这些含义差异提供了思路。首先，他提出“变焦理论”（varifocal theory）：近距离观察时，一个概念可被视为由相关观点构成的图式；而从整体视角观察时，一个图式又可被视为一个概念。这一观点（由美国研究生罗伯特·齐默（Robert Zimmer）于1972年华威教育研讨会首次提出）揭示了“概念”与“图式”之间显著的二元性——这比杜宾斯基提出“图式可被封装为概念”的观点早了十年。然而，变焦理论并未明确概念如何从图式中构建，反之亦然。针对概念构建，斯肯普提出了一套更宽泛的理论，该理论考虑了心理结构形成的背景。这套理论包含三种构建与检验概念的模式：第一种是通过与具体物体的互动，第二种是通过与他人的互动，第三种是通过个体思维内部数学逻辑的一致性。这三种模式均可被视为构建和完善特定心理概念的方法，且每种模式在斯法德的理论框架下都兼具操作层面与结构层面的特征。不过，第三种模式与前文提及的专业数学家的工作方式（Sfard, 1994）存在明确关联。1985年，我们中的一位学者与斯肯普交谈时，他提出一个假设：阿基米德“把曲面看作由线构成”的“方法”（Method）是推导面积公式的策略，而“穷竭法”（proof by exhaustion）这一形式化方法则用于检验公式，并在正式出版物中公布。同样，许多数学家在公布供公众审视的形式化理论之前，会将直觉性概念作为私人构建的工具。

This suggests a clear distinction between the building of conceptions from visual and kinesthetic senses described verbally on the one hand and the more sophisticated construction of a formal theory of the properties of a concept which is specified by set-theoretic axioms. This in turn suggests that Sfard’s notion of “structural” can usefully be subdivided into two. On the one hand there is the focus on properties of observed or conceived objects (which usually involve visual and enactive elements). On the other, some of these properties are specified as set-theoretic axioms and definitions to give a formal theory that is “structural” in the sense of Bourbaki. In our perception, we would prefer to use the term “structural” to relate to the Bourbaki sense of the term, because we believe that this involves a more sophisticated form of mental construction than the visual and enactive construction of meaning from mental images.
这表明，一方面，通过视觉和动觉感知（并以语言描述）构建概念，另一方面，通过集合论公理明确概念属性并构建形式化理论，二者之间存在明显区别。这进而意味着，斯法德提出的“结构的”这一概念可有益地细分为两类：一类聚焦于观察到或构想的对象的属性（通常涉及视觉和动作元素）；另一类则是将部分属性明确为集合论公理和定义，从而形成在布尔巴基（Bourbaki）意义上具有“结构性”的形式化理论。在我们看来，更倾向于将“结构性”一词限定为布尔巴基意义上的含义，因为我们认为，相较于从心理图像中通过视觉和动作构建意义，这种形式的心理构建更为复杂。

Categorizing mathematical objects and their construction

数学对象及其构建的分类

Faced with several different versions of “object construction” in mathematics, we now come to the question of categorizing objects and their constructions in a manner that sharpens the relationships and differences between the theories. Dubinsky offers a broad vision which encompasses all the various methods of construction as process-object encapsulation. Sfard divides her theory into two (operational and structural) where the first is concerned with operations which may be reified into objects and the second focuses more on the properties of the objects themselves. Gray and Tall offer a theory more focused on the relationship between mathematical processes, objects and symbols that dually evoke both.
面对数学中“对象构建”的多种不同理论版本，我们接下来需要探讨如何对数学对象及其构建方式进行分类，以明晰各理论之间的关联与差异。杜宾斯基提出了一个宽泛的框架，将所有不同的构建方法都归为“过程-对象封装”。斯法德则将其理论分为两类（操作型与结构型）：前者关注可被物化为对象的操作，后者更聚焦于对象本身的属性。格雷与托尔提出的理论则更侧重于数学过程、对象与符号之间的关系——符号可同时唤起过程与概念。

Different classifications can operate at different levels and have different uses. Our quest here is to seek a classification that is useful for distinguishing cases that involve differing cognitive constructions and necessitate different strategies in learning. Sfard makes a step in this direction by hypothesizing that operational constructions are more primitive than structural so that that an operational approach should usually precede a structural. Other levels of categorizations are also possible, for instance, subdividing the notion of structural into distinct categories as proposed earlier.
不同的分类方式可在不同层面发挥作用，且具有不同用途。我们此处的目标是找到一种分类方式，能够有效区分涉及不同认知构建过程的情况，并为不同的学习策略提供依据。斯法德在这一方向上迈出了一步，她假设操作型构建比结构型构建更基础，因此操作型方法通常应先于结构型方法。当然，也存在其他层面的分类可能，例如如前文所提议的，将“结构型”这一概念细分为不同类别。

To seek a cognitively based categorization at an appropriate level of generality, we return to Piaget’s distinction between empirical and pseudo-empirical abstraction which we see operating in different cognitive ways. Empirical abstraction arises through a focus on the objects being acted upon, while pseudo-empirical abstraction focuses on actions and their subsequent symbolization and conception as encapsulated objects. Tall (1995) hypothesized that these begin distinct sequences of construction, one in geometry which follows a broad development
为了在恰当的普适性层面建立基于认知的分类，我们回归皮亚杰对“经验抽象”与“准经验抽象”的区分——我们认为这两种抽象在认知层面的运作方式不同。经验抽象源于对“被作用对象”的关注，而准经验抽象则聚焦于“动作”及其后续的符号化过程，以及将动作构想为封装对象的过程。托尔（Tall, 1995）提出假设：这两种抽象开启了不同的构建序列，一种出现在几何学中，遵循范·海勒（Van Hiele, 1986）所描述的广泛发展路径；另一种则出现在算术与代数中，基于“符号同时代表过程与概念”的特征。

described by Van Hiele (1986) and another in arithmetic and algebra based on symbols dually representing process and concept. After the initial use of empirical and pseudo-empirical abstraction, constructions proceed also by reflective abstraction that occurs through reflections on the manipulation of mental objects already constructed. However, we see differing kinds of reflective abstraction depending on the focus of attention of the construction process.
在初步运用经验抽象与准经验抽象之后，对象构建还会通过“反思抽象”继续推进——反思抽象源于对已构建的心理对象进行操作后的反思。不过，根据构建过程中关注焦点的不同，我们发现反思抽象也存在不同类型。

In geometry, although there are many processes involved, including the formal processes of geometric construction, we hypothesize that the main focus is initially on objects. This leads to a sequence of development from teasing out the properties of the objects, making verbal descriptions, thinking about relationships, verbalizing inferences, formulating verbal proofs, leading to a broad development after the fashion described by Van Hiele (1986). As the development becomes more sophisticated, the form of object construction becomes more subtle. In the initial stages the child is building a conceptual meaning for real-world objects and their properties by perceiving and acting upon them to construct percepts (through empirical abstraction). The next Van Hiele stage involves classifying these percepts into separate classes that are not yet hierarchical, for instance, squares (four sided figures with right angles and all four sides equal) are not special cases of rectangles (which only have opposite sides equal). The next stage builds up hierarchies by using verbal descriptions. The focus switches from the objects described to the nature of the descriptions themselves that are then used to define objects. At this stage mental objects are constructed as imaginary manifestations of the perfection of the definition, for instance, lines with no thickness that can be extended arbitrarily in either direction. Thus the construction of perceived geometric objects leads later to conceived geometric objects which, though imagined in the mind’s eye, and discussed verbally between individuals, are perfect entities that have no real-world equivalent. Tall (1995) referred to these conceived objects as platonic objects, and quoted the observations of G. H. Hardy explaining his conceptions as a professional mathematician:
在几何学中，尽管涉及多种过程（包括形式化的几何作图过程），但我们假设其初始焦点主要是“对象”。这使得几何认知呈现出一系列发展阶段：从梳理对象的属性、进行语言描述，到思考对象间的关系、用语言表述推论、构建语言化证明，最终形成符合范·海勒（Van Hiele, 1986）所描述的广泛发展路径。随着认知发展逐渐复杂，对象构建的形式也变得更为精妙。在初始阶段，儿童通过感知现实世界中的物体并对其施加动作（借助经验抽象），构建感知体（percept），进而为这些物体及其属性建立概念意义。在范·海勒理论的下一阶段，儿童会将这些感知体分类为互不包含的类别（尚未形成层级），例如，正方形（四条边相等且四个角为直角的四边形）不会被视为长方形（仅对边相等的四边形）的特例。再下一阶段，儿童通过语言描述建立类别层级，关注焦点从“被描述的对象”转向“描述本身的性质”——这些描述随后被用于定义对象。在此阶段，心理对象被构建为“定义所蕴含的完美属性的想象体现”，例如，没有厚度且可向任意方向无限延伸的直线。因此，从“感知到的几何对象”出发，最终会发展出“构想的几何对象”：这类对象虽存在于人的想象中，可通过语言与他人交流，但却是具有完美属性、无现实对应物的实体。托尔（Tall, 1995）将这类构想对象称为“柏拉图式对象”（platonic objects），并引用G·H·哈代（G. H. Hardy）的表述来阐释专业数学家的这类认知：

… I draw figures on the blackboard to stimulate the imagination of my audience, rough drawings of straight lines or circles or ellipses. It is plain, first, that the truth of the theorems which I prove is in no way affected by the quality of my drawings. Their function is merely to bring home my meaning to my hearers, and, if I can do that, there would be no gain in having them redrawn by the most skilful draughtsman. They are pedagogical illustrations, not part of the real subject-matter of the lecture.(Hardy, 1940, p. 125.)
……我在黑板上画图是为了激发听众的想象力，这些不过是直线、圆或椭圆的粗略草图。首先很明显的是，我所证明的定理的真实性，丝毫不受草图质量的影响。这些图的作用仅仅是让听众理解我的意图；若能达到此目的，即便让最精湛的绘图员重新绘制，也不会有额外的益处。它们只是教学辅助插图，并非讲座真正的核心内容。

The mental objects of higher geometrical thinking are therefore no longer the physical drawings of the individuals, but shared platonic mental conceptions with perfect properties that underlie them.
因此，高等几何思维中的心理对象，不再是个体绘制的物理图形，而是潜藏于这些图形之下、具有完美属性的共享柏拉图式心理概念。

In arithmetic, the initial focus is on processes, such as counting,addition, subtraction, and on the symbols used to represent them. The encapsulation of processes into arithmetic procepts (using symbols as a pivot between the two) also change in their cognitive sophistication as the conceptions are built in later developments. In arithmetic all the symbols have built-in computational processes to “give an answer”. In algebra the symbols are now algebraic expressions which only have a potential process of inner computation— the evaluation of the expression when the variables are given numerical values. Despite this, the symbols themselves can be manipulated algebraically and a finite number of such manipulations can be used to solve linear and quadratic equations. In the calculus the situation changes again, with limit processes that are now potentially infinite and so we may have a limit concept which has an infinite process attached with no finite procedure of computation. Thus, although the symbols all have a process-concept framework, which enables them all to be classified as procepts, the changes in the nature of the procept cause major cognitive difficulties in the transition from arithmetic to algebra and algebra to calculus. This also explains why students are so much more able to compute with the finite procedural rules for differentiation in the calculus than conceptualize the infinite process of limit.
在算术中，初始焦点则是“过程”，例如计数、加法、减法，以及用于表示这些过程的符号。随着后续认知发展，“过程被封装为算术概念过程体”（以符号为二者的枢纽）的认知复杂程度也会随之变化。在算术中，所有符号都内置了“得出答案”的计算过程；在代数中，符号表现为代数表达式，仅具有“潜在的内部计算过程”——即当变量被赋予数值时对表达式进行求值。尽管如此，这些符号本身仍可进行代数变形，且通过有限次变形可求解一次方程与二次方程。到了微积分领域，情况再次发生变化：极限过程具有潜在的无限性，因此“极限概念”虽关联着无限过程，却不存在有限的计算步骤。因此，尽管这些符号都遵循“过程-概念”框架（均可被归为概念过程体），但概念过程体本质的变化，导致学生在从算术到代数、从代数到微积分的过渡中面临重大认知困难。这也解释了为何学生更擅长运用微积分中有限的求导程序规则进行计算，却难以理解极限所涉及的无限过程。

Focusing on both operational processes and the properties of objects—either in turn or at the same time—gives a versatile approach (Tall & Thomas, 1991). This proves particular valuable when computer software is available to carry out the processes internally, allowing the individual to focus either on the study of the processes, which they may carry out (or program) for themselves, or on the concepts produced by the computer (Tall & Thomas, 1989). Versatile approaches have proved successful in research studies. Thomas (1988) used visual properties, such as studying the evaluation of expressions (carried out by the student) separately from the properties of (equivalent) expressions, evaluated by the computer, producing significant improvements in symbols as process and concept. Tall (1985) used the “local straightness” of graphs represented by computer software to complement the process of symbolic differentiation and produced significant improvements in visual interpretations of concepts of the slope of a graph. Hong & Thomas (1997) used a versatile approach to the integral calculus to produce corresponding improvements in conceptualization. A versatile approach has also been shown to give flexible insight into the relationship between figures in trigonometry and their corresponding symbolic trigonometric functions (Blackett & Tall, 1991). This involves the proceptual structure of the symbolism relating the process of computing the trigonometric ratio (as “opposite over hypotenuse”) to the concept of trigonometric function (in this case “sine”).
同时或交替关注“操作过程”与“对象属性”，形成了一种“灵活方法”（Tall & Thomas, 1991）。当有计算机软件可在内部执行这些过程时，这种方法尤为珍贵——它允许个体要么聚焦于“过程研究”（可自行执行或编程实现），要么聚焦于“计算机生成的概念”（Tall & Thomas, 1989）。灵活方法已在多项研究中被证实有效：托马斯（Thomas, 1988）利用视觉属性（如让学生独立完成表达式求值，同时与计算机计算的等价表达式属性进行对比），显著提升了学生对“符号兼具过程与概念属性”的理解；托尔（Tall, 1985）利用计算机软件呈现的图像“局部平直性”，补充符号求导过程，大幅改善了学生对“图像斜率概念”的视觉理解；洪与托马斯（Hong & Thomas, 1997）将灵活方法应用于积分学，同样提升了学生的概念理解水平；此外，灵活方法还能让学生灵活理解三角学图形与其对应符号三角函数之间的关系（Blackett & Tall, 1991），这涉及符号的概念过程体结构——即“计算三角函数比值的过程”（如“对边比斜边”）与“三角函数概念”（此处为“正弦”）之间的关联。

In elementary mathematics a versatile approach links proceptual symbolism of arithmetic and algebra to visual representations using a number line and coordinate plane. As visual and symbolic aspects grow in sophistication, they may be linked together in different ways, with a curriculum designed to build from those aspects more easily grasped by the individual to building up more sophisticated aspects. This occurs in the “locally straight” approach to the calculus where the visual and enactive notion of local straightness (with its dynamic computer pictures) can be explored before, or in tandem with, the numerical or algebraic facets which have a more sophisticated notion of “locally linear” (with its ensuing symbolism). Likewise the inverse process of solving differential equations can be performed visually and enactively supported by computer software to give a perceived solution before the numeric or symbolic processes are studied in detail.
在初等数学中，灵活方法通过数轴与坐标平面，将算术和代数的概念过程体符号与视觉表征关联起来。随着视觉与符号层面的认知逐渐复杂，二者可通过多种方式建立关联；课程设计应遵循“从个体易掌握的层面逐步过渡到复杂层面”的原则。例如，在微积分的“局部平直”教学法中，可先探索“局部平直”的视觉与动作概念（借助动态计算机图像），再引入或同时讲解“局部线性”的数值与代数层面（及其对应的符号体系）——后者的概念更为复杂。同样，在详细学习微分方程的数值解法或符号解法之前，可借助计算机软件，通过视觉与动作操作执行“求解微分方程”这一逆过程，从而获得对“解”的感知理解。

A further case of “process-object” construction is the notion of a defined object. For Dubinsky this is again a case of encapsulation of process into object. In the sense of Sfard, it has structural overtones because the definition specifies certain structural criteria. However, just as Sfard’s notion of “structural” has two different senses, there are two corresponding meanings to an object specified by a definition. Our earlier discussion of Euclidean geometry noted that the individual has in mind certain mental objects that are then defined linguistically, such as a line, a point, a triangle, and a circle. This notion of “definition” is consonant with the notion of definition in a dictionary. It specifies an object by referring to other familiar ideas encountered earlier by the individual. There may be well-known examples of such a defined object that all seem to have a common property, and yet this property may not follow from the specified criteria. For instance, the notion of triangle carries with it the properties inherent in two-dimensional space. Its angles add up to 180°. Yet if a triangle is defined formally in some other type of geometry, for instance the geometry on the surface of a sphere (on which we happen to live—“geometry” means “earth measurement”)—then the sum of its angles may be different. Just try cutting the skin of an orange and see. Two cuts through the “north pole” at right angles to one another meeting a cut round the equator will give a spherical triangle with three right angles, adding up to 270°.
“过程-对象”构建的另一种情况是“定义对象”（defined object）。在杜宾斯基看来，这仍是“过程封装为对象”的案例；在斯法德的理论框架下，由于定义明确了特定结构标准，因此它带有结构层面的含义。但正如斯法德的“结构的”概念存在两种含义，“通过定义明确的对象”也具有两种对应含义。前文对欧几里得几何的讨论指出，个体先在脑海中形成特定心理对象（如直线、点、三角形、圆），再通过语言对其进行定义——这种“定义”与词典中的定义类似，即通过个体已熟悉的其他概念来明确对象。这类被定义的对象可能存在广为人知的实例，且这些实例看似具有某一共同属性，但该属性未必能从定义的标准中推导得出。例如，“三角形”的概念隐含了二维空间的固有属性（内角和为180°）；但如果在其他几何体系（如球面几何——我们恰好生活在球面上，“几何”一词原意即为“大地测量”）中对三角形进行形式化定义，其内角和可能会不同。只需切开橘子皮观察即可：从“北极”切出两条互相垂直的切口，再沿赤道切出一条切口，三者形成的球面三角形具有三个直角，内角和为270°。

Euclidean constructions may carry with them certain implications that may not follow from the definitions. (One famous case in Euclid concerns the meeting of the diagonals of a rhombus “inside” the figure, when the idea of “inside” had not been defined. Although the notion of “inside” appears to be a primitive intuition, when a triangle is drawn on a sphere, the “inside” is but the smaller of the two regions into which the triangle divides the circle. Hence, as a small triangle increases smoothly in size until it is greater than half the surface area of the sphere, what may be termed the “inside” becomes the “outside” and vice-versa.)
欧几里得几何中的作图可能隐含一些无法从定义中推导的结论。（欧几里得几何中有一个著名案例：菱形的对角线相交于图形“内部”，但“内部”这一概念并未被定义。尽管“内部”看似是一种原始直觉，但当在球面上绘制三角形时，“内部”仅是三角形将球面分割成的两个区域中较小的那一个。因此，当一个小球面三角形逐渐扩大，直至其面积超过球面面积的一半时，原本所谓的“内部”会变成“外部”，反之亦然。）

The shift from Euclidean geometry, with its platonic objects based on the refined perceptions of the senses, to axiomatic geometries where everything is defined purely in terms of explicit axioms and logical deductions was noted as a further stage in cognitive development by Van Hiele. This confirms the need to distinguish between two different constructions of defined objects—platonic objects found, for example, in Euclidean geometry, and axiomatic objects constructed within a theory founded on axioms.
范·海勒指出，从“基于精细化感官感知的柏拉图式对象的欧几里得几何”，向“一切通过明确公理与逻辑推导定义的公理几何”的过渡，是认知发展的又一阶段。这一观点印证了我们有必要区分“定义对象”的两种不同构建方式：一种是如欧几里得几何中的柏拉图式对象，另一种是在基于公理的理论中构建的公理对象。

In the first case a platonic object begins with a mental image refined from experience of physical forms and then given a verbal definition to focus on the perfection of the ideas within it. In the second case, an axiomatic object may arise from experiences of explicit situations that have certain properties in common. Historically, the definitions used in axiomatic theories have only arisen after long periods of study of the phenomena to be axiomatized. But once these properties have been enshrined in axioms and set-theoretic definitions the only properties possessed by the axiomatic object are those that can be deduced by formal proof. The formal part of an axiomatic theory (e.g. group theory, vector space theory, topology) begins with an explicit set of axioms (for group, vector space, topological space) and further objects may be defined by formal definitions in each theory (e.g. sub-group, normal sub-group, coset, quotient group; spanning set, linearly independent set, basis, subspace; continuous function, uniformly continuous function, compact set, etc.).
对于柏拉图式对象，构建始于从物理形态经验中提炼出的心理图像，随后通过语言定义聚焦于图像所蕴含的完美属性。对于公理对象，其构建可能源于对“具有共同属性的明确情境”的经验；从历史角度看，公理理论中的定义往往是在对“待公理化的现象”进行长期研究后才形成的。但一旦这些属性被确立为公理和集合论定义，公理对象仅具有通过形式化证明可推导得出的属性。公理理论的形式化部分（如群论、向量空间理论、拓扑学）以一组明确的公理（群公理、向量空间公理、拓扑空间公理）为起点，且在各理论中，可通过形式化定义进一步构建对象（如子群、正规子群、陪集、商群；生成集、线性无关集、基、子空间；连续函数、一致连续函数、紧集等）。

The essential difference between platonic and axiomatic objects is that in the first the object is the initial focus of attention and the definition is constructed from the object. In the axiomatic theory of the second, the definition is the initial focus of attention and the object is constructed from the definition, determining its properties by proving theorems. Gray et al. (in press) distinguish these by referring to “object→definition” construction of platonic objects and “definition→object” construction of axiomatic objects.
柏拉图式对象与公理对象的核心区别在于：前者以“对象”为初始关注焦点，定义从对象中提炼而来；后者以“定义”为初始关注焦点，对象从定义中构建而来，其属性需通过证明定理来确定。格雷等人（Gray et al., in press）通过“对象→定义”（柏拉图式对象的构建方式）与“定义→对象”（公理对象的构建方式）的表述，对二者进行了区分。

This raises the question of the nature of the objects constructed through “definition→object” construction. For instance, if we write down the axioms for a mathematical group, in what sense is there an object that we call “a group”? We arrive at the same conundrum as before. We speak of “an animal” without necessarily having in our mind’s eye a visual mental image for a “general animal”, we know how to manipulate “5” without having a specific mental image for it as an object. Likewise we can build up the properties of an “axiomatic object” by deduction from the given axiom or definition. For instance, we might deduce that “there is only
这引发了一个问题：通过“定义→对象”方式构建的对象，其本质是什么？例如，当我们写下数学群的公理时，我们所说的“群”作为一个对象，究竟具有何种意义？我们再次面临此前的困境：谈论“动物”时，脑海中未必有“通用动物”的视觉心理图像；运用“5”时，也无需有“5作为对象”的特定心理图像。同样，我们可通过从给定公理或定义中推导，逐步明确“公理对象”的属性。例如，我们可推导出“群中存在唯一的单位元”。

one, unique, identity element in a group”. Theorems deduced in this way give properties shared by all structures that satisfy the criteria. We use the same linguistic conventions in speaking of “a group” as we do in speaking of “an animal”. For instance, we can say treat it as a “thing” and say “a group has a unique identity element.”
通过这种方式推导的定理，给出了所有满足公理标准的结构所共有的属性。我们谈论“群”时所使用的语言习惯，与谈论“动物”时一致——例如，我们可将“群”视为一个“事物”，并表述为“群具有唯一的单位元”。

Summary

总结

In this paper we have discussed the ways in which mental objects can be constructed in mathematics and have distinguished three types of object construction, each of which operates in an increasingly sophisticated manner:
本文探讨了数学中心理对象的构建方式，并区分出三种对象构建类型，其复杂程度依次递增：

• perceived objects, arising through empirical abstraction from objects in the environment (and later may be given successively subtler meaning through focusing also on verbal descriptions and definitions to construct platonic objects),
感知对象（perceived objects）：通过对环境中物体的经验抽象形成，后续可通过关注语言描述与定义，进一步赋予其更精妙的意义，进而构建出柏拉图式对象；

• procepts, which first involve processes on real-world objects, using symbols which can then be manipulated as objects, upon which operations may be performed and symbolized in the same way,
概念过程体（procepts）：最初涉及对现实世界物体的操作过程，借助符号实现“过程与对象”的转换——符号可被视为对象进行操作，且操作本身也可被符号化；

and
以及

• axiomatic objects, conceived by specifying criteria (axioms or definitions) from which properties are deduced by formal proof.
公理对象（axiomatic objects）：通过明确标准（公理或定义）构建，其属性需通过形式化证明从标准中推导得出。

These are related to Piaget’s notions of abstraction as follows. Perceived objects arise through empirical abstraction, and more sophisticated platonic objects may be later constructed through reflective abstraction. Procepts arise first through pseudo-empirical abstraction from actions on real-world objects and then by higher level reflective abstraction on the resulting conceived objects that represented by symbols enabling us to pivot between process and concept. Defined objects occur by reflective abstraction from the properties of perceived or conceived objects (including both platonic and proceptual). Selected properties are formulated as axioms and definitions and other properties are constructed as theorems through logical processes of deduction. Platonic objects, procepts and defined objects are all categories of conceived objects.
这些对象构建类型与皮亚杰的抽象概念存在如下关联：感知对象通过经验抽象形成，更复杂的柏拉图式对象可通过反思抽象后续构建；概念过程体首先通过对“现实世界物体的动作”的准经验抽象形成，随后通过对“符号所代表的构想对象”的更高层次反思抽象发展——符号使我们能够在过程与概念之间灵活转换；定义对象（包括柏拉图式对象与概念过程体）通过对感知对象或构想对象属性的反思抽象形成，其中部分属性被确立为公理与定义，其他属性则通过逻辑推导过程构建为定理。柏拉图式对象、概念过程体与定义对象均属于“构想对象”（conceived objects）的范畴。

The “object” of the encapsulation of a process is a way of thinking which uses a rich concept image to allow it to be a manipulable entity, in part by using mental processes and relationships to do mathematics and in part to use a name or symbol to mentally manipulate, and to think about mathematics.
过程封装所形成的“对象”，本质上是一种思维方式：它借助丰富的概念意象（concept image），使对象成为可操作的实体——部分通过运用心理过程与关系来开展数学活动，部分通过运用名称或符号来进行心理层面的操作与数学思考。

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• What is the object of the encapsulation of a process?
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