📚 Pre-U OCR Mathematics: Essay Writing Framework and Sample Essays | Pre-U OCR 数学：论文写作框架与范文
The Personal Investigation for OCR Pre-U Mathematics (9768) challenges you to explore an area of mathematics beyond the standard syllabus, developing independent research and communication skills. A well-structured essay is crucial for attaining the top marks, as it demonstrates clarity of thought, mathematical rigour, and thoughtful reflection. This article provides a proven framework and a detailed sample extract to help you craft an outstanding submission.
OCR Pre-U 数学（9768）的个人探究项目要求你超越标准大纲，独立探索一个数学领域，并培养研究与沟通能力。结构清晰的论文对获取高分至关重要，因为它能展现你思维清晰、数学严谨和深度反思。本文提供一套经过验证的写作框架和详细的范文摘录，帮助你完成出色的作品。
1. Understanding the Personal Investigation | 理解个人探究项目
The Personal Investigation is an internally assessed, externally moderated piece of coursework that contributes 20% of the total Pre-U Mathematics marks. You are expected to identify a mathematical problem or area of interest, formulate a research question, and conduct an in-depth investigation. The final output is a report that must evidence advanced mathematical thinking, appropriate use of technology, and personal engagement.
个人探究是一项内部评估、外部审核的课程作业，占 Pre-U 数学总分的 20%。你需要识别一个数学问题或感兴趣的领域，拟定研究问题，并进行深入探究。最终成果是一份报告，必须体现高等数学思维、恰当运用技术以及个人投入。
2. Choosing a Suitable Topic | 选择合适的主题
A good topic sits at the intersection of personal interest and mathematical depth. It should be neither too trivial nor impossibly advanced. Typical areas include mathematical modelling (e.g. SIR epidemic models, projectile motion with air resistance), number theory (e.g. RSA encryption), geometry (e.g. non-Euclidean spaces), calculus applications (e.g. optimisation of splash patterns), or statistics (e.g. Bayesian inference). Ensure your topic allows for meaningful mathematical manipulation and data analysis.
好的选题位于个人兴趣与数学深度的交汇点。既不能太简单，也不应过于高深。常见领域包括数学建模（如 SIR 传染病模型、考虑空气阻力的抛体运动）、数论（如 RSA 加密）、几何（如非欧空间）、微积分应用（如飞溅模式优化）或统计学（如贝叶斯推断）。确保所选主题允许有意义的数学推导和数据分析。
3. Structural Framework Overview | 整体结构框架
A coherent structure is essential. The recommended framework comprises: Title Page, Abstract, Introduction, Mathematical Background, Methodology/Model Development, Analysis and Discussion, Conclusion and Evaluation, References, and Appendices. The Abstract (about 200 words) summarises the aim, methods, key findings, and conclusion. The body should flow logically from establishing context to presenting results and reflection.
条理清晰的结构至关重要。推荐框架包括：标题页、摘要、引言、数学背景、方法论/模型构建、分析与讨论、结论与评估、参考文献和附录。摘要（约 200 词）概括目标、方法、关键发现和结论。正文应逻辑流畅地从建立背景到呈现结果和反思。
4. Writing the Introduction | 撰写引言
The introduction must engage the reader and clearly state your research question. Begin with a contextual hook – perhaps a real-world problem or a historical note. Then narrow down to the specific mathematical focus, state your aim(s), and briefly outline the essay’s structure. Avoid going into mathematical details here; save that for later sections. End the introduction with a clear thesis statement or research question.
引言必须吸引读者并清晰陈述研究问题。以一个情境钩子开始——可以是现实世界问题或历史注脚。随后缩小至具体的数学焦点，说明研究目标，并简要概述论文结构。避免在此处展开数学细节，留待后续章节。以清晰的研究陈述或问题结束引言。
Example Research Question: “How can a modified SIR model with vaccination rate account for the 2014 Ebola outbreak dynamics?”
示例研究问题：“考虑接种率的修正 SIR 模型如何解释 2014 年埃博拉疫情的动态？”
5. Developing Mathematical Rigour | 发展数学严谨性
This section forms the core of your investigation. Present the mathematical theories and techniques you will use, ensuring all variables are defined. If you are working with a model, derive the governing equations step by step. For an SIR model, you might write:
这是你探究的核心部分。呈现你将要使用的数学理论和技巧，确保定义所有变量。若使用模型，需一步步推导控制方程。对于 SIR 模型，可写出：
dS/dt = -β S I / N , dI/dt = β S I / N – γ I , dR/dt = γ I
where S, I, R represent susceptible, infected, and recovered populations, β is the transmission rate, γ is the recovery rate, and N is the total population. Explain the assumptions (homogeneous mixing, no births/deaths) and how they affect validity. Show mathematical manipulations such as non-dimensionalisation or stability analysis. Every step must be justified and linked to your research question.
其中 S, I, R 分别代表易感、感染和康复人数，β 为传播率，γ 为康复率，N 为总人口。解释假设（均匀混合、无出生死亡）及其对有效性的影响。展示数学处理，如无量纲化或稳定性分析。每一步都必须有依据并关联研究问题。
6. Presenting Analysis and Discussion | 展示分析与讨论
Once the mathematical groundwork is laid, apply it to your specific problem. Use technology (e.g. Excel, Python, MATLAB, Desmos) to solve equations, produce graphs, and carry out simulations. Present your results clearly: labelled figures, tables with descriptive captions. Do not simply dump data; interpret what the results mean in context. For instance, if the basic reproduction number R₀ is computed, discuss its significance for outbreak control.
在数学基础奠定后，将其应用于具体问题。使用技术工具（如 Excel、Python、MATLAB、Desmos）求解方程、绘制图表并进行模拟。清晰地呈现结果：有标注的图形、带说明的表格。不要只堆砌数据；要结合情境解读结果意义。例如，若计算了基本再生数 R₀，讨论其对疫情控制的意义。
A table helps to organise parameter values and their origins, enhancing the report’s transparency.
表格有助于整理参数值及其来源，提升报告的透明度。
| Parameter | Value | Source |
|---|---|---|
| β | 0.45 day⁻¹ | Literature |
| γ | 0.2 day⁻¹ | Estimated |
| R₀ | 2.25 | β/γ |
7. Crafting a Conclusion and Reflection | 构思结论与反思
Your conclusion should directly answer the research question, summarising key findings without introducing new ideas. Then critically evaluate your investigation: discuss limitations of the model, reliability of data, and potential improvements. Reflection demonstrates personal engagement – mention what you learned, challenges faced, and how your mathematical understanding deepened.
结论应直接回答研究问题，总结关键发现，而不引入新内容。然后批判性地评估你的探究：讨论模型局限、数据可靠性和可能的改进方向。反思体现个人投入——提及你所学到的、面临的挑战，以及数学理解如何加深。
8. Referencing and Formatting | 参考文献与格式
Use a consistent citation style (e.g. Harvard, APA) throughout. Every source, including data sets and software, must be cited both in-text and in the reference list. Formatting guidelines: 12 pt readable font, 1.5 line spacing, numbered pages, and clear section headings. Appendices should contain raw data, extended calculations, or code snippets – only include material essential to understanding the main text.
全文使用一致的引用格式（如哈佛、APA）。每个来源，包括数据集和软件，都必须在正文和文献列表中注明。格式指南：12 磅可读字体，1.5 倍行距，页码标注，以及清晰的章节标题。附录应包含原始数据、扩展计算或代码片段——仅放入理解正文所必需的材料。
9. Common Pitfalls to Avoid | 常见错误规避
- Choosing a topic that is too broad or too descriptive without genuine mathematical analysis.
- Neglecting to define variables, leaving the reader confused.
- Over-reliance on screenshots of calculators without explaining the mathematical steps.
- Writing a passive, impersonal account; the Pre-U expects personal voice and engagement.
- Ignoring the word count limit – excessive length can signal poor organisation.
避免以下常见错误：
- 选题过宽或过于描述性，缺乏真正的数学分析。
- 忽略定义变量，使读者困惑。
- 过度依赖计算器截图，而不解释数学步骤。
- 写作被动、不带个人色彩；Pre-U 期望个人声音和参与。
- 忽视字数限制——过长篇幅可能显示组织不力。
10. Sample Essay Extract with Commentary | 范文摘录及点评
Below is an annotated excerpt from a high-scoring Personal Investigation titled “Modelling the Spread of Misinformation on Social Media Using a Modified SEIR Framework”. The extract demonstrates how to integrate mathematical rigour with clear exposition. (Note: This is an invented example for illustrative purposes.)
以下是选自高分个人探究 “运用修正 SEIR 框架建模社交媒体虚假信息传播” 的带批注摘录。摘录展示了如何将数学严谨性与清晰阐述相结合。（注：此为说明性虚构示例。）
English extract:
“To capture the latency period during which an individual has been exposed to misinformation but is not yet sharing it, we introduce a fourth compartment E (exposed). The resulting SEIR model is governed by the system of ordinary differential equations: dS/dt = μ N – β S I / N – μ S, dE/dt = β S I / N – σ E – μ E, dI
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