📚 IB & CIE Mathematics: Essay Writing Template | IB CIE 数学：论文写作模板
Writing a mathematical essay in the IB Diploma Programme, or for a CIE extended project, requires a blend of rigorous logic, clear communication, and personal engagement. This template guides you through every stage of crafting an investigation—from posing a precise research question to reflecting on your conclusions—while ensuring you meet assessment criteria such as mathematical presentation, personal engagement, and critical reflection.
撰写IB文凭课程中的数学论文，或为剑桥国际考试（CIE）拓展项目准备数学探究，需要严谨的逻辑、清晰的表达与个人思考的深度融合。本模板将带你走过数学论文写作的每一个阶段——从提出精准的研究问题到对结论进行反思，同时帮助你满足数学表达、个人投入与批判性反思等评分标准。
1. Understanding the Essay Assignment | 理解论文任务
The IB Mathematics Internal Assessment (IA) is an individual exploration of a mathematical topic of your choice, typically 12–20 pages. For CIE candidates, a similar extended essay or project may be required within Further Mathematics or as an independent qualification. Familiarise yourself with the specific mark scheme: for IB, the criteria are Presentation, Mathematical Communication, Personal Engagement, Reflection, and Use of Mathematics.
IB数学内部评估（IA）是对自选数学主题的独立探究，篇幅通常为12至20页。对于CIE考生，进阶数学中可能需要完成类似的拓展论文，或者作为独立资质项目提交。务必熟悉详细的评分标准：IB数学IA的评分维度包括表达、数学交流、个人投入、反思与数学运用。
Begin by collecting official subject guides and exemplar essays. Notice how high-scoring papers interweave mathematical notation with explanatory paragraphs. Your essay is not simply a set of equations; it is an argument that shows your personal inquiry and reasoning.
开始时先收集官方学科指南和范文。注意高分文章如何将数学符号与解释性段落交织在一起。你的论文不是一组方程式的堆砌，而是一次展现个人探究与推理过程的论述。
2. Selecting an Appropriate Topic | 选择合适的主题
A strong topic sits at the crossroads of syllabus content and personal curiosity. Consider areas such as modelling the spread of a disease using SIR differential equations, optimising a sports strategy with game theory, or analysing the harmonics of a musical instrument with Fourier series. The topic must be accessible yet offer depth for genuine mathematical exploration.
好的主题位于课程内容与个人兴趣的交汇点。可以考虑诸如运用SIR微分方程模型化疾病传播、借助博弈论优化体育策略，或利用傅里叶级数分析乐器谐波等方向。主题必须既能驾驭，又能为真正的数学探索提供足够深度。
Avoid topics that are too broad (e.g., 'Calculus in real life') or merely descriptive (e.g., 'The life of Pythagoras'). Instead, narrow down: 'How does the shape of a roller coaster loop minimise g‑force?' or 'Modelling carbon dating accuracy using exponential decay with error propagation'. The research question should emerge naturally from this initial brainstorming.
避免过于宽泛（如“微积分在生活中的应用”）或仅仅是描述性的主题（如“毕达哥拉斯生平”）。应当聚焦：例如“过山车环圈的形状如何最小化g力？”或“利用指数衰减与误差传播模型评估碳定年法的精度”。研究问题应从这样的头脑风暴中自然浮现。
3. Formulating a Focused Research Question | 制定聚焦的研究问题
Transform your idea into a precise, arguable question. A well‑crafted question contains the variables, mathematical tools, and scope. Compare these: a weak question—'Is the golden ratio beautiful?'—versus a strong one—'To what extent does the golden ratio appear in the dimensions of the Parthenon, and how can a χ² test quantify deviation from φ?'
将你的想法转化为精准、可论证的问题。好的研究问题应包含变量、数学工具与范围。对比：弱问题——“黄金分割是否美丽？”——与强问题——“黄金分割在帕特农神庙尺寸中的体现在多大程度上符合理论值，且卡方检验如何量化偏离φ的程度？”
The question dictates the mathematical level. If you aim for high marks in Use of Mathematics, the question should invite calculus, probability distributions, or proof. Write a question that you do not yet fully know the answer to; genuine inquiry drives personal engagement.
问题决定了数学水平。若要在数学运用上获高分，问题应引入微积分、概率分布或证明。写一个你尚未完全知道答案的问题；真实的未知感将激发个人投入。
4. Structuring Your Essay: The Blueprint | 构建论文结构：蓝图
A standard structure for a mathematical exploration includes: Title page, Introduction, Mathematical Background, Methodology, Analysis, Discussion/Evaluation, Conclusion, and References. Appendices may include raw data or lengthy proofs. Stick to this skeleton while allowing each section to breathe.
数学探究的标准结构包括：标题页、引言、数学背景、方法学、分析、讨论/评价、结论与参考文献。附录可包含原始数据或冗长证明。遵循这一骨架的同时，让每一部分有充分展开的空间。
| Section | Purpose | Approx. Length |
| Introduction | State research question, rationale, and brief overview. | 1–2 pages |
| Background | Define all necessary mathematical concepts and notation. | 2–4 pages |
| Methodology | Describe models, assumptions, data collection. | 1–2 pages |
| Analysis | Present calculations, derivations, graphs, and tables. | 5–8 pages |
| Discussion | Interpret results, critique model, suggest improvements. | 2–3 pages |
| Conclusion | Direct answer to the question, summary of finding. | 1 page |
5. Writing the Introduction | 撰写引言
The introduction must hook the reader and establish the essay’s purpose. Start with a real‑world context or a personal anecdote that reveals why the topic matters to you. Then narrow down to your central research question, stated clearly and in bold. Briefly outline the structure of the essay, indicating the mathematical techniques you will employ.
引言必须抓住读者并确立论文目的。从现实背景或个人轶事开始，揭示你为何关注该主题。然后缩小到中心研究问题，清晰陈述并以粗体标示。简要概述文章结构，指明你将使用的数学技巧。
Indicate the scope and any key assumptions upfront. For example: 'This exploration assumes no air resistance for the projectile model; a sensitivity analysis will test the validity of this assumption.' Clear boundaries prevent the essay from veering off track.
预先说明范围和关键假设。例如：“本探究假设抛体运动无空气阻力；将进行敏感性分析以检验该假设的合理性。”清晰的边界可防止文章偏离主题。
6. Mathematical Background and Preliminaries | 数学背景与预备知识
terest in mathematics but may not recall every detail. Include derivations for key results—e.g., deriving the logistic growth equation dP/dt = rP(1 − P/K) from first principles—but avoid simply copying textbook proofs.
在此介绍那些并非浅显但对分析至关重要的定理、公式与定义。写作时应假设读者对数学有浓厚兴趣，但未必记得每一个细节。为关键结果加入推导——例如，从基本原理推导逻辑斯蒂增长方程 dP/dt = rP(1 − P/K)——但避免直接复制教科书上的证明。
Use consistent notation throughout. Introduce variables with a clear table:
| Symbol | Meaning |
| P(t) | Population size at time t |
| r | Intrinsic growth rate |
| K | Carrying capacity |
用统一的符号贯穿全文。用一个清晰的表格介绍变量。
7. Methodology and Data Handling | 方法学与数据处理
Explain how you are going to answer the research question. If empirical data is used, describe the source, collection method, and any sampling techniques. If you are running simulations (e.g., Monte Carlo), detail the algorithm and parameters. Always justify the chosen mathematical model: why is a linear regression appropriate? Why do you assume a Poisson distribution for the arrival of customers?
解释你将如何回答研究问题。若使用实证数据，描述来源、采集方法与采样技术。若进行模拟（如蒙特卡洛），详述算法与参数。始终论证所选数学模型的合理性：为什么线性回归是合适的？为什么假设顾客到达服从泊松分布？
Mention ethical considerations where relevant, and note any limitations of the data at this stage. A thorough methodology earns high marks for mathematical communication and personal engagement.
提及相关的伦理考量，并在此阶段注明数据的任何局限。严谨的方法学会为数学交流与个人投入赢得很高的评价。
8. Presentation of Analysis and Results | 分析与结果展示
This is the core of your essay. Present calculations step by step, using bullet‑point style only when comparing options; otherwise, write in full sentences. For instance: ' to the velocity function v(t)=−9.8t+25 yields v(10)=−73 m s⁻¹, indicating downward motion.' Support every claim with graphs, tables, or algebraic manipulations.
这是论文的核心。分步展示计算过程，只在选项对比时使用项目符号；其他情况下使用完整句子。例如：“将t=10代入速度函数 v(t)=−9.8t+25 得到 v(10)=−73 m s⁻¹，表明向下运动。”用图、表或代数运算支撑每一个论断。
Place all graphs and diagrams within the text, labeled as Figure 1, Figure 2, with captions. Ensure axes are labeled with units and scales are appropriate. Use software such as GeoGebra or Desmos to generate high‑quality plots, and reference it.
将所有图形与图表嵌入正文，标注为图1、图2，并配标题。确保坐标轴标有单位和合适的刻度。使用GeoGebra或Desmos等软件生成高质量图像并注明。
9. Discussion and Critical Evaluation | 讨论与批判性评价
Now interpret your results: What mathematical patterns emerge? Do the findings support the hypothesis? This is also where you reflect on the validity of your model. Critically assess the assumptions—e.g., 'Assuming a constant contact rate β overestimates infection speed in the later stage of the outbreak.' Use error bounds or sensitivity analysis to quantify uncertainty.
现在解释结果：出现了哪些数学模式？发现是否支持假设？这也是你反思模型有效性的地方。批判性地评估假设——例如，“假设恒定接触率β 高估了疫情后期的感染速度。”使用误差界限或敏感性分析来量化不确定性。
If a quadratic fit yields R²=0.92 but a sinusoidal model gives R²=0.78, discuss the trade‑ equent data, extending the time interval, or relaxing a linearity assumption. Genuine self‑reflection is the key to the 'Reflection' criterion.
比较替代模型。若二次拟合得到R²=0.92，而正弦模型R²=0.78，讨论其权衡。提出改进建议：更密集地收集数据、延长时间区间，或放宽线性假设。真实的自我反思是“反思”评分点的核心。
10. Conclusion and Final Reflections | 结论与最终反思
Directly answer your research question: 'In conclusion, the golden ratio appears in an average of 68% of the measured dimensions, with a χ² value of 3.4 (p>0.05), suggesting no statistically significant deviation—thus the Parthenon’s adherence to φ cannot be rejected.' Summarise the main mathematical journey without introducing new content.
直接回答你的研究问题：“结论表明，黄金比例平均出现在68%的测量尺寸中，卡方值为3.4（p>0.05），显示无统计显著性偏差——因此不能排除帕特农神庙遵循φ的可能性。”总结主要数学历程，不引入新内容。
Link back to your personal engagement: what did the process teach you about mathematics? Mention any surprising insights and how your initial understanding has deepened. A powerful concluding paragraph leaves a lasting impression.
回归个人投入：这个过程让你对数学有了怎样的认识？提及任何意外的洞见，以及你的初始理解如何深化。有力的结尾段落会留下持久印象。
11. Citations, Referencing, and Academic Honesty | 引用、参考文献与学术诚信
Use a consistent citation style (APA, MLA, or Chicago) as specified by your programme. Every mathematical formula sourced from a textbook, every data set, and every image must be acknowledged. Failure to do so constitutes plagiarism. Include in‑text citations like (Stewart, 2021, p.143) and a final reference list.
使用符合项目要求的一贯引用格式（APA、MLA或Chicago）。教科书中的每条数学公式、每一个数据集和每一张图片都必须注明出处。未注明即构成剽窃。文中加入引用如（Stewart, 2021, p.143），并附录最终参考文献列表。
For IB, the guide for the Extended Essay provides detailed referencing examples. Acknowledge any use of AI tools as a brainstorming aid, but your writing and analysis must be your own. Proofread your reference list meticulously—formatting errors can cost the Presentation mark.
对于IB，拓展论文指南提供了详细的参考文献示例。若使用AI工具辅助头脑风暴，应予以说明，但你的写作与分析必须是自己完成的。仔细校对你参考文献列表——格式错误可能使表达分流失。
12. Submission Checklist and Sample Outline | 提交检查清单与大纲示例
Before submitting, verify the following:
- Word count (or page count) within the specified limit.
- Research question stated exactly as on the title page.
- All graphs labeled and referenced in the text.
- No appendices left unreferenced.
- Mathematical symbols consistent throughout.
- Personal voice evident in the introduction and reflection.
- Reference list complete and cross‑checked.
提交前请核实以下各项：
- 字数（或页数）在限制范围内。
- 研究问题与标题页上陈述完全一致。
- 所有图形均标注并在正文中提到。
- 附录未被遗漏引用。
- 全文数学符号统一。
- 引言和反思中能听到个人声音。
- 参考文献列表完整并交叉核对。
A sample outline for an IA on 'Modelling the spread of a rumour using logistic and SIR models' could be: Title – Introduction (personal experience with viral news) – Background (derivation of logistic and SIR) – Methodology (simulation parameters) – Analysis (comparison graphs, parameter estimation) – Discussion (which model fits better, limitations) – Conclusion – References. This template can be adapted for any topic.
一份关于“利用逻辑斯蒂与SIR模型模拟谣言传播”的IA示例大纲可以是：标题 – 引言（个人对病毒式新闻的体验） – 背景（逻辑斯蒂与SIR的推导） – 方法学（模拟参数） – 分析（比较图、参数估计） – 讨论（哪个模型拟合更好，局限） – 结论 – 参考文献。此模板可适配任何主题。
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