IB数学 IA 探究报告 选题 评分标准 反思
IB 的数学两门课，分析数学（Analysis and approaches）与应用数学（Applications and interpretation），都要交一份内部评估，官方叫数学探究报告（mathematical exploration）。这份报告 20 分，占这门课最终成绩的 20%，与一整套外部试卷的分量相当。IB 官方指南写明，没有提交探究报告的学生，这门数学课不会拿到成绩。它由任课老师评分，再由 IB 外部抽样审核。现行考纲 2021 年 5 月首次评估，两门课用同一套分项标准。很多学生以为 IA 就是写一篇自己喜欢的数学，实际上它有明确的问题、明确的方法与明确的分项档位。下面按官方标准的五个分项，把选题、研究问题、写作结构到时间表拆开讲。
The IB Diploma mathematics courses, analysis and approaches and applications and interpretation, both require an internally assessed mathematical exploration, and that piece of work carries 20 marks, which is 20 per cent of the final subject grade. Students who do not submit an exploration do not receive a grade for the course, so the deadline matters as much as the mathematics. The report is marked by your teacher and then moderated externally by the IB, under the syllabus first assessed in May 2021. The sections below follow the five criteria in order, from choosing a topic to the timetable that keeps a draft moving.
IA 的 20 分在数学总成绩里的位置
现行考纲（2021 年 5 月首次评估）给 SL 的两张卷各 40% 与 IA 的 20%，给 HL 的两张卷各 30%、Paper 3 的 20% 与 IA 的 20%。SL 两张卷各 80 分、各 90 分钟；HL 两张卷各 110 分、各 120 分钟，再加一张 55 分的 Paper 3。换算下来，IA 的 20 分与 HL 那张 Paper 3 的分量相同，比 SL 卷面上任何一道大题都重。同一个 20 分标准，SL 与 HL 的档位描述分列，对数学深度的要求也不同。很多学生以为 IA 的分差顶多一两分，其实从 12 分到 18 分这一段，只要数学层次对得上，完全可控。外部卷靠一年半的积累，IA 的 20 分集中在最后那几周里。阅卷时看到的差距，多数不在题目大小，而在方法有没有做透。
The current syllabus, first assessed in May 2021, gives SL candidates two papers at 40 per cent each plus the exploration at 20 per cent; HL candidates sit two papers at 30 per cent each, a Paper 3 worth 20 per cent, and the same exploration at 20 per cent. SL papers carry 80 marks in 90 minutes, while the HL papers carry 110 marks in 120 minutes and Paper 3 carries 55 marks. The exploration therefore weighs as much as the whole of HL Paper 3 and more than any single question on an SL paper. Teachers mark it against one 20-mark scheme with separate SL and HL descriptors, and the gap between a 12 and an 18 usually comes down to whether the mathematics was pushed far enough.
五项评分标准的分值分布
A 表现 4 分，看报告的组织与连贯，包括引言、正文、结论是否扣住同一个目标。B 数学表达 4 分，看数学语言、符号、术语与变量定义，以及公式、表格、图与模型的配合是否恰当。C 个人参与 3 分，官方档位只写「有一些」「显著」「突出」三档。D 反思 3 分，看对结果的回顾、评价与改进。E 数学运用 6 分，是五项里最大的一块。20 分里 E 加上 A 与 B 共 14 分，数学本身与数学表达占了大头。常见的错误是把力气全花在选题的新奇上，五项标准里有三项与选题无关。
Five criteria decide the 20 marks: presentation (4), mathematical communication (4), personal engagement (3), reflection (3) and use of mathematics (6). Presentation covers organisation and coherence, so an introduction, a body and a conclusion have to serve one stated aim, while mathematical communication covers notation, symbols, terminology, defined variables and the choice of formulae, tables, graphs or models. Personal engagement is worth only 3 marks and the official descriptors stop at some, significant and outstanding evidence, whereas reflection asks for critical review, analysis and evaluation of what was found. Use of mathematics carries 6 marks and is where most of the differentiation happens, so a safe topic handled with strong mathematics beats a striking topic handled thinly.
12 到 20 页双倍行距与参考书目不计入
现行指南给的篇幅是 12 到 20 页双倍行距，图与表算在页数里，参考书目不算。网上流传的 6 到 12 页来自 2014 年首次考试那一版课程，与现行口径不同。官方同时写明简洁与相关比页数更重要，重复的计算不会因为占版面而得分。把 20 页当上限用，意味着每一页都要有信息量，而不是靠截图撑满。图表要有标题、坐标轴要有名称与单位，从软件里截出来的图也应该重新画一遍或至少标注清楚。很多学生把附录当仓库，数据全塞进去，正文却读不出方法的链条。正文留一段样品计算，附录放完整数据，这是一条省心的分工线。
The current guide suggests roughly 12 to 20 double-spaced pages, with diagrams and graphs counted inside that range and the bibliography excluded. Guidance of 6 to 12 pages belongs to the earlier course first examined in 2014, and mixing the two standards is a common way to under-write an exploration. Conciseness is rewarded, so calculations copied out of software output add pages without adding marks; every graph needs a title and labelled axes, and screenshots should be redrawn or at least annotated. Keep one sample calculation in the body and move the full data set into an appendix, so the reader can follow the method without hunting for it.
选题的三条筛选线
选题先过三道筛：数据或情境能不能拿到，数学能不能落到课程内容上，问题能不能用一句话回答。数据可得这一条最容易被低估，气温记录、运动数据、问卷结果与公开统计都能用，但要在开工前确认拿得到手。数学可做指的是报告里的方法与课程内容对得上，AA 的学生用函数、微积分、向量与复数，AI 的学生用统计、概率与建模，两条路都能出好题。问题可回答意味着结果能算出一个具体判断，而不是「探讨数学与音乐的关系」这样的题目。可行的例子包括用卡方检验判断某校学生的生日是否均匀分布，用牛顿冷却定律拟合一杯茶的降温过程，用旋转体求一个瓶子的容积，用 Dijkstra 算法比较校园里两条路线的耗时。选题阶段最省时间的做法是先写三行：研究对象、要算的量、用的方法。三行里任何一行写不出来，就说明这个题还要再收窄。
Three filters settle a topic: whether the data or context is genuinely available, whether the mathematics sits inside your course, and whether the question can be answered in one sentence. Data availability is the filter students underestimate, since temperature records, sports measurements, survey responses and public statistics all work, but only if access is confirmed before writing starts. Analysis and approaches students lean on functions, calculus, vectors and complex numbers, while applications and interpretation students lean on statistics, probability and modelling; both routes produce strong explorations. Worked examples include a chi-squared test on whether birthdays at a school are evenly spread, Newton's law of cooling fitted to a cooling drink, a solid of revolution used to find the volume of a bottle, and Dijkstra's algorithm used to compare two routes across a campus.
AA 与 AI 的数学深度差异
E 分的档位描述按 SL 与 HL 分列，AA 与 AI 的侧重也不同：AA 看重代数与推理的严谨，AI 看重把情境转成模型并解释结果。AA 的高分报告常常是一个方法用到深处，比如用一个微分方程描出变化的规律，再把解代回原始情境检验。AI 的高分报告常见的是数据口径清楚，比如问卷样本写成 120 份，用卡方检验在自由度 5 与 5% 显著水平下对照临界值 11.07 做判断，再说清检验的假设是否成立。两门课都不奖励超出课程范围的炫技，把不该出现的旧方法搬进来反而会被扣分。IB 官方在 E 分的说明里用的是「与课程水平相称」这个说法。选方法时先问一句：这是大纲里的哪一条？答案写不出来，就换一个能写出来的方法。
Criterion E has separate SL and HL descriptors, and the two courses expect different mathematics: analysis and approaches rewards algebraic rigour and deduction, while applications and interpretation rewards turning a context into a model and interpreting the output. A strong analysis report often pushes one method a long way, such as a differential equation that describes a rate of change before the solution is checked against the original situation. A strong applications report usually keeps the data honest, reporting a sample of 120 responses and testing with a chi-squared statistic against the 5 per cent critical value of 11.07 at five degrees of freedom. Neither course rewards mathematics beyond its syllabus, so ask which syllabus item each method belongs to before writing it up.
研究问题怎么收成一个可量化的目标
研究问题写得好不好，看它有没有给出研究对象、要算的量与判断标准。反例是「探索数学与音乐」，正例是「用 2014 到 2024 年某地的逐日气温，比较正弦拟合与线性回归的残差平方和」。题干里出现的每个名词，报告里都要有对应的一段数学处理，做不到就说明问题太大。很多学生以为题目越大越显得有水平，阅卷时看的是这条线能不能一直拉到结论。把问题收窄还有一个好处：误差与小样本的影响都能算出来，D 分的反思自然有东西可写。题目定稿后不要急着写正文，先用一页纸把数学路线画出来，卡住的地方就是后面要补的地方。一份报告只回答一个主问题，第二问留在讨论里当延伸。
A usable research question names the object of study, the quantity to be calculated and the standard for judging the answer. A weak version reads as exploring mathematics and music, while a workable version compares the residual sum of squares from a sinusoidal fit and a linear regression using daily temperatures from 2014 to 2024. Every noun in the question needs a section of mathematics that handles it, so a question that cannot be answered with the methods you already have is too large. Narrowing the question also makes uncertainty and small-sample effects visible, which gives criterion D something concrete to evaluate.
个人参与 3 分从哪里来
C 分的三档写得很直白：1 分有一些个人参与，2 分有显著的个人参与，3 分有突出的个人参与，官方同时写明这不是努力的度量。能被看出来的是你自己的判断：为什么选这个题、为什么换掉那个模型、数据为什么可信。自己实测的数据在这条标准上最占便宜，比如自己记录 30 天的跑步配速，比引用一份网上表格更容易写出取舍。IB 官方在 C 分的说明里举的例子包括独立思考、从不同角度探索、提出并检验预测。这三件事都能在正文里留下痕迹：一处自己的预测、一次换角度、一次被数据推翻的判断。检查自己有没有做到，可以看报告里有没有出现「我原本以为，后来发现」这样的叙述。不要用一句「我对这个题目很感兴趣」交代个人参与，那属于陈述，不是证据。
Criterion C has only three levels: some personal engagement, significant personal engagement and outstanding personal engagement, and the guide states plainly that it is not a measure of effort. What examiners look for is evidence of your own decisions, such as why this topic was chosen, why one model was abandoned, or why one data source was trusted more than another. Data collected by you works hardest here, because a month of your own running times carries reasoning that a borrowed spreadsheet cannot. The official examples are thinking independently, exploring from different perspectives, and making and testing predictions, so put a prediction in the report and then say what the data did to it.
数学表达 4 分：变量定义与图表标注
B 分的官方口径看四件事：数学语言是否恰当，术语与变量有没有定义，表示形式是否用对，以及推导与证明有没有按逻辑写。四档描述从「相关但只是部分恰当」到「相关、恰当且通篇一致」，一致指的是符号不要中途换写法。正文里不能出现 * 或 ^ 这类计算器记法，除非那是软件直接输出的内容。图要有标题、坐标轴要有名称与单位，表格要有编号并说明每一列是什么。常见的错误是圆心不在原点却不交代，或者把两个来源不同的数据画在同一张图上却不加区分。变量第一次出现时写清含义与单位，后面就不要再改符号。写完后通读一遍符号表，把同一个量在正文与图里写成一样。
Criterion B asks whether the mathematical language is appropriate, whether key terms and variables are defined, whether the choice of representation fits the argument, and whether derivations are set out logically. The descriptors run from relevant but partially appropriate communication up to communication that is relevant, appropriate and consistent throughout, so symbols cannot change halfway down the page. Calculator notation is acceptable only where it comes straight from software output, and every graph needs a title with labelled axes carrying units. One common loss of a mark comes from a circle drawn away from the origin without explanation, or two data sets plotted on one set of axes without any way of telling them apart.
反思 3 分要写进正文
D 分官方给出的三档是「有一些回顾」「有意义的回顾」与「批判性的回顾」，评价、分析、综合都算在内，而且要求通篇可见。做法是把局限写在相关结论的旁边，而不是堆到结尾一段：忽略了空气阻力、问卷只有 30 份、样本集中在同一所学校，这些都要在用到它们的段落里说明影响。误差能量化就量化，比如测量误差 0.5 摄氏度会让拟合出的变化率偏多少，这样反思就有了数字。如果结果与预测相反，那是最好的素材，把偏差写清楚比藏起来更得分。很多学生把反思写成「本次探索让我学到了很多」，阅卷时看到的是没有内容的句子。改进方向也要具体，写出换一组数据会怎样、换一个模型会怎样，比写「以后会更加严谨」有用得多。一份报告里有两到三处实质反思就够，密度胜过数量。
Criterion D rewards review, analysis, evaluation and synthesis at three levels, from some evidence of reflection up to critical reflection throughout the report. The practical move is to place each limitation beside the result it affects rather than in a closing paragraph, so air resistance ignored in a projectile model or a survey of only 30 responses is discussed where it changes the answer. Where error can be quantified, quantify it, because an uncertainty of 0.5 degrees Celsius becomes evidence once you say how far it shifts the fitted rate. Reflecting on how much was learned is not reflection and earns nothing, so write instead about what a different data set or a different model would change.
数学运用 6 分：把课程内的方法推到新情境
E 分 0 到 6 分，官方要求数学与题目相关、与课程水平相称，并且能看出理解。一个方法做透比三个方法并列更容易拿高分，因为阅卷人看的是每一步为什么这么做。举例来说，AA 的学生可以先用量纲与几何把瓶子的轮廓写成函数，再用旋转体公式算容积，最后讨论分段拟合带来的偏差。AI 的学生可以对同一组降温数据先做指数回归，再用分离变量解出 T(t) = Ta + (T0 - Ta)e^(-kt)，比较两者的参数含义。把课程里的方法用到新情境，比把新方法硬塞进报告更容易得分。数学量级也要合理，算出 3 千克的茶杯或 10 米宽的校园小路，第一反应应该是检查单位。报告里每一步数学最好都能用一句话交代来源与目的。
Criterion E runs from 0 to 6 marks and asks for mathematics that is relevant to the topic, commensurate with the course, and understood rather than copied. One method carried to depth beats three methods mentioned in passing, so each step should have a reason you can state in a sentence. An analysis student might model the outline of a bottle as a function, integrate it as a solid of revolution to find the volume, then discuss the error introduced by fitting the profile in pieces. An applications student might fit an exponential model to cooling data, solve the same problem by separating variables to reach T(t) = Ta + (T0 - Ta)e^(-kt), and compare what each parameter means.
时间账：10 到 15 小时与四周节点
官方指南给的课堂时间是 10 到 15 小时，包含讲解、课堂工作与咨询，真正写一份报告通常在 20 小时上下。四周的排法可以这样切：第一周定题并把数据拿到手，第二周把数学做完，第三周写正文，第四周交初稿并按老师反馈修改。初稿必须在期限前交，官方明写草稿不是删减版，要包含引言、结论与参考书目。如果第二周末数学还没落地，就把题目收窄，而不是继续找资料。数据整理、画图与写作的时间常常被低估，排期时给文字留出与数学相当的时间。时间投入要清醒：IA 只占 20%，外部卷占 80%，把两周全部押在 IA 上往往得不偿失。每周留一次半小时的复盘，看看数据、数学与文字哪一块落后，再决定下一周往哪投时间。
The guide recommends 10 to 15 hours of class time for explanation, work and consultation, while most students spend around 20 hours of their own writing time on the report. A four-week plan works well: fix the topic and secure the data in week one, finish the mathematics in week two, write the body in week three, and submit a draft for feedback in week four. A draft has to be complete, since the guide is explicit that it is not an abbreviated version of the exploration, and it must contain an introduction, a conclusion and a bibliography. Keep the balance in mind as well, because the exploration is 20 per cent while the external papers carry 80, and three weeks spent chasing a single mark is rarely the best use of time.
2029 年 5 月起换成四项标准
IB 官网写明新版数学课程 2027 年 8 月首次施教、2029 年 5 月首次评估，探究报告仍是内部评估，但评分标准换成四项：A 问题界定 4 分、B 抽象 6 分、C 计算 4 分、D 解释 6 分，SL 与 HL 用同一套标准。同一份更新说明里，SL 两张卷降到各 75 分，HL 两张卷降到各 100 分，HL 的 Paper 3 降到 50 分并缩短到一小时。内容也有删减：SL 去掉财务应用与双变量数据，HL 再去掉反例证明与欧拉法。新标准的四项都能对上探究的过程：先界定问题，再抽象成数学形式，然后算，最后解释并修正。2029 年 5 月之前参加考试的学生按现行五项标准评分，两套标准不要混用。核对口径最可靠的办法是看 IBO 官网对应课程的更新页与当年的指南版本。
The IB website states that the new mathematics courses are launched in February 2027 for first teaching in August 2027, with first assessment in May 2029, and that the exploration remains an internal assessment but is marked against four criteria instead of five. The new criteria are problem specification (4), abstraction (6), computation (4) and interpretation (6), used at both SL and HL, and the same update reduces the SL papers to 75 marks each, the HL papers to 100 marks each, and HL Paper 3 to 50 marks in one hour. Content is trimmed rather than extended, with financial applications and bivariate data removed at SL, and proof by counter-example and Euler's method removed at HL as well. Candidates assessed before May 2029 keep the current five criteria, and the safest check is the subject page on ibo.org rather than any third-party summary.
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