📚 Preface to the Fourth Edition | 第四版序言
It is with great pleasure and a profound sense of responsibility that we present the fourth edition of our IB Mathematics series. Since the publication of the first edition, the landscape of mathematics education within the International Baccalaureate Diploma Programme has undergone a remarkable transformation. The introduction of the new curriculum, with its two distinct pathways — Mathematics: Analysis and Approaches and Mathematics: Applications and Interpretation — demanded not only a fresh instructional approach but also a complete reimagining of how students encounter, master, and apply mathematical ideas. This fourth edition is our response to that challenge, crafted meticulously over several years of classroom practice, teacher feedback, and deep engagement with the IB philosophy.
我们怀着极大的喜悦和深切的责任感，呈上 IB 数学系列的第四版。自第一版问世以来，国际文凭大学预科项目（IBDP）中的数学教育格局经历了显著变革。新课程引入了两个截然不同的路径——数学：分析与方法和数学：应用与解释——这不仅需要全新的教学方法，也需要彻底重构学生接触、掌握和应用数学思想的方式。第四版正是我们对这一挑战的回应，是经过多年课堂实践、教师反馈以及对 IB 理念的深度参与而精心编写的成果。
1. A New Architecture for Two Distinct Journeys | 为两条独特路径设计的新架构
The most visible change in this edition is the structural separation of the two IB Mathematics courses. Rather than forcing a single textbook to serve both pathways superficially, we have designed dedicated sections with clear signposting for Analysis and Approaches (AA) and Applications and Interpretation (AI). Each chapter now opens with a pathway indicator and a list of syllabus content addressed, ensuring that students and teachers can navigate the material with confidence. Core topics common to both routes are presented first, followed by pathway-specific extension material that delves deeper into theoretical rigour for AA or contextual problem-solving for AI. This architecture respects the integrity of each course while allowing learners to appreciate the interconnected nature of mathematics.
本版最显著的变化是对两门 IB 数学课程做了结构性区分。我们没有让一本教材勉强服务于两条路径，而是设计了专门的单元，并对分析与方法（AA）和应用与解释（AI）进行了清晰的标识。每章开头都设有路径指示符和对应的教学大纲内容清单，确保学生和教师能够自信地使用材料。两条路径共同的核心主题放在前面讲解，随后是路径专属的拓展内容，或是对 AA 的理论严谨性进行深入挖掘，或是对 AI 的情境问题解决进行延伸。这种架构尊重了每门课程的完整性，同时也让学习者领略到数学的内在联系。
2. Strengthening Conceptual Understanding Through Inquiry | 通过探究强化概念性理解
At the heart of the IB approach lies the principle that mathematics is not a collection of isolated procedures but a coherent body of knowledge built upon fundamental concepts. In this fourth edition, we have significantly expanded the inquiry-based opening activities. Each topic begins with an open-ended problem or a thought-provoking question that invites students to explore before being taught the formal mathematics. These investigations are designed to be accessible yet challenging, encouraging learners to formulate conjectures, test ideas, and reflect on their findings. By constructing meaning for themselves, students develop a deeper and more durable understanding that transfers readily to unfamiliar contexts — a skill essential for the higher-order assessment objectives of the IB.
IB 理念的核心在于，数学并非孤立过程步骤的集合，而是建立在基本概念之上的连贯知识体系。在第四版中，我们大幅扩展了基于探究的开篇活动。每个主题都从一个开放性问题或发人深省的提问开始，邀请学生在学习正式数学之前先进行探索。这些探究活动既容易入手又有一定挑战，鼓励学习者提出猜想、检验想法并反思自己的发现。通过自己构建意义，学生能形成更深刻、更持久的理解，这种理解能顺畅地迁移到陌生情境中——这正是 IB 高阶评估目标所必需的技能。
3. Rigour and Accessibility in Tandem | 严谨性与可及性并行不悖
Mathematical rigour remains non-negotiable, especially for the Analysis and Approaches course at Higher Level. However, rigour does not mean obscurity. We have worked tirelessly to ensure that formal definitions, theorems, and proofs are presented with clarity and supported by concrete examples before generalisation. In the Applications and Interpretation sections, formal notation is introduced gradually, always tethered to real-world meaning. Diagrams, graphs, and visual models have been refined to enhance accessibility. Each concept is segmented into manageable steps, and side notes offer linguistic support for students whose first language is not English, without compromising the precision of mathematical language.
数学的严谨性不容妥协，尤其是针对高水平（HL）的分析与方法课程。然而，严谨并不意味着晦涩难懂。我们不遗余力地确保形式化的定义、定理和证明表述清晰，并在推广之前用具体例子加以支撑。在应用与解释部分，形式符号逐步引入，始终紧扣现实意义。图表、图形和可视化模型经过精心打磨，以提升可及性。每个概念被拆分为可轻松掌握的步骤，侧栏注释还为非英语母语学生提供语言支持，同时不损害数学语言的精确性。
4. Real-World Applications Grounded in Authentic Data | 扎根真实数据的现实应用
Mathematics comes alive when it is connected to the world outside the classroom. For this edition, we have collaborated with professionals in fields such as epidemiology, climatology, finance, and engineering to source up-to-date, authentic datasets. Students studying exponential models will analyse genuine COVID-19 transmission data, while those exploring probability distributions will examine insurance claim records. The Applications and Interpretation pathway in particular benefits from these real-world contexts, but even Analysis and Approaches learners will find that seeing abstract structures reflected in reality deepens their appreciation. All datasets are provided in the accompanying digital platform, along with guidance on using technology to manipulate and visualise them.
当数学与课堂之外的世界连接时，它便焕发出生命力。在本版中，我们与流行病学、气候学、金融和工程等领域的专业人士合作，获取了最新、真实的数据集。学习指数模型的学生将分析真实的 COVID-19 传播数据，而探索概率分布的学生则会研究保险理赔记录。应用与解释路径尤其从这些现实情境中受益，但即便是分析与方法的学习者也会发现，观察到抽象结构在现实中得以映射，能加深他们的理解。所有数据集都提供在配套的数字平台上，并附有使用技术工具进行处理和可视化的指导。
5. A Modern Approach to Technology Integration | 技术整合的现代法门
Graphical display calculators and mathematical software are no longer optional in IB Mathematics; they are essential tools for inquiry, problem-solving, and communication. The fourth edition embeds technology guidance directly within the text, rather than relegating it to appendices. Step-by-step instructions for TI‑Nspire, Casio, and NumWorks calculators are included alongside more generic approaches for spreadsheet software and Desmos. We have also introduced Python-based explorations for students wishing to extend their computational thinking, particularly relevant for the AI Higher Level internal assessment. These coding snippets are presented in a way that requires no prior programming experience, aligning with the IB’s emphasis on algorithmic reasoning.
图形显示计算器和数学软件在 IB 数学中不再是可选项，而是探究、解决问题和交流的必要工具。第四版将技术指导直接嵌入正文，而不是放在附录中。书中提供了 TI‑Nspire、Casio 和 NumWorks 计算器的逐步操作说明，以及针对电子表格软件和 Desmos 的更通用方法。我们还引入了基于 Python 的探索活动，供希望拓展计算思维的学生使用，这对 AI 高水平课程的内部评估尤为相关。这些代码片段以无需任何编程基础的方式呈现，与 IB 强调算法推理的理念一致。
6. Internal Assessment Support from Conception to Completion | 从构思到完成的内部评估支持
The Internal Assessment (IA) is a defining feature of the IB Mathematics course, requiring students to undertake an independent mathematical exploration. For many, this represents the first encounter with authentic mathematical research. Our fourth edition devotes an entire chapter to the IA journey, broken down into phases: choosing a topic, formulating a research question, gathering and analysing data, structuring the write-up, and reflecting on the process. The chapter includes annotated exemplar extracts that highlight both strengths and common pitfalls. Furthermore, we have embedded IA‑style open-ended problems throughout the chapters, so that students can gradually build the skills needed to initiate and sustain their own investigation with confidence.
内部评估（IA）是 IB 数学课程的一个标志性特征，要求学生开展独立的数学探究。对于许多人来说，这是他们初次接触真正的数学研究。我们的第四版用整整一章来阐释 IA 的历程，将其分解为各个阶段：选题、拟定研究问题、收集与分析数据、撰写报告结构和反思过程。该章包含带有评注的范例摘录，突出显示了优点和常见误区。此外，我们在各章中嵌入了 IA 式的开放性问题，让学生能够逐步培养起自信地启动并持续开展自己的探究所需的技能。
7. Examination Preparation Reimagined | 重新构想的考试备考
Performing well under timed conditions requires more than subject knowledge; it demands exam technique, time management, and familiarity with the style of questioning unique to the IB. We have expanded our end-of-chapter review sections to include a balanced mix of short-response, extended-response, and investigation-style questions, each tagged with the relevant assessment objective. A dedicated exam‑strategy chapter at the end of the book analyses the command terms — ‘show that’, ‘hence’, ‘determine whether’ — and explains what examiners look for in each case. Full practice papers for both SL and HL are provided online, complete with markschemes and examiner commentary, mirroring the format students will face on the day.
在限时条件下考出好成绩，需要的不仅仅是学科知识，它还需要考试技巧、时间管理以及对 IB 独特提问风格的熟悉。我们扩充了章末复习部分，包含简答题、拓展题和探究题型的均衡组合，每道题都标注了相关的评估目标。书末专门设有一章考试策略，分析了指令词——如“证明”、“由此”、“判断是否”——并解释了考官在每种情况下期望看到什么。线上提供了 SL 和 HL 的完整模拟试卷，配有评分方案和考官评注，如实反映了学生考试时将会遇到的试卷格式。
8. Language, Inclusion and International‑Mindedness | 语言、包容性与国际情怀
IB learners come from a rich variety of linguistic and cultural backgrounds, and a mathematics textbook should reflect this diversity. In this edition, we have deliberately chosen names, contexts, and examples that represent a broad spectrum of global cultures. Problems are set in locations from Buenos Aires to Nairobi, and historical notes feature mathematicians from Africa, Asia, the Middle East, and South America alongside the traditionally cited European figures. The language of instruction has been reviewed by EAL specialists to reduce unnecessary linguistic load while preserving mathematical accuracy. Key terms are provided in a multilingual glossary that accompanies the digital resources, supporting learners who may be studying mathematics in a non‑native language.
IB 学习者来自丰富的语言和文化背景，一本数学教材也应反映这种多样性。在本版中，我们有意识地选择了代表广泛全球文化的名字、情境和例子。题目背景从布宜诺斯艾利斯到内罗毕，历史注释中的数学家不仅包括传统引用的欧洲人物，还有来自非洲、亚洲、中东和南美洲的学者。教学语言经过了英语作为附加语言（EAL）专家的审查，以减少不必要的语言负担，同时保持数学准确性。数字资源附有多语种词汇表，为可能用非母语学习数学的学习者提供支持。
9. Sustainability and Ethical Applications | 可持续与伦理应用
Aligning with the IB’s commitment to education for a sustainable future, we have woven themes of global significance into the fabric of the mathematical content. Chapters on sequences and series use case studies on solar panel output and recycling rates; statistics sections explore carbon footprint data and biodiversity indices; calculus problems examine the rate of glacier retreat. These contexts are not decorative — they are integral to the mathematics being taught, and they are accompanied by reflective prompts that encourage students to consider the ethical implications of the models they use and the conclusions they draw. Such tasks also serve as excellent springboards for TOK discussions, bridging mathematics and the wider aims of the diploma programme.
与 IB 对可持续未来教育的承诺相一致，我们将具有全球意义的主题编织进了数学内容的经纬之中。数列级数部分使用了太阳能板发电量和回收率的案例研究；统计章节探讨了碳足迹数据和生物多样性指数；微积分问题则分析冰川退缩的速率。这些情境并非装饰——它们是所授数学的有机组成部分，并配有反思提示，鼓励学生思考所用模型和所得结论的伦理意涵。这类任务也是知识论（TOK）讨论的绝佳跳板，为数学和大学预科项目的更广泛目标之间架起桥梁。
10. Responding to Teacher and Student Voice | 倾听师生之声
No textbook can hope to meet the needs of a global classroom without genuine dialogue with its users. During the development of this fourth edition, we conducted extensive surveys and focus groups with IB Mathematics teachers from over thirty countries. The feedback was instrumental in shaping our revisions: many teachers requested more scaffolded proof exercises for AA HL, while others asked for additional spreadsheet tutorials for AI SL. Students expressed a desire for clearer linkages between different syllabus topics, which we addressed through cross‑referenced margin notes and conceptual flow diagrams at the start of each unit. Every change you see in this edition has been tested in real classrooms, and we are deeply grateful to all who contributed their time and insight.
一本教材如果未与使用者进行真诚对话，便难以期待它能满足全球课堂的需求。在第四版的编写过程中，我们对来自三十多个国家的 IB 数学教师进行了广泛的调查和焦点小组访谈。这些反馈对我们的修订起到了关键作用：许多教师要求为 AA HL 增加更多有支架引导的证明练习，还有的则要求为 AI SL 增设更多电子表格教程。学生们希望不同大纲主题之间有更清晰的联系，我们则通过交叉引用的页边注释和每单元开头的概念流程图来回应这一需求。本版中您所见的每一处改动都已在真实课堂中经受检验，我们深深感谢所有贡献了时间和见解的人们。
11. The Digital Companion: Beyond the Printed Page | 数字伴侣：超越印刷页
While the printed book remains our foundation, the fourth edition is fully integrated with a rich digital environment. Students gain access to interactive applets for exploring concepts such as limits, transformations, and the central limit theorem. Teachers are supported by a comprehensive online resource pack containing lesson plans, editable worksheets, and auto‑graded quizzes with analytic dashboards. The AI‑enhanced tutorial system can recommend personalised revision paths based on quiz performance, adapting to the individual learner’s pace. Importantly, all digital resources are designed to function seamlessly on smartphones and tablets, acknowledging the reality of today’s learning environments.
虽然印刷版依然是我们的基础，但第四版已与丰富的数字环境充分整合。学生可访问用于探索极限、变换以及中心极限定理等概念的交互式小程序。教师则享有一整套在线资源包，其中包括教案、可编辑的练习单，以及带有分析仪表盘的自动评分测验。人工智能增强的辅导系统能够根据测验表现推荐个性化复习路径，以适应每位学习者的节奏。重要的是，所有数字资源都设计为能在智能手机和平板电脑上无缝运行，充分考虑了当今学习环境的实际状况。
12. A Note of Thanks and a Look Forward | 致谢与前瞻
Publishing a textbook of this scope is a collective endeavour, and we wish to express our heartfelt gratitude to the editorial team at TutorHao, whose dedication to educational excellence is evident on every page. We are indebted to the copyeditors, proofreaders, typesetters, and illustrators who transformed manuscript drafts into a polished and coherent whole. Above all, we thank our students past and present, whose curiosity and perseverance continually remind us why we write. Mathematics, at its core, is a human activity — a way of making sense of the world. Our deepest hope is that this fourth edition will not only prepare you for examination success but also ignite a lifelong appreciation for the beauty and power of mathematical thought. We invite you to turn the page and begin your journey.
这样一本涵盖广泛内容的教材，其出版是一项集体事业，我们要向 TutorHao 的编辑团队致以由衷的感谢，他们对教育卓越的执着在每一页上都显而易见。我们还要感谢文字编辑、校对、排印人员和插画师，他们把手稿草稿转变为一部精美连贯的整体。最重要的是，我们感谢过去和现在的学生，他们的好奇心与毅力不断提醒我们为何而写。数学，归根结底，是一种人类活动——是理解世界的一种方式。我们最深切的期盼是，这第四版不仅能助您取得考试成功，更能点燃您对数学思想之美与力量的终生欣赏。我们诚邀您翻开书页，开启自己的旅程。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply