📚 Pre-U CIE Mathematics: Experimental/Practical Assessment Key Points | Pre-U CIE 数学：实验/实践考核要点
In the Cambridge Pre-U Mathematics qualification (9794), the ‘experimental/practical’ component is the Mathematical Exploration – an internally assessed, externally moderated investigation. This coursework element counts for 25% of the total marks and requires candidates to independently explore a mathematical topic, demonstrating skills in modelling, reasoning, and communication. The following article unpacks every key point you need to master for success.
在剑桥 Pre-U 数学资格证书 (9794) 中，“实验/实践”部分指的是数学探究——一项内部评估、外部审核的调查作业。这部分课程作业占总分的 25%，要求考生独立探索一个数学主题，展现建模、推理和沟通能力。下文将深入解析成功完成该考核所需掌握的每一个关键要点。
1. Understanding the Nature of the Mathematical Exploration | 理解数学探究的本质
The Mathematical Exploration is not a timed practical exam but an extended piece of written work, typically 12–20 pages, produced over several weeks under teacher supervision. It allows you to investigate an area of mathematics that interests you, using techniques from Pure Mathematics, Statistics, Mechanics or Discrete Mathematics, depending on your Paper 2 option.
数学探究不是限时实操考试，而是一份较长的书面作业（通常 12–20 页），在教师监督下利用数周时间完成。它让你可以根据自己的兴趣，运用纯数学、统计学、力学或离散数学中的技巧进行探究，具体方向取决于你试卷二所选的模块。
2. Formal Assessment Criteria and Weighting | 正式评估标准与权重
The exploration is marked internally by your teacher and moderated by Cambridge. The assessment criteria are divided into five strands: Presentation (4 marks), Mathematical Communication (4 marks), Personal Engagement (4 marks), Reflection (4 marks), and Use of Mathematics (14 marks). The total is 30 marks, scaled to 25% of the overall qualification.
探究由你的教师内部评分，剑桥进行外部审核。评分标准分为五个方面：呈现 (4 分)、数学交流 (4 分)、个人投入 (4 分)、反思 (4 分) 以及数学运用 (14 分)。总分为 30 分，按比例换算后占整体资格分的 25%。
| Criterion / 标准 | Marks / 分数 | Focus / 关注点 |
|---|---|---|
| Presentation | 4 | Structure, readability, use of technology |
| Mathematical Communication | 4 | Clear and precise use of mathematical language, notation, graphs |
| Personal Engagement | 4 | Evidence of genuine curiosity, independent thought, ownership |
| Reflection | 4 | Critical evaluation of the process, results, and limitations |
| Use of Mathematics | 14 | Sophistication, accuracy, relevance of mathematical methods |
3. Choosing a Suitable Topic: The Starting Point | 选择合适的主题：起点
Your topic must be mathematically rich and go beyond standard syllabus content. It should allow for an investigation rather than a simple report. Good topics often arise from modelling real-world data, exploring a puzzle, or extending a known theorem. Examples include ‘Modelling the spread of a rumour using differential equations’, ‘Investigating the irrationality of √2 using continued fractions’, or ‘Optimising the shape of a soda can using calculus and surface area minimisation’.
你的主题必须具有数学深度，并超越常规大纲内容。它应当允许进行探究，而不是简单的报告。好的主题通常源自对现实世界数据的建模、探索一个谜题或对已知定理的拓展。例如：“使用微分方程对谣言传播的建模”、“利用连分数探究 √2 的无理性”，或“使用微积分和表面积最小化优化铝罐形状”。
4. Structuring Your Exploration Like a Scientific Report | 以科学报告结构组织探究
An effective exploration follows a clear logical flow: Introduction (purpose, rationale), Background mathematics, Main investigation (methods, development, data analysis), Conclusion and reflection. Use section headings to guide the reader. Tables, graphs, and diagrams must be numbered, titled, and referred to in the text.
一个有效的探究遵循清晰的逻辑流程：引言（目的、理由）、数学背景、主体探究（方法、展开、数据分析）、结论与反思。使用小标题引导读者。表格、图表和图形必须编号、加标题，并在正文中引用。
5. Demonstrating Personal Engagement Authentically | 真实展现个人投入
Personal engagement is not just stating ‘I found this interesting’. It is evidenced by the choice of a unique angle, by testing multiple approaches, by programming a simulation, or by connecting mathematics to a personal hobby such as music or sport. For example, if you are a runner, you could model your own 5k splits using a parametric curve, showing genuine ownership of the data.
个人投入不仅仅体现在说“我觉得这很有趣”。它通过选择独特的角度、尝试多种方法、编写模拟程序，或将数学与个人爱好（如音乐或运动）联系起来加以证明。例如，如果你是一名跑步者，你可以用参数曲线建模自己的 5 公里分段成绩，展示对数据的真正拥有感。
6. Mathematical Communication: Precision is Everything | 数学交流：精准至上
Use accurate mathematical notation throughout. Define all variables clearly at first use. Avoid vague phrases like ‘the graph goes up’; instead write ‘the gradient of f(x) is positive for x > 2’. Equations should be displayed on separate lines, centred, and numbered if referred to later. Consistent units and significant figures are vital.
始终使用准确的数学符号。首次使用变量时明确定义。避免模糊用语，如“图形往上走”；应写“对于 x > 2，f(x) 的梯度为正”。公式应单独成行、居中，若后文引用则编号。单位一致和有效数字至关重要。
7. The Role of Technology: Calculators, Spreadsheets, and Coding | 技术的作用：计算器、电子表格与编程
Technology can elevate your exploration. You may use graphical calculators, Desmos, GeoGebra, Excel, or Python. However, mere screen captures without explanation are not enough. You must interpret outputs, analyse the mathematical implications, and reflect on the limitations of the technology used, such as rounding errors or restricted domain limits.
技术可以提升你的探究。你可以使用图形计算器、Desmos、GeoGebra、Excel 或 Python。然而，仅在文章里贴截图而不加解释是不够的。你必须解读输出结果，分析数学意义，并反思所用技术的局限性，例如舍入误差或有限的定义域限制。
8. Embedding Critical Reflection Throughout | 全程嵌入批判性反思
Reflection is not an afterthought paragraph at the end. It should appear at multiple points: comparing different models, discussing why an initial assumption failed, or evaluating whether a result is realistic. For example, ‘The linear model initially predicted a negative population after 10 years, which is impossible, so a logistic model was introduced.’ This shows ongoing mathematical thinking.
反思不是文末附加的一个段落。它应该出现在多个节点：比较不同模型，讨论初始假设为何失败，或评估结果是否现实。例如，“线性模型最初预测 10 年后人口为负，这不可能，因此引入了逻辑模型。”这展示了持续的数学思考。
9. Use of Mathematics: Sophistication and Rigour | 数学的运用：精湛与严谨
The highest marks in this criterion demand mathematics that is precise, appropriate to the problem, and goes beyond routine calculations. This might involve finding an analytical solution to a differential equation, performing a hypothesis test on collected data, or proving a conjecture you developed. Simply applying a formula from the formula sheet will not score highly. Aim for thorough understanding and correct execution.
该标准的高分要求数学精确、与问题相适，并且超越常规计算。这可能涉及求解微分方程的解析解、对收集的数据进行假设检验，或证明你提出的猜想。仅仅套用公式表里的公式不会得高分。追求透彻的理解和正确的执行。
10. Common Pitfalls and How to Avoid Them | 常见误区及规避方法
Many students lose marks by being overambitious with a topic they do not fully understand, resulting in mathematical errors. Others forget to define a non-standard function or mislabel axes. Plagiarism, even of a single paragraph, leads to disqualification. Poor time management often causes rushed conclusions. Plan a timeline with your teacher, and have a clear, focused research question from the start.
许多学生因选择自己并不完全理解的过大题目而丢分，导致数学错误。还有人忘记定义非标准函数或未给坐标轴标注。抄袭，哪怕只是一个段落，也会导致取消资格。时间管理不善通常导致草率收尾。与老师一起制定时间表，并从一开始就有一个明确、聚焦的研究问题。
11. The Teacher’s Role and Submission Deadlines | 教师的角色与提交截止日期
Your teacher can provide general guidance and feedback on one draft only. They cannot correct detailed errors or rewrite sections for you. All work must be your own. Cambridge sets an annual submission deadline, usually in May for the June series. Internal deadlines for drafts are set by your centre; missing them can seriously jeopardise your overall grade.
你的老师可以提供一般性指导，并且只能对一份草稿给出反馈。他们不能替你纠正具体错误或重写段落。所有工作必须由你独立完成。剑桥设定年度提交截止日期，通常六月考季在五月。草稿的内部截止日期由学校设定；错过这些期限会严重影响你的整体成绩。
12. Final Checklist for a Top-Scoring Exploration | 高分探究终检清单
Before submission, verify that: the title is clear and precise; the exploration is within the 12–20 page limit (excluding appendices); all sources are cited in a bibliography; any data used is referenced; the mathematics is correct and well-explained; personal engagement is evident; and the reflection critically evaluates the strengths and weaknesses of your approach.
提交前，请确认：标题清晰准确；探究篇幅在 12–20 页之内（不含附录）；所有来源均在参考书目中列出；所使用的任何数据都有出处；数学内容正确且解释充分；个人投入显而易见；反思批判性地评价了所用方法的优缺点。
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