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Constructing mathematical examinations to assess a range of knowledge and skills 构建数学考试以评估一系列知识和技能
Geoff Smith ^(a){ }^{a}, Leigh Wood ^("a "){ }^{\text {a }}, Mary Coupland ^("a "){ }^{\text {a }}, Brian Stephenson ^("a "){ }^{\text {a }}, Kathryn Crawford ^("b "){ }^{\text {b }} & Geoff Ball ^("c "){ }^{\text {c }} 杰夫·史密斯 ^(a){ }^{a} ，利·伍德 ^("a "){ }^{\text {a }} ，玛丽·库普兰 ^("a "){ }^{\text {a }} ，布赖恩·史蒂芬森 ^("a "){ }^{\text {a }} ，凯瑟琳·克劳福德 ^("b "){ }^{\text {b }} 和杰夫·巴尔 ^("c "){ }^{\text {c }} ^(a){ }^{a} School of Mathematical Sciences, University of Technology, Sydney, P.O. Box 123, Broadway, NSW 2007, Australia 悉尼科技大学数学科学学院，邮政信箱 123，百老汇，纽南州 2007，澳大利亚 ^("b "){ }^{\text {b }} Faculty of Education, University of Sydney, Australia 悉尼大学教育学院，澳大利亚 ^("c "){ }^{\text {c }} School of Mathematics and Statistics, University of Sydney, Australia 悉尼大学数学与统计学院，澳大利亚
Version of record first published: 30 Jul 2010. 首次发布的版本记录：2010 年 7 月 30 日。
To cite this article: Geoff Smith , Leigh Wood , Mary Coupland , Brian Stephenson , Kathryn Crawford & Geoff Ball (1996): Constructing mathematical examinations to assess a range of knowledge and skills, International Journal of Mathematical Education in Science and Technology, 27:1, 65-77 引用本文：Geoff Smith, Leigh Wood, Mary Coupland, Brian Stephenson, Kathryn Crawford & Geoff Ball (1996)：构建数学考试以评估多种知识和技能，《国际科学与技术数学教育杂志》，27:1，65-77
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Constructing mathematical examinations to assess a range of knowledge and skills 构建数学考试以评估一系列知识和技能
by GEOFF SMITH, LEIGH WOOD, MARY COUPLAND, BRIAN STEPHENSON 由杰夫·史密斯、利·伍德、玛丽·库普兰、布赖恩·斯蒂芬森撰写School of Mathematical Sciences, University of Technology, Sydney, P.O. Box 123, Broadway, NSW 2007, Australia 悉尼科技大学数学科学学院，邮政信箱 123，百老汇，新南威尔士州 2007，澳大利亚KATHRYN CRAWFORD 凯瑟琳·克劳福德Faculty of Education, University of Sydney, Australia 悉尼大学教育学院，澳大利亚and GEOFF BALL 和杰夫·鲍尔School of Mathematics and Statistics, University of Sydney, Australia 悉尼大学数学与统计学院，澳大利亚
(Received and reviewed by an AEBA E B member) （由 AEBA E B 成员接收和审核）
Abstract 摘要
In this paper, we describe aspects of a programme to enhance student learning in undergraduate mathematics. We present ways of constructing formal examinations which assess a range of knowledge and skills and which encourage students to reflect on their learning. To assist with this process, we propose a taxonomy to classify assessment tasks ordered by the nature of the activity required to complete each task successfully, rather than in terms of difficulty. An extensive list of university level examples is given to illustrate descriptors in the taxonomy and to provide ideas for those interested in implementing an alternative approach to writing examination questions. 在本文中，我们描述了一个旨在增强本科数学学生学习的项目的各个方面。我们提出了构建正式考试的方法，这些考试评估一系列知识和技能，并鼓励学生反思他们的学习。为了帮助这一过程，我们提出了一种分类法，以根据完成每项任务所需的活动性质对评估任务进行分类，而不是根据难度进行分类。我们提供了一个广泛的大学水平示例列表，以说明分类法中的描述符，并为那些有兴趣实施替代考试问题编写方法的人提供思路。
1. Introduction 1. 引言
This paper describes aspects of a programme to improve student learning in undergraduate mathematics. We demonstrate ways of constructing formal examinations which assess a range of knowledge and skills and which encourage students to reflect on their learning. Analyses of students’ responses, marking methods and other types of assessment will be published later. We have compiled a list of questions which illustrate alternative approaches to examination questions. We have found it useful to classify these questions in a taxonomy, but other readers may not wish to follow this approach. These readers can simply treat this paper as a list of novel and, hopefully, useful approaches to examination questions. 本文描述了一个旨在提高本科数学学生学习效果的项目的各个方面。我们展示了构建正式考试的方法，这些考试评估一系列知识和技能，并鼓励学生反思他们的学习。关于学生反应、评分方法和其他类型评估的分析将稍后发布。我们编制了一份问题清单，展示了对考试问题的替代方法。我们发现将这些问题分类为一个分类法是有用的，但其他读者可能不希望采用这种方法。这些读者可以简单地将本文视为对考试问题的新颖且希望有用的方法的列表。
Logistics and tradition have meant that assessment in mathematics has relied heavily on formal examinations. There are significant difficulties in changing this in early undergraduate years and little educational reason to do so. However, examinations often test a narrow range of skills and we have been investigating ways to broaden the skills tested. In addition, textbooks are virtual clones of each other, with long sets of repetitive exercises. These factors all encourage a surface approach to learning. To quote from Ramsden [1] 物流和传统意味着数学评估在很大程度上依赖于正式考试。在早期本科阶段改变这一点存在重大困难，并且没有太多教育理由去这样做。然而，考试往往测试的技能范围较窄，我们一直在研究扩展测试技能的方法。此外，教科书几乎是彼此的克隆，包含大量重复的练习。这些因素都鼓励了表面学习的方法。引用 Ramsden 的话[1]
It has become clear from numerous investigations that: 从众多调查中已经明确：
Many students are accomplished at complex routine skills is science, mathematics and humanities, including problem solving algoritk ms. 许多学生在科学、数学和人文学科的复杂常规技能方面表现出色，包括问题解决算法。
Many have appropriated enormous amounts of detailed ku owledge, including knowledge-specific terminology. 许多人已经获取了大量详细的知识，包括特定领域的术语。
Many are able to reproduce large quantities of factual inforr ation on demand. 许多人能够根据需要再现大量的事实信息。
Many are able to pass examinations. 许多人能够通过考试。
But many are unable to show that they understand what they hav: learned, when asked simple yet searching questions that test their gra ip of the content. 但许多人在被问到简单却深刻的问题时，无法表明他们理解所学的内容。
In summary, the research seems to indicate that, at least for a sho tt period, students retain vast quantities of information. On the other hand, mm any seem to forget much of it and do not appear to make good use of wha; they do remember [1[1, pp. 30-31]30-31]. 总之，研究似乎表明，至少在短期内，学生能够保留大量信息。另一方面，似乎许多人会忘记其中的很多内容，并且似乎没有很好地利用他们所记得的内容。
This is not new. Over the years, teachers and lecturers have recognized tt lese facts and have attempted to influence student learning by a number of method; such as changing the style of their teaching, by attempting to give clearer explan tions, by giving more examples, by preparing better lecture notes and so on, all on the assumption that mathematics need only be presented logically in order to b : learned. However, research has shown that students are often more motivatec to learn material or methods that are of direct relevance to passing, and will ac apt their learning styles and do what they perceive is necessary to pass assessment ts sks. This means that changing teaching methods without due attention to assessmen methods is not sufficient [1]. 这并不是新鲜事。多年来，教师和讲师们已经认识到这些事实，并试图通过多种方法影响学生的学习；例如，改变教学风格，尝试提供更清晰的解释，提供更多的例子，准备更好的讲义等等，所有这些都是基于数学只需以逻辑方式呈现即可学习的假设。然而，研究表明，学生往往更有动力学习与通过考试直接相关的材料或方法，并会调整他们的学习风格，做他们认为必要的事情以通过评估任务。这意味着，在没有充分关注评估方法的情况下改变教学方法是不够的。
Our assumptions are: 我们的假设是：
(i) In the interests of higher quality in education, a deep approach tc learning mathematics is to be valued over a surface approach, and student: entering university with a surface approach should be given every encourą ement to progress to a deep approach. It has been shown [2] that students abl : to adopt a deep approach to study tended to achieve at a higher level after a year of university study. 为了提高教育质量，深入学习数学的方法应被重视，而表面学习的方法则应受到限制。进入大学的学生如果采用表面学习方法，应给予充分的鼓励以转向深入学习。研究表明，能够采用深入学习方法的学生在大学学习一年后往往能取得更高的成绩。
(ii) The adoption of a surface approach to learning is more widesp: ead than might be expected in university students [2] and is a learned respon: e, gained from previous experience in school [3-5] and often reinforced by a niversity practices. (ii) 对学习采取表面学习方法在大学生中比预期的更为普遍[2]，这是一种通过在学校的以往经验获得的学习反应[3-5]，并且常常受到大学实践的强化。
(iii) Students are capable of changing their approach to learning, from a surface approach towards a deeper approach, and will do so if they see it as lecessary in order to succeed. 学生能够改变他们的学习方式，从表层学习转向深层学习，如果他们认为这样做是成功所必需的，他们会这样做。
(iv) Assessment drives what students learn. It controls their approach to learning by directing them to take either a surface approach or a deep ap roach to learning [1]. The types of questions that we set show students what we value and how we expect them to direct their time. Good questions are thi se which help to build concepts, alert students to misconceptions and introduce applications and theoretical ideas. （四）评估驱动学生的学习。它通过引导学生采取表层学习或深层学习的方式来控制他们的学习方法[1]。我们设定的问题类型向学生展示了我们重视什么，以及我们期望他们如何分配时间。好的问题是那些有助于构建概念、提醒学生误解并引入应用和理论思想的问题。
(v) It is possible to use a taxonomy to classify a set of tasks ordered by t e nature of the activity required to complete each task successfully, rathe rr than in 可以使用分类法对一组任务进行分类，这些任务按完成每项任务所需活动的性质进行排序，而不是按其他方式
Justifying and
interpreting| Justifying and |
| :---: |
| interpreting |
Comprehension 理解
在新情况下的应用
Application in new
situations
Application in new
situations| Application in new |
| :---: |
| situations |
含义、推测和比较
Implications,
conjectures and
comparisons
Implications,
conjectures and
comparisons| Implications, |
| :---: |
| conjectures and |
| comparisons |
常规使用程序
Routine use of
procedures
Routine use of
procedures| Routine use of |
| :---: |
| procedures |
Evaluation 评估
Group A Group B Group C
"Factual
knowledge" Information transfer "Justifying and
interpreting"
Comprehension "Application in new
situations" "Implications,
conjectures and
comparisons"
"Routine use of
procedures" Evaluation
| Group A | Group B | Group C |
| :---: | :---: | :---: |
| Factual <br> knowledge | Information transfer | Justifying and <br> interpreting |
| Comprehension | Application in new <br> situations | Implications, <br> conjectures and <br> comparisons |
| Routine use of <br> procedures | | Evaluation |
| | | |
Figure 1. 图 1。
terms of difficulty. Activities which need only a surface approach appear at one end, while those requiring a deeper approach appear at the other end. 在难度方面。只需要表面处理的活动出现在一端，而那些需要更深入处理的活动则出现在另一端。
There are several taxonomies that one could use, depending on the purpose. One of the best known is Bloom’s taxonomy [6], which gives a hierarchy of concepts. 有几种分类法可以使用，具体取决于目的。其中最著名的是布鲁姆分类法[6]，它提供了一个概念的层次结构。
Bloom’s taxonomy is quite good for structuring assessment tasks, but does have some limitations in the mathematical context. We propose a modification of Bloom’s taxonomy, the MATH taxonomy (mathematical assessment task hierarchy) for the structuring of assessment tasks. The categories in the taxonomy are summarized in Figure 1. A detailed list of descriptors is given in section 2. 布鲁姆的分类法在构建评估任务方面相当有效，但在数学背景下确实存在一些局限性。我们提出了一种布鲁姆分类法的修改版，即数学分类法（MATH 分类法），用于构建评估任务。该分类法中的类别在图 1 中进行了总结。第 2 节提供了详细的描述符列表。
We have found it helpful to use a grid (Figure 2) which combines subject topics with the descriptors of the taxonomy. The grid entries are references to particular questions on the examination paper. This enables us to more readily determine the balance of assessment tasks on the examination paper. Most of the mathematics examination papers we have analysed are heavily biased towards group A tasks. 我们发现使用一个网格（图 2）将主题与分类法的描述符结合起来是很有帮助的。网格条目是对考试试卷中特定问题的引用。这使我们能够更容易地确定考试试卷上评估任务的平衡。我们分析的大多数数学考试试卷严重偏向于 A 组任务。
Students enter tertiary institutions with most of their mathematical learning experience in group A tasks, with some experience with group B tasks [2]. Their experience in group C tasks in mathematics is severely limited or non-existent. One of the aims of tertiary education in mathematics should be to develop skills at all three levels. 学生进入高等院校时，大部分数学学习经验来自 A 组任务，部分经验来自 B 组任务[2]。他们在 C 组任务中的数学经验非常有限或不存在。高等教育在数学方面的目标之一应该是培养所有三个层次的技能。
It is easy to have mistaken assumptions about the skills that a parti ular task assesses. If a student is asked to prove a theorem given in lectures, a corre it answer may be given by a student who understands the theorem and its significanc e and can apply it in relevant situations or prove similar theorems. However, the mr st we can assume is that the student can reproduce the theorem on demand; thi; style of assessment cannot discriminate between different types of learning whicl can lead to the same response. If we are content with this, then the question is sat sfactory, but if we wish to be sure that the student understands the theorem and has nn st merely learned it by rote, then we need to ask more probing questions. It is esser tial to be clear about the desired outcomes of our assessment and to be able to identify the types of assessment tasks which are reliable indicators of these outcomes. 关于某个任务评估的技能，容易产生错误的假设。如果要求学生证明课堂上给出的定理，能够给出正确答案的学生可能是理解该定理及其重要性，并能在相关情况下应用它或证明类似定理的学生。然而，我们最多只能假设学生能够按要求复述该定理；这种评估方式无法区分不同类型的学习，而这些学习可能导致相同的反应。如果我们对此感到满意，那么这个问题是令人满意的，但如果我们希望确保学生理解该定理，而不仅仅是死记硬背，那么我们需要提出更深入的问题。明确我们评估的期望结果，并能够识别出哪些评估任务是这些结果的可靠指标，这是至关重要的。
The list of descriptors given below attempts to force all types of mat iematical examination assessment into one of eight categories. There will cer ainly be borderline cases or cases which do not fit comfortably in any category or ca es which fit into more than one category. However, it is not our aim to be able to uniquely characterize every conceivable assessment task. Rather, the aim of the di scriptors is to assist with writing examination questions, and to allow the e: aminer’s judgement, objectives and experience to determine the final evaluati in of an assessment task. 下面给出的描述符列表试图将所有类型的数学考试评估强制归入八个类别之一。肯定会有边界案例或不适合任何类别的案例，或者适合多个类别的案例。然而，我们的目标并不是能够唯一地描述每一个可以想象的评估任务。相反，描述符的目的是帮助编写考试问题，并让考官的判断、目标和经验来决定评估任务的最终评价。
We have also (section 3) given a list of examples to illustrate these de criptors. Some of these depend on the prior knowledge of the student. A student who succeeds in proving an unseen theorem is demonstrating an ability to apply knowled re to new situations, but may only be demonstrating factual recall when proving it for a second time. 我们还在（第 3 节）提供了一些示例来说明这些描述符。其中一些依赖于学生的先前知识。成功证明一个未见定理的学生展示了将知识应用于新情境的能力，但在第二次证明时可能仅仅是在展示事实回忆。
There is a loose hierarchy in this list of descriptors; finding the area ( f a circle requires factual knowledge (the formula pir^(2)\pi r^{2} ), comprehension (knowing v hat each of the three symbols represent, knowing that multiplication and exponent ation are implied) and the use of a routine procedure (evaluation after substitution) Finding the area without previous knowledge of the method (as was the case for th >=\geq ancient Greeks) would require other skills. The assumptions being made in each group of questions are given. 在这个描述符列表中存在一个松散的层次结构；计算圆的面积需要事实知识（公式 pir^(2)\pi r^{2} ）、理解（知道三个符号分别代表什么，知道乘法和指数运算是隐含的）以及使用常规程序（替换后的评估）。在没有先前方法知识的情况下计算面积（就像古希腊人 >=\geq 的情况一样）将需要其他技能。每组问题中所做的假设已给出。
It is important to understand, however, that no hierarchy of difficulty i implied as we move down the list. It is the nature of the activity we are interested it, not the degree of difficulty, which in any case is a very subjective and elusive concer tt. (Some definitions: Easy-I can do it. Difficult-I can’t do it.) 然而，重要的是要理解，随着我们向下移动列表，并没有暗示任何难度的等级。我们感兴趣的是活动的性质，而不是难度的程度，后者在任何情况下都是一个非常主观和难以捉摸的问题。 tt （一些定义：简单-我能做到。困难-我做不到。）
2.1. Group AA 2.1. 组 AA
Factual knowledge and fact systems. The difficulty and depth of the mat :rial may cover a wide range from remembering a specific formula or definition (factual knowledge) [Examples 1, 2 and 3 in section 3] to learning a complex theort mm (a fact system) [Examples 4 and 5 in section 3], but the only skill required is to bring to mind previously learned information in the form that it was given. 事实知识和事实系统。材料的难度和深度可能涵盖从记住特定公式或定义（事实知识）[第 3 节中的示例 1、2 和 3]到学习复杂理论（事实系统）[第 3 节中的示例 4 和 5]的广泛范围，但所需的唯一技能是以所给的形式回忆起先前学习的信息。
Comprehension. It is quite possible to reproduce knowledge withou: understanding. To demonstrate comprehension of factual knowledge, student: should: 理解。完全有可能在没有理解的情况下再现知识。为了展示对事实知识的理解，学生应该：
be able to decide whether or not conditions of a simple definition are satisfied [Examples 6-9]. By a simple definition, we mean one which is a r tatter of terminology, making use of previously acquired knowledge or skil s, e.g. aa 能够决定简单定义的条件是否满足[示例 6-9]。简单定义是指一种术语问题，利用先前获得的知识或技能，例如 aa 。
linear first-order differential equation is one which satisfies… . The student has merely learned a new term, but not one which requires a significant conceptual change in their mathematical understanding; 线性一阶微分方程是满足……的方程。学生仅仅学到了一个新术语，但这并不需要他们在数学理解上进行重大概念上的改变；
understand the significance of the symbols in a formula (both implicit and explicit) and show an ability to substitute in a formula; 理解公式中符号的意义（包括隐含和显含）并展示在公式中替换的能力；
be able to recognize examples and counterexamples [Example 10]. 能够识别例子和反例 [例子 10]。
Routine procedures. This requires the ability to use material in a way which goes beyond simple factual recall. The essential feature is that when the procedure or algorithm is properly used, all people solve the problem correctly and in the same way. This does not preclude the possibility that there may be more than one routine procedure applicable to a given problem. Examples include pattern recognition, such as evaluating a formula after substitution, or solving a differential equation [Examples 11 and 12]. Students would have been expected to have worked on problems using these procedures in drill exercises. In some cases, there may be several distinct processes underlying a particular procedure and although students may be able to state the general procedure to be followed and understand its principles, they may be unable to carry out the detail. As an example, a student may know that the area under a curve can be obtained by integration and may be able to set up the integral correctly, but be unable to do all but the simplest integrations. 常规程序。这需要能够以超越简单事实回忆的方式使用材料。其基本特征是，当程序或算法被正确使用时，所有人都能以相同的方式正确解决问题。这并不排除对特定问题可能适用多种常规程序的可能性。例子包括模式识别，例如在替换后评估公式，或求解微分方程[例子 11 和 12]。学生们应该在训练练习中使用这些程序解决问题。在某些情况下，特定程序可能有几个不同的过程，尽管学生可能能够陈述要遵循的一般程序并理解其原理，但他们可能无法执行细节。例如，学生可能知道曲线下的面积可以通过积分获得，并且能够正确设置积分，但无法完成除最简单的积分以外的所有积分。
2.2. Group BB 2.2. 组 BB
Information transfer. This may be shown by the ability to perform the following tasks: 信息传递。这可以通过执行以下任务的能力来体现：
transformation of information from one form to another-verbal to numerical or vice versa [Example 13]; 信息从一种形式转化为另一种形式——口头到数字或反之 [示例 13];
deciding whether or not conditions of a conceptual definition are satisfied. A conceptual definition is one whose understanding requires a significant change in the student’s mode of thought or mathematical knowledge, for example, the definition of a limit or of linear independence. Deciding whether or not a definition is simple or conceptual will often be a subjective judgement, but, given our aims, we do not regard this as important; 决定概念定义的条件是否得到满足。概念定义是指理解它需要学生在思维方式或数学知识上发生重大变化的定义，例如极限或线性独立的定义。判断一个定义是简单的还是概念性的通常是主观判断，但鉴于我们的目标，我们并不认为这很重要；
recognizing the applicability of a formula or method in different or unusual contexts [Examples 14 and 15]; 识别公式或方法在不同或不寻常的上下文中的适用性 [示例 14 和 15];
recognizing the inapplicability of a generic formula in particular contexts [Example 16]; 认识到通用公式在特定上下文中的不适用性 [示例 16];
summarizing in non-technical terms for a different audience, or paraphrasing; 用非技术性术语为不同的受众进行总结或改述；
framing a mathematical argument from a verbal outline of the method; 从方法的口头大纲构建数学论证；
explaining the relationships between component parts of the material; 解释材料各组成部分之间的关系；
explaining processes; 解释过程；
reassembling the [given] component parts of an argument in their logical order [Example 17]. 重新按照逻辑顺序重新组合论证的[给定]组成部分[示例 17]。
Application in new situations. Ability to choose and apply appropriate methods or information in new situations, including the following: 在新情况下的应用。能够在新情况下选择和应用适当的方法或信息，包括以下内容：
modelling real life settings; 建模现实生活场景；
proving a previously unseen theorem or result which goes beyond the routine use of procedures [Examples 18 and 20]; 证明一个之前未见的定理或结果，超越常规程序的使用 [示例 18 和 20];
extrapolation of known procedures to new situations [Example 19]; 已知程序对新情况的推断 [示例 19];
choosing and applying appropriate statistical techniques; 选择和应用适当的统计技术；
choosing and applying appropriate algorithms. 选择和应用适当的算法。
2.3. Group CC 2.3. 组 CC
fustifying and interpreting. Ability to justify and/or interpret a giver result or a result derived by the student. This includes: 证明和解释。能够证明和/或解释给定的结果或学生得出的结果。这包括：
proving a theorem in order to justify a result, method or model; 证明一个定理以证明一个结果、方法或模型；
the ability to find errors in reasoning [Examples 21 and 22]; 发现推理错误的能力 [示例 21 和 22];
recognizing the limitations in a model and being able to decide if epsi\varepsilon model is appropriate [Example 23]; 识别模型的局限性并能够决定 epsi\varepsilon 模型是否合适[示例 23];
recognition of computational limitations and sources of error; 计算限制和误差来源的识别；
interpreting a regression model; 解释回归模型；
discussing the significance of given examples and counterexample: ; 讨论给定例子和反例的重要性；
recognition of unstated assumptions. 未表述假设的识别。
Implications, conjectures, and comparisons. Given or having found a r epsilon\epsilon sult/situation, the student has the ability to draw implications and make conjecturı s and the ability to justify or prove these. The student also has the ability to make corr parisons, with justification, in various mathematical contexts. Examples are: 含义、推测和比较。给定或发现一个结果/情况，学生能够推导出含义并进行推测，并能够证明或证明这些。学生还能够在各种数学背景下进行比较，并提供理由。示例包括：
the ability to make conjectures based, for example, on inductive or heuristic arguments, and then to prove these conjectures by rigorous methods [Examples 24 and 25]; 基于例如归纳或启发式论证进行推测的能力，然后通过严格的方法证明这些推测 [示例 24 和 25];
comparisons between algorithms [Example 27]; 算法之间的比较 [示例 27];
the ability to deduce the implications of a given result; 推断给定结果的含义的能力；
the construction of examples and counterexamples. 例子和反例的构建。
Evaluation. Evaluation is concerned with the ability to judge the value of material for a given purpose based on definite criteria. The students may be given the criteria or may have to determine them. This includes the following 评估。评估涉及根据明确的标准判断材料在特定目的下的价值的能力。学生可能会被提供标准，或者需要自己确定标准。这包括以下内容：
the ability to make judgements [Examples 29 and 30]; 做出判断的能力 [示例 29 和 30];
the ability to select for relevance [Example 29]; 选择相关性的能力 [示例 29];
the ability to coherently argue the merits of an algorithm [Examp] 31]; 能够连贯地论证算法的优点 [Examp] 31];
organizational skills; 组织技能；
creativity, which includes going beyond what is given, restruct ring the information into a new whole and seeing implications of the informat: on which is not apparent to others. 创造力，包括超越已有的事物，将信息重组为一个新的整体，并看到信息的含义，这些含义对他人并不明显。
3. Examples 3. 示例
A series of sample questions which illustrates the above list of descriptos ss is given below. We assume that the questions would be asked of students currently studying the topic under consideration, since, in all these examples, the actual cate gory into which the question fits could depend on the prior experience of the stud nt; what may be factual knowledge to a graduate mathematician could be incompr hensible to a clever high school student. There can also be various levels of diff culty for particular skills. It is more difficult, for example, to memorize a proof of a theorem than to memorize a simple definition. If the exact question (or very clost pattern) has been seen before, then the student can answer questions entirely by 1 ote. It is up to us to be clever and creative enought to design questions that discou age rote 一系列示例问题如下，说明了上述描述列表 ss 。我们假设这些问题将问及当前正在学习相关主题的学生，因为在所有这些例子中，问题适合的实际类别可能取决于学生的先前经验；对一位研究生数学家来说，可能是事实知识的内容对一位聪明的高中生来说可能是难以理解的。特定技能的难度也可能有不同的层次。例如，记忆定理的证明比记忆简单的定义要困难。如果确切的问题（或非常相似的模式）之前见过，那么学生可以完全通过死记硬背来回答问题。我们需要足够聪明和有创意，设计出能够阻止死记硬背的问题。
learning and that test the skills we wish to assess. You will know best what your students’ prior experience has been, so it is your classification that matters. 学习以及测试我们希望评估的技能。您最了解学生的先前经验，因此您的分类才是重要的。
With minor exceptions, the examples are taken from a pure mathematics core which would be common knowledge in areas such as statistics, operations research and applied mathematics. People from these areas should have no difficulty in understanding our ideas and adapting them to their own disciplines. 除少数例外，例子均来自于纯数学核心，这在统计学、运筹学和应用数学等领域是常识。这些领域的人士应该能够轻松理解我们的想法并将其应用到自己的学科中。
3.1. Factual knowledge and fact systems 3.1. 事实知识和事实系统
The assumption in these questions is that students have met the material in the form required for the answer. 这些问题的假设是学生已经以回答所需的形式接触过相关材料。
Factual knowledge 事实知识
Example 1 - What is the formula for the area of a circle? 例子 1 - 圆的面积公式是什么？
Example 2 - State Cramer’s rule for solving a system of equations. 例子 2 - 陈述克拉默法则以求解方程组。
Example 3 - What is meant by the term linear differential equation? 例子 3 - 线性微分方程这个术语是什么意思？
Fact systems 事实系统
Example 4 - (a) State the comparison test for series of non-negative terms. 例 4 - (a) 说明非负项级数的比较测试。
(b) State and prove the ratio test for the convergence of a series of positive terms. (b) 陈述并证明正项级数的比值测试收敛性。
Example 5 - State and prove the Hahn-Banach theorem. 例 5 - 陈述并证明汉-巴拿赫定理。
3.2. Comprehension of factual knowledge 3.2. 事实知识的理解
We assume that various routine skills needed to do the problems (such as partial differentiation in the second example) are familiar to the students, and that they have done drill work in similar but not identical problems. 我们假设学生对解决问题所需的各种常规技能（例如第二个例子中的偏微分）是熟悉的，并且他们已经在类似但不完全相同的问题上进行了练习。
Example 6 - Decide, given reasons, whether or not the following differential equation is linear 例 6 - 给出理由，判断以下微分方程是否是线性的
xy^(')+y=exp(x)x y^{\prime}+y=\exp (x)
Example 7 - Show that x^(3)-3xy^(2)x^{3}-3 x y^{2} is a harmonic function. 例子 7 - 证明 x^(3)-3xy^(2)x^{3}-3 x y^{2} 是一个调和函数。
Example 8 - Answer True or False [7]. 示例 8 - 答案是对还是错 [7]。
[ ] All continuous functions are differentiable 所有连续函数都是可微的
[ ] Some continuous functions are not differentiable 一些连续函数是不可微的
[ ] All differentiable functions are continuous 所有可微函数都是连续的
Example 9 - Indicate whether the following statements are true or false. 示例 9 - 指出以下陈述是正确还是错误。
[ ] A function may be differentiable at z_(0)z_{0}, but not analytic at z_(0)z_{0}. 一个函数在 z_(0)z_{0} 处可能是可微的，但在 z_(0)z_{0} 处却不是解析的。
[ ] A function may be differentiable at z_(0)z_{0}, and also analytic at z_(0)z_{0}. 一个函数在 z_(0)z_{0} 处可能是可微的，并且在 z_(0)z_{0} 处也是解析的。
[ ] A function may be analytic at z_(0)z_{0}, but not differentiable at z_(0)z_{0}. 一个函数在 z_(0)z_{0} 处可能是解析的，但在 z_(0)z_{0} 处可能不可微。
[ ] A function may be analytic everywhere in the complex plane. 一个函数在复平面上可能处处解析。
[ ] A function may be analytic nowhere in the complex plane. 一个函数在复平面上可能处处不解析。
Trials of questions such as examples 8 and 9 above show that students who can correctly quote the relevant definitions may nevertheless be unable to answer these questions correctly. This shows that the ability to quote a definition may be a meaningless skill. 如上面的例子 8 和 9 所示的试验表明，能够正确引用相关定义的学生仍然可能无法正确回答这些问题。这表明，引用定义的能力可能是一种无意义的技能。
given that Q(tau)=Q_(0)//2Q(\tau)=Q_{0} / 2, where tau=5568\tau=5568. 鉴于 Q(tau)=Q_(0)//2Q(\tau)=Q_{0} / 2 ，其中 tau=5568\tau=5568 。
Example 12∙12 \bullet Let CC be the circle |z-1|=1|z-1|=1. Evaluate the integral 例子 12∙12 \bullet 让 CC 为圆 |z-1|=1|z-1|=1 。计算积分
Any capable high school calculus student would have the technical skills, nat rely the theory of maxima and minima, to solve the next two problems, but in far tt many university students have difficulty with them. 任何有能力的高中微积分学生都具备解决接下来的两个问题所需的技术技能，而不依赖于极大值和极小值的理论，但许多大学生对此却感到困难。
Example 14 - Find all the real roots of the equation 3x^(4)+4x^(3)-12x^(2)+31=03 x^{4}+4 x^{3}-12 x^{2}+31=0, or explain why no solutions exist. 例 14 - 找出方程 3x^(4)+4x^(3)-12x^(2)+31=03 x^{4}+4 x^{3}-12 x^{2}+31=0 的所有实根，或解释为什么不存在解。
Example 15 - Show that x^(3)+cx+d=0x^{3}+c x+d=0 has only one real root if c >= 0c \geq 0. 例 15 - 如果 c >= 0c \geq 0 ，则证明 x^(3)+cx+d=0x^{3}+c x+d=0 只有一个实根。
Example 16∙16 \bullet A function is defined by 示例 16∙16 \bullet 函数由以下定义
f(x)={[sin x,x!=0],[1,x=0]:}f(x)=\left\{\begin{array}{rl}
\sin x & x \neq 0 \\
1 & x=0
\end{array}\right.
Find f^(')(x)f^{\prime}(x). 找到 f^(')(x)f^{\prime}(x) 。
This last question comes from a study by Harel and Kaput [8, p. 85]. A c ımmon response was: 这个最后的问题来自 Harel 和 Kaput 的一项研究[8, 第 85 页]。一个常见的回应是：
f^(')(x)={[cos x,x!=0],[0,x=0]:}f^{\prime}(x)=\left\{\begin{array}{rr}
\cos x & x \neq 0 \\
0 & x=0
\end{array}\right.
This shows that the students were blithely applying an algorithm to each point rather than considering the real meaning of differentiation, which is not a pointwise concept. It is a good question because it addresses vividly a common student misconception. The following example appears in [9]. 这表明学生们在对每个点轻松地应用算法，而不是考虑微分的真实含义，这并不是一个逐点的概念。这是一个好问题，因为它生动地揭示了一个常见的学生误解。以下示例出现在[9]中。
Example 17∙17 \bullet For any finite set SS, a field FF of subsets of SS and a real-valued function PP on FF, a probability space is defined by the following three axioms: 例子 17∙17 \bullet 对于任何有限集合 SS ，一个子集的域 FF 以及一个在 FF 上的实值函数 PP ，一个概率空间由以下三个公理定义：
(i) P(A) >= 0P(A) \geq 0 for all A in FA \in F (i) P(A) >= 0P(A) \geq 0 对于所有 A in FA \in F
(ii) P(S)=1P(S)=1
(iii) P(uuu_(i=1)^(n)A_(i))=sum_(i=1)^(n)P(A_(i))P\left(\bigcup_{i=1}^{n} A_{i}\right)=\sum_{i=1}^{n} P\left(A_{i}\right) for all A_(i)in FA_{i} \in F with A_(i)nnA_(j)=O/quad i!=jA_{i} \cap A_{j}=\varnothing \quad i \neq j (iii) P(uuu_(i=1)^(n)A_(i))=sum_(i=1)^(n)P(A_(i))P\left(\bigcup_{i=1}^{n} A_{i}\right)=\sum_{i=1}^{n} P\left(A_{i}\right) 对于所有 A_(i)in FA_{i} \in F 具有 A_(i)nnA_(j)=O/quad i!=jA_{i} \cap A_{j}=\varnothing \quad i \neq j
Our assumption in this section is that the students have not met any of the results they are asked to prove. The first example was an examination question given to students who had not previously encountered Liouville’s theorem or Cauchy’s inequalities. 我们在这一部分的假设是学生们没有接触过他们被要求证明的任何结果。第一个例子是一个考试题，给那些之前没有接触过李乌维尔定理或柯西不等式的学生。
Example 18∙18 \bullet Suppose f is analytic within and on the circle |z-z_(0)|=R\left|z-z_{0}\right|=R, denoted by CC, and let M_(R)M_{R} denote the maximum value of |f(z)||f(z)| on CC. Use Cauchy’s integral formula to derive Cauchy’s inequalities: 例子 18∙18 \bullet 假设 f 在圆 |z-z_(0)|=R\left|z-z_{0}\right|=R 内部及其上是解析的，记作 CC ，并且让 M_(R)M_{R} 表示 |f(z)||f(z)| 在 CC 上的最大值。使用柯西积分公式推导柯西不等式：
The above result can be used in the proof of the following important theorem: 上述结果可用于证明以下重要定理：
If a function ff is entire and bounded in the complex plane, then f(z)f(z) is constant throughout the plane (Liouville’s theorem). Prove this result by the following method: 如果一个函数 ff 在复平面上是全纯且有界的，那么 f(z)f(z) 在整个平面上是常数（李乌维尔定理）。通过以下方法证明这一结果：
Step 1 Use the fact that ff is entire to show that inequality ( {:^(**))\left.{ }^{*}\right) above with n=1n=1 holds for any choice of z_(0)z_{0} and RR. 步骤 1 利用 ff 是整个函数的事实，证明不等式 ( {:^(**))\left.{ }^{*}\right) 上的 n=1n=1 对于任何选择的 z_(0)z_{0} 和 RR 都成立。
Step 2 Use the fact that ff is bounded to show that there is a constant MM such that 步骤 2 利用 ff 是有界的事实，证明存在一个常数 MM 使得
where z_(0)z_{0} is any fixed point in the plane and RR is arbitrarily large. 在平面中， z_(0)z_{0} 是任何固定点，而 RR 是任意大的。
Step 3 Show that the inequality in step 2 can only hold if f^(')(z_(0))=0f^{\prime}\left(z_{0}\right)=0 and hence show that ff is a constant. unction. 步骤 3：证明步骤 2 中的不等式只有在 f^(')(z_(0))=0f^{\prime}\left(z_{0}\right)=0 时才成立，因此证明 ff 是一个常数。
The next example assumes students have only met linear differential equ tions in their studies. 下一个例子假设学生在学习中只接触过线性微分方程。
Example 19 - Solve the following two equations by showing that the :ndicated substitution transforms the equation to one which is linear in xx and v. 示例 19 - 通过显示所示的替换将以下两个方程求解为线性方程，变量为 xx 和 v。
y^(')-5y=-(5)/(2)xy^(3)quad v=y^(-2)y^{\prime}-5 y=-\frac{5}{2} x y^{3} \quad v=y^{-2}
Generalize the method in 1 and 2 above to solve 将上述 1 和 2 中的方法推广以解决
y^(')+P(x)y=Q(x)y^(n)y^{\prime}+P(x) y=Q(x) y^{n}
The next example could be given in a class test or as a tutorial problem to itudents early in a complex variables course before they had met the Cauchy-]iemann equations. 下一个例子可以在复杂变量课程的课堂测试中或作为教程问题提供给学生，早于他们接触柯西-黎曼方程之前。
Example 20 - Use the method outlined below to show that if the function ff defined by f(z)=u(x,y)+iv(x,y)f(z)=u(x, y)+i v(x, y) has a derivative at z_(0)=x_(0)+iy_(0)z_{0}=x_{0}+i y_{0}, then the first partial derivatives of uu and vv exist at z_(0)z_{0} and 示例 20 - 使用下面概述的方法来表明，如果由 f(z)=u(x,y)+iv(x,y)f(z)=u(x, y)+i v(x, y) 定义的函数 ff 在 z_(0)=x_(0)+iy_(0)z_{0}=x_{0}+i y_{0} 处具有导数，则 uu 和 vv 的一阶偏导数在 z_(0)z_{0} 处存在
at z_(0)z_{0}. These are the Cauchy-Riemann equations. 在 z_(0)z_{0} 。这些是柯西-黎曼方程。
Step 1 Write the derivative of ff as a limit. 步骤 1 将 ff 的导数写成极限形式。
Step 2 Express this limit in terms of uu and vv. 步骤 2 将此极限用 uu 和 vv 表示。
Step 3 Evaluate this limit in two ways and compare th. @\circ results. 步骤 3 以两种方式评估此极限并比较结果。
3.6. Fustifying and interpreting 3.6. 证明和解释
Example 21 - Here is an attempted proof of Cauchy’s mean value theor ^(m)\stackrel{m}{ } : 例 21 - 这是对柯西均值定理的一个尝试证明 ^(m)\stackrel{m}{ } ：
Statement: If functions fandg are continuous on the close’ interval [a,b][a, b] and differentiable on the open interval (a,b)(a, b) then th ire exists a value cc in (a,b)(a, b) such that: 声明：如果函数 f 和 g 在闭区间 [a,b][a, b] 上连续，并且在开区间 (a,b)(a, b) 上可微分，则在 (a,b)(a, b) 中存在一个值 cc
Cancel (b-a)(b-a) on the right and cross multiply to obtain t ie result. The attempted proof is actually invalid. Examine it care ully and write a short criticism of it [8]. 取消右侧的 (b-a)(b-a) 并交叉相乘以获得结果。所尝试的证明实际上是无效的。仔细检查并写出对其的简短批评[8]。
Example 22 - Here are two arguments, one to show that sin^(-1)x-cos^(-1)x=pi//2\sin ^{-1} x-\cos ^{-1} x=\pi / 2 and the other that sin^(-1)x+cos^(-1)x=pi//2\sin ^{-1} x+\cos ^{-1} x=\pi / 2. They cannot both be correct (and may both be wrong). Find and explain the error(s) in reasoning. 示例 22 - 这里有两个论点，一个是为了证明 sin^(-1)x-cos^(-1)x=pi//2\sin ^{-1} x-\cos ^{-1} x=\pi / 2 ，另一个是为了证明 sin^(-1)x+cos^(-1)x=pi//2\sin ^{-1} x+\cos ^{-1} x=\pi / 2 。它们不可能都是正确的（也可能都是错误的）。找出并解释推理中的错误。
We know that 我们知道
cos y=sin(y+pi//2)\cos y=\sin (y+\pi / 2)
for all yy, so suppose 对于所有 yy ，假设
x=cos y=sin(y+pi//2)x=\cos y=\sin (y+\pi / 2)
Then 然后
y=cos^(-1)xy=\cos ^{-1} x
and 和
y+pi//2=sin^(-1)xy+\pi / 2=\sin ^{-1} x
Subtraction of equation (1) from equation (2) gives the result 从方程（2）中减去方程（1）得到结果
Example 23 - One way of modelling population growth is to assume that the population increases at a rate proportional to the size of the population, that is P^(')prop PP^{\prime} \propto P. When would this be an appropriate model for population growth? What assumptions are being made and what are the limitations of the model? 示例 23 - 一种建模人口增长的方法是假设人口以与人口规模成比例的速度增长，即 P^(')prop PP^{\prime} \propto P 。在什么情况下这将是一个适合的人口增长模型？有哪些假设被提出，模型的局限性是什么？
3.7. Implications, conjectures, and comparisons 3.7. 含义、推测和比较
The following problem required students to use a computer program to multiply given matrices and then to make conjectures based on the results they obtained. 以下问题要求学生使用计算机程序来相乘给定的矩阵，然后根据他们获得的结果进行推测。
Example 24 - This problem investigates the similarity of nth powers of matrices. You are given two square matrices AA and BB and a nonsingular matrix PP that satisfy the relationship B=P^(-1)APB=P^{-1} A P. 示例 24 - 这个问题研究矩阵的 n 次幂的相似性。给定两个方阵 AA 和 BB 以及一个满足关系 B=P^(-1)APB=P^{-1} A P 的非奇异矩阵 PP 。
(i) Check that B=P^(-1)APB=P^{-1} A P as claimed. 检查 B=P^(-1)APB=P^{-1} A P 是否如所声称的那样。
(ii) Calculate B^(2)B^{2} and P^(-1)A^(2)PP^{-1} A^{2} P. (ii) 计算 B^(2)B^{2} 和 P^(-1)A^(2)PP^{-1} A^{2} P 。
(iii) Calculate B^(3)B^{3} and P^(-1)A^(3)PP^{-1} A^{3} P. (iii) 计算 B^(3)B^{3} 和 P^(-1)A^(3)PP^{-1} A^{3} P 。
(iv) Calculate B^(4)B^{4} and P^(-1)A^(4)PP^{-1} A^{4} P. (iv) 计算 B^(4)B^{4} 和 P^(-1)A^(4)PP^{-1} A^{4} P 。
(v) Let CC and DD be any two similar matrices. Make a cinjecture about the similarity of C^(n)C^{n} and D^(n)D^{n}, for n=1,2,3,dotsn=1,2,3, \ldots 让 CC 和 DD 是任何两个相似矩阵。对 C^(n)C^{n} 和 D^(n)D^{n} 的相似性做一个猜想，适用于 n=1,2,3,dotsn=1,2,3, \ldots 。
(vi) Prove your conjecture. 证明你的猜想。
The next two examples appear in [10]. 下两个例子出现在[10]中。
Example 25 - Take the expression n^(2)+n+17n^{2}+n+17, let n=1n=1, and evaluate tt te result. Is it a prime number? Substitute n=2n=2. Is the result a primt number? Substitute values of n from 3 to 10 . Are the results all prime iumbers? Can you come to a general conclusion? Are you using dea uctive or inductive arguments? Are you certain of your conclusio t? Is the conclusion actually true? 示例 25 - 取表达式 n^(2)+n+17n^{2}+n+17 ，设 n=1n=1 ，并评估 tt 的结果。它是一个质数吗？替换 n=2n=2 。结果是一个质数吗？将 n 的值从 3 替换到 10。结果都是质数吗？你能得出一个一般结论吗？你是在使用演绎还是归纳论证？你对你的结论确定吗？结论实际上是正确的吗？
Example 26 - Prove or disprove the following: The expression n^(2)+n+41n^{2}+n+41 represents a prime number for any natural number nn. 例 26 - 证明或反驳以下说法：表达式 n^(2)+n+41n^{2}+n+41 对于任何自然数 nn 代表一个质数。
Example 27 - Compare the method of undetermined coefficients with va:iation of parameters for second order linear differential equations. 例 27 - 比较不确定系数法与变参数法在二阶线性微分方程中的应用。
3.8. Evaluation 3.8. 评估
Example 28 - Is it possible to prove the result e^(ix)=cos x+i sin xe^{i x}=\cos x+i \sin x ? Give rc asons for your answer. 示例 28 - 是否可以证明结果 e^(ix)=cos x+i sin xe^{i x}=\cos x+i \sin x ？请给出理由。
Example 29 - Here are two definitions of a complex number: 示例 29 - 这里有两个复数的定义：
1 The equation x^(2)=-1x^{2}=-1 has no real roots, but we may : nvent an imaginary unit ii for which i^(2)=-1i^{2}=-1. We then define c complex number as a combination p+iqp+i q formed from two reaı numbers pp and qq and the imaginary unit ii. 1 这个方程 x^(2)=-1x^{2}=-1 没有实根，但我们可以发明一个虚数单位 ii ，使得 i^(2)=-1i^{2}=-1 。然后我们将复数 c 定义为由两个实数 pp 和 qq 以及虚数单位 ii 组合而成的 p+iqp+i q 。
2 The complex numbers can be defined as the set C={(x,y):x,y inR}\mathbb{C}=\{(x, y): x, y \in \mathbb{R}\} together with certain standard a ithmetical operations defined on this set. 复数可以定义为集合 C={(x,y):x,y inR}\mathbb{C}=\{(x, y): x, y \in \mathbb{R}\} 以及在该集合上定义的某些标准算术运算。
Compare the two definitions. Your answer could include: 比较这两个定义。你的答案可以包括：
The circumstances under which each definition would be apt ropriate. The relative merits of each definition from a mathematica point of view. 每个定义适用的情况。从数学的角度看每个定义的相对优缺点。
Historical aspects of these definitions. 这些定义的历史方面。 AA demonstration of the equivalence of the definitions. AA 定义等价性的演示。
Example 30 - Write a short exposition evaluating the relative merits of 'eeibniz’s and Newton’s notation for differentiation. 例 30 - 写一篇简短的论文，评估‘eeibniz’和牛顿的微分符号的相对优缺点。
Example 31 - Explain why the method of Laplace transforms works sc well for linear differential equations with constant coefficients ana integrodifferential equations involving a convolution [11]. 例 31 - 解释为什么拉普拉斯变换法对具有常系数的线性微分方程和涉及卷积的积分微分方程效果如此好 [11]。
References 参考文献
[1] Ramsden, P., 1992, Learning to Teach in Higher Education (London: Routl dge). 拉姆斯登，P.，1992，《高等教育教学学习》（伦敦：劳特利奇）。
[2] Crawford, K., Gordon, S., Nicholas, J., and Prosser, M., 1993, Learning mathematics at university level: initial conceptions of mathematics, in A1 weh, W., Kanes, C., Carss, M., and Booker, G. (ed) Contexts in Mathematics I ducation: Proceedings of the Sixteenth Annual Conference of the Mathematics Educatior Research Group of Australasia (MERGA), Brisbane. 克劳福德，K.，戈登，S.，尼古拉斯，J.，和普罗瑟，M.，1993 年，在大学水平学习数学：数学的初步概念，收录于 A1 weh，W.，凯恩斯，C.，卡斯，M.，和布克，G.（编辑）《数学教育的背景：澳大利亚数学教育研究小组第十六届年会论文集》，布里斯班。
[3] Crawford, K., 1983, Past Experiences and Mathematical Problem Solving. Proceedings of the Seventh Annual Conference of the International Group for the Psychology of Mathematics Education, Israel. 克劳福德，K.，1983 年，《过去的经验与数学问题解决》。国际数学教育心理学小组第七届年会论文集，以色列。
[4] Crawford, K., 1986, Cognitive and Social Factors in Problem Solving Behaviour, Proceedings of the Tenth Annual Conference of the International Group for the Psychology of Mathematics Education, London. 克劳福德，K.，1986 年，问题解决行为中的认知和社会因素，《国际数学教育心理学组第十届年会论文集》，伦敦。
[5] Crawford, K., 1990, School Definitions of Work: Their Impact on the Use of Technology in Mathematics Education, Proceedings of the Conference on the Political Dimensions of Mathematics Education, London Institute of Education. 克劳福德，K.，1990 年，《学校对工作的定义：它们对数学教育中技术使用的影响》，《数学教育政治维度会议论文集》，伦敦教育学院。
[6] Bloom, B. S., Engelhart, M. D., Furst, E. J., Hill, W. H., and Krathwohl, D. R., 1956, Taxonomy of Educational Objectives: Cognitive Domain (New York: McKay). [6] 布鲁姆, B. S., 恩格尔哈特, M. D., 费尔斯特, E. J., 希尔, W. H., 和 克拉斯沃尔, D. R., 1956, 教育目标分类学：认知领域 (纽约: 麦凯).
[7] Wood, L. N., and Smith, G. H., 1993, Calculus Questions in Mathematics: Of Primary Importance. Proceedings of the Thirtieth Annual Conference of the Mathematical Association of Victoria, Melbourne. [7] Wood, L. N. 和 Smith, G. H., 1993, 数学中的微积分问题：至关重要。维多利亚数学协会第三十届年会论文集，墨尔本。
[8] Harel, G., and Kaput, J., 1991, Conceptual entities and symbols. In Tall, D. O., (ed.), Advanced Mathematical Thinking (Dordrecht: Kluwer Academic Publishers). [8] Harel, G. 和 Kaput, J.，1991，概念实体和符号。在 Tall, D. O.（编），《高级数学思维》（多德雷赫特：克鲁威尔学术出版社）。
[9] Smith, G. H., and Petocz, P., 1994, Int. F. Math. Educ. Sci. Technol., 25, 139. [9] 史密斯，G. H.，和佩托茨，P.，1994，国际数学教育科学与技术杂志，25，139。
[10] Petocz, P., Petocz, D., and Wood, L. N., 1992, Introductory Mathematics - (Melbourne: Thomas Nelson). [10] Petocz, P., Petocz, D., 和 Wood, L. N., 1992, 《基础数学》 - （墨尔本：托马斯·尼尔森）。
[11] Nagle, R. K., and Saff, E. B., 1993, Differential Equations and Boundary Value Problems (New York: Addison Wesley). [11] Nagle, R. K. 和 Saff, E. B., 1993, 《微分方程与边值问题》（纽约：阿迪森-韦斯利）。
