📚 IGCSE CAIE History: Formula & Theorem Quick Reference | IGCSE CAIE 历史：公式定理速查手册
This is not a mathematics cheat sheet. In IGCSE CAIE History, "formulas" mean repeatable reasoning patterns for causation, source evaluation and essay structure. They help you turn scattered factual knowledge into exam-ready analytical arguments.
这不是数学公式表。在 IGCSE CAIE 历史中，"公式"指的是因果关系、史料评估和论文结构的可复现推理框架。它们能帮你把零散的事实知识转化为符合考试要求的分析性论证。
The article below collects the reasoning frameworks that examiners at Cambridge reward most often: how to build a layered explanation of why something happened, how to judge the usefulness of a source in four steps, and how to write an analytical paragraph that ends in a verdict rather than a summary. Each "theorem" is followed by a worked example drawn from the Core Content and Depth Studies of the CAIE 0470 syllabus.
下文汇集的是剑桥考官最常给分的推理框架：如何分层解释某一事件为何发生、如何在四个步骤内判断一份史料的价值、以及如何写出一段以"结论"而非"复述"收尾的分析段落。每一条"定理"都附带一个取自 CAIE 0470 考纲核心内容与深度研究主题的实操示例。
1. What Counts as a "Formula" in History | 历史学科里的"公式"是什么
In a science paper a formula produces one correct numerical answer. In history a formula produces a defensible pattern of reasoning. The examiner is not checking whether you reached "the" cause of the First World War; the examiner is checking whether your explanation is layered, evidenced and weighed. That is why historical formulas are written as word equations rather than numbers.
在理科试卷中，公式导出唯一正确的数值答案。在历史学科中，公式导出的是可辩护的推理模式。考官并不检查你是否找到了第一次世界大战"那个"原因，而是检查你的解释是否有层次、有证据、有权衡。这就是为什么历史公式写成文字等式而非数字等式。
Three families of formulas cover almost every question on Paper 1, Paper 2 and the coursework component. The first family explains causes and consequences (why did it happen, what did it change). The second family handles sources (what is this, how useful is it, why was it made). The third family builds written answers (how to paragraph, how to argue, how to judge). Learn the three families and you will stop staring at a blank page.
三大类公式几乎可以覆盖 Paper 1、Paper 2 和课程作业部分的每一道题。第一类解释原因与后果（为何发生、改变了什么）；第二类处理史料（这是什么、有多大用、为何制作）；第三类搭建书面答案（如何分段、如何论证、如何下判断）。掌握这三类公式，你就不会再对着空白的答题纸发呆。
These frameworks also map neatly onto the assessment objectives: AO1 rewards accurate knowledge, AO2 rewards explanation and evaluation, AO3 rewards the use of sources. A formula is simply a way of guaranteeing that all three objectives appear in every answer you write.
这些框架也恰好对应评分目标：AO1 奖励准确的知识，AO2 奖励解释与评价，AO3 奖励史料运用。公式的作用，就是保证你写下的每一个答案都同时体现这三个目标。
2. The Causation Theorem | 因果推理定理
For most "why" or "explain" questions, use the three-layer cause formula: long-term causes set the background, medium-term causes create the fault line, and a short-term trigger starts the event.
大多数"为什么"或"解释"类题目，使用三层原因公式：长期原因构成背景，中期原因形成断层线，短期导火索引发事件。
Cause = Background (long-term) + Fault line (medium-term) + Trigger (short-term)
The long-term layer explains why the situation was unstable in the first place: ideologies, economic structures, rivalries, social tensions. The medium-term layer explains why the tension became organised and dangerous: alliances, plans, institutions, decisions taken months or years earlier. The short-term layer explains why it happened in that particular week: an assassination, a vote, a telegram, a border incident.
长期层解释局势起初为何就不稳定：意识形态、经济结构、宿敌关系、社会张力。中期层解释紧张为何被组织化并变得危险：同盟、计划、制度、数月或数年前做出的决定。短期层解释它为何偏偏发生在那一周：一次刺杀、一次表决、一封电报、一次边境冲突。
The most common mistake is to spend three paragraphs on the trigger and one sentence on everything else. The trigger decides the timing; the deeper layers decide whether the event was possible at all. Always weight the layers - a good answer states which layer mattered most and why.
最常见的错误是用三段写导火索，却用一句话打发其余部分。导火索决定时间，深层原因决定事件是否可能发生。务必给各层分配权重——好的答案会明确说明哪一层最重要以及为什么。
| Layer 层级 | Function 功能 | Useful connectives 常用连接语 |
| Long-term 长期 | Creates conditions 创造条件 | "This made conflict likely because..." "这使冲突变得可能，因为..." |
| Medium-term 中期 | Turns tension into machinery 把紧张变成机制 | "This created the fault line along which..." "这形成了断裂线，沿此..." |
| Short-term 短期 | Sets the timing 决定发生时间 | "This triggered, but did not cause..." "这引爆了，但并未造成..." |
3. The Categorising Formula | 原因分类公式
Layers deal with time; categories deal with type. Use the four-fold formula Political, Economic, Social, Cultural (PESC) to make sure your explanation is broad enough for a high mark.
分层处理的是时间维度；分类处理的是类型维度。使用政治、经济、社会、文化四分法（PESC），确保你的解释足够全面从而拿到高分。
Explanation = Political + Economic + Social + Cultural / Religious
Political causes cover government decisions, treaties, leadership and power struggles. Economic causes cover trade, industry, unemployment, inflation, resources and taxation. Social causes cover class, gender, migration, living conditions and popular attitudes. Cultural or religious causes cover nationalism, ideology, propaganda, education and belief. A typical weak answer uses only the political category; a strong answer combines at least three.
政治原因包括政府决策、条约、领导人与权力斗争。经济原因包括贸易、工业、失业、通胀、资源与税收。社会原因包括阶级、性别、移民、生活条件与大众心态。文化或宗教原因包括民族主义、意识形态、宣传、教育与信仰。典型的中等答案只用政治一类；高分答案至少结合三类。
When a question asks how far one factor was the most important, PESC gives you a ready-made comparison grid. You can say that the economic factor outweighed the political one because its effects were longer lasting and affected more people - that sentence is a judgement, not a list.
当题目问某个因素在多大程度上最重要时，PESC 就是现成的比较框架。你可以说经济因素重于政治因素，因为其影响持续更久、波及人群更广——这句话是判断，不是罗列。
4. Applying the Causation Formula: The First World War | 因果公式实操：第一次世界大战
(long-term) + Fault line (medium-term) + Spark (short-term). For the First World War: nationalism and imperialism (long-term) + alliance system and war plans (medium-term) + the assassination at Sarajevo (short-term).
示例公式：原因 = 背景（长期）+ 断层线（中期）+ 导火索（短期）。以第一次世界大战为例：民族主义与帝国主义（长期）+ 同盟体系与战争计划（中期）+ 萨拉热窝刺杀（短期）。
Long-term: rival nationalisms in the Balkans and great-power imperialism created a Europe in which states expected to gain from a war and feared losing from peace. Medium-term: the Triple Alliance and the Triple Entente turned a regional quarrel into a continental one, while detailed mobilisation timetables meant that a decision taken in Vienna or Berlin could not be reversed once the machinery started. Short-term: the assassination of Archduke Franz Ferdinand in Sarajevo on 28 June 1914 provided the occasion, and the July Crisis turned it into war.
长期：巴尔干地区相互对立的民族主义与列强的帝国主义，塑造出一个各国都指望从战争中获利、又害怕在和平中受损的欧洲。中期：三国同盟与三国协约把地区争执升级为大陆冲突，而详尽的动员时间表意味着机器一旦启动，在维也纳或柏林做出的决定就无法撤回。短期：1914 年 6 月 28 日萨拉热窝的斐迪南大公遇刺提供了导火索，七月危机把它变成了战争。
Now apply weighting. The assassination explained the timing of the war, but not its scale; the alliance system explained the scale, but not the hostility behind it; nationalism and imperialism explained the hostility, but not the week. This three-sentence chain is exactly what a level-4 explanation looks like on Paper 1.
现在进行权衡。刺杀解释了战争的时间，却解释不了其规模；同盟体系解释了规模，却解释不了其背后的敌意；民族主义与帝国主义解释了敌意，却解释不了那一周。这三句话构成的链条，正是 Paper 1 中第四等级解释的样子。
5. The Consequence and Significance Theorem | 后果与意义定理
Consequences questions ("What were the effects of...?") and significance questions ("How significant was...?") need a measuring formula rather than another causal chain.
后果题（"……产生了哪些影响？"）与意义题（"……有多重要？"）需要的是一把衡量尺，而不是再写一条因果链。
Significance = Breadth × Depth × Duration
Breadth asks how many people, regions or states were affected. Depth asks how fundamentally structures changed - a treaty that redraws borders is deep, a treaty that only changes a tariff is shallow. Duration asks how long the effect lasted, whether it was reversed, and whether later generations still felt it. Multiplying rather than adding is deliberate: an event that is broad but shallow is usually not significant.
广度问有多少人、地区或国家受影响。深度问结构被改变得多么根本——重划边界的条约是深层的，只改关税的条约是浅层的。时长问影响持续多久、是否被逆转、后代是否仍感受到它。用乘法而非加法是刻意的：影响广但浅的事件，通常并不重要。
Also separate immediate from long-term consequences. The immediate consequence of the Treaty of Versailles was German disarmament and territorial loss; the long-term consequence is often argued to be resentment that fed political extremism in the 1920s and 1930s. Naming both, then stating which mattered more, converts a description into an evaluation.
还要区分直接后果与长期后果。《凡尔赛条约》的直接后果是德国解除武装与领土丧失；长期后果则常被论证为滋生了怨恨，进而助长了 20 世纪二三十年代的政治极端主义。两者都点出，再说明哪个更重要，就把描述变成了评价。
6. The Change and Continuity Theorem | 变与不变定理
Questions about change over time are answered with a four-part formula rather than a story.
关于"随时间变化"的题目，要用四要素公式回答，而不是讲故事。
Change = Direction + Pace + Extent + Reversal
Direction: what changed and in which direction (more democratic, more centralised, more equal). Pace: was it gradual reform or sudden revolution? Extent: did it affect law only, or also daily life? Reversal: was the change later undone, and by whom? Continuity must be named explicitly too - many essay answers lose marks for describing change in detail while never stating what stayed the same.
方向：什么发生了变化，朝哪个方向变（更民主、更集权、更平等）。速度：是渐进改良还是突进革命？程度：只改变了法律，还是也改变了日常生活？逆转：这种变化后来是否被撤销，被谁撤销？不变同样必须明确写出——许多论述题答案丢分，正是因为详细描述了变化，却从未说明什么保持不变。
This formula is especially useful for Depth Studies such as Russia 1905-41, Germany 1918-45, the USA 1919-41 or China 1930-90, where examiners frequently ask whether a period was a turning point. A turning point is simply a change with high extent and low reversal.
这一公式对深度研究主题尤其有用，例如 1905-1941 年的俄国、1918-1945 年的德国、1919-1941 年的美国或 1930-1990 年的中国，考官常问某段时期是否是转折点。所谓转折点，就是程度高、逆转低的变化。
7. The Source Evaluation Formula: OPCT | 史料评估公式：OPCT
Paper 2 rewards a four-step reading of every source. The formula is OPCT: Origin, Purpose, Content, Tone.
Paper 2 奖励对每份史料进行四步解读。公式是 OPCT：来源、目的、内容、语气。
| Step 步骤 | Question to ask 要问的问题 | Exam value 考试价值 |
| Origin 来源 | Who made it, when, where, and in what position? 谁制作、何时、何地、身份如何？ | Reveals the viewpoint and possible bias 揭示立场与可能的偏见 |
| Purpose 目的 | Why was it produced and for which audience? 为何制作、给谁看？ | Explains exaggeration, omission and selection 解释夸大、省略与取舍 |
| Content 内容 | What does it claim, and what does it leave out? 它主张什么、遗漏什么？ | Supplies the evidence you quote 提供你引用的证据 |
| Tone 语气 | How is it said - emotive, statistical, official, satirical? 如何表述——情绪化、数据化、官方、讽刺？ | Signals the attitude of the producer 透露制作者的态度 |
(Origin and Purpose as limits or support) + (Tone as evidence of attitude). The examiners' hat provenance should modify what the content proves, never replace it. A sentence such as "Source A is biased because it is a German newspaper" earns little; " 4, so its claim that Russia alone mobilised serves the purpose of justifying German action" earns a great deal.
写成等式：史料价值 = 内容 +（来源与目的作为限制或支撑）+（语气作为态度的证据）。考官的规则是：史料出处应修饰内容所证明的东西，而绝不能取代内容。像"史料 A 有偏见，因为它是德国报纸"这样的句子得分很低；而"史料 A 是 1914 年 7 月的德国报纸，因此其'只有俄国在动员'的说法服务于为德国行动辩护的目的"则得分很高。
8. Reliability and Usefulness: Two Different Questions | 可靠性与有用性：两个不同问题
Reliability and usefulness are not the same thing, and confusing them is one of the most common errors on Paper 2.
可靠性与有用性不是一回事，把二者混为一谈是 Paper 2 上最常见的错误之一。
Usefulness = What it tells you + What limits it + What it does not tell you
A Nazi propaganda poster is not reliable as a description of Jewish life, but it is extremely useful as evidence of how the regime wanted Germans to think. A single soldier's diary is unreliable about strategy, yet useful about morale. Usefulness depends on the historian's question; reliability depends on the source's accuracy.
一张纳粹宣传海报，作为对犹太人生活的描述并不可靠，但作为"该政权希望德国人如何思考"的证据却极其有用。一名士兵的日记在战略问题上不可靠，却对了解士气很有用。有用性取决于历史学者的问题，可靠性取决于史料的准确性。
When answering "How useful is Source A to a historian studying X?", use three moves: state what the source does show, using its content; explain how origin and purpose shape it; and state what else the historian would need - a second source, statistics, or government records. The third move is what separates a top-band answer from a competent one.
回答"史料 A 对研究 X 的历史学者有多大用处？"时，走三步：用其内容说明它确实展示了什么；解释来源与目的如何塑造它；说明历史学者还需要什么——第二份史料、统计数据或政府档案。第三步是高分答案与合格答案
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