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。
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。
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。
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。
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。
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。
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。
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。
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 then ). Adolescents are able to understand and apply complex concepts such as permutations and combinations, proportions, correlations, and probability; and they can conceive of the infinitely large and the infinitesimally small. 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 那么 ）。青少年能够理解和应用复杂的概念，如排列和组合、比例、相关性和概率；他们还 能想象无限大和无限小的事物。
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。
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。
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。
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。
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）。
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
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)。
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。
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。
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。
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, , from examples such as and . These children will not be able to handle several variables simultaneously, and complex relationships such as proportions and functions of several variables are inappropriate for many middle school children. Mathematical symbols and manipulations involve formal operations, and many students learn algebra by memorizing rules for combining and manipulating symbols with little understanding of the meaning of algebraic techniques. For example, , and are perfectly sensible statements for many algebra sludents. Even numerical counterexamples to illustrate the fallacy of these statements are not meaningful to students who are merely manipulating 's, and 's according to arbitrary rules. 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 和 等例子中得出加法的交换律 。这些孩子无法同时处理多个变量，而比例和多个变量的函数等复杂关系也不适合许多初中生。数学符号和运算涉及形式运算，许多学生在学习代数时只记住符号的组合和运算规则，而对代数技巧的含义却不甚了解。例如， 和 对许多代数学生来说是完全合理的语句。对于那些只是按照任意规则操作 和 的学生来说，即使用数字反例来说明这些语句的谬误也毫无意义。
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。
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。
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。
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 -when asked to give the sine of is using his or her memory. A child who can separate a mixed pile of squares and triangles into separate piles of squares and triangles is exercising a degree of cognition. When a member of a jury sits through a trial, deliberates in a closed session with other jury members, and concludes that the defendant is guilty as charged, that person has used his or her mental ability of evaluation. An lgebra student who finds the correct solution to a set of three linear equations in three unknowns has used his or her convergent production ability. A mathematician who discovers and proves a new and important mathematical theorem is exhibiting considerable ability in divergent production. 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 的正弦值时，他（她）会立即回答 ，这就是利用了他（她）的记忆力。一个孩子能把一堆混杂的正方形和三角形分开，变成一堆独立的正方形和三角形，这就是在进行一定程度的认知。当陪审团成员参加审判，与其他陪审团成员一起进行闭门商议，并得出被告有罪的结论时，这个人就使用了他或她的心理评估能力。一个代数学学生如果找到了三个未知数中三个线性方程组的正确解法，他或她就运用了自己的收敛生成能力。一个数学家发现并证明了一个新的重要数学定理，他（她）表现出了相当高的发散思维能力。
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. is a symbolic representation for a woman, and + is the mathematical symbol for the operation of addition Semantic contents of leaming are those words and ideas which evoke a mental image when they are presented as stimuli. Tree, dog, sun, war, fear, and red are words which evoke images in people's minds when they hear or read them. The behavioral contents of learning are the manifestations of stimuli and responses in people; that is, the way people behave as a consequence of their own desires and the actions of other people. The concrete shapes and forms (figures), the character representations (symbols), the spoken and written words (semantics), and the actions of people (behaviors) combine to make up the information that we discem in our environment. 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。 是一个女人的符号表示，+是加法运算的数学符号。 语言的语义内容是那些在作为刺激呈现时能唤起心理形象的词语和观念。树、狗、太阳、战争、恐惧和红色这些词在人们听到或读到时会在脑海中唤起形象。学习的行为内容是刺激和反应在人身上的表现，即人的行为方式是其自身欲望和他人行为的结果。具体的形状和形式（图形）、字符表征（符号）、口头和书面语言（语义）以及人们的行为（行为）共同构成了我们在环境中识别的信息。
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。
The distinct intellectual abilities defined in Guilford's Structure of Intellect Model result from taking all possible combinations of the five operations, four contents, and six products. For instance, one intellectual aptitude, memory for figural units, is the ability of a person to remember figural objects which he or she has seen. An example of this aptitude in mathematics is a student's ability to reproduce a geometric figure after he or she has been shown an example of that particular figure. The following list of operations, contents, and products indicates how the 120 intellectual aptitudes can be formed by combining any operation, with any content, with any product, to form an ordered triple: 吉尔福德的智力结构模型所定义的不同智力能力，是由五种操作、四种内容和六种产物的所有可能组合而成的。例如，一种智力能力--对形象单位的记忆，是指一个人记住他或她所见过的形象物体的能力。数学中这种能力的一个例子是，学生在看到一个几何图形的示例后，能够再现该几何图形。下面的运算、内容和乘积列表说明了 120 种智力能力如何通过将任何运算、任何内容和任 何乘积结合起来，形成一个有序的三元组：
Guilford's Factors of Intellectual Ability 吉尔福特的智力因素
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
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
5. transformation
6. implications
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。
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：
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。
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。
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
Robert Gagné's Theory of Learning 罗伯特-盖尼耶的学习理论
The research of the psychologist Robert . 'Gagné into the phases of a leaming sequence and the types of leaming is particularly relevent for teaching mathematics. Professor Gagné has used mathematics as a medium for testing and applying his theories about learming and has collaborated with the University of Maryland Mathematics Project in studies of mathematics learning and curriculum development. 心理学家罗伯特 .盖尼耶对学习序列的阶段和学习类型的研究对数学教学尤其有意义。盖尼耶教授将数学作为测试和应用其学习理论的媒介，并与马里兰大学数学项目合作开展数学学习和课程开发研究。
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。
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
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
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）
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 as a variable for the quadratic equation is demonstrating acquisition of a concept. And, a person who can derive (or prove) the quadratic formula and explain his derivation to someone else has mastered a principle. Consequently, the quadratic formula which is a principle may be regarded as either a fact, a skill, or a concept by a student whose viewpoint of the quadratic formula is not as sophisticated as that of a mathematician. 将 作为一元二次方程 的变量，是对概念的掌握。而能够推导（或证明）一元二次方程式并向他人解释其推导过程的人，就掌握了一个原理。因此，作为原理的二次方程式可以被学生视为事实、技能或概念，因为他们对二次方程式的理解还没有数学家那么深刻。
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。
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。
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。
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。
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。
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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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's desired 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。
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。
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。
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。
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。
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。
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 " is determined by " as a cue for the word function, another person may code function symbolically as " ," and someone else may visualize two sets of elements enclosed in circles with arrows extending from the elements of one set to the elements of the other set. Other codes can be taught. For instance, a commonly used memory code for the order in which arithmetic operations in algebra are carried out is " y Dear Aunt Sally" which is a code for "Multiply, Divide, Add, then Subtract." Research and observation suggest that an efficient method for memorizing long verbal passages such as poetry is to progressively learn each new part by rehearsing the previously leamed older parts up to the new part and then rehearse the new part. For instance, the fifth line of a poem may best be leamed by repeating the first four lines in sequence and then including line five. 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" 由 决定 "这一语言心理代码作为功能一词的提示，另一个人可能会将功能象征性地编码为" "，还有人可能会将两组元素围成圆圈，箭头从一组元素延伸到另一组元素。还可以教授其他代码。例如，代数中算术运算顺序的常用记忆代码是" y 亲爱的莎莉阿姨"，这是 "乘、除、加、再减 "的代码。研究和观察表明，记忆诗歌等长篇语言段落的有效方法是，通过排练以前学过的旧部分到新部分，然后再排练新部分，循序渐进地学习每一个新部分。例如，学习诗歌第五行的最佳方法是依次重复前四行，然后再学习第五行。
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。
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 . Through a simple stimulus-response chain the child leams to recoghize (not, in this case, the name " wo" for the concept of two), but the physical appearance of the numeral 2 . This is an example of single discrimination where the child can recognize the numeral 2 . At the same time the child may be learning to recognize the numerals , and 9 and to discriminate among them, which is an example of multiple discrimination. On Tuesday the child may work with only the numeral 6 and on Wednestay he or she may learn to discriminate a 9 . However, when all of the single digit numerals are presented together, the same child may have trouble discriminating between the 6 and the 9. If the child has previously leamed each of the chains making up each numeral to be learned, can identify each numeral by itself, can say the names of each numeral, and has appropriate mental codes for the names and numeral symbols, then he or she is ready to leam to discriminate among the numerals. 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 来练习辨认数字 2。通过一个简单的刺激-反应链，幼儿学会了识别（在这种情况下，不是识别 "2 "概念的名称" wo"），而是识别数字 2 的物理外观。这是一个单一辨别的例子，孩子可以认出数字 2。与此同时，孩子可能正在学习识别数字 和 9，并对它们进行区分，这是多重辨别的一个例子。星期二，孩子可能只学习数字 6；星期三，他或她可能学习辨别数字 9。但是，当所有的个位数字放在一起时，同一个孩子可能就难以区分 6 和 9 了。 如果孩子已经学会了组成待学数字的每一个数字链，能够辨认出每一个数字本身，能够说出每一个数字的名称，并对数字名称和数字符号有了适当的心理编码，那么他或她就可以学习辨别数字了。
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。
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。
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。
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。
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。
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。
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。
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。
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 and that ; however without knowing the rule that can be represented by , we would not be able to generalize beyond those few specific multiplication problems which we have already attempted. Most people first leam and use the rule that multiplication is commutative without being able to state it, and usually without realizing that they know and apply the rule. In order to discuss this rule, it must be given either a verbal or a symbolic formulation such as "the order in which multiplication is done doesn't make any difference in the answer" or "for all numbers and ." This particular rule, and rules in general, can be thought of as sets of relations among sets of concepts. 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 和 ；但是，如果不知道 所代表的规则，我们就无法将其推广到我们已经尝试过的几个特定乘法问题之外。大多数人最初学习和使用乘法交换律时，都无法说明这一规则，通常也没有意识到他们知道并应用了这一规则。要讨论这一规则，必须用语言或符号来表述，如 "乘法运算的顺序对答案没有任何影响 "或 "对于所有数字 和 "。这个特定的规则，以及一般的规则，都可以看作是概念集之间的关系集。
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 is a rule explaining how to treat the symbol ! Other rules are chains of connected concepts, such as the rule that in the absence of symbols of grouping arithmetic operations should be carried out in the ordered sequence . Still other mathematical rules provide sets of responses for sets of stimuli. The quadratic formula provides for an infinite set of responses, one response for each of an infinite set of quadratic equations. Each particular quadratic equation is a stimulus consisting of a concept chain, and each solution is a response made up of a chain of concepts. 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 是一条解释如何处理符号 的规则！其他规则是连接概念的链条，例如，在没有分组符号的情况下，算术运算应按 的有序顺序进行。还有一些数学规则为一组刺激提供了一组反应。一元二次方程式提供了无穷多的反应集，每个反应集对应一个无穷多的一元二次方程。每个特定的一元二次方程都是由概念链组成的刺激，而每个解法都是由概念链组成的反应。
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 , but without additional learning few people could apply it correctly. Most people use the rule that multiplication is commutative, but few people can state this rule as "multiplication is commutative" or . 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 的符号序列，但如果没有额外的学习，很少有人能正确应用它。大多数人都会使用 "乘法具有交换性 "这一规则，但很少有人能将这一规则表述为 "乘法具有交换性 "或 。
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：
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)
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。
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 . Such a student would have to select the skill of completing the square of a trinomial from his stock of skills and apply that skill in the proper way to develop the quadratic formula. A student who derives the quadratic formula by carrying out a set of instructions from his or her teacher is learning a rule. The criterion for problem-solving is that the student has not previously solved that particular problem, even though the problem may have been solved previously by many other people. 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 建立了一元二次方程式。这样的学生必须从他的技能库中选择完成三项式平方的技能，并以适当的方式应用该技能来推导二次公式。通过执行教师的一系列指令而推导出二次公式的学生是在学习一种规则。解决问题的标准是学生以前没有解决过该特定问题，即使该问题以前可能有很多人解 决过。
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。
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。
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]。
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。
PROBLEM TO BE SOLVED Derive the Quadratic Formula 待解决的问题 推导二次公式
Step 1. Write the general form of a quadratic equation 步骤 1.写出一元二次方程 的一般形式
Step 2. Add negalive to both sides of the equation. 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。
Step 4. Complete the square of by adding to both sides of the equation. 步骤 4.在等式两边加上 完成 的平方。
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。
Step 6. Take the square root of both sides of the equation. 步骤 6.取等式两边的平方根。
Step 7. Add to both sides and simplify the right side. 第 7 步在两边加上 并简化右边。
Flgure 3.1 .
Figure 3.2 is a leaming hierarchy, because boith superordinate and subordinate Robert Gagne's bilities are specified in their appropriate relationships to each other. Figure 3.2 can be thought of as a first approximation to the leaming hierarchy for solving a quadratic equation. A more careful consideration of prerequisite abilities and research with students might result in a more precise hierarchy for this problemsolving ability. However, the hierarchy shown in this figure, as well as other hierarchies, can easily be developed by mathematics teachers and can be helpful in deiermining sludent readiness for this and other problem-solving-activities 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。
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：
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)
Mathematic Concepts 数学 概念
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。
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 are all examples of the concept of even number; however each is a different way of representing a particular even number. 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 都是偶数概念的例子；但每种概念都是表示特定偶数的不同方式。
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。
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 implies , and are attempting to apply pure and notational concepts which they have not adequately leaned. 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 暗示