📚 Pre-U WJEC Mathematics: Essay Writing Framework & Model Essays | Pre-U WJEC 数学：论文写作框架与范文
The Pre-U WJEC Mathematics investigation (Unit 3) is a unique opportunity to demonstrate independent research, structured reasoning and mathematical communication. Writing a high-scoring essay requires a clear framework, a well-chosen topic and the ability to blend rigorous mathematics with coherent narrative.
Pre-U WJEC 数学探究（Unit 3）是一个展示独立研究、结构化推理和数学交流能力的独特机会。写出一篇高分论文需要清晰的框架、精心选择的题目，以及将严谨数学与连贯叙述融合的能力。
This guide provides a complete writing framework and a model essay excerpt, helping you understand what examiners look for and how to structure your own submission for maximum marks.
本指南提供完整的写作框架和一篇范文节选，帮助你理解考官关注的重点，以及如何构建自己的论文以获得最高分数。
1. Understanding the Pre-U Mathematics Investigation | 理解 Pre-U 数学探究
The Unit 3 investigation counts for a significant proportion of your Pre-U Mathematics award. You are expected to produce a 1500–2500 word essay on a mathematical topic of your choice, demonstrating research, problem-solving and evaluative skills.
Unit 3 探究在 Pre-U 数学总分中占很大比重。你需要选择一个数学话题，撰写一篇 1500–2500 字的论文，展示研究、解决问题的能力与评价能力。
The essay is not a standard textbook exercise. It requires you to explore beyond the syllabus, consult academic sources, and present your findings in a well-structured academic style.
这篇论文不是标准课本练习。它要求你探索考纲以外的内容，查阅学术资料，并以结构良好的学术风格呈现研究结果。
Examiners assess five key areas: Mathematical Content, Use of Sources, Organisation and Communication, Personal Engagement, and Critical Evaluation.
考官评估五个关键领域：数学内容、资料使用、组织与交流、个人投入以及批判性评价。
2. Selecting a Suitable Topic | 选择合适的题目
Choose a topic that genuinely interests you and allows for mathematical depth. It should be narrow enough to be manageable, yet rich enough to sustain a full essay. Avoid topics that are purely computational or limited to A-level standard techniques.
选择一个你真正感兴趣且具备数学深度的题目。题目应该窄到可以驾驭，但又要足够丰富以支撑整篇论文。避免纯计算类或仅限于 A-level 标准技巧的题目。
Good topics often bridge pure and applied mathematics, such as fractals, dynamical systems, cryptography, optimisation or the mathematics of voting systems. A historical angle can also work, provided the mathematics is central.
好的题目往往连接纯数学与应用数学，例如分形、动力系统、密码学、优化或投票系统的数学。历史角度也可行，但必须确保数学内容处于核心地位。
Before committing, check the availability of accessible academic sources. A preliminary search on Google Scholar or JSTOR can confirm whether you have enough material to reference.
在确定题目之前，检查是否有可获取的学术资料。在 Google Scholar 或 JSTOR 上初步搜索，可以确认是否有足够的参考文献。
3. Structuring Your Essay: The Classic Framework | 论文结构：经典框架
Your essay should follow a logical progression, guiding the reader from motivation to conclusion. The recommended structure includes: Abstract, Introduction, Background Theory, Methodology/Analysis, Results, Discussion, Conclusion, and References.
论文应遵循逻辑递进，引导读者从动机走向结论。推荐的结构包括：摘要、引言、背景理论、方法论/分析、结果、讨论、结论和参考文献。
An abstract is optional but recommended. It summarises the aim, method and main findings in about 150 words. The introduction should clearly state the research question and set the scope.
摘要虽为可选但建议包含。它用约 150 字概括研究目的、方法和主要发现。引言应清晰陈述研究问题并设定范围。
Each section must have a clear heading, and paragraphs should be short and focused. Use transitions to show how each part contributes to the whole argument.
每一部分必须有清晰的标题，段落应简短且聚焦。使用过渡语句展示每一部分如何为整体论证服务。
4. Introduction: Setting the Scene | 引言：设定场景
The introduction should hook the reader by explaining why the topic is interesting or important. Begin with a broad context, then narrow down to your specific research question. State your objectives clearly.
引言应通过解释话题为何有趣或重要来吸引读者。从广泛背景入手，然后收窄到具体的研究问题。清晰地陈述你的目标。
For example, if your essay explores the SIR epidemic model, you might open with the relevance of mathematical modelling in public health before specifying that you will analyse the threshold condition R₀.
例如，如果你的论文探讨 SIR 流行病模型，可以在开篇提到数学建模在公共卫生中的相关性，然后具体说明你将分析阈值条件 R₀。
Avoid vague statements. Instead, write something like: “This investigation aims to derive the conditions under which an epidemic will die out, and to evaluate the sensitivity of the model to parameter changes.”
避免模糊陈述。可以写成：“本研究旨在推导流行病消亡的条件，并评估模型对参数变化的敏感性。”
5. Literature Review or Background Theory | 文献综述或背景理论
This section demonstrates your reading and situates your work in the existing body of knowledge. Summarise key results from textbooks or papers that you will use or extend. Do not simply copy equations; explain their significance.
这一部分展示你的阅读量，并将你的工作置于现有知识体系中。总结你将要使用或拓展的教科书或论文中的关键结果。不要只是照搬方程，要解释其意义。
For instance, when discussing the logistic differential equation dP/dt = rP(1 − P/K), you might mention its origins in population ecology and how the carrying capacity K stabilises the solution.
例如，在讨论逻辑斯蒂微分方程 dP/dt = rP(1 − P/K) 时，你可以提及它在种群生态学中的起源，以及承载容量 K 如何使解稳定。
Cite every source using a consistent referencing style, such as APA. The bibliography should be a working list, not an afterthought.
使用一致的引用格式（如 APA）引用每个来源。参考文献列表应当伴随写作过程，而非事后补充。
6. Methodology and Approach | 方法论与途径
Describe the mathematical methods you will use and justify your choices. If you are solving equations analytically, state the techniques (separation of variables, eigenvalue analysis, etc.). If using numerical methods, explain the algorithms and your choice of software or coding language.
描述你将使用的数学方法，并论证你的选择。如果你是解析求解方程，说明所用技巧（如分离变量法、特征值分析等）。若用数值方法，解释算法以及选择软件或编程语言的理由。
Be precise. Instead of writing “I simulated the system,” write: “I implemented a fourth-order Runge-Kutta method in Python to solve the system of coupled ODEs, with a time step of 0.01.”
力求精确。与其写“我对系统进行了模拟”，不如写：“我在 Python 中实现了四阶 Runge-Kutta 方法求解耦合常微分方程组，时间步长取 0.01。”
This section also allows you to show personal engagement by discussing any programming challenges you overcame or analytical insights you developed.
这一部分还能让你展示个人投入，比如你克服的编程难题或发展出的解析见解。
7. Results and Mathematical Derivation | 结果与数学推导
Present your findings in a logical order. State conjectures, derive key formulas step by step, and illustrate results with tables, graphs or diagrams. Keep the algebra tidy and use notation consistently.
按照逻辑顺序展示你的发现。提出猜想，逐步推导关键公式，并用表格、图形或图表来说明结果。保持代数推导整洁，符号使用一致。
For example, when analysing the logistic map xₙ₊₁ = r xₙ (1 − xₙ), you might show how fixed points are found by solving x = r x (1 − x), yielding x∗ = 0 and x∗ = (r − 1)/r.
例如，在分析逻辑斯蒂映射 xₙ₊₁ = r xₙ (1 − xₙ) 时，可以展示如何通过求解 x = r x (1 − x) 找到不动点，得到 x∗ = 0 和 x∗ = (r − 1)/r。
Use center-aligned equations for key results, like:
x∗ = (r − 1)/r, valid for r > 1
关键结果使用居中对齐方程，如：
x∗ = (r − 1)/r，对 r > 1 成立
Do not hide algebraic steps; the examiner wants to see your reasoning. Number equations for easy reference, e.g., (1), (2).
不要隐藏代数步骤；考官希望看到你的推理过程。为方程编号以便引用，如 (1)、(2)。
8. Discussion and Analysis | 讨论与分析
Interpret your results and connect them back to your research question. Explain what the mathematical outcomes mean in the real-world context of your topic. Discuss any limitations or surprises.
解读你的结果，并将它们与你的研究问题联系起来。解释这些数学结果在你的话题的现实世界语境中意味着什么。讨论任何局限性或意外发现。
In the logistic map investigation, you could discuss the onset of period-doubling and chaos as r increases beyond 3.56995…, relating this to the Feigenbaum constants and universality.
在逻辑斯蒂映射探究中，你可以讨论当 r 增大超过 3.56995… 时出现的倍周期分岔和混沌，将这与费根鲍姆常数和普遍性联系起来。
Critical evaluation is essential. Acknowledge assumptions made, such as the model ignoring migration or seasonal variation, and suggest how they could be relaxed.
批判性评价至关重要。承认所作假设，例如模型忽略了迁徙或季节性变化，并建议如何放宽这些假设。
9. Conclusion and Evaluation | 结论与评价
Summarise your main findings without introducing new material. Restate the answer to your research question and reflect on the investigation process. What did you learn? What would you do differently next time?
总结主要发现，不要引入新内容。重申研究问题的答案，并反思探究过程。你学到了什么？下次你会有什么不同的做法？
A strong conclusion links back to the introduction, giving a sense of closure. For instance: “This investigation confirmed that the basic reproduction number R₀ = β/γ determines the long-term behaviour of the SIR model, but also revealed limitations in assuming a homogeneously mixing population.”
一个有力的结论能呼应引言，带来完结感。例如：“本研究证实，基本再生数 R₀ = β/γ 决定了 SIR 模型的长期行为，但也暴露了假设人群均匀混合的局限性。”
10. Referencing and Academic Integrity | 引用与学术诚信
All sources must be acknowledged. Plagiarism is treated very seriously by WJEC. If you quote a definition or paraphrase an idea, cite the author and year. Your bibliography should list every source you consulted, formatted consistently.
所有来源必须致谢。WJEC 对剽窃行为处理非常严格。如果你引用定义或转述观点，需注明作者和年份。参考文献列表应按统一格式列出所有你查阅过的来源。
A typical book reference: Strogatz, S.H. (2015). Nonlinear Dynamics and Chaos. 2nd ed. Westview Press.
典型的书籍引用格式：Strogatz, S.H. (2015). 非线性动力学与混沌. 第 2 版. Westview 出版社.
For websites, include the URL and access date. Use a reference manager like Zotero if possible.
对于网站，需包含网址和访问日期。可能的话，使用 Zotero 等参考文献管理工具。
11. A Worked Example: Investigating the Logistic Map | 范文示例：探究逻辑斯蒂映射
Below is a condensed extract from a high-scoring Pre-U essay on the logistic map. It illustrates how the framework is applied in practice. The full essay would include more diagrams and extended derivations.
以下是一篇高分 Pre-U 论文关于逻辑斯蒂映射的浓缩节选，展示了如何在实践中应用该框架。完整论文将包含更多图表和扩展推导。
Excerpt: Introduction
The logistic map, xₙ₊₁ = r xₙ (1 − xₙ), is a deceptively simple recurrence relation that exhibits a stunning range of dynamical behaviour, from stable fixed points to deterministic chaos. Despite its discrete nature, it has been used to model population growth with resource limitation. This investigation aims to characterise the bifurcation structure of the map as the control parameter r varies, and to verify the Feigenbaum constant δ numerically.
节选：引言
逻辑斯蒂映射 xₙ₊₁ = r xₙ (1 − xₙ) 是一个看似简单的递推关系，却展现出从稳定不动点到确定性混沌的丰富动力学行为。尽管是离散形式，它已被用于模拟具有资源限制的种群增长。本研究旨在刻画随控制参数 r 变化时分岔结构的特征，并从数值上验证费根鲍姆常数 δ。
Excerpt: Derivation of Fixed Points
Setting xₙ₊₁ = xₙ = x∗ yields the algebraic equation x∗ = r x∗ (1 − x∗). Factoring gives x∗ (1 − r + r x∗) = 0, producing the trivial fixed point x∗ = 0 and the nontrivial fixed point x∗ = (r − 1)/r. The stability of these points can be determined by examining the derivative f'(x) = r − 2 r x. At x∗ = 0, f'(0) = r, so the origin is stable for 0 < r < 1. At x∗ = (r − 1)/r, f' = 2 − r, indicating stability for 1 < r < 3.
节选：不动点推导
令 xₙ₊₁ = xₙ = x∗ 得到代数方程 x∗ = r x∗ (1 − x∗)。因式分解得 x∗ (1 − r + r x∗) = 0，产生平凡不动点 x∗ = 0 和非平凡不动点 x∗ = (r − 1)/r。这些点的稳定性可通过导数 f'(x) = r − 2 r x 来判断。在 x∗ = 0 处 f'(0) = r，因此当 0 < r < 1 时原点稳定。在 x∗ = (r − 1)/r 处 f' = 2 − r，表明当 1 < r < 3 时稳定。
Excerpt: Bifurcation Diagram and Feigenbaum Constant
Numerically iterating the map and plotting the asymptotic values of x against r reveals a cascade of period doublings. By measuring the r-values at which successive bifurcations occur (r₁ ≈ 3, r₂ ≈ 3.44949, r₃ ≈ 3.54409, r₄ ≈ 3.56441), the ratio (rₙ − rₙ₋₁)/(rₙ₊₁ − rₙ) approaches the Feigenbaum constant δ ≈ 4.669. This universal constant appears not only in the logistic map but also in many other chaotic systems, indicating a deep mathematical structure.
节选：分岔图与费根鲍姆常数
通过数值迭代该映射，并绘制 x 的渐近值随 r 的变化图，可看到一连串倍周期分岔。测量相继分岔发生的 r 值（r₁ ≈ 3, r₂ ≈ 3.44949, r₃ ≈ 3.54409, r₄ ≈ 3.56441），比值 (rₙ − rₙ₋₁)/(rₙ₊₁ − rₙ) 趋近于费根鲍姆常数 δ ≈ 4.669。这一普适常数不仅出现在逻辑斯蒂映射中，也出现在许多其他混沌系统中，表明深层的数学结构。
12. Final Tips for Top Marks | 获得高分的关键提示
Start early and keep a research diary. Record your ideas, dead ends and breakthroughs; this material can enrich your reflection and personal engagement marks.
尽早开始并写研究日志。记录你的想法、死胡同和突破；这些材料可以丰富你的反思，提高个人投入的得分。
Use graphs and tables generated by your own code or spreadsheets; these demonstrate technical skill. Ensure all figures are numbered, captioned and referred to in the text.
使用你自己编写代码或电子表格生成的图表；这能展示技术技能。确保所有图表都有编号、标题，并在正文中被引用。
Ask a teacher or peer to read your draft for clarity. Does the argument flow logically? Is the mathematics correct and well explained? Proofread for spelling, grammar and consistent notation.
请老师或同学阅读你的草稿以确保清晰。论证是否逻辑流畅？数学内容是否正确且解释清楚？仔细校阅拼写、语法和符号的一致性。
Finally, adhere strictly to the word limit and formatting guidelines provided by WJEC. A well-presented essay creates a positive first impression.
最后，严格遵守 WJEC 规定的字数和格式要求。排版良好的论文会给阅卷人留下积极的第一印象。
Published by TutorHao | Pre-U WJEC 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