Using Learning and Instructional Theories in Teaching Mathematics
在数学教学中运用学习和教学理论
TUnderstanding of theories about how people leam and the ability to apply these theories in teaching mathematics are important prerequisites for effective mathematics teaching.j Many people have approached the study of intellectual development and the nature of leaming in different ways; this has resulted in several theories of learning. Although there is still some disagreement among psychologists, leaming theorists, and educators about how people learn and the most effective methods for promoting leaming, there are many areas of agreement. The different theories of learning should not be viewed as a set of competing theories, one of which is true and the others false. Each theory can be regarded as a method of organizing and studying some of the many variables in learning and intellectual development, and teachers can, select and apply elements of each theory in their own classes. You may find that some theories are more applicable to you and your students because they seem to be appropriate models for the learning environment and the students with whom you interact. However, a perceptive teacher will find some applications of each leaming theory for his or her students. As a consequence of being able to appreciate the learning theoretic reasons for various forms of behavior exhibited by each student, he or she will be a more understanding and sympathetic teachier.
了解有关人们如何学习的理论以及在数学教学中应用这些理论的能力，是有效开展数学教学的重要前提。虽然心理学家、学习理论家和教育工作者对人们如何学习以及促进学习的最有效方法仍存在一些分歧，但在许多方面达成了一致。不同的学习理论不应被视为一系列相互竞争的理论，其中一种理论是正确的，而其他理论则是错误的。每种理论都可以被视为组织和研究学习和智力发展中许多变量的一种方法，教师可以在自己的课堂上选择和应用每种理论的要素。你可能会发现，有些理论更适用于你和你的学生，因为它们似乎是适合学习环境和与你互动的学生的模式。然而，敏锐的教师会发现每种学习理论都有一些适用于自己学生的地方。由于能够从学习理论上理解每个学生的各种行为表现的原因，他或她就会成为一个更善解人意、更有同情心的教师。
In the past many mathematics teachers and teacher educators neglected the application of theories about the nature of leaming and centered their teaching methods around knowledge of the subject. Recent findings in leaming theory, better understanding of mental development, and new applications of theory to classroom teaching now enable teachers to choose teaching strategies according to information about the nature of learning. The purpose of this chapter is to present several of the major theories about the nature of intellectual development, to discuss theorics about leaming, and to illustrate applications of each theory to teaching and leaming mathematics.
过去，许多数学教师和师范教育工作者忽视了学习本质理论的应用，他们的教学方法以学科知识为中心。现在，有关学习理论的最新研究成果、对智力发展的更深入理解，以及理论在课堂教学中的新应用，使教师能够根据有关学习本质的信息来选择教学策略。本章旨在介绍有关智力发展性质的几种主要理论，讨论有关学习的理论，并说明每种理论在数学教学和学习中的应用。
了解有关人们如何学习的理论以及在数学教学中应用这些理论的能力，是有效开展数学教学的重要前提。虽然心理学家、学习理论家和教育工作者对人们如何学习以及促进学习的最有效方法仍存在一些分歧，但在许多方面达成了一致。不同的学习理论不应被视为一系列相互竞争的理论，其中一种理论是正确的，而其他理论则是错误的。每种理论都可以被视为组织和研究学习和智力发展中许多变量的一种方法，教师可以在自己的课堂上选择和应用每种理论的要素。你可能会发现，有些理论更适用于你和你的学生，因为它们似乎是适合学习环境和与你互动的学生的模式。然而，敏锐的教师会发现每种学习理论都有一些适用于自己学生的地方。由于能够从学习理论上理解每个学生的各种行为表现的原因，他或她就会成为一个更善解人意、更有同情心的教师。
In the past many mathematics teachers and teacher educators neglected the application of theories about the nature of leaming and centered their teaching methods around knowledge of the subject. Recent findings in leaming theory, better understanding of mental development, and new applications of theory to classroom teaching now enable teachers to choose teaching strategies according to information about the nature of learning. The purpose of this chapter is to present several of the major theories about the nature of intellectual development, to discuss theorics about leaming, and to illustrate applications of each theory to teaching and leaming mathematics.
过去，许多数学教师和师范教育工作者忽视了学习本质理论的应用，他们的教学方法以学科知识为中心。现在，有关学习理论的最新研究成果、对智力发展的更深入理解，以及理论在课堂教学中的新应用，使教师能够根据有关学习本质的信息来选择教学策略。本章旨在介绍有关智力发展性质的几种主要理论，讨论有关学习的理论，并说明每种理论在数学教学和学习中的应用。
First, we will look at the theory of Jean Piaget who has determined and studied the various stages through which humans progress in their intellectual growth from birth to adulthood. Next we will consider the work of J. P. Guilford, who bas developed and lested a theoretical model of human intellectual
首先，我们将研究让-皮亚杰的理论，他确定并研究了人类从出生到成年的智力成长所经历的各个阶段。接下来，我们将研究 J. P. 吉尔福德的工作，他在此基础上发展并构建了人类智力发展的理论模型。
structure. Guilford and his associates have identified one hundred twenty intellectual aptitudes which encompass many of the mental abilities which are capable of being measured and evaluated. We will also consider the work of Robert Gagné who has identified four phases of a leaming sequence. These phases are the apprehending phase, the acquisition phase, the storage phase, and the retrieval phase. Gagné also has specified eight types of leaming which can be distinguished from each other according to the necessary conditions for the occurrence of each leaming type; they are signal learning, stimulus-response learning, chaining, verbal association, discrimination learning, concept leaming, rule learning, and problem-solving. The theories and work of Zoltan Dienes are also relevant to teaching mathematics. Dienes regards mathematics as the study of structures and relationships among structures and has developed a system for mathematics education which is based upon a theory of leaming and a process for teaching mathematics. David Ausubel has made significant contributions to the study of verbal learning, which he believes can be accomplished through careful consideration of the structure of the discipline and by using appropriate principles to order the subject matter for presentation to students. The psychologist Jerome Bruner has listed general theorems for instruction and has developed a philosophy of education centered around the structural framework essential for learning, student readiness for learning, intuition, and motivation to learn. His general theories are also relevant for mathematics teachers. B. F. Skinner has conducted extensive studies of behavior and has developed a science of human behavior based upon his work in behavioral analysis. His writings suggest ways in which teachers can create more effective learning situations by using appropriate techniques to elicit desirable behaviors from students.
结构。吉尔福德和他的同事们确定了 120 种智力倾向，其中包括许多可以测量和评估的心智能力。我们还将考虑罗伯特-盖尼耶的研究成果，他确定了学习序列的四个阶段。这些阶段是理解阶段、习得阶段、储存阶段和检索阶段。盖尼耶还指出了八种学习类型，根据每种学习类型发生的必要条件，可以将它们相互区分开来，它们是信号学习、刺激-反应学习、连锁学习、言语联想、辨别学习、概念学习、规则学习和问题解决。Zoltan Dienes 的理论和著作也与数学教学相关。迪内斯认为数学是对结构和结构间关系的研究，并根据学习理论和数学教学过程建立了一套数学教育体系。戴维-奥苏贝尔（David Ausubel）对语言学习的研究做出了重大贡献，他认为通过仔细考虑学科的结构，并使用适当的原则将学科内容有序地呈现给学生，可以实现语言学习。心理学家杰罗姆-布鲁纳（Jerome Bruner）列出了教学的一般定理，并围绕学习所必需的结构框架、学生的学习准备、直觉和学习动机提出了教育哲学。他的一般理论也适用于数学教师。B. F. Skinner 对行为进行了广泛的研究，并在其行为分析工作的基础上发展了人类行为科学。 他的著作提出了一些方法，让教师可以通过使用适当的技巧来激发学生的理想行为，从而创造更有效的学习环境。
首先，我们将研究让-皮亚杰的理论，他确定并研究了人类从出生到成年的智力成长所经历的各个阶段。接下来，我们将研究 J. P. 吉尔福德的工作，他在此基础上发展并构建了人类智力发展的理论模型。
structure. Guilford and his associates have identified one hundred twenty intellectual aptitudes which encompass many of the mental abilities which are capable of being measured and evaluated. We will also consider the work of Robert Gagné who has identified four phases of a leaming sequence. These phases are the apprehending phase, the acquisition phase, the storage phase, and the retrieval phase. Gagné also has specified eight types of leaming which can be distinguished from each other according to the necessary conditions for the occurrence of each leaming type; they are signal learning, stimulus-response learning, chaining, verbal association, discrimination learning, concept leaming, rule learning, and problem-solving. The theories and work of Zoltan Dienes are also relevant to teaching mathematics. Dienes regards mathematics as the study of structures and relationships among structures and has developed a system for mathematics education which is based upon a theory of leaming and a process for teaching mathematics. David Ausubel has made significant contributions to the study of verbal learning, which he believes can be accomplished through careful consideration of the structure of the discipline and by using appropriate principles to order the subject matter for presentation to students. The psychologist Jerome Bruner has listed general theorems for instruction and has developed a philosophy of education centered around the structural framework essential for learning, student readiness for learning, intuition, and motivation to learn. His general theories are also relevant for mathematics teachers. B. F. Skinner has conducted extensive studies of behavior and has developed a science of human behavior based upon his work in behavioral analysis. His writings suggest ways in which teachers can create more effective learning situations by using appropriate techniques to elicit desirable behaviors from students.
结构。吉尔福德和他的同事们确定了 120 种智力倾向，其中包括许多可以测量和评估的心智能力。我们还将考虑罗伯特-盖尼耶的研究成果，他确定了学习序列的四个阶段。这些阶段是理解阶段、习得阶段、储存阶段和检索阶段。盖尼耶还指出了八种学习类型，根据每种学习类型发生的必要条件，可以将它们相互区分开来，它们是信号学习、刺激-反应学习、连锁学习、言语联想、辨别学习、概念学习、规则学习和问题解决。Zoltan Dienes 的理论和著作也与数学教学相关。迪内斯认为数学是对结构和结构间关系的研究，并根据学习理论和数学教学过程建立了一套数学教育体系。戴维-奥苏贝尔（David Ausubel）对语言学习的研究做出了重大贡献，他认为通过仔细考虑学科的结构，并使用适当的原则将学科内容有序地呈现给学生，可以实现语言学习。心理学家杰罗姆-布鲁纳（Jerome Bruner）列出了教学的一般定理，并围绕学习所必需的结构框架、学生的学习准备、直觉和学习动机提出了教育哲学。他的一般理论也适用于数学教师。B. F. Skinner 对行为进行了广泛的研究，并在其行为分析工作的基础上发展了人类行为科学。 他的著作提出了一些方法，让教师可以通过使用适当的技巧来激发学生的理想行为，从而创造更有效的学习环境。
Piaget's Theory of Intellectual Development
皮亚杰的智力发展理论
According to the theory of the noted Swiss psychologist Jean Piaget, human intellectual development progresses chronologically through four sequential stages. The order in which the stages occur has been found to be invariant among people; however the ages at which people enter each higher order stage vary according to each person's unique hereditary and environmental characteristics.
根据瑞士著名心理学家让-皮亚杰（Jean Piaget）的理论，人类的智力发展按时间顺序依次经历四个阶段。各阶段的发生顺序在人与人之间是不变的，但每个人进入每个高阶阶段的年龄却因每个人独特的遗传和环境特征而异。
根据瑞士著名心理学家让-皮亚杰（Jean Piaget）的理论，人类的智力发展按时间顺序依次经历四个阶段。各阶段的发生顺序在人与人之间是不变的，但每个人进入每个高阶阶段的年龄却因每个人独特的遗传和环境特征而异。
Sensary-Mfotor Stage 感知-运动阶段
The first period of intellectual development, called the sensory-motor stage, extends from birth until about two years of age. In this period the infant's leaming consists of developing and organizing his or her physical and mental activities into well-defined sequences of actions called schemas. From birth to two years of age children leam to coordinate their senses and movements, learn that an object which is removed from sight does not.cease to exist, and learn to attach word symbols to physical objects. For example, near the end of this stage a child can recognize the sound of father closing the front door to leave for work, can totter to the window and watch him get on the bus, and understands that he will return later. In this period children progress from having only reflex abilities at birth to being able to walk and talk at two years of age.
智力发展的第一阶段称为感觉运动阶段，从出生到两岁左右。在这一阶段，婴儿的学习包括发展和组织他或她的身体和心理活动，使之成为被称为图式的明确的动作序列。从出生到两岁，婴儿开始学习如何协调自己的感官和动作，学习物体离开视线并 不意味着不存在，并学习把文字符号附加到实物上。例如，在此阶段快结束时，小孩可辨认出父亲关上前门去上班的声音，可蹒跚地走到窗前看父亲上车，并懂得父亲稍后会回来。在这一阶段，儿童从出生时只有反射能力发展到两岁时能走路和说话。
The first period of intellectual development, called the sensory-motor stage, extends from birth until about two years of age. In this period the infant's leaming consists of developing and organizing his or her physical and mental activities into well-defined sequences of actions called schemas. From birth to two years of age children leam to coordinate their senses and movements, learn that an object which is removed from sight does not.cease to exist, and learn to attach word symbols to physical objects. For example, near the end of this stage a child can recognize the sound of father closing the front door to leave for work, can totter to the window and watch him get on the bus, and understands that he will return later. In this period children progress from having only reflex abilities at birth to being able to walk and talk at two years of age.
智力发展的第一阶段称为感觉运动阶段，从出生到两岁左右。在这一阶段，婴儿的学习包括发展和组织他或她的身体和心理活动，使之成为被称为图式的明确的动作序列。从出生到两岁，婴儿开始学习如何协调自己的感官和动作，学习物体离开视线并 不意味着不存在，并学习把文字符号附加到实物上。例如，在此阶段快结束时，小孩可辨认出父亲关上前门去上班的声音，可蹒跚地走到窗前看父亲上车，并懂得父亲稍后会回来。在这一阶段，儿童从出生时只有反射能力发展到两岁时能走路和说话。
Preoperational Stage 前运算阶段
The second period, the preoperational stage, extends from approximately age two to age seven. In this stage children are very egocentric; that is, they assimi- late most experiences in the world at large into schemas developed from thei immediate environment and view everything in relation to themselves. Young children believe that all their thoughts and experiences are shared by everyone else, that inanimate objects have animate characteristics, and that the distinction between one and many is of little consequence. This explains why a young child does not question a different Santa Claus on every street comer and Santa Claus mannequins in every department store window. The preoperational thinker has difficulty reversing thoughts and reconstructing actions, can not consider two aspects of an object or a situation simultaneously, and does not reason inductively (from specific to general) or deductively (from general to specific). The young child reasons transductively; that is, from specific instances to specific instances. In this stage children can not differentiate fact and fancy, which is why their "lies" are not a consequence of any moral deficiency, but result from their inability to separate real events from the world of their imaginations. Through physical maturation and interacting with his or her environment, the child in the pre-conceptual stage is developing the necessary mental schemas to operate at a higher intellectual level. Near the end of this stage children become capable of giving reasons for their beliefs, can classify sets of objects according to a single specified characteristic, and begin to attain some actual concepts.
第二个阶段是前运算阶段，大约从两岁延续到七岁。在这一阶段，儿童非常以自我为中心；也就是说，他们会把世界上的大多数经验同化为从周围环境中形成的图式，并把一切都与自己联系起来看待。幼儿认为，他们的所有想法和经验都与其他人相同，无生命的物体也有生命特征，一个人和许多人之间的区别并不重要。这就解释了为什么幼儿不会质疑每条街上都有不同的圣诞老人，也不会质疑每个百货商店橱窗里的圣诞老人人体模型。前运算思维者很难逆转思维和重构行动，不能同时考虑一个物体或一个情境的两个方面，不会进行归纳推理（从具体到一般）或演绎推理（从一般到具体）。幼儿的推理是反推式的，即从具体事例推理到具体事例。在这一阶段，幼儿无法区分事实和幻想，这就是为什么他们的 "谎言 "不是道德缺失的结果，而是因为他们无法将真实事件与想象世界区分开来。通过身体的成熟和与环境的互动，前概念阶段的儿童正在形成必要的心智模式，以便在更高的智力水平上运作。在这一阶段接近尾声时，儿童开始能够为自己的信念提供理由，能够根据单一的特定特征对一组物体进行分类，并开始获得一些实际的概念。
第二个阶段是前运算阶段，大约从两岁延续到七岁。在这一阶段，儿童非常以自我为中心；也就是说，他们会把世界上的大多数经验同化为从周围环境中形成的图式，并把一切都与自己联系起来看待。幼儿认为，他们的所有想法和经验都与其他人相同，无生命的物体也有生命特征，一个人和许多人之间的区别并不重要。这就解释了为什么幼儿不会质疑每条街上都有不同的圣诞老人，也不会质疑每个百货商店橱窗里的圣诞老人人体模型。前运算思维者很难逆转思维和重构行动，不能同时考虑一个物体或一个情境的两个方面，不会进行归纳推理（从具体到一般）或演绎推理（从一般到具体）。幼儿的推理是反推式的，即从具体事例推理到具体事例。在这一阶段，幼儿无法区分事实和幻想，这就是为什么他们的 "谎言 "不是道德缺失的结果，而是因为他们无法将真实事件与想象世界区分开来。通过身体的成熟和与环境的互动，前概念阶段的儿童正在形成必要的心智模式，以便在更高的智力水平上运作。在这一阶段接近尾声时，儿童开始能够为自己的信念提供理由，能够根据单一的特定特征对一组物体进行分类，并开始获得一些实际的概念。
Concrete Operational Stage
混凝土运行阶段
The concrete operational stage of mental development extends from age seven to age twelve, thitteen or even later. At the beginning of this stage there is a substantial decrease in children's egocentricity; play with other children replaces isolated play and individualized play in the presence of other children. In this stage children become able to classify objects having several characteristics into sets and subsets according to specified characteristics, and they can simultaneously consider several characteristics of an object. They begin to understand jokes; however they still have trouble explaining proverbs and fail to see hidden meanings. They are now able to deal with complex relationships between classes, can reverse operations and procedures, and can understand and visualize intermediate states of a transformation such as the sun rising and setting. In the concrete operational period children become able to see another person's viewpoint and near the end of this period begin to reason inductively and deductively; however many still tend to regard successive examples of a general principle as unrelated events.
心理发展的具体操作阶段从 7 岁开始，到 12 岁、13 岁甚至更晚。在这一阶段的初期，儿童的自我中心意识大大减弱；与其他儿童一起玩耍取代了在有其他儿童在场的情况下单独玩耍和个性化玩耍。在此阶段，幼儿开始懂得把具有若干特征的物体按特定的特征分为集合和子集合，并能同时考虑一个物体的若干特征。他们开始听懂笑话，但在解释谚语时仍有困难，看不出其中隐藏的含义。现在，他们已懂得处理类与类之间的复杂关系，懂得逆向运算和程序，懂得并可视 化变换的中间状态，如太阳升起和落下。在具体运算阶段，幼儿开始懂得从他人的角度看问题，并在此阶段快结束时开始进 行归纳和演绎推理；但许多幼儿仍倾向于把一般原理的连续例子视为毫不相干的事件。
心理发展的具体操作阶段从 7 岁开始，到 12 岁、13 岁甚至更晚。在这一阶段的初期，儿童的自我中心意识大大减弱；与其他儿童一起玩耍取代了在有其他儿童在场的情况下单独玩耍和个性化玩耍。在此阶段，幼儿开始懂得把具有若干特征的物体按特定的特征分为集合和子集合，并能同时考虑一个物体的若干特征。他们开始听懂笑话，但在解释谚语时仍有困难，看不出其中隐藏的含义。现在，他们已懂得处理类与类之间的复杂关系，懂得逆向运算和程序，懂得并可视 化变换的中间状态，如太阳升起和落下。在具体运算阶段，幼儿开始懂得从他人的角度看问题，并在此阶段快结束时开始进 行归纳和演绎推理；但许多幼儿仍倾向于把一般原理的连续例子视为毫不相干的事件。
Although children in this stage do develop many of the intellectual abilities found in adults, they have difficulties understanding verbal abstractions. They can perform complex operations such as reversibility, substitution, unions and intersections of sets, and serial orderings on concrete objects, but may not be able to carry out these same operations with verbal symbols. Their powers of judgment and logical reasoning are not well developed, and they rarely can solve a problem such as: Jane is taller than Bill; Jane is shorter than Susan; who is shortest of the three? However, children in this stage can order a pile of sticks from shortest to longest. Before the end of this period children are seldom able to formulate a precise, descriptive definition; although they can memorize another person's definition and reproduce what they have memorized. In this stage children learn to differentiate between deliberate wrongdoing and inadvertant mistakes. Even after developing a conception of rules and morality, they still attach
虽然此阶段的幼儿已具备成人的许多智力，但他们对语言抽象概念的理解仍有困难。他们懂得对具体物体进行还原、替换、集合的联合和相交以及序列排序等复杂运算， 但却无法用语言符号进行这些运算。他们的判断力和逻辑推理能力都不发达，很少能解决诸如以下的问题：简比比尔高；简比苏珊矮；三人中谁最矮？不过，此阶段的幼儿懂得把一堆木棒按从短到长的顺序排列。在这一阶段结束前，幼儿很少能提出准确的、描述性的定义，但他们能记住别人的 定义，并能把记住的东西再现出来。在这一阶段，儿童学会区分故意的错误行为和无意的错误。即使在形成了规则和道德的概念后，他们仍然会把 "错 "和 "不对 "挂钩。
a mystical aura to the origin of rules, morals, laws, and conventions, as well as the origin of names. To preadolescent children, a rose is called a rose because it is a rose, not because someone named it a rose.
规则、道德、法律和惯例的起源以及名称的起源都带有神秘色彩。对于青春期前的孩子来说，玫瑰之所以被称为玫瑰，是因为它就是玫瑰，而不是因为有人给它起了个玫瑰的名字。
虽然此阶段的幼儿已具备成人的许多智力，但他们对语言抽象概念的理解仍有困难。他们懂得对具体物体进行还原、替换、集合的联合和相交以及序列排序等复杂运算， 但却无法用语言符号进行这些运算。他们的判断力和逻辑推理能力都不发达，很少能解决诸如以下的问题：简比比尔高；简比苏珊矮；三人中谁最矮？不过，此阶段的幼儿懂得把一堆木棒按从短到长的顺序排列。在这一阶段结束前，幼儿很少能提出准确的、描述性的定义，但他们能记住别人的 定义，并能把记住的东西再现出来。在这一阶段，儿童学会区分故意的错误行为和无意的错误。即使在形成了规则和道德的概念后，他们仍然会把 "错 "和 "不对 "挂钩。
a mystical aura to the origin of rules, morals, laws, and conventions, as well as the origin of names. To preadolescent children, a rose is called a rose because it is a rose, not because someone named it a rose.
规则、道德、法律和惯例的起源以及名称的起源都带有神秘色彩。对于青春期前的孩子来说，玫瑰之所以被称为玫瑰，是因为它就是玫瑰，而不是因为有人给它起了个玫瑰的名字。
This developmental period is called concrete operational because psychologists have found that children between seven and twelve have trouble applying formal intellectual processes to verbal symbols and abstract ideas; even though by age twelve most children have become quite adept at using their intellect to manipulate concrete physical objects. In this period children like to build things, manipulate objects, and make mechanical gadgets operate.
心理学家发现，7 到 12 岁的儿童很难把正式的智力过程应用于语言符号和抽象概念，因此把这一发展阶段称为具体运算阶段。在此阶段，儿童喜欢造东西、摆弄物体和使机械小工具运转。
心理学家发现，7 到 12 岁的儿童很难把正式的智力过程应用于语言符号和抽象概念，因此把这一发展阶段称为具体运算阶段。在此阶段，儿童喜欢造东西、摆弄物体和使机械小工具运转。
Formal Operational Stage 正式运行阶段
When adolescents reach the formal operational stage, they no longer need to rely upon concrete operations to represent or illustrate mental abstractions. They are now able to simultaneously consider many viewpoints, to regard their own actions objectivety, and to reflect upon their own thought processes. The formal operational thinker can formulate theories, generate hypotheses, and test various hypotheses. People who have reached this intellectual stage can appreciate degrees of good and evil and can view definitions, rules and laws in a proper, objective context. They can also think inductively and deductively and can argue by implication (i.e., if
当青少年进入形式运算阶段时，他们不再需要依靠具体运算来表示或说明思维抽象。他们现在能够同时考虑多种观点，客观地看待自己的行为，并反思自己的思维过程。形式运算思维者可以提出理论、产生假设并检验各种假设。达到这一智力阶段的人能够理解善恶的程度，能够在正确、客观的背景下看待定义、规则和法律。他们还能进行归纳和演绎思维，并能通过暗示进行论证（即，如果
当青少年进入形式运算阶段时，他们不再需要依靠具体运算来表示或说明思维抽象。他们现在能够同时考虑多种观点，客观地看待自己的行为，并反思自己的思维过程。形式运算思维者可以提出理论、产生假设并检验各种假设。达到这一智力阶段的人能够理解善恶的程度，能够在正确、客观的背景下看待定义、规则和法律。他们还能进行归纳和演绎思维，并能通过暗示进行论证（即，如果
Factors in Intellectual Development
智力发展的因素
Piagetian theory explains intellectual development as a process of assimilation and accommodation of information into the mental structure. Assimilation is the process through which new information and experiences are incorporated into mental structure, and accommodation is the resulting restructuring of the mind as a consequence of new information and experiences. The mind not only receives new information but it restructures its old information to accommodate the new. For example, new information about a political personality is not only added to the mind's old information about that person. This information may also alter the individual's viewpoint of politics, politicians, and govemment in general, and may even change his or her moral and ethical values. Leaming is not merely adding new information to the stack of old information, because every piece of new information causes the stack of old information to be modified to accommodate the assimilation of the new information.
皮亚杰理论将智力发展解释为信息与心理结构的同化和调适过程。同化是将新信息和经验纳入心理结构的过程，而调适则是由于新信息和经验而导致的心理结构重组。思维不仅会接收新信息，还会重组旧信息以适应新信息。例如，有关政治人物的新信息不仅会添加到头脑中有关此人的旧信息中。这些信息还可能改变个人对政治、政治家和政府的总体看法，甚至可能改变其道德和伦理价值观。阅读并不仅仅是在旧信息堆中添加新信息，因为每一条新信息都会导致旧信息堆被修改，以适应新信息的吸收。
皮亚杰理论将智力发展解释为信息与心理结构的同化和调适过程。同化是将新信息和经验纳入心理结构的过程，而调适则是由于新信息和经验而导致的心理结构重组。思维不仅会接收新信息，还会重组旧信息以适应新信息。例如，有关政治人物的新信息不仅会添加到头脑中有关此人的旧信息中。这些信息还可能改变个人对政治、政治家和政府的总体看法，甚至可能改变其道德和伦理价值观。阅读并不仅仅是在旧信息堆中添加新信息，因为每一条新信息都会导致旧信息堆被修改，以适应新信息的吸收。
According to Piaget's theory, there are several factors influencing intellectual development. First, the physiological growth of the brain and nervous system is an important factor in general intellectual progress. This growth process is called maturation. Piaget also recognizes the importance of experience in mental development and identifies two types of experience. Physical experience is the interaction of each person with objects in his or her environment, and logicomathematical experiences are those mental actions performed by individuals as their mental schemas are restructured according to their experiences. Another factor, social transmission, is the interaction and cooperation of a person with other people and is quite important for the development of logic in a child's mind. Piaget believes that formal operations would not develop in the mind without an exchange and coordination of viewpoints among people. The last factor, equilibration, is the process whereby a person's mental structure loses its stability as a consequence of new experiences and returns to equilibrium through the processes of assimilation and accommodation. As a result of equilibration, mental structures develop and mature. Piaget believes that these live factors (maturation, physical experience, logico-mathematical experience, social transmission, and equilibration) account for intellectual development and that each one must be present if a person is to progress through the four stages of intellectual development.
根据皮亚杰的理论，影响智力发展的因素有几个。首先，大脑和神经系统的生理成长是智力普遍进步的一个重要因素。这一成长过程被称为成熟。皮亚杰还认识到经验在智力发展中的重要性，并确定了两种类型的经验。物理经验是指每个人与周围环境中的物体的相互作用，而逻辑数学经验则是指个人根据自己的经验对心理图式进行重组时所产生的心理行为。另一个因素，即社会传递，是一个人与其他人的互动与合作，对儿童思维中逻辑的发展相当重要。皮亚杰认为，如果没有人与人之间观点的交流与协调，形式运算就不会在头脑中形成。最后一个因素是 "平衡"（equilibration），是指一个人的心理结构因新经验而失去稳定性，并通过同化和调适过程恢复平衡的过程。平衡的结果是心理结构的发展和成熟。皮亚杰认为，这些活生生的因素（成熟、身体经验、逻辑数学经验、社会传递和平衡）是智力发展的原因，一个人要想在智力发展的四个阶段中取得进步，就必须具备每一个因素。
根据皮亚杰的理论，影响智力发展的因素有几个。首先，大脑和神经系统的生理成长是智力普遍进步的一个重要因素。这一成长过程被称为成熟。皮亚杰还认识到经验在智力发展中的重要性，并确定了两种类型的经验。物理经验是指每个人与周围环境中的物体的相互作用，而逻辑数学经验则是指个人根据自己的经验对心理图式进行重组时所产生的心理行为。另一个因素，即社会传递，是一个人与其他人的互动与合作，对儿童思维中逻辑的发展相当重要。皮亚杰认为，如果没有人与人之间观点的交流与协调，形式运算就不会在头脑中形成。最后一个因素是 "平衡"（equilibration），是指一个人的心理结构因新经验而失去稳定性，并通过同化和调适过程恢复平衡的过程。平衡的结果是心理结构的发展和成熟。皮亚杰认为，这些活生生的因素（成熟、身体经验、逻辑数学经验、社会传递和平衡）是智力发展的原因，一个人要想在智力发展的四个阶段中取得进步，就必须具备每一个因素。
The four stages of development (sensory-motor, preoperational, concrete operational, and formal operational) while sequential in nature, do not have welldefined starting and ending points. The progression from one stage to the next occurs over a period of time and each individual may vacillate in his or her ability to exhibit the higher order mental processes throughout this transitional period. Even after a person has completed the transition from one stage to the next, he or she may still use mental processes associated with the earlier stages. An adolescent who has developed his or her intellectual capabilities to the formal operational stage has the mental structures necessary to carry out formal operations, but will not always do so. Many formal operational adults frequently count on their fingers which is a preoperational trait. A young person who has entered the formal operational stage will continue to improve his or her formal operational skills for many years.
发展的四个阶段（感觉运动阶段、前运算阶段、具体运算阶段和形式运算阶段）虽 然是有先后顺序的，但并没有明确的起点和终点。从一个阶段发展到下一个阶段需要一段时间，在整个过渡时期，每个人表现高阶心智过程的能力可能会出现波动。即使一个人已经完成了从一个阶段到下一个阶段的过渡，他或她仍然可以使用与早期阶段相关的心理过程。智力已发展到正规运算阶段的青少年拥有进行正规运算所需的心理结构，但并不总能这样做。许多正式运算阶段的成年人经常数手指，这属于前运算阶段的特征。进入正规运算阶段的青少年，其正规运算能力会在多年内不断提高。
发展的四个阶段（感觉运动阶段、前运算阶段、具体运算阶段和形式运算阶段）虽 然是有先后顺序的，但并没有明确的起点和终点。从一个阶段发展到下一个阶段需要一段时间，在整个过渡时期，每个人表现高阶心智过程的能力可能会出现波动。即使一个人已经完成了从一个阶段到下一个阶段的过渡，他或她仍然可以使用与早期阶段相关的心理过程。智力已发展到正规运算阶段的青少年拥有进行正规运算所需的心理结构，但并不总能这样做。许多正式运算阶段的成年人经常数手指，这属于前运算阶段的特征。进入正规运算阶段的青少年，其正规运算能力会在多年内不断提高。
Piaget's Theory and Teaching Mathematics
皮亚杰理论与数学教学
Several years ago while discussing teaching methods with a young mathematics teacher, she remarked that she was appalled because most of her seventh graders could not understand even a simple proof. I asked if she had studied Piaget's learning theories in college, and she replied that she had but didn't see what that had to do with her seventh graders doing mathematical proofs. This incident illustrates the need for teachers to see the applications in their own teaching of the theories which they learn in college, and for teacher educators to show prospective teachers the applications of learning theory.
几年前，在与一位年轻的数学教师讨论教学方法时，她说她感到很震惊，因为她的七年级学生中的大多数甚至连一个简单的证明都无法理解。我问她是否在大学里学习过皮亚杰的学习理论，她回答说学习过，但不明白这与七年级学生做数学证明有什么关系。这件事说明，教师需要看到他们在大学学习的理论在自己教学中的应用，教师教育工作者也需要向未来的教师展示学习理论的应用。
皮亚杰理论与数学教学
Several years ago while discussing teaching methods with a young mathematics teacher, she remarked that she was appalled because most of her seventh graders could not understand even a simple proof. I asked if she had studied Piaget's learning theories in college, and she replied that she had but didn't see what that had to do with her seventh graders doing mathematical proofs. This incident illustrates the need for teachers to see the applications in their own teaching of the theories which they learn in college, and for teacher educators to show prospective teachers the applications of learning theory.
几年前，在与一位年轻的数学教师讨论教学方法时，她说她感到很震惊，因为她的七年级学生中的大多数甚至连一个简单的证明都无法理解。我问她是否在大学里学习过皮亚杰的学习理论，她回答说学习过，但不明白这与七年级学生做数学证明有什么关系。这件事说明，教师需要看到他们在大学学习的理论在自己教学中的应用，教师教育工作者也需要向未来的教师展示学习理论的应用。
Since seventh graders are twelve or thirteen years of age, some of them are ill in the concrete operational stage, others have just entered the stage of formal operations, and still others are in transition between these two stages of intellectual development. Consequently, many seventh grade students' intellectual development has not yet progressed to the point where they have the mental structures necessary for constructing formal mathematical proofs. Some of these students do not yet see the difference between a single instance of a general principle and a proof of that principle. This is not to say that a seventh grade teacher should not explore the nature of intuitive and formal mathematical proofs with students; however he or she should realize that a twelve year old adolescent has a different mental structure (as well as an obviously different physical structure) than a twenty-two year old teacher.
由于七年级学生已经十二三岁了，他们有的还处在具体运算阶段，有的刚刚进入形式运算阶段，还有的正处于这两个智力发展阶段之间的过渡阶段。因此，许多七年级学生的智力发展尚未达到具备构建正式数学证明所需的心理结构的程度。其中有些学生还看不出一般原理的单个实例与该原理的证明之间的区别。这并不是说，七年级的教师不应该与学生探讨直观和形式数学证明的本质；但是，他或她应该认识到，一个 12 岁的青少年与一个 22 岁的教师有着不同的心理结构（以及明显不同的生理结构）。
由于七年级学生已经十二三岁了，他们有的还处在具体运算阶段，有的刚刚进入形式运算阶段，还有的正处于这两个智力发展阶段之间的过渡阶段。因此，许多七年级学生的智力发展尚未达到具备构建正式数学证明所需的心理结构的程度。其中有些学生还看不出一般原理的单个实例与该原理的证明之间的区别。这并不是说，七年级的教师不应该与学生探讨直观和形式数学证明的本质；但是，他或她应该认识到，一个 12 岁的青少年与一个 22 岁的教师有着不同的心理结构（以及明显不同的生理结构）。
Since secondary level mathematics teachers are expected to be able to teach students in middle schools, junior high schools and high schools, they must
由于中学数学教师应能教授初中、初中和高中学生，因此他们必须
由于中学数学教师应能教授初中、初中和高中学生，因此他们必须
and eighth grade teachers can expect to find many concrete operational students in their classes, and even some high school juniors and seniors are still in this stage of intellectual development. Therefore, it is appropriate for us to examine the intellectual attributes which some secondary school students do not have, but which are required to carry out many standard school mathematics learning activities.)八年级教师可以在班上发现许多具体操作能力强的学生，甚至一些高三和高四学生也仍处于这一智力发展阶段。因此，我们应该研究一些中学生所不具备的、但开展许多标准的学校数学学习活动所需要的智力特质（informal school mathematics learning activities)。
A teacher should expect certain complex abilities, skills and behaviors from a student who is in the formal operational stage and should be concemed if formal operational mental processes are not exhibited. However, at every secondary school grade level there are students who have not completely entered the formal operational stage, and teachers should be aware of the behaviors that can be expected from these students. Such students merely illustrate the fact that people mature mentally at different ages which is analogous to the different rates of physical maturation which we have come to expect. No teacher would regard a seventh grader who is small for his age group as a physical cripple, and neither should teachers regard children who mature intellectually at a later age as being mentally retarded. Every mathematics teacher, especially those who teach in. grades six through nine, should expect many students to be in the concrete operational stage, should be understanding of students' mental inabilities in this stage, should provide leaming strategies appropriate for concrete operations, and should plan activities to help students progress to the stage of formal operations.).
教师应该期待处于正规操作阶段的学生具备某些复杂的能力、技能和行为，如果他们没有表现出正规操作的心理过程，教师就应该感到担忧。然而，在中学的每一个年级，都有一些尚未完全进入正规运算阶段的学生，教师应该了解这些学生的行为表现。这些学生只是说明了一个事实，即人在不同年龄段的心智成熟程度是不同的，这就好比我们所期望的身体成熟的速度是不同的一样。没有哪个教师会把一个在同龄人中身材矮小的七年级学生视为身体瘸子，教师也不应该把智力成熟较晚的儿童视为智力迟钝。每一位数学教师，尤其是六至九年级的数学教师，都应预期许多学生处于具体运算阶段，应理解学生在这一阶段的智力缺陷，应提供适合具体运算的学习策略，并应计划活动帮助学生进入形式运算阶段（formal operations)。
教师应该期待处于正规操作阶段的学生具备某些复杂的能力、技能和行为，如果他们没有表现出正规操作的心理过程，教师就应该感到担忧。然而，在中学的每一个年级，都有一些尚未完全进入正规运算阶段的学生，教师应该了解这些学生的行为表现。这些学生只是说明了一个事实，即人在不同年龄段的心智成熟程度是不同的，这就好比我们所期望的身体成熟的速度是不同的一样。没有哪个教师会把一个在同龄人中身材矮小的七年级学生视为身体瘸子，教师也不应该把智力成熟较晚的儿童视为智力迟钝。每一位数学教师，尤其是六至九年级的数学教师，都应预期许多学生处于具体运算阶段，应理解学生在这一阶段的智力缺陷，应提供适合具体运算的学习策略，并应计划活动帮助学生进入形式运算阶段（formal operations)。
Students in grades six through nine are difficult to teach because they are still testing their recently discovered concrete operational abilities while they are entering the formal operational stage. Concrete operational students have discovered that rules are not absolute, but are arbitrary. These students are trying out their own rules and challenging the teacher's rules, which results in what we usually call discipline problems. In this period children need to associate and talk to other children as an aid to entering the formal operational stage through the process of social transmission. As a result junior high school students may appear to teachers to be talkative, noisy, rowdy, and undisciplined. What seems to adults to be a lot of fooling around on the part of students is partly a means of fostering their intellectual development.
六至九年级的学生比较难教，因为他们在进入正式运算阶段的同时，还在检验自己刚刚发现的具体运算能力。具体运算能力强的学生发现规则不是绝对的，而是任意的。这些学生在尝试自己的规则的同时，也在挑战老师的规则，这就造成了我们通常所说的纪律问题。在这一阶段，孩子们需要与其他孩子交往和交谈，通过社会传递过程帮助自己进入正式操作阶段。因此，在老师看来，初中生可能爱说话、爱吵闹、爱闹腾、不守纪律。在成人看来，学生们的胡闹在某种程度上是促进智力发展的一种手段。
六至九年级的学生比较难教，因为他们在进入正式运算阶段的同时，还在检验自己刚刚发现的具体运算能力。具体运算能力强的学生发现规则不是绝对的，而是任意的。这些学生在尝试自己的规则的同时，也在挑战老师的规则，这就造成了我们通常所说的纪律问题。在这一阶段，孩子们需要与其他孩子交往和交谈，通过社会传递过程帮助自己进入正式操作阶段。因此，在老师看来，初中生可能爱说话、爱吵闹、爱闹腾、不守纪律。在成人看来，学生们的胡闹在某种程度上是促进智力发展的一种手段。
These students do not want to accept statements based only upon the teacher's authority and do not care to accept new concepts which are outside their ability to visualize and conceptualize. Consequently they would be unlikely to either believe or accept on faith the concept of different orders of infinities or the fact that the cardinal number of the set of counting numbers is the same as the cardinal number of the set of even counting numbers which is a proper subset of the counting numbers. In fact, most concrete operational students have trouble with the concept of infinity and indefinite subdivisions of a line segment into arbitrarily small segments.
这些学生不愿意只接受基于教师权威的说法，也不愿意接受超出其形象化和概念化能 力的新概念。因此，他们不可能相信或接受无穷大的不同阶数的概念，也不可能相信或接受这样的事实：计数数集合的底数与偶数集合的底数相同，而偶数集合是计数数的适当子集。事实上，大多数具体运算学生都很难理解无穷大的概念，也很难将线段无限细分为任意小的线段。
(Junior high school students enjoy working with diagrams, models, and other physical devices; they need to relate new abstract concepts to physical reality and their own experiences. New topics in mathematics should be introduced through concrete examples, and intuition and experimentation should play a large part in teaching strategies for new principles and concepts) In geometry one should ex- pect that many students will have trouble visualizing three-dimensional objects and relationships among objects. They will need to construct and manipulate models of geometric figures. Geometry in the junior high school should be presented informally and intuitively and formal geometric proof should wait until students are well into their formal operational stage of intellectual development. For a few people this will not happen until their freshman or sophomore year in college.
(初中生喜欢使用图表、模型和其他实物设备；他们需要将新的抽象概念与实物现实 和自身经验联系起来。新的数学课题应通过具体实例来介绍，直觉和实验应在新原理和新概念的教学策略中 发挥重要作用。他们需要构建和操作几何图形的模型。初中阶段的几何学习应该以非正式和直观的方式进行，而正式的几何证明应该等到学生进入智力发展的正式运算阶段后再进行。对于少数人来说，这要到大学一、二年级才能实现。
这些学生不愿意只接受基于教师权威的说法，也不愿意接受超出其形象化和概念化能 力的新概念。因此，他们不可能相信或接受无穷大的不同阶数的概念，也不可能相信或接受这样的事实：计数数集合的底数与偶数集合的底数相同，而偶数集合是计数数的适当子集。事实上，大多数具体运算学生都很难理解无穷大的概念，也很难将线段无限细分为任意小的线段。
(Junior high school students enjoy working with diagrams, models, and other physical devices; they need to relate new abstract concepts to physical reality and their own experiences. New topics in mathematics should be introduced through concrete examples, and intuition and experimentation should play a large part in teaching strategies for new principles and concepts) In geometry one should ex- pect that many students will have trouble visualizing three-dimensional objects and relationships among objects. They will need to construct and manipulate models of geometric figures. Geometry in the junior high school should be presented informally and intuitively and formal geometric proof should wait until students are well into their formal operational stage of intellectual development. For a few people this will not happen until their freshman or sophomore year in college.
(初中生喜欢使用图表、模型和其他实物设备；他们需要将新的抽象概念与实物现实 和自身经验联系起来。新的数学课题应通过具体实例来介绍，直觉和实验应在新原理和新概念的教学策略中 发挥重要作用。他们需要构建和操作几何图形的模型。初中阶段的几何学习应该以非正式和直观的方式进行，而正式的几何证明应该等到学生进入智力发展的正式运算阶段后再进行。对于少数人来说，这要到大学一、二年级才能实现。
Although concrete operational students can formulate and use concepts correctly they have trouble explaining concepts using mathematical and verbal symbols. As a result of this deficiency, many students (maybe even most younger students) can not solve mathematical word problems, and resort to memorizing pattems and trial and error problem-solving. Their trial and error attempts are so unsystematic that they may keep repeating incorrect trials. As might be expected, many younger high school students are unable to make meaningful definitions of mathematical terms and merely memorize definitions.
虽然具体运算能力强的学生能正确地提出和使用概念，但他们在使用数学和语言符 号解释概念时会遇到困难。由于这种缺陷，许多学生（甚至可能是大多数低年级学生）无法解决数学文字问题，只能靠记忆公式和试错来解决问题。他们的试错尝试缺乏系统性，可能会不断重复错误的试验。正如所料，许多低年级高中生无法为数学术语下有意义的定义，而只是死记定义。
虽然具体运算能力强的学生能正确地提出和使用概念，但他们在使用数学和语言符 号解释概念时会遇到困难。由于这种缺陷，许多学生（甚至可能是大多数低年级学生）无法解决数学文字问题，只能靠记忆公式和试错来解决问题。他们的试错尝试缺乏系统性，可能会不断重复错误的试验。正如所料，许多低年级高中生无法为数学术语下有意义的定义，而只是死记定义。
Concrete thinkers can not be expected to solve logical puzzles or to resolve mathematical paradoxes. Also they tend not to be able to arrive at generalizations based upon a number of similar instances. For instance, they would not arrive at the commutative principle for addition,
不能指望具体思维者解决逻辑难题或数学悖论。而且，他们往往无法根据大量类似的事例得出概括性的结论。例如，他们无法从
不能指望具体思维者解决逻辑难题或数学悖论。而且，他们往往无法根据大量类似的事例得出概括性的结论。例如，他们无法从
In conclusion, it should be pointed out that Piaget and his close associates have been concemed with studying and defining the nature and development of human thought and have not attempted to specify methods for improving teaching and learning. It has been left to others to apply the theories and findings of the Piagetians to classroom teaching. Many of the experiments which were developed to determine the stages of intellectual development involve observing and recording children's responses when they are given tasks of a mathematical nature. Consequently, some of the types of mathematical problems which children can handle at different ages and intellectual levels have been specified by the Piagetians. Even though much work on Piaget's theory of intellectual development remains to be done, his theory has gained wide acceptance among psychologists, leaming theorists, and educators. Every mathematics teacher should be familiar with Piaget's work and should apply his discoveries about mental readiness for various learning lasks to his or her own teaching. Heed the example at the beginning of this lopic of the teacher who knew the theory but never thought to apply it in her own classes. Some of the Things to Do and
总之，应该指出的是，皮亚杰和他的亲密伙伴们一直致力于研究和界定人类思维的本质和发展，并没有试图明确改进教学和学习的方法。将皮亚杰的理论和研究成果应用于课堂教学的工作留给了其他人。为确定智力发展阶段而开发的许多实验都涉及观察和记录儿童在完成数学任务时的反应。因此，皮亚杰学派明确指出了儿童在不同年龄和智力水平时可以处理的一些数学问题类型。尽管皮亚杰的智力发展理论还有许多工作要做，但他的理论已被心理学家、学习理论家和教育家广泛接受。每一位数学教师都应该熟悉皮亚杰的研究成果，并将他在各种学习任务的心理准备方面的发现应用到自己的教学中。请注意本专题开头所举的例子：有一位教师知道皮亚杰的理论，却从未想过将其应用于自己的课堂。要做的一些事情和
references in the Selected Bibliography at the end of this chapter will help you leam more about applications of Piaget's theory in mathematics teaching.
本章最后的 "参考书目选编 "中的参考文献将帮助你了解皮亚杰理论在数学教学中的更多应用。
总之，应该指出的是，皮亚杰和他的亲密伙伴们一直致力于研究和界定人类思维的本质和发展，并没有试图明确改进教学和学习的方法。将皮亚杰的理论和研究成果应用于课堂教学的工作留给了其他人。为确定智力发展阶段而开发的许多实验都涉及观察和记录儿童在完成数学任务时的反应。因此，皮亚杰学派明确指出了儿童在不同年龄和智力水平时可以处理的一些数学问题类型。尽管皮亚杰的智力发展理论还有许多工作要做，但他的理论已被心理学家、学习理论家和教育家广泛接受。每一位数学教师都应该熟悉皮亚杰的研究成果，并将他在各种学习任务的心理准备方面的发现应用到自己的教学中。请注意本专题开头所举的例子：有一位教师知道皮亚杰的理论，却从未想过将其应用于自己的课堂。要做的一些事情和
references in the Selected Bibliography at the end of this chapter will help you leam more about applications of Piaget's theory in mathematics teaching.
本章最后的 "参考书目选编 "中的参考文献将帮助你了解皮亚杰理论在数学教学中的更多应用。
J. P. Guilford's Structure of Intellect Model
J.P. 吉尔福德的智力结构模型
While Jean Piaget and others have studied the stages of intellectual development, J. P. Guilford and his colleagues have developed a three-dimensional model containing 120 distinct types of intellectual abilities. These 120 intellectual factors appear to encompass most of the human mental abilities which can be specified and measured. In formulating this model, Guilford and his associates have attempted to define and structure general intelligence into a variety of very specific mental aptitudes. Their findings verify what many perceptive teachers have observed: even very intelligent students may have difficulty carrying out certain mental tasks; whereas other students who have attained low scores on general intelligence tests may do surpisingly well at some types of mental activities. It is quite important for teachers to understand that individual students may possess a variety of specific mental strengths and weaknesses. Tests have been designed to measure many of these factors of intelligence, and it is possible to select appropriate tasks to assist people in strengthening their specific cognitive inadequacies.
让-皮亚杰（Jean Piaget）和其他人研究了智力发展的各个阶段，而 J. P. 吉尔福德（J. P. Guilford）和他的同事们则建立了一个包含 120 种不同智力类型的三维模型。这 120 种智力因素似乎涵盖了大部分可以具体化和测量的人类心智能力。吉尔福德和他的同事们在建立这个模型时，试图把一般智力定义为各种非常具体的心智能力，并把它们结构化。他们的研究结果验证了许多有洞察力的教师所观察到的现象：即使是非常聪明的学生，在完成某些智力任务时也会遇到困难；而在一般智力测验中得分较低的其他学生，在某些类型的智力活动中却表现得出奇的好。教师必须明白，每个学生都可能有各种具体的智力强项和弱项。我们已经设计了一些测试来测量其中的许多智力因素，并且可以选择适当的任务来帮助人们加强他们在认知方面的不足。
让-皮亚杰（Jean Piaget）和其他人研究了智力发展的各个阶段，而 J. P. 吉尔福德（J. P. Guilford）和他的同事们则建立了一个包含 120 种不同智力类型的三维模型。这 120 种智力因素似乎涵盖了大部分可以具体化和测量的人类心智能力。吉尔福德和他的同事们在建立这个模型时，试图把一般智力定义为各种非常具体的心智能力，并把它们结构化。他们的研究结果验证了许多有洞察力的教师所观察到的现象：即使是非常聪明的学生，在完成某些智力任务时也会遇到困难；而在一般智力测验中得分较低的其他学生，在某些类型的智力活动中却表现得出奇的好。教师必须明白，每个学生都可能有各种具体的智力强项和弱项。我们已经设计了一些测试来测量其中的许多智力因素，并且可以选择适当的任务来帮助人们加强他们在认知方面的不足。
When a teacher finds that a student seems to be unable to attain even a minimal level of mastery of certain skills, the school psychologist may be able to determine which intellectual abilities are poorly developed in that student, and may suggest activities to improve those abilities. Even a teacher who works in a school where the services of a psychologist are unavailable, or are available only for students with severe intellectual or emotional handicaps, can recognize certain inadequately developed mental skills in some students and can assist them in developing these skills. Teachers can have a significant positive influence upon the formation of each student's self image, and every teacher should recognize and encourage those unique talents which each individual possesses. Teachers can also negatively affect students. Some teachers indicate through covert and overt actions that students who are not particularly proficient and interested in the teacher's specialty have little prospect of leading a useful and happy life. Every mathematics teacher should appreciate the value of mathematics and should encourage students to learm and enjoy mathematics; however each teacher should be objective enough to understand that mathematics is only one small, and in some cases unimportant, concem in the lives of many successful people.
当教师发现学生似乎无法达到掌握某些技能的最低水平时，学校心理学家可能能够确定该学生的哪些智力能力发展较差，并建议开展一些活动来提高这些能力。即使是在没有心理学家服务的学校工作的教师，或者只为有严重智力或情感障碍的学生提供服务的教师，也能认识到一些学生的某些心智技能发展不足，并能帮助他们发展这些技能。教师可以对每个学生自我形象的形成产生重要的积极影响，每位教师都应承认并鼓励每个人所拥有的独特才能。教师也可能对学生产生负面影响。有些教师会通过或明或暗的行动来暗示那些对教师的专业不特别精通和不感兴趣的学生，他们几乎没有希望过上有用而幸福的生活。每一位数学教师都应该重视数学的价值，鼓励学生学习数学，享受数学；但是，每一位教师都应该客观地认识到，数学只是许多成功人士生活中的一个小插曲，在某些情况下甚至并不重要。
Intellectual Yariables 知识分子
Guilford's model of intellectual aptitudes, which is called The Structure of Intellect Model, was developed at the University of Califomia using a statistical procedure called factor analysis to identify and classify various mental abilities. The model was substantiated by testing people varying in age from two years through adulthood. The Structure of Intellect Model, which has been used as a tool by researchers studying the variables in intelligence, characterizes leaming and intellectual development as being composed of three variables. The first of these variables, operations, is the set of mental processes used in learning. The second variable, contents, categorizes the nature of the material being learned.
吉尔福德的智力倾向模型被称为 "智力结构模型"，是加利福利亚大学利用一种名为 "因子分析 "的统计程序，对各种智力能力进行识别和分类而开发出来的。该模型通过对从两岁到成年的不同年龄段的人进行测试而得到证实。智力结构模型 "一直被研究智力变量的研究人员用作工具，它认为学习和智力发展由三个变量组成。第一个变量 "操作 "是指学习中使用的一系列心理过程。第二个变量 "内容 "是指学习材料的性质。
Products, the third variable in intelligence, refers to the manner in which inforJ.F. Guilj Structure of Int mation is organized in the mind.
产品是智力的第三个变量，指的是信息在头脑中的组织方式。
当教师发现学生似乎无法达到掌握某些技能的最低水平时，学校心理学家可能能够确定该学生的哪些智力能力发展较差，并建议开展一些活动来提高这些能力。即使是在没有心理学家服务的学校工作的教师，或者只为有严重智力或情感障碍的学生提供服务的教师，也能认识到一些学生的某些心智技能发展不足，并能帮助他们发展这些技能。教师可以对每个学生自我形象的形成产生重要的积极影响，每位教师都应承认并鼓励每个人所拥有的独特才能。教师也可能对学生产生负面影响。有些教师会通过或明或暗的行动来暗示那些对教师的专业不特别精通和不感兴趣的学生，他们几乎没有希望过上有用而幸福的生活。每一位数学教师都应该重视数学的价值，鼓励学生学习数学，享受数学；但是，每一位教师都应该客观地认识到，数学只是许多成功人士生活中的一个小插曲，在某些情况下甚至并不重要。
Intellectual Yariables 知识分子
Guilford's model of intellectual aptitudes, which is called The Structure of Intellect Model, was developed at the University of Califomia using a statistical procedure called factor analysis to identify and classify various mental abilities. The model was substantiated by testing people varying in age from two years through adulthood. The Structure of Intellect Model, which has been used as a tool by researchers studying the variables in intelligence, characterizes leaming and intellectual development as being composed of three variables. The first of these variables, operations, is the set of mental processes used in learning. The second variable, contents, categorizes the nature of the material being learned.
吉尔福德的智力倾向模型被称为 "智力结构模型"，是加利福利亚大学利用一种名为 "因子分析 "的统计程序，对各种智力能力进行识别和分类而开发出来的。该模型通过对从两岁到成年的不同年龄段的人进行测试而得到证实。智力结构模型 "一直被研究智力变量的研究人员用作工具，它认为学习和智力发展由三个变量组成。第一个变量 "操作 "是指学习中使用的一系列心理过程。第二个变量 "内容 "是指学习材料的性质。
Products, the third variable in intelligence, refers to the manner in which inforJ.F. Guilj Structure of Int mation is organized in the mind.
产品是智力的第三个变量，指的是信息在头脑中的组织方式。
Operations of the Mind 心灵的运作
Guilford has identified five types of mental operations which he calls memory. cognition, evaltation, convergent production, and divergent production. Memory is the ability to store information in the mind and to call out stored information in response to certain stimuli. Cognition is the ability to recognize various forms of information and to understand information. Evaluation is the ability to procesis information in order to make judgments, draw conclusions, and arrive at decisions. Convergent production is the ability to take a specified set of information and draw a universally accepted conclusion or response based upon the given information. Divergent production is the creative ability to view given information in a new way so that unique and unexpected conclusions are the consequence. A student who immediately answers
吉尔福特确定了五种心理运作类型，他称之为记忆、认知、评估、聚合生成和发散生成。记忆是将信息储存在头脑中，并在特定刺激下调出储存信息的能力。认知是识别各种形式的信息和理解信息的能力。评价是处理信息以做出判断、得出结论和做出决定的能力。聚合生成是指根据给定的信息，获取一组特定信息并得出普遍接受的结论或反应的能力。发散思维是一种创造性的能力，它能以一种新的方式看待给定的信息，从而得出独特的、出乎意料的结论。当要求学生给出
吉尔福特确定了五种心理运作类型，他称之为记忆、认知、评估、聚合生成和发散生成。记忆是将信息储存在头脑中，并在特定刺激下调出储存信息的能力。认知是识别各种形式的信息和理解信息的能力。评价是处理信息以做出判断、得出结论和做出决定的能力。聚合生成是指根据给定的信息，获取一组特定信息并得出普遍接受的结论或反应的能力。发散思维是一种创造性的能力，它能以一种新的方式看待给定的信息，从而得出独特的、出乎意料的结论。当要求学生给出
Contents of Learning 学习内容
Guilford, in his Structure of Intellect Model, identifies four types of conten involved in learning. He calls the things that are leained figural, symbolic, semanic, and behavioral contents. Figural contents are shapes and forms such as triangles, cubes, parabolas, etc. Symbolic contents are symbols or codes representing concrete objects or abstract concepts.
吉尔福德在《智力结构模型》中指出，学习涉及四种类型的内容。他把被学习的东西称为具象内容、符号内容、半具象内容和行为内容。形象内容是指三角形、立方体、抛物线等形状和形式。符号内容是代表具体物体或抽象概念的符号或代码。
吉尔福德在《智力结构模型》中指出，学习涉及四种类型的内容。他把被学习的东西称为具象内容、符号内容、半具象内容和行为内容。形象内容是指三角形、立方体、抛物线等形状和形式。符号内容是代表具体物体或抽象概念的符号或代码。
Products of Learning 学习产品
In Guilford's Model, the six products of leaming (the way information is identified and organized in the mind) are units, classes, relations, systems, transfor mations, and implications. A unit is a single symbol, figure, word, object, or idea. Sets of units are called classes, and one mental ability is that of classifying units. Relations are connections among units and classes. In our minds we or-
在吉尔福特模式中，学习（信息在头脑中识别和组织的方式）的六种产物是单位、类别、关系、系统、转换和影响。单位是指单一的符号、图形、单词、物体或概念。单位的集合称为类，一种思维能力就是对单位进行分类。关系是单位和类之间的联系。在我们的头脑中，我们或
lationships among these two products of leaming A system is a composition of units, classes, and relationships into a larger and more meaningful structure. Transformation is the process of modifying, reinterpreting, and restructuring existing information into new information. The transformation abiliy is usually thought to be a characteristic of creative people. An implication is a prediction or a conjecture about the consequences of interactions among units, classes, relations, systems, and transformations. The way in which the real number system is structured illustrates how the mind organizes information into the six products of learning. Each real number can be considered as a unit, and the entire set of real numbers is a class. Equality and inequality are relations in the set of real numbers. The set of real numbers together with the operations of addition, subtraction, multiplication, and division and the algebraic properties of these operations is a mathematical system. Functions defined on the real number system are transformations, and each theorem about functions on the real numbers is an implication.
系统是由单元、类和关系组成的一个更大、更有意义的结构。转换是将现有信息修改、重新解释和重组为新信息的过程。转化能力通常被认为是创造性人才的特征。蕴涵是对单位、类、关系、系统和变换之间相互作用的后果的预测或猜想。实数系统的结构方式说明了思维如何将信息组织成六种学习产品。每个实数都可视为一个单位，而整个实数集则是一个类。相等和不相等是实数集合中的关系。实数集连同加、减、乘、除运算以及这些运算的代数性质是一个数学系统。定义在实数系统上的函数是变换，关于实数上函数的每个定理都是蕴涵。
In Guilford's Model, the six products of leaming (the way information is identified and organized in the mind) are units, classes, relations, systems, transfor mations, and implications. A unit is a single symbol, figure, word, object, or idea. Sets of units are called classes, and one mental ability is that of classifying units. Relations are connections among units and classes. In our minds we or-
在吉尔福特模式中，学习（信息在头脑中识别和组织的方式）的六种产物是单位、类别、关系、系统、转换和影响。单位是指单一的符号、图形、单词、物体或概念。单位的集合称为类，一种思维能力就是对单位进行分类。关系是单位和类之间的联系。在我们的头脑中，我们或
lationships among these two products of leaming A system is a composition of units, classes, and relationships into a larger and more meaningful structure. Transformation is the process of modifying, reinterpreting, and restructuring existing information into new information. The transformation abiliy is usually thought to be a characteristic of creative people. An implication is a prediction or a conjecture about the consequences of interactions among units, classes, relations, systems, and transformations. The way in which the real number system is structured illustrates how the mind organizes information into the six products of learning. Each real number can be considered as a unit, and the entire set of real numbers is a class. Equality and inequality are relations in the set of real numbers. The set of real numbers together with the operations of addition, subtraction, multiplication, and division and the algebraic properties of these operations is a mathematical system. Functions defined on the real number system are transformations, and each theorem about functions on the real numbers is an implication.
系统是由单元、类和关系组成的一个更大、更有意义的结构。转换是将现有信息修改、重新解释和重组为新信息的过程。转化能力通常被认为是创造性人才的特征。蕴涵是对单位、类、关系、系统和变换之间相互作用的后果的预测或猜想。实数系统的结构方式说明了思维如何将信息组织成六种学习产品。每个实数都可视为一个单位，而整个实数集则是一个类。相等和不相等是实数集合中的关系。实数集连同加、减、乘、除运算以及这些运算的代数性质是一个数学系统。定义在实数系统上的函数是变换，关于实数上函数的每个定理都是蕴涵。
The
| Guilford's Factors of Intellectual Ability 吉尔福特的智力因素 |
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| Operations 业务 | Contents 目录 | Products 产品 | ||||
| 1. memory 1. 记忆 | 1. figural 1. 具象 | 1. units 1. 单位 | ||||
| 2. cognition 2. 认知 | 2. symbolic 2. 象征性 |
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2. classes 2. 班级 | |||
| 3. evaluation 3. 评估 | 7 | 3. semantic 3. 语义 | 7 | 3. relations 3. 关系 | ||
| 4. convergent production 4. 趋同生产 | 4. behavioral 4. 行为 | 4. systeris 4. 系统 | ||||
| 5. divergent production 5. 分歧生产 | 5. 转变 6. 影响
|
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Although this model of human intelligence is useful in identifying variables in leaming and helps to explain various learning aptitudes and abilities, one limitation of the Structure of Intellect Model should be noted. Any attempt to structure and categorize complex human abilities into a model must result in an oversimplification of reality. Most of the facts, skills, principles, and concepts which teachers teach and students leam require complex combinations of intellectual abilities. When a student is unable to construct proofs in plane geometry, it may be quite difficult to determine which mental aptitude (or set of aptitudes) is causing this leaming problem. Proving theorems in plane geometry may require a unique combination of a large subset of the 120 intellectual abilities, and most mathematics teachers have neither the skills nor resources to identify and
尽管这一人类智力模型有助于确定学习中的变量，有助于解释各种学习本领和能力， 但智力结构模型的一个局限性值得注意。任何试图将复杂的人类能力结构化并归类为一种模式的尝试，其结果必然是对现实的过度简化。教师教授和学生学习的大多数事实、技能、原理和概念都需要复杂的智力组合。当学生无法构建平面几何的证明时，可能很难确定是哪种（或哪组）智力能力导致了这一学习问题。证明平面几何中的定理可能需要 120 种智力中很大一部分的独特组合。
尽管这一人类智力模型有助于确定学习中的变量，有助于解释各种学习本领和能力， 但智力结构模型的一个局限性值得注意。任何试图将复杂的人类能力结构化并归类为一种模式的尝试，其结果必然是对现实的过度简化。教师教授和学生学习的大多数事实、技能、原理和概念都需要复杂的智力组合。当学生无法构建平面几何的证明时，可能很难确定是哪种（或哪组）智力能力导致了这一学习问题。证明平面几何中的定理可能需要 120 种智力中很大一部分的独特组合。
ices of a trained psychologist may be required to determine precisely the intellectual deficiencies in a particular student and prescribe remedial activities, every teacher should leam to recognize certain general learning insufficiences and assist students in overcoming some of their leaming problems. The first step in dealing with these natural human intellectual variations is to recognize that every student's intellect is comprised of many different factors which may be present in varying degrees in eacli student. The next ștep is to observe each student's individual performance in specified areas of mathematics and altempt to identify his or her distinct strengths and weaknesses. The third step is to provide individualized work (as students' needs require and time permits) for students so that they can both apply their stronger intellectual abilities in learning mathematics and improve their weaker intellectual aptitudes. This step suggests that there are wo approaches to overcoming learning handicaps. One approach is for the learner to bypass his or her weaknesses and apply his or her intellectual strengths to each task. Another approach is to attempt to strengthen intellectual deficiencies. Both methods of attacking intellectual shortcomings are useful and both can be employed simultaneously in the classroom. Finally every teacher should strive to leam more about the nature of intelligence and leaming by reading professional joumals and participating in inservice workshops, college courses, and postbaccalaureate programs.每个教师都应学会识别某些一般性的学习缺陷，并帮助学生克服一些学习问题。处理这些人类自然的智力差异的第一步是认识到每个学生的智力是由许多不同的因素组成的，这些因素可能在每个学生身上都有不同程度的存在。下一步是观察每个学生在特定数学领域的个人表现，并试图找出其明显的优势和劣势。第三步是根据学生的需要和时间的允许，为学生提供个性化的作业，使他们既能在数学学习中运用自己较强的智力，又能改善自己较弱的智力。这一步表明，克服学习障碍有两种方法。一种方法是让学习者绕过自己的弱点，在每项任务中运用自己的智力优势。另一种方法是试图加强智力缺陷。这两种弥补智力缺陷的方法都很有用，而且在课堂上可以同时使用。最后，每位教师都应努力通过阅读专业刊物，参加在职讲习班、大学课程和学士后课程，更多地了解智力和学习的本质。
A good source for further study of Guilford's Structure of Intellect Model and its interpretation and applications in teaching is Mary Meeker's book (1969) The Structure of Intellect. In this book Dr. Meeker defines each one of the 120 intellectual factors, cites tests to measure most of the factors, and suggests classroom activities and experiences which may be useful in strengthening each coghitive factor. To illustrate the format in which Dr. Meeker presents each intellectual factor, her discussion of cognition of symbolic classes is quoted below:
要进一步研究吉尔福德的智力结构模型及其在教学中的解释和应用，玛丽-米克尔的著作（1969 年）《智力结构》是一个很好的资料来源。在这本书中，米克尔博士对 120 个智力因素逐一进行了定义，列举了测量大多数智力因素的测试方法，并提出了有助于加强每个智力因素的课堂活动和经验。为了说明米克尔博士介绍每个智力因素的格式，下面引用她对符号类认知的论述：
要进一步研究吉尔福德的智力结构模型及其在教学中的解释和应用，玛丽-米克尔的著作（1969 年）《智力结构》是一个很好的资料来源。在这本书中，米克尔博士对 120 个智力因素逐一进行了定义，列举了测量大多数智力因素的测试方法，并提出了有助于加强每个智力因素的课堂活动和经验。为了说明米克尔博士介绍每个智力因素的格式，下面引用她对符号类认知的论述：
COGNITION OF SYMBOLIC CLASSES (CSC) is̀ the ability to recognize common properties in sets of symbolic information.
符号类认知（CSC）是̀ 在符号信息集合中识别共同属性的能力。
符号类认知（CSC）是̀ 在符号信息集合中识别共同属性的能力。
Tests 测试
Number-Group Naming. State what it is that three given numbers have in common.
数组命名。说出三个给定数字的共同点。
Number Classification. Select one of five altemative numbers to fit into each of four classes of three given numbers each. .
数字分类。从五个备选数字中选择一个，分别归入四个类别（每个类别有三个给定的数字）。
Best Number Pairs. Choose one of three number pairs that makes the most exclusive (best) class.
最佳数对。从三个数对中选出一个最独特（最佳）的数对。
Other than the factor tests, few group-achievement tests include items in which symbols are classified.
除因子测试外，很少有群体成就测试包含对符号进行分类的项目。
数组命名。说出三个给定数字的共同点。
Number Classification. Select one of five altemative numbers to fit into each of four classes of three given numbers each. .
数字分类。从五个备选数字中选择一个，分别归入四个类别（每个类别有三个给定的数字）。
Best Number Pairs. Choose one of three number pairs that makes the most exclusive (best) class.
最佳数对。从三个数对中选出一个最独特（最佳）的数对。
Other than the factor tests, few group-achievement tests include items in which symbols are classified.
除因子测试外，很少有群体成就测试包含对符号进行分类的项目。
Curriculum Suggestions 课程建议
Using the above tests as models, teachers at any grade level can develop exercises within the context of their arithmetic tasks. Classifications in algebraic symbols will differ from classifications in multiplication or geometry. The primary goal would be the recognition of common properties in the sub ject matter. Chemistry, which is composed primarily of symbolic information, is predicated upon a classification model. Even here, though, the symbols can be classified in other unique ways. A close visual inspection of the
以上述测试为范本，任何年级的教师都可以根据自己的算术任务编制练习。代数符号的分类将不同于乘法或几何的分类。主要目标是认识子课题的共同性质。化学主要由符号信息组成，它以分类模式为基础。不过，即使在这里，符号也可以以其他独特的方式进行分类。仔细观察
以上述测试为范本，任何年级的教师都可以根据自己的算术任务编制练习。代数符号的分类将不同于乘法或几何的分类。主要目标是认识子课题的共同性质。化学主要由符号信息组成，它以分类模式为基础。不过，即使在这里，符号也可以以其他独特的方式进行分类。仔细观察
Robert Gagné's Theory of Learning
罗伯特-盖尼耶的学习理论
The research of the psychologist Robert
心理学家罗伯特
心理学家罗伯特
The Objects of Mathematics Learning
数学学习的对象
Before examining Gagné's four phases of a leaming sequence and eight types of learming, it is appropriate to discuss the objects of mathematics learning, which are considered in his theory. These objects of mathematics learning are those direct and indirect things which we want students to learn in mathematics. The direct objects of mathematics leaming are facts, skills, concepts, and principles; some of the many indirect objects are transfer of leaming, inquiry ability, problem-solving ability, self-discipline, and appreciation for the structure of mathematics. The direct objects of mathematics leaming-facts, skills, concepts, and principles-are the four categories into which mathematical content can be separated.
在研究盖尼耶的四阶段学习顺序和八种学习类型之前，我们应该先讨论一下他的理论中所考虑的数学学习对象。这些数学学习的对象是我们希望学生学习数学的直接和间接事物。数学学习的直接对象是事实、技能、概念和原理；许多间接对象中的一些是学习迁移、探究能力、解决问题的能力、自律和对数学结构的欣赏。数学学习的直接对象--事实、技能、概念和原理--是数学内容可分为的四个类别。
Mathematical facts are those arbitrary conventions in mathematics such as the symbols of mathematics. It is a fact that 2 is the symbol for the word two, that + is the symbol for the operation of addition, and that sine is the name given to a special function in trigonometry. Facts are learried through various techniques of rote learming such as memorization, drill, practice, timed tests, games, and conlests. People are considered to have leamed a fact when they can state the fact and make appropriate use of it in a number of different situations.
数学事实是数学中的任意约定，如数学符号。例如，"2 "是 "二 "的符号，"+"是加法运算的符号，"正弦 "是三角函数中一个特殊函数的名称。事实是通过各种死记硬背的方法学习的，如记忆、操练、练习、计时测验、游戏和竞赛。当人们能够说出一个事实，并能在许多不同的情况下适当地使用这个事实时，就可以认为他们已经学会了这个事实。
Mathematical skills are those operations and procedures which students and mathematicians are expected to carry out with speed and accuracy. Many skills can be specified by sets of rules and instructions or by ordered sequences of specific procedures called algorithms. Among the mathematical skills which most people are expected to master in school are long division, addition of fractions and multiplication of decimal fractions. Constructing right angles, bisecting angles, and finding unions or intersections of sets of objects and events are examples of other useful mathematical skills. Skills are leamed through demonstrations and various types of drill and practice such as worksheets, work at the chalkboard, group activities and games. Students have mastered a skill when they can correctly demonstrate the skill by solving different types of problems requiring the skill or by applying the skill in various situations.
数学技能是指学生和数学家需要快速准确地完成的操作和程序。许多技能可以通过一组规则和指令或称为算法的特定程序的有序序列来规定。大多数人在学校要掌握的数学技能包括长除法、分数加法和十进制分数乘法。其他有用的数学技能还包括直角的构造、角的平分以及寻找物体和事件集合的结合点或交叉点。技能的学习是通过示范和各种类型的练习，如作业纸、黑板作业、小组活动和游戏等。当学生能够通过解决不同类型的需要技能的问题或在各种情况下应用技能来正确展示技能时，他们就掌握了技能。
在研究盖尼耶的四阶段学习顺序和八种学习类型之前，我们应该先讨论一下他的理论中所考虑的数学学习对象。这些数学学习的对象是我们希望学生学习数学的直接和间接事物。数学学习的直接对象是事实、技能、概念和原理；许多间接对象中的一些是学习迁移、探究能力、解决问题的能力、自律和对数学结构的欣赏。数学学习的直接对象--事实、技能、概念和原理--是数学内容可分为的四个类别。
Mathematical facts are those arbitrary conventions in mathematics such as the symbols of mathematics. It is a fact that 2 is the symbol for the word two, that + is the symbol for the operation of addition, and that sine is the name given to a special function in trigonometry. Facts are learried through various techniques of rote learming such as memorization, drill, practice, timed tests, games, and conlests. People are considered to have leamed a fact when they can state the fact and make appropriate use of it in a number of different situations.
数学事实是数学中的任意约定，如数学符号。例如，"2 "是 "二 "的符号，"+"是加法运算的符号，"正弦 "是三角函数中一个特殊函数的名称。事实是通过各种死记硬背的方法学习的，如记忆、操练、练习、计时测验、游戏和竞赛。当人们能够说出一个事实，并能在许多不同的情况下适当地使用这个事实时，就可以认为他们已经学会了这个事实。
Mathematical skills are those operations and procedures which students and mathematicians are expected to carry out with speed and accuracy. Many skills can be specified by sets of rules and instructions or by ordered sequences of specific procedures called algorithms. Among the mathematical skills which most people are expected to master in school are long division, addition of fractions and multiplication of decimal fractions. Constructing right angles, bisecting angles, and finding unions or intersections of sets of objects and events are examples of other useful mathematical skills. Skills are leamed through demonstrations and various types of drill and practice such as worksheets, work at the chalkboard, group activities and games. Students have mastered a skill when they can correctly demonstrate the skill by solving different types of problems requiring the skill or by applying the skill in various situations.
数学技能是指学生和数学家需要快速准确地完成的操作和程序。许多技能可以通过一组规则和指令或称为算法的特定程序的有序序列来规定。大多数人在学校要掌握的数学技能包括长除法、分数加法和十进制分数乘法。其他有用的数学技能还包括直角的构造、角的平分以及寻找物体和事件集合的结合点或交叉点。技能的学习是通过示范和各种类型的练习，如作业纸、黑板作业、小组活动和游戏等。当学生能够通过解决不同类型的需要技能的问题或在各种情况下应用技能来正确展示技能时，他们就掌握了技能。
A concept in mathematics is an abstract idea which enables people to classify objects or events and to specify whether the objects and events are examples or nonexamples of the abstract idea. Sets. subsets, equality, inequality, triangle, cube, radius and exponent are all examples of concepts. A person who has learned the concept of triangle is able to classify sets of figures into subsets of triangles and non-triangles. Concepts can be leamed either through definitions or
数学中的概念是一种抽象概念，它使人们能够对物体或事件进行分类，并明确物体和事件是抽象概念的例子还是非例子。集合、子集、相等、不等式、三角形、立方体、半径和指数都是概念的例子。学习过三角形概念的人能够将图形集合分为三角形子集和非三角形子集。概念可以通过定义或
数学中的概念是一种抽象概念，它使人们能够对物体或事件进行分类，并明确物体和事件是抽象概念的例子还是非例子。集合、子集、相等、不等式、三角形、立方体、半径和指数都是概念的例子。学习过三角形概念的人能够将图形集合分为三角形子集和非三角形子集。概念可以通过定义或
learn to classify plane objects into sets of triangles, circles, or squares; however few young children would be able to define the concept of a triangle. A concept is learned by hearing, seeing, handling, discussing, or thinking about a variely of examples and non-examples of the concept and by contrasting the examples and nonexamples. Younger children who are in Piaget's stage of concrete operations usually need to see or handle physical representations of a concept to leam it; whereas older formal operational people may be able to learn concepts through discussion and contemplation. A person has learned a concept when he or she is able to separate examples of the concept from nonexamples.然而，很少有幼儿能够定义三角形的概念。一个概念是通过听、看、操作、讨论或思考该概念的各种实例和非实例，并通过实例和非实例的对比来学习的。处于皮亚杰具体操作阶段的年幼儿童通常需要看到或操作概念的实物来学习概念；而年长的正式操作人员则可以通过讨论和思考来学习概念。当一个人能够将概念的例子与非例子区分开来时，他或她就学会了一个概念。
Principles are the most complex of the mathematical objects. Principles are sequences of concepts logether with relationships among these concepts. The statements, "two triangles are congruent if two sides and the included angle of one triangle are equal to two sides and the included angle of the other" and "the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides" are examples of principles. Each of these principles involves several concepts and relationships among these concepts. To understand the principle about congruent triangles, one must know the concepts triangle, angle, and side. 7 According to Gagne (1966) in a chapter appearing in the book
原理是最复杂的数学对象。原理是一连串的概念以及这些概念之间的关系。例如，"如果一个三角形的两条边和所包含的角等于另一个三角形的两条边和所包含的角，则两个三角形全等 "和 "直角三角形斜边的平方等于其他两条边的平方和 "就是原理的例子。每条原理都涉及多个概念和这些概念之间的关系。要理解全等三角形的原理，必须知道三角形、角和边的概念。7 根据加涅（1966 年）在《全等三角形》一书中的一章所述
原理是最复杂的数学对象。原理是一连串的概念以及这些概念之间的关系。例如，"如果一个三角形的两条边和所包含的角等于另一个三角形的两条边和所包含的角，则两个三角形全等 "和 "直角三角形斜边的平方等于其他两条边的平方和 "就是原理的例子。每条原理都涉及多个概念和这些概念之间的关系。要理解全等三角形的原理，必须知道三角形、角和边的概念。7 根据加涅（1966 年）在《全等三角形》一书中的一章所述
Harris: 哈里斯It would appear, then, that principles can be distinguished from what have previously been called concepts in two ways. First, the performance required to demonstrate that a concept has been learned is simply an identification, that is, a choice from a number of alternatives; a principle, in contrast, must be demonstrated by means of performances that identify its component concepts and the operation relating them to one another. Second, this means that the inference to be made about mediating processes is different in the two cases. A concept is a single mediator that represents a class of stimuli (or objects), whereas a principle is a sequence of mediators, each one of which is itself a concept. (pp. 86-87)
由此看来，原则与以前所谓的概念有两方面的区别。首先，要证明一个概念已经学会，所需要的表现仅仅是一种识别，即从一系列备选方案中做出选择；相反，原则则必须通过表演来证明，即识别其组成概念以及将它们相互联系起来的操作。其次，这意味着在两种情况下对中介过程的推断是不同的。概念是代表一类刺激物（或对象）的单一中介物，而原理则是一连串中介物，每一个中介物本身都是一个概念。(第 86-87 页）
由此看来，原则与以前所谓的概念有两方面的区别。首先，要证明一个概念已经学会，所需要的表现仅仅是一种识别，即从一系列备选方案中做出选择；相反，原则则必须通过表演来证明，即识别其组成概念以及将它们相互联系起来的操作。其次，这意味着在两种情况下对中介过程的推断是不同的。概念是代表一类刺激物（或对象）的单一中介物，而原理则是一连串中介物，每一个中介物本身都是一个概念。(第 86-87 页）
- Principles can be leamed through processes of scientific inquiry, guided discovery lessons, group discussions, the use of problem solving strategies and denonstrations. A student has leamed a principle when he or she can identify the concepts included in the principle, put the concepts in their correct relation to one another, and apply the principle to a particular situation.J
可以通过科学探究过程、引导式发现课程、小组讨论、使用问题解决策略和演示来学习原理。当学生能够识别原理中包含的概念，正确处理概念之间的关系，并将原理应用于特定情境时，他或她就掌握了原理。
It probably would not be a very precise or useful activity to classify all the objects of secondary school mathematics into the four object categories-facts, skills, concepts and principles. Even the experts in mathematics and learning theory would disagree about the proper category for many mathematical objects. In general, the objects progress in order of complexity from simple facts, to skills and concepts, through complex principles. Also the classification of many (maybe even most) mathematical objects is relative to the observer's own viewpoint, which is an important fact (or is that a principle!) for every mathematics teacher to know. A student who merely memorizes the quadratic formula knows a fact. A student who can plug numbers into the quadratic formula and come up with two answers has leamed a skill. A student who can classify 5,3 , and 4 as
要把中学数学的所有对象归入事实、技能、概念和原理这四个对象类别，可能并不是 一项非常精确或有用的活动。即使是数学和学习理论方面的专家，也会对许多数学对象的适当类别产生分歧。一般来说，数学对象的复杂程度依次从简单的事实、技能和概念到复杂的原理。而且，许多（甚至大多数）数学对象的分类是相对于观察者自己的观点而言的，这是每个数学教师都必须知道的重要事实（或者说是原则！）。仅仅记住二次方程式的学生知道一个事实。能把数字输入二次方程式并得出两个答案的学生已经掌握了一种技能。能把 5、3 和 4 归为
constanis and
将
I As a mathematics teacher, fou should develop testing and observation techniques to assist in recognizing students viewpoints of the concepts and principles which you are teaching. All ofushave at times memorized the proofs of theorems, with no understanding of the concepts and principles involved in the proof, in order to pass tests. While this subterfuge is a form of learning, it is not what teachers hope to have students leam by proving theorems: The point to recognize here is that many times when teachers are teaching what they view as mathematical principles, students are internalizing as facts or skills the information which is being presented:
作为一名数学教师，应发展测试和观察技巧，以帮助认识学生对所教概念和原理的观点。我们每个人都曾为了通过考试而死记硬背定理的证明，却对证明中涉及的概念和原理一无所知。虽然这种 "潜规则 "也是一种学习形式，但它并不是教师希望学生通过证明定理来学习的内容：这里需要认识到的一点是，很多时候，当教师在讲授他们所认为的数学原理时，学生只是将教师所提供的信息内化为事实或技能：
The Pltases of A Learning Sequence.
学习序列的各个阶段
Gagne has identified eight sets of conditions that distinguish eight learning types which he calls signal learning, stimulus-response leaming, chaining, verbal association, discrimination learning, concept learning, rule leaming, and problem solving. Gagné believes that each one of these eight learning types occurs in the learner in four sequential phases. He calls these phases the apprehending phase, the acquisition phase, the storage phase, and the retrieval phase.
加涅确定了八组条件，将八种学习类型区分开来，他称之为信号学习、刺激-反应学习、连锁学习、言语联想、辨别学习、概念学习、规则学习和问题解决。盖尼耶认为，这八种学习类型中的每一种都会在学习者身上依次经历四个阶段。他将这四个阶段分别称为领会阶段、习得阶段、储存阶段和检索阶段。
要把中学数学的所有对象归入事实、技能、概念和原理这四个对象类别，可能并不是 一项非常精确或有用的活动。即使是数学和学习理论方面的专家，也会对许多数学对象的适当类别产生分歧。一般来说，数学对象的复杂程度依次从简单的事实、技能和概念到复杂的原理。而且，许多（甚至大多数）数学对象的分类是相对于观察者自己的观点而言的，这是每个数学教师都必须知道的重要事实（或者说是原则！）。仅仅记住二次方程式的学生知道一个事实。能把数字输入二次方程式并得出两个答案的学生已经掌握了一种技能。能把 5、3 和 4 归为
constanis and
将
I As a mathematics teacher, fou should develop testing and observation techniques to assist in recognizing students viewpoints of the concepts and principles which you are teaching. All ofushave at times memorized the proofs of theorems, with no understanding of the concepts and principles involved in the proof, in order to pass tests. While this subterfuge is a form of learning, it is not what teachers hope to have students leam by proving theorems: The point to recognize here is that many times when teachers are teaching what they view as mathematical principles, students are internalizing as facts or skills the information which is being presented:
作为一名数学教师，应发展测试和观察技巧，以帮助认识学生对所教概念和原理的观点。我们每个人都曾为了通过考试而死记硬背定理的证明，却对证明中涉及的概念和原理一无所知。虽然这种 "潜规则 "也是一种学习形式，但它并不是教师希望学生通过证明定理来学习的内容：这里需要认识到的一点是，很多时候，当教师在讲授他们所认为的数学原理时，学生只是将教师所提供的信息内化为事实或技能：
The Pltases of A Learning Sequence.
学习序列的各个阶段
Gagne has identified eight sets of conditions that distinguish eight learning types which he calls signal learning, stimulus-response leaming, chaining, verbal association, discrimination learning, concept learning, rule leaming, and problem solving. Gagné believes that each one of these eight learning types occurs in the learner in four sequential phases. He calls these phases the apprehending phase, the acquisition phase, the storage phase, and the retrieval phase.
加涅确定了八组条件，将八种学习类型区分开来，他称之为信号学习、刺激-反应学习、连锁学习、言语联想、辨别学习、概念学习、规则学习和问题解决。盖尼耶认为，这八种学习类型中的每一种都会在学习者身上依次经历四个阶段。他将这四个阶段分别称为领会阶段、习得阶段、储存阶段和检索阶段。
The first phase of leaming, the apprehending phase, is the leamer's awareness of a stimulus or a set of stimuli which are present in the learning situation. Awareness, or altending, will lead the leamer to perceive characteristics of the set of stimuli. What the leamer perceives will be uniquely coded by each individual and will be registered in his or her mind. This idiosyncratic way in which each learner apprehends a given stimulus results in a common problem in teaching and leaming. When a teacher presents a lesson (stimuli) he or she may perceive different characteristics of the content of the lesson than are perceived by students, and each student may have a somewhat different perception than every other student. This is to say that learning is a unique process within each ștudent, and as a consequence each student is responsible for his or her own learning because of the unique way in which he or she perceives each learning situation. The uniqueness of individual perceptions explains why students will interpret facts, concepts, and principles differently from the way a teacher meant for them to be interpreted. Although this situation may make teaching and leaming somewhat imprecise and unpredictable, it does have many advantages for society. Each person is able to apply his or her unique perceptions of a problem and its solution to a group discussion of the problem, which results in more appropriate solutions of problems in our society.
学习的第一阶段，即领会阶段，是指学习者意识到学习情境中存在的一个或一组刺激。这种意识（或称 "觉察"）会使学习者感知到这组刺激的特征。学习者的感知将被每个人独特地编码，并记录在他或她的头脑中。每个学习者理解特定刺激的这种独特方式，导致了教学和学习中的一个常见问题。当教师呈现课程（刺激）时，他或她对课程内容的感知可能与学生对课程内容的感知不同，而每个学生对课程内容的感知也可能与其他学生有所不同。这就是说，学习是每个学生内心的一个独特过程，因此，每个学生都要对自己的学习负责，因为他（她）对每个学习情境的感知都是独特的。个人感知的独特性解释了为什么学生对事实、概念和原理的理解会与教师的理解不同。尽管这种情况可能会使教学和学习变得有些不精确和不可预测，但它对社会确实有许多好处。每个人都能在小组讨论中运用自己对问题及其解决方案的独特认识，从而为我们的社会带来更恰当的问题解决方案。
学习的第一阶段，即领会阶段，是指学习者意识到学习情境中存在的一个或一组刺激。这种意识（或称 "觉察"）会使学习者感知到这组刺激的特征。学习者的感知将被每个人独特地编码，并记录在他或她的头脑中。每个学习者理解特定刺激的这种独特方式，导致了教学和学习中的一个常见问题。当教师呈现课程（刺激）时，他或她对课程内容的感知可能与学生对课程内容的感知不同，而每个学生对课程内容的感知也可能与其他学生有所不同。这就是说，学习是每个学生内心的一个独特过程，因此，每个学生都要对自己的学习负责，因为他（她）对每个学习情境的感知都是独特的。个人感知的独特性解释了为什么学生对事实、概念和原理的理解会与教师的理解不同。尽管这种情况可能会使教学和学习变得有些不精确和不可预测，但它对社会确实有许多好处。每个人都能在小组讨论中运用自己对问题及其解决方案的独特认识，从而为我们的社会带来更恰当的问题解决方案。
The next phase in learning, the acquisition phase, is attaining or possessing the fact, skill, concept, or principle which is to be learned. Acquisition of mathematical knowledge can be determined by observing or measuring the fact that a person does not possess the required knowledge or behayior before an appropriate stimulus is presented, and that he or she has allained the required knowledge or behavior immediately after presentation of the stimulus.
学习的下一阶段是习得阶段，即达到或掌握所要学习的事实、技能、概念或原理。数学知识的习得可以通过观察或测量以下事实来确定：在适当的刺激出现之前，一个人并不具备所需的知识或行为，而在刺激出现之后，他或她立即获得了所需的知识或行为。
Alter a person has acquired a new capability, it must be retained or remembered. This is the storage phase of learning. The human storage facility is the memory, and research indicates that there are two types of memory. Short-term memory has a limited capacity for information and lasts for a short period of time. Most people can retain seven or eight distinct pieces of information in their short-term memories for up to thirty seconds. An example of how short-term memory operates is our ability to look up a seven digit telephone number, remember it for a few seconds while we are dialing, and forget the number as soon as someone answers our call. Long-term memory is our ability to remember information for a longer period of time than thirty seconds, and much of what we learn is stored permanently in our minds.
一个人掌握了一种新能力后，必须将其保留或记忆下来。这就是学习的储存阶段。人类的存储设备是记忆，研究表明记忆有两种类型。短时记忆的信息容量有限，持续时间很短。大多数人的短时记忆可以保留七八条不同的信息，最长可达三十秒。短时记忆运作的一个例子是，我们能够查找一个七位数的电话号码，在拨号时能记住几秒钟，而一旦有人接听电话，我们就会忘记这个号码。长时记忆是指我们对信息的记忆时间超过三十秒的能力，我们所学到的大部分知识都会永久储存在我们的脑海中。
学习的下一阶段是习得阶段，即达到或掌握所要学习的事实、技能、概念或原理。数学知识的习得可以通过观察或测量以下事实来确定：在适当的刺激出现之前，一个人并不具备所需的知识或行为，而在刺激出现之后，他或她立即获得了所需的知识或行为。
Alter a person has acquired a new capability, it must be retained or remembered. This is the storage phase of learning. The human storage facility is the memory, and research indicates that there are two types of memory. Short-term memory has a limited capacity for information and lasts for a short period of time. Most people can retain seven or eight distinct pieces of information in their short-term memories for up to thirty seconds. An example of how short-term memory operates is our ability to look up a seven digit telephone number, remember it for a few seconds while we are dialing, and forget the number as soon as someone answers our call. Long-term memory is our ability to remember information for a longer period of time than thirty seconds, and much of what we learn is stored permanently in our minds.
一个人掌握了一种新能力后，必须将其保留或记忆下来。这就是学习的储存阶段。人类的存储设备是记忆，研究表明记忆有两种类型。短时记忆的信息容量有限，持续时间很短。大多数人的短时记忆可以保留七八条不同的信息，最长可达三十秒。短时记忆运作的一个例子是，我们能够查找一个七位数的电话号码，在拨号时能记住几秒钟，而一旦有人接听电话，我们就会忘记这个号码。长时记忆是指我们对信息的记忆时间超过三十秒的能力，我们所学到的大部分知识都会永久储存在我们的脑海中。
The fourth phase of leaming, the retrieval phase, is the ability to call out the information that has been acquired and stored in memory. The process of information retrieval is very imprecise, disorganized, and even mystical. At times desired information such as a name can not be retrieved from memory upon demand, but will "pop up" later when one is thinking about something that appears to be completely unrelated to the moment when the name was wanted. Other information is stored so deeply in memory that special techniques such as electrical stimulation of the brain or hypnosis are required to initiate retrieval.
学习的第四个阶段，即检索阶段，是调出已获得并存储在记忆中的信息的能力。信息检索的过程是非常不精确的、无序的，甚至是神秘的。有时，想要的信息（如名字）并不能按要求从记忆中检索出来，而是会在稍后思考与想要名字的那一刻似乎完全无关的事情时 "突然出现"。还有一些信息被深深地储存在记忆中，需要使用特殊的技术，如对大脑进行电刺激或催眠，才能开始检索。
学习的第四个阶段，即检索阶段，是调出已获得并存储在记忆中的信息的能力。信息检索的过程是非常不精确的、无序的，甚至是神秘的。有时，想要的信息（如名字）并不能按要求从记忆中检索出来，而是会在稍后思考与想要名字的那一刻似乎完全无关的事情时 "突然出现"。还有一些信息被深深地储存在记忆中，需要使用特殊的技术，如对大脑进行电刺激或催眠，才能开始检索。
These four phases of human leaming-apprehending, acquisition, storage, and retrieval-have been incorporated into the design of computer systems, although in a much less complex form than they appear in human beings. A computer apprehends electronic stimuli from the computer user, acquires these stimuli in its central processing unit, stores the information present in the stimuli in one of its memory devices, and retrieves the information upon demand. The infinitely (?) more complex leaming process in people is illustrated every day in mathematics classrooms. If students are to learn a procedure for finding an approximation to the square root of any number which is not a perfect square, they must apprehend the method, acquire the method, store it in memory, and retrieve the square root algorithm when it is needed. To aid students in progressing through these four stages in learning the square root algorithm, the teacher evokes apprehension by working through an example on the chalkboard, facilitates acquisition by having each student work an example by following, step-bystep, a list of instructions, assists storage by giving problems for homework, and evokes retrieval by giving a quiz the next day.
人类学习的这四个阶段--领会、获取、存储和检索--已被纳入计算机系统的设计中，尽管其复杂程度远不及人类。计算机接收来自计算机用户的电子刺激，在中央处理单元中获取这些刺激，将刺激中的信息存储在其中一个存储设备中，并根据需要检索信息。在数学课堂上，人们的学习过程无限（？如果学生要学习求任何非完全平方数的平方根近似值的程序，他们必须理解方法，掌握方法，将其存储在记忆中，并在需要时检索平方根算法。为了帮助学生在学习平方根算法的这四个阶段中取得进展，教师通过在黑板上板书一个例题来唤起学生的理解；通过让每个学生按照指令表一步一步地板书一个例题来促进学生的掌握；通过布置家庭作业中的问题来帮助学生记忆；以及通过第二天的测验来唤起学生的检索。
人类学习的这四个阶段--领会、获取、存储和检索--已被纳入计算机系统的设计中，尽管其复杂程度远不及人类。计算机接收来自计算机用户的电子刺激，在中央处理单元中获取这些刺激，将刺激中的信息存储在其中一个存储设备中，并根据需要检索信息。在数学课堂上，人们的学习过程无限（？如果学生要学习求任何非完全平方数的平方根近似值的程序，他们必须理解方法，掌握方法，将其存储在记忆中，并在需要时检索平方根算法。为了帮助学生在学习平方根算法的这四个阶段中取得进展，教师通过在黑板上板书一个例题来唤起学生的理解；通过让每个学生按照指令表一步一步地板书一个例题来促进学生的掌握；通过布置家庭作业中的问题来帮助学生记忆；以及通过第二天的测验来唤起学生的检索。
Types of Learning 学习类型
The eight types of leaming which Gagné has identified and studied (signal learning, stimulus-response leaming, chaining, verbal association, discrimination learning, concept leaming, rule learning, and problem-solving) will be presented and explained below. Some of the conditions appropriate for facilitating each leaming type will be discussed.
下面将介绍和解释盖尼耶所确定和研究的八种学习类型（信号学习、刺激-反应学习、连锁学习、言语联想、辨别学习、概念学习、规则学习和问题解决）。此外，还将讨论促进每种游戏类型的一些适当条件。
下面将介绍和解释盖尼耶所确定和研究的八种学习类型（信号学习、刺激-反应学习、连锁学习、言语联想、辨别学习、概念学习、规则学习和问题解决）。此外，还将讨论促进每种游戏类型的一些适当条件。
Signal Learning 信号学习
Signal learning is involuntary learning resulting from either a single instance or a number of repetitions of a stimulus which will evoke an emotional response in an individual. When a person says "I can't eat shrimp anymore because I once had a traumatic experience while eating them," that person is describing an example
信号学习是指通过单次或多次重复刺激而产生的非自愿学习，这种刺激会唤起个体的情绪反应。当一个人说："我不能再吃虾了，因为我曾经在吃虾的时候有过痛苦的经历。
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信号学习是指通过单次或多次重复刺激而产生的非自愿学习，这种刺激会唤起个体的情绪反应。当一个人说："我不能再吃虾了，因为我曾经在吃虾的时候有过痛苦的经历。
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of undesirable signal leaming. Signal learning is emotional learning and just as emotions can be either positive or negative, so also can the outcomes of signal learning be pleasant or unpleasant. Driving past your childhood home may evoke a flood of pleasant memories, while walking into your high school chemistry laboratory could be rather unpleasant if chemistry was difficult and frustrating for you. The examples in the previous sentence illustrate that signal learning can occur over a long period of time with a number of stimuli evoking a variety of pleasant or unpleasant responses. Signal leaming can also occur from a single instance of an event which evoked intense emotional response; that was the case for the person who disliked shrimp. Another example of signal leaming happening from a single event is that of a person who will not sing with other people present as a consequence of a first grade music teacher shouting at a little girl and slapping her with a ruler because she violated a rule during a group sing. The reason why many high school students dislike mathematics may be that they have experienced a set of unpleasant events in elementary school which they associate with a mathematics classroom. The cliché that "success breeds success and failre breeds failure" is a statement of the consequences of signal learning.
不良信号学习。信号学习是一种情感学习，正如情感可以是积极或消极的一样，信号学习的结果也可以是令人愉快或不愉快的。开车经过儿时的家，可能会唤起你大量愉快的回忆；而走进高中化学实验室，如果化学对你来说是困难和令人沮丧的，你可能会感到相当不愉快。前一句话中的例子说明，信号学习可以在很长一段时间内发生，通过一些刺激唤起各种令人愉快或不愉快的反应。信号学习也可能发生在某一事件引起强烈情绪反应的单一实例中；不喜欢吃虾的人就是这种情况。另一个由单一事件引起信号传递的例子是，一个人不会和其他人一起唱歌，因为一年级的音乐老师对一个小女孩大喊大叫，并用尺子打她，因为她在集体唱歌时违反了规则。许多高中生不喜欢数学的原因，可能是他们在小学经历了一系列不愉快的事件，并将其与数学课堂联系在一起。"成功孕育成功，失败孕育失败 "这句俗语道出了信号学习的后果。
In order for signal leaming to occur, there must be a neutral signal stimulus and a second, unexpected stimulus that will evoke an emotional response in the learner which he or she will associate with the neutral stimulus. In the example of the person who leamed to fear group signing in a first grade music class, the geutral signal stimulus was singing in a group and the unexpected stimuli were a shout and a slap. People who have a high anxiety level tend to acquire responses through signal learning more rapidly than do nonanxious people. A few harsh remarks by the teacher to a shy, nervous seventh grader sitting in a mathematics classroom may condition a dislike for mathematics in that person. Signal learning cannot be easily controlled by the leamer and can have considerable influence on his or her actions. Consequently, you as a mathematics teacher, should attempt to generate unconditioned stimuli which will evoke pleasant emotions in your students and hope that they will associate some of these pleasant sensations with the neutral signal which is your mathematics classroom. While many conscious attempts to generate positive, unexpected stimuli may fail to evoke the desired positive associations with neutral signals, inadvertently generated nega'tive stimuli can at times do considerable damage to a student's desire to leam the subject which you teach.
要使信号学习发生，必须有一个中性信号刺激和第二个意外刺激，后者会唤起学习者的情绪反应，学习者会将其与中性刺激联系起来。在害怕一年级音乐课上集体签名的人的例子中，中性信号刺激是集体唱歌，意外刺激是一声喊叫和一记耳光。与不焦虑的人相比，焦虑水平高的人往往更快地通过信号学习获得反应。在数学课堂上，老师对一个害羞、紧张的七年级学生说几句严厉的话，可能会让这个学生对数学产生厌恶感。信号学习不容易被学习者控制，而且会对其行为产生相当大的影响。因此，作为数学教师，你应该尝试产生非条件刺激，唤起学生的愉快情绪，并希望他们将这些愉快的感觉与数学课堂这一中性信号联系起来。虽然许多有意识地尝试产生积极的、意想不到的刺激可能无法唤起学生对中性信号的积极联想，但无意中产生的消极刺激有时会对学生学习你所教学科的愿望造成相当大的伤害。
Stimulus-Response Learning
刺激-反应学习
Stimulus-response learning is also leaming to respond to a signal; however, this form of learning differs in two ways from signal learning. Signal leaming is involuntary and emotional; whereas stimulus-response learning is voluntary and physical. Stimulus-response leaming involves voluntary movements of the learner's skeletal muscles in response to stimuli so that the learner can carry out an action when he or she wants to. This form of leaming requires an external stimulus, which causes an internal muscular stimulation, followed by the desired response with a single, direct connection between the stimulus and the response. In stimulus-response learning a stimulus is presented to an individual who may react to the stimulus in several different ways. Each time the desired response occurs, the individual receives a positive reinforcement, which may be a word of praise or a satisfying experience. As a result of a senies of reinforcements for a Robert Gagne'sdesired response, the individual learns to discriminate the appropriate response from a set of other less desirable responses which could also follow the occurrence of the stimulus.
刺激-反应学习也是对信号做出反应的学习，但这种学习形式与信号学习有两点不同。信号学习是非自愿的、情绪化的；而刺激-反应学习则是自愿的、物理性的。刺激-反应学习涉及学习者骨骼肌肉的自愿运动，以对刺激做出反应，这样学习者就能在想做的时候做动作。这种学习方式需要一个外部刺激，引起内部肌肉刺激，然后做出想要的反应，刺激和反应之间只有一个直接的联系。在刺激-反应学习中，刺激会呈现给一个人，而这个人可能会对刺激做出几种不同的反应。每次出现期望的反应时，个体都会得到积极的强化，可能是一句表扬的话，也可能是一次令人满意的经历。由于罗伯特-加涅的理想反应得到了一系列的强化，个体学会了从一系列其他不太理想的反应中分辨出适当的反应，这些反应也可能在刺激发生后出现。
Most examples of pure stimulus-response leaming in people are found in young children. They are leaming to say words, carry out various life-supporting functions, use simple tools, and display socially acceptable behaviors. Learning to say the appropriate names of people and inanimate objects, holding a bottle at the proper angle so that milk can be sucked from it, and picking up a block are examples of stimulus-response learning. In order to leam a desired response, the learner must be physically capable of carrying out the appropriate muscular acts, and the correct response must result in an immediate reinforcement of the response from the leamer's surroundings. Of course undesirable responses can be learned if they result in satisfying reinforcements, and desirable actions can be suppressed if their occurrence is accompanied by punishment.
人类纯粹的刺激-反应学习大多出现在幼儿身上。他们正在学习说词语、执行各种维持生命的功能、使用简单的工具以及表现出社会可接受的行为。学习说出适当的人名和无生命物体的名称、以适当的角度握住奶瓶以便吸奶、捡起积木等都是刺激-反应学习的例子。为了学习所需的反应，学习者必须有身体能力实施适当的肌肉行为，而且正确的反应必须导致学习者周围环境对其反应的立即强化。当然，如果不理想的反应会导致令人满意的强化，那么这种反应也可以被学习；如果理想的动作会伴随着惩罚，那么这种动作也可以被抑制。
人类纯粹的刺激-反应学习大多出现在幼儿身上。他们正在学习说词语、执行各种维持生命的功能、使用简单的工具以及表现出社会可接受的行为。学习说出适当的人名和无生命物体的名称、以适当的角度握住奶瓶以便吸奶、捡起积木等都是刺激-反应学习的例子。为了学习所需的反应，学习者必须有身体能力实施适当的肌肉行为，而且正确的反应必须导致学习者周围环境对其反应的立即强化。当然，如果不理想的反应会导致令人满意的强化，那么这种反应也可以被学习；如果理想的动作会伴随着惩罚，那么这种动作也可以被抑制。
Chaining 连锁
Chaining is the sequential connection of two or more previously leamed nonverbal stimulus-response actions.) Although stimulus-response leaming can involve either verbal or nonyerbal muscular responses, Gagné chooses to call sequences of nonverbal stimulus-response actions chaining and sequences of verbal stimulus-response actions verbal association which will be discussed as a separate learning type. Tying a shoe, opening a door, starting an automobile, throwing a ball, sharpening a pencil, and painting a ceiling are examples of chaining. In each of these situations it is necessary to chain an ordered sequence of previously leaned stimulus-response skills in order to complete the task. Opening a door involves the four separate stimulus-response muscular actions of grasping the doorknob, turning the knob, holding the knob in the turned position, and pulling the door open.
连锁是将两个或更多先前已连锁的非语言刺激-反应动作按顺序连接起来）。虽然刺激-反应连锁可以涉及语言或非语言肌肉反应，但盖尼耶选择将非语言刺激-反应动作序列称为连锁，而将语言刺激-反应动作序列称为语言联想，后者将作为一种单独的学习类型进行讨论。系鞋带、开门、发动汽车、扔球、削铅笔和粉刷天花板都是连锁的例子。在上述每种情况下，都需要将先前掌握的刺激-反应技能有序地串联起来，才能完成任务。开门需要四个独立的刺激-反应肌肉动作：抓住门把手、转动门把手、将门把手保持在转动位置以及拉开门。
连锁是将两个或更多先前已连锁的非语言刺激-反应动作按顺序连接起来）。虽然刺激-反应连锁可以涉及语言或非语言肌肉反应，但盖尼耶选择将非语言刺激-反应动作序列称为连锁，而将语言刺激-反应动作序列称为语言联想，后者将作为一种单独的学习类型进行讨论。系鞋带、开门、发动汽车、扔球、削铅笔和粉刷天花板都是连锁的例子。在上述每种情况下，都需要将先前掌握的刺激-反应技能有序地串联起来，才能完成任务。开门需要四个独立的刺激-反应肌肉动作：抓住门把手、转动门把手、将门把手保持在转动位置以及拉开门。
In order for chaining to occur, the leamer must have previously leamed each stimulus-response link required in the chain. If each link has been leamed, chaining can be facilitated by helping the leamer establish the correct sequence of stimulus-response acts for the chain. Pulling on a doorknob before tuming it is not the proper sequence of stimulus-responce actions for opening a door. Also for most chaining, the leamer must be taught to execute the links in close time succession. For example, the chain of activities required for shifting gears in an automobile with a standard transmission requires a very close time sequence. It is usually necessary to practice a chain of stimulus-response actions in order for the chain to be completely mastered and remembered. If chaining is not accompanied by a satisfying reinforcement, learning can become more difficult and will take longer. Since chaining requires complex physical and mental interactions, fear of ridicule or punishment for failure to properly execute the chain may block these interactions and interfere with chain leaming. In chaining, the completion of one stimulus-responce link may provide an intermediate stimulus to evoke the next stimulus-response link.
为了实现连锁，学习者必须已经学习了连锁所需的每个刺激-反应环节。如果每个环节都已掌握，那么就可以通过帮助学习者建立正确的刺激-反应行为顺序来促进连锁。先拉门把手再拧门把手并不是正确的开门刺激-反应行为顺序。此外，对于大多数连锁活动，必须教会学习者在很短的时间内连续执行各个环节。例如，使用标准变速箱的汽车在换挡时所需的一连串动作就需要非常紧密的时间顺序。通常需要练习一连串的刺激-反应动作，才能完全掌握和记住这一连串动作。如果没有令人满意的强化，学习就会变得更加困难，所需的时间也会更长。由于连锁学习需要复杂的身体和心理互动，如果害怕因不能正确执行连锁动作而受到嘲笑或惩罚，就会阻碍这些互动，影响连锁学习。在连锁学习中，一个刺激-反应环节的完成可能会提供一个中间刺激，以唤起下一个刺激-反应环节。
为了实现连锁，学习者必须已经学习了连锁所需的每个刺激-反应环节。如果每个环节都已掌握，那么就可以通过帮助学习者建立正确的刺激-反应行为顺序来促进连锁。先拉门把手再拧门把手并不是正确的开门刺激-反应行为顺序。此外，对于大多数连锁活动，必须教会学习者在很短的时间内连续执行各个环节。例如，使用标准变速箱的汽车在换挡时所需的一连串动作就需要非常紧密的时间顺序。通常需要练习一连串的刺激-反应动作，才能完全掌握和记住这一连串动作。如果没有令人满意的强化，学习就会变得更加困难，所需的时间也会更长。由于连锁学习需要复杂的身体和心理互动，如果害怕因不能正确执行连锁动作而受到嘲笑或惩罚，就会阻碍这些互动，影响连锁学习。在连锁学习中，一个刺激-反应环节的完成可能会提供一个中间刺激，以唤起下一个刺激-反应环节。
Most activities in mathematics which entail manipulation of physical devices such as rulers, compasses, and geometric models require chaining. Leaming to bisect an angle with a straightedge and a compass requires proper sequencing and
数学中的大多数活动都需要对直尺、圆规和几何模型等物理装置进行操作，这些活动都需要连锁。要学会用直尺和圆规将一个角一分为二，需要正确的顺序和方法。
implementing of a set of previously learned stimulus-response type skills. Among these skills are the ability to use a compass to strike an arc and the ability to construct a straight line between two points.
实施一套先前学习过的刺激-反应型技能。这些技能包括使用圆规画弧线的能力和在两点之间画直线的能力。
数学中的大多数活动都需要对直尺、圆规和几何模型等物理装置进行操作，这些活动都需要连锁。要学会用直尺和圆规将一个角一分为二，需要正确的顺序和方法。
implementing of a set of previously learned stimulus-response type skills. Among these skills are the ability to use a compass to strike an arc and the ability to construct a straight line between two points.
实施一套先前学习过的刺激-反应型技能。这些技能包括使用圆规画弧线的能力和在两点之间画直线的能力。
In teaching mathematical skills requiring muscular activities, two characteris-tics of stimulus-responce leaming and chaining should be understood and exploited. First, chaining, which involves appropriate sequencing of a set of individual stimuli and responses, cannot be accomplished by students who have not mastered the separate skills through appropriate stimulus-response leaming situations. A student who can not learn to carry out a chain of stimulus-response activities may not have leamed some of the links in the chain. Second, stimulusresponse learning and chaining can be facilitated by a teacher who provides rewards and reinforcement for desired behaviors. Even though punishment can be used to promote certain types of stimulus-response leaming, it can interfere with chaining and can negatively influence emotional development, attitudes and motivation to learn.
在教授需要肌肉活动的数学技能时，应了解和利用刺激-反应连锁和连锁的两个特 点。首先，链式教学涉及对一组单独的刺激和反应进行适当的排序，如果学生没有通过适当的刺激-反应学习情境掌握单独的技能，就无法完成链式教学。一个学生如果不能学会进行一连串的刺激-反应活动，就可能没有学会链条中的某些环节。其次，刺激-反应学习和链式学习可以通过教师对所需行为的奖励和强化来促进。尽管惩罚可以用来促进某些类型的刺激--反应学习，但它会干扰链式学习，并对情感发展、学习态度和学习动机产生负面影响。
在教授需要肌肉活动的数学技能时，应了解和利用刺激-反应连锁和连锁的两个特 点。首先，链式教学涉及对一组单独的刺激和反应进行适当的排序，如果学生没有通过适当的刺激-反应学习情境掌握单独的技能，就无法完成链式教学。一个学生如果不能学会进行一连串的刺激-反应活动，就可能没有学会链条中的某些环节。其次，刺激-反应学习和链式学习可以通过教师对所需行为的奖励和强化来促进。尽管惩罚可以用来促进某些类型的刺激--反应学习，但它会干扰链式学习，并对情感发展、学习态度和学习动机产生负面影响。
Werbal Association Werbal 协会
Verbal association is chaining of verbal stimuli; that is, the sequential connection of two or more previously leamed verbal stimulus-response actions. The simplest type of verbal chain is the association of an object with its name which involves chaining the stimulus-response of connecting the appearance of an object with its characteristics and the stimulus-response of observing the object and responding by saying its name. More complicated chains of verbal associations are forming sentences, learning poetry, memorizing the lines of a character in a play and leaming a foreign language.
言语联想是言语刺激的连锁，即把两个或两个以上以前学过的言语刺激-反应动作依次联系起来。最简单的言语联想链是物体与名称的联想，包括将物体的外观与特征联系起来的刺激-反应链，以及观察物体并说出其名称的刺激-反应链。更复杂的语言联想链包括造句、学习诗歌、记住剧中人物的台词以及学习外语。
言语联想是言语刺激的连锁，即把两个或两个以上以前学过的言语刺激-反应动作依次联系起来。最简单的言语联想链是物体与名称的联想，包括将物体的外观与特征联系起来的刺激-反应链，以及观察物体并说出其名称的刺激-反应链。更复杂的语言联想链包括造句、学习诗歌、记住剧中人物的台词以及学习外语。
The mental processes involved in verbal association are very complex and not completely understood at present. Most researchers do agree that efficient verbal association requires the use of intervening mental links which act as codes and which can be either verbal, auditory, or visual images. These codes usually occur in the leamer's mind and will vary from leamer to leamer according to each person's unique mental storehouse of codes. For example, one person may use the verbal mental code "
言语联想所涉及的心理过程非常复杂，目前还不完全清楚。大多数研究人员都认为，有效的言语联想需要使用作为代码的中间心理联系，这些代码可以是言语、听觉或视觉图像。这些代码通常出现在学习者的头脑中，并根据每个人独特的心理代码库而因人而异。例如，一个人可能会使用"
言语联想所涉及的心理过程非常复杂，目前还不完全清楚。大多数研究人员都认为，有效的言语联想需要使用作为代码的中间心理联系，这些代码可以是言语、听觉或视觉图像。这些代码通常出现在学习者的头脑中，并根据每个人独特的心理代码库而因人而异。例如，一个人可能会使用"
The most important use of the verbal association type of leaming is in verbal dialogue. Good oratory and writing depend upon a yast store of memorized verbal associations in the mind of the orator or writer. To express ideas and rationa arguments in mathematics it is necessary to have a large store of verbal associations about mathematics. You can assist students in improving their verbal as- sociations in mathematics by encouraging them to express facts, definitions, Robert Gagne's concepts, and principles correctly and concisely and to discuss mathematical ideas with each other. Many teachers inadvertently discourage verbal associations in their students by rephrasing every student's answers and comments. Students should be encouraged (even required) to communicate important mathematical concepts and processes to each other without having to use the teacher as an intermediary or interpreter. In so doing they will improve their mathematical verbal associations and will leam to influence others through effeciive communication.
语言联想学习法最重要的应用是在口头对话中。好的演说和写作有赖于演说家或作家头脑中大量记忆性的语言联想。要表达数学思想和论证，就必须有大量的数学语言联想。你可以通过鼓励学生正确、简洁地表达事实、定义、罗伯特-加涅的概念和原理，以及相互讨论数学思想，来帮助他们提高数学语言表达能力。许多教师对学生的答案和评论进行重新措辞，无意中打击了学生的口头表达能力。应鼓励（甚至要求）学生相互交流重要的数学概念和过程，而无需教师作为中介或翻译。这样，他们将提高数学语言联想能力，并学会通过有效的交流影响他人。
语言联想学习法最重要的应用是在口头对话中。好的演说和写作有赖于演说家或作家头脑中大量记忆性的语言联想。要表达数学思想和论证，就必须有大量的数学语言联想。你可以通过鼓励学生正确、简洁地表达事实、定义、罗伯特-加涅的概念和原理，以及相互讨论数学思想，来帮助他们提高数学语言表达能力。许多教师对学生的答案和评论进行重新措辞，无意中打击了学生的口头表达能力。应鼓励（甚至要求）学生相互交流重要的数学概念和过程，而无需教师作为中介或翻译。这样，他们将提高数学语言联想能力，并学会通过有效的交流影响他人。
Biscrimination Learning 双证学习
As you may have observed, each successive leaming type which we have discussed is more complex than the type preceding it. Characteristics of the simpler types of learning are found in the more complex types. Discrimination learning is exception to this building block pattem of growth and increasing complexity. After stimulus-response connections have been learned, they can be sequenced into chains of more complex learning behaviors. Discrimination learning is learning to differentiate among chains; that is, to recognize various physical and conceptual objects. There are two kinds of discrimination-single discrimination and multiple discrimination. As an illustration, a young child may be given practice in recognizing the numeral 2 by viewing fifty 2 's on a page and by drawing a page of
正如你可能已经注意到的，我们所讨论的每一种学习类型都比前一种类型更加复杂。较简单的学习类型的特征在较复杂的学习类型中也能找到。辨别学习是这种积木式增长和复杂性递增模式的一个例外。在学习了刺激与反应之间的联系之后，就可以将其排列成更复杂的学习行为链。辨别学习就是学习如何区分不同的行为链，即辨别各种物理和概念对象。辨别有两种--单一辨别和多重辨别。举例来说，幼儿可以通过观察一页纸上的 50 个 2 和画一页
正如你可能已经注意到的，我们所讨论的每一种学习类型都比前一种类型更加复杂。较简单的学习类型的特征在较复杂的学习类型中也能找到。辨别学习是这种积木式增长和复杂性递增模式的一个例外。在学习了刺激与反应之间的联系之后，就可以将其排列成更复杂的学习行为链。辨别学习就是学习如何区分不同的行为链，即辨别各种物理和概念对象。辨别有两种--单一辨别和多重辨别。举例来说，幼儿可以通过观察一页纸上的 50 个 2 和画一页
As students are learning various discriminations among chains, they may also e forming these stimulus-response chains at the same time. This somewhat disorganized leaming situation can, and usually does, result in several phenomena of multiple discrimination leaming-generalization, extinction, and interference.
在学生学习各种辨别链的同时，他们可能也在形成这些刺激-反应链。这种有些无序的学习情况可能会，而且通常也会导致多重辨别学习的几种现象--泛化、消退和干扰。
在学生学习各种辨别链的同时，他们可能也在形成这些刺激-反应链。这种有些无序的学习情况可能会，而且通常也会导致多重辨别学习的几种现象--泛化、消退和干扰。
Generalization is the tendency for the leamer to classify a set of similar but distinct chains into a single category and fail to discriminate or differentiate among the chains. The greater the similarity among chains, the more difficult is multiple discrimination among the chains. For example, a one-to-one mapping and an onto mapping have enough common characteristics so that many algebra students have trouble discriminating one from the other.
泛化是指学习者倾向于将一组相似但不同的链归入单一类别，而无法区分或区别这些链。链之间的相似性越高，就越难对链进行多重区分。例如，"一一映射 "和 "到映射 "有足够多的共同特征，因此许多代数学生很难区分它们。
泛化是指学习者倾向于将一组相似但不同的链归入单一类别，而无法区分或区别这些链。链之间的相似性越高，就越难对链进行多重区分。例如，"一一映射 "和 "到映射 "有足够多的共同特征，因此许多代数学生很难区分它们。
If appropriate reinforcement is absent from the learning of a chain of stimuli and responses, extinction or elimination of that chain occurs. Incorrect responses can be eliminated by withholding reinforcement; however, the occurtence of
如果在一连串刺激和反应的学习过程中缺乏适当的强化，就会出现该刺激和反应的消亡或消除。不正确的反应可以通过停止强化来消除；但是，如果出现以下情况
incorrect responses (even without reinforcement) can extinguish correct responses which must then be releamed. The problem of extinction is apparent in some methods of dealing with homework assignments. If students are not told whether their solutions to homework problems are appropriate, correct responses may become extinct and incorrect responses may interfere with learning of correct responses. Consequently, for many less complex types of learning, immediate teacher feedback concerning the correctness of student solutions of problems is desirable.
不正确的反应（即使没有强化）会使正确的反应消失，而正确的反应又必须被重新唤醒。在一些处理家庭作业的方法中，熄灭问题十分明显。如果不告诉学生他们对家庭作业问题的解答是否恰当，正确的反应可能会消失，而不正确的反应可能会干扰对正确反应的学习。因此，对于许多不太复杂的学习类型来说，教师最好能就学生解决问题的正确性提供即时反馈。
如果在一连串刺激和反应的学习过程中缺乏适当的强化，就会出现该刺激和反应的消亡或消除。不正确的反应可以通过停止强化来消除；但是，如果出现以下情况
incorrect responses (even without reinforcement) can extinguish correct responses which must then be releamed. The problem of extinction is apparent in some methods of dealing with homework assignments. If students are not told whether their solutions to homework problems are appropriate, correct responses may become extinct and incorrect responses may interfere with learning of correct responses. Consequently, for many less complex types of learning, immediate teacher feedback concerning the correctness of student solutions of problems is desirable.
不正确的反应（即使没有强化）会使正确的反应消失，而正确的反应又必须被重新唤醒。在一些处理家庭作业的方法中，熄灭问题十分明显。如果不告诉学生他们对家庭作业问题的解答是否恰当，正确的反应可能会消失，而不正确的反应可能会干扰对正确反应的学习。因此，对于许多不太复杂的学习类型来说，教师最好能就学生解决问题的正确性提供即时反馈。
Forgetting previously learned chains of stimuli and responses can result from interference generated by learning new chains. The new information may also interact with the old information causing some previously leamed responses to be forgotten and making it more difficult to leam the new responses. Interference can be a problem in learning a foreign language such as French, which has many words similar in meaning and spelling to English words. In learning to read and write French some people forget how to spell many English words and have trouble learning to spell some French words due to interference. These generalization and interference factors can creale leaming problems in algebra when students are taught a number of similar, but slightly different, techniques in close succession for simplifying different types of algebraic expressions containing exponents and radical signs. Many students can apply each technique for simplifying a particular type expression when that technique and those expressions are studied in isolation from the other techniques and problem types. However, on a unit test where each of forty different problems must be solved by selecting the correct procedure from the ten previously leamed procedures, many students have little success because their leaming of the ten different techniques interferes with their attempts to discriminate among the different problem types. Some students generalize the ten different techniques into several hybrid methods which they use indiscriminately (and improperiy) in attempting to simplify different types of algebraic expressions.
遗忘以前学习过的刺激和反应链可能是由于学习新的刺激和反应链产生的干扰造成的。新信息还可能与旧信息相互作用，导致一些以前学过的反应被遗忘，从而增加学习新反应的难度。在学习外语（如法语）时，干扰可能是一个问题，因为法语中有许多单词的意思和拼写与英语单词相似。在学习读写法语的过程中，有些人由于受到干扰而忘记了许多英语单词的拼写，在学习拼写一些法语单词时也会遇到困难。在代数学习中，如果接二连三地向学生传授一些相似但略有不同的技巧，以简化含有指数和根号的不同类型的代数表达式，这些泛化和干扰因素就会造成学习问题。如果将每种技巧和这些表达式与其他技巧和问题类型分开来学习，许多学生都能运用这些技巧来简化某一特定类型的表达式。然而，在一次单元测试中，40 个不同的问题都必须通过从以前学过的 10 个程序中选择正确的程序来解决，许多学生都没有取得什么成功，因为他们对 10 种不同技巧的学习干扰了他们区分不同问题类型的尝试。有些学生将这十种不同的技巧归纳为几种混合方法，在尝试简化不同类型的代数表达式时，他们不加区分地（和不适当地）使用这些方法。
遗忘以前学习过的刺激和反应链可能是由于学习新的刺激和反应链产生的干扰造成的。新信息还可能与旧信息相互作用，导致一些以前学过的反应被遗忘，从而增加学习新反应的难度。在学习外语（如法语）时，干扰可能是一个问题，因为法语中有许多单词的意思和拼写与英语单词相似。在学习读写法语的过程中，有些人由于受到干扰而忘记了许多英语单词的拼写，在学习拼写一些法语单词时也会遇到困难。在代数学习中，如果接二连三地向学生传授一些相似但略有不同的技巧，以简化含有指数和根号的不同类型的代数表达式，这些泛化和干扰因素就会造成学习问题。如果将每种技巧和这些表达式与其他技巧和问题类型分开来学习，许多学生都能运用这些技巧来简化某一特定类型的表达式。然而，在一次单元测试中，40 个不同的问题都必须通过从以前学过的 10 个程序中选择正确的程序来解决，许多学生都没有取得什么成功，因为他们对 10 种不同技巧的学习干扰了他们区分不同问题类型的尝试。有些学生将这十种不同的技巧归纳为几种混合方法，在尝试简化不同类型的代数表达式时，他们不加区分地（和不适当地）使用这些方法。
Concept Learning 概念学习
Concept learning is learning to recognize common properties of concrete objects or events and responding to these objects or events as a class. In one sense concept leaming is the opposite of discrimination leaming. Whereas discrimination learning requires that the leamer distinguish among objects according to their different characteristics, concept learning involves classifying objects into sets according to a common characteristic and responding to the common property.
概念学习是指学习识别具体物体或事件的共同属性，并将这些物体或事件作为一个类别作出反应。从某种意义上说，概念学习与辨别学习正好相反。辨别学习要求学习者根据物体的不同特征对其进行区分，而概念学习则是根据物体的共同特征对其进行分类，并对共同属性做出反应。
概念学习是指学习识别具体物体或事件的共同属性，并将这些物体或事件作为一个类别作出反应。从某种意义上说，概念学习与辨别学习正好相反。辨别学习要求学习者根据物体的不同特征对其进行区分，而概念学习则是根据物体的共同特征对其进行分类，并对共同属性做出反应。
In order for students to learn a concept, simpler types of prerequisite leaming must have occurred. Acquisition of any specific concept must be accompanied by prerequisite stimulus-response chains, appropriate verbal associations, and multiple discrimination of distinguishing characteristics. For example, the first step in acquiring the concept circle might be leaming to say the word circle as a self-generated stimulus-response connection, so that students can repeat the word. Then students may leam to identify several different objects as circles by acquiring individual verbal associations. Next, students may learn to discriminate between circles and other objects such as squares and triangles. It is also important for students to be exposed to circles in a wide variety of representative ituations so that they learn to recognize circles which are imbedded in more Robert Gugne's 7 complex objects. When the students are able to spontaneously identify circles in unfamiliar contexts, they have acquired the concept of circle. This ability to generalize a concept to new situations is the ability which distinguishes concept learning from other forms of leaming. When students have leamed a concept, they no longer need specific and familiar stimuli in order to identify and react to new instances of the concept. Consequently, the way to show that a concept has been learned is to demonstrate that the leamer can generalize the concept in an unfamiliar situation.
学生要学习一个概念，必须先进行较简单的先决学习。任何特定概念的学习都必须伴随着先决的刺激-反应链、适当的言语联想以及对区别特征的多重辨别。例如，学习圆这一概念的第一步可能是学习说圆这个词，作为一种自我产生的刺激-反应联系，这样学生就能重复这个词。然后，学生可以通过获得单个的语言联想，学会把几个不同的物体识别为圆。接下来，学生可以学习区分圆形和其他物体，如正方形和三角形。同样重要的是，让学生接触到各种具有代表性的圆，这样他们就能学会识别蕴含在更多罗伯特-古格内的 7 个复杂物体中的圆。当学生能够自发地在陌生的情境中识别圆时，他们就掌握了圆的概念。这种将概念推广到新情境中的能力是概念学习与其他形式学习的区别所在。当学生掌握了一个概念后，他们就不再需要特定的、熟悉的刺激来识别概念的新实例并作出反应。因此，证明概念已经学会的方法，就是证明学习者能够在不熟悉的情境中概括概念。
学生要学习一个概念，必须先进行较简单的先决学习。任何特定概念的学习都必须伴随着先决的刺激-反应链、适当的言语联想以及对区别特征的多重辨别。例如，学习圆这一概念的第一步可能是学习说圆这个词，作为一种自我产生的刺激-反应联系，这样学生就能重复这个词。然后，学生可以通过获得单个的语言联想，学会把几个不同的物体识别为圆。接下来，学生可以学习区分圆形和其他物体，如正方形和三角形。同样重要的是，让学生接触到各种具有代表性的圆，这样他们就能学会识别蕴含在更多罗伯特-古格内的 7 个复杂物体中的圆。当学生能够自发地在陌生的情境中识别圆时，他们就掌握了圆的概念。这种将概念推广到新情境中的能力是概念学习与其他形式学习的区别所在。当学生掌握了一个概念后，他们就不再需要特定的、熟悉的刺激来识别概念的新实例并作出反应。因此，证明概念已经学会的方法，就是证明学习者能够在不熟悉的情境中概括概念。
When new mathematics concepts are being taught to students it is important to (1) present a variety of dissimilar instances of the concept to facilitate generalizing, (2) show examples of different but related concepts to aid in discrimination, (3) present non-examples of the concept to promote discrimination and generalization, and (4) avoid presenting instances of the concept all of which have some common characteristic that may interfere with proper classification of examples of the concept. The importance of these four procedures in teaching a concept can be illustrated by discussing some of the pitfalls in teaching and learning the triangle concept. First, if all the examples of triangles are of the same variety (for instance, if all examples are drawn on the chalkboard), then students may not be able to identify triangular faces of solids or recognize triangular shapes outside of the classroom. If this is the case, the triangle concept has not been learned. Second, if students are not shown examples of otier geometric objects such as trapezoids and pyramids, they may have trouble discriminating among different objects having some common characteristics. Third, plane objects which are not triangles should be presented and discussed to assist students in identifying the characteristics of triangles and the features of other objects which distinguish them from triangles. Fourth, if all of the triangle models shown to students happen to be colored red, then some students mày associate the property of redness with the triangle concept and fail to recognize triangles which are not colored red.
在向学生传授新的数学概念时，重要的是：(1) 展示概念的各种不同实例，以促进概括；(2) 展示不同但相关概念的实例，以帮助辨别；(3) 展示概念的非实例，以促进辨别和概括；(4) 避免展示概念的所有实例，因为所有实例都有一些共同特征，可能会影响对概念实例的正确分类。通过讨论三角形概念教学中的一些误区，可以说明这四个程序在概念教学中的重要性。首先，如果所有三角形的例子都是同一种类的（例如，如果所有例子都画在黑板上），那么学生可能无法辨认出固体的三角形面，也无法在课堂外认识三角形图形。如果是这种情况，三角形的概念就没有学到。其次，如果不向学生展示梯形和金字塔等其他几何物体的实例，他们可能难以区分具有某些共同特征的不同物体。第三，应展示和讨论非三角形的平面物体，以帮助学生识别三角形的特征和其他物体区别于三角形的特征。第四，如果向学生展示的所有三角形模型都是红色的，那么有些学生可能会将红色的特性与三角形的概念联系起来，而无法识别不是红色的三角形。
在向学生传授新的数学概念时，重要的是：(1) 展示概念的各种不同实例，以促进概括；(2) 展示不同但相关概念的实例，以帮助辨别；(3) 展示概念的非实例，以促进辨别和概括；(4) 避免展示概念的所有实例，因为所有实例都有一些共同特征，可能会影响对概念实例的正确分类。通过讨论三角形概念教学中的一些误区，可以说明这四个程序在概念教学中的重要性。首先，如果所有三角形的例子都是同一种类的（例如，如果所有例子都画在黑板上），那么学生可能无法辨认出固体的三角形面，也无法在课堂外认识三角形图形。如果是这种情况，三角形的概念就没有学到。其次，如果不向学生展示梯形和金字塔等其他几何物体的实例，他们可能难以区分具有某些共同特征的不同物体。第三，应展示和讨论非三角形的平面物体，以帮助学生识别三角形的特征和其他物体区别于三角形的特征。第四，如果向学生展示的所有三角形模型都是红色的，那么有些学生可能会将红色的特性与三角形的概念联系起来，而无法识别不是红色的三角形。
All people acquire many concepts through teaching and learning strategies employing verbal chains; however if an acquired concept is to be of much use to a person, it must be identifiable in real-world stimulus situations. Students can memorize the verbal chain "a triangle is a three-sided closed plane figure having straight sides," but this definition will be of little value if they can not use it to classify triangles into the triangle concept category. Also, if students do not have a large repertoire of words and sentences (verbal chains) available for use in concept learning, their facility to acquire concepts will be lessened and the time required to leam each concept may be greatly extended. Even though concept leaming is usually based upon verbal cues, the value of a leamed concept in thought and communication comes from the concrete references that people have for each concept name. One problem in communication and interpretation, and in teaching and learning, is that various people may have different viewpoints (verbal stimulus-response chains and concrete references) of the same concept, which can lead to misunderstanding, argument, and even conflict. If it were not for concept learning with the ability to generalize, all leaming in the formal educational system would be extremely inefficient and of little practical use be-
所有的人都会通过采用语言链的教学和学习策略获得许多概念；但是，如果要使获得的概念对一个人有很大的用处，就必须能在现实世界的刺激情境中加以识别。学生可以记住 "三角形是有直边的三边封闭平面图形 "这一语言链，但如果他们不能利用这一定义将三角形归入三角形概念类别，那么这一定义就没有什么价值。此外，如果学生在概念学习中没有大量的单词和句子（言语链）可用，他们掌握概念的能力就会降低，学习每个概念所需的时间也会大大延长。尽管概念学习通常以语言提示为基础，但所学概念在思维和交流中的价值来自于人们对每个概念名称的具体参照。交流和解释以及教学中的一个问题是，不同的人对同一概念可能有不同的观点（语言刺激-反应链和具体参照），这可能导致误解、争论甚至冲突。如果没有具有概括能力的概念学习，正规教育系统中的所有学习都将是极其低效的，几乎没有任何实际用处。
cause every instance of each concept would have to be experienced directly in order for that instance to be leamed.
因为每个概念的每个实例都必须直接体验过，才能学会。
所有的人都会通过采用语言链的教学和学习策略获得许多概念；但是，如果要使获得的概念对一个人有很大的用处，就必须能在现实世界的刺激情境中加以识别。学生可以记住 "三角形是有直边的三边封闭平面图形 "这一语言链，但如果他们不能利用这一定义将三角形归入三角形概念类别，那么这一定义就没有什么价值。此外，如果学生在概念学习中没有大量的单词和句子（言语链）可用，他们掌握概念的能力就会降低，学习每个概念所需的时间也会大大延长。尽管概念学习通常以语言提示为基础，但所学概念在思维和交流中的价值来自于人们对每个概念名称的具体参照。交流和解释以及教学中的一个问题是，不同的人对同一概念可能有不同的观点（语言刺激-反应链和具体参照），这可能导致误解、争论甚至冲突。如果没有具有概括能力的概念学习，正规教育系统中的所有学习都将是极其低效的，几乎没有任何实际用处。
cause every instance of each concept would have to be experienced directly in order for that instance to be leamed.
因为每个概念的每个实例都必须直接体验过，才能学会。
Rule Learning 规则学习
The six types of leaming which we have just discussed (signal leaming, stimulus-response leaming, chaining, verbal association, discrimination leaming, and concept learning) are basic leaming types that must precede the two higher order learning types (rule leaming and problem solving) which are the primary concern of formal education. Rule learning is the ability to respond to an entire set of situations (stimuli) with a whole set of actions (responses). Rule leaming appears to be the predominant type of learning to facilitate efficient and coherent human functioning. Our speech, writing, routine daily activities, and many of our behaviors are governed by rules which we have learned. In order for people to communicate and interact, and for society to function in any form except anarchy, a huge and complex set of rules must be leamed and observed by a large majority of people. Much of mathematics leaming is rule learning. For example, we know that
我们刚才讨论的六种学习类型（信号学习、刺激-反应学习、连锁学习、言语联想、辨别学习和概念学习）是基本的学习类型，必须先于正规教育主要关注的两种高阶学习类型（规则学习和问题解决）。规则学习是用一整套行动（反应）来应对一整套情况（刺激）的能力。规则学习似乎是促进人类高效、协调运作的主要学习类型。我们的语言、书写、日常活动和许多行为都受我们所学规则的支配。为了让人们进行交流和互动，为了让社会以除无政府状态之外的任何形式运转，必须让绝大多数人学习和遵守一整套庞大而复杂的规则。数学学习的大部分内容就是规则学习。例如，我们知道
我们刚才讨论的六种学习类型（信号学习、刺激-反应学习、连锁学习、言语联想、辨别学习和概念学习）是基本的学习类型，必须先于正规教育主要关注的两种高阶学习类型（规则学习和问题解决）。规则学习是用一整套行动（反应）来应对一整套情况（刺激）的能力。规则学习似乎是促进人类高效、协调运作的主要学习类型。我们的语言、书写、日常活动和许多行为都受我们所学规则的支配。为了让人们进行交流和互动，为了让社会以除无政府状态之外的任何形式运转，必须让绝大多数人学习和遵守一整套庞大而复杂的规则。数学学习的大部分内容就是规则学习。例如，我们知道
Rules may be of different types and of different degrees of complexity. Some rules are definitions and may be regarded as defined concepts. The defined concept
规则可以有不同的类型和复杂程度。有些规则是定义，可视为已定义概念。定义概念
规则可以有不同的类型和复杂程度。有些规则是定义，可视为已定义概念。定义概念
As was noted previously, there is also a distinction between stating a rule and correctly using the rule. Just because a student can state a rule does not mean that he or she has learned the rule in the sense that the capability to use the rule is present in the person. Conversely, it is quite possible to correctly apply a rule without being able to state it. Nearly everyone can memorize the sequence of symbols
如前所述，陈述规则与正确使用规则之间也有区别。学生能够说出一条规则，并不意味着他（她）已经学会了这条规则，也就是说，他（她）已经具备了使用这条规则的能力。相反，不能够说出规则也完全有可能正确运用规则。几乎每个人都能记住
如前所述，陈述规则与正确使用规则之间也有区别。学生能够说出一条规则，并不意味着他（她）已经学会了这条规则，也就是说，他（她）已经具备了使用这条规则的能力。相反，不能够说出规则也完全有可能正确运用规则。几乎每个人都能记住
Mathematics teachers need to be aware that being able to state a definition or write a rule on a sheet of paper is little indication of whether a student has learned the rule. If students are to learn a rule they must have previously learned the Robert Gagne's The chains of concepts that constitute the rule. The conditions of rule leaming begin by specifying the behavior expected of the leamer in order to verify that the rule has been learned. A rule has been learned when the learner can appropriately and correctly apply the rule in a number of different situations. In his book The Conditions of Learning, Robert Gagné (1970) gives a five step instructional sequence for teaching rules:
数学教师需要意识到，能够说出一个定义或在纸上写出一条规则，并不能说明学 生是否学会了这条规则。如果学生要学习一条规则，他们必须事先学习过罗伯特-加涅的 "构成规则的概念链"。规则学习的条件首先要明确学习者的预期行为，以验证规则是否已经学会。当学习者能在多种不同情况下恰当、正确地运用规则时，就说明已经学会了规则。罗伯特-盖尼耶（Robert Gagné，1970 年）在《学习的条件》一书中提出了规则教学的五步教学顺序：
数学教师需要意识到，能够说出一个定义或在纸上写出一条规则，并不能说明学 生是否学会了这条规则。如果学生要学习一条规则，他们必须事先学习过罗伯特-加涅的 "构成规则的概念链"。规则学习的条件首先要明确学习者的预期行为，以验证规则是否已经学会。当学习者能在多种不同情况下恰当、正确地运用规则时，就说明已经学会了规则。罗伯特-盖尼耶（Robert Gagné，1970 年）在《学习的条件》一书中提出了规则教学的五步教学顺序：
Step 1: Inform the leamer about the form of the performance to be expected when learning is completed.
步骤 1：告知学习者完成学习后的预期表现形式。
Step 2: Question the learner. in a way that requires the reinstatement (recall) of the previously leamed concepts that make up the rule.
步骤 2：向学习者提问，要求恢复（回忆）以前学过的构成规则的概念。
Step 3: Use verbal statements (cues) that will lead the leamer to put the rule together, as a chain of concepts, in the proper order.
步骤 3：使用口头陈述（提示），引导学习者将规则作为一个概念链，按照正确的顺序组合起来。
Step 4: By means of a question, ask the leamer to "demonstrate" one of (sic) more concrete instances of the rule.
步骤 4：通过提问，要求学习者 "演示 "该规则的一个（原文如此）更具体的实例。
Step 5: (Optional, but useful for later instruction): By a suitable question, require the learner to make a verbal statement of the rule. (p. 203)
步骤 5：（可选，但对以后的教学有用）：通过一个适当的问题，要求学习者口头陈述规则。(p. 203)
步骤 1：告知学习者完成学习后的预期表现形式。
Step 2: Question the learner. in a way that requires the reinstatement (recall) of the previously leamed concepts that make up the rule.
步骤 2：向学习者提问，要求恢复（回忆）以前学过的构成规则的概念。
Step 3: Use verbal statements (cues) that will lead the leamer to put the rule together, as a chain of concepts, in the proper order.
步骤 3：使用口头陈述（提示），引导学习者将规则作为一个概念链，按照正确的顺序组合起来。
Step 4: By means of a question, ask the leamer to "demonstrate" one of (sic) more concrete instances of the rule.
步骤 4：通过提问，要求学习者 "演示 "该规则的一个（原文如此）更具体的实例。
Step 5: (Optional, but useful for later instruction): By a suitable question, require the learner to make a verbal statement of the rule. (p. 203)
步骤 5：（可选，但对以后的教学有用）：通过一个适当的问题，要求学习者口头陈述规则。(p. 203)
Problem-Solving 解决问题
As one might expect, problem-solving is a higher order and more complex type learning than rule-leaming, and rule acquisition is prerequisite to problemsolving. Problem solving involves selecting and chaining sets of rules in a manner unique to the leamer which results in the establishment of a higher order set of rules which was previously unknown to the leamer. Words like discovery and creativity are often associated with problem-solving. In rule-learning, the rule to be learned is known in a precise form by the teacher who structures activities for the student so that he or she will learn the rule in the form in which the teacher knows it and will apply it in the correct manner at the proper time. The rule exists outside the leamer who attempts to intemalize the existing rule. In problemsolving the leamer attempts to select and use previously leamed rules to formulate a solution to a novel (at least novel for the leamer) problem. Routine substitution of numerical values into the quadratic formula is not regarded by Gagné, and most other learning theorists, as an example of problem-solving. Such routine activities involve merely using a previously learned rule.
正如我们所预料的那样，解决问题是一种比学习规则更高阶、更复杂的学习类型，而规则的掌握是解决问题的先决条件。问题解决涉及以学习者特有的方式选择和串联规则集，从而建立起学习者以前不知道的高阶规则集。发现和创造等词常常与解决问题联系在一起。在规则学习中，要学习的规则是由教师以精确的形式知道的，教师为学生安排活动，使学生以教师知道的形式学习规则，并在适当的时候以正确的方式应用规则。规则存在于学习者之外，学习者试图将现有规则内化。在解决问题的过程中，学习者试图选择和使用以前学过的规则来制定新问题（至少对学习者来说是新问题）的解决方案。盖尼耶和其他大多数学习理论家都不认为将数值例行代入二次方程式是解决问题的例子。这种例行活动仅仅涉及使用以前学过的规则。
As one might expect, problem-solving is a higher order and more complex type learning than rule-leaming, and rule acquisition is prerequisite to problemsolving. Problem solving involves selecting and chaining sets of rules in a manner unique to the leamer which results in the establishment of a higher order set of rules which was previously unknown to the leamer. Words like discovery and creativity are often associated with problem-solving. In rule-learning, the rule to be learned is known in a precise form by the teacher who structures activities for the student so that he or she will learn the rule in the form in which the teacher knows it and will apply it in the correct manner at the proper time. The rule exists outside the leamer who attempts to intemalize the existing rule. In problemsolving the leamer attempts to select and use previously leamed rules to formulate a solution to a novel (at least novel for the leamer) problem. Routine substitution of numerical values into the quadratic formula is not regarded by Gagné, and most other learning theorists, as an example of problem-solving. Such routine activities involve merely using a previously learned rule.
正如我们所预料的那样，解决问题是一种比学习规则更高阶、更复杂的学习类型，而规则的掌握是解决问题的先决条件。问题解决涉及以学习者特有的方式选择和串联规则集，从而建立起学习者以前不知道的高阶规则集。发现和创造等词常常与解决问题联系在一起。在规则学习中，要学习的规则是由教师以精确的形式知道的，教师为学生安排活动，使学生以教师知道的形式学习规则，并在适当的时候以正确的方式应用规则。规则存在于学习者之外，学习者试图将现有规则内化。在解决问题的过程中，学习者试图选择和使用以前学过的规则来制定新问题（至少对学习者来说是新问题）的解决方案。盖尼耶和其他大多数学习理论家都不认为将数值例行代入二次方程式是解决问题的例子。这种例行活动仅仅涉及使用以前学过的规则。
An example of novel problem-solving is that of a student, who has never seen he quadratic formula, developing this formula for the solution of the genera quadratic equation
解决新问题的一个例子是，一个从未见过一元二次方程式的学生，为解一元二次方程
解决新问题的一个例子是，一个从未见过一元二次方程式的学生，为解一元二次方程
Real-world problem solving usually involves five steps-(1) presentation of the problem in a general form, (2) restatement of the problem into an operational definition, (3) formulation of alternative hypotheses and procedures which may be appropriate means of attacking the problern, (4) testing hypotheses and cartying out procedures to obtain a solution or a set of altemative solutions, and
现实世界中的问题解决通常包括五个步骤--（1）以一般形式提出问题，（2）将问题重述为可操作的定义，（3）提出备选假设和程序，这些假设和程序可能是解决问题的适当方法，（4）检验假设和制定程序，以获得一个或一组备选解决方案，以及（5）将问题重述为可操作的定义。
(5) deciding which possible solution is most appropriate or verifying that a single solution is correct. A novel problem for most people would be that of determining how much water flows from the Mississippi River in a year which is step 1, a general statement of the problem. Assuming one would attempt solving the probIem rather than looking for the answer in a book of trivia, the second step is to restate the problem in a more precise, operational manner which may sugges how to solve the problem. After considering the general problem for a time, the problem solver may decide to carry out step 2 by restating the problem as "What is the approximate area of the land mass drained by the Mississippi River and what is the approximate average yearly rainfall over this land mass?" Another operational definition is "What is the approximate area of a cross-section of the Mississippi near its mouth and what is its approximate rate of flow at that point?" Now the problem is stated in terms which suggest methods of solution. In step 3, the problem-solver may decide to estimate the cross-section of the river to be one mile wide by an average of thirty feet deep and the rate of flow to be one and one-half miles per hour. He or she may also make an estimate of the area of the river's watershed and the average yearly rainfall over the watershed. It may be decided that other variables are negligible or will average each other out and have no significant influence on the problem. Step 4 is to solve the problem using each operational definition; this necessitates using previously leamed measurement conversion rules, rules for finding volume, and several different rules of arithmetic. In this example, step 5 could be carried out by comparing the solutions obtained through using each of the operational definitions. If these two solutions are close to each other, the problem-solver may decide that the solution is acceptable for non-lechnical purposes.
(5) 决定哪种可能的解决方案最合适，或验证单一解决方案是否正确。对大多数人来说，一个新颖的问题是确定密西西比河一年的水流量，这是第一步，即问题的一般陈述。假设人们会尝试解决这个问题，而不是在琐事书中寻找答案，那么第二步就是以更精确、更可操作的方式重述问题，这可能会提示如何解决问题。在对一般问题考虑一段时间后，问题解决者可能会决定执行第二步，将问题重述为："密西西比河排水的陆地面积大约是多少，这片陆地的年平均降雨量大约是多少？另一个可操作的定义是："密西西比河河口附近横截面的面积大约是多少，该处的流速大约是多少？现在，问题的表述提出了解决问题的方法。在第 3 步，解题者可以决定估计河流的横截面宽 1 英里，平均深 30 英尺，流速为每小时 1.5 英里。他或她还可以估算出河流的流域面积和流域的年平均降雨量。可以认为其他变量可以忽略不计，或者相互取平均值，不会对问题产生重大影响。第 4 步是使用每个操作定义来解决问题；这就需要使用以前学过的测量转换规则、求体积的规则和几种不同的运算规则。 在本例中，第 5 步可以通过比较使用每个操作定义得到的解决方案来完成。如果这两个解决方案彼此接近，那么问题解决者就可以判定，就非技术目的而言，该解决方案是可以接受的。
现实世界中的问题解决通常包括五个步骤--（1）以一般形式提出问题，（2）将问题重述为可操作的定义，（3）提出备选假设和程序，这些假设和程序可能是解决问题的适当方法，（4）检验假设和制定程序，以获得一个或一组备选解决方案，以及（5）将问题重述为可操作的定义。
(5) deciding which possible solution is most appropriate or verifying that a single solution is correct. A novel problem for most people would be that of determining how much water flows from the Mississippi River in a year which is step 1, a general statement of the problem. Assuming one would attempt solving the probIem rather than looking for the answer in a book of trivia, the second step is to restate the problem in a more precise, operational manner which may sugges how to solve the problem. After considering the general problem for a time, the problem solver may decide to carry out step 2 by restating the problem as "What is the approximate area of the land mass drained by the Mississippi River and what is the approximate average yearly rainfall over this land mass?" Another operational definition is "What is the approximate area of a cross-section of the Mississippi near its mouth and what is its approximate rate of flow at that point?" Now the problem is stated in terms which suggest methods of solution. In step 3, the problem-solver may decide to estimate the cross-section of the river to be one mile wide by an average of thirty feet deep and the rate of flow to be one and one-half miles per hour. He or she may also make an estimate of the area of the river's watershed and the average yearly rainfall over the watershed. It may be decided that other variables are negligible or will average each other out and have no significant influence on the problem. Step 4 is to solve the problem using each operational definition; this necessitates using previously leamed measurement conversion rules, rules for finding volume, and several different rules of arithmetic. In this example, step 5 could be carried out by comparing the solutions obtained through using each of the operational definitions. If these two solutions are close to each other, the problem-solver may decide that the solution is acceptable for non-lechnical purposes.
(5) 决定哪种可能的解决方案最合适，或验证单一解决方案是否正确。对大多数人来说，一个新颖的问题是确定密西西比河一年的水流量，这是第一步，即问题的一般陈述。假设人们会尝试解决这个问题，而不是在琐事书中寻找答案，那么第二步就是以更精确、更可操作的方式重述问题，这可能会提示如何解决问题。在对一般问题考虑一段时间后，问题解决者可能会决定执行第二步，将问题重述为："密西西比河排水的陆地面积大约是多少，这片陆地的年平均降雨量大约是多少？另一个可操作的定义是："密西西比河河口附近横截面的面积大约是多少，该处的流速大约是多少？现在，问题的表述提出了解决问题的方法。在第 3 步，解题者可以决定估计河流的横截面宽 1 英里，平均深 30 英尺，流速为每小时 1.5 英里。他或她还可以估算出河流的流域面积和流域的年平均降雨量。可以认为其他变量可以忽略不计，或者相互取平均值，不会对问题产生重大影响。第 4 步是使用每个操作定义来解决问题；这就需要使用以前学过的测量转换规则、求体积的规则和几种不同的运算规则。 在本例中，第 5 步可以通过比较使用每个操作定义得到的解决方案来完成。如果这两个解决方案彼此接近，那么问题解决者就可以判定，就非技术目的而言，该解决方案是可以接受的。
It can be seen from this example of problem-solving that previously learned rules are needed to solve problems, but that the problem-solver also formulates a unique (for that person) higher-order rule which is the method of proceeding from the general statement of the problem to a reasonable solution. If the person who solved the Mississippi River problem were asked to determine the amount of water flowing from the Ohio River in one year, he or she could use the general problem-solving strategy, developed in solving the Mississippi River problem, o solve the Ohio River problem. Solving this second problem about the Ohio River would be a problem solving-situation for another student who never had been confronted with this type problem, but would be a routine application of a previously learned skill for the first problem-solver.
从这个解决问题的例子中可以看出，解决问题需要以前学过的规则，但解决问题的人也会制定一个独特的（对该人而言的）高阶规则，即从问题的一般陈述到合理解决方案的方法。如果密西西比河问题的解题者被要求确定俄亥俄河一年的水流量，他或她可以使用在解决密西西比河问题时形成的一般解题策略来解决俄亥俄河问题。解决有关俄亥俄河的第二个问题，对于另一个从未遇到过此类问题的学生来说，是一种解决问题的情境，但对于第一个问题的解决者来说，则是对以前所学技能的例行应用。
从这个解决问题的例子中可以看出，解决问题需要以前学过的规则，但解决问题的人也会制定一个独特的（对该人而言的）高阶规则，即从问题的一般陈述到合理解决方案的方法。如果密西西比河问题的解题者被要求确定俄亥俄河一年的水流量，他或她可以使用在解决密西西比河问题时形成的一般解题策略来解决俄亥俄河问题。解决有关俄亥俄河的第二个问题，对于另一个从未遇到过此类问题的学生来说，是一种解决问题的情境，但对于第一个问题的解决者来说，则是对以前所学技能的例行应用。
Learning Hierarchies 学习层次
rGagné has applied his theory, parts of which have been discussed in this section, to structuring specific mathematics learning hierarchies for problem-solving and rule-learning. A learning hierarchy for problem-solving or rule-leaming is a structure containing a sequence of subordinate and prerequisite abilities which a student must master before he or she can leam the higher order task. Gagné describes leaming as observable changes in people's behaviors, and his learning hierarchies are composed of abilities which can be observed or measured. According to Gagné, if a person has leamed, then that person can carry out some activity that he or she could not do previously. Since most activities in mathemat- Robert Gagne's:; ics require definable and observable prerequisite leaming, mathematics topics lend themselves to hierarchical analyses. When specifying a learning hierarchy for a mathematical skill, it is usually not necessary to consider all of the subordinate skills. Usually, but not always, a mathematics teacher is correct in assuming that all students in the class have acquired certain basic mathematics abilities that are prerequisite to mastering higher order skills.]
盖尼耶将其部分理论应用于构建解决问题和规则学习的具体数学学习层次结构。问题解决或规则学习的学习层次结构是一个包含一系列从属和先决能力的结构，学生在学习高阶任务之前必须掌握这些能力。盖尼耶将 "学习 "描述为人们行为中可观察到的变化，而他的学习层次结构是由可以观察或测量的能力组成的。根据盖尼耶的观点，如果一个人已经掌握了学习方法，那么这个人就可以进行一些他或她以前无法完成的活动。由于数学中的大多数活动都需要可定义、可观察的先决学习，因此数学主题适合进行分层分析。在确定一种数学技能的学习层次时，通常不必考虑所有的从属技能。通常情况下，数学教师假定班上所有学生都已掌握了某些基本的数学能力，这些能力是掌握高阶技能的先决条件，这种假定是正确的，但并非总是如此]。
rGagné has applied his theory, parts of which have been discussed in this section, to structuring specific mathematics learning hierarchies for problem-solving and rule-learning. A learning hierarchy for problem-solving or rule-leaming is a structure containing a sequence of subordinate and prerequisite abilities which a student must master before he or she can leam the higher order task. Gagné describes leaming as observable changes in people's behaviors, and his learning hierarchies are composed of abilities which can be observed or measured. According to Gagné, if a person has leamed, then that person can carry out some activity that he or she could not do previously. Since most activities in mathemat- Robert Gagne's:; ics require definable and observable prerequisite leaming, mathematics topics lend themselves to hierarchical analyses. When specifying a learning hierarchy for a mathematical skill, it is usually not necessary to consider all of the subordinate skills. Usually, but not always, a mathematics teacher is correct in assuming that all students in the class have acquired certain basic mathematics abilities that are prerequisite to mastering higher order skills.]
盖尼耶将其部分理论应用于构建解决问题和规则学习的具体数学学习层次结构。问题解决或规则学习的学习层次结构是一个包含一系列从属和先决能力的结构，学生在学习高阶任务之前必须掌握这些能力。盖尼耶将 "学习 "描述为人们行为中可观察到的变化，而他的学习层次结构是由可以观察或测量的能力组成的。根据盖尼耶的观点，如果一个人已经掌握了学习方法，那么这个人就可以进行一些他或她以前无法完成的活动。由于数学中的大多数活动都需要可定义、可观察的先决学习，因此数学主题适合进行分层分析。在确定一种数学技能的学习层次时，通常不必考虑所有的从属技能。通常情况下，数学教师假定班上所有学生都已掌握了某些基本的数学能力，这些能力是掌握高阶技能的先决条件，这种假定是正确的，但并非总是如此]。
Constructing a leaming hierarchy for a mathematical topic is more than merely listing the steps in learning the rule or solving the problem. Preparing a list of steps is a good starting point; however the distinguishing characteristic of a learning hierarchy is an up-side-down tree diagram of subordinate and superordinate abilities which can be demonstrated by students or measured by teachers. Figure 3.1 contains an ordered list of steps which can be used to derive the quadratic formula, and Figure 3.2 is a learning hierarchy of prerequisite abilities needed for deriving the quadratic formula. You will notice that Figure 3.1 is nothing more than a list of steps. Neither the abilities necessary for implementing the steps nor the prerequisite abilities for these superordinate abilities are given in this list.
为数学主题构建学习层次结构，不仅仅是列出学习规则或解决问题的步骤。编制步骤列表是一个很好的起点；然而，学习层次结构的显著特征是一个由下至上的树状图，其中包含可由学生展示或由教师测量的能力。图 3.1 列出了推导二次公式的有序步骤，而图 3.2 则是推导二次公式所需的先决能力的学习层次。你会发现，图 3.1 只不过是一个步骤列表。图 3.2 是推导二次公式所需的先决能力的学习层次。
为数学主题构建学习层次结构，不仅仅是列出学习规则或解决问题的步骤。编制步骤列表是一个很好的起点；然而，学习层次结构的显著特征是一个由下至上的树状图，其中包含可由学生展示或由教师测量的能力。图 3.1 列出了推导二次公式的有序步骤，而图 3.2 则是推导二次公式所需的先决能力的学习层次。你会发现，图 3.1 只不过是一个步骤列表。图 3.2 是推导二次公式所需的先决能力的学习层次。
Step 1. Write the general form of a quadratic equation
步骤 1.写出一元二次方程
Step 2. Add negalive
步骤 2.在等式两边加上负数
步骤 1.写出一元二次方程
Step 2. Add negalive
步骤 2.在等式两边加上负数
Step 3. Divide both sides of the equation by a.
第 3 步用 a 除等式两边。
第 3 步用 a 除等式两边。
Step 4. Complete the square of
步骤 4.在等式两边加上
步骤 4.在等式两边加上
Step 5. Factor the left side of the equation and add the terms on the right side.
步骤 5.将方程左边的因式分解，然后将右边的项相加。
步骤 5.将方程左边的因式分解，然后将右边的项相加。
Step 6. Take the square root of both sides of the equation.
步骤 6.取等式两边的平方根。
步骤 6.取等式两边的平方根。
Step 7. Add
第 7 步在两边加上
第 7 步在两边加上
Flgure 3.1 .
Figure 3.2 is a leaming hierarchy, because boith superordinate and subordinate Robert Gagne's
图 3.2 可以看作是解一元二次方程的学习层次结构的第一近似值。如果能更仔细地考虑学生的先决能力，并对学生进行研究，可能会为这种解题能 力提供更精确的层次结构。然而，数学教师可以很容易地建立图中所示的层次结构以及其他层次结构，并有助于 确定学生是否准备好进行这一活动和其他解决问题的活动。
图 3.2 可以看作是解一元二次方程的学习层次结构的第一近似值。如果能更仔细地考虑学生的先决能力，并对学生进行研究，可能会为这种解题能 力提供更精确的层次结构。然而，数学教师可以很容易地建立图中所示的层次结构以及其他层次结构，并有助于 确定学生是否准备好进行这一活动和其他解决问题的活动。
Good learning hierarchies, even very informal ones, canturate student readiness for learners for preparing preasse良好的学习分层，即使是非常非正式的学习分层，也能为学习者准备课前预习提供准确的学生准备情况
ing mathematics topics. 数学课题。
A Final Nole on Gagné
关于盖尼耶的最后一个诺尔
Gagnè's division of learning into eight types from the simplest (signal learning), through the progressively more complex types (stimulus-response leaming, chaining, verbal association, discrimination leaming, and concept learning), to the higher order types (rule leaming and problem-solving) is a useful and valid way to view leaming. However, leaming does not usually progress in a sequence fices and the various learning types do not
盖尼耶将学习分为八种类型，从最简单的（信号学习），到逐渐复杂的类型（刺激-反应学习、连锁学习、言语联想学习、辨别学习和概念学习），再到更高阶的类型（规则学习和问题解决学习）。然而，学习通常不是按顺序进行的，各种学习类型也不是按顺序进行的。
关于盖尼耶的最后一个诺尔
Gagnè's division of learning into eight types from the simplest (signal learning), through the progressively more complex types (stimulus-response leaming, chaining, verbal association, discrimination leaming, and concept learning), to the higher order types (rule leaming and problem-solving) is a useful and valid way to view leaming. However, leaming does not usually progress in a sequence fices and the various learning types do not
盖尼耶将学习分为八种类型，从最简单的（信号学习），到逐渐复杂的类型（刺激-反应学习、连锁学习、言语联想学习、辨别学习和概念学习），再到更高阶的类型（规则学习和问题解决学习）。然而，学习通常不是按顺序进行的，各种学习类型也不是按顺序进行的。
ment. All of these eight leaming types can, and do, occur nearly simultaneously in all but a few people through most of their lives. As a teacher you should understand Gagne's different types of learning and select teaching strategies and classroom activities which promote each learning type when that particular type seems to be appropriate for leaming the mathenatics topic that you are teaching. Most leachinglleaming sequences will require several of these eight lypes of learning which may interact in very complex ways.这八种沥青类型几乎同时出现在除少数人之外的所有人的一生中。除少数人外，所有这八种学习类型几乎同时出现在他们的一生中。作为一名教师，你应该了解加涅的不同学习类型，并选择能够促进每种学习类型的教学策略和课堂活动，当这种特定的学习类型似乎适合你正在教授的数学课题时。大多数学习序列都需要这八种学习类型中的几种，它们可能以非常复杂的方式相互影响。
Dienes on Learning Mathematics
Dienes 谈数学学习
Zoltan P. Dienes, who was educated in Hungary, France and England, has used his interest and experience in mathematics education and leaming psychology to develop a system for teaching mathematics. His system, which is based in part upon the learning psychology of Jean Piaget, was developed in an attempt to make mathematics more interesting and easier to learn. In his book Building up Mathematics, Professor Dienes summarized his view of mathematics education as follows:
Zoltan P. Dienes 曾在匈牙利、法国和英国接受教育，他利用自己在数学教育和学习心理学方面的兴趣和经验，开发了一套数学教学系统。他的教学体系部分基于让-皮亚杰（Jean Piaget）的学习心理学，旨在使数学更有趣、更易学。在《建立数学》一书中，第尼斯教授将他的数学教育观总结如下：
Zoltan P. Dienes 曾在匈牙利、法国和英国接受教育，他利用自己在数学教育和学习心理学方面的兴趣和经验，开发了一套数学教学系统。他的教学体系部分基于让-皮亚杰（Jean Piaget）的学习心理学，旨在使数学更有趣、更易学。在《建立数学》一书中，第尼斯教授将他的数学教育观总结如下：
At the present time there can hardly be a single member of the teaching rofession and at and from ifonts upwards, who can honestly say to himself that all is well with the and aching of mathematics. There are firth hatics, more so as they get older, and many whith children never succeed in hat is very simple. Let us face it. he majority of chical at best they be understanding the real meanings of mainemate come deft lechnicians in the art of manipulaling complicated which the present at worst they are baffled by the impossible situations into which the permon mathematical requirements in schools tend to place them. An all thought is given to mathematics. With relatively few exceptions, this situation is quite general
目前，几乎没有一个教师职业的成员可以坦率地对自己说，数学教学一切顺利。有许多孩子是数学白痴，随着年龄的增长，这种现象愈演愈烈。让我们面对现实吧，大多数孩子充其量只是理解了数学的真正含义，但他们在处理复杂的数学问题时却显得十分灵巧。所有的思想都集中在数学上。除了极少数例外，这种情况是相当普遍的。
atu malu tu be ben tor granted. Whematics is generally regarded as difficult and tricky, except in a few isolated cases where enthusiastic teachers have infused life into the subject, making it exciting and so less difficult. (p.1)
数学被认为是一门困难而棘手的学科。人们普遍认为数学是一门困难而棘手的学科，只有个别情况下，热心的教师为这门学科注入了活力，使其变得令人兴奋，从而降低了难度。(p.1)
目前，几乎没有一个教师职业的成员可以坦率地对自己说，数学教学一切顺利。有许多孩子是数学白痴，随着年龄的增长，这种现象愈演愈烈。让我们面对现实吧，大多数孩子充其量只是理解了数学的真正含义，但他们在处理复杂的数学问题时却显得十分灵巧。所有的思想都集中在数学上。除了极少数例外，这种情况是相当普遍的。
atu malu tu be ben tor granted. Whematics is generally regarded as difficult and tricky, except in a few isolated cases where enthusiastic teachers have infused life into the subject, making it exciting and so less difficult. (p.1)
数学被认为是一门困难而棘手的学科。人们普遍认为数学是一门困难而棘手的学科，只有个别情况下，热心的教师为这门学科注入了活力，使其变得令人兴奋，从而降低了难度。(p.1)
Mathematic
数学
Dienes regards mathematics as the study of structures, the classification of structures, sorting out relationships within structures, and categorizing relationships among structures. He believes that each mathematical concept (or principle) can be properly understood only if it is first presented to students through a variety of concrete, physical representations. Dienes uses the term concept to mean a mathematical structure, which is a much broader definition of concept than Gagne's definition. According to Dienes there are three types of mathematics concepts-pure mathematical concepts, notational concepts, and applied concepts.
迪尼斯认为数学就是研究结构、对结构进行分类、理清结构内部的关系以及对结构之间的关系进行分类。他认为，每个数学概念（或原理）只有首先通过各种具体的物理表象呈现给学生，才能使学生正确理解。Dienes 使用 "概念 "一词来指数学结构，这是比 Gagne 的定义更宽泛的概念定义。Dienes 认为数学概念有三种类型--纯数学概念、符号概念和应用概念。
迪尼斯认为数学就是研究结构、对结构进行分类、理清结构内部的关系以及对结构之间的关系进行分类。他认为，每个数学概念（或原理）只有首先通过各种具体的物理表象呈现给学生，才能使学生正确理解。Dienes 使用 "概念 "一词来指数学结构，这是比 Gagne 的定义更宽泛的概念定义。Dienes 认为数学概念有三种类型--纯数学概念、符号概念和应用概念。
Pure nathematical concepts deal with classifications of numbers and relationships among numbers, and are completely independent of the way in which the numbers are represented. For instance, six, 8, XII, 1110 (base two), and
纯数概念涉及数的分类和数之间的关系，与表示数的方式完全无关。例如，6、8、XII、1110（以二为基数）和
纯数概念涉及数的分类和数之间的关系，与表示数的方式完全无关。例如，6、8、XII、1110（以二为基数）和
Notational concepts are those properties of numbers which are a direct consequence of the manner in which numbers are represented. The fact that in base ten, 275 means 2 hundreds, plus 7 tens, plus 5 units is a consequence of our positional notation for representing numbers based upon a powers-of-ten system. The selection of an appropriate notational system for various branches of mathematics is an important factor in the subsequent development and extension of mathematics. The fact that arithmetic developed so slowly is due in large part to the cumbersome way in which the ancients represented numbers. We have already mentioned the problems which occurred in the development of mathematical analysis in England as a consequence of the English mathematicians insistence upon using Newton's cumbersome notational systemi for calculus, rather than the more efficient system of Leibniz.
符号概念是数字的属性，是数字表示方式的直接结果。在十进制中，275 表示 2 个百，加 7 个十，再加 5 个单位，这就是我们根据十的幂次系统来表示数的位置符号的结果。为数学的各个分支选择合适的符号系统，是数学后续发展和延伸的一个重要因素。算术之所以发展如此缓慢，在很大程度上是因为古人表示数字的方法过于繁琐。我们已经提到过，由于英国数学家坚持使用牛顿繁琐的微积分符号系统i ，而不是使用莱布尼茨更有效的系统，英国数学分析的发展出现了一些问题。
符号概念是数字的属性，是数字表示方式的直接结果。在十进制中，275 表示 2 个百，加 7 个十，再加 5 个单位，这就是我们根据十的幂次系统来表示数的位置符号的结果。为数学的各个分支选择合适的符号系统，是数学后续发展和延伸的一个重要因素。算术之所以发展如此缓慢，在很大程度上是因为古人表示数字的方法过于繁琐。我们已经提到过，由于英国数学家坚持使用牛顿繁琐的微积分符号系统i ，而不是使用莱布尼茨更有效的系统，英国数学分析的发展出现了一些问题。
Applied concepts are the applications of pure and notational mathematical concepts to problem solving in mathematics and related fields. Length, area and volume are applied mathematical concepts. Applied concepts should be taught to students after they have leamed the prerequisite pure and notational mathematical concepts. Pure concepts should be leamed by students before notational concepts are presented, otherwise students will merely memorize pattems for manipulating symbols without understanding the underlying pure mathematical concepts. Students who make symbol manipulation errors such as
应用概念是纯数学概念和符号数学概念在数学和相关领域解决问题中的应用。长度、面积和体积就是应用数学概念。应用概念应在学生学习了先决的纯数学概念和符号数学概念之后教授。学生应在掌握纯数学概念后再学习符号概念，否则学生只会记住符号的操作步骤，而不会理解其背后的纯数学概念。学生在符号运算中出现
应用概念是纯数学概念和符号数学概念在数学和相关领域解决问题中的应用。长度、面积和体积就是应用数学概念。应用概念应在学生学习了先决的纯数学概念和符号数学概念之后教授。学生应在掌握纯数学概念后再学习符号概念，否则学生只会记住符号的操作步骤，而不会理解其背后的纯数学概念。学生在符号运算中出现
Dienes regards concept leaming as a creative art which can not be explained by any stimulus-response theory such as Gagné's stages of learning. Dienes believes that all abstractions are based upon intuition and concrete experiences; consequently his system for teaching mathematics emphasizes mathematics laboratories, manipulative objects, and mathematical games. He thinks that in order to learn muthematics (hat is, to be able to ciasily shullmis anducm. relationships) students must learn to:
迪内斯认为概念学习是一门创造性的艺术，不能用任何刺激-反应理论（如盖尼耶的学习阶段理论）来解释。迪内斯认为，所有抽象概念都建立在直觉和具体经验的基础上；因此，他的数学教学体系强调数学实验室、操作对象和数学游戏。他认为，为了学好数学（即能够轻松地理解数学和数学关系），学生必须学会：
(1) analyze mathematical structures and their logical relationships,
(1) 分析数学结构及其逻辑关系、
2) abstract a common property from a number of different structures or events and classify the structures or events as belonging together,
2) 从一些不同的结构或事件中抽象出一个共同属性，并将这些结构或事件归为一类、
generalize previously leamed classes of mathematical structures by enlarg ing them to broader classes which have properties similar to those found in the more nartowly defined classes, and
通过将数学结构扩大到更宽泛的类别来概括先前已定义的数学结构类别，这些类别具有与更精确定义的类别类似的特性；以及
use previously leamed abstractions to construct more complex, highe order abstractions.
使用以前学过的抽象概念来构建更复杂的高阶抽象概念。
Stages in Learning Mathematical Concepts.
数学概念的学习阶段。
Dienes believes that mathematical concepts are leamed in progressive stages which are, somewhat analogous to Piaget's stages of intellectual development. He postulates six stages in teaching and leaming mathematical concepts: (1) free lay, (2) games, (3) searching for communalities, (4) representation, (5) symbolization, and (6) formalization.
Dienes 认为数学概念的学习是循序渐进的，这与皮亚杰的智力发展阶段有些类似。他提出了数学概念教学和学习的六个阶段：(1) 自由铺垫，(2) 游戏，(3) 寻找共性，(4) 表象，(5) 符号化，(6) 形式化。
Stage 1, Free Play 第 1 阶段，自由游戏
The free play stage of concept learning consists of unstructured and undirected activities which permit students to experiment with and manipulate physical and 觔 . This stage of concept learning should provide a rich variety of materials for students to manipulate. Even though this unregulated period of free play may appear to be the point of view of a teacher who is accustomed to teaching 化 mathematics using very structured conlearning. Here students first experience many of the which contains concrete cept through interacting with a learning environment which contalsing representations of the concept. In this stage students form mental structures and them to understand the mathematical structure of the concept.
概念学习的自由游戏阶段包括无组织和无指导的活动，允许学生尝试和操作实物和觔。这一阶段的概念学习应提供丰富多样的材料供学生操作。儘管這段無規管的自由遊戲時期，看起來可能是習慣以非常有系統的概念 學習來教授化數學的教師的觀點。在这个阶段，学生通过与概念表征的学习环境互动，首次体验到许多包含具体概念的数学知识。在这一阶段，学生形成心理结构，并理解概念的数学结构。
Stage 2. Games 第 2 阶段：游戏游戏
After a period of free play with representations of a concept, students will begin they will notice that certain rules govern events, that some things are possible and that other things are impossible. Once students have found the rules and properties which determine events, they are ready to play games, experiment with altering the rules of teacher-made games and make up their own games. Games permit students to experiment with the parameters and variations within the concept and to begin analyzing the mathematical structure of the concept. Various games with different representations of the concept will help students discover the logical and mathematical elements of the concept
在对概念的表象进行一段时间的自由游戏后，学生开始会注意到事件有一定的规 则，有些事情是可能的，有些事情是不可能的。一旦学生找到了决定事件的规则和属性，他们就可以玩游戏，尝试改变教师自制游戏的规则，并自编游戏。游戏允许学生尝试概念中的参数和变化，并开始分析概念的数学结构。概念的不同表现形式的各种游戏将帮助学生发现概念的逻辑和数学元素
Stage 3. Searching for Communalities Even after playing several games using different physical represtich is common concept, students may not discover the mathematical structure wist of the com-
第三阶段。寻找共性 即使在使用不同的物理概念玩了几个游戏之后，学生可能也不会发现共性的数学结构。
to all representations of that concept. Until students become aware
到该概念的所有表征。直到学生意识到
mon properties in the representations, they will not be able to classify examples and nonexamples of the concept. Dienes suggests that teachers can help students see the communality of structure in the examples of the concept by showing them how each example can be translated into every other example without altering the abstract properties which are common to all the examples. This amounts to pointing out the common properties found in each example by considering several examples at the same time.
如果学生不了解表象中的抽象属性，就无法对概念的实例和非实例进行分类。Dienes 建议，教师可以通过向学生展示如何在不改变所有示例所共有的抽象属性的情况下，将每个示例转化为其他示例，从而帮助学生看到概念示例中结构的共通性。这相当于通过同时考虑几个例子来指出每个例子中的共同属性。
Stage 4, Representation 第 4 阶段，代表
After students have observed the common elements in each example of the concept, they need to develop, or receive from the teacher, a single representation of the concept which embodies all the common elements found in each of the examples. This representation could be either a diagrammatic representation of the concept, a verbal representation, or an inclusive example. Students need a representation in order to sort out the common elements which are present in all examples of the concept. A representation of the concept will usually be more abstract than the examples and will bring students closer to understanding the abstract mathematical structure underlying the concept.
在学生观察了每个概念示例中的共同要素之后，他们需要建立或从教师那里获得该概念的单一表征，该表征应体现每个示例中的所有共同要素。这种表征可以是概念的图解表征，也可以是口头表征，还可以是包罗万象的示例。学生需要一个表征，以便理清概念的所有示例中存在的共同要素。概念的表征通常比示例更抽象，能使学生更接近理解概念背后的抽象数学结构。
迪内斯认为概念学习是一门创造性的艺术，不能用任何刺激-反应理论（如盖尼耶的学习阶段理论）来解释。迪内斯认为，所有抽象概念都建立在直觉和具体经验的基础上；因此，他的数学教学体系强调数学实验室、操作对象和数学游戏。他认为，为了学好数学（即能够轻松地理解数学和数学关系），学生必须学会：
(1) analyze mathematical structures and their logical relationships,
(1) 分析数学结构及其逻辑关系、
2) abstract a common property from a number of different structures or events and classify the structures or events as belonging together,
2) 从一些不同的结构或事件中抽象出一个共同属性，并将这些结构或事件归为一类、
generalize previously leamed classes of mathematical structures by enlarg ing them to broader classes which have properties similar to those found in the more nartowly defined classes, and
通过将数学结构扩大到更宽泛的类别来概括先前已定义的数学结构类别，这些类别具有与更精确定义的类别类似的特性；以及
use previously leamed abstractions to construct more complex, highe order abstractions.
使用以前学过的抽象概念来构建更复杂的高阶抽象概念。
Stages in Learning Mathematical Concepts.
数学概念的学习阶段。
Dienes believes that mathematical concepts are leamed in progressive stages which are, somewhat analogous to Piaget's stages of intellectual development. He postulates six stages in teaching and leaming mathematical concepts: (1) free lay, (2) games, (3) searching for communalities, (4) representation, (5) symbolization, and (6) formalization.
Dienes 认为数学概念的学习是循序渐进的，这与皮亚杰的智力发展阶段有些类似。他提出了数学概念教学和学习的六个阶段：(1) 自由铺垫，(2) 游戏，(3) 寻找共性，(4) 表象，(5) 符号化，(6) 形式化。
Stage 1, Free Play 第 1 阶段，自由游戏
The free play stage of concept learning consists of unstructured and undirected activities which permit students to experiment with and manipulate physical and 觔 . This stage of concept learning should provide a rich variety of materials for students to manipulate. Even though this unregulated period of free play may appear to be the point of view of a teacher who is accustomed to teaching 化 mathematics using very structured conlearning. Here students first experience many of the which contains concrete cept through interacting with a learning environment which contalsing representations of the concept. In this stage students form mental structures and them to understand the mathematical structure of the concept.
概念学习的自由游戏阶段包括无组织和无指导的活动，允许学生尝试和操作实物和觔。这一阶段的概念学习应提供丰富多样的材料供学生操作。儘管這段無規管的自由遊戲時期，看起來可能是習慣以非常有系統的概念 學習來教授化數學的教師的觀點。在这个阶段，学生通过与概念表征的学习环境互动，首次体验到许多包含具体概念的数学知识。在这一阶段，学生形成心理结构，并理解概念的数学结构。
Stage 2. Games 第 2 阶段：游戏游戏
After a period of free play with representations of a concept, students will begin they will notice that certain rules govern events, that some things are possible and that other things are impossible. Once students have found the rules and properties which determine events, they are ready to play games, experiment with altering the rules of teacher-made games and make up their own games. Games permit students to experiment with the parameters and variations within the concept and to begin analyzing the mathematical structure of the concept. Various games with different representations of the concept will help students discover the logical and mathematical elements of the concept
在对概念的表象进行一段时间的自由游戏后，学生开始会注意到事件有一定的规 则，有些事情是可能的，有些事情是不可能的。一旦学生找到了决定事件的规则和属性，他们就可以玩游戏，尝试改变教师自制游戏的规则，并自编游戏。游戏允许学生尝试概念中的参数和变化，并开始分析概念的数学结构。概念的不同表现形式的各种游戏将帮助学生发现概念的逻辑和数学元素
Stage 3. Searching for Communalities Even after playing several games using different physical represtich is common concept, students may not discover the mathematical structure wist of the com-
第三阶段。寻找共性 即使在使用不同的物理概念玩了几个游戏之后，学生可能也不会发现共性的数学结构。
to all representations of that concept. Until students become aware
到该概念的所有表征。直到学生意识到
mon properties in the representations, they will not be able to classify examples and nonexamples of the concept. Dienes suggests that teachers can help students see the communality of structure in the examples of the concept by showing them how each example can be translated into every other example without altering the abstract properties which are common to all the examples. This amounts to pointing out the common properties found in each example by considering several examples at the same time.
如果学生不了解表象中的抽象属性，就无法对概念的实例和非实例进行分类。Dienes 建议，教师可以通过向学生展示如何在不改变所有示例所共有的抽象属性的情况下，将每个示例转化为其他示例，从而帮助学生看到概念示例中结构的共通性。这相当于通过同时考虑几个例子来指出每个例子中的共同属性。
Stage 4, Representation 第 4 阶段，代表
After students have observed the common elements in each example of the concept, they need to develop, or receive from the teacher, a single representation of the concept which embodies all the common elements found in each of the examples. This representation could be either a diagrammatic representation of the concept, a verbal representation, or an inclusive example. Students need a representation in order to sort out the common elements which are present in all examples of the concept. A representation of the concept will usually be more abstract than the examples and will bring students closer to understanding the abstract mathematical structure underlying the concept.
在学生观察了每个概念示例中的共同要素之后，他们需要建立或从教师那里获得该概念的单一表征，该表征应体现每个示例中的所有共同要素。这种表征可以是概念的图解表征，也可以是口头表征，还可以是包罗万象的示例。学生需要一个表征，以便理清概念的所有示例中存在的共同要素。概念的表征通常比示例更抽象，能使学生更接近理解概念背后的抽象数学结构。
Stage S, Symbolization S阶段，符号化
In this stage the student needs to formulate appropriate verbal and mathematical symbols to describe his or her representation of the concept. It is good for students to invent their own symbolic representations of each concept; however, for the sake of consistency with the textbook, teachers probably should intervene in students' selections of symbol systems. It may be well to permit students to first make up their own symbolic representations, and then have them compare their symbolizations with those in the textbook. Students should be shown the value of good symbol systems in solving problems, proving theorems, and explaining concepts. For example, the Pythagorean theorem may be easier to remember and use when it is represented symbolically as
在这一阶段，学生需要制定适当的语言和数学符号来描述他或她对概念的表述。学生为每个概念发明自己的符号表征是件好事；但是，为了与教科书保持一致，教师或许应该干预学生对符号系统的选择。不妨先让学生编造自己的符号表征，然后让他们将自己的符号表征与教科书中的符号表征进行比较。应向学生展示好的符号系统在解决问题、证明定理和解释概念方面的价值。例如，勾股定理如果用符号表示为
In this stage the student needs to formulate appropriate verbal and mathematical symbols to describe his or her representation of the concept. It is good for students to invent their own symbolic representations of each concept; however, for the sake of consistency with the textbook, teachers probably should intervene in students' selections of symbol systems. It may be well to permit students to first make up their own symbolic representations, and then have them compare their symbolizations with those in the textbook. Students should be shown the value of good symbol systems in solving problems, proving theorems, and explaining concepts. For example, the Pythagorean theorem may be easier to remember and use when it is represented symbolically as
在这一阶段，学生需要制定适当的语言和数学符号来描述他或她对概念的表述。学生为每个概念发明自己的符号表征是件好事；但是，为了与教科书保持一致，教师或许应该干预学生对符号系统的选择。不妨先让学生编造自己的符号表征，然后让他们将自己的符号表征与教科书中的符号表征进行比较。应向学生展示好的符号系统在解决问题、证明定理和解释概念方面的价值。例如，勾股定理如果用符号表示为
Stage 6, Formalization 第 6 阶段，正式化
After students have learned a concept and the related mathematical structures, they must order the properties of the concept and consider the consequences. The fundamental properties in a mathematical structure are the axioms of the system. Derived properties are the theorems, and the procedures for going from axioms to theorems are the mathematical proofs. In this stage students examine the consequences of the concept and use the concept to solve pure and applied mathematics problems.
学生在学习了概念和相关的数学结构后，必须对概念的性质进行排序并考虑其结果。数学结构中的基本性质是系统的公理。派生性质是定理，从公理到定理的过程是数学证明。在这一阶段，学生将研究概念的后果，并利用概念解决纯数学和应用数学问题。
Gaines 盖恩斯
Dienes believes that games are useful vehicles for learning mathematical conepts throughout the six stages of concept development. He calls the games played in the undirected play.stage, where students are doing things for their own enjoyment, preliminary games. Preliminary games are usually informal and unstructured and may be made up by students and played individually or in groups. In the middle stages of concept learning, where students are sorting out the elements of the concept, structured games are useful. Structured games are designed for specific leaming objectives and may be developed by the teacher or purchased from companies which produce mathematics curriculum materials. In the final stages of concept development, when students are solidifying and applying the concept, practice games are useful. Practice games can be used as dril and practice exercises, for reviewing concepts, or as ways to develop applications of concepts.
Dienes 认为，在概念发展的六个阶段中，游戏是学习数学概念的有用工具。他把无指导游戏阶段的游戏称为初步游戏，在这一阶段，学生为自己的乐趣而做事情。初步游戏通常是非正式和无组织的，可以由学生自己编排，单独或分组进行。在概念学习的中期阶段，学生正在梳理概念的要素，这时结构化游戏就很有用。结构化游戏是为特定的学习目标而设计的，可以由教师开发，也可以从制作数学课程材料的公司购买。在概念发展的最后阶段，即学生巩固和应用概念的阶段，练习游戏非常有用。练习游戏可用作操练和练习、复习概念或开发概念应用的方法。
After students have learned a concept and the related mathematical structures, they must order the properties of the concept and consider the consequences. The fundamental properties in a mathematical structure are the axioms of the system. Derived properties are the theorems, and the procedures for going from axioms to theorems are the mathematical proofs. In this stage students examine the consequences of the concept and use the concept to solve pure and applied mathematics problems.
学生在学习了概念和相关的数学结构后，必须对概念的性质进行排序并考虑其结果。数学结构中的基本性质是系统的公理。派生性质是定理，从公理到定理的过程是数学证明。在这一阶段，学生将研究概念的后果，并利用概念解决纯数学和应用数学问题。
Gaines 盖恩斯
Dienes believes that games are useful vehicles for learning mathematical conepts throughout the six stages of concept development. He calls the games played in the undirected play.stage, where students are doing things for their own enjoyment, preliminary games. Preliminary games are usually informal and unstructured and may be made up by students and played individually or in groups. In the middle stages of concept learning, where students are sorting out the elements of the concept, structured games are useful. Structured games are designed for specific leaming objectives and may be developed by the teacher or purchased from companies which produce mathematics curriculum materials. In the final stages of concept development, when students are solidifying and applying the concept, practice games are useful. Practice games can be used as dril and practice exercises, for reviewing concepts, or as ways to develop applications of concepts.
Dienes 认为，在概念发展的六个阶段中，游戏是学习数学概念的有用工具。他把无指导游戏阶段的游戏称为初步游戏，在这一阶段，学生为自己的乐趣而做事情。初步游戏通常是非正式和无组织的，可以由学生自己编排，单独或分组进行。在概念学习的中期阶段，学生正在梳理概念的要素，这时结构化游戏就很有用。结构化游戏是为特定的学习目标而设计的，可以由教师开发，也可以从制作数学课程材料的公司购买。在概念发展的最后阶段，即学生巩固和应用概念的阶段，练习游戏非常有用。练习游戏可用作操练和练习、复习概念或开发概念应用的方法。
Principles of Concept Learning
概念学习的原则
Dienes (1971), in his book Building up Mathematics, summarizes his system of eaching mathematics in four general principles for teaching concepts. His six stages in concept leaming are refinements of these four principles:
Dienes (1971)在他的《建立数学》（Building up Mathematics）一书中，将他的数学教学体系概括为概念教学的四个一般原则。他的概念教学六阶段是对这四项原则的细化：
Dienes (1971)在他的《建立数学》（Building up Mathematics）一书中，将他的数学教学体系概括为概念教学的四个一般原则。他的概念教学六阶段是对这四项原则的细化：
Dynamic Principle. Preliminary, structured and practice and/or reflecve type of games must be provided as necessary experiences from which mathematical concepts can eventually be built, so long as each type of game is mathematical concepts can ventured at the appropriate time. We shall see that this break-up can be introduced at the
动态原则。必须提供初步的、有条理的、练习和/或反思类型的游戏，作为最终建立数学概念的必 要经验，只要每种类型的游戏都能在适当的时候大胆地提出数学概念。我们将看到，这种分解可以在以下阶段引入
further refined. 进一步完善。
动态原则。必须提供初步的、有条理的、练习和/或反思类型的游戏，作为最终建立数学概念的必 要经验，只要每种类型的游戏都能在适当的时候大胆地提出数学概念。我们将看到，这种分解可以在以下阶段引入
further refined. 进一步完善。
Although while children are young these games must be played with concrete material, mental games can gradually be introduced to
虽然这些游戏在儿童年幼时必须使用具体的材料，但智力游戏可以逐渐引入到他们的日常生活中。
hat most fascinating of all games, mathematical research.
最吸引人的游戏，数学研究。
2. Constructivity Principle. In the structuring of the games, construction should always precede analysis, which is almost altogether absent from children's learning until the age of 12 .
2.建构原则。在游戏结构中，建构总是先于分析，而儿童在 12 岁之前的学习中几乎完全没有建构。
3. Mathematical Variability Principle. Concepts involving variables hould be leamed by experiences involving the largest possible number of variables.
3.数学变量原则。涉及变量的概念应通过涉及尽可能多变量的经验来学习。
4. Perceptual Variabiliry Principle or Multiple Embodinent Principle.
4.感知变异原则或多重体现原则。
虽然这些游戏在儿童年幼时必须使用具体的材料，但智力游戏可以逐渐引入到他们的日常生活中。
hat most fascinating of all games, mathematical research.
最吸引人的游戏，数学研究。
2. Constructivity Principle. In the structuring of the games, construction should always precede analysis, which is almost altogether absent from children's learning until the age of 12 .
2.建构原则。在游戏结构中，建构总是先于分析，而儿童在 12 岁之前的学习中几乎完全没有建构。
3. Mathematical Variability Principle. Concepts involving variables hould be leamed by experiences involving the largest possible number of variables.
3.数学变量原则。涉及变量的概念应通过涉及尽可能多变量的经验来学习。
4. Perceptual Variabiliry Principle or Multiple Embodinent Principle.
4.感知变异原则或多重体现原则。
To allow as much scope as possible for individual variations in concept-formation, as well as to induce children to gather the mathematical essence of an mation, as well as to induce conceptual structure should be presented in the form of abstraction, the same conceptual perceptual equivalents as possible. (pp. 30-31)
为了让概念形成过程中的个体差异有尽可能大的余地，同时也为了诱导儿童收集一个数 据的数学本质，以及诱导概念结构，应尽可能以抽象的形式呈现相同的概念感知等价物。(第 30-31 页）
为了让概念形成过程中的个体差异有尽可能大的余地，同时也为了诱导儿童收集一个数 据的数学本质，以及诱导概念结构，应尽可能以抽象的形式呈现相同的概念感知等价物。(第 30-31 页）
Applying Dienes' Theory in a Lesson
在课程中应用第尼斯理论
In applying Dienes' six slages for concept learning to planning a mathematics lesson, you may find that one stage (possibly the free play stage) is not appropriate for your students or that the activities for two or three stages could be combined into a single activity. It may be necessary to plan unique leaming activities for each stage when teaching younger elementary school students; however olde secondary school students may be able to omit certain stages in leaming some concepts. Dienes' model for teaching mathematics should serve as a guide, and not a set of regulations to be followed slavishly.
在运用 Dienes 的六个概念学习阶段来规划数学课程时，你可能会发现其中一个阶段 （可能是自由游戏阶段）不适合你的学生，或者两个或三个阶段的活动可以合并成一个 活动。在教授低年级小学生时，可能有必要为每个阶段制定独特的学习活动计划；但是，高年级中学生在学习某些概念时，可能可以省略某些阶段。Dienes 的数学教学模式应作为一种指导，而不是一套必须严格遵守的规定。
在运用 Dienes 的六个概念学习阶段来规划数学课程时，你可能会发现其中一个阶段 （可能是自由游戏阶段）不适合你的学生，或者两个或三个阶段的活动可以合并成一个 活动。在教授低年级小学生时，可能有必要为每个阶段制定独特的学习活动计划；但是，高年级中学生在学习某些概念时，可能可以省略某些阶段。Dienes 的数学教学模式应作为一种指导，而不是一套必须严格遵守的规定。
The concept of multiplying negative integers will be discussed here as an example of how Dienes' stages can be used as a guide in planning teaching/learning activities. Since nearly all students learn to add, subtract, multiply and divide ing activities. Since nearty ali sumbers, and to add and subtract integers before leaming to multiply natural numbers, and to add and subtract integers beills and he ween mastered by our hypothetical students.
在此，我们将以负整数乘法的概念为例，讨论如何以第尼斯阶段为指导来规划教 学活动。由于几乎所有的学生都在学习加减乘除。在学习自然数乘法和整数加减法之前，我们假设的学生已经掌握了整数加减法。
在此，我们将以负整数乘法的概念为例，讨论如何以第尼斯阶段为指导来规划教 学活动。由于几乎所有的学生都在学习加减乘除。在学习自然数乘法和整数加减法之前，我们假设的学生已经掌握了整数加减法。
For students who are in sixth or seventh grade, one could begin the free play session by informally discussing the arithmetic operations on the natural numbers and the algebraic properties of natural numbers. The teacher might also discuss adding and subtracting integers and the commutative and associative properties for addition. He or she may even choose to substitute an informal review for.free play. Or the free play and game stages could be combined into several games such as the following simple card game: The teacher should prepare enough decks of standard playing cards with the face cards removed so that there is one deck for every five students in the class. Students playing in groups of five would each be dealt four cards. Each student would group his or her cards into pairs, then take the product of the numbers showing on the cards in each pair, and then add the two products. The student who is able to pair his or her cards to obtain the greatest product-sum is the winner of that hand in his or her group. The numbers on black cards (clubs and spades) are considered to be positive numbers, and numbers on red cards (hearts and diamonds) are negative numbers. Consequently students would immediately be confronted with the problem of how to group negative cards to get large positive products and sums. Various groups may agree upon different rules for handling the product of two negative numbers. For instance, a black 2 and 4 and a red 7 and 5 could be used to make
对于六年级或七年级的学生，可以在自由游戏环节开始时非正式地讨论自然数的 算术运算和自然数的代数性质。教师还可以讨论整数的加法和减法，以及加法的交换性质和联立性质。教师甚至可以选择用非正式复习代替自由游戏。或者将自由游戏和游戏阶段合并为几个游戏，如下面的简单纸牌游戏：教师应准备足够的去掉面牌的标准扑克牌，使全班每五名学生有一副牌。五人一组的学生每人发四张牌。每个学生将自己的牌组成一对，然后求出每对牌上数字的乘积，再将两个乘积相加。谁能将自己的牌配成对，得到最大的乘积和，谁就是该组的赢家。黑牌（梅花和黑桃）上的数字被认为是正数，红牌（红心和方块）上的数字是负数。因此，学生们马上就会面临如何将负数牌分组，以得到大的正数积和的问题。不同的小组可能会商定不同的规则来处理两个负数的乘积。例如，如果制定了两个负整数的积是正整数的正确规则，那么黑色的 2 和 4 以及红色的 7 和 5 就可以用来组成
对于六年级或七年级的学生，可以在自由游戏环节开始时非正式地讨论自然数的 算术运算和自然数的代数性质。教师还可以讨论整数的加法和减法，以及加法的交换性质和联立性质。教师甚至可以选择用非正式复习代替自由游戏。或者将自由游戏和游戏阶段合并为几个游戏，如下面的简单纸牌游戏：教师应准备足够的去掉面牌的标准扑克牌，使全班每五名学生有一副牌。五人一组的学生每人发四张牌。每个学生将自己的牌组成一对，然后求出每对牌上数字的乘积，再将两个乘积相加。谁能将自己的牌配成对，得到最大的乘积和，谁就是该组的赢家。黑牌（梅花和黑桃）上的数字被认为是正数，红牌（红心和方块）上的数字是负数。因此，学生们马上就会面临如何将负数牌分组，以得到大的正数积和的问题。不同的小组可能会商定不同的规则来处理两个负数的乘积。例如，如果制定了两个负整数的积是正整数的正确规则，那么黑色的 2 和 4 以及红色的 7 和 5 就可以用来组成
To decide how to handle the product of two negative numbers, the tencher could present a series of problems involving a search for communalities. For instance, these problems could be discussed in class:
为了决定如何处理两个负数的乘积，教员可以提出一系列涉及寻找共性的问题。例如，可以在课堂上讨论这些问题：
为了决定如何处理两个负数的乘积，教员可以提出一系列涉及寻找共性的问题。例如，可以在课堂上讨论这些问题：
- Assume that bad people are negative and good people are positive. Also assume that moving into a community is a positive act and leaving a community is a negative act. What is the net effect of five bad people leaving wo different communities? The class should decide that these events constitute ten positive happenings.
假设坏人是消极的，好人是积极的。还假设迁入一个社区是积极行为，离开一个社区是消极行为。五个坏人离开两个不同的社区会产生什么净影响？全班应认定这些事件构成十个积极事件。 - What is the effect on your cash balance of subtracting four, three dollar debts from your newspaper route account book? Several students should immediately observe that the effect is positive twelve dollars.
从你的报路账簿中减去四笔三美元的债务，对你的现金余额有什么影响？几个学生应该马上就会发现，影响是正 12 美元。 - Finish this table: 完成这张桌子：
? -
but 但
so 那么
What number is?? 什么号码？
As a mathematics teacher, you may be able to construct other examples showing that the product of two negative intergers is a positive integer.
作为一名数学教师，你也许还能举出其他例子来说明两个负整数的乘积是一个正整数。
In the representation stage of forming the concept of multiplying negative integers, students should be able to observe a diagram representing the concept and describe the general property of multiplication of two negative integers. The following diagram, shown in Figure 3.3, is one way to represent that the product of two negative integers is a positive integer.
在形成负整数乘法概念的表象阶段，学生应能观察表示这一概念的图，并描述 两个负整数相乘的一般性质。下图（如图 3.3 所示）是表示两个负整数的乘积是正整数的一种方法。
在形成负整数乘法概念的表象阶段，学生应能观察表示这一概念的图，并描述 两个负整数相乘的一般性质。下图（如图 3.3 所示）是表示两个负整数的乘积是正整数的一种方法。
In the symbolization stage, each student should be able to explain the diagram in Figure 3.3 and use it to show examples of the concept. Each student should also explain that the diagram shows that the product of two negative integers must be a positive integer in order for the distributive property to be true for multiplication and addition of integers. Finally, the class should adopt the symbol system that for any natural numbers
在符号化阶段，每个学生都应能解释图 3.3 中的示意图，并用它来举例说明这一概念。每个学生还应该解释，该图表明，两个负整数的乘积必须是正整数，整数乘法和加法的分配性质才能成立。最后，全班应采用这样的符号系统：对于任何自然数
在符号化阶段，每个学生都应能解释图 3.3 中的示意图，并用它来举例说明这一概念。每个学生还应该解释，该图表明，两个负整数的乘积必须是正整数，整数乘法和加法的分配性质才能成立。最后，全班应采用这样的符号系统：对于任何自然数
is any natural number. 是任何自然数。
is any natural number. 是任何自然数。
is any natural number larger than
是大于
is the natural number 是自然数
Flgure 3.3. A representation of the concept that the product of two negalive integers is a positive integer.
图 3.3.两个负整数的乘积是正整数这一概念的表示。
This concept can be formalized by recognizing that the statement, "the product of two negative integers is a positive integer,"' is an axiom. Theorems such as
通过认识到 "两个负整数的乘积是一个正整数 "是一个公理，这一概念可以形式化。
通过认识到 "两个负整数的乘积是一个正整数 "是一个公理，这一概念可以形式化。
Diene's approach to teaching and leaming mathematics can be summarized in the following list of subprinciples which are inherent in his four principles for concept leaming.
Diene 的数学教学和学习方法可以概括为以下列出的子原则，这些子原则是他的概念学习四原则的内在组成部分。
Diene 的数学教学和学习方法可以概括为以下列出的子原则，这些子原则是他的概念学习四原则的内在组成部分。
- All of mathematics is based upon experience and students leari mathematics by abstracting mathematical concepts and structures out of real experiences.
所有的数学都以经验为基础，学生通过从实际经验中抽象出数学概念和结构来学习数学。 - There is a fixed natural process that students must carry out in order to leam mathematical concepts. The process must include:
学生学习数学概念必须有一个固定的自然过程。这个过程必须包括
a. A play and experimental period involving concrete materials and abstract ideas.
a.涉及具体材料和抽象概念的游戏和实验阶段。
b. An ordering of experiences into a meaningful whole.
b.将各种经验排列成一个有意义的整体。
c. A flash of insight and understanding when the student suddenly comprehends the concept.
c.当学生突然理解概念时，会有一种恍然大悟的感觉。
d. A practice stage to anchor the new concept so that the student can apply it and use it in new mathematical learning experiences.
d.巩固新概念的练习阶段，以便学生能在新的数学学习体验中应用新概念。 - Mathematics is a creative art and it must be taught and learned as an art.
数学是一门创造性的艺术，必须把它作为一门艺术来教、来学。 - New mathematics concepts must be related to previously learned concepts and structures so that there is a transfer of old leaming to new learning.
新的数学概念必须与以前学过的概念和结构相关联，这样才能实现旧知识向新知识的迁移。 - In order to learn mathematics, students must be able to translate a concrete situation or event into an abstract symbolic formulation.
为了学好数学，学生必须能够将具体情境或事件转化为抽象的符号表述。
Ausubel's Theory of Meaningful Yerbal Learning
奥苏贝尔的有意义语言学习理论
During the nineteen fifties many mathematics educators came to believe that the prevailing lecture method for teaching mathematics was resulting in rote learning which was not meaningful to students. As new mathematics prograris with an emphasis upon understanding of concepts were developed and implemented in schools during the nineteen sixties, verbal expository teaching began to fall into disrepute. Many people felt that expository teaching resulted in rote leaming, and teaching models such as discovery learning, inquiry, and mathematics laboratories were thought to be more appropriate methods for fostering meaningful learning. However, there were people who still believed that since the lecture method of teaching had worked reasonably well in the past, it should not be discarded as a bad teaching strategy. Throughout this period, the learming theorist David P. Ausubel argued that expository teaching was the only efficient way to transmit the accumulated discoveries of countless generations to each succeeding generation, and that many of the recently popular methods were not only inefficient, but were also ineffective in promoting meaningful learming. Ausubel's theory of meaningful verbal learning contains a procedure for effective expository teaching resulting in meaningful learning. To Ausubel, the lecture or expository method is a very effective teaching strategy, and he believes that educators should devote more effort toward developing effective expository teaching techniques.
十九世纪五十年代，许多数学教育工作者认为，当时流行的数学讲授法导致学生死记硬背，没有意义。十九世纪六十年代，随着以理解概念为重点的新数学课程的开发和在学校的实施，口头讲解式教学开始不受重视。许多人认为，讲解式教学导致死记硬背，而发现学习、探究和数学实验室等教学模式被认为是更适合培养有意义学习的方法。不过，仍有人认为，既然讲授法在过去的教学中取得了不错的效果，就不应该将其作为一种糟糕的教学策略而加以抛弃。在这一时期，启发式教学理论家戴维-P-奥苏贝尔（David P. Ausubel）认为，论述式教学是将无数代人积累的发现传授给每一代后人的唯一有效方法，而最近流行的许多方法不仅效率低下，而且不能有效促进有意义的启发式教学。奥苏贝尔的 "有意义的言语学习 "理论包含了一种有效的说明性教学程序，可促成有意义的学习。奥苏贝尔认为，讲授法或论述法是一种非常有效的教学策略，他认为教育工作者应投入更多精力，开发有效的论述教学技巧。
奥苏贝尔的有意义语言学习理论
During the nineteen fifties many mathematics educators came to believe that the prevailing lecture method for teaching mathematics was resulting in rote learning which was not meaningful to students. As new mathematics prograris with an emphasis upon understanding of concepts were developed and implemented in schools during the nineteen sixties, verbal expository teaching began to fall into disrepute. Many people felt that expository teaching resulted in rote leaming, and teaching models such as discovery learning, inquiry, and mathematics laboratories were thought to be more appropriate methods for fostering meaningful learning. However, there were people who still believed that since the lecture method of teaching had worked reasonably well in the past, it should not be discarded as a bad teaching strategy. Throughout this period, the learming theorist David P. Ausubel argued that expository teaching was the only efficient way to transmit the accumulated discoveries of countless generations to each succeeding generation, and that many of the recently popular methods were not only inefficient, but were also ineffective in promoting meaningful learming. Ausubel's theory of meaningful verbal learning contains a procedure for effective expository teaching resulting in meaningful learning. To Ausubel, the lecture or expository method is a very effective teaching strategy, and he believes that educators should devote more effort toward developing effective expository teaching techniques.
十九世纪五十年代，许多数学教育工作者认为，当时流行的数学讲授法导致学生死记硬背，没有意义。十九世纪六十年代，随着以理解概念为重点的新数学课程的开发和在学校的实施，口头讲解式教学开始不受重视。许多人认为，讲解式教学导致死记硬背，而发现学习、探究和数学实验室等教学模式被认为是更适合培养有意义学习的方法。不过，仍有人认为，既然讲授法在过去的教学中取得了不错的效果，就不应该将其作为一种糟糕的教学策略而加以抛弃。在这一时期，启发式教学理论家戴维-P-奥苏贝尔（David P. Ausubel）认为，论述式教学是将无数代人积累的发现传授给每一代后人的唯一有效方法，而最近流行的许多方法不仅效率低下，而且不能有效促进有意义的启发式教学。奥苏贝尔的 "有意义的言语学习 "理论包含了一种有效的说明性教学程序，可促成有意义的学习。奥苏贝尔认为，讲授法或论述法是一种非常有效的教学策略，他认为教育工作者应投入更多精力，开发有效的论述教学技巧。
Now that studies of mathematics skills in children and young adults (for example, studies conducted by the National Assessment of Educational Prog- ress, NAEP) indicate that all is not well in applying arithmetic skills, many people are beginning to question the new mathematics programs and the new teaching methods. A study completed in 1975 ty the NAEP and reported in the August, 1975 NAEP Newsletter showed that fewer than half of the 17 -year-olds and young adults between the ages of
现在，对儿童和青少年数学技能的研究（例如，由国家教育进展评估（NAEP）进行的研究）表明，在应用算术技能方面并不尽如人意，许多人开始质疑新的数学课程和新的教学方法。1975 年完成的一项研究显示，17 岁的青少年和年龄在
现在，对儿童和青少年数学技能的研究（例如，由国家教育进展评估（NAEP）进行的研究）表明，在应用算术技能方面并不尽如人意，许多人开始质疑新的数学课程和新的教学方法。1975 年完成的一项研究显示，17 岁的青少年和年龄在
Reception and Discovery Learning, Meaningful and Rote Learning
启蒙和发现式学习、有意义学习和背诵式学习
Ausubel's theory of meaningful verbal learning contains a rationale for expository teaching and shows how lecture-type lessons can be organized to teach the structure of a discipline to make leaming more meaningful to students. As a proponent of expository teaching and verbal learning, Ausubel shows how reception learning can be both efficient and meaningful. However, some critics of reception learning and some proponents of discovery leaming claim that reception learning usually is rote learning and discovery leaming usually is meaningful for students. Consequently, many of Ausubel's writings contain a discussion of reception learning versus discovery leaming and meaningful leaming versus rote learning, in which he refutes these claims.
奥苏贝尔的 "有意义的言语学习 "理论包含了说明性教学的基本原理，并展示了如何组织讲授型课程来教授学科结构，从而使学习对学生更有意义。作为论述式教学和言语学习的倡导者，奥苏贝尔展示了接受式学习如何既高效又有意义。然而，一些对接受式学习的批评者和一些发现式学习的支持者声称，接受式学习通常是死记硬背，而发现式学习通常对学生有意义。因此，奥苏贝尔的许多著作都讨论了接受学习与发现学习、有意义学习与死记硬背的问题，并对这些说法进行了反驳。
奥苏贝尔的 "有意义的言语学习 "理论包含了说明性教学的基本原理，并展示了如何组织讲授型课程来教授学科结构，从而使学习对学生更有意义。作为论述式教学和言语学习的倡导者，奥苏贝尔展示了接受式学习如何既高效又有意义。然而，一些对接受式学习的批评者和一些发现式学习的支持者声称，接受式学习通常是死记硬背，而发现式学习通常对学生有意义。因此，奥苏贝尔的许多著作都讨论了接受学习与发现学习、有意义学习与死记硬背的问题，并对这些说法进行了反驳。
In an article in the February 1968 Arithmetic Teacher, Ausubel describes reception learning and discovery learning as follows:
奥苏贝尔在 1968 年 2 月的《算术教师》杂志上发表了一篇文章，对接受式学习和发现式学习作了如下描述：
奥苏贝尔在 1968 年 2 月的《算术教师》杂志上发表了一篇文章，对接受式学习和发现式学习作了如下描述：
The distinction between reception and discovery learning is not difficult to understand. In reception learning the principal content of what is to be learned is presented to the learner in more or less final form. The leaming does not involve any discovery on his part. He is required only to internalize the material or incorporate it into his cognitive structure so that it is available fo reproduction or other use at some future date. The essential feature of discovery learning on the other hand, is that the principal content of what is to be learned is not given but must be discovered by the leamer before he can intemalize it; the distinctive and prior leaming task, in other words, is to discover something. After this phase is completed, the discovered content is internalized just as in receptive learning. (p. 126)
接受式学习和发现式学习之间的区别并不难理解。在接受式学习中，要学习的主要内容或多或少是以最终形式呈现给学习者的。学习者不需要进行任何发现。学习者只需将材料内化，或将其纳入自己的认知结构，以便在将来的某一天进行复制或其他使用。另一方面，发现式学习的基本特征是，要学习的主要内容并不是给定的，而是必须由学习者在将其内化之前发现的；换句话说，独特的、先行的学习任务就是发现一些东西。在这一阶段完成后，被发现的内容就会像接受性学习一样被内化。(p. 126)
The following explanation of Ausubel's distinction between rote and meaningful learning is taken from his article in the January 1961 issue of Educational Theory.
以下对奥苏贝尔区分死记硬背和有意义学习的解释摘自他在 1961 年 1 月号的《教育理论》上发表的文章。
接受式学习和发现式学习之间的区别并不难理解。在接受式学习中，要学习的主要内容或多或少是以最终形式呈现给学习者的。学习者不需要进行任何发现。学习者只需将材料内化，或将其纳入自己的认知结构，以便在将来的某一天进行复制或其他使用。另一方面，发现式学习的基本特征是，要学习的主要内容并不是给定的，而是必须由学习者在将其内化之前发现的；换句话说，独特的、先行的学习任务就是发现一些东西。在这一阶段完成后，被发现的内容就会像接受性学习一样被内化。(p. 126)
The following explanation of Ausubel's distinction between rote and meaningful learning is taken from his article in the January 1961 issue of Educational Theory.
以下对奥苏贝尔区分死记硬背和有意义学习的解释摘自他在 1961 年 1 月号的《教育理论》上发表的文章。
The distinction between rote and meaningful leaming is frequently confused with the reception-discovery distinction
死记硬背和有意义学习之间的区别经常与接受-发现之间的区别相混淆。造成这种混淆的部分原因是，人们普遍认为接受式学习总是死记硬背的，而发现式学习总是有意义的。实际上，每种区别都构成了学习的一个完全独立的维度。因此，接受学习和发现学习都可以是死记硬背的，也可以是有意义的，这取决于学习发生的条件。
By "meaningful learning" we also refer primarily to a distinctive kind of learning process, and only secondarily to a meaningful leaming outcome-at
我们所说的 "有意义的学习 "主要是指一种独特的学习过程，其次才是指有意义的学习结果--即
tamment of menning-that necessarily rellects the completion of such a process. Meaningful learning as a process presupposes, in tun, both that the leamer employs a meaningful leaming set and that the material he leams is potentially meaningful to him. Thus, regardless of how much potential meaning may inhere in a given proposition, if the learner's intention is to memorize it verbatim, i.e., as a series of arbitrarily related words, both the leaming process and the learning outcome must necessarily be rote and meaningless. And conversely, no matter how meaningful the leamer's set may be, neither the process nor outcome of leaming can possibly be meaningful if the learning task itself is devoid of potential meaning. (pp. 17-18.).
有意义的学习作为一种过程，其前提条件是学习者使用了一套有意义的学习材料，而且他所学习的材料具有潜在的意义。有意义学习作为一个过程，其前提条件是学习者使用一套有意义的学习方法，而且他所学习的材料对他来说是有潜在意义的。因此，无论某个命题的潜在意义有多大，如果学习者的目的是逐字逐句地记住它，即记住一系列任意相关的单词，那么学习过程和学习结果必然都是死记硬背和毫无意义的。反过来说，无论学习者的词组多么有意义，如果学习任务本身没有潜在的意义，那么学习的过程和结果都不可能是有意义的。(第 17-18 页）。
Ausubel has observed that discovery leaming and problem-solving teaching techniques can result in rote learning, just as poor expository teaching can cause students to memorize material which has no meaning to them. When leaming to solve statement problems in algebra, many students memorize problem types and sets of rules for solving each type, with little meaningful understanding of why the rules lead to solutions. Good expository teaching, whereby a teacher structures and explains a mathematics topic so that students can organize the topic and relate it to previous meaningfully leamed topics, can result in efficient and effective learning. Since people are able to memorize and retain limited quantities of arbitrary verbatim materials and must devote a great deal of time and effort to rote learning, the most efficient learning is meaningful leaming. Ausubel believes that good expository teaching is the only efficient way to promote meaningful learning. He thinks that inquiry models, discovery lessons, and laboratory exercises are very inefficient leaming strategies and should be used infrequently in schools. Although Ausubel concedes that some attention should be given to teaching problem-solving, inquiry methods and creative and critical thinking. However, he believes that schools should concentrate on teaching specific information which is useful for social survival and cultural progress, and the basic skills which are teachable to and leamable by the majority of students.
奥苏贝尔（Ausubel）观察到，发现式学习和解决问题的教学技巧会导致死记硬背，就像拙劣的说明式教学会导致学生死记硬背对他们毫无意义的材料一样。在学习解决代数中的语句问题时，许多学生只记住了问题类型和每种类型的解题规则，而对这些规则为什么会导致问题的解决却缺乏有意义的理解。好的说明式教学，即教师对数学题目进行结构化的讲解，使学生能够对题目进行组织，并将其与以前学过的有意义的题目联系起来，可以提高学习效率和效果。由于人们能够记忆和保留的任意逐字材料数量有限，而且必须投入大量的时间和精力进行死记硬背，因此最有效的学习就是有意义的学习。奥苏贝尔认为，好的说明性教学是促进有意义学习的唯一有效方法。他认为，探究模式、发现课和实验练习都是非常低效的学习策略，在学校里应该少用。尽管 Ausubel 承认，在教学中应重视问题解决、探究方法以及创造性和批判性思维。但他认为，学校应集中精力教授对社会生存和文化进步有用的具体信息，以及大多数学生都能学到的基本技能。
死记硬背和有意义学习之间的区别经常与接受-发现之间的区别相混淆。造成这种混淆的部分原因是，人们普遍认为接受式学习总是死记硬背的，而发现式学习总是有意义的。实际上，每种区别都构成了学习的一个完全独立的维度。因此，接受学习和发现学习都可以是死记硬背的，也可以是有意义的，这取决于学习发生的条件。
By "meaningful learning" we also refer primarily to a distinctive kind of learning process, and only secondarily to a meaningful leaming outcome-at
我们所说的 "有意义的学习 "主要是指一种独特的学习过程，其次才是指有意义的学习结果--即
tamment of menning-that necessarily rellects the completion of such a process. Meaningful learning as a process presupposes, in tun, both that the leamer employs a meaningful leaming set and that the material he leams is potentially meaningful to him. Thus, regardless of how much potential meaning may inhere in a given proposition, if the learner's intention is to memorize it verbatim, i.e., as a series of arbitrarily related words, both the leaming process and the learning outcome must necessarily be rote and meaningless. And conversely, no matter how meaningful the leamer's set may be, neither the process nor outcome of leaming can possibly be meaningful if the learning task itself is devoid of potential meaning. (pp. 17-18.).
有意义的学习作为一种过程，其前提条件是学习者使用了一套有意义的学习材料，而且他所学习的材料具有潜在的意义。有意义学习作为一个过程，其前提条件是学习者使用一套有意义的学习方法，而且他所学习的材料对他来说是有潜在意义的。因此，无论某个命题的潜在意义有多大，如果学习者的目的是逐字逐句地记住它，即记住一系列任意相关的单词，那么学习过程和学习结果必然都是死记硬背和毫无意义的。反过来说，无论学习者的词组多么有意义，如果学习任务本身没有潜在的意义，那么学习的过程和结果都不可能是有意义的。(第 17-18 页）。
Ausubel has observed that discovery leaming and problem-solving teaching techniques can result in rote learning, just as poor expository teaching can cause students to memorize material which has no meaning to them. When leaming to solve statement problems in algebra, many students memorize problem types and sets of rules for solving each type, with little meaningful understanding of why the rules lead to solutions. Good expository teaching, whereby a teacher structures and explains a mathematics topic so that students can organize the topic and relate it to previous meaningfully leamed topics, can result in efficient and effective learning. Since people are able to memorize and retain limited quantities of arbitrary verbatim materials and must devote a great deal of time and effort to rote learning, the most efficient learning is meaningful leaming. Ausubel believes that good expository teaching is the only efficient way to promote meaningful learning. He thinks that inquiry models, discovery lessons, and laboratory exercises are very inefficient leaming strategies and should be used infrequently in schools. Although Ausubel concedes that some attention should be given to teaching problem-solving, inquiry methods and creative and critical thinking. However, he believes that schools should concentrate on teaching specific information which is useful for social survival and cultural progress, and the basic skills which are teachable to and leamable by the majority of students.
奥苏贝尔（Ausubel）观察到，发现式学习和解决问题的教学技巧会导致死记硬背，就像拙劣的说明式教学会导致学生死记硬背对他们毫无意义的材料一样。在学习解决代数中的语句问题时，许多学生只记住了问题类型和每种类型的解题规则，而对这些规则为什么会导致问题的解决却缺乏有意义的理解。好的说明式教学，即教师对数学题目进行结构化的讲解，使学生能够对题目进行组织，并将其与以前学过的有意义的题目联系起来，可以提高学习效率和效果。由于人们能够记忆和保留的任意逐字材料数量有限，而且必须投入大量的时间和精力进行死记硬背，因此最有效的学习就是有意义的学习。奥苏贝尔认为，好的说明性教学是促进有意义学习的唯一有效方法。他认为，探究模式、发现课和实验练习都是非常低效的学习策略，在学校里应该少用。尽管 Ausubel 承认，在教学中应重视问题解决、探究方法以及创造性和批判性思维。但他认为，学校应集中精力教授对社会生存和文化进步有用的具体信息，以及大多数学生都能学到的基本技能。
Preconditions for Meaningful Reception Learning
有意义的接收学习的先决条件
According to Ausubel there are two preconditions for meaningful reception learning. First, meaningful reception leaming can only occur in a student who has a meaningful learning set. This is to say that the student's conditioning and attitudes are such that he or she approaches the learning task with the appropriate intentions. If a student approaches the leaming task with the attitude that he or she intends to understand the leaming material, and apply the new learning and relate it to previous learning, that student is likely to learn the new task in a meaningful manner. However, a student who regards the new learning task as an arbitrary, verbatim set of words having little inherent meaning or value will merely attempt to memorize the new material as an isolated set of verbal symbols. If the leamer does not want to translate the new information into terminology consistent with his or her own vocabulary, does not attempt to evaluate how well he or she understands the information, and does not relate it to previously learned information, then meaningful leaming will not occur. There are several reasons why students do not have appropriate leaming sets for meaningful learning of mathematics. Many students have given up hope for ever understanding mathematics due to chronic failure and frustration in mathematics classes. Other students have found that their mathematics teachers expect definitions to be reproduced in a verbatim manner, steps in solving homework problems to be carried out in a strict, unalterable sequence, and rules to be obeyed without question.. For these students, attempts to relate new mathematics concepts to their own unique mental structures result in failure to satisfy teachers; so they memorize the material in exactly the same form as their teachers present it. Other students with good memories may temporarily find that memorizing new information and processes is easier than attempting to understand underlying concepts. Eventually these students will forget much of the mathematics which they have memorized and will confuse new information with previously memorized mathematical structures. For example, a bright student can get part way through a Euclidean plane geometry course by memorizing theorems and their proofs; however this "straw house of geometry' usually collapses long before the final test.
奥苏贝尔认为，有意义的接受学习有两个先决条件。首先，有意义的接受学习只能发生在具有有意义学习设定的学生身上。这就是说，学生的条件和态度使他或她以适当的意图去完成学习任务。如果一个学生带着这样的态度去完成学习任务，即他或她打算理解学习材料，应用新的学习内容，并将其与以前的学习内容联系起来，那么这个学生就有可能以有意义的方式学习新的任务。然而，如果一个学生把新的学习任务看作是一组任意的、逐字逐句的单词，没有什么内在意义或价值，那么他就只会试图把新材料当作一组孤立的语言符号来记忆。如果学习者不想把新信息转化为与自己词汇相一致的术语，不试图评估自己对信息的理解程度，也不把它与以前学过的信息联系起来，那么有意义的学习就不会发生。学生没有适当的学习方法来进行有意义的数学学习有几个原因。许多学生由于在数学课上长期的失败和挫折而放弃了理解数学的希望。还有一些学生发现，他们的数学老师希望定义能被逐字逐句地复制，解决作业问题的步骤能按照严格的、不可更改的顺序进行，规则能被毫无疑问地遵守。 对这些学生来说，试图将新的数学概念与自己独特的心理结构联系起来的做法无法令教师满意；因此，他们会以与教师所呈现的完全相同的形式来记忆教材。其他记忆力好的学生可能会暂时发现，记忆新信息和过程比试图理解基本概念更容易。最终，这些学生会忘记他们所记忆的大部分数学知识，并将新信息与先前记忆的数学结构混淆。例如，聪明的学生可以通过记忆定理及其证明来完成欧几里得平面几何课程的部分学习；然而，这种 "几何草房 "通常会在期末考试之前很久就倒塌。
有意义的接收学习的先决条件
According to Ausubel there are two preconditions for meaningful reception learning. First, meaningful reception leaming can only occur in a student who has a meaningful learning set. This is to say that the student's conditioning and attitudes are such that he or she approaches the learning task with the appropriate intentions. If a student approaches the leaming task with the attitude that he or she intends to understand the leaming material, and apply the new learning and relate it to previous learning, that student is likely to learn the new task in a meaningful manner. However, a student who regards the new learning task as an arbitrary, verbatim set of words having little inherent meaning or value will merely attempt to memorize the new material as an isolated set of verbal symbols. If the leamer does not want to translate the new information into terminology consistent with his or her own vocabulary, does not attempt to evaluate how well he or she understands the information, and does not relate it to previously learned information, then meaningful leaming will not occur. There are several reasons why students do not have appropriate leaming sets for meaningful learning of mathematics. Many students have given up hope for ever understanding mathematics due to chronic failure and frustration in mathematics classes. Other students have found that their mathematics teachers expect definitions to be reproduced in a verbatim manner, steps in solving homework problems to be carried out in a strict, unalterable sequence, and rules to be obeyed without question.. For these students, attempts to relate new mathematics concepts to their own unique mental structures result in failure to satisfy teachers; so they memorize the material in exactly the same form as their teachers present it. Other students with good memories may temporarily find that memorizing new information and processes is easier than attempting to understand underlying concepts. Eventually these students will forget much of the mathematics which they have memorized and will confuse new information with previously memorized mathematical structures. For example, a bright student can get part way through a Euclidean plane geometry course by memorizing theorems and their proofs; however this "straw house of geometry' usually collapses long before the final test.
奥苏贝尔认为，有意义的接受学习有两个先决条件。首先，有意义的接受学习只能发生在具有有意义学习设定的学生身上。这就是说，学生的条件和态度使他或她以适当的意图去完成学习任务。如果一个学生带着这样的态度去完成学习任务，即他或她打算理解学习材料，应用新的学习内容，并将其与以前的学习内容联系起来，那么这个学生就有可能以有意义的方式学习新的任务。然而，如果一个学生把新的学习任务看作是一组任意的、逐字逐句的单词，没有什么内在意义或价值，那么他就只会试图把新材料当作一组孤立的语言符号来记忆。如果学习者不想把新信息转化为与自己词汇相一致的术语，不试图评估自己对信息的理解程度，也不把它与以前学过的信息联系起来，那么有意义的学习就不会发生。学生没有适当的学习方法来进行有意义的数学学习有几个原因。许多学生由于在数学课上长期的失败和挫折而放弃了理解数学的希望。还有一些学生发现，他们的数学老师希望定义能被逐字逐句地复制，解决作业问题的步骤能按照严格的、不可更改的顺序进行，规则能被毫无疑问地遵守。 对这些学生来说，试图将新的数学概念与自己独特的心理结构联系起来的做法无法令教师满意；因此，他们会以与教师所呈现的完全相同的形式来记忆教材。其他记忆力好的学生可能会暂时发现，记忆新信息和过程比试图理解基本概念更容易。最终，这些学生会忘记他们所记忆的大部分数学知识，并将新信息与先前记忆的数学结构混淆。例如，聪明的学生可以通过记忆定理及其证明来完成欧几里得平面几何课程的部分学习；然而，这种 "几何草房 "通常会在期末考试之前很久就倒塌。
The second precondition for meaningful reception learning is that the leaming task be potentially meaningful through its relation to the learner's existing cognitive structure. By relating new mathematical concepts and principles to previously learned (in a meaningful way) mathematical structures, the student can assimilatc new materials into older cognitive structures. Earlier meaningful learning provides an anchor for new learning so that new learning and retention do not require rote learning of arbitrary associations. According to Ausubel, this anchoring process keeps newly leamed material from interfering with previously learned similar materials, which is a hazard of rote leaming. The new mathematics leaming task must be nonarbitrarily and substantively related to the leamer's existing structure of mathematical knowledge. Whether a learning task is potentially meaningful depends upon the nature of the material to be leamed, the way in which the teacher structures his or her presentation of the mathematics topic, and the leamer's unique cognitive structure, which is the manner in which the learner's existing knowledge is organized.
有意义的接受学习的第二个先决条件是，学习任务通过与学习者现有认知结构的关 系而具有潜在的意义。通过将新的数学概念和原理与以前学过的（有意义的）数学结构联系起来，学生可以将新材料同化到旧的认知结构中。先前有意义的学习为新的学习提供了一个锚，这样新的学习和保持就不需要死记硬背任意的联想。奥苏贝尔认为，这种锚定过程能使新学的材料不与以前学过的类似材料发生干扰，而这正是死记硬背的危害。新的数学学习任务必须与学习者现有的数学知识结构有非任意的实质性联系。学习任务是否有潜在意义，取决于学习材料的性质、教师呈现数学主题的结构方式，以及学习者独特的认知结构，即学习者现有知识的组织方式。
有意义的接受学习的第二个先决条件是，学习任务通过与学习者现有认知结构的关 系而具有潜在的意义。通过将新的数学概念和原理与以前学过的（有意义的）数学结构联系起来，学生可以将新材料同化到旧的认知结构中。先前有意义的学习为新的学习提供了一个锚，这样新的学习和保持就不需要死记硬背任意的联想。奥苏贝尔认为，这种锚定过程能使新学的材料不与以前学过的类似材料发生干扰，而这正是死记硬背的危害。新的数学学习任务必须与学习者现有的数学知识结构有非任意的实质性联系。学习任务是否有潜在意义，取决于学习材料的性质、教师呈现数学主题的结构方式，以及学习者独特的认知结构，即学习者现有知识的组织方式。
There are several factors that may hinder meaningful verbal leaming. First, the learner may not possess the necessary level of mental development for meaningful learning of some mathematical concepts to occur. Those students in junior high school who are still in Piaget's stage of concrete operations may not be able to learn highly abstract mathematical principles and concepts without concurrent concrete examples of the principles and concepts. Second, students may not be sufficiently motivated to attempt to leam mathematics in a meaningful way. Poorly motivated students may not consciously resort to rote leaming, but may delude themselves and their teachers into believing that their vague and imprecise verbal statements about mathematical concepts and principles are genuinely meaningful. Third, some teachers delude themselves into believing that their lists of definitions, problem solving rules and steps in the proofs of theorems are meaningful to students. Being able to define congruence and prove three theorems about congruent triangles does not necessarily mean that a student understands congruence and geometric proof in any meaningful way. Definitions and proofs can be memorized as sequences of words which have little meaning for students. The teacher who usually insists that students define a concept in the teacher's words, use the teacher's sequence of steps to solve problems, and give the teacher's sequence of statements and reasons in the proof of a theorem is inadvertently promoting meaningless rote learning.
有几个因素可能会阻碍有意义的口语学习。首先，学习者的心智发展水平可能不足以对某些数学概念进行有意义的学习。那些仍处于皮亚杰的具体运算阶段的初中学生，可能无法在没有相关具体例子的情况下学习高度抽象的数学原理和概念。其次，学生可能没有足够的动力去尝试有意义地学习数学。学习动机不强的学生可能不会有意识地采用死记硬背的方法，但他们可能会欺骗自己和老师，让自己相信他们对数学概念和原理的模糊和不精确的口头陈述是真正有意义的。第三，有些教师自欺欺人地认为，他们所列举的定义、解题规则和定理证明步骤对学生是有意义的。能够定义全等和证明有关全等三角形的三个定理，并不一定意味着学生对全等和几何证明有任何有意义的理解。定义和证明可以作为对学生意义不大的单词序列来记忆。如果教师通常坚持让学生用教师的话来定义概念，用教师的步骤序列来解决问题，在定理证明中给出教师的陈述和理由序列，那么他就在无意中促进了无意义的死记硬背。
有几个因素可能会阻碍有意义的口语学习。首先，学习者的心智发展水平可能不足以对某些数学概念进行有意义的学习。那些仍处于皮亚杰的具体运算阶段的初中学生，可能无法在没有相关具体例子的情况下学习高度抽象的数学原理和概念。其次，学生可能没有足够的动力去尝试有意义地学习数学。学习动机不强的学生可能不会有意识地采用死记硬背的方法，但他们可能会欺骗自己和老师，让自己相信他们对数学概念和原理的模糊和不精确的口头陈述是真正有意义的。第三，有些教师自欺欺人地认为，他们所列举的定义、解题规则和定理证明步骤对学生是有意义的。能够定义全等和证明有关全等三角形的三个定理，并不一定意味着学生对全等和几何证明有任何有意义的理解。定义和证明可以作为对学生意义不大的单词序列来记忆。如果教师通常坚持让学生用教师的话来定义概念，用教师的步骤序列来解决问题，在定理证明中给出教师的陈述和理由序列，那么他就在无意中促进了无意义的死记硬背。
Strategies for Meaningful Verbal Learning
有意义的口头学习策略
Ausubel regards each academic discipline as having a distinct organizational and methodological structure and each individual as having a distinct cognitive structure. He conceptualizes the information-processing structure of the discipline and the information-processing structure of the mind as analogous. Both a discipline such as mathematics and a human mind contain a hierarchical structure of ideas in which the most inclusive ideas are at the top of the structure and subsume progressively less inclusive and more highly differentiated sub-ideas. Since each discipline has its unique structure, Ausubel thinks that disciplines should not be taught using an interdisciplinary approach; rather, each subject should be taught separately. Ausubel does not approve of unified science courses in which biology, physics, and chemistry are taught together; neither does he regard unified mathematics-science programs as an appropriate way to teach these two subjects. He regards the structure as the most important part of a discipline, and combining the teaching of two disciplines will cause the unique structure of each one to be obscured from the learner. Since geometry, algebra and analysis have different structures, it is doubtful that Ausubel would approve of unified mathematics programs such as Howard Fehr's unified modern mathematics textbook series for secondary school students.
奥苏贝尔认为每个学科都有独特的组织和方法结构，每个人都有独特的认知结构。他认为，学科的信息处理结构与思维的信息处理结构具有相似性。数学等学科和人的思维都包含一个层次分明的思想结构，其中最具包容性的思想位于结构的顶端，并逐步包含包容性较弱、差异化程度较高的子思想。由于每门学科都有其独特的结构，奥苏贝尔认为各学科不应采用跨学科的教学方法，而应分别进行教学。奥苏贝尔不赞成将生物、物理和化学放在一起讲授的统一科学课程；他也不认为统一的数学-科学课程是讲授这两门学科的适当方式。他认为结构是一门学科最重要的部分，将两门学科的教学结合起来，会使学习者看不清每一门学科的独特结构。由于几何、代数和分析具有不同的结构，奥苏贝尔是否会赞同统一的数学课程，如霍华德-费尔（Howard Fehr）为中学生编写的统一的现代数学教科书系列，令人怀疑。
有意义的口头学习策略
Ausubel regards each academic discipline as having a distinct organizational and methodological structure and each individual as having a distinct cognitive structure. He conceptualizes the information-processing structure of the discipline and the information-processing structure of the mind as analogous. Both a discipline such as mathematics and a human mind contain a hierarchical structure of ideas in which the most inclusive ideas are at the top of the structure and subsume progressively less inclusive and more highly differentiated sub-ideas. Since each discipline has its unique structure, Ausubel thinks that disciplines should not be taught using an interdisciplinary approach; rather, each subject should be taught separately. Ausubel does not approve of unified science courses in which biology, physics, and chemistry are taught together; neither does he regard unified mathematics-science programs as an appropriate way to teach these two subjects. He regards the structure as the most important part of a discipline, and combining the teaching of two disciplines will cause the unique structure of each one to be obscured from the learner. Since geometry, algebra and analysis have different structures, it is doubtful that Ausubel would approve of unified mathematics programs such as Howard Fehr's unified modern mathematics textbook series for secondary school students.
奥苏贝尔认为每个学科都有独特的组织和方法结构，每个人都有独特的认知结构。他认为，学科的信息处理结构与思维的信息处理结构具有相似性。数学等学科和人的思维都包含一个层次分明的思想结构，其中最具包容性的思想位于结构的顶端，并逐步包含包容性较弱、差异化程度较高的子思想。由于每门学科都有其独特的结构，奥苏贝尔认为各学科不应采用跨学科的教学方法，而应分别进行教学。奥苏贝尔不赞成将生物、物理和化学放在一起讲授的统一科学课程；他也不认为统一的数学-科学课程是讲授这两门学科的适当方式。他认为结构是一门学科最重要的部分，将两门学科的教学结合起来，会使学习者看不清每一门学科的独特结构。由于几何、代数和分析具有不同的结构，奥苏贝尔是否会赞同统一的数学课程，如霍华德-费尔（Howard Fehr）为中学生编写的统一的现代数学教科书系列，令人怀疑。
Since, as Ausubel believes, the major job of education is to teach the disciplines, two conditions must be satisfied. First, the discipline must be presented to students so that the structure of the discipline is stabilized within each student's cognitive configuration and not absorbed and obliterated as a unique structure. According to Bruce Joyce and Marsha Weil (1972) in their book Models of Teaching:
奥苏贝尔认为，教育的主要工作是教授学科，因此必须满足两个条件。首先，必须将学科呈现给学生，使学科结构稳定在每个学生的认知结构中，而不是作为一种独特的结构被吸收和湮没。布鲁斯-乔伊斯（Bruce Joyce）和玛莎-韦尔（Marsha Weil）（1972 年）在《教学模式》一书中指出："学科的结构必须稳定在每个学生的认知结构中，而不是作为独特的结构被吸收和抹去：
奥苏贝尔认为，教育的主要工作是教授学科，因此必须满足两个条件。首先，必须将学科呈现给学生，使学科结构稳定在每个学生的认知结构中，而不是作为一种独特的结构被吸收和湮没。布鲁斯-乔伊斯（Bruce Joyce）和玛莎-韦尔（Marsha Weil）（1972 年）在《教学模式》一书中指出："学科的结构必须稳定在每个学生的认知结构中，而不是作为独特的结构被吸收和抹去：
Ausubel's insistence on stabilization of new ideas rather than integrating and absorbing them comes from his position that the hierarchical organization of ideas within each discipline is extremely powerful and that the learner can make maximal use of these ideas by having them stabilized within his structure, rather than by integrating them with his old ideas and making new kinds of structures. (p. 168)
奥苏贝尔坚持稳定新思想，而不是整合和吸收新思想，这是因为他认为每门学科中思想的层次组织是极其强大的，学习者可以通过将这些思想稳定在自己的结构中，而不是通过将它们与旧思想整合并建立新型结构，最大限度地利用这些思想。(p. 168)
The second condition in teaching a discipline is to make the material meaningful to the learner. To insure meaningful learning, thie teacher must help students build linkages between their own cognitive strúctures and the structure of the discipline being taught. Each new concept or principle within the discipline must be related to relevant, previously learned concepts and principles which are in the learner's cognitive structure?
学科教学的第二个条件是使教材对学习者有意义。为了确保有意义的学习，教师必须帮助学生在自己的认知结构和所教学科的结构之间建立联系。学科中的每一个新概念或原理都必须与学习者认知结构中相关的、以前学过的概念和原理相关联。
奥苏贝尔坚持稳定新思想，而不是整合和吸收新思想，这是因为他认为每门学科中思想的层次组织是极其强大的，学习者可以通过将这些思想稳定在自己的结构中，而不是通过将它们与旧思想整合并建立新型结构，最大限度地利用这些思想。(p. 168)
The second condition in teaching a discipline is to make the material meaningful to the learner. To insure meaningful learning, thie teacher must help students build linkages between their own cognitive strúctures and the structure of the discipline being taught. Each new concept or principle within the discipline must be related to relevant, previously learned concepts and principles which are in the learner's cognitive structure?
学科教学的第二个条件是使教材对学习者有意义。为了确保有意义的学习，教师必须帮助学生在自己的认知结构和所教学科的结构之间建立联系。学科中的每一个新概念或原理都必须与学习者认知结构中相关的、以前学过的概念和原理相关联。
Ausubel has developed two principles for presenting content in a subject field-progressive differentiation and integrative reconciliation. The principle of progressive differentiation is described by Ausubel (1963) in his book The Psychology of Meaningful Verbal Learning:
奥苏贝尔提出了在学科领域中呈现内容的两个原则--渐进分化和综合调和。奥苏贝尔（1963 年）在其著作《有意义语言学习心理学》中描述了渐进分化原则：
奥苏贝尔提出了在学科领域中呈现内容的两个原则--渐进分化和综合调和。奥苏贝尔（1963 年）在其著作《有意义语言学习心理学》中描述了渐进分化原则：
When subject matter is programed in accordance with the principles of progressive differentiation, the most general and inclusive ideas of the discipline are presented first, and are then progressively differentiated in terms of detail and specificity. . . . The assumption we are making here, in other words, is that an individual's organization of the content of a particular subject-matter discipline in his own mind, consists of a hierarchical structure in which the most inclusive concepts occupy a position at the apex of the structure and subsume progressively less inclusive and more highly differentiated subconcepts and factual data. (p. 79)
在按照渐进分化原则编排学科课程时，首先呈现的是学科中最普遍、最具包容性的观点，然后再逐步分化出细节和具体内容。. . .换句话说，我们在这里所做的假设是，一个人在自己的头脑中对特定学科内容的组织是由一个层次结构构成的，在这个结构中，包容性最强的概念占据了结构的顶点，并逐渐包含了包容性较弱、分化程度较高的子概念和事实数据。(p. 79)
The integrative reconciliation principle implies that new information about the discipline being studied should be reconciled and integrated with previously leamed information from that discipline. The leaching/learning sequence should be structured so that each new lesson is carefully related to previously leamed materials. New learning in a discipline should be related to and built upon previous learning. It should be noted that although Ausubel says that each part of a discipline should be integrated with other parts, he does not support integrating the structures of various disciplines, thus obscuring the unique structure of each discipline.
综合调和原则是指有关所学学科的新信息应与以前所学的该学科信息进行调和和整 合。浸入式学习/学习顺序的安排应使每节新课都与以前学习过的材料密切相关。一门学科的新学习内容应与以前的学习内容相关联，并以以前的学习内容为基础。值得注意的是，虽然奥苏贝尔说一门学科的每一部分都应与其他部分相整合，但他并不支持整合不同学科的结构，从而掩盖了每门学科的独特结构。
在按照渐进分化原则编排学科课程时，首先呈现的是学科中最普遍、最具包容性的观点，然后再逐步分化出细节和具体内容。. . .换句话说，我们在这里所做的假设是，一个人在自己的头脑中对特定学科内容的组织是由一个层次结构构成的，在这个结构中，包容性最强的概念占据了结构的顶点，并逐渐包含了包容性较弱、分化程度较高的子概念和事实数据。(p. 79)
The integrative reconciliation principle implies that new information about the discipline being studied should be reconciled and integrated with previously leamed information from that discipline. The leaching/learning sequence should be structured so that each new lesson is carefully related to previously leamed materials. New learning in a discipline should be related to and built upon previous learning. It should be noted that although Ausubel says that each part of a discipline should be integrated with other parts, he does not support integrating the structures of various disciplines, thus obscuring the unique structure of each discipline.
综合调和原则是指有关所学学科的新信息应与以前所学的该学科信息进行调和和整 合。浸入式学习/学习顺序的安排应使每节新课都与以前学习过的材料密切相关。一门学科的新学习内容应与以前的学习内容相关联，并以以前的学习内容为基础。值得注意的是，虽然奥苏贝尔说一门学科的每一部分都应与其他部分相整合，但他并不支持整合不同学科的结构，从而掩盖了每门学科的独特结构。
The teaching strategy that Ausubel suggests in order to promote meaningful verbal leaming through progressive differentiation and integrative reconciliation is the use of advance organizers. An advance organizer is a preliminary statement, discussion, or other activity which introduces new material at a higher level of generality, inclusiveness, and abstraction than the actual new leaming task. The organizer is selected for its appropriateness in explaining and integrat ing the new material. Its purpose is to provide the learner with a conceptual structure into which he or she will integrate the new material. Advance organizers set the stage for meaningful reception leaming and provide a top-down approach to leaming new concepts and principles. The advance organizer is structured as an anchoring vehicle which subsumes the new material to be learned as a consequence of its high degree of generality and inclusiveness. Advance or ganizers are not merely outlines, overviews, or summaries which are usually presented at the same level of abstraction and generality as the material to be learned. Rather, they are inclusive subsumers which prepare students to meaningfully learn new materials by helping to organize abstract cognitive structures in their minds.
为了通过渐进分化和综合调和来促进有意义的言语学习，奥苏贝尔提出的教学策略是使用先行组织者。先行组织者是一种初步的陈述、讨论或其他活动，它以比实际的新学习任务更高的概括性、包容性和抽象性水平来介绍新材料。选择组织者的标准是其在解释和整合新材料方面的适当性。其目的是为学习者提供一个概念结构，使其能够将新材料融会贯通。先行组织者为有意义的接收学习奠定了基础，并为学习新概念和原理提供了一种自上而下的方法。先行组织者的结构是一个锚定载体，由于其高度的概括性和包容性，它包含了要学习的新材料。先行组织者不仅仅是提纲、概述或摘要，它们通常与要学习的材料具有相同的抽象性和概括性。相反，它们是包容性的子集，通过帮助学生在头脑中组织抽象的认知结构，为他们有意义地学习新教材做好准备。
为了通过渐进分化和综合调和来促进有意义的言语学习，奥苏贝尔提出的教学策略是使用先行组织者。先行组织者是一种初步的陈述、讨论或其他活动，它以比实际的新学习任务更高的概括性、包容性和抽象性水平来介绍新材料。选择组织者的标准是其在解释和整合新材料方面的适当性。其目的是为学习者提供一个概念结构，使其能够将新材料融会贯通。先行组织者为有意义的接收学习奠定了基础，并为学习新概念和原理提供了一种自上而下的方法。先行组织者的结构是一个锚定载体，由于其高度的概括性和包容性，它包含了要学习的新材料。先行组织者不仅仅是提纲、概述或摘要，它们通常与要学习的材料具有相同的抽象性和概括性。相反，它们是包容性的子集，通过帮助学生在头脑中组织抽象的认知结构，为他们有意义地学习新教材做好准备。
The purposes of advance organizers as stated by Ausubel (1968) in his book Educational Psychology, A Cognitive View are:
奥苏贝尔（Ausubel，1968 年）在其著作《教育心理学，一种认知观点》中指出，提前组织者的目的是
奥苏贝尔（Ausubel，1968 年）在其著作《教育心理学，一种认知观点》中指出，提前组织者的目的是
Advance organizers probably facilitate the incorporability and longevity of meaningfully leamed material in three different ways. First, they explicitly draw upon and mobilize whatever relevant anchoring concepts are alread established in the leamer's cognitive structure and make them part of the subsuming entity. Thus, not only is new material rendered more familiar and potentially meaningful, but the most relevant ideational antecedents in cognitive structure are also selected and utilized in integrated fashion. Second, advance organizers at an appropriate level of inclusiveness, by making sub sumption under specifically relevant propositions possible (and drawing on other advantages of subsumptive leaming), provide optimal anchorage. This promotes both initial learning and later resistance to obliterative subsumption. Third, the use of advance organizers renders unnecessary much of the rote memorization to which students often resort because they are required to learm the details of an unfamiliar discipline before having available a sufficient number of key anchoring ideas. Because of the unavailability of such ideas in cognitive structure to which the details can be nonarbitrarily and substantively related, the material, although logically meaningful, lacks potential meaningfulness. (pp. 137-138)
先行组织者可能通过三种不同的方式促进有意义学习材料的可融入性和持久性。首先，先行组织者会明确利用和调动学习者认知结构中已经建立起来的相关锚定概念，并使其成为子实体的一部分。这样，不仅新材料变得更熟悉、更有意义，而且认知结构中最相关的表意先行概念也被综合地选择和利用。其次，具有适当包容性的先行组织者，通过使在具体相关命题下的子归纳成为可能（并利用子归纳学习的其他优势），提供最佳的锚定。这既促进了最初的学习，也增强了后来对湮没性归纳的抵抗力。第三，预先组织者的使用使学生不必进行死记硬背，因为学生在掌握足够数量的关键锚定思想之前，就必须掌握一门陌生学科的细节。由于在认知结构中没有与细节建立非任意和实质性联系的思想，因此材料虽然在逻辑上有意义，但却缺乏潜在的意义。(第 137-138 页）
先行组织者可能通过三种不同的方式促进有意义学习材料的可融入性和持久性。首先，先行组织者会明确利用和调动学习者认知结构中已经建立起来的相关锚定概念，并使其成为子实体的一部分。这样，不仅新材料变得更熟悉、更有意义，而且认知结构中最相关的表意先行概念也被综合地选择和利用。其次，具有适当包容性的先行组织者，通过使在具体相关命题下的子归纳成为可能（并利用子归纳学习的其他优势），提供最佳的锚定。这既促进了最初的学习，也增强了后来对湮没性归纳的抵抗力。第三，预先组织者的使用使学生不必进行死记硬背，因为学生在掌握足够数量的关键锚定思想之前，就必须掌握一门陌生学科的细节。由于在认知结构中没有与细节建立非任意和实质性联系的思想，因此材料虽然在逻辑上有意义，但却缺乏潜在的意义。(第 137-138 页）
Ans Advance Organizer Lesson
Ans 提前组织课程
The following Model for Teaching
下面的
下面的
Let's Make a Computer 让我们制作一台电脑
This nodel will describe a lesson that illustrates the process used by computers to solve mathematical prob ems, by having students play the roles of components in a computer system. It will also motivate students into urther inquiry about specific methods of computer programming and the devices needed in a computer system.
本节课将介绍一堂课，通过让学生扮演计算机系统中的部件，说明计算机解决数学问题的过程。它还将激励学生进一步探究计算机编程的具体方法和计算机系统所需的设备。
本节课将介绍一堂课，通过让学生扮演计算机系统中的部件，说明计算机解决数学问题的过程。它还将激励学生进一步探究计算机编程的具体方法和计算机系统所需的设备。
Performance Objective 绩效目标
Students will list in writing the general components needed for a computer system and describe the finction of each component. They will also orally describe the prom.
学生将以书面形式列出计算机系统所需的一般组件，并描述每个组件的功能。他们还将口头描述舞会。
学生将以书面形式列出计算机系统所需的一般组件，并描述每个组件的功能。他们还将口头描述舞会。
Preassessment 预评估
The students should be able to solve mathematical problems by following specified algorithmic procedures (sets of instructions). Have then solve the following problems, each of which requires the use of an al-
学生应能按照指定的算法程序（指令集）解决数学问题。让学生解决下列问题，其中每个问题都需要使用一种算法------。
gorithm:
算法：
The students should be able to solve mathematical problems by following specified algorithmic procedures (sets of instructions). Have then solve the following problems, each of which requires the use of an al-
学生应能按照指定的算法程序（指令集）解决数学问题。让学生解决下列问题，其中每个问题都需要使用一种算法------。
gorithm:
算法：
Teaching Strategies 教学策略
Explain to the class that a computer solves mathemat ical problems not by inventing or learning a method of solution, but by very rapidly and without error following a set of instructions called a computer program. These instructions were written and given to the computer by a human being who had to know the procedure for solving the problem, but did not have sufficient time to carry ou the many mathematical operations necessary to finding the solution. Tell the students that several of them will be selected to act as parts of a computer to solve a prob lem the way a computer would.
向全班同学解释，计算机不是通过发明或学习一种解题方法来解决数学问题的， 而是通过非常迅速地、无差错地遵循一套称为计算机程序的指令来解决数学问题的。这些指令是由人编写并提供给计算机的，而人必须知道解题的程序，但没有足够的时间来完成求解所需的许多数学运算。告诉学生们，他们中的几个人将被选中作为计算机的一部分，以计算机的方式来解决一个问题。
向全班同学解释，计算机不是通过发明或学习一种解题方法来解决数学问题的， 而是通过非常迅速地、无差错地遵循一套称为计算机程序的指令来解决数学问题的。这些指令是由人编写并提供给计算机的，而人必须知道解题的程序，但没有足够的时间来完成求解所需的许多数学运算。告诉学生们，他们中的几个人将被选中作为计算机的一部分，以计算机的方式来解决一个问题。
Ditto, distribute and briefly explain the following parts of a computer: naned COUNTER. Again, according to the instrucions in BOX 3 the curent numbers stored in FACTO RIAL and COUNTER are to be multipled, the produc o replace the cuirent number in FACTORIAL. Not the equal sign is not used here in the sense of equivalence. In the instructions for BOX4, when COUNTER contain an 8 skip BOXS and BOX6 and obtain the next instruction from BOX7; otherwise the next instruction comes rom BOXS. BOXS causes the contents of COUNTE to be replaced witli one more than its current contents.
同上，分发并简要说明计算机的下列部件：NANED COUNTER。同样，根据方框 3 中的说明，将存储在 FACTO RIAL 和 COUNTER 中的现有数字相乘，产生的结果取代 FACTORIAL 中的现有数字。这里使用的不是等号。在 BOX4 的指令中，当 COUNTER 中包含 8 时，跳过 BOXS 和 BOX6，从 BOX7 中获取下一条指令；否则，下一条指令从 BOXS 中获取。BOXS 使 COUNTE 的内容被替换为比当前内容多一个的内容。
BOX6 sends you back to BOX3 for the next instruction. When BO 7 is reached, the contents of FACTORIAL which will be 31 are printed and
BOX6 将返回 BOX3，执行下一条指令。当到达 BO7 时，将打印 FACTORIAL 中的内容 31，
Ask for student volunteers to play the roles and give ach one a sign. Since you are the chief, wear the CHIEF sign around your neck. The problem your human computer will solve is that of computing 8 ! (eigh factorial).
请学生志愿者扮演这些角色，并给每个人一个标志。既然你是酋长，就把酋长的标志挂在脖子上。人类计算机要解决的问题是计算 8 ！(八阶乘）。
Now position the student computer components in the same configuration as shown on the drawing. You and the five components should executive the instruc ions until BOX8 is reached and the problem is solved.
现在将学生计算机组件按图上所示的相同配置进行定位。您和五个组件应执行指令，直到达到 BOX8 并解决问题。
As CHIEF be sure to instruct each student compute part in precisely what he is to do and when he is to act READER's job is to read each insiruction from the boxes, one at a ime, when CFIEF lells him to do so Then ARIF FACTORIAL and COUNTER and tell CHIEF IC FACT ANSWER WRITER's only job is to write the contents of FACTORIAL when BOX7 is finally reached. (CHIEF gives the instruction to write.) Ench time CHEF needs to know what numbers are in eilher FACTORIAL or COUNTER, he instructs LOOK-AND-TELL to look into the appropriate box and lell him what is there. When an instuct COUNTER is the number in elther FACToRmber, CHIEF tells RUN-AND-CHANGE to make the appropriale re pacement. CHIEF writes the number on a
作为总负责人，一定要准确地指示每个计算部分的学生他应该做什么，以及他什么时候应该行动 阅读者的任务是在总负责人让他这样做的时候，一个一个地读出方框中的每一个引号，然后 ARIF FACTORIAL 和 COUNTER，并告诉总负责人 IC FACT ANSWER WRITER 的唯一任务是在最后读到第 7 个方框时写出 FACTORIAL 中的内容。(如果 CHEF 需要知道 FACTORIAL 或 COUNTER 中的数字，他就会指示 LOOK-AND-TELL 查看相应的盒子，并告诉他里面有什么。当一个指示 COUNTER 是另一个 FACTORIAL 中的数字时，CHIEF 会告诉 RUN-AND-CHANGE 进行适当的替换。CHIEF 将数字写在
You may have enough time in a single class period to un through the computer in action twice. This will give a second group of student volunteers the opportunity to be active participants. To capitalize upon the motiva tional aspects of the model, continue discussion abou computers in the following class meeting. or invite guest computer expert to takk with the students and anield trip to see a computer in operation.
在一节课的时间里，您可能有足够的时间让计算机实际操作两次。这将为第二批学生志愿者提供积极参与的机会。为了充分利用该模型的激励作用，可在下一次班会上继续讨论有关计算机的问题，或邀请特邀计算机专家与学生一起实地参观计算机的运行情况。
同上，分发并简要说明计算机的下列部件：NANED COUNTER。同样，根据方框 3 中的说明，将存储在 FACTO RIAL 和 COUNTER 中的现有数字相乘，产生的结果取代 FACTORIAL 中的现有数字。这里使用的不是等号。在 BOX4 的指令中，当 COUNTER 中包含 8 时，跳过 BOXS 和 BOX6，从 BOX7 中获取下一条指令；否则，下一条指令从 BOXS 中获取。BOXS 使 COUNTE 的内容被替换为比当前内容多一个的内容。
BOX6 sends you back to BOX3 for the next instruction. When BO 7 is reached, the contents of FACTORIAL which will be 31 are printed and
BOX6 将返回 BOX3，执行下一条指令。当到达 BO7 时，将打印 FACTORIAL 中的内容 31，
Ask for student volunteers to play the roles and give ach one a sign. Since you are the chief, wear the CHIEF sign around your neck. The problem your human computer will solve is that of computing 8 ! (eigh factorial).
请学生志愿者扮演这些角色，并给每个人一个标志。既然你是酋长，就把酋长的标志挂在脖子上。人类计算机要解决的问题是计算 8 ！(八阶乘）。
Now position the student computer components in the same configuration as shown on the drawing. You and the five components should executive the instruc ions until BOX8 is reached and the problem is solved.
现在将学生计算机组件按图上所示的相同配置进行定位。您和五个组件应执行指令，直到达到 BOX8 并解决问题。
As CHIEF be sure to instruct each student compute part in precisely what he is to do and when he is to act READER's job is to read each insiruction from the boxes, one at a ime, when CFIEF lells him to do so Then ARIF FACTORIAL and COUNTER and tell CHIEF IC FACT ANSWER WRITER's only job is to write the contents of FACTORIAL when BOX7 is finally reached. (CHIEF gives the instruction to write.) Ench time CHEF needs to know what numbers are in eilher FACTORIAL or COUNTER, he instructs LOOK-AND-TELL to look into the appropriate box and lell him what is there. When an instuct COUNTER is the number in elther FACToRmber, CHIEF tells RUN-AND-CHANGE to make the appropriale re pacement. CHIEF writes the number on a
作为总负责人，一定要准确地指示每个计算部分的学生他应该做什么，以及他什么时候应该行动 阅读者的任务是在总负责人让他这样做的时候，一个一个地读出方框中的每一个引号，然后 ARIF FACTORIAL 和 COUNTER，并告诉总负责人 IC FACT ANSWER WRITER 的唯一任务是在最后读到第 7 个方框时写出 FACTORIAL 中的内容。(如果 CHEF 需要知道 FACTORIAL 或 COUNTER 中的数字，他就会指示 LOOK-AND-TELL 查看相应的盒子，并告诉他里面有什么。当一个指示 COUNTER 是另一个 FACTORIAL 中的数字时，CHIEF 会告诉 RUN-AND-CHANGE 进行适当的替换。CHIEF 将数字写在
You may have enough time in a single class period to un through the computer in action twice. This will give a second group of student volunteers the opportunity to be active participants. To capitalize upon the motiva tional aspects of the model, continue discussion abou computers in the following class meeting. or invite guest computer expert to takk with the students and anield trip to see a computer in operation.
在一节课的时间里，您可能有足够的时间让计算机实际操作两次。这将为第二批学生志愿者提供积极参与的机会。为了充分利用该模型的激励作用，可在下一次班会上继续讨论有关计算机的问题，或邀请特邀计算机专家与学生一起实地参观计算机的运行情况。
Postassessment 评估后
Have the students list the general components of computcr system and describe the function of each Then have them describe the procedure a computer follows in solving a math problem
让学生列出计算机系统的一般组件，并描述每个组件的功能 然后让学生描述计算机在解决数学问题时所遵循的程序
Two additional operations that must be done by a computer are carried out in the circuits. LOOK-AND TELL is the ability to look into the mailoxes and end CHANGE is the ability to ake information out of mait boxes and replace it with new information.
电路中还有两个必须由计算机完成的操作。LOOK-AND TELL（看和说）是指查看邮箱的能力，而 CHANGE（换）是指从邮箱中提取信息并用新信息替换的能力。
让学生列出计算机系统的一般组件，并描述每个组件的功能 然后让学生描述计算机在解决数学问题时所遵循的程序
Two additional operations that must be done by a computer are carried out in the circuits. LOOK-AND TELL is the ability to look into the mailoxes and end CHANGE is the ability to ake information out of mait boxes and replace it with new information.
电路中还有两个必须由计算机完成的操作。LOOK-AND TELL（看和说）是指查看邮箱的能力，而 CHANGE（换）是指从邮箱中提取信息并用新信息替换的能力。
Before the next class session prepare six cardboard
在下一堂课之前，准备六张纸板
Before the next class session prepare six carcboard name signs for students to wear around their necks: IN
在下一堂课之前，准备六个硬纸板名字标志，让学生戴在脖子上：在
STRUCTION READER, ARITHMETICKER, AN STRUCTIONREADER, ARITHMETICKER, ANRUN-AND-CHANGE.
结构读取器、计算器、运行和更改。
在下一堂课之前，准备六张纸板
Before the next class session prepare six carcboard name signs for students to wear around their necks: IN
在下一堂课之前，准备六个硬纸板名字标志，让学生戴在脖子上：在
STRUCTION READER, ARITHMETICKER, AN STRUCTIONREADER, ARITHMETICKER, ANRUN-AND-CHANGE.
结构读取器、计算器、运行和更改。
Also prepare a large poster or ditto and distribute as Hustrated at the conclusion of this model, the set of in ilfuctrated at the compler program) which your human computer will use to find 81 . Each instruction is boxed in a "mathox." You will also need two small boxes withou lops and a package of
同时，准备一张大海报或同上，并作为本模型的结论分发，这组指令将在完成程序中显示），你的人类计算机将使用它来找到 81 个指令。每条指令都装在一个 "数学盒 "中。你还需要两个小盒子和一包
同时，准备一张大海报或同上，并作为本模型的结论分发，这组指令将在完成程序中显示），你的人类计算机将使用它来找到 81 个指令。每条指令都装在一个 "数学盒 "中。你还需要两个小盒子和一包
Post the instruction poster on a wall in the front of the lassroom and place the boxes on your desk so that the mane cards face the class. According to the instructions FACTORIAL ind the card numbered 2 in the mailbox
将说明海报张贴在教室前面的墙上，将盒子放在桌上，使鬃毛卡面向全班。根据说明 FACTORIAL 将编号为 2 的卡片放入信箱中
将说明海报张贴在教室前面的墙上，将盒子放在桌上，使鬃毛卡面向全班。根据说明 FACTORIAL 将编号为 2 的卡片放入信箱中
Jerome Bruner on Learning and Instruction
杰罗姆-布鲁纳谈学习与教学
The well-known psychologist, Jerome Bruner, has written extensively on learning theory, the instructional process and educational philosophy. Since he has modified his position on the nature of instruction and his philosophy of education between 1960 and 1970, any comprehensive consideration of Bruner's work must include a comparison of his changing attitudes. In the late nineteen fifties Bruner and many other educators, notably those people who were begining to develop the new curricula in mathematics and science, appeared to regard the structure of the disciplines as a very important factor (maybe even the most important factor) in education. At least, it would not be incorrect to say that the content issue was of major concem to many of the developers of the several variations of a modern mathematics curriculum. Bruner's highly acclaimed book, The Process of Education, which was written in 1959-60, reflects the then current thinking of the scholarly community with regard to primary and secondary education. This book is a synthesis of the discussions and perceptions of 34 mathematicians, scientists, psychologists and educators who met for ten days at Woods Hole on Cape Cod to discuss ways to improve education in schools in the United States. Their discussions centered around the importance of teaching the structure of disciplines, readiness for leaming, intuitive and analytic thinking, and motives for learming. General principles such as those stated in the following list emerged from the Woods Hole conference:
著名心理学家杰罗姆-布鲁纳在学习理论、教学过程和教育哲学方面著述颇丰。由于他在 1960 年至 1970 年间改变了自己对教学本质和教育哲学的立场，因此对布鲁纳作品的任何全面思考都必须包括对他态度变化的比较。在 1950 年代末，布鲁纳和许多其他教育家，特别是那些开始制定数学和科学新课程的人，似乎把学科结构视为教育中一个非常重要的因素（甚至可能是最重要的因素）。至少可以这样说，内容问题是现代数学课程几种变体的许多开发者所关心的主要问题。布鲁纳于 1959-60 年撰写的《教育过程》一书备受赞誉，它反映了当时学术界对中小学教育的看法。这本书综合了 34 位数学家、科学家、心理学家和教育家的讨论和看法，他们在科德角的伍兹霍尔举行了为期 10 天的会议，讨论如何改进美国学校的教育。他们的讨论围绕学科结构教学的重要性、学习准备、直觉和分析思维以及学习动机展开。伍兹霍尔会议提出了一些一般性原则，如以下所列：
杰罗姆-布鲁纳谈学习与教学
The well-known psychologist, Jerome Bruner, has written extensively on learning theory, the instructional process and educational philosophy. Since he has modified his position on the nature of instruction and his philosophy of education between 1960 and 1970, any comprehensive consideration of Bruner's work must include a comparison of his changing attitudes. In the late nineteen fifties Bruner and many other educators, notably those people who were begining to develop the new curricula in mathematics and science, appeared to regard the structure of the disciplines as a very important factor (maybe even the most important factor) in education. At least, it would not be incorrect to say that the content issue was of major concem to many of the developers of the several variations of a modern mathematics curriculum. Bruner's highly acclaimed book, The Process of Education, which was written in 1959-60, reflects the then current thinking of the scholarly community with regard to primary and secondary education. This book is a synthesis of the discussions and perceptions of 34 mathematicians, scientists, psychologists and educators who met for ten days at Woods Hole on Cape Cod to discuss ways to improve education in schools in the United States. Their discussions centered around the importance of teaching the structure of disciplines, readiness for leaming, intuitive and analytic thinking, and motives for learming. General principles such as those stated in the following list emerged from the Woods Hole conference:
著名心理学家杰罗姆-布鲁纳在学习理论、教学过程和教育哲学方面著述颇丰。由于他在 1960 年至 1970 年间改变了自己对教学本质和教育哲学的立场，因此对布鲁纳作品的任何全面思考都必须包括对他态度变化的比较。在 1950 年代末，布鲁纳和许多其他教育家，特别是那些开始制定数学和科学新课程的人，似乎把学科结构视为教育中一个非常重要的因素（甚至可能是最重要的因素）。至少可以这样说，内容问题是现代数学课程几种变体的许多开发者所关心的主要问题。布鲁纳于 1959-60 年撰写的《教育过程》一书备受赞誉，它反映了当时学术界对中小学教育的看法。这本书综合了 34 位数学家、科学家、心理学家和教育家的讨论和看法，他们在科德角的伍兹霍尔举行了为期 10 天的会议，讨论如何改进美国学校的教育。他们的讨论围绕学科结构教学的重要性、学习准备、直觉和分析思维以及学习动机展开。伍兹霍尔会议提出了一些一般性原则，如以下所列：
- Proper leaming under optimum conditions leads students to "learm how to leam."
在最佳条件下进行适当的学习，引导学生 "如何学习"。 - Any topic from any subject can be taught to any student in some intellectually honest form at any stage in the student's intellectual development.
在学生智力发展的任何阶段，任何学科的任何主题都可以以某种智力诚实的形式教授给任何学生。 - Intellectual activity is the same anywhere, whether the person is a third grader or a research scientist.
智力活动在任何地方都是一样的，无论是三年级的小学生还是研究科学家。 - The best form of motivation is interest in the subject.
兴趣是最好的动力。
Studying the structure of each subject was thought to be so important that four reasons for teaching structure were formulated. First, it was thought that an examination of the fundamental structure of a subject makes the subject more comprehensible to students. Second, in order to remember details of a subject, the details must be placed in a structured pattern. Third, the optimum way to promote transfer of specific learning to general applications of leaming is through understanding of concepts, principles and the structure of each subject. Fourth, if the fundamental structures of subjects are studied early in school, the lag between current research findings and what is taught in school will be reduced.
研究各学科的结构被认为非常重要，因此提出了结构教学的四个理由。首先，研究学科的基本结构可使学生更容易理解该学科。其次，为了记住一个学科的细节，必须将这些细节置于一个结构化的模式中。第三，通过理解概念、原理和各学科的结构，是促进具体学习向一般学习应用迁移的最佳途径。第四，如果在学校早期就学习学科的基本结构，就会减少当前研究成果与学校教学内容之间的差距。
研究各学科的结构被认为非常重要，因此提出了结构教学的四个理由。首先，研究学科的基本结构可使学生更容易理解该学科。其次，为了记住一个学科的细节，必须将这些细节置于一个结构化的模式中。第三，通过理解概念、原理和各学科的结构，是促进具体学习向一般学习应用迁移的最佳途径。第四，如果在学校早期就学习学科的基本结构，就会减少当前研究成果与学校教学内容之间的差距。
These general principles of instruction and the more specific arguments for teaching structure were thought to constitute the basic rational for the curriculum changes which were under way in 1960. However, in his article "The Process of Education Revisited, " which appeared in 1971 in the Phi Delta Kappan journal, Bruner assessed the major notions about education which were prevelant ten years previous and found them to be quite inadequate. In reference to the educa- tional thinking of 1959, Bruner, in comments critical of that type of thought, stated in 1971 that:
这些一般性的教学原则和关于教学结构的具体论点被认为是 1960 年课程改革的基本依据。然而，布鲁纳在 1971 年发表在《Phi Delta Kappan》杂志上的文章《教育过程的重新审视》中，对十年前盛行的主要教育观念进行了评估，发现这些观念并不充分。在谈到 1959 年的教育思想时，布鲁纳在 1971 年对这种思想提出了批评，他说：
这些一般性的教学原则和关于教学结构的具体论点被认为是 1960 年课程改革的基本依据。然而，布鲁纳在 1971 年发表在《Phi Delta Kappan》杂志上的文章《教育过程的重新审视》中，对十年前盛行的主要教育观念进行了评估，发现这些观念并不充分。在谈到 1959 年的教育思想时，布鲁纳在 1971 年对这种思想提出了批评，他说：
The prevailing notion was that if you understood the structure of knowledge that understanding would then permit you to go ahead on your own; you did not need to encounter everything in nature in order to know nature, but by understanding some deep principles you could extrapolate to the particulars as needed. Knowing was a canny strategy whereby you could know a great deal about a lot of things while keeping very little in mind. (p. 18)
人们普遍认为，如果了解了知识的结构，就可以独立前行；不需要接触自然界的一切来了解自然界，只要了解一些深奥的原理，就可以根据需要推断出具体的知识。认识是一种聪明的策略，通过这种策略，你可以对很多事情了如指掌，同时又只需记住很少的东西。(p. 18)
and 和
The movement of which The Process of Education was a part was based on a formula of faith: that learning was what students wanted to do, that they wanted to achieve an expertise in some paticular subject matter. Their motivation was taken for granted. It also accepted the tacit assumption that everybody who came to these curricula in the schools already had been the beneficiary of the middle-class hidden curricula that taught them analytic skills and launched them in the traditionally intellectual use of mind.
《教育过程》是这场运动的一部分，而这场运动的基础是一种信念：学习是学生们想要做的事，他们想要在某些特定的学科领域获得专业知识。他们的动机是理所当然的。它还接受了一个默认的假设，即学校里的每个人都已经是中产阶级隐性课程的受益者，这些课程教给他们分析技能，让他们开始传统的智力运用。
Failure to question these assumptions has, of course, caused much grief to all of us. (p. 19)
当然，如果不对这些假设提出质疑，就会给我们所有人带来许多痛苦。(p. 19)
In this same Phi Delta Kappan article, Bruner states his more recent viewpoint of the school curriculum as follows:
在这篇《Phi Delta Kappan》文章中，布鲁纳阐述了他对学校课程的最新观点如下：
人们普遍认为，如果了解了知识的结构，就可以独立前行；不需要接触自然界的一切来了解自然界，只要了解一些深奥的原理，就可以根据需要推断出具体的知识。认识是一种聪明的策略，通过这种策略，你可以对很多事情了如指掌，同时又只需记住很少的东西。(p. 18)
and 和
The movement of which The Process of Education was a part was based on a formula of faith: that learning was what students wanted to do, that they wanted to achieve an expertise in some paticular subject matter. Their motivation was taken for granted. It also accepted the tacit assumption that everybody who came to these curricula in the schools already had been the beneficiary of the middle-class hidden curricula that taught them analytic skills and launched them in the traditionally intellectual use of mind.
《教育过程》是这场运动的一部分，而这场运动的基础是一种信念：学习是学生们想要做的事，他们想要在某些特定的学科领域获得专业知识。他们的动机是理所当然的。它还接受了一个默认的假设，即学校里的每个人都已经是中产阶级隐性课程的受益者，这些课程教给他们分析技能，让他们开始传统的智力运用。
Failure to question these assumptions has, of course, caused much grief to all of us. (p. 19)
当然，如果不对这些假设提出质疑，就会给我们所有人带来许多痛苦。(p. 19)
In this same Phi Delta Kappan article, Bruner states his more recent viewpoint of the school curriculum as follows:
在这篇《Phi Delta Kappan》文章中，布鲁纳阐述了他对学校课程的最新观点如下：
If I had my choice now, in terms of a curriculum project for the seventies, it would be to find a means whereby we could bring society back to its sense of values and priorities in life. I believe I would be quite satisfied to declare, if not a moratorium, then something of a de-emphasis on matters that have to do with the structure of history, the structure of physics, the nature of mathematical consistency, and deal with it rather in the context of the problems that face us. We might better concern ourselves with how those problems can be face
如果现在让我来选择七十年代的课程计划，我的选择是找到一种方法，使我们的社 会重新认识生活中的价值观和优先事项。我相信，如果不是宣布暂停，我也会非常满意地宣布不再强调与历史结构、物理结构、数学一致性的性质有关的问题，而是在我们面临的问题的背景下处理这些问题。我们可以更好地关注如何面对这些问题
solved, not just by practical action, but by putting knowledge, wherever we find it and in whatever form we find it, to work in these massive tasks. (p. 21)
要解决这些问题，不仅要靠实际行动，而且要靠知识，无论我们在哪里找到的知识，无论我们以何种形式找到的知识，都要在这些艰巨的任务中发挥作用。(p. 21)
如果现在让我来选择七十年代的课程计划，我的选择是找到一种方法，使我们的社 会重新认识生活中的价值观和优先事项。我相信，如果不是宣布暂停，我也会非常满意地宣布不再强调与历史结构、物理结构、数学一致性的性质有关的问题，而是在我们面临的问题的背景下处理这些问题。我们可以更好地关注如何面对这些问题
solved, not just by practical action, but by putting knowledge, wherever we find it and in whatever form we find it, to work in these massive tasks. (p. 21)
要解决这些问题，不仅要靠实际行动，而且要靠知识，无论我们在哪里找到的知识，无论我们以何种形式找到的知识，都要在这些艰巨的任务中发挥作用。(p. 21)
Bruner's Theory of Instruction
布鲁纳的教学理论
In his book Toward a Theory of Instruction, Bruner presents his viewpoint of the nature of intellectual growth and discusses six characteristics of growth. He also gives two general characteristics which he believes should form the basis of a general theory of instruction and discusses four specific major features which he thinks should be present in any theory of instruction.
布鲁纳在《走向教学理论》一书中提出了他对智力成长本质的观点，并论述了智力成长的六个特征。他还给出了他认为应构成一般教学理论基础的两个一般特征，并讨论了他认为任何教学理论都应具备的四个具体的主要特征。
布鲁纳在《走向教学理论》一书中提出了他对智力成长本质的观点，并论述了智力成长的六个特征。他还给出了他认为应构成一般教学理论基础的两个一般特征，并讨论了他认为任何教学理论都应具备的四个具体的主要特征。
Characteristics of Intellectual Growth
智力成长的特点
According to Bruner, intellectual growth is characterized by a person's increasing ability to separate his or her responses from immediate and specific stimuli. As people develop intellectually, they leam to delay, restructure and control their responses to particular sets of stimuli. One might understand, and even expect, a seventh grader's uncontrolled, angry response in the form of harsh, vulgar words and unacceptable physical actions to criticisms from his or her teacher. However, one would neither expect nor tolerate a teacher's swearing at or striking a student in response to the student's criticisms of the teacher. One of the general
布鲁纳认为，智力成长的特点是一个人越来越有能力将自己的反应从直接和特定的刺激中分离出来。随着智力的发展，人们学会了延迟、重组和控制自己对特定刺激的反应。一个七年级学生对老师的批评做出无节制的、愤怒的反应，表现为尖刻、粗俗的言语和令人无法接受的肢体动作，这是可以理解的，甚至是可以预期的。然而，人们既不会期望也不会容忍教师因学生批评教师而对学生破口大骂或殴打学生。一般来说
bjectives of education is to assist students in leaming to control their responses and to make socially acceprable responses to a variety of stimuli.
教育的目的是帮助学生学会控制自己的反应，并对各种刺激做出社会可接受的反应。
A second characteristic of growth is development of the ability to internalize xternal events into a mental structure which corresponds to the learner's environment and which aids the leamer in generalizing from specific instances. People learn to make predictions and to extrapolate information by structuring sets of events and data. In one sense, the totality of a person's capabilities to extend and apply his or her previous leaming is greater than the sum of that person's specific leaming activities. Mathematical theorem-proving and problem-solving require this somewhat intuitive and creative ability to generalize specific learning.
成长的第二个特点是发展将外部事件内化为心理结构的能力，这种心理结构与学习者所处的环境相一致，有助于学习者从具体事例中归纳总结。人们通过构建事件和数据的结构，学会预测和推断信息。从某种意义上说，一个人扩展和应用其以往学习能力的总和要大于其具体学习活动的总和。数学定理的推导和问题的解决都需要这种直观的、创造性的能力来概括具体的学习内容。
布鲁纳认为，智力成长的特点是一个人越来越有能力将自己的反应从直接和特定的刺激中分离出来。随着智力的发展，人们学会了延迟、重组和控制自己对特定刺激的反应。一个七年级学生对老师的批评做出无节制的、愤怒的反应，表现为尖刻、粗俗的言语和令人无法接受的肢体动作，这是可以理解的，甚至是可以预期的。然而，人们既不会期望也不会容忍教师因学生批评教师而对学生破口大骂或殴打学生。一般来说
bjectives of education is to assist students in leaming to control their responses and to make socially acceprable responses to a variety of stimuli.
教育的目的是帮助学生学会控制自己的反应，并对各种刺激做出社会可接受的反应。
A second characteristic of growth is development of the ability to internalize xternal events into a mental structure which corresponds to the learner's environment and which aids the leamer in generalizing from specific instances. People learn to make predictions and to extrapolate information by structuring sets of events and data. In one sense, the totality of a person's capabilities to extend and apply his or her previous leaming is greater than the sum of that person's specific leaming activities. Mathematical theorem-proving and problem-solving require this somewhat intuitive and creative ability to generalize specific learning.
成长的第二个特点是发展将外部事件内化为心理结构的能力，这种心理结构与学习者所处的环境相一致，有助于学习者从具体事例中归纳总结。人们通过构建事件和数据的结构，学会预测和推断信息。从某种意义上说，一个人扩展和应用其以往学习能力的总和要大于其具体学习活动的总和。数学定理的推导和问题的解决都需要这种直观的、创造性的能力来概括具体的学习内容。
A third characteristic of mental development is the increasing ability. to use words and symbols to represent things which have been done or will be done in the future. The use of words and mathematical symbols permits people to go beyond intuition and empirical adaptation and to use logical and analytical modes of thought. The importance to mathematics of appropriate symbol systems has already been illustrated. Without symbolic notation, mathematics would develop very slowly and would have limited applications for modeling physical and conceptual situations.
智力发展的第三个特点是，使用文字和符号来表示已经做过或将来要做的事情的能 力不断提高。文字和数学符号的使用使人们能够超越直觉和经验适应，使用逻辑和分析思维模式。适当的符号系统对数学的重要性已经说明。如果没有符号记号，数学的发展将非常缓慢，在物理和概念情景建模方面的应用也将十分有限。
The fourth growth characteristic is that mental development depends upon systematic and structured interactions between the learner and teachers; a student's "teachers" are other students, parents, school teachers, or anyone who chooses to instruct the leamer. According to both Bruner and Piaget, intellectual development will be severely retarded if children do not have a variety of contacts with other people. One thing that many school teachers tend not to do is to exploit the unique abilities which students have for teaching each other. On many occasions, students are better able to learn concepts by discussing them with each other and explaining them to each other than through exclusive instruction from the teacher.
学生的 "老师 "是其他学生、家长、学校老师或任何选择指导学习者的人。布鲁纳和皮亚杰都认为，如果儿童不与其他人进行各种接触，智力发展就会严重滞后。许多学校教师往往不善于利用学生的独特能力进行教学。在许多情况下，学生通过相互讨论和相互解释来学习概念，比通过教师的独家指导更有效果。
智力发展的第三个特点是，使用文字和符号来表示已经做过或将来要做的事情的能 力不断提高。文字和数学符号的使用使人们能够超越直觉和经验适应，使用逻辑和分析思维模式。适当的符号系统对数学的重要性已经说明。如果没有符号记号，数学的发展将非常缓慢，在物理和概念情景建模方面的应用也将十分有限。
The fourth growth characteristic is that mental development depends upon systematic and structured interactions between the learner and teachers; a student's "teachers" are other students, parents, school teachers, or anyone who chooses to instruct the leamer. According to both Bruner and Piaget, intellectual development will be severely retarded if children do not have a variety of contacts with other people. One thing that many school teachers tend not to do is to exploit the unique abilities which students have for teaching each other. On many occasions, students are better able to learn concepts by discussing them with each other and explaining them to each other than through exclusive instruction from the teacher.
学生的 "老师 "是其他学生、家长、学校老师或任何选择指导学习者的人。布鲁纳和皮亚杰都认为，如果儿童不与其他人进行各种接触，智力发展就会严重滞后。许多学校教师往往不善于利用学生的独特能力进行教学。在许多情况下，学生通过相互讨论和相互解释来学习概念，比通过教师的独家指导更有效果。
Bruner's fifth characteristic of growth is that teaching and learning are vastly facilitated through the use of language. Not only is language used by teachers to communicate information to students, language is necessary for the complete formulation of most concepts and principles. In mathematics classrooms, one of the primary ways for students to demonstrate knowledge and understanding of mathematical ideas is through the use of language to express their conceptions of the ideas.
布鲁纳的第五个成长特征是，语言的使用极大地促进了教学。语言不仅是教师向学生传递信息的工具，也是完整表述大多数概念和原理所必需的。在数学课堂上，学生展示对数学思想的认识和理解的主要途径之一，就是使用语言表达他们对数学思想的概念。
布鲁纳的第五个成长特征是，语言的使用极大地促进了教学。语言不仅是教师向学生传递信息的工具，也是完整表述大多数概念和原理所必需的。在数学课堂上，学生展示对数学思想的认识和理解的主要途径之一，就是使用语言表达他们对数学思想的概念。
The sixth characteristic is that intellectual growth is demonstrated by the increasing ability to handle several variables simultaneously. People who are intellectually mature can consider several alternatives simultaneously and can give attention to multiple, and even conflicting, demands at the same time. The influnce of Piaget's work upon Bruner's thinking is apparent in Bruner's formulation of this characteristic of intellectual growth. You will recall that Piaget's research has shown that small children who are still intellectually immature are able to deal with only a single characteristic of an object at one time.
第六个特点是智力成长表现为同时处理多个变量的能力不断增强。智力成熟的人可以同时考虑多个备选方案，可以同时关注多种甚至相互冲突的需求。从布鲁纳对智力成长这一特征的表述中，可以明显看出皮亚杰的研究对布鲁纳思想的影响。大家应该还记得，皮亚杰的研究表明，智力尚未成熟的幼儿在同一时间只能处理对象的单一特征。
第六个特点是智力成长表现为同时处理多个变量的能力不断增强。智力成熟的人可以同时考虑多个备选方案，可以同时关注多种甚至相互冲突的需求。从布鲁纳对智力成长这一特征的表述中，可以明显看出皮亚杰的研究对布鲁纳思想的影响。大家应该还记得，皮亚杰的研究表明，智力尚未成熟的幼儿在同一时间只能处理对象的单一特征。
Features of a Theory of Instruction
教学理论的特点
According to Bruner, a theory of instruction should be prescriptive and normative. A theory of instruction is prescriptive if it contains principles for the most effective procedures for teaching and learning facts, skills, concepts, and principles. That is, within the theory there are prescribed processes and methods for attaining the learning objectives of instruction. In addition, the theory should contain processes for evaluating and modifying teaching and leaming strategies. A theory of instruction is normative if it contains general criteria of leaming and states the conditions for meeting the criteria. That is, the theory should contain general learning objectives or goals and should specify how these objectives can be met.
布鲁纳认为，教学理论应具有规定性和规范性。如果教学理论包含了教与学事实、技能、概念和原理的最有效程序的原则，那么它就是规定性的。也就是说，在教学理论中，有实现教学学习目标的规定过程和方法。此外，该理论还应包含评估和修改教与学策略的过程。如果教学理论包含一般的学习标准，并说明达到标准的条件，那么它就是规范性的。也就是说，该理论应包含一般的学习目标或目的，并具体说明如何达到这些目标。
教学理论的特点
According to Bruner, a theory of instruction should be prescriptive and normative. A theory of instruction is prescriptive if it contains principles for the most effective procedures for teaching and learning facts, skills, concepts, and principles. That is, within the theory there are prescribed processes and methods for attaining the learning objectives of instruction. In addition, the theory should contain processes for evaluating and modifying teaching and leaming strategies. A theory of instruction is normative if it contains general criteria of leaming and states the conditions for meeting the criteria. That is, the theory should contain general learning objectives or goals and should specify how these objectives can be met.
布鲁纳认为，教学理论应具有规定性和规范性。如果教学理论包含了教与学事实、技能、概念和原理的最有效程序的原则，那么它就是规定性的。也就是说，在教学理论中，有实现教学学习目标的规定过程和方法。此外，该理论还应包含评估和修改教与学策略的过程。如果教学理论包含一般的学习标准，并说明达到标准的条件，那么它就是规范性的。也就是说，该理论应包含一般的学习目标或目的，并具体说明如何达到这些目标。
Bruner distinguishes between a theory of learming, or a theory of intellectual development, and a theory of instruction. Leaming theories are descriptive, not prescriptive. A theory of leaming is a description of what has happened and what can be expected to happen. For example, Piaget's theory of intellectual development describes the stages through which mental growth progresses and even identifies mental activities which people are or are not able to carry out in each stage. However, Piaget's leaming theory does not prescribe teaching procedures. A theory of instruction is prescriptive and does have leaming objectives. A theory of learning will describe those mental activities which children tives. A theory of theory of instruction will prescribe how to teach students certain capabilities when they are intellectually ready to leam them. For example, Piaget's leaming theory cescribes the fact that young children can not understand one-to-one correspondence; however, an instructional theory might prescribe methods for teaching one-to-one correspondence to students who are intellectually ready to master this concept.
布鲁纳将 "学习理论 "或 "智力发展理论 "与 "教学理论 "区分开来。学习理论是描述性的，而不是规定性的。学习理论是对已经发生的事情和可能发生的事情的描述。例如，皮亚杰的智力发展理论描述了智力成长所经历的各个阶段，甚至确定了人们在每个阶段能够或不能进行的智力活动。然而，皮亚杰的学习理论并没有规定教学程序。教学理论是规定性的，并且有学习目标。学习理论将描述儿童的心理活动。教学理论会规定，当学生在智力上准备好学习某些能力时，如何教给他们这些能力。例如，皮亚杰的学习理论描述了幼儿无法理解一一对应关系的事实；然而，教学理论可能会规定向智力上准备好掌握这一概念的学生教授一一对应关系的方法。
布鲁纳将 "学习理论 "或 "智力发展理论 "与 "教学理论 "区分开来。学习理论是描述性的，而不是规定性的。学习理论是对已经发生的事情和可能发生的事情的描述。例如，皮亚杰的智力发展理论描述了智力成长所经历的各个阶段，甚至确定了人们在每个阶段能够或不能进行的智力活动。然而，皮亚杰的学习理论并没有规定教学程序。教学理论是规定性的，并且有学习目标。学习理论将描述儿童的心理活动。教学理论会规定，当学生在智力上准备好学习某些能力时，如何教给他们这些能力。例如，皮亚杰的学习理论描述了幼儿无法理解一一对应关系的事实；然而，教学理论可能会规定向智力上准备好掌握这一概念的学生教授一一对应关系的方法。
Theories of leaming and theories of instruction are important in education and are, in fact, inseparable. While Piaget's major research efforts are designed to describe the nature of learning, he is not unconcerned with theories of instrucion. Much of Bruner's work has been devoted to developing theories of instrucion, but his theories of instruction are related to and compatible with elements of certain learning theories.
学习理论和教学理论在教育中非常重要，而且事实上密不可分。虽然皮亚杰的主要研究工作旨在描述学习的本质，但他并非不关心教学理论。布鲁纳的大部分工作都致力于发展教学理论，但他的教学理论与某些学习理论的要素相关并相容。
学习理论和教学理论在教育中非常重要，而且事实上密不可分。虽然皮亚杰的主要研究工作旨在描述学习的本质，但他并非不关心教学理论。布鲁纳的大部分工作都致力于发展教学理论，但他的教学理论与某些学习理论的要素相关并相容。
Bruner believes that any theory of instruction should have four major features which prescribe the nature of the instructional process.
布鲁纳认为，任何教学理论都应具备四大特征，它们规定了教学过程的性质。
布鲁纳认为，任何教学理论都应具备四大特征，它们规定了教学过程的性质。
The first feature is that a theory of instruction should specify the experiences which predispose or motivate various types of students to leam; that is, to learn in general and to learn a specific subject such as mathematics. The theory should specify how the student's environment, social status, early childhood, self imand other factors influence his or her altitudes about learning. Predisposition for learning is an important aspect of any theory of instruction.
第一个特点是，教学理论应具体说明各种类型的学生学习的倾向或动机，即学习一般知识和学习数学等具体学科的经验。该理论应具体说明学生所处的环境、社会地位、幼年时期、自我及其他因素是如何影响其学习态度的。学习倾向是任何教学理论的一个重要方面。
第一个特点是，教学理论应具体说明各种类型的学生学习的倾向或动机，即学习一般知识和学习数学等具体学科的经验。该理论应具体说明学生所处的环境、社会地位、幼年时期、自我及其他因素是如何影响其学习态度的。学习倾向是任何教学理论的一个重要方面。
Second, the theory should specify the manner in which general knowledge and particular disciplines must be organized and structured so that they can be most readily leamed by different types of students. Before it is presented to nost readily shoulated be organized so that it relates to the characteristics of learners and embodies the specific structure of the subject. Bruner believes that the structure of any body of knowledge can be described in three ways: its mode the structure of any body of know economy, and its power; each of which varies according to leamer characteristics and disciplines.
其次，理论应明确常识和特定学科的组织和结构方式，以便不同类型的学生最容易掌握。在呈现给学生之前，应先对其进行组织，使其与学习者的特点相关联，并体现出学科的具体结构。布鲁纳认为，任何知识体系的结构都可以从三个方面来描述：知识体系的模式、知识体系的结构和知识体系的力量。
其次，理论应明确常识和特定学科的组织和结构方式，以便不同类型的学生最容易掌握。在呈现给学生之前，应先对其进行组织，使其与学习者的特点相关联，并体现出学科的具体结构。布鲁纳认为，任何知识体系的结构都可以从三个方面来描述：知识体系的模式、知识体系的结构和知识体系的力量。
The mode of representation of a body of knowledge can be either sets of examples or images of the concepts and principles contained in the body of knowledge, or sets of symbolic and logical propositions together with rules for
知识体系的表征方式可以是知识体系中所包含的概念和原理的实例集或图像集，也可以是符号和逻辑命题集，同时还包括以下规则
transforming them. For seventh graders, the concept of a function could be represented quite appropriately by sets of actions such as adding 2 to a specified set of numbers, halving each measurement in a set of measurements, or converting a set of Fahrenheit measurements to the Centigrade scale. High school sophomores could be given examples of functions such as sets of ordered pairs of objects, or could be shown linear relations such as
转换它们。对于七年级的学生来说，函数的概念可以用一组动作来表示，如将一组指定的数加上 2，将一组测量值中的每个测量值减半，或将一组华氏测量值转换成摄氏度。可以向高中二年级学生举例说明函数，如一组有序的成对物体，或展示线性关系，如
知识体系的表征方式可以是知识体系中所包含的概念和原理的实例集或图像集，也可以是符号和逻辑命题集，同时还包括以下规则
transforming them. For seventh graders, the concept of a function could be represented quite appropriately by sets of actions such as adding 2 to a specified set of numbers, halving each measurement in a set of measurements, or converting a set of Fahrenheit measurements to the Centigrade scale. High school sophomores could be given examples of functions such as sets of ordered pairs of objects, or could be shown linear relations such as
转换它们。对于七年级的学生来说，函数的概念可以用一组动作来表示，如将一组指定的数加上 2，将一组测量值中的每个测量值减半，或将一组华氏测量值转换成摄氏度。可以向高中二年级学生举例说明函数，如一组有序的成对物体，或展示线性关系，如
Economy in representing the structure of a discipline is the quantity of information which must be stored in memory in order to understand elements of the discipline. The less information one must remember in order to understand a concept, principle, or process in mathematics, the more economical is the representation of that particular idea or procedure. It is more economical to remember the formula for converting a Fahrenheit scale measurement to a Centigrade scale measurement than it is to remember a table of specific conversions. Economy of representation depends upon the way in which information is organized and sequenced, the manner in which it is presented to students, and the unique learning style of each student.
表示学科结构的经济性是指为了理解学科要素而必须存储在记忆中的信息量。要理解数学中的某个概念、原理或过程，必须记住的信息越少，表示该特定概念或过程的方式就越经济。记住将华氏度量程转换为摄氏度量程的公式，比记住具体的转换表更经济。表述的经济性取决于信息的组织和排序方式、向学生展示信息的方式以及每个学生独特的学习风格。
表示学科结构的经济性是指为了理解学科要素而必须存储在记忆中的信息量。要理解数学中的某个概念、原理或过程，必须记住的信息越少，表示该特定概念或过程的方式就越经济。记住将华氏度量程转换为摄氏度量程的公式，比记住具体的转换表更经济。表述的经济性取决于信息的组织和排序方式、向学生展示信息的方式以及每个学生独特的学习风格。
The power of the structure of a body of knowledge for each learner is related the mental structure which he or she forms in learning the information and is the learner's capacity to organize, connect, and apply information which has been learned. A learner who has structured his or her leaming of the mathematical concepts group, ring, and field in such a way that he or she sees no relationship among these three mathematical ideas, has mentally structured the concepts in a manner which is not very powerful.
对每个学习者来说，知识体系结构的力量与学习者在学习信息时形成的心理结构有关，是学习者组织、连接和应用所学信息的能力。如果一个学习者在学习数学概念群、环和场时，认为这三个数学概念之间没有任何关系，那么他（她）对这些概念的心智结构就不会很强大。
对每个学习者来说，知识体系结构的力量与学习者在学习信息时形成的心理结构有关，是学习者组织、连接和应用所学信息的能力。如果一个学习者在学习数学概念群、环和场时，认为这三个数学概念之间没有任何关系，那么他（她）对这些概念的心智结构就不会很强大。
The third feature of a theory of instruction is that the theory should specify the most effective ways of sequencing material and presenting it to students in order to facilitate learning. Dienes believes that material in mathematics should be sequenced so that students manipulate concrete representations of the concepis in the form of games before they proceed to more abstract representations. Gagné's hierarchical sequencing of mathematics topics suggests that some material should be sequenced using a bottom-to-top approach with prerequisite and simple material being presented first. In contrast to Gagné's sequencing of material, Ausubel suggests a top-down approach which begins with an advance organizer to subsume subordinate material and provide an anchoring mental structure. The problem of sequencing material in mathematics is very complex and is closely related to each student's individual leaming characteristics.
教学理论的第三个特点是，该理论应明确规定最有效的教材排序方式，并将其呈现给学生，以促进学习。Dienes 认为，数学教材的排序应先让学生以游戏的形式操作具体的概念表征，然后再进行更抽象的表征。盖尼耶的数学主题分层排序法认为，有些材料应采用从下到上的排序方法，先呈现先决条件和简单的材料。与盖尼耶的材料排序法不同，奥苏贝尔建议采用一种自上而下的方法，即从先行组织者开始，将从属材料归入其中，并提供一个锚定的心理结构。数学教材的排序问题非常复杂，与每个学生的学习特点密切相关。
教学理论的第三个特点是，该理论应明确规定最有效的教材排序方式，并将其呈现给学生，以促进学习。Dienes 认为，数学教材的排序应先让学生以游戏的形式操作具体的概念表征，然后再进行更抽象的表征。盖尼耶的数学主题分层排序法认为，有些材料应采用从下到上的排序方法，先呈现先决条件和简单的材料。与盖尼耶的材料排序法不同，奥苏贝尔建议采用一种自上而下的方法，即从先行组织者开始，将从属材料归入其中，并提供一个锚定的心理结构。数学教材的排序问题非常复杂，与每个学生的学习特点密切相关。
Bruner's fourth feature of a theory of instruction is that the theory should specify the nature, selection, and sequencing of appropriate rewards and punishments in teaching and leaming a discipline. Certain students, especially younger children, may require immediate teacher-centered rewards such as praise and grades on a frequent basis; whereas many older students may learn more effectively when the rewards are intrinsic, such as self-satisfaction and the joy of learning a new skill. Some high school students regard grades and school awards as artificial and not very meaningful; however other students are moti- vated to a large extent through their desire to obtain high grades and teacher approval.
布鲁纳的教学理论的第四个特点是，该理论应明确规定在教学和学习一门学科时，适当奖惩的性质、选择和顺序。某些学生，尤其是年龄较小的儿童，可能需要以教师为中心的直接奖励，如经常性的表扬和分数；而许多年龄较大的学生，如果奖励是内在的，如自我满足感和学习新技能的喜悦，学习效果可能会更好。有些高中生认为成绩和学校奖励是人为的，没有多大意义；但有些学生则在很大程度上是出于对获得高分和教师认可的渴望。
布鲁纳的教学理论的第四个特点是，该理论应明确规定在教学和学习一门学科时，适当奖惩的性质、选择和顺序。某些学生，尤其是年龄较小的儿童，可能需要以教师为中心的直接奖励，如经常性的表扬和分数；而许多年龄较大的学生，如果奖励是内在的，如自我满足感和学习新技能的喜悦，学习效果可能会更好。有些高中生认为成绩和学校奖励是人为的，没有多大意义；但有些学生则在很大程度上是出于对获得高分和教师认可的渴望。
These four features of a theory of instruction (developing a predisposition to learn, structuring knowledge, sequencing the presentation of materials, and providing rewards and reinforcement) suggest corresponding activities which mathematics teachers should engage in when preparing to teach courses, units, topics and lessons in mathematics. Motivating students to learn mathematics, while not within the exclusive control of the teacher, usually is the responsibility of the teacher. Structuring of knowledge and sequencing of topics in mathematics has been done, in part for teachers, by the writers of mathematics textbooks. However, many perceptive teachers find that student learming can be improved by some judicious resequencing of textbook topics, by selecting supplementary topics, and even by changing textbooks. The primary extrinsic reward system in schools is the grading system; although many good teachers encourage students to learn mathematics by developing leaming activities which provide intemal rewards such as satisfaction in work well done and appreciation of the nature and structure of mathematics as an interesting intellectual activity.
教學理論的這四個特點（建立學習傾向、組織知識結構、編排教材次序，以及 提供獎勵和鞏固）建議了數學教師在準備教授數學課程、單元、課題和課堂時應 進行的相應活動。激发学生学习数学的兴趣，虽然不是教师所能完全控制的，但通常是教师的责任。数学教科书的编写者已经部分地为教师完成了数学知识的结构化和题目的序列化。然而，许多有洞察力的教师发现，通过对教科书题目进行一些明智的重新排序，通过选择补充题目，甚至通过更换教科书，可以提高学生的学习兴趣。學校的主要外在獎勵制度是評分制度；雖然許多優秀的教師會透過發展學習活動來鼓勵學 生學習數學，而這些活動可提供內在獎勵，例如對學習成果的滿足感，以及欣賞數 學作為有趣智力活動的本質和結構。
教學理論的這四個特點（建立學習傾向、組織知識結構、編排教材次序，以及 提供獎勵和鞏固）建議了數學教師在準備教授數學課程、單元、課題和課堂時應 進行的相應活動。激发学生学习数学的兴趣，虽然不是教师所能完全控制的，但通常是教师的责任。数学教科书的编写者已经部分地为教师完成了数学知识的结构化和题目的序列化。然而，许多有洞察力的教师发现，通过对教科书题目进行一些明智的重新排序，通过选择补充题目，甚至通过更换教科书，可以提高学生的学习兴趣。學校的主要外在獎勵制度是評分制度；雖然許多優秀的教師會透過發展學習活動來鼓勵學 生學習數學，而這些活動可提供內在獎勵，例如對學習成果的滿足感，以及欣賞數 學作為有趣智力活動的本質和結構。
Theorems on Learning Mathematics
数学学习定理
In order to identify factors involved in teaching and leaming mathematics: Bruner and his associates have observed a large number of mathematics classes and have conducted experiments on teaching and learning mathematics. As a consequence of these observations and experiments, Bruner and Kenney (April, 1963) formulated four general "theorems" about learning mathematics which they have named the construction theorem, the notation theorem, the theorem of contrast and variation, and the theorem of connectivity.
为了确定数学教学和学习中的相关因素：布鲁纳和他的同事们观察了大量的数学课堂，并进行了数学教与学的实验。作为这些观察和实验的结果，布鲁纳和肯尼（1963 年 4 月）提出了有关数学学习的四个一般性 "定理"，并将其命名为构造定理、符号定理、对比和变化定理以及连通性定理。
为了确定数学教学和学习中的相关因素：布鲁纳和他的同事们观察了大量的数学课堂，并进行了数学教与学的实验。作为这些观察和实验的结果，布鲁纳和肯尼（1963 年 4 月）提出了有关数学学习的四个一般性 "定理"，并将其命名为构造定理、符号定理、对比和变化定理以及连通性定理。
Construction Theorem 构造定理
The construction theorem says that the best way for a student to begin to learn a mathematical concept, principle, or rule is by constructing a representation of it Older students may be able to grasp a mathematical idea by analyzing a representation which is presented by the teacher; however Bruner believes that most students, especially younger children, should construct their own representations of ideas. He also thinks that it is better for students to begin with concrete representations which they have a hand in formulating. If students are permitted to help in formulating and constructing rules in mathematics, they will be more inclined to remember rules and apply them correctly in appropriate situations Bruner has found that giving students finished mathematical rules tends to decrease motivation for leaming and causes many students to become confused. In the early stages of concept leaming, understanding appears to depend upon the concrete activities which students carry out as they construct representations of each concept.
建构定理认为，学生开始学习数学概念、原理或规则的最佳方法是建构其表象。他还认为，最好让学生从自己参与形成的具体表象开始学习。布鲁纳发现，给学生提供完备的数学规则，往往会降低学生学习的积极性，并使许多学生感到困惑。在概念学习的早期阶段，对概念的理解似乎取决于学生在构建每个概念的表象时所开展的具体活动。
建构定理认为，学生开始学习数学概念、原理或规则的最佳方法是建构其表象。他还认为，最好让学生从自己参与形成的具体表象开始学习。布鲁纳发现，给学生提供完备的数学规则，往往会降低学生学习的积极性，并使许多学生感到困惑。在概念学习的早期阶段，对概念的理解似乎取决于学生在构建每个概念的表象时所开展的具体活动。
Notation Theorem 符号定理
The notation theorem states that early constructions or representations can be made cognitively simpler and can be better understood by students if they contain notation which is appropriate for the students' levels of mental development. Efficient notational systems in mathematics make possible the extension of principles and the creation of new principles. Until efficient notational systems for
符号定理指出，如果早期的建构或表象包含适合学生心智发展水平的符号，就能使学生的认知更简单，理解更透彻。高效的数学符号系统使数学原理的扩展和新原理的创造成为可能。高效的数学符号系统
representing equations were formulated, the development of general methods for solving polynomial equations and systems of linear equations progressed very slowly. Students should have a say in creating and selecting notational representations for mathematical ideas; simpler and more transparent notations should be used when concepts are being leamed by younger students. Since seventh and. eighth graders have just leamed to use parentheses as symbols of grouping in arithmetic representations such as
由于学生对方程的表述能力有限，解多项式方程和线性方程组的一般方法的发展十分 缓慢。学生在创建和选择数学概念的符号表示法时应有发言权；在低年级学生学习概念时， 应使用更简单、更透明的符号。由于七年级和八年级的学生刚刚学会在算术表示中使用小括号作为分组符号，如
Contrast and Variation Theorem
对比和变异定理
Bruner's theorem of contrast and variation states that the procedure of going from concrete representations of concepts to more abstract representations involves the operations of contrast and variation. Most mathematical concepts have little meaning for students until they are contrasted to other concepts. In geometry, arcs, radii, diameters and chords of circles all become more meaningful to students when they are contrasted to each other. In fact, many mathematical concepts are defined according to their contrasting properties. Prime numbers are defined as numbers which are neither units nor composite numbers, and irrational numbers are defined as numbers which are not rational. In order for any new concept or principle to be fully understood, it is necessary that its contrasting ideas be presented and considered. Contrast is one of the most useful ways to help students establish an intuitive understanding of a new mathematical topic and to aid them in progressing to more abstract representations of each topic.
布鲁纳的对比和变异定理指出，从概念的具体表象到更抽象的表象的过程涉及对比和变异操作。大多数数学概念在与其他概念对比之前，对学生来说意义不大。在几何学中，圆的弧线、半径、直径和弦都是在相互对比后才对学生更有意义的。事实上，许多数学概念都是根据它们的对比性质来定义的。质数被定义为既不是单位数也不是合数的数，无理数被定义为非有理数。为了让人们充分理解任何新概念或新原理，有必要提出并考虑其对比思想。对比是帮助学生建立对新数学课题的直观理解，并帮助他们逐步对每个课题进行更抽象表述的最有用方法之一。
If students are to learn general concepts in mathematics, each new concept must be represented by a variety of examples of that concept. If not, a general concept may be learned in close association with specific representations of itself. There have been cases in elementary school where children leamed the concept of a set through examples of sets, all of which were represented in the textbook and by the teacher as being enclosed in braces, i.e. {} . Consequently, students who were shown sets of objects such as
如果学生要学习数学中的一般概念，每个新概念都必须由该概念的各种实例来表示。否则，一般概念的学习可能会与概念本身的具体表示密切相关。小学时，孩子们曾通过集合的例子学习集合的概念，而课本和老师都把所有的集合都用大括号表示，即 {} 。.因此，学生在看到
符号定理指出，如果早期的建构或表象包含适合学生心智发展水平的符号，就能使学生的认知更简单，理解更透彻。高效的数学符号系统使数学原理的扩展和新原理的创造成为可能。高效的数学符号系统
representing equations were formulated, the development of general methods for solving polynomial equations and systems of linear equations progressed very slowly. Students should have a say in creating and selecting notational representations for mathematical ideas; simpler and more transparent notations should be used when concepts are being leamed by younger students. Since seventh and. eighth graders have just leamed to use parentheses as symbols of grouping in arithmetic representations such as
由于学生对方程的表述能力有限，解多项式方程和线性方程组的一般方法的发展十分 缓慢。学生在创建和选择数学概念的符号表示法时应有发言权；在低年级学生学习概念时， 应使用更简单、更透明的符号。由于七年级和八年级的学生刚刚学会在算术表示中使用小括号作为分组符号，如
Contrast and Variation Theorem
对比和变异定理
Bruner's theorem of contrast and variation states that the procedure of going from concrete representations of concepts to more abstract representations involves the operations of contrast and variation. Most mathematical concepts have little meaning for students until they are contrasted to other concepts. In geometry, arcs, radii, diameters and chords of circles all become more meaningful to students when they are contrasted to each other. In fact, many mathematical concepts are defined according to their contrasting properties. Prime numbers are defined as numbers which are neither units nor composite numbers, and irrational numbers are defined as numbers which are not rational. In order for any new concept or principle to be fully understood, it is necessary that its contrasting ideas be presented and considered. Contrast is one of the most useful ways to help students establish an intuitive understanding of a new mathematical topic and to aid them in progressing to more abstract representations of each topic.
布鲁纳的对比和变异定理指出，从概念的具体表象到更抽象的表象的过程涉及对比和变异操作。大多数数学概念在与其他概念对比之前，对学生来说意义不大。在几何学中，圆的弧线、半径、直径和弦都是在相互对比后才对学生更有意义的。事实上，许多数学概念都是根据它们的对比性质来定义的。质数被定义为既不是单位数也不是合数的数，无理数被定义为非有理数。为了让人们充分理解任何新概念或新原理，有必要提出并考虑其对比思想。对比是帮助学生建立对新数学课题的直观理解，并帮助他们逐步对每个课题进行更抽象表述的最有用方法之一。
If students are to learn general concepts in mathematics, each new concept must be represented by a variety of examples of that concept. If not, a general concept may be learned in close association with specific representations of itself. There have been cases in elementary school where children leamed the concept of a set through examples of sets, all of which were represented in the textbook and by the teacher as being enclosed in braces, i.e. {} . Consequently, students who were shown sets of objects such as
如果学生要学习数学中的一般概念，每个新概念都必须由该概念的各种实例来表示。否则，一般概念的学习可能会与概念本身的具体表示密切相关。小学时，孩子们曾通过集合的例子学习集合的概念，而课本和老师都把所有的集合都用大括号表示，即 {} 。.因此，学生在看到
Connectivity Theorem 连接性定理
The connectivity itreorem can be stated as follows: each concept, principle, and skill in mathematics is connected to other concepts, principles, or skills. The structured connections among the elements in each branch of mathematics permit analytic and synthetic mathematical reasoning, as well as intuitive jumps in mathematical thought. The result is mathematical progress. One of the most important activities of mathematicians is the search for connections and relationships among mathematical structures. In teaching mathematics it is not only necessary for teachers to help students observe the contrasts and variations among mathematical structures, but students also need to become aware of connections between various mathematical structures. Gagne's development of learning hierarchies for structuring the teaching of mathematical content involves searching for connections in mathematics. The structure of mathematics is condensed and simplified and learning mathematics is made easier by identifying connections such as one-to-one correspondences and isomorphisms. In fact, many of the modem mathematics curriculum projects have attempted to illustrate the connections within each branch of mathematics and connections among various branches such as algebra, geometry, and analysis. Not only are connections important for the progress of mathematics, but awareness of connections is also important in leaming mathematics. Since very few mathematics topics exist in isolation from all other mathematics topics, connections among topics must be illustrated and understood if progressive, meaningful learning is to be accomplished by students.
连通性定理可以表述为：数学中的每一个概念、原理和技能都与其他概念、原理或技能相关联。数学各分支中各要素之间的结构性联系允许进行分析和综合数学推理，以及数学思维中的直觉跳跃。结果就是数学的进步。数学家最重要的活动之一就是寻找数学结构之间的联系和关系。在数学教学中，教师不仅要帮助学生观察数学结构之间的对比和变化，还要让学生意识到各种数学结构之间的联系。加涅为构建数学教学内容而提出的学习分层，就是要寻找数学中的联系。通过找出一一对应和同构等联系，将数学结构浓缩和简化，使数学学习变得更容易。事实上，许多现代数学课程项目都试图说明数学各分支内部的联系以及代数、几何和分析等各分支之间的联系。联系不仅对数学的发展很重要，而且对联系的认识对学习数学也很重要。由于很少有数学课题是孤立于所有其他数学课题而存在的，因此必须说明和理解各课题之间的联系，才能让学生循序渐进地完成有意义的学习。
Applications of Bruner's Work
布鲁纳作品的应用
Bruner's earlier works, as well as his recent writings, are relevant and useful for teachers and students of mathematics. His viewpoint regarding the importance of intuition and discovery learning for meaningful learning provides mathematics teachers with a balanced contrast to the structured, expository approach to teaching and learning which Ausubel has promoted. In closing our discussion of Bruner's contributions to mathematics education, we will consider an illustration showing how his four theorems for teaching and learning mathematics can be applied to a topic in mathematics-the topic of limits.
布鲁纳的早期作品和近期著作对数学教师和学生都有现实意义和实用价值。他认为直觉和发现式学习对有意义学习的重要性，这与奥苏贝尔所提倡的结构化、讲解式的教学方法形成了平衡的对比。在结束布鲁纳对数学教育的贡献的讨论时，我们将通过一个例子来说明他的四个数学教学定理如何应用于数学中的一个课题--极限课题。
连通性定理可以表述为：数学中的每一个概念、原理和技能都与其他概念、原理或技能相关联。数学各分支中各要素之间的结构性联系允许进行分析和综合数学推理，以及数学思维中的直觉跳跃。结果就是数学的进步。数学家最重要的活动之一就是寻找数学结构之间的联系和关系。在数学教学中，教师不仅要帮助学生观察数学结构之间的对比和变化，还要让学生意识到各种数学结构之间的联系。加涅为构建数学教学内容而提出的学习分层，就是要寻找数学中的联系。通过找出一一对应和同构等联系，将数学结构浓缩和简化，使数学学习变得更容易。事实上，许多现代数学课程项目都试图说明数学各分支内部的联系以及代数、几何和分析等各分支之间的联系。联系不仅对数学的发展很重要，而且对联系的认识对学习数学也很重要。由于很少有数学课题是孤立于所有其他数学课题而存在的，因此必须说明和理解各课题之间的联系，才能让学生循序渐进地完成有意义的学习。
Applications of Bruner's Work
布鲁纳作品的应用
Bruner's earlier works, as well as his recent writings, are relevant and useful for teachers and students of mathematics. His viewpoint regarding the importance of intuition and discovery learning for meaningful learning provides mathematics teachers with a balanced contrast to the structured, expository approach to teaching and learning which Ausubel has promoted. In closing our discussion of Bruner's contributions to mathematics education, we will consider an illustration showing how his four theorems for teaching and learning mathematics can be applied to a topic in mathematics-the topic of limits.
布鲁纳的早期作品和近期著作对数学教师和学生都有现实意义和实用价值。他认为直觉和发现式学习对有意义学习的重要性，这与奥苏贝尔所提倡的结构化、讲解式的教学方法形成了平衡的对比。在结束布鲁纳对数学教育的贡献的讨论时，我们将通过一个例子来说明他的四个数学教学定理如何应用于数学中的一个课题--极限课题。
Calculus is a difficult subject for many students, and some people who complete high school or college calculus courses do so by memorizing rules and problem types and have little understanding of the conceptual nature of the subject. The creation of calculus was motivated by the need for mathematical techniques to handle continuous processes in nature, such as the movement of bodies in our universe. While algebraic concepts and skills are quite satisfactory for dealing with discrete, finite processes, the concept of a limit is needed to attack those continuous and infinite processes in nature which are now commonly studied in calculus and related subjects. The fundamental concept of calculus, that of limits, is also the fundamental source of many difficulties in leaming and applying the subject. Throughout school, until calculus comes along, little is said about limits in spite of the fact that the limit concept is indispensable to any serious consideration of continuous natural processes.
微积分对许多学生来说都是一门难学的学科，有些人在完成高中或大学微积分课程后，只是记住了一些规则和题型，而对这门学科的概念本质却知之甚少。微积分的产生是由于人们需要数学技术来处理自然界中的连续过程，如宇宙中的天体运动。虽然代数概念和技巧对于处理离散、有限的过程是相当令人满意的，但要处理自然界中那些连续、无限的过程，就需要极限的概念，而这正是微积分和相关学科现在通常研究的内容。微积分的基本概念 "极限"，也是学习和应用微积分的许多困难的根本原因。在学校里，在微积分出现之前，人们对极限的概念知之甚少，尽管极限概念对于认真研究连续的自然过程是不可或缺的。
微积分对许多学生来说都是一门难学的学科，有些人在完成高中或大学微积分课程后，只是记住了一些规则和题型，而对这门学科的概念本质却知之甚少。微积分的产生是由于人们需要数学技术来处理自然界中的连续过程，如宇宙中的天体运动。虽然代数概念和技巧对于处理离散、有限的过程是相当令人满意的，但要处理自然界中那些连续、无限的过程，就需要极限的概念，而这正是微积分和相关学科现在通常研究的内容。微积分的基本概念 "极限"，也是学习和应用微积分的许多困难的根本原因。在学校里，在微积分出现之前，人们对极限的概念知之甚少，尽管极限概念对于认真研究连续的自然过程是不可或缺的。
For each year in school from seventh grade on, Bruner's theorems of con-
从七年级开始的每一年，布鲁纳的 con-
从七年级开始的每一年，布鲁纳的 con-
In an advance mathenathes counse sum as semui analysts, these sequences would be appropriate for students to consider:
在作为 Semui 分析师的高等数学课程中，学生可以考虑这些序列：
(m)
(m)
(n)
(n)
In an advanced mathematics course, notation such as
在高等数学课程中，诸如
在作为 Semui 分析师的高等数学课程中，学生可以考虑这些序列：
(m)
(m)
(n)
(n)
In an advanced mathematics course, notation such as
在高等数学课程中，诸如
is appropriate for students to use, provided they understand and can explain this type of symbolic notation.
适合学生使用，前提是他们理解并能解释这类符号记号。
适合学生使用，前提是他们理解并能解释这类符号记号。
Of course, in calculus, limits such as the limiting value of a sequence of secant lines through a point on a curve, which is a representation of the concept of a derivative, and the sums of sequences of rectangles bounded by a curve, which represents the definite integral concept, are appropriate representations for the limit concept. Students in more advanced mathematics courses, such as real nalysis, deal with representations and notational systems for limits of sequence of functions and limits of sums of sequences of functions.
当然，在微积分中，极限概念的适当表示方法包括：表示导数概念的通过曲线上某点的正割直线序列的极限值，以及表示定积分概念的以曲线为界的矩形序列之和。更高级数学课程（如实数分析）的学生将学习函数序列极限和函数序列之和极限的表示方法和符号系统。
当然，在微积分中，极限概念的适当表示方法包括：表示导数概念的通过曲线上某点的正割直线序列的极限值，以及表示定积分概念的以曲线为界的矩形序列之和。更高级数学课程（如实数分析）的学生将学习函数序列极限和函数序列之和极限的表示方法和符号系统。
Even though this example merely contains a set of representations and does not have specifications for teaching strategies and leaming activities, we can observe that Bruner's four theorems for teaching mathematics are exemplified in the sequence of examples irrespective of the specific teaching/learning strategies which teachers may choose to use. A variety of concrete and abstract examples of the limit concept are given above, and the notation becomes more abstract and symbolic as the examples progress from seventh grade level to college level. The representations are varied enough so that the limit concept is freed from any particular type of representation or subject in mathematics; the concept of limit is connected to concepts from algebra, geometry, trigonometry, and higher mathematics. When teaching the limit concept, it also would be necessary to encourage students to construct their own representations of limits, to present them to each other, and to discuss the differences and similarities among the various embodiments of the concept.
儘管這個例子只包含一系列的表象，並沒有對教學策略和學習活動作出規範， 但我們可以看到，無論教師選擇採用何種具體的教與學策 略，布魯納的四個數學教學定理都能在一連串的例子中得到體現。上面给出了极限概念的各种具体和抽象示例，随着示例从七年级水平上升到大学水平，符号也变得更加抽象和符号化。这些示例形式多样，使极限概念摆脱了数学中任何特定类型的示例或科目；极限概念与代数、几何、三角和高等数学中的概念相关联。在讲授极限概念时，也有必要鼓励学生构建自己的极限表象，相互介绍这些表象，并讨论极限概念各种表象之间的异同。
儘管這個例子只包含一系列的表象，並沒有對教學策略和學習活動作出規範， 但我們可以看到，無論教師選擇採用何種具體的教與學策 略，布魯納的四個數學教學定理都能在一連串的例子中得到體現。上面给出了极限概念的各种具体和抽象示例，随着示例从七年级水平上升到大学水平，符号也变得更加抽象和符号化。这些示例形式多样，使极限概念摆脱了数学中任何特定类型的示例或科目；极限概念与代数、几何、三角和高等数学中的概念相关联。在讲授极限概念时，也有必要鼓励学生构建自己的极限表象，相互介绍这些表象，并讨论极限概念各种表象之间的异同。
B. F. Skinner on Teaching and Learning
B.F. 斯金纳论教与学
For many years philosophers and psychologists have formulated and debated altemative viewpoints about the nature of human beings. There now are two generally accepted models of human actions-the behaviorist model and the
多年来，哲学家和心理学家就人类的本质提出了各种不同的观点，并展开了激烈的辩论。目前，有两种普遍接受的人类行为模式--行为主义模式和行为主义模式。
whenomenological model. Those philosophers and psychologists who subscribe thenomenological molel. Those philosophers and being somewhat passive organisms to the behaviorist viewpoint regard people as being somewhat ats who are primarily controlled by stimuli from their environments. Proponents of this behaviorist model of human nature believe that people's behaviors can be controlled by properly controlling their environments, and that scientific methods are appropriate for the study of human behavior.
现象学模式。那些赞同现象学模式的哲学家和心理学家认为，人是一种被动的有机体。行为主义观点认为，人在某种程度上是被动的有机体，主要受环境刺激的控制。这种行为主义人性模式的支持者认为，可以通过适当控制环境来控制人的行为，而且科学方法适合于研究人的行为。
多年来，哲学家和心理学家就人类的本质提出了各种不同的观点，并展开了激烈的辩论。目前，有两种普遍接受的人类行为模式--行为主义模式和行为主义模式。
whenomenological model. Those philosophers and psychologists who subscribe thenomenological molel. Those philosophers and being somewhat passive organisms to the behaviorist viewpoint regard people as being somewhat ats who are primarily controlled by stimuli from their environments. Proponents of this behaviorist model of human nature believe that people's behaviors can be controlled by properly controlling their environments, and that scientific methods are appropriate for the study of human behavior.
现象学模式。那些赞同现象学模式的哲学家和心理学家认为，人是一种被动的有机体。行为主义观点认为，人在某种程度上是被动的有机体，主要受环境刺激的控制。这种行为主义人性模式的支持者认为，可以通过适当控制环境来控制人的行为，而且科学方法适合于研究人的行为。
The phenomenological viewpoint proposes that people are inherently and primarily in control of their own actions. People are regarded as being free to make their own choices and to control their own behaviors. A philosopher or psychologist who agrees with the phenomenological model of mankind would center a study of human behavior around human consciousness, awareness, and self expression.
现象学观点认为，人天生就能控制自己的行为。人们被认为可以自由地做出自己的选择并控制自己的行为。认同现象学人类模式的哲学家或心理学家会围绕人的意识、认知和自我表达来研究人类行为。
现象学观点认为，人天生就能控制自己的行为。人们被认为可以自由地做出自己的选择并控制自己的行为。认同现象学人类模式的哲学家或心理学家会围绕人的意识、认知和自我表达来研究人类行为。
Although these two viewpoints of human behavior stem from contrasting philosophies, they do have some common elements: perhaps hurman behavior is, paradoxically, both capable and incapable of being studied scientifically. It may be that much of human behavior is subject to certain laws which are fixed for the time being, but which change through an evolutionary process as humanity gain new information and knowledge about itself. Frank Milhollan and Bill Forisha (1972), in their book From Skinner to Rogers, develop these contrasting models of human behavior by contrasting two psychologists' (B. F. Skinner and Carl Rogers) approaches to human behavior and education.
尽管这两种关于人类行为的观点源于截然不同的哲学，但它们确实有一些共同点：也许人类的行为既能够又不能够用科学的方法来研究。也许人类的许多行为都受制于某些规律，这些规律暂时是固定不变的，但随着人类获得关于自身的新信息和新知识，这些规律会在进化过程中发生变化。弗兰克-米尔霍兰（Frank Milhollan）和比尔-福里沙（Bill Forisha，1972 年）在《从斯金纳到罗杰斯》（From Skinner to Rogers）一书中，通过对比两位心理学家（B. F. Skinner 和卡尔-罗杰斯）对人类行为和教育的研究方法，提出了这些截然不同的人类行为模式。
In this section we are going to study the scientific behavioral approach to teaching and learning which B. F. Skinner has described and researched. Skinner is regarded as one of the most influential of the modern psychologists. His work has provided a basis for many programmed instruction and individualized leaming packages, and, more recently, for some computer-based instructional systems. Skinner's work has also had considerable impact upon society in general, through his development and promotion of strategies for the effective and efficient modification of human behavior. One of Skinner's major contributions to education is his experimental and scientific analysis of behavior, which has important implications for teaching and learning. In fact, according to Milhollan and Forisha (1972):
在本节中，我们将学习 B. F. Skinner 所描述和研究的科学行为教学法。斯金纳被认为是现代心理学家中最有影响力的一位。他的研究成果为许多程序化教学和个性化学习软件包提供了基础，最近还为一些基于计算机的教学系统提供了基础。斯金纳的工作还对整个社会产生了相当大的影响，因为他开发并推广了有效和高效地改变人类行为的策略。斯金纳对教育的主要贡献之一是他对行为的实验和科学分析，这对教学和学习有着重要的影响。事实上，根据 Milhollan 和 Forisha（1972 年）的说法
尽管这两种关于人类行为的观点源于截然不同的哲学，但它们确实有一些共同点：也许人类的行为既能够又不能够用科学的方法来研究。也许人类的许多行为都受制于某些规律，这些规律暂时是固定不变的，但随着人类获得关于自身的新信息和新知识，这些规律会在进化过程中发生变化。弗兰克-米尔霍兰（Frank Milhollan）和比尔-福里沙（Bill Forisha，1972 年）在《从斯金纳到罗杰斯》（From Skinner to Rogers）一书中，通过对比两位心理学家（B. F. Skinner 和卡尔-罗杰斯）对人类行为和教育的研究方法，提出了这些截然不同的人类行为模式。
In this section we are going to study the scientific behavioral approach to teaching and learning which B. F. Skinner has described and researched. Skinner is regarded as one of the most influential of the modern psychologists. His work has provided a basis for many programmed instruction and individualized leaming packages, and, more recently, for some computer-based instructional systems. Skinner's work has also had considerable impact upon society in general, through his development and promotion of strategies for the effective and efficient modification of human behavior. One of Skinner's major contributions to education is his experimental and scientific analysis of behavior, which has important implications for teaching and learning. In fact, according to Milhollan and Forisha (1972):
在本节中，我们将学习 B. F. Skinner 所描述和研究的科学行为教学法。斯金纳被认为是现代心理学家中最有影响力的一位。他的研究成果为许多程序化教学和个性化学习软件包提供了基础，最近还为一些基于计算机的教学系统提供了基础。斯金纳的工作还对整个社会产生了相当大的影响，因为他开发并推广了有效和高效地改变人类行为的策略。斯金纳对教育的主要贡献之一是他对行为的实验和科学分析，这对教学和学习有着重要的影响。事实上，根据 Milhollan 和 Forisha（1972 年）的说法
One of the most influential positions regarding the nature of psychology One of the most influential position is exemplified by the work of B. F. and how it can be applied ofremably represents the most complete and sysSkinner. Skinner's system probably represents the most complete and sys-
关于心理学的本质，最有影响力的立场之一 B. F. 斯金纳的工作是最有影响力的立场之一，他的工作体现了这一立场以及如何应用这一立场。斯金纳的体系可能是最完整、最系统的心理学体系。
tematic statement of the associationist, behaviorist, environmentalist, determinist position in psychology today. (p. 44)
对当今心理学界的关联主义、行为主义、环境主义和决定论立场的专题阐述。(p. 44)
While Piaget, Guilford, and Ausubel are primarily concerned with the development of the mind or the way the mind receives and structures information (that is, what goes on in the mind), Skinner believes that a study of leaching and learning depends primarily upon the observable behaviors of teachers and students. Since the scientific method has been quite successful in advancing knowledge in the physical sciences, Skinner thinks that a scientific approach can be used equally well for studying the social sciences. Milhollan and Forisha interpret Skinner's thoughts on this issue as follows:
皮亚杰、吉尔福特和奥苏贝尔主要关注的是心智的发展或心智接收和构建信息的方式（即心智中发生了什么），而斯金纳则认为，对浸润和学习的研究主要取决于教师和学生的可观察行为。由于科学方法在推动物理科学知识方面取得了相当大的成功，斯金纳认为科学方法同样可以用于研究社会科学。米尔霍兰和福里沙对斯金纳的这一观点做了如下解释：
关于心理学的本质，最有影响力的立场之一 B. F. 斯金纳的工作是最有影响力的立场之一，他的工作体现了这一立场以及如何应用这一立场。斯金纳的体系可能是最完整、最系统的心理学体系。
tematic statement of the associationist, behaviorist, environmentalist, determinist position in psychology today. (p. 44)
对当今心理学界的关联主义、行为主义、环境主义和决定论立场的专题阐述。(p. 44)
While Piaget, Guilford, and Ausubel are primarily concerned with the development of the mind or the way the mind receives and structures information (that is, what goes on in the mind), Skinner believes that a study of leaching and learning depends primarily upon the observable behaviors of teachers and students. Since the scientific method has been quite successful in advancing knowledge in the physical sciences, Skinner thinks that a scientific approach can be used equally well for studying the social sciences. Milhollan and Forisha interpret Skinner's thoughts on this issue as follows:
皮亚杰、吉尔福特和奥苏贝尔主要关注的是心智的发展或心智接收和构建信息的方式（即心智中发生了什么），而斯金纳则认为，对浸润和学习的研究主要取决于教师和学生的可观察行为。由于科学方法在推动物理科学知识方面取得了相当大的成功，斯金纳认为科学方法同样可以用于研究社会科学。米尔霍兰和福里沙对斯金纳的这一观点做了如下解释：
Skinner believes that the methods of science should be applied to the field of human affairs. We are all controlled by the world, part of which is constructed human affairs. We are all controlled accident, by tyrants, or by ourselves? A scientific society should reject accidental manipulation. He asserts that a specific plan is needed to promote fully the development of man and society. We cannot make wise decisions if we continue to pretend that we are not controlled.
斯金纳认为，科学的方法应该应用于人类事务领域。我们都受到世界的控制，而世界的一部分就是构建出来的人类事务。我们是被意外控制，被暴君控制，还是被我们自己控制？一个科学的社会应该拒绝偶然的操纵。他断言，需要一个具体的计划来全面促进人类和社会的发展。如果我们继续假装自己不受控制，就无法做出明智的决定。
斯金纳认为，科学的方法应该应用于人类事务领域。我们都受到世界的控制，而世界的一部分就是构建出来的人类事务。我们是被意外控制，被暴君控制，还是被我们自己控制？一个科学的社会应该拒绝偶然的操纵。他断言，需要一个具体的计划来全面促进人类和社会的发展。如果我们继续假装自己不受控制，就无法做出明智的决定。
As Skinner points out, the possibility of behavioral control is offensive to many people. We have traditionally regarded man as a free agent whose behavior occurs by virtue of spontaneous inner changes. We are reluctant to abandon the internal "will" which makes prediction and control of behavior impossible. (p. 45)
正如斯金纳指出的那样，行为控制的可能性让很多人感到反感。传统上，我们认为人是一个自由的主体，其行为是由自发的内在变化产生的。我们不愿意放弃内在的 "意志"，因为它使得预测和控制行为成为不可能。(p. 45)
and 和
Skinner notes that a scientific conception of human behavior dictates one practice and a philosophy of personal freedom another. Until we adopt a consistent view we are likely to remain ineffective in solving our social problems. A scientific conception entails the acceptance of an assumption of determinism, the doctrine that behavior is caused and that the behavior which appears is the only one which could have appeared. (p. 46)
斯金纳指出，人类行为的科学概念决定了一种做法，而个人自由哲学则决定了另一种做法。除非我们采取一致的观点，否则在解决社会问题方面，我们很可能仍然是无效的。科学的观念要求我们接受决定论的假设，即行为是由原因造成的，出现的行为是唯一可能出现的行为。(p. 46)
正如斯金纳指出的那样，行为控制的可能性让很多人感到反感。传统上，我们认为人是一个自由的主体，其行为是由自发的内在变化产生的。我们不愿意放弃内在的 "意志"，因为它使得预测和控制行为成为不可能。(p. 45)
and 和
Skinner notes that a scientific conception of human behavior dictates one practice and a philosophy of personal freedom another. Until we adopt a consistent view we are likely to remain ineffective in solving our social problems. A scientific conception entails the acceptance of an assumption of determinism, the doctrine that behavior is caused and that the behavior which appears is the only one which could have appeared. (p. 46)
斯金纳指出，人类行为的科学概念决定了一种做法，而个人自由哲学则决定了另一种做法。除非我们采取一致的观点，否则在解决社会问题方面，我们很可能仍然是无效的。科学的观念要求我们接受决定论的假设，即行为是由原因造成的，出现的行为是唯一可能出现的行为。(p. 46)
Types of Behavior and Learning
行为和学习类型
According to Skinner, nearly all identifiable human behavior falls into two categories, respondent behavior and operant behavior. Respondent behaviors are involuntary (reflex) behaviors and result from special environmental stimuli. In order for a respondent behavior to occur, it is first necessary that a stimulus be applied to the organism. The stimulus of a bug flying toward your eyes will cause you to blink, an embarrassing event may cause you to blush, and a bright flash of light will result in your blinking your eyes. Only a few of our behaviors are respondent behaviors.
斯金纳认为，几乎所有可识别的人类行为都可分为两类，即应答行为和操作行为。反应行为是一种非自主（条件反射）行为，由特殊的环境刺激引起。要发生应答行为，首先需要对生物体施加刺激。小虫子飞向你眼睛的刺激会让你眨眼，尴尬的事件可能会让你脸红，强光闪烁会让你眨眼。我们的行为中只有少数是反应行为。
斯金纳认为，几乎所有可识别的人类行为都可分为两类，即应答行为和操作行为。反应行为是一种非自主（条件反射）行为，由特殊的环境刺激引起。要发生应答行为，首先需要对生物体施加刺激。小虫子飞向你眼睛的刺激会让你眨眼，尴尬的事件可能会让你脸红，强光闪烁会让你眨眼。我们的行为中只有少数是反应行为。
Most of our behaviors are operant behaviors, which are neither automatic, predictable, nor related in any known manner to easily identifiable stimuli. Skinner believes that certain behaviors merely happen, and even if they are caused by specific (but hard to identify) stimuli, these stimuli are inconsequential to the study of behavior. The word "operant"' describes an entire set of specific instances of behaviors which operate upon the environment to generate events or responses within the environment. If these events or responses are satisfying, the probability that the operant behavior will be repeated is usually increased.
我们的大多数行为都是操作行为，这些行为既不是自动的、可预测的，也不是以任何已知的方式与易于识别的刺激相关联的。斯金纳认为，某些行为只是发生了，即使它们是由特定的（但难以识别的）刺激引起的，这些刺激对于行为研究来说也是无关紧要的。操作性 "一词描述的是一整套具体的行为实例，这些行为对环境产生作用，从而在环境中产生事件或反应。如果这些事件或反应令人满意，那么操作行为重复出现的概率通常就会增加。
我们的大多数行为都是操作行为，这些行为既不是自动的、可预测的，也不是以任何已知的方式与易于识别的刺激相关联的。斯金纳认为，某些行为只是发生了，即使它们是由特定的（但难以识别的）刺激引起的，这些刺激对于行为研究来说也是无关紧要的。操作性 "一词描述的是一整套具体的行为实例，这些行为对环境产生作用，从而在环境中产生事件或反应。如果这些事件或反应令人满意，那么操作行为重复出现的概率通常就会增加。
Both respondent and operant behaviors can be taught and learned. Teaching and learning a respondent behavior requires the presentation of a stimulus which will cause the desired behavior to occur; whereas an operant behavior is leamed through an appropriate reinforcement (either a positive or a negative reinforcement) which is adninistered immediately or shortly after the spontaneous occurrence of the operant behavior. The administration of a reinforcement to a person following the occurrence of a desirable behavior usually increases the probability that he or she will repeat the behavior. If the reinforcement is a punishment, it is hoped that the individual will learn to refrain from the undesirable behavior which evoked the punishment.
反应行为和操作行为都可以教和学。教导和学习应答行为需要刺激，刺激会导致期望行为的发生；而操作行为则是通过适当的强化物（正强化物或负强化物）来学习的，强化物是在操作行为自发发生后立即或很快给予的。在一个人出现期望行为后给予强化物，通常会增加他或她重复该行为的可能性。如果强化物是一种惩罚，则希望当事人学会避免诱发惩罚的不良行为。
反应行为和操作行为都可以教和学。教导和学习应答行为需要刺激，刺激会导致期望行为的发生；而操作行为则是通过适当的强化物（正强化物或负强化物）来学习的，强化物是在操作行为自发发生后立即或很快给予的。在一个人出现期望行为后给予强化物，通常会增加他或她重复该行为的可能性。如果强化物是一种惩罚，则希望当事人学会避免诱发惩罚的不良行为。
For each type of behavior, respondent behavior and operant behavior, Skinner has identified a type of conditioning, a generalized teaching/leaming strategy, which will facilitate leaming the desired behavior. Classical respondent conditioning for respondent learning (which is similar to what Gagné calls signal learning) results when a new stimulus is presented simultaneously with an older
对于每一种行为，即应答行为和操作行为，斯金纳都确定了一种条件反射类型，即一种通用的教学/驯化策略，这种策略将有助于驯化所需的行为。当新刺激与旧刺激同时出现时，就会产生应答学习的经典应答条件反射（类似于盖尼耶所说的信号学习）。
对于每一种行为，即应答行为和操作行为，斯金纳都确定了一种条件反射类型，即一种通用的教学/驯化策略，这种策略将有助于驯化所需的行为。当新刺激与旧刺激同时出现时，就会产生应答学习的经典应答条件反射（类似于盖尼耶所说的信号学习）。
B.F. Skin! B.F. 皮肤
此
放
stimulus which elicits the expected response. After a variable number of pairings, the new stimulus will elicit the response without being paired with the old stimulus. The classical example of respondent conditioning is provided in the work of the Russian physiologist Ivan Pavlov. Pavloy conditioned dogs in his laboratory to salivate at the sound of a tone by first sounding the tone simultaneously with the presentation of food to the dogs. After a number of paired presentations of food and tone sounding, the dogs salivated upon hearing the tone, even though they were not given food when the tone was sounded.
刺激，从而引起预期的反应。经过不同次数的配对后，新的刺激就会引起反应，而无需与旧的刺激配对。俄罗斯生理学家伊万-巴甫洛夫（Ivan Pavlov）的研究提供了应答条件反射的经典范例。巴甫洛夫在他的实验室里让狗在听到音调时产生流口水的条件反射，方法是首先在向狗展示食物的同时发出音调。在多次将食物和音调配对后，狗在听到音调时会流口水，即使在音调响起时并没有给它们食物。
Operant conditioning, as specified by Skinner, can be used to promote operant learming Operan conditioning for operanLlearning is controlled by following a behavior with a stimulus. This stimulus, which is presented after the response, is usually called a reinforcement. It can be either a positive or negative reinforcement, since both positive and negative reinforcements can be used to increase the likelihood that a behavior will be repeated. As an example of operant conditioning and operant learming, consider a hypothetical student sitting in the rear of the classroom who usually is shy, quiet, and unresponsive in class. The following altemative dialogs might occur between this student and his teacher:
斯金纳提出的操作性条件反射可以用来促进操作性学习。这种在反应之后出现的刺激通常被称为强化物。强化物可以是正强化物，也可以是负强化物，因为正强化物和负强化物都可以用来增加行为重复的可能性。举一个操作性条件反射和操作性强化的例子，假设有一个学生坐在教室后面，平时在课堂上害羞、安静、反应迟钝。这名学生和他的老师之间可能会发生以下替代对话：
(1) Teacher: "Jim, what does
(1) 教师："吉姆，
放
stimulus which elicits the expected response. After a variable number of pairings, the new stimulus will elicit the response without being paired with the old stimulus. The classical example of respondent conditioning is provided in the work of the Russian physiologist Ivan Pavlov. Pavloy conditioned dogs in his laboratory to salivate at the sound of a tone by first sounding the tone simultaneously with the presentation of food to the dogs. After a number of paired presentations of food and tone sounding, the dogs salivated upon hearing the tone, even though they were not given food when the tone was sounded.
刺激，从而引起预期的反应。经过不同次数的配对后，新的刺激就会引起反应，而无需与旧的刺激配对。俄罗斯生理学家伊万-巴甫洛夫（Ivan Pavlov）的研究提供了应答条件反射的经典范例。巴甫洛夫在他的实验室里让狗在听到音调时产生流口水的条件反射，方法是首先在向狗展示食物的同时发出音调。在多次将食物和音调配对后，狗在听到音调时会流口水，即使在音调响起时并没有给它们食物。
Operant conditioning, as specified by Skinner, can be used to promote operant learming Operan conditioning for operanLlearning is controlled by following a behavior with a stimulus. This stimulus, which is presented after the response, is usually called a reinforcement. It can be either a positive or negative reinforcement, since both positive and negative reinforcements can be used to increase the likelihood that a behavior will be repeated. As an example of operant conditioning and operant learming, consider a hypothetical student sitting in the rear of the classroom who usually is shy, quiet, and unresponsive in class. The following altemative dialogs might occur between this student and his teacher:
斯金纳提出的操作性条件反射可以用来促进操作性学习。这种在反应之后出现的刺激通常被称为强化物。强化物可以是正强化物，也可以是负强化物，因为正强化物和负强化物都可以用来增加行为重复的可能性。举一个操作性条件反射和操作性强化的例子，假设有一个学生坐在教室后面，平时在课堂上害羞、安静、反应迟钝。这名学生和他的老师之间可能会发生以下替代对话：
(1) Teacher: "Jim, what does
(1) 教师："吉姆，
Jim: (no response) 吉姆： （没有回应）
Teacher: "Well students, Jim must have forgotten how to talk." (Loud, laughter from the class, and Jim turns red with embarrassment.)
"同学们，吉姆一定是忘了怎么说话了"(全班哄堂大笑，吉姆尴尬得满脸通红）。
(2) Teacher: "Jim, what does
(2) 教师："吉姆，
Teacher: "Well students, Jim must have forgotten how to talk." (Loud, laughter from the class, and Jim turns red with embarrassment.)
"同学们，吉姆一定是忘了怎么说话了"(全班哄堂大笑，吉姆尴尬得满脸通红）。
(2) Teacher: "Jim, what does
(2) 教师："吉姆，
Jim: "It means take four factors of
吉姆："这意味着取
Teacher: "That's quite good Jim, it's obvious that you have read the assignment and understand the meaning of exponents. Thank you." (Several students turm and give Jim looks of approval.)
教师："很好，吉姆，很明显你已经阅读了作业并理解了指数的含义。谢谢。"(几名学生转过头，向吉姆投去赞许的目光）。
Now, what type of operant learning might take place in Jim as a consequence of a series of events, over a two month period between Jim and his teacher, where most of the events are like (1)? Where most are like (2)? Obviously, the situation in case (1) is likely to result in a negative outcome for Jim. In this situation which of Jim's behaviors was made more likely to occur? The teacher probably hoped that Jim was more likely to respond to questions. However, Jim's embarrassment, together with his shyness, probably caused him to dislike mathematics class even more and made it less likely that he would respond to future questions from the teacher. So the teacher's negative response had an undesirable effect upon Jim and further conditioned Jim to exhibit the "undesirable behavior" of not responding to questions. A series of events such as those stated in case (2), where the teacher and the class presented Jim with a positive stimulus following his behavior, would be likely to improve Jim's attitude toward mathematics class, reduce his shyness, and cause him to volunteer answers in class-all of which are "desirable" behaviors.
现在，吉姆和他的老师在两个月的时间里发生了一系列事件，其中大多数事件都像（1）那样，吉姆可能会进行哪种类型的操作性学习？大多数情况下像（2）？显然，案例（1）中的情况很可能会给吉姆带来负面结果。在这种情况下，吉姆的哪种行为更有可能发生？老师可能希望吉姆更有可能回答问题。然而，吉姆的尴尬加上他的害羞，很可能使他更加不喜欢数学课，也使他以后更不可能回答老师的问题。因此，老师的消极反应对吉姆产生了不良影响，进一步促使吉姆表现出不回答问题的 "不良行为"。在案例（2）中，老师和全班同学在吉姆的行为之后给了他一个积极的刺激，这一系列事件可能会改善吉姆对数学课的态度，减少他的害羞，使他在课堂上主动回答问题--所有这些都是 "理想的 "行为。
吉姆："这意味着取
Teacher: "That's quite good Jim, it's obvious that you have read the assignment and understand the meaning of exponents. Thank you." (Several students turm and give Jim looks of approval.)
教师："很好，吉姆，很明显你已经阅读了作业并理解了指数的含义。谢谢。"(几名学生转过头，向吉姆投去赞许的目光）。
Now, what type of operant learning might take place in Jim as a consequence of a series of events, over a two month period between Jim and his teacher, where most of the events are like (1)? Where most are like (2)? Obviously, the situation in case (1) is likely to result in a negative outcome for Jim. In this situation which of Jim's behaviors was made more likely to occur? The teacher probably hoped that Jim was more likely to respond to questions. However, Jim's embarrassment, together with his shyness, probably caused him to dislike mathematics class even more and made it less likely that he would respond to future questions from the teacher. So the teacher's negative response had an undesirable effect upon Jim and further conditioned Jim to exhibit the "undesirable behavior" of not responding to questions. A series of events such as those stated in case (2), where the teacher and the class presented Jim with a positive stimulus following his behavior, would be likely to improve Jim's attitude toward mathematics class, reduce his shyness, and cause him to volunteer answers in class-all of which are "desirable" behaviors.
现在，吉姆和他的老师在两个月的时间里发生了一系列事件，其中大多数事件都像（1）那样，吉姆可能会进行哪种类型的操作性学习？大多数情况下像（2）？显然，案例（1）中的情况很可能会给吉姆带来负面结果。在这种情况下，吉姆的哪种行为更有可能发生？老师可能希望吉姆更有可能回答问题。然而，吉姆的尴尬加上他的害羞，很可能使他更加不喜欢数学课，也使他以后更不可能回答老师的问题。因此，老师的消极反应对吉姆产生了不良影响，进一步促使吉姆表现出不回答问题的 "不良行为"。在案例（2）中，老师和全班同学在吉姆的行为之后给了他一个积极的刺激，这一系列事件可能会改善吉姆对数学课的态度，减少他的害羞，使他在课堂上主动回答问题--所有这些都是 "理想的 "行为。
Each of these situations is an example of operant conditioning resulting in operant leaming. In the first case the operant learning was undesirable, and in the second case it was desirable. Note that in both of these operant learning situations the stimulus (the teacher's and students' reactions to Jim's response or lack of response) came after Jim's behavior (Jim's action in response to the teacher's question).
这两种情况都是操作性条件反射导致操作性学习的例子。在第一种情况下，操作性学习是不可取的，而在第二种情况下，操作性学习是可取的。请注意，在这两种操作性学习情境中，刺激（教师和学生对吉姆的反应或无反应的反应）都出现在吉姆的行为（吉姆对教师提问的反应）之后。
这两种情况都是操作性条件反射导致操作性学习的例子。在第一种情况下，操作性学习是不可取的，而在第二种情况下，操作性学习是可取的。请注意，在这两种操作性学习情境中，刺激（教师和学生对吉姆的反应或无反应的反应）都出现在吉姆的行为（吉姆对教师提问的反应）之后。
Let us reiterate the distinction between respondent conditioning and operant conditioning. Respondent conditioning results in the leamer being conditioned to exhibit a particular behavior in response to a specific stimulus. In respondent conditioning a new stimulus is presented together with an old stimulus which causes a reflexive reaction. After a series of simultaneous presentations of the two stimuli, the learner gives the same reaction to the new stimulus (in the absence of the old stimulus) which he or she previously gave in response to the old stimulus by itself. In respondent learning, the leamer responds to environmental stimuli.
让我们重申一下应答性条件反射和操作性条件反射之间的区别。应答性条件反射的结果是，学习者在对特定刺激做出反应时，会条件反射地表现出特定行为。在应答性条件反射中，新刺激与旧刺激同时出现，旧刺激会引起反射性反应。在同时呈现一系列这两种刺激后，学习者对新刺激（在没有旧刺激的情况下）的反应与之前对旧刺激本身的反应相同。在反应式学习中，学习者对环境刺激做出反应。
让我们重申一下应答性条件反射和操作性条件反射之间的区别。应答性条件反射的结果是，学习者在对特定刺激做出反应时，会条件反射地表现出特定行为。在应答性条件反射中，新刺激与旧刺激同时出现，旧刺激会引起反射性反应。在同时呈现一系列这两种刺激后，学习者对新刺激（在没有旧刺激的情况下）的反应与之前对旧刺激本身的反应相同。在反应式学习中，学习者对环境刺激做出反应。
In operant conditioning, the leamer's unpredictable response (action) is followed by a stimulus. It is hoped that the stimulus will either help to suppress the response if it is undesirable or will increase the future likelihood of the response if it is desirable. Respondent learning is stimulus-response leaming; whereas operant learning is response-stimulus leaming. In respondent leaming the leamer responds to environmental stimuli; whereas in operant leaming the leamer operates on his or her environment and has these operations reinforced through appropriate stimuli or changes in the environment as a consequence of his or her. actions.
在操作性条件反射中，学习者在做出不可预测的反应（动作）之后会受到刺激。如果反应是不可取的，则希望刺激能够帮助抑制反应；如果反应是可取的，则希望刺激能够增加未来发生反应的可能性。反应式学习是刺激-反应的学习；而操作式学习是反应-刺激的学习。在反应式学习中，学习者对环境刺激做出反应；而在操作式学习中，学习者对环境进行操作，并通过适当的刺激或环境变化来强化这些操作，作为其行为的结果。
在操作性条件反射中，学习者在做出不可预测的反应（动作）之后会受到刺激。如果反应是不可取的，则希望刺激能够帮助抑制反应；如果反应是可取的，则希望刺激能够增加未来发生反应的可能性。反应式学习是刺激-反应的学习；而操作式学习是反应-刺激的学习。在反应式学习中，学习者对环境刺激做出反应；而在操作式学习中，学习者对环境进行操作，并通过适当的刺激或环境变化来强化这些操作，作为其行为的结果。
Promtoting Learning and Changing Betavior
促进学习和改变行为
Reinforcement 加固
Reinforcers, which are happenings or stimuli that follow a response and which tend to increase the probability of that response, can. Facilitate learming and changes in behavior. In a school leaming environment with classrooms and teachers, we find that.grades, teacher and peecapproval, punishment, and various means of recognizing and rewarding certain behaviors function as reinforcers. The many different environmental stimuli which act as reinforcers fall into twe-general categories, positive reinforcers and negative reinforcers.
强化物是在某种反应之后发生的事情或刺激，往往会增加这种反应的可能性，它可以促进强化和行为改变。在有教室和教师的学校学习环境中，我们会发现，成绩、教师和学生的认可、惩罚以及认可和奖励某些行为的各种手段都是强化物。作为强化物的许多不同环境刺激可分为两大类，即正强化物和负强化物。
Skinner defines positive reinforcers as stimuli which, when presented following a behavior by the learner, tend to increase the probability that that particular behavior will be repeated; that is, the behavior is strengthened. When our hypotheiical student Jim answered correctly in class, the teacher's praise increased the likelihood that Jim would again respond to the teacher's questions; consequently the teacher's pleasant reaction functioned as a positive reinforcer for Jim. The teacher's unpleasant remark following Jim's failure to respond to the teacher's question also acted as a positive reinforcer, because it reinforced Jim's behavior which was to remain silent when questioned by the teacher. Any stimulus, pleasant or unpleasant, which follows the behavior that elicited it, and strengthens that behavior, is considered to be a positive reinforcer by Skinner.
斯金纳将正强化物定义为：当学习者做出某种行为后，刺激物会增加该行为重复发生的可能性；也就是说，该行为会得到强化。当我们假设的学生吉姆在课堂上回答正确时，老师的表扬增加了吉姆再次回答老师问题的可能性；因此，老师的愉快反应对吉姆起到了正强化物的作用。吉姆没有回答老师的问题后，老师不愉快的话语也起到了正强化作用，因为它强化了吉姆的行为，即在老师提问时保持沉默。斯金纳认为，任何刺激，不管是令人愉快的还是令人不愉快的，只要是在引起刺激的行为之后出现并强化了该行为，都是正强化物。
强化物是在某种反应之后发生的事情或刺激，往往会增加这种反应的可能性，它可以促进强化和行为改变。在有教室和教师的学校学习环境中，我们会发现，成绩、教师和学生的认可、惩罚以及认可和奖励某些行为的各种手段都是强化物。作为强化物的许多不同环境刺激可分为两大类，即正强化物和负强化物。
Skinner defines positive reinforcers as stimuli which, when presented following a behavior by the learner, tend to increase the probability that that particular behavior will be repeated; that is, the behavior is strengthened. When our hypotheiical student Jim answered correctly in class, the teacher's praise increased the likelihood that Jim would again respond to the teacher's questions; consequently the teacher's pleasant reaction functioned as a positive reinforcer for Jim. The teacher's unpleasant remark following Jim's failure to respond to the teacher's question also acted as a positive reinforcer, because it reinforced Jim's behavior which was to remain silent when questioned by the teacher. Any stimulus, pleasant or unpleasant, which follows the behavior that elicited it, and strengthens that behavior, is considered to be a positive reinforcer by Skinner.
斯金纳将正强化物定义为：当学习者做出某种行为后，刺激物会增加该行为重复发生的可能性；也就是说，该行为会得到强化。当我们假设的学生吉姆在课堂上回答正确时，老师的表扬增加了吉姆再次回答老师问题的可能性；因此，老师的愉快反应对吉姆起到了正强化物的作用。吉姆没有回答老师的问题后，老师不愉快的话语也起到了正强化作用，因为它强化了吉姆的行为，即在老师提问时保持沉默。斯金纳认为，任何刺激，不管是令人愉快的还是令人不愉快的，只要是在引起刺激的行为之后出现并强化了该行为，都是正强化物。
Negative reinforcers are stimuli whose removal tends to strengthen behaviors. Many times the student behavior of attentiveness to appropriate classroom activities can be increased by removing distracting stimuli such as undesirable noise, a disruptive student, or distracting teacher mannerisms.
负强化物是一些刺激物，消除这些刺激物往往会强化学生的行为。很多时候，通过消除干扰性刺激，如不良噪音、捣乱的学生或干扰性的教师举止，可以增强学生专心于适当课堂活动的行为。
负强化物是一些刺激物，消除这些刺激物往往会强化学生的行为。很多时候，通过消除干扰性刺激，如不良噪音、捣乱的学生或干扰性的教师举止，可以增强学生专心于适当课堂活动的行为。
Forgeting and Extinction 遗忘与消亡
If a learned behavior is not used for a long period of time it will be forgotten and will have to be relearned. In forgetting, the effect of operant conditioning is simply lost with the passage of time. Many students forget many of their algebraic skills if they do not practice them between their freshman year in high school and graduation from school. Even though most secondary school teachers learn calculus in college, we may forget many of the details and skills of this subject if we work in a school where we are not assigned to teach a calculus course for several years.
如果长时间不使用已学会的行为，它就会被遗忘，必须重新学习。在遗忘的过程中，操作性条件反射的效果会随着时间的流逝而消失。许多学生从高一到毕业期间，如果不进行代数练习，就会忘记许多代数技能。尽管大多数中学教师在大学里学习微积分，但如果我们在一所学校工作，几年都没有被分配教授微积分课程，我们可能会忘记这门学科的许多细节和技能。
If a learned behavior is not used for a long period of time it will be forgotten and will have to be relearned. In forgetting, the effect of operant conditioning is simply lost with the passage of time. Many students forget many of their algebraic skills if they do not practice them between their freshman year in high school and graduation from school. Even though most secondary school teachers learn calculus in college, we may forget many of the details and skills of this subject if we work in a school where we are not assigned to teach a calculus course for several years.
如果长时间不使用已学会的行为，它就会被遗忘，必须重新学习。在遗忘的过程中，操作性条件反射的效果会随着时间的流逝而消失。许多学生从高一到毕业期间，如果不进行代数练习，就会忘记许多代数技能。尽管大多数中学教师在大学里学习微积分，但如果我们在一所学校工作，几年都没有被分配教授微积分课程，我们可能会忘记这门学科的许多细节和技能。
Skinner considers extinction to be a process of "unlearning' conditioned responses, this is distinct from forgetting. Many times students initially learn incorrect responses and behaviors and need to "unlearn" them later in school, while other conditioned responses are naturally "unlearned" as a consequence of the withdrawal of expected reinforcements. Skinner defines extinction as the process through which conditioned responses become less and less frequent when reinforcements are no longer forthcoming. Fortunately for learning desirable conditioned responses and behaviors (but unfortunately for "unleaming" undesirable behaviors), research studies have shown that operant extinction takes place much more slowly than operant conditioning. Several reinforcements may suffice in learning a response; however hundreds of unreinforced instances of the response may be necessary in order to "unlearn" the response; that is, to refrain from exhibiting the behavior.
斯金纳认为消退是一个 "解除学习 "条件反射的过程，这与遗忘是不同的。很多时候，学生最初学会的是不正确的反应和行为，需要在以后的学习中 "解除学习"，而其他条件反射则会因为预期强化物的撤销而自然 "解除学习"。斯金纳将 "消退 "定义为当强化物不再出现时，条件反射的频率越来越低的过程。幸运的是，对于学习理想的条件反射和行为（不幸的是，对于 "解除 "不良行为）来说，研究表明，操作性消退的速度比操作性条件反射慢得多。在学习一个反应的过程中，几次强化可能就足够了；但是，为了 "解除 "该反应，即避免表现出该行为，可能需要数百次未强化的反应。
斯金纳认为消退是一个 "解除学习 "条件反射的过程，这与遗忘是不同的。很多时候，学生最初学会的是不正确的反应和行为，需要在以后的学习中 "解除学习"，而其他条件反射则会因为预期强化物的撤销而自然 "解除学习"。斯金纳将 "消退 "定义为当强化物不再出现时，条件反射的频率越来越低的过程。幸运的是，对于学习理想的条件反射和行为（不幸的是，对于 "解除 "不良行为）来说，研究表明，操作性消退的速度比操作性条件反射慢得多。在学习一个反应的过程中，几次强化可能就足够了；但是，为了 "解除 "该反应，即避免表现出该行为，可能需要数百次未强化的反应。
Extinction of desirable student behaviors such as reviewing for quizzes and completing homework assignments may occur if quiz grades are not averaged into period grade reports and assignments are not graded by the teacher and returned to the students shortly after they have been handed to the teacher. In some secondary schools and colleges where letter grading systems have been replaced with Satisfactory/Unsatisfactory student evaluations, students, who had been conditioned to cxpect letter grades as reinforcements, became frustrated and lacked motivation as a consequence of being denied their previously expected letter grade reinforcements.
如果测验成绩没有被平均计入期末成绩报告，作业没有在交到教师手中后立即由教师评分并返还给学生，那么就可能会导致学生复习测验和完成家庭作业等理想行为的消失。在一些中学和大学，字母评分制度已被 "满意/不满意 "的学生评价所取代，学生已习惯于将字母评分作为一种强化手段，但由于无法获得以前期望的字母评分强化手段，他们变得沮丧并缺乏动力。
如果测验成绩没有被平均计入期末成绩报告，作业没有在交到教师手中后立即由教师评分并返还给学生，那么就可能会导致学生复习测验和完成家庭作业等理想行为的消失。在一些中学和大学，字母评分制度已被 "满意/不满意 "的学生评价所取代，学生已习惯于将字母评分作为一种强化手段，但由于无法获得以前期望的字母评分强化手段，他们变得沮丧并缺乏动力。
Extinction of undesirable behaviors, such as using incorrect mathematics techniques or indulging in cigarette smoking, which have been repeated many times with occasional reinforcement, is very difficult to accomplish. I once tutored a young man who was having trouble in higher level mathematics courses because he had leamed a number of incorrect algebra techniques such as
消除不良行为，例如使用不正确的数学技巧或沉迷于吸烟，这些行为已经重复了很多次，偶尔会得到强化，但要做到这一点却非常困难。我曾经辅导过一个在高等数学课程中遇到困难的年轻人，因为他学会了许多错误的代数技巧，比如
Aversion and Avoidance 厌恶和回避
A negative reinforcer is a stimulus whose withdrawal results in the strengthening of a response. An unpleasant, annoying, or frustrating negative reinforcer is called an aversive stimulus by Skinner. There are two ways to deal with aversive stimuli. One can escape an aversive stimulus either by removing the stimulus after he or she has come in contact with it, or by leaving the environment where the aversive stimulus exists. An aversive-stimulus can be avoided by anticipating its occurrence and staying away from it. Note that avoidance is accomplished by never contacting the aversive stimulus and that escape is accomplished by removing the aversive stimulus after coming in contact with it. Many people avoid art upset stomach by refraining from over-eating spicy fried foods, while others go ahead and eat improperly and escape the resulting indigestion by taking an antacid preparation.
负强化物是一种刺激物，它的撤消会导致反应的加强。斯金纳把不愉快、烦人或令人沮丧的负强化物称为厌恶刺激。应对厌恶刺激有两种方法。人们可以在接触到厌恶刺激后将其移开，或者离开厌恶刺激存在的环境，从而逃避厌恶刺激。避开厌恶刺激的方法是预测到它的出现并远离它。需要注意的是，避免是通过永远不接触厌恶刺激来实现的，而逃避则是通过在接触厌恶刺激后将其移开来实现的。许多人通过避免过量食用辛辣油炸食品来避免胃部不适，而另一些人则继续不当进食，并通过服用抗酸制剂来逃避由此导致的消化不良。
消除不良行为，例如使用不正确的数学技巧或沉迷于吸烟，这些行为已经重复了很多次，偶尔会得到强化，但要做到这一点却非常困难。我曾经辅导过一个在高等数学课程中遇到困难的年轻人，因为他学会了许多错误的代数技巧，比如
Aversion and Avoidance 厌恶和回避
A negative reinforcer is a stimulus whose withdrawal results in the strengthening of a response. An unpleasant, annoying, or frustrating negative reinforcer is called an aversive stimulus by Skinner. There are two ways to deal with aversive stimuli. One can escape an aversive stimulus either by removing the stimulus after he or she has come in contact with it, or by leaving the environment where the aversive stimulus exists. An aversive-stimulus can be avoided by anticipating its occurrence and staying away from it. Note that avoidance is accomplished by never contacting the aversive stimulus and that escape is accomplished by removing the aversive stimulus after coming in contact with it. Many people avoid art upset stomach by refraining from over-eating spicy fried foods, while others go ahead and eat improperly and escape the resulting indigestion by taking an antacid preparation.
负强化物是一种刺激物，它的撤消会导致反应的加强。斯金纳把不愉快、烦人或令人沮丧的负强化物称为厌恶刺激。应对厌恶刺激有两种方法。人们可以在接触到厌恶刺激后将其移开，或者离开厌恶刺激存在的环境，从而逃避厌恶刺激。避开厌恶刺激的方法是预测到它的出现并远离它。需要注意的是，避免是通过永远不接触厌恶刺激来实现的，而逃避则是通过在接触厌恶刺激后将其移开来实现的。许多人通过避免过量食用辛辣油炸食品来避免胃部不适，而另一些人则继续不当进食，并通过服用抗酸制剂来逃避由此导致的消化不良。
If a person is always successful in avoiding an aversive situation, the situation may lose its aversion for that person. Eventually the person may fail to emit the avoidance response to stimuli preceding the aversive stimulus and the aversive situation is not avoided. A person may avoid eating oysters due to a previously diagnosed allergic reaction to oysters long enough to be tempted into ealing some more oysters. The resulting illness will reestablish the aversive response to oysters, and that person will then avoid oysters for some time until his or her aversion to oysters has weakened once more.
如果一个人总能成功地避开厌恶的情境，那么这种情境对他来说就会失去厌恶感。最终，这个人可能无法对厌恶刺激之前的刺激做出回避反应，厌恶情境也就无法避免了。一个人可能会因为之前被诊断出对生蚝过敏而避免吃生蚝，久而久之，他就会被诱惑再吃一些生蚝。由此产生的疾病会重新建立对生蚝的厌恶反应，然后这个人会在一段时间内避免吃生蚝，直到他或她对生蚝的厌恶感再次减弱。
如果一个人总能成功地避开厌恶的情境，那么这种情境对他来说就会失去厌恶感。最终，这个人可能无法对厌恶刺激之前的刺激做出回避反应，厌恶情境也就无法避免了。一个人可能会因为之前被诊断出对生蚝过敏而避免吃生蚝，久而久之，他就会被诱惑再吃一些生蚝。由此产生的疾病会重新建立对生蚝的厌恶反应，然后这个人会在一段时间内避免吃生蚝，直到他或她对生蚝的厌恶感再次减弱。
Many examples of aversion, escape, and avoidance are found in students in mathematics classes. For example, after unsuccessfully trying to solve the problems on a test, a student may attempt to escape failure by copying answers from the paper of a student seated nearby. Some students avoid failing tests by staying away from school on test days. Of course from the teacher's point of view, the desired method for avoiding failure may be to complete all the homework assignments and to prepare for tests through concentrated review sessions. Many so-called discipline problems are in fact students' attempts to escape from the boredom and failures which they associate with mathematics classes. Some notso-subtle attempts to escape aversive classroom stimuli are student actions which disrupt the teacher's planned activities. More subtle escape attempts are questions from students which are asked in order to "get the teacher off the subject."
在数学课上，我们可以在学生身上发现许多厌恶、逃避和回避的例子。例如，学生在尝试解决考试中的问题未果后，可能会试图通过抄袭邻座学生的试卷答案来逃避失败。有些学生为了避免考试不及格，在考试当天不上学。当然，从教师的角度来看，避免失败的理想方法可能是完成所有的家庭作业，并通过集中复习备考。许多所谓的纪律问题实际上是学生试图逃避数学课的无聊和失败。一些并不明显的试图逃避厌恶性课堂刺激的行为是学生扰乱教师计划活动的行为。更隐蔽的逃避企图是学生为了 "让老师离开主题 "而提出的问题。
在数学课上，我们可以在学生身上发现许多厌恶、逃避和回避的例子。例如，学生在尝试解决考试中的问题未果后，可能会试图通过抄袭邻座学生的试卷答案来逃避失败。有些学生为了避免考试不及格，在考试当天不上学。当然，从教师的角度来看，避免失败的理想方法可能是完成所有的家庭作业，并通过集中复习备考。许多所谓的纪律问题实际上是学生试图逃避数学课的无聊和失败。一些并不明显的试图逃避厌恶性课堂刺激的行为是学生扰乱教师计划活动的行为。更隐蔽的逃避企图是学生为了 "让老师离开主题 "而提出的问题。
Punishment. 惩罚。
Throughout history punishment has been a common technique for attempting to control behavior, and Skinner (1953) discusses the effects and by-products of punishment in his book Science and Human Behavior. His general viewpoint. regarding the use of punishment to control behavior is summarized in the following quotation from this book:
纵观历史，惩罚一直是试图控制行为的常用手段，斯金纳（1953 年）在《科学与人类行为》一书中讨论了惩罚的效果和副产品。斯金纳（1953 年）在《科学与人类行为》一书中论述了惩罚的效果和副产品。他对使用惩罚控制行为的总体观点概括在该书的以下引文中：
. . concem [about punishment] may be due to the realization that the technique has unfortunate by-products. In the long run, punishment, unlike reinforcement, works to the disadvantage of both the punished organism and
.......[对惩罚]的疑虑可能是由于人们认识到，惩罚这种手段会产生不幸的副产品。从长远来看，惩罚与强化不同，它对被惩罚的生物和其他人都不利。
ring and :al Theories the punishing agency. The aversive stimuli which are needed generate emothe punishing agency. The aversive stimul whiction retaliate, and disabling anxieties. (pp.
惩罚机构理论。惩罚机构需要厌恶刺激来产生情绪。需要报复的厌恶性刺激和失能性焦虑。(pp.）
Skinner regards punishment as the deliberate presentation of a negative reinforcer (a negative reinforcer is a stimulus whose removal will strengthen a behavior) or the deliberate removal of a positive reinforcer (a positive reinforcer is a stimulus whose presentation will strengthen a behavior). Skinner and others have shown in laboratory experiments with both animals and humans that punishment does not have the opposite effect of reward. An equal number of punishments will not extinguish the effects of a given number of rewards. While punishment, and even prolonged punishment, can be effective in suppressing "unwanted" behaviors, this supression is usually only temporary. After a time the punished behaviors tend to reappear at a level not much lower than if no punishment had been administered. Even if punishment were effective in supressing or removing undesirable behaviors, it can generate unpredictable social and emotional consequences.
斯金纳认为，惩罚是指故意出示负强化物（负强化物是一种刺激物，将其移除会强化某种行为）或故意移除正强化物（正强化物是一种刺激物，将其出示会强化某种行为）。斯金纳等人通过对动物和人类的实验室实验表明，惩罚不会产生与奖励相反的效果。同等数量的惩罚不会消除一定数量奖励的效果。虽然惩罚，甚至是长时间的惩罚，可以有效地抑制 "不想要的 "行为，但这种抑制通常只是暂时的。一段时间后，被惩罚的行为往往会再次出现，其水平不会比不实施惩罚时低多少。即使惩罚能有效地抑制或消除不良行为，它也会产生难以预料的社会和情感后果。
纵观历史，惩罚一直是试图控制行为的常用手段，斯金纳（1953 年）在《科学与人类行为》一书中讨论了惩罚的效果和副产品。斯金纳（1953 年）在《科学与人类行为》一书中论述了惩罚的效果和副产品。他对使用惩罚控制行为的总体观点概括在该书的以下引文中：
. . concem [about punishment] may be due to the realization that the technique has unfortunate by-products. In the long run, punishment, unlike reinforcement, works to the disadvantage of both the punished organism and
.......[对惩罚]的疑虑可能是由于人们认识到，惩罚这种手段会产生不幸的副产品。从长远来看，惩罚与强化不同，它对被惩罚的生物和其他人都不利。
ring and :al Theories the punishing agency. The aversive stimuli which are needed generate emothe punishing agency. The aversive stimul whiction retaliate, and disabling anxieties. (pp.
惩罚机构理论。惩罚机构需要厌恶刺激来产生情绪。需要报复的厌恶性刺激和失能性焦虑。(pp.）
Skinner regards punishment as the deliberate presentation of a negative reinforcer (a negative reinforcer is a stimulus whose removal will strengthen a behavior) or the deliberate removal of a positive reinforcer (a positive reinforcer is a stimulus whose presentation will strengthen a behavior). Skinner and others have shown in laboratory experiments with both animals and humans that punishment does not have the opposite effect of reward. An equal number of punishments will not extinguish the effects of a given number of rewards. While punishment, and even prolonged punishment, can be effective in suppressing "unwanted" behaviors, this supression is usually only temporary. After a time the punished behaviors tend to reappear at a level not much lower than if no punishment had been administered. Even if punishment were effective in supressing or removing undesirable behaviors, it can generate unpredictable social and emotional consequences.
斯金纳认为，惩罚是指故意出示负强化物（负强化物是一种刺激物，将其移除会强化某种行为）或故意移除正强化物（正强化物是一种刺激物，将其出示会强化某种行为）。斯金纳等人通过对动物和人类的实验室实验表明，惩罚不会产生与奖励相反的效果。同等数量的惩罚不会消除一定数量奖励的效果。虽然惩罚，甚至是长时间的惩罚，可以有效地抑制 "不想要的 "行为，但这种抑制通常只是暂时的。一段时间后，被惩罚的行为往往会再次出现，其水平不会比不实施惩罚时低多少。即使惩罚能有效地抑制或消除不良行为，它也会产生难以预料的社会和情感后果。
Skinner (1953) has identified three effects of punishment upon the person being punished. First, punishment suppresses behavior. Since a person's response to punishment is usually incompatible with the behavior being punished, the behavior is changed, at least temporarily. A student who is reprimanded by his or her teacher for arguing in class with another student may stop talking or may direct the argument toward the teacher.
斯金纳（1953 年）指出了惩罚对被惩罚者的三种影响。首先，惩罚会抑制行为。由于人对惩罚的反应通常与被惩罚的行为不相容，因此行为至少会暂时改变。一个学生因为在课堂上与另一个学生争吵而受到老师的训斥，他可能会停止说话，或者把争吵的矛头指向老师。
The second effect of punishment is to evoke incompatible behavior resulting in anxiety and accompanying physiological changes such as increased heart rate, hioher blood pressure, and muscle tension. People who know when they are lying and who have been punished for lying, will exhibit such physiological changes when lying, even in the absence of punishment. This suggests the principles which were used in developing lie-detecting devices.
惩罚的第二个作用是唤起不相容的行为，导致焦虑和伴随的生理变化，如心率加快、血压升高和肌肉紧张。那些知道自己在撒谎并曾因撒谎而受到惩罚的人，即使没有受到惩罚，在撒谎时也会表现出这种生理变化。这表明了开发测谎设备所使用的原理。
斯金纳（1953 年）指出了惩罚对被惩罚者的三种影响。首先，惩罚会抑制行为。由于人对惩罚的反应通常与被惩罚的行为不相容，因此行为至少会暂时改变。一个学生因为在课堂上与另一个学生争吵而受到老师的训斥，他可能会停止说话，或者把争吵的矛头指向老师。
The second effect of punishment is to evoke incompatible behavior resulting in anxiety and accompanying physiological changes such as increased heart rate, hioher blood pressure, and muscle tension. People who know when they are lying and who have been punished for lying, will exhibit such physiological changes when lying, even in the absence of punishment. This suggests the principles which were used in developing lie-detecting devices.
惩罚的第二个作用是唤起不相容的行为，导致焦虑和伴随的生理变化，如心率加快、血压升高和肌肉紧张。那些知道自己在撒谎并曾因撒谎而受到惩罚的人，即使没有受到惩罚，在撒谎时也会表现出这种生理变化。这表明了开发测谎设备所使用的原理。
The third and most important effect of punishment is to condition the punished person to do something other than the act for which he or she is being punished. Whatever this alternative behavior may be, it will be reinforced and it may be as undesirable in its long-range effects as the behavior being suppressed. At times students who are severely punished in school will develop an aversion to schools and structured learning in general. They may even perpetrate acts of vandalism against the school or the teacher's property. In extreme cases a studen who has been punished by a teacher may even physically assault that teacher.
惩罚的第三种也是最重要的一种效果是让被惩罚者做出与惩罚行为不同的行为。无论这种替代行为是什么，它都会得到强化，其长期效果可能与被抑制的行为一样不可取。有时，在学校受到严厉惩罚的学生会对学校和有组织的学习产生厌恶感。他们甚至会对学校或教师的财产进行破坏。在极端情况下，受到老师惩罚的学生甚至会对老师进行人身攻击。
惩罚的第三种也是最重要的一种效果是让被惩罚者做出与惩罚行为不同的行为。无论这种替代行为是什么，它都会得到强化，其长期效果可能与被抑制的行为一样不可取。有时，在学校受到严厉惩罚的学生会对学校和有组织的学习产生厌恶感。他们甚至会对学校或教师的财产进行破坏。在极端情况下，受到老师惩罚的学生甚至会对老师进行人身攻击。
There is little evidence to suggest that the consequences of severe or repeated unishment are desirable or are even predictable. Even relatively mild punishment such as assigning extra homework or making students stay after school may not necessarily suppress the behavior which is being punished. It may even reinforce an aversion for mathematics and for school in general. Mild punishment may be temporarily effective is suppressing certain responses; however most types of punishment appear to be relatively ineffective in promoting permanent modification of behavior. Furthermore, severe or frequent punishment may produce emotional side effects which may prove to be even less desirable than the original behavior.
几乎没有证据表明，严厉的或反复的惩罚所产生的后果是可取的，甚至是可以预测的。即使是相对温和的惩罚，如布置额外的家庭作业或让学生放学后留校，也不一定能抑制被惩罚的行为。它甚至可能强化学生对数学和学校的厌恶。轻微的惩罚可能会暂时有效地抑制某些反应，但大多数类型的惩罚在促进行为的永久性改变方面似乎相对无效。此外，严厉或频繁的惩罚可能会产生情绪上的副作用，这些副作用可能比原来的行为更不可取。
General Conditions of Learning
一般学习条件
In Skinner's view, three variables make up the contingencies under which leaming takes place. First there must be a situation in which a behavior occurs. The second contingency is the behavior itself, and the third contingency is the consequence of the behavior. If in a certain situation a person exhibits a particular behavior or response from a class of responses called an operant, and if he or she is reinforced as a consequence of the response, then it is likely that leaming will take place. That is, it becomes more probable that a similar response from the same class of responses will be given by that person in a similar situation. Even though the situation did not initially act as a stimulus for the response, the learner, after receiving reinforcement for the response, will tend to associate the behavior which evoked the reinforcement with the initial situation. Operants (classes of responses) do acquire relationships to previous sets of stimuli (called discriminated stimuli); but the relationships between stimuli and responses are different from those found in classical stimulus-response conditioning. The prior situation (set of stimuli), when encountered again, becomes the occasion for the operant behavior, but does not cause the behavior as is the case in respondent learning. An unexpected finger in the ribs (stimulus) will usually cause most people to jump (response). A word of praise or extra credit toward a final grade for solving mathematics problems which were not assigned by the teacher may cause a student to continue to do extra work in mathematics. In the future, the more difficult textbook problems (which may not usually be assigned for homework by the teacher) act as discriminated stimuli for the student and become the occasion for his or her response of doing extra work.
在斯金纳看来，有三个变量构成了发生学习行为的条件。首先，必须有发生行为的情境。第二个或然因素是行为本身，第三个或然因素是行为的后果。如果一个人在某种情况下表现出一种特定的行为或反应，而这种行为或反应又被称为操作性反应，如果他或她的行为或反应的结果得到了强化，那么就很有可能发生 "跃迁"。也就是说，在类似的情况下，该人更有可能做出同一类反应中的类似反应。即使最初的情境并没有对反应起到刺激作用，但学习者在对反应进行强化后，会倾向于把引起强化的行为与最初的情境联系起来。操作者（反应类别）确实会获得与先前刺激（称为辨别刺激）的关系；但刺激与反应之间的关系不同于经典刺激-反应条件反射中的关系。先前的情境（一组刺激）再次出现时，会成为操作行为的契机，但不会像反应学习那样导致行为的发生。一根突如其来的手指戳在肋骨上（刺激）通常会让大多数人跳起来（反应）。如果学生解决了老师没有布置的数学问题，老师的一句表扬或期末成绩的额外加分，可能会让学生继续做额外的数学作业。 今后，难度较大的课本问题（教师通常可能不会布置家庭作业）会成为学生的辨别刺激，并成为其做额外作业的反应契机。
几乎没有证据表明，严厉的或反复的惩罚所产生的后果是可取的，甚至是可以预测的。即使是相对温和的惩罚，如布置额外的家庭作业或让学生放学后留校，也不一定能抑制被惩罚的行为。它甚至可能强化学生对数学和学校的厌恶。轻微的惩罚可能会暂时有效地抑制某些反应，但大多数类型的惩罚在促进行为的永久性改变方面似乎相对无效。此外，严厉或频繁的惩罚可能会产生情绪上的副作用，这些副作用可能比原来的行为更不可取。
General Conditions of Learning
一般学习条件
In Skinner's view, three variables make up the contingencies under which leaming takes place. First there must be a situation in which a behavior occurs. The second contingency is the behavior itself, and the third contingency is the consequence of the behavior. If in a certain situation a person exhibits a particular behavior or response from a class of responses called an operant, and if he or she is reinforced as a consequence of the response, then it is likely that leaming will take place. That is, it becomes more probable that a similar response from the same class of responses will be given by that person in a similar situation. Even though the situation did not initially act as a stimulus for the response, the learner, after receiving reinforcement for the response, will tend to associate the behavior which evoked the reinforcement with the initial situation. Operants (classes of responses) do acquire relationships to previous sets of stimuli (called discriminated stimuli); but the relationships between stimuli and responses are different from those found in classical stimulus-response conditioning. The prior situation (set of stimuli), when encountered again, becomes the occasion for the operant behavior, but does not cause the behavior as is the case in respondent learning. An unexpected finger in the ribs (stimulus) will usually cause most people to jump (response). A word of praise or extra credit toward a final grade for solving mathematics problems which were not assigned by the teacher may cause a student to continue to do extra work in mathematics. In the future, the more difficult textbook problems (which may not usually be assigned for homework by the teacher) act as discriminated stimuli for the student and become the occasion for his or her response of doing extra work.
在斯金纳看来，有三个变量构成了发生学习行为的条件。首先，必须有发生行为的情境。第二个或然因素是行为本身，第三个或然因素是行为的后果。如果一个人在某种情况下表现出一种特定的行为或反应，而这种行为或反应又被称为操作性反应，如果他或她的行为或反应的结果得到了强化，那么就很有可能发生 "跃迁"。也就是说，在类似的情况下，该人更有可能做出同一类反应中的类似反应。即使最初的情境并没有对反应起到刺激作用，但学习者在对反应进行强化后，会倾向于把引起强化的行为与最初的情境联系起来。操作者（反应类别）确实会获得与先前刺激（称为辨别刺激）的关系；但刺激与反应之间的关系不同于经典刺激-反应条件反射中的关系。先前的情境（一组刺激）再次出现时，会成为操作行为的契机，但不会像反应学习那样导致行为的发生。一根突如其来的手指戳在肋骨上（刺激）通常会让大多数人跳起来（反应）。如果学生解决了老师没有布置的数学问题，老师的一句表扬或期末成绩的额外加分，可能会让学生继续做额外的数学作业。 今后，难度较大的课本问题（教师通常可能不会布置家庭作业）会成为学生的辨别刺激，并成为其做额外作业的反应契机。
The Art of Teaching 教学艺术
Skinner's research on the science of leaming and the art of teaching suggests several reasons why many students leave elementary school without having leamed simple arithmetic skills, and why they fail to leam these skills after repeated attempts in secondary school. First, some "reinforcements" for leaming mathematics skills are still aversive. That is, many students still learn (or attempt to leam) arithmetic to escape punishment or the threat of punishment. Instead of studying and learning arithmetic in order to obtain positive reinforcements, many students do their schoolwork to avoid negative consequences: the teacher's displeasure, ridicule from classmates, poor grades resulting in punishment from parents, or poor results in competition with other students. Second, even when appropriate positive reinforcements are used in attempts to promote learning in arithmetic, the reinforcements are usually not optimized. In schools, reinforcement from teachers for students' correct written solutions to problems may occur infrequently in large classes. Or, when it does occur, may be given several minutes following a student's response. Skinner has found that in leam ing certain types of skills, such as basic arithmetic, some students require immediate reinforcement of their responses. Even a time lapse of a minute or two between a response and a reinforcement can, at times, remove much of the positive effects of an immediate reinforcement. Since homework assignments and tests are marked by the teacher and returned a day or more after students com-
斯金纳对数学学习的科学性和教学的艺术性的研究表明，许多学生在小学毕业时没有掌握简单的算术技能，而且在中学阶段反复尝试也未能掌握这些技能，这有几个原因。首先，学习数学技能的某些 "强化 "仍然是厌恶性的。也就是说，许多学生学习（或试图学习）算术仍然是为了逃避惩罚或惩罚的威胁。许多学生不是为了获得正强化而学习算术，而是为了避免消极后果而做功课：老师的不满、同学的嘲笑、成绩差而受到家长的惩罚，或在与其他同学的竞争中成绩不佳。其次，即使在试图促进算术学习的过程中使用了适当的积极强化手段，强化效果通常也不会达到最佳。在学校里，教师对学生正确的书面解题方法的强化，在大班中可能很少发生。或者，即使有强化，也可能是在学生回答后几分钟才进行。斯金纳发现，在学习某些类型的技能（如基本算术）时，有些学生需要教师立即强化他们的回答。即使在反应和强化之间相隔一两分钟，有时也会使立即强化的积极效果大打折扣。由于家庭作业和测验由教师批改，并在学生完成作业一天或更长时间后才发回。
plete them, much of the leaming value of these activities can be lost for many students. A third reason why so many students fail to learn in schools, even where immediate reinforcement is given to them, is that the frequency of reinforcement is inadequate. Skinner (1968, p. 17) in his book The Technology of Teaching estimates that a student throughout his or her first four years of school requires something on the order of 25,000 reinforcements and may be given only a few thousand.
因此，对许多学生来说，这些活动的学习价值可能会丧失殆尽。很多学生在学校学不好的第三个原因是，即使对他们进行了即时强化，强化的频率也不够。斯金纳（Skinner，1968 年，第 17 页）在他的著作《教学技术》中估计，一个学生在他或她在学校的头四年里需要大约 25,000 次强化，而可能只得到几千次。
斯金纳对数学学习的科学性和教学的艺术性的研究表明，许多学生在小学毕业时没有掌握简单的算术技能，而且在中学阶段反复尝试也未能掌握这些技能，这有几个原因。首先，学习数学技能的某些 "强化 "仍然是厌恶性的。也就是说，许多学生学习（或试图学习）算术仍然是为了逃避惩罚或惩罚的威胁。许多学生不是为了获得正强化而学习算术，而是为了避免消极后果而做功课：老师的不满、同学的嘲笑、成绩差而受到家长的惩罚，或在与其他同学的竞争中成绩不佳。其次，即使在试图促进算术学习的过程中使用了适当的积极强化手段，强化效果通常也不会达到最佳。在学校里，教师对学生正确的书面解题方法的强化，在大班中可能很少发生。或者，即使有强化，也可能是在学生回答后几分钟才进行。斯金纳发现，在学习某些类型的技能（如基本算术）时，有些学生需要教师立即强化他们的回答。即使在反应和强化之间相隔一两分钟，有时也会使立即强化的积极效果大打折扣。由于家庭作业和测验由教师批改，并在学生完成作业一天或更长时间后才发回。
plete them, much of the leaming value of these activities can be lost for many students. A third reason why so many students fail to learn in schools, even where immediate reinforcement is given to them, is that the frequency of reinforcement is inadequate. Skinner (1968, p. 17) in his book The Technology of Teaching estimates that a student throughout his or her first four years of school requires something on the order of 25,000 reinforcements and may be given only a few thousand.
因此，对许多学生来说，这些活动的学习价值可能会丧失殆尽。很多学生在学校学不好的第三个原因是，即使对他们进行了即时强化，强化的频率也不够。斯金纳（Skinner，1968 年，第 17 页）在他的著作《教学技术》中估计，一个学生在他或她在学校的头四年里需要大约 25,000 次强化，而可能只得到几千次。
Skinner's proposed solution for overcoming the impossiblity of teachers' being able to provide immediate reinforcement to every student on a regular basis, is to use programmed instructional materials and teaching machines to assist the teacher in reinforcing students. Printed programmed instructional modules and textbooks are usually structured so that information is presented in very small pieces. After each piece of information is given, the reader is asked a question, after which he or she immediately compares his or her answer to the correct answer which is printed foHowing the question. The student may slide a card down the page in order to hide new information and answers to questions until each piece of information is read, and a corresponding question about that particular bit of information is answered. The principles which are used in preparing programmed materials are that information should be presented in small pieces and that the leamer should show that he or she has learned each piece of information by answering a question, followed by immediate feedback conceming the student's answer. Some programmed materials are linear; that is, regardless of the student's response, he or she continues on. Other materials contain branches so that each student's next step in the program is determined by his or her response to a question, or responses to a sequence of questions. There are a number of programmed instructional materials for mathematics on the market. Samples can be obtained for previewing by contacting various publishing companies. The following passage is an example of a small part of a programmed instructional sequence in mathematics:
斯金纳提出的解决方案是，利用程序化教学材料和教学机器来协助教师强化学生，以克服教师无法定期为每个学生提供即时强化的问题。印刷的程序化教学模块和教科书的结构通常是将信息分成很小的片段。每给出一条信息后，都会向读者提出一个问题，然后读者会立即将自己的答案与印在问题下方的正确答案进行比较。学生可以在页面上滑动卡片，以隐藏新信息和问题答案，直到读完每条信息并回答了有关该信息的相应问题为止。编制程序教材的原则是，信息应小块呈现，学习者应通过回答问题来表明他或她已学会了每条信息，随后应立即反馈学生的答案。有些编程教材是线性的，即无论学生的回答如何，他或她都会继续学习。其他材料则包含分支，每个学生在程序中的下一步由其对某个问题的回答或对一连串问题的回答决定。市场上有许多数学编程教材。您可以与各出版公司联系，索取样本进行预览。下面的段落是数学编程教学序列中一小部分的示例：
斯金纳提出的解决方案是，利用程序化教学材料和教学机器来协助教师强化学生，以克服教师无法定期为每个学生提供即时强化的问题。印刷的程序化教学模块和教科书的结构通常是将信息分成很小的片段。每给出一条信息后，都会向读者提出一个问题，然后读者会立即将自己的答案与印在问题下方的正确答案进行比较。学生可以在页面上滑动卡片，以隐藏新信息和问题答案，直到读完每条信息并回答了有关该信息的相应问题为止。编制程序教材的原则是，信息应小块呈现，学习者应通过回答问题来表明他或她已学会了每条信息，随后应立即反馈学生的答案。有些编程教材是线性的，即无论学生的回答如何，他或她都会继续学习。其他材料则包含分支，每个学生在程序中的下一步由其对某个问题的回答或对一连串问题的回答决定。市场上有许多数学编程教材。您可以与各出版公司联系，索取样本进行预览。下面的段落是数学编程教学序列中一小部分的示例：
Frame 59. A prime number is a number whose only divisors are one and the number itself. For example 7 is a prime number because only one and seven divide 7 without a remainder.
第 59 帧质数是指被除数只有 1 和质数本身的数。例如，7 是一个质数，因为只有 1 和 7 除以 7 没有余数。
Question 1. Is 29 a prime number?
问题 1.29 是质数吗？
You should have said yes. 29 is a prime number because its only divisors are one and twenty-nine.
29 是质数，因为它的除数只有 1 和 29。
Question 2. What are the only divisors of a prime number?
问题 2.质数的唯一除数是什么？
You should have answered that the only divisors of a prime number are one and the number itself.
你应该回答质数的除数只有 1 和质数本身。
If you answered both Question 1 and Question 2 correctly go on to Frame 60. If either of your answers was incorrect go back to Frame 59.
如果您正确回答了问题 1 和问题 2，请进入第 60 画格。如果其中一个答案不正确，请返回第 59 画格。
第 59 帧质数是指被除数只有 1 和质数本身的数。例如，7 是一个质数，因为只有 1 和 7 除以 7 没有余数。
Question 1. Is 29 a prime number?
问题 1.29 是质数吗？
You should have said yes. 29 is a prime number because its only divisors are one and twenty-nine.
29 是质数，因为它的除数只有 1 和 29。
Question 2. What are the only divisors of a prime number?
问题 2.质数的唯一除数是什么？
You should have answered that the only divisors of a prime number are one and the number itself.
你应该回答质数的除数只有 1 和质数本身。
If you answered both Question 1 and Question 2 correctly go on to Frame 60. If either of your answers was incorrect go back to Frame 59.
如果您正确回答了问题 1 和问题 2，请进入第 60 画格。如果其中一个答案不正确，请返回第 59 画格。
In the past Skinner and others developed and used ingenious mechanical teaching machines which presented printed sequences of instructional materials to students and which branched to new or previous material according to the responses which students selected to questions. Although these teaching machines did provide a medium for testing and evaluating Skinner's theories of instruction, they have become historical relics. Students now can interact with a computer-based instructional program at a remote teaching terminal and can obtain a variety of instructional "frames" based upon a computer analysis of their responses to computer generated questions and problems. The flexibility of PLATO computer terminals, which were developed at the University of Illinois as part of a very complex and sophisticated computer-based instructional system, is so great that students can select and use both audio and video programs at the computer terminal. They can "talk" to the computer by either typing commands or responses at the remote terminal or touching objects displayed on a televisionlike screen. They can also communicate with human teachers and other students via the computer network when they have a problem that the computer is not programmed to deal with. Rather than serving as a replacement for teachers, complex educational computing systems can free teachers to give more attention to the leaming problems of individual students: problems which are complex enough to require the assistance of a human being who can be much more flexible, perceptive, and sympathetic than a computer.
过去，斯金纳和其他人开发并使用了巧妙的机械教学机器，这些机器向学生展示印刷好的教学材料序列，并根据学生对问题的回答选择新的或以前的材料。尽管这些教学机器确实为测试和评估斯金纳的教学理论提供了媒介，但它们已成为历史的遗迹。现在，学生可以在远程教学终端上与基于计算机的教学程序进行交互，并可以根据计算机对他们对计算机生成的问题和难题的回答进行的分析，获得各种教学 "框架"。PLATO 计算机终端是伊利诺伊大学开发的一个非常复杂和精密的计算机辅助教学系统的一部分，其灵活性非常大，学生可以在计算机终端上选择和使用音频和视频节目。他们可以通过在远程终端上键入命令或响应，或触摸电视屏幕上显示的物体，与计算机 "对话"。当他们遇到计算机无法处理的问题时，还可以通过计算机网络与人类教师和其他学生交流。复杂的教育计算机系统不是要取代教师，而是可以让教师腾出手来，更多地关注个别学生的学习问题：这些问题非常复杂，需要人类的帮助，而人类的灵活性、洞察力和同情心要比计算机强得多。
In conclusion, much of Skinner's research on the science of learning and the art of teaching is useful to mathematics teachers. His principles of teaching and learning are particularly helpful in developing strategies for teaching facts and basic arithmetic skills to both elementary and secondary school students. Some of Gagné's work in structuring hierarchies of mathematical facts, skills, concepts, and principles to aid students in learning mathematics has .its theoretical foundation in the theories of Burmus Frederic Skinner. Even though individual chapters of Skinner's book, The Technology of Teaching, were written between 1954 and 1968, his analysis of teaching is still relevant-even for teaching "modern mathematics" in modern secondary schools.
总之，斯金纳在学习科学和教学艺术方面的许多研究对数学教师很有帮助。他的教学原则尤其有助于为中小学生制定数学事实和基本算术技能的教学策略。盖尼耶在构建数学事实、技能、概念和原理的层次结构以帮助学生学习数学方面所做的一些工作，其理论基础来自于伯穆斯-弗雷德里克-斯金纳（Burmus Frederic Skinner）的理论。尽管斯金纳的《教学技术》一书中的个别章节是在 1954 年至 1968 年间撰写的，但他对教学的分析仍然具有现实意义--甚至对现代中学的 "现代数学 "教学也是如此。
过去，斯金纳和其他人开发并使用了巧妙的机械教学机器，这些机器向学生展示印刷好的教学材料序列，并根据学生对问题的回答选择新的或以前的材料。尽管这些教学机器确实为测试和评估斯金纳的教学理论提供了媒介，但它们已成为历史的遗迹。现在，学生可以在远程教学终端上与基于计算机的教学程序进行交互，并可以根据计算机对他们对计算机生成的问题和难题的回答进行的分析，获得各种教学 "框架"。PLATO 计算机终端是伊利诺伊大学开发的一个非常复杂和精密的计算机辅助教学系统的一部分，其灵活性非常大，学生可以在计算机终端上选择和使用音频和视频节目。他们可以通过在远程终端上键入命令或响应，或触摸电视屏幕上显示的物体，与计算机 "对话"。当他们遇到计算机无法处理的问题时，还可以通过计算机网络与人类教师和其他学生交流。复杂的教育计算机系统不是要取代教师，而是可以让教师腾出手来，更多地关注个别学生的学习问题：这些问题非常复杂，需要人类的帮助，而人类的灵活性、洞察力和同情心要比计算机强得多。
In conclusion, much of Skinner's research on the science of learning and the art of teaching is useful to mathematics teachers. His principles of teaching and learning are particularly helpful in developing strategies for teaching facts and basic arithmetic skills to both elementary and secondary school students. Some of Gagné's work in structuring hierarchies of mathematical facts, skills, concepts, and principles to aid students in learning mathematics has .its theoretical foundation in the theories of Burmus Frederic Skinner. Even though individual chapters of Skinner's book, The Technology of Teaching, were written between 1954 and 1968, his analysis of teaching is still relevant-even for teaching "modern mathematics" in modern secondary schools.
总之，斯金纳在学习科学和教学艺术方面的许多研究对数学教师很有帮助。他的教学原则尤其有助于为中小学生制定数学事实和基本算术技能的教学策略。盖尼耶在构建数学事实、技能、概念和原理的层次结构以帮助学生学习数学方面所做的一些工作，其理论基础来自于伯穆斯-弗雷德里克-斯金纳（Burmus Frederic Skinner）的理论。尽管斯金纳的《教学技术》一书中的个别章节是在 1954 年至 1968 年间撰写的，但他对教学的分析仍然具有现实意义--甚至对现代中学的 "现代数学 "教学也是如此。
Summary 摘要
The seven theories which are presented and discussed in this chapter are attempts by their developers to structure and explain the very complex processes of instruction-and leaming. No single theory provides a complete model of either teaching or leaming, and there are areas of disagreement among the several theories. In spite of the limitations of these theories, each has applications for teaching and learning secondary school mathematics.
本章介绍和讨论的七种理论，是理论提出者试图构建和解释非常复杂的教学和学习过程的尝试。没有一种理论能提供完整的教学或学习模式，而且几种理论之间也存在分歧。尽管这些理论有其局限性，但每种理论都适用于中学数学的教与学。
本章介绍和讨论的七种理论，是理论提出者试图构建和解释非常复杂的教学和学习过程的尝试。没有一种理论能提供完整的教学或学习模式，而且几种理论之间也存在分歧。尽管这些理论有其局限性，但每种理论都适用于中学数学的教与学。
Piaget and Skinner have formulated two very different models of human leaming. Piaget has developed a theory of intellectual maturation and development, whereas Skinner has studied the conditions under which human behaviors take place. Although they are different approaches to the study of learning and behavior, these two theories do complement each other; each has many applications in teaching mathematics.
皮亚杰和斯金纳提出了两种截然不同的人类学习模式。皮亚杰提出了智力成熟和发展的理论，而斯金纳则研究了人类行为发生的条件。虽然它们是研究学习和行为的不同方法，但这两种理论确实是相辅相成的；每种理论在数学教学中都有许多应用。
Guilford has determined what he believes to be the 120 mental abilities which comprise general intelligence, and his findings can be of considerable use to teachers in identifying and dealing with specific leaming problems in individual students.
吉尔福德确定了他认为构成一般智力的 120 种心智能力，他的发现对教师识别和处理个别学生的具体学习问题有很大帮助。
皮亚杰和斯金纳提出了两种截然不同的人类学习模式。皮亚杰提出了智力成熟和发展的理论，而斯金纳则研究了人类行为发生的条件。虽然它们是研究学习和行为的不同方法，但这两种理论确实是相辅相成的；每种理论在数学教学中都有许多应用。
Guilford has determined what he believes to be the 120 mental abilities which comprise general intelligence, and his findings can be of considerable use to teachers in identifying and dealing with specific leaming problems in individual students.
吉尔福德确定了他认为构成一般智力的 120 种心智能力，他的发现对教师识别和处理个别学生的具体学习问题有很大帮助。
Bruner's theory of instruction is useful to teachers in helping them formulate general approaches to teaching, and much of his work has been shown to be directly applicable to teaching mathematics.
布鲁纳的教学理论对教师很有帮助，可以帮助他们制定一般的教学方法，而且他的许多研究成果已被证明可直接用于数学教学。
Based, in part, upon Piaget's theory of intellectual development, Dienes has
Dienes 部分以皮亚杰的智力发展理论为基础，认为
B.F. Skini Teaching and
B.F.斯基尼教学和
developed a theory of teaching mathematics which contains a sequence of strategies for teaching mathernatics concepts. He has also described how specific topics from secondary school mathematics can be approached by using his six stages in concept development as a general model for teaching and leaming mathematics.
他提出了一套数学教学理论，其中包含一系列数学概念教学策略。他还介绍了如何通过将概念发展的六个阶段作为数学教学和学习的一般模式来处理中学数学的具体课题。
布鲁纳的教学理论对教师很有帮助，可以帮助他们制定一般的教学方法，而且他的许多研究成果已被证明可直接用于数学教学。
Based, in part, upon Piaget's theory of intellectual development, Dienes has
Dienes 部分以皮亚杰的智力发展理论为基础，认为
B.F. Skini Teaching and
B.F.斯基尼教学和
developed a theory of teaching mathematics which contains a sequence of strategies for teaching mathernatics concepts. He has also described how specific topics from secondary school mathematics can be approached by using his six stages in concept development as a general model for teaching and leaming mathematics.
他提出了一套数学教学理论，其中包含一系列数学概念教学策略。他还介绍了如何通过将概念发展的六个阶段作为数学教学和学习的一般模式来处理中学数学的具体课题。
Gagné and Ausubel, while concemed with refining theories of leaming and instruction, have developed techniques and strategies for classroom teaching. Both of these men have formulated models for structuring the content of a discipline such as mathematics. Gagné has taken a bottom-to-top approach to structuring content into learning hierarchies which build upon simpler, prerequisite facts, skills, and concepts to leam more complex skills, concepts and principles. Ausubel has developed a theory of meaningful verbal learning which can be used by teachers when presenting material in a lecture or expository mode to students. Since a large proportion of mathematics teaching is carried out in a lecture mode, Ausubel's procedures for structuring information so that it can be learned in an efficient and meaningful way can be very useful to secondary school mathematics teachers.
盖尼耶和奥苏贝尔在完善教学理论的同时，还发展了课堂教学的技巧和策略。他们两人都制定了数学等学科的内容结构模型。盖尼耶采用自下而上的方法，将教学内容编排成学习层次，在较简单的先决事实、技能和概念的基础上，学习较复杂的技能、概念和原理。奥苏贝尔（Ausubel）提出了一种有意义的言语学习理论，教师在以讲授或说明的方式向学生介绍教材时，可以利用这一理论。由于大部分数学教学都是以讲授的方式进行的，奥苏贝尔的信息结构化程序对中学数学教师非常有用，使他们能够以高效和有意义的方式学习信息。
盖尼耶和奥苏贝尔在完善教学理论的同时，还发展了课堂教学的技巧和策略。他们两人都制定了数学等学科的内容结构模型。盖尼耶采用自下而上的方法，将教学内容编排成学习层次，在较简单的先决事实、技能和概念的基础上，学习较复杂的技能、概念和原理。奥苏贝尔（Ausubel）提出了一种有意义的言语学习理论，教师在以讲授或说明的方式向学生介绍教材时，可以利用这一理论。由于大部分数学教学都是以讲授的方式进行的，奥苏贝尔的信息结构化程序对中学数学教师非常有用，使他们能够以高效和有意义的方式学习信息。
The various theories of teaching and leaming can be used as a basis for designing and presenting mathematics lessons and also provide a rich background of information which teachers can use in developing and improving the effectiveness of their classroom strategies for teaching mathematics to students in secondary schools.
各种教学和学习理论可以作为设计和展示数学课程的基础，也为教师提供了丰富的信息背景，教师可以利用这些信息来制定和改进课堂教学策略，提高中学学生数学学习的有效性。
各种教学和学习理论可以作为设计和展示数学课程的基础，也为教师提供了丰富的信息背景，教师可以利用这些信息来制定和改进课堂教学策略，提高中学学生数学学习的有效性。
Things To Do 可做之事
- List and define the stages of intellectual development identified by Piaget and discuss the mathematical abilities which people can be expected to attain in each stage. What differences in leaming abilities exist between junior high school and senior high school students? Discuss teaching strategies which would be appropriate to use with junior high school students; with senior high school students.
列出并定义皮亚杰确定的智力发展阶段，并讨论人们在每个阶段可望达到的数学能力。初中生和高中生的学习能力有哪些差异？讨论适合初中生和高中生的教学策略。 - Define these terms which are used in Piaget's theory of intellectual development: assimilation, accommodation, maturation, physical experiences, logico-mathematical experiences, social transmission, and equilibration. Discuss each term according to its application to learning mathematics.
定义皮亚杰智力发展理论中的这些术语：同化、调适、成熟、身体经验、逻辑数学经验、社会传递和平衡。讨论每个术语在数学学习中的应用。 - J. P. Guilford has identified five operations of learning, four contents of learning, and six products of learning. Define and give an example from secondary school mathematics of each of these fifteen characteristics of intelligence.
J.P. Guilford 提出了五种学习操作、四种学习内容和六种学习产品。请定义这十五种智力特征，并举例说明中学数学中的每一种特征。 - From the set of operations, contents, and products of leaming identified in Guilford's Structure of Intellect Model, select ten specific intellectual aptitudes (ten ordered triples, each consisting of an operation, a content, and a product) which you think are important aptitudes for learning mathematics. Then, give examples of ten learning tasks in mathematics, each o which requires one of the ten intellectual aptitudes which you selected.
从吉尔福德的智力结构模型中确定的一系列学习操作、内容和结果中，选择十种你认为是学习数学的重要能力的具体智力能力（十个有序三元组，每个由一个操作、一个内容和一个结果组成）。然后，举例说明十个数学学习任务，每个任务都需要你所选择的十种智力能力中的一种。 - How can a mathematics teacher who is not a trained psychologist apply Guilford's Structure of Intellect Model in his or her own teaching?
数学教师如果不是训练有素的心理学家，如何在自己的教学中应用吉尔福德的智力结构模型？ - The four direct objects of mathematics leaming are facts, skills, concepts, and principles. Define and give several examples of each object. Choose and discuss four different teaching strategies, each one of which would be appropriate for teaching one of the four direct objects of mathematics.
数学学习的四个直接对象是事实、技能、概念和原理。定义并举例说明每种对象。选择并讨论四种不同的教学策略，每种策略都适合教授数学四个直接对象中的一个。 - Define each of the eight types of leaming which Robert Gagné has identified, and give an example from mathematics education of each leaming type. Suggest some teaching strategies which would be appropriate for promoting each one of the eight leaming types.
分别定义罗伯特-盖尼耶提出的八种学习类型，并举例说明每种学习类型。建议一些适合促进八种学习类型的教学策略。 - Select a mathematical skill (for example, solving two equations in two unknowns) and write a list of steps required in applying the skill. Then, develop a learning hierarchy (see Figure 3.2 for an example of a learning hierarchy) for teaching and learning that skill.
选择一项数学技能（例如，解两个未知数的方程），并写出应用该技能所需的步骤 清单。然后，为该技能的教与学制定一个学习层次结构（学习层次结构示例见图 3.2）。 - Zoltan Dienes has categorized three types of mathematics concepts-pure concepts, notational concepts, and applied concepts. Define and give several examples of each type of concept, and suggest teaching/leaming activities which would be appropriate for each type.
Zoltan Dienes 将数学概念分为三类--纯概念、符号概念和应用概念。请定义每种类型的概念并举出几个例子，并建议适合每种类型的教学/启发活动。 - Analyze and illustrate Dienes' six stages in teaching and learning mathematics by selecting a secondary school mathematics topic and preparing a six-stage teaching/learning strategy which illustrates the applications of the six stages to classroom teaching.
通过选择一个中学数学课题并准备一个六阶段教学/学习策略来分析和说明 Dienes 的数学教学六阶段，从而说明六阶段在课堂教学中的应用。 - Explain the relationship between Dienes' six stages in concept leaming and his four. general principles and five subprinciples for teaching concepts.
解释 Dienes 提出的概念学习的六个阶段与他提出的概念教学的四项一般原则和五项次原则之间的关系。 - Discuss Ausubel's two preconditions for meaningful reception leaming and explain how they can be applied to meaningful teaching and learning.
讨论奥苏贝尔提出的有意义接受学习的两个先决条件，并解释如何将其应用于有意义的教与学。 - Select a topic from secondary school mathematics and, based upon Ausubel's strategies for meaningful verbal leaming, design a teaching/leaming plan for presenting that topic to students.
从中学数学中选择一个主题，并根据奥苏贝尔的有意义的语言学习策略，设计一个向学生展示该主题的教学/学习计划。 - Choose a secondary school mathematics topic and develop an advance organizer for use in introducing the topic.
选择一个中学数学主题，并制作一个用于介绍该主题的预组织器。 - List and describe Bruner's six characteristics of intellectual growth, and discuss the implications of each characteristic for teaching secondary school mathematics.
列出并描述布鲁纳提出的智力成长的六个特征，并讨论每个特征对中学数学教学的影响。 - According to Bruner, a theory of instruction should be both prescriptive and normative. Explain what he means by these two terms and why it is necessary for instructional theories to be both prescriptive and normative. Discuss the four major features which prescribe the nature of the instructional process, and which Bruner believes should be contained in any theory of instruction.
布鲁纳认为，教学理论应该既是规定性的，又是规范性的。请解释这两个术语的含义，以及为什么教学理论必须同时具有规定性和规范性。讨论布鲁纳认为任何教学理论都应包含的规定教学过程性质的四个主要特征。 - Explain the distinction between a leaming theory and a theory of instruction. Why is it necessary for teachers to be familiar with learning theories and instructional theories? What is the relationship between leaming
解释学习理论与教学理论的区别。为什么教师必须熟悉学习理论和教学理论？学习理论与教学理论之间的关系是什么？ - State the four general "theorems' about leaming mathematics which were developed by Bruner and Kenneys and explain how these theorems can be used in teaching and learning mathematics.
说明布鲁纳和肯尼斯提出的有关学习数学的四个一般 "定理"，并解释如何在数学教 学中使用这些定理。 - Skinner uses the terms respondent conditioning and operant conditioning in his theory of leaming. Define each of these terms and give an example of each type of conditioning in mathematics education.
斯金纳在其学习理论中使用了反应性条件反射和操作性条件反射这两个术语。请定义这两个术语，并举例说明每种条件反射在数学教育中的应用。 - Explain the difference between positive and negative reinforcers. and give several examples of each type of reinforcement.
解释正强化物和负强化物的区别，并举出几例说明每种强化物。 - Discuss the difference between forgetting and extinction. Explain how extinction occurs. Give several examples of the operations of forgetting and extinction of previously leamed mathematics facts, skills, concepts, or principles.
讨论遗忘和消亡的区别。解释遗忘是如何发生的。举例说明遗忘和淡忘以前学过的数学事实、技能、概念或原理的操作。 - Discuss the relative advantages and disadvantages of rewards and punishment in a mathematics classroom. What do you consider to be appropriate methods of rewarding and punishing students?
讨论数学课堂中奖惩的相对利弊。您认为奖励和惩罚学生的适当方法是什么？ - Discuss the difference between Gagne's and Dienes' definitions of concept and the difference between their approaches to leaming.
讨论加涅和迪内斯对概念定义的不同，以及他们学习方法的不同。 - Show how Dienes' theories of teaching and learning mathematics are related to Piaget's theory of intellectual development.
说明 Dienes 的数学教学理论与皮亚杰的智力发展理论有何联系。 - Analyze and discuss the similarities and differenes between Gagné's and Skinner's theories of leaming.
分析并讨论盖尼耶和斯金纳的学习理论的异同。 - Select two or three of the theories presented in this chapter and research Select two or three of the theories presented in fortaphy for this chapter provides a
从本章介绍的理论中选择两三个理论进行研究 从本章介绍的理论中选择两三个理论进行研究，为本章提供了一个理论框架。
them in more detail. The Selected Bibliograph good starting point for locating additional references for this activity.
《更详细地了解它们。书目选编》是查找本活动其他参考资料的良好起点。 - Synthesize a composite theory of leaming and instruction based upon the theories of teaching and leaming which are presented in this chapter.
在本章介绍的教学和学习理论的基础上，总结出学习和教学的综合理论。
Selected Bibliography 书目选编
Aichele, Douglas B. and Reys, Robert E. (Editors). Readings in Secondary School Mathematics. Boston, Mass.: Prindle, Weber & Schmidt, Inc., 1971.
Aichele, Douglas B. and Reys, Robert E. (Editors).中学数学读本》。马萨诸塞州波士顿：Prindle、Weber & Schmidt 公司，1971 年。
Aichele, Douglas B. and Reys, Robert E. (Editors).中学数学读本》。马萨诸塞州波士顿：Prindle、Weber & Schmidt 公司，1971 年。
This book of readings in mathematics education contains eight articles on contemporary theories about how people leam mathematics. Among the authors of these articles are Robert Gagné, Jerome Bruner, David Ausubel, and Zoltan Dienes.
这本数学教育读本收录了八篇文章，介绍了有关人们如何学习数学的当代理论。这些文章的作者包括罗伯特-盖尼耶（Robert Gagné）、杰罗姆-布鲁纳（Jerome Bruner）、戴维-奥苏贝尔（David Ausubel）和佐尔坦-迪尼斯（Zoltan Dienes）。
Ausubel, David P. "In Defense of Verbal Learning." Educational Theory XI (1961):
Ausubel, David P. "为口头学习辩护"。教育理论 XI》（1961 年）：
-The Psychology of Meaningful Verbal Learning. New York: Grune & Stratton, 1963.
-有意义的语言学习心理学》。纽约：Grune & Stratton，1963 年。
A comprehensive exposition of Ausubel's theory of meaningful verbal learning through expository teaching, this book is useful to the person who wants to carry out an in-depth study of the work of Professor Ausubel.
本书全面阐述了奥苏贝尔关于通过说明性教学进行有意义的言语学习的理论，对于想要深入研究奥苏贝尔教授作品的人来说非常有用。
Educational Psychology, A Cognitive Viev. New York: Holt, Rinehart and Winston, Inc., 1968.
《教育心理学，认知维度》。纽约：Holt, Rinehart and Winston, Inc.，1968 年。
这本数学教育读本收录了八篇文章，介绍了有关人们如何学习数学的当代理论。这些文章的作者包括罗伯特-盖尼耶（Robert Gagné）、杰罗姆-布鲁纳（Jerome Bruner）、戴维-奥苏贝尔（David Ausubel）和佐尔坦-迪尼斯（Zoltan Dienes）。
Ausubel, David P. "In Defense of Verbal Learning." Educational Theory XI (1961):
Ausubel, David P. "为口头学习辩护"。教育理论 XI》（1961 年）：
-The Psychology of Meaningful Verbal Learning. New York: Grune & Stratton, 1963.
-有意义的语言学习心理学》。纽约：Grune & Stratton，1963 年。
A comprehensive exposition of Ausubel's theory of meaningful verbal learning through expository teaching, this book is useful to the person who wants to carry out an in-depth study of the work of Professor Ausubel.
本书全面阐述了奥苏贝尔关于通过说明性教学进行有意义的言语学习的理论，对于想要深入研究奥苏贝尔教授作品的人来说非常有用。
Educational Psychology, A Cognitive Viev. New York: Holt, Rinehart and Winston, Inc., 1968.
《教育心理学，认知维度》。纽约：Holt, Rinehart and Winston, Inc.，1968 年。
A general work on the psychology of learning, this book is a presentation of learning psychology based upon Ausubel's theories of leaming. Among the topics which Ausubel discusses are meaningful reception learning, retention, transfer of leaming, practice, individual differences, motivation, personality, social factors in leaming, discovery learning, concept acquisition, problem solving, and creativity.
这是一本学习心理学的通俗读物，以奥苏贝尔的学习理论为基础，介绍了学习心理学。奥苏贝尔讨论的主题包括有意义的接受学习、保持、学习迁移、练习、个体差异、动机、个性、学习中的社会因素、发现学习、概念习得、问题解决和创造力。
" "Facilitating Meaningful Verbal Learning in the Classroom." The Arithmetic Teacher 15 (1968): 126-32.
""促进课堂上有意义的口语学习"。算术教师》第 15 期（1968 年）：126-32.
这是一本学习心理学的通俗读物，以奥苏贝尔的学习理论为基础，介绍了学习心理学。奥苏贝尔讨论的主题包括有意义的接受学习、保持、学习迁移、练习、个体差异、动机、个性、学习中的社会因素、发现学习、概念习得、问题解决和创造力。
" "Facilitating Meaningful Verbal Learning in the Classroom." The Arithmetic Teacher 15 (1968): 126-32.
""促进课堂上有意义的口语学习"。算术教师》第 15 期（1968 年）：126-32.
This article is recommended reading for the mathematics teacher who is concemed with improving his or her expository teaching.
推荐阅读这篇文章，以帮助数学教师提高论述式教学水平。
Beard, Ruth M. An Outline of Piaget's Developmental Psychology. London: Routledge & Kegan Paiu, 1969.
Beard, Ruth M. 《皮亚杰发展心理学概要》。伦敦：Routledge & Kegan Paiu, 1969.
推荐阅读这篇文章，以帮助数学教师提高论述式教学水平。
Beard, Ruth M. An Outline of Piaget's Developmental Psychology. London: Routledge & Kegan Paiu, 1969.
Beard, Ruth M. 《皮亚杰发展心理学概要》。伦敦：Routledge & Kegan Paiu, 1969.
This book is a very readable discussion of the Piagetian stages of cognitive - development with some implications for classroom teaching.
这本书对皮亚杰认知发展阶段的论述具有很强的可读性，对课堂教学也有一些启示。
这本书对皮亚杰认知发展阶段的论述具有很强的可读性，对课堂教学也有一些启示。
Bell, Frederick H. "Let's Make a Computer." Model for Teaching, a component of the Croft Teacher's Service, New London, Conn., February, 1973.
Bell, Frederick H. "Let's Make a Computer"。教学模式》，克罗夫特教师服务机构的一个组成部分，康涅狄格州新伦敦，1973 年 2 月。
Brearley, Molly, and Hitchfield, Elizabeth. A Guide to Reading Piaget. New York: Shocken Books, 1972.
《布雷尔利、莫莉和希菲尔德、伊丽莎白。皮亚杰阅读指南》。New York：Shocken Books, 1972.
Bell, Frederick H. "Let's Make a Computer"。教学模式》，克罗夫特教师服务机构的一个组成部分，康涅狄格州新伦敦，1973 年 2 月。
Brearley, Molly, and Hitchfield, Elizabeth. A Guide to Reading Piaget. New York: Shocken Books, 1972.
《布雷尔利、莫莉和希菲尔德、伊丽莎白。皮亚杰阅读指南》。New York：Shocken Books, 1972.
An interpretation of Piaget's research, this book is a good starting point for the novice to Piaget's developmental psychology. Several Piagetian experiments relating to each developmental stage are presented and discussed.
本书是对皮亚杰研究的诠释，是皮亚杰发展心理学新手的良好起点。书中介绍并讨论了与每个发展阶段相关的几个皮亚杰实验。
Bruner, Jerome S. "On Learning Mathematics." The Mathematics Teacher LIII (1960): 610-619.
Bruner, Jerome S. "论数学学习"。The Mathematics Teacher LIII (1960)：610-619.
本书是对皮亚杰研究的诠释，是皮亚杰发展心理学新手的良好起点。书中介绍并讨论了与每个发展阶段相关的几个皮亚杰实验。
Bruner, Jerome S. "On Learning Mathematics." The Mathematics Teacher LIII (1960): 610-619.
Bruner, Jerome S. "论数学学习"。The Mathematics Teacher LIII (1960)：610-619.
In this article Bruner presents his thoughts about discovery leaming, intuition, and readiness for leaming.
在这篇文章中，布鲁纳阐述了他对发现式学习、直觉和学习准备的看法。
. "Observations on the Learning of Mathematics." Science Education News April, 1963: 1-5.
."数学学习观察"。科学教育新闻》，1963 年 4 月：1-5.
在这篇文章中，布鲁纳阐述了他对发现式学习、直觉和学习准备的看法。
. "Observations on the Learning of Mathematics." Science Education News April, 1963: 1-5.
."数学学习观察"。科学教育新闻》，1963 年 4 月：1-5.
In this article Bruner states, discusses, and illustrates his four theorems about leaming mathematics.
在这篇文章中，布鲁纳阐述、讨论并说明了他关于学习数学的四个定理。
在这篇文章中，布鲁纳阐述、讨论并说明了他关于学习数学的四个定理。
- The Process of Education. Cambridge, Mass.: Harvard University Press, 1966.
《教育的过程》。马萨诸塞州剑桥市：哈佛大学出版社，1966 年。
This book is Bruner's synthesis of the themes which were debated by 34 educators and scientists from the fields of mathematics, science, history, education, psychology and medicine at a conference in Woods Hole, Massachusetts. The purpose of the conference was to discuss the basic processes involved in teaching the content and methods of science to school students. The major themes which were considered at the Woods Hole conference and which are presented in this book are (1) the importance of studying the structure of each subject, (2) student readiness for learning, (3) intuitive and analytic thinking, (4) motivation to leam, and (5) aids to teaching.
本书是布鲁纳对在马萨诸塞州伍兹霍尔举行的一次会议上，来自数学、科学、历史、教育、心理学和医学领域的 34 位教育家和科学家所讨论的主题的总结。会议的目的是讨论向在校学生教授科学内容和方法的基本过程。伍兹霍尔会议上讨论的主要议题在本书中有所介绍，这些议题包括：(1) 研究各学科结构的重要性；(2) 学生的学习准备；(3) 直觉和分析思维；(4) 学习动机；(5) 教学辅助工具。
本书是布鲁纳对在马萨诸塞州伍兹霍尔举行的一次会议上，来自数学、科学、历史、教育、心理学和医学领域的 34 位教育家和科学家所讨论的主题的总结。会议的目的是讨论向在校学生教授科学内容和方法的基本过程。伍兹霍尔会议上讨论的主要议题在本书中有所介绍，这些议题包括：(1) 研究各学科结构的重要性；(2) 学生的学习准备；(3) 直觉和分析思维；(4) 学习动机；(5) 教学辅助工具。
- Toward a Theory of Instruction. Cambridge, Mass.: The Belknap Press of Harvard University Press, 1966.
Toward a Theory of Instruction.马萨诸塞州剑桥：哈佛大学出版社贝尔纳普出版社，1966 年。
This book which contains some of Bruner's views of teaching and learning has a chapter "Notes on a Theory of Instruction" which is of particular interest and use to mathematics teachers.
这本书收录了布鲁纳对教学和学习的一些看法，其中有一章 "教学理论笔记 "对数学教师特别有用。
这本书收录了布鲁纳对教学和学习的一些看法，其中有一章 "教学理论笔记 "对数学教师特别有用。
OnKnowing: Essays for the Left Hand. New York: Atheneum, 1971. A collection of Bruner's previous writings, this book contains Bruncr's general thoughts on education and his paper "On Learning Mathematics" which summarizes some of his views that are related to mathematics education.
OnKnowing：Essays for the Left Hand.纽约：Atheneum，1971 年。本书是布鲁纳以往著作的结集，收录了布鲁纳对教育的总体思考，以及他的论文《论数学学习》，其中总结了他与数学教育有关的一些观点。
."The Process of Education Revisited." Phi Delta Kappan 53 (Septernber, 1971): 18-21.
重新审视教育过程"。Phi Delta Kappan 53 (Septernber, 1971)：18-21.
OnKnowing：Essays for the Left Hand.纽约：Atheneum，1971 年。本书是布鲁纳以往著作的结集，收录了布鲁纳对教育的总体思考，以及他的论文《论数学学习》，其中总结了他与数学教育有关的一些观点。
."The Process of Education Revisited." Phi Delta Kappan 53 (Septernber, 1971): 18-21.
重新审视教育过程"。Phi Delta Kappan 53 (Septernber, 1971)：18-21.
In this article Bruner reflects upon the views of education held in 1960 and finds his own earlier book, The Process of Education, lacking in relevance for education in the nineteen seventies. Any student of Bruner's works should. read this article.
在这篇文章中，布鲁纳对1960年的教育观点进行了反思，发现自己早先出版的《教育过程》一书对70年代的教育缺乏现实意义。任何布鲁纳作品的学生都应该阅读这篇文章。
Dienes, Z. P. Building up Mathematics (Fourth Edition). London: Hutchinson Educational Led., 1971.
Dienes, Z. P. Building up Mathematics (Fourth Edition).London: Hutchinson Educational Led., 1971.
在这篇文章中，布鲁纳对1960年的教育观点进行了反思，发现自己早先出版的《教育过程》一书对70年代的教育缺乏现实意义。任何布鲁纳作品的学生都应该阅读这篇文章。
Dienes, Z. P. Building up Mathematics (Fourth Edition). London: Hutchinson Educational Led., 1971.
Dienes, Z. P. Building up Mathematics (Fourth Edition).London: Hutchinson Educational Led., 1971.
In this book Dienes presents his theory of mathematics learning and discusses specific techniques for teaching several arithmetic concepts, algebraic concepts, topics in linear algebra, and other topics in geometry.
在这本书中，Dienes 提出了他的数学学习理论，并讨论了一些算术概念、代数概念、线性代数主题和其他几何主题的具体教学技巧。
Gagné, Robert M. The Conditions of Learning (Second Edition). New York: Holt, Rinehart and Winston, Inc., 1970.
Gagné，Robert M.《学习的条件》（第二版）。纽约：Holt, Rinehart and Winston, Inc.，1970 年。
在这本书中，Dienes 提出了他的数学学习理论，并讨论了一些算术概念、代数概念、线性代数主题和其他几何主题的具体教学技巧。
Gagné, Robert M. The Conditions of Learning (Second Edition). New York: Holt, Rinehart and Winston, Inc., 1970.
Gagné，Robert M.《学习的条件》（第二版）。纽约：Holt, Rinehart and Winston, Inc.，1970 年。
A comprehensive presentation of his theories about the conditions under which learning occurs, in this book Gagné defines and gives examples of his eight types of learning. He also describes leaming hierarchies, discusses procedures of instruction, and considers appropriate uses for various learning resources.
本书全面阐述了盖尼耶关于学习发生条件的理论，定义并举例说明了他提出的八种学习类型。他还描述了学习的层次结构，讨论了教学程序，并考虑了各种学习资源的适当用途。
Guiford, J. P. The Nature of Human Intelligence. New York: McGraw-Hill Book Company, 1967.
Guiford, J. P. The Nature of Human Intelligence.New York：麦格劳-希尔图书公司，1967 年。
本书全面阐述了盖尼耶关于学习发生条件的理论，定义并举例说明了他提出的八种学习类型。他还描述了学习的层次结构，讨论了教学程序，并考虑了各种学习资源的适当用途。
Guiford, J. P. The Nature of Human Intelligence. New York: McGraw-Hill Book Company, 1967.
Guiford, J. P. The Nature of Human Intelligence.New York：麦格劳-希尔图书公司，1967 年。
This book is indispensable to the person who is undertaking an extensive study of Guilford's conceptual model of the structure of human intelligence.
如果要对吉尔福德关于人类智力结构的概念模型进行广泛研究，本书是不可或缺的。
Higgins, John L. Mathematics Teaching and Learning. Worthington, Ohio: Charles A. Jones Publishing Company, 1973.
Higgins, John L. Mathematics Teaching and Learning.俄亥俄州沃辛顿：Charles A. Jones Publishing Company, 1973.
如果要对吉尔福德关于人类智力结构的概念模型进行广泛研究，本书是不可或缺的。
Higgins, John L. Mathematics Teaching and Learning. Worthington, Ohio: Charles A. Jones Publishing Company, 1973.
Higgins, John L. Mathematics Teaching and Learning.俄亥俄州沃辛顿：Charles A. Jones Publishing Company, 1973.
In this book Higgins presents a discussion of several general theories of learning as they apply to teaching and learning mathematics. He organizes his material into five units-(1) intelligence and the structure of mathematical abilities, (2) Piaget's analysis of intelligence, (3) stimulus-response leaming in mathematics, (4) Gestalt learning and heuristic teaching and (5) concept leaming and cognitive structure.
在本书中，希金斯讨论了几种适用于数学教学的一般学习理论。他将材料分为五个单元--(1) 智力和数学能力结构，(2) 皮亚杰的智力分析，(3) 数学中的刺激-反应学习，(4) 格式塔学习和启发式教学，以及 (5) 概念学习和认知结构。
Hilgard, Emest R. (Editor) Theories of Learning and Instruction (Sixty-third Yearbook). Chicago: National Society for the Study of Education, 1964.
Hilgard, Emest R. (Editor) Theories of Learning and Instruction (Sixty-third Yearbook).芝加哥：全美教育研究学会，1964 年。
在本书中，希金斯讨论了几种适用于数学教学的一般学习理论。他将材料分为五个单元--(1) 智力和数学能力结构，(2) 皮亚杰的智力分析，(3) 数学中的刺激-反应学习，(4) 格式塔学习和启发式教学，以及 (5) 概念学习和认知结构。
Hilgard, Emest R. (Editor) Theories of Learning and Instruction (Sixty-third Yearbook). Chicago: National Society for the Study of Education, 1964.
Hilgard, Emest R. (Editor) Theories of Learning and Instruction (Sixty-third Yearbook).芝加哥：全美教育研究学会，1964 年。
This yearbook contains articles on learning theories, psychology, motivation, learning readiness, theories of teaching and educational practice. It also has an article on stimulus-response learning and an article titled "Some Theorems on Instruction Illustrated with Reference to Mathematics" by Jerome Bruner.
本年鉴收录了有关学习理论、心理学、学习动机、学习准备、教学理论和教育实践的文章。其中还有一篇关于刺激-反应学习的文章，以及杰罗姆-布鲁纳撰写的题为 "以数学为例说明教学的一些定理 "的文章。
Isaacs, Nathan. Piaget: Some Answers to Teachers' Questions. London: National Froebel Foundation, 1965
Isaacs, Nathan.皮亚杰：对教师问题的一些回答》。伦敦：国家福禄贝尔基金会，1965 年
In this pamphlet, Isaacs answers nine questions about Piaget's theory and its applications in teaching arithmetic.
在这本小册子中，艾萨克斯回答了有关皮亚杰理论及其在算术教学中的应用的九个问题。
Joyce, Bruce and Weil, Marsha. Models of Teaching. Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1972.
《乔伊斯、布鲁斯和韦尔、玛莎。教学模式》。Englewood Cliffs, New Jersey：Prentice-Hall, Inc., 1972.
The authors discuss approximately fifteen general models of teaching, many of which are applicable to teaching mathematics. Each teaching/leaming model is presented according to its nature and structure and its classroom applications. Chapter 10 is an excellent introduction to David Ausubel's advance organizer approach to meaningful expository teaching.
作者讨论了大约十五种一般教学模式，其中许多适用于数学教学。每种教学/启发式教学模式都根据其性质、结构和课堂应用进行了介绍。第 10 章很好地介绍了大卫-奥苏贝尔（David Ausubel）的 "先行组织者"（prevanced organizer）方法，用于有意义的说明性教学。
Klausmeier, Herbert J. and Häris, Chester W. (Editors). Analyses of Concept Learning. New York: Academic Press, 1966.
Klausmeier, Herbert J. and Häris, Chester W. (Editors).概念学习分析》。New York：学术出版社，1966 年。
本年鉴收录了有关学习理论、心理学、学习动机、学习准备、教学理论和教育实践的文章。其中还有一篇关于刺激-反应学习的文章，以及杰罗姆-布鲁纳撰写的题为 "以数学为例说明教学的一些定理 "的文章。
Isaacs, Nathan. Piaget: Some Answers to Teachers' Questions. London: National Froebel Foundation, 1965
Isaacs, Nathan.皮亚杰：对教师问题的一些回答》。伦敦：国家福禄贝尔基金会，1965 年
In this pamphlet, Isaacs answers nine questions about Piaget's theory and its applications in teaching arithmetic.
在这本小册子中，艾萨克斯回答了有关皮亚杰理论及其在算术教学中的应用的九个问题。
Joyce, Bruce and Weil, Marsha. Models of Teaching. Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1972.
《乔伊斯、布鲁斯和韦尔、玛莎。教学模式》。Englewood Cliffs, New Jersey：Prentice-Hall, Inc., 1972.
The authors discuss approximately fifteen general models of teaching, many of which are applicable to teaching mathematics. Each teaching/leaming model is presented according to its nature and structure and its classroom applications. Chapter 10 is an excellent introduction to David Ausubel's advance organizer approach to meaningful expository teaching.
作者讨论了大约十五种一般教学模式，其中许多适用于数学教学。每种教学/启发式教学模式都根据其性质、结构和课堂应用进行了介绍。第 10 章很好地介绍了大卫-奥苏贝尔（David Ausubel）的 "先行组织者"（prevanced organizer）方法，用于有意义的说明性教学。
Klausmeier, Herbert J. and Häris, Chester W. (Editors). Analyses of Concept Learning. New York: Academic Press, 1966.
Klausmeier, Herbert J. and Häris, Chester W. (Editors).概念学习分析》。New York：学术出版社，1966 年。
A general reference on concept learning, the book contains sections on schemes for classifying concepts, learning of concepts, teaching and learming processes, and teaching mathematics concepts. Included among the sixteen chapters are articles written by Gagné, Ausubel, and Howard Fehr.
本书是一本关于概念学习的通用参考书，包含概念分类方案、概念学习、教学和学习过程以及数学概念教学等章节。全书共十六章，其中包括盖尼耶、奥苏贝尔和霍华德-费尔撰写的文章。
Lamon, William E. (Editor). Learning and the Nature of Mathematics. Palo Alto, Califomia: Science Research Associates, Inc., 1972.
Lamon, William E. (Editor).学习与数学的本质》。加州帕洛阿尔托：科学研究联营公司，1972 年。
本书是一本关于概念学习的通用参考书，包含概念分类方案、概念学习、教学和学习过程以及数学概念教学等章节。全书共十六章，其中包括盖尼耶、奥苏贝尔和霍华德-费尔撰写的文章。
Lamon, William E. (Editor). Learning and the Nature of Mathematics. Palo Alto, Califomia: Science Research Associates, Inc., 1972.
Lamon, William E. (Editor).学习与数学的本质》。加州帕洛阿尔托：科学研究联营公司，1972 年。
This book is a collection of articles written by mathematics educaiors, learning theorists, and psychologists. Among the contributors to this work are Dienes, Piaget, and Gagné. A brief biography of each writer is included as an introduction to his article.
本书汇集了数学教育家、学习理论家和心理学家撰写的文章。本书的作者包括迪尼斯、皮亚杰和盖尼耶。每位作者的简介都作为其文章的引言。
Martorella, Peter H. Concept Learning: Designs for Instruction. Scranton, Pennsylvania: Intext Educational Publishers, 1972.
Martorella, Peter H. Concept Learning：教学设计》。Scranton, Pennsylvania: Intext Educational Publishers, 1972.
本书汇集了数学教育家、学习理论家和心理学家撰写的文章。本书的作者包括迪尼斯、皮亚杰和盖尼耶。每位作者的简介都作为其文章的引言。
Martorella, Peter H. Concept Learning: Designs for Instruction. Scranton, Pennsylvania: Intext Educational Publishers, 1972.
Martorella, Peter H. Concept Learning：教学设计》。Scranton, Pennsylvania: Intext Educational Publishers, 1972.
A general work on concept leaming, this book contains a chapter titled "Concept Learning in the Mathematics Curriculum, K-12: Issues and Approaches." Piaget's, Bruner's, Dienes', Gagné's, and Ausubel's contributions to mathematics education are presented in this chapter, and several important mathematics curriculum projects, such as the School Mathematics Study Group and the University of Illinois Committee on School Mathematics, are discussed.
本书是一本关于概念学习的综合性著作，其中有一章题为 "K-12 数学课程中的概念学习：问题与方法"。本章介绍了皮亚杰、布鲁纳、第尼斯、盖尼耶和奥苏贝尔对数学教育的贡献，并讨论了几个重要的数学课程项目，如学校数学研究小组和伊利诺伊大学学校数学委员会。
Meeker, Mary Nacol. The Structure of Intellect, Its Interpretation and Uses. Columbus, Ohio: Charles E. Merrill Publishing CCompany, 1969.
Meeker, Mary Nacol.智力的结构、解释和用途》。俄亥俄州哥伦布市：查尔斯-E-梅里尔出版公司，1969 年。
本书是一本关于概念学习的综合性著作，其中有一章题为 "K-12 数学课程中的概念学习：问题与方法"。本章介绍了皮亚杰、布鲁纳、第尼斯、盖尼耶和奥苏贝尔对数学教育的贡献，并讨论了几个重要的数学课程项目，如学校数学研究小组和伊利诺伊大学学校数学委员会。
Meeker, Mary Nacol. The Structure of Intellect, Its Interpretation and Uses. Columbus, Ohio: Charles E. Merrill Publishing CCompany, 1969.
Meeker, Mary Nacol.智力的结构、解释和用途》。俄亥俄州哥伦布市：查尔斯-E-梅里尔出版公司，1969 年。
Dr. Meeker's book describes and interprets J. P. Guilford's Structure of Intellect Model and explains how it can be applied in curriculum planning and classroom teaching. She also presents examples of most of the 120 factors in the structure of intellect cube and references tests and test items which can be used to assess students' capabilities on individual intellectual factors.
米克尔博士在书中描述并解释了 J. P. 吉尔福德的智力结构模型，并说明了如何将其应用于课程规划和课堂教学。她还举例说明了智力结构立方体中 120 个因素中的大多数因素，并参考了可用于评估学生在各个智力因素方面能力的测试和测试项目。
Milhollan, Frank and Forisha, Bill E. From Skinner to Rogers: Contrasting Approaches to Education. Lincoln, Nebraska: Professional Educators Publications, Inc., 1972.
Milhollan, Frank and Forisha, Bill E. From Skinner to Rogers：从斯金纳到罗杰斯：截然不同的教育方法》。Lincoln, Nebraska：Professional Educators Publications, Inc., 1972.
米克尔博士在书中描述并解释了 J. P. 吉尔福德的智力结构模型，并说明了如何将其应用于课程规划和课堂教学。她还举例说明了智力结构立方体中 120 个因素中的大多数因素，并参考了可用于评估学生在各个智力因素方面能力的测试和测试项目。
Milhollan, Frank and Forisha, Bill E. From Skinner to Rogers: Contrasting Approaches to Education. Lincoln, Nebraska: Professional Educators Publications, Inc., 1972.
Milhollan, Frank and Forisha, Bill E. From Skinner to Rogers：从斯金纳到罗杰斯：截然不同的教育方法》。Lincoln, Nebraska：Professional Educators Publications, Inc., 1972.
This book contains an overview of B. F. Skinner's analysis of the process of learning and his scientific analysis of behavior. Skinner's theory of learning is contrasted to Carl R. Roger's humanistic approach to human psychology and learning.
本书概述了 B. F. 斯金纳对学习过程的分析及其对行为的科学分析。斯金纳的学习理论与卡尔-罗杰（Carl R. Roger）对人类心理和学习的人本主义方法进行了对比。
National Assessment of Educational Progress. NAEP Newsletter. VII (August, 1975):1,3.
国家教育进展评估。NAEP Newsletter.VII (August, 1975):1,3.
本书概述了 B. F. 斯金纳对学习过程的分析及其对行为的科学分析。斯金纳的学习理论与卡尔-罗杰（Carl R. Roger）对人类心理和学习的人本主义方法进行了对比。
National Assessment of Educational Progress. NAEP Newsletter. VII (August, 1975):1,3.
国家教育进展评估。NAEP Newsletter.VII (August, 1975):1,3.
This pamphlet contains a summary of the findings of the NAEP study of consumer mathematics skills of 17 -year-olds and young adults between ages 26-35.
本手册概述了 NAEP 对 17 岁青少年和 26-35 岁青年消费者数学技能的研究结果。
Piaget, Jean. Science of Education and the Psychology of the Child (Translated from the French by Derek Coltman). New York: Orion Press, 1970.
《皮亚杰，让教育科学与儿童心理学》（德里克-科尔特曼译自法文）。New York：Orion Press, 1970.
This is a general child psychology book which deals with developments in
这是一本儿童心理学普及读物，涉及以下方面的发展
leaming theory, adolescent psychology, tcaching methods, teacher education and de velopmental psychology.
学习理论、青少年心理学、教学方法、教师教育和发展心理学。
Raths, James, Pancella, John R., and Van Ness, James S. (Editors) Studying Teaching (Second Edition). Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1971.
Raths, James, Pancella, John R., and Van Ness, James S. (Editors) Studying Teaching (Second Edition).Englewood Cliffs, New Jersey：Prentice-Hall, Inc., 1971.
本手册概述了 NAEP 对 17 岁青少年和 26-35 岁青年消费者数学技能的研究结果。
Piaget, Jean. Science of Education and the Psychology of the Child (Translated from the French by Derek Coltman). New York: Orion Press, 1970.
《皮亚杰，让教育科学与儿童心理学》（德里克-科尔特曼译自法文）。New York：Orion Press, 1970.
This is a general child psychology book which deals with developments in
这是一本儿童心理学普及读物，涉及以下方面的发展
leaming theory, adolescent psychology, tcaching methods, teacher education and de velopmental psychology.
学习理论、青少年心理学、教学方法、教师教育和发展心理学。
Raths, James, Pancella, John R., and Van Ness, James S. (Editors) Studying Teaching (Second Edition). Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1971.
Raths, James, Pancella, John R., and Van Ness, James S. (Editors) Studying Teaching (Second Edition).Englewood Cliffs, New Jersey：Prentice-Hall, Inc., 1971.
Studying Teaching is a collection of articles about teaching and leaming which is organized into sections on teaching methods, interactions in classrooms, planning in education, testing, discipline, motivation, and other topics. The book contains articles by Bruner, Bloom, Gagné, Piaget, and Ausubel.
《研究教学》是一本关于教学和学习的文章集，分为教学方法、课堂互动、教育规划、考试、纪律、动机和其他主题等部分。书中收录了布鲁纳、布鲁姆、盖尼耶、皮亚杰和奥苏贝尔的文章。
Rosskopf, Myron F., Steffe, Leslie P., and Taback, Stanley (Editors) Piagetian Cognitive-Development Research and Mathematical Education. Washington, D.C.: National Council of Teachers of Mathematics, 1971.
Rosskopf, Myron F., Steffe, Leslie P., and Taback, Stanley (Editors) Piagetian Cognitive-Development Research and Mathematical Education.华盛顿特区：全国数学教师委员会，1971 年。
《研究教学》是一本关于教学和学习的文章集，分为教学方法、课堂互动、教育规划、考试、纪律、动机和其他主题等部分。书中收录了布鲁纳、布鲁姆、盖尼耶、皮亚杰和奥苏贝尔的文章。
Rosskopf, Myron F., Steffe, Leslie P., and Taback, Stanley (Editors) Piagetian Cognitive-Development Research and Mathematical Education. Washington, D.C.: National Council of Teachers of Mathematics, 1971.
Rosskopf, Myron F., Steffe, Leslie P., and Taback, Stanley (Editors) Piagetian Cognitive-Development Research and Mathematical Education.华盛顿特区：全国数学教师委员会，1971 年。
This book is a collection of papers which were presented at Teachers College, Columbia University as part of a conference on Piaget sponsored by the National Council of Teachers of Mathematics and the Department of Mathematical Education of Teachers College. Many of the papers illustrate implications of Piaget's theory for teaching topics in secondary school mathematics such as mathematical proof, functions, proportionality, and probability. The book also contains lists of references related to Piaget's work and its applications to mathematics education. This book is recommended reading for all secondary school mathematics teachers.
本书收录了在哥伦比亚大学师范学院发表的论文，这些论文是由美国国家数学教师委员会和师范学院数学教育系主办的皮亚杰会议的一部分。许多论文阐述了皮亚杰理论对中学数学教学主题的影响，如数学证明、函数、比例和概率。本书还包含与皮亚杰著作及其在数学教育中的应用相关的参考文献列表。本书是所有中学数学教师的推荐读物。
Skinner, B. F. Science and Human Behavior. New York: The Macmillan Company, 1953.
Skinner, B. F. Science and Human Behavior.纽约：麦克米伦公司，1953 年。
本书收录了在哥伦比亚大学师范学院发表的论文，这些论文是由美国国家数学教师委员会和师范学院数学教育系主办的皮亚杰会议的一部分。许多论文阐述了皮亚杰理论对中学数学教学主题的影响，如数学证明、函数、比例和概率。本书还包含与皮亚杰著作及其在数学教育中的应用相关的参考文献列表。本书是所有中学数学教师的推荐读物。
Skinner, B. F. Science and Human Behavior. New York: The Macmillan Company, 1953.
Skinner, B. F. Science and Human Behavior.纽约：麦克米伦公司，1953 年。
Skinner, in this book which is representative of his early work in human behavior, presents his analysis of the behavior of individuals and people in groups, and discusses problems in controlling human behavior.
这本书是斯金纳早期研究人类行为的代表作，书中介绍了他对个人和群体行为的分析，并讨论了控制人类行为的问题。
. The Technology of Teaching. Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1968.
.教学技术》。Englewood Cliffs, New Jersey：Prentice-Hall, Inc., 1968.
这本书是斯金纳早期研究人类行为的代表作，书中介绍了他对个人和群体行为的分析，并讨论了控制人类行为的问题。
. The Technology of Teaching. Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1968.
.教学技术》。Englewood Cliffs, New Jersey：Prentice-Hall, Inc., 1968.
In this widely acclaimed book, Skinner presents his views on learning as a science and teaching as an art. He also discusses reasons for teacher failure and methods for motivating students. In this book, Skinner shows how his theories of teaching and leaming can be applied in the classroom.
在这本广受赞誉的书中，斯金纳提出了他的观点：学习是一门科学，教学是一门艺术。他还讨论了教师失败的原因和激励学生的方法。在本书中，斯金纳展示了如何在课堂上应用他的教学和学习理论。
Weisgerber, Robert A. (Editor) Perspectives in Individualized Learning. Itasca, Illinois: F. E. Peacock Publishers, Inc., 1971.
Weisgerber, Robert A. (Editor) Perspectives in Individualized Learning.伊利诺伊州伊塔斯卡市：F. E. Peacock Publishers, Inc., 1971.
在这本广受赞誉的书中，斯金纳提出了他的观点：学习是一门科学，教学是一门艺术。他还讨论了教师失败的原因和激励学生的方法。在本书中，斯金纳展示了如何在课堂上应用他的教学和学习理论。
Weisgerber, Robert A. (Editor) Perspectives in Individualized Learning. Itasca, Illinois: F. E. Peacock Publishers, Inc., 1971.
Weisgerber, Robert A. (Editor) Perspectives in Individualized Learning.伊利诺伊州伊塔斯卡市：F. E. Peacock Publishers, Inc., 1971.
Here is a general reference book which describes how learning theories and teaching strategies can be used to promote individualized learning in students. This book contains chapters written by J. P. Guilford and Robert Gagné, and also has a chapter on Benjamin Bloom's cognitive and affective taxonomies.
这是一本介绍如何利用学习理论和教学策略促进学生个性化学习的通用参考书。本书包含 J. P. Guilford 和 Robert Gagné 撰写的章节，还有一章介绍本杰明-布鲁姆的认知和情感分类法。
这是一本介绍如何利用学习理论和教学策略促进学生个性化学习的通用参考书。本书包含 J. P. Guilford 和 Robert Gagné 撰写的章节，还有一章介绍本杰明-布鲁姆的认知和情感分类法。