功能性技能数学 五级内容 无计算器考试
功能性技能数学（Functional Skills maths）是英格兰在 GCSE 之外的另一条数学资格路线，从入门级一到二级共五级。内容由教育部（DfE）制定，Ofqual 负责监管，2024 年 11 月更新的官方内容文件逐级列明每级要会什么。三个内容领域贯穿五个等级，难度靠问题解决的复杂程度与知识之间的连接数量往上走。这篇文章按等级把内容清单与考试结构摊开讲。文章还会说明它在 16 岁以后资助规则里的位置，以及 2025 年课程与评估审查对它的意见。
Functional Skills maths is England's other maths route below GCSE, running across five levels from Entry Level 1 to Level 2, with the subject content set by the Department for Education and regulated by Ofqual. The official content document, updated in November 2024, lists what each level covers; this article walks through those lists, the assessment structure, the place of the qualification in post-16 funding and the reservations raised by the 2025 curriculum review.
五个等级与官方内容框架
功能性技能数学共分五个等级，从入门级一、入门级二、入门级三，到一级与二级。官方内容文件写明，等级每往上走一步，问题解决的难度与内容之间的连接数量都会增加。二级在 16 至 19 岁资助口径里被认作 level 2 pass，与 GCSE 数学 4 分及以上同等看待。很多学生以为入门级只是小学算术，翻开清单会发现时间、货币、度量与图表都在里面。常见的错误是按年龄挑等级，实际上该按现在的能力挑。
Functional Skills maths sits at five levels: Entry Levels 1 to 3, then Level 1 and Level 2; the content at each step raises the difficulty of problem solving and the number of connections inside the material, and nothing is dropped, since the guidance states that each level subsumes and builds on the level below. A Level 2 pass counts as a level 2 pass in the 16 to 19 funding rules, on a par with a GCSE grade 4 in maths.
三大内容领域与问题解决的三个难度词
内容在每个等级都分成三块：数与数系、常用度量与形状空间、信息与数据处理。Ofqual 的数学条件文件要求考试合理覆盖这三块，并把与无计算器卷、计算器卷各自相称的内容放进去。文件还要求考试尽量在少数几轮里采样完整内容，所以不同场次的覆盖面会比较整齐。问题解决在官方文件里有三个固定说法：入门级是 simple，一级是 straightforward，二级是 complex。二级的 complex 被定义成需要多步骤、至少两个相扣环节的问题。很多学生以为升级只是数字变大，其实真正多出来的是步骤之间的连接。常见的错误是把三块当成三张独立清单来背，考试里的题几乎都会跨两块。
Officially the content of every level is organised into three areas: numbers and the number system, common measures, shape and space, and handling information and data, and Ofqual's conditions require assessments to sample a reasonable balance across them. Problem solving carries three fixed labels in the DfE content, simple at entry level, straightforward at Level 1 and complex at Level 2, where a complex problem needs a multi-step process and at least two connected stages. Many students read the higher level as bigger numbers, when the real change is the number of links between topics; questions rarely stay inside one content area.
入门级一 数字 20 以内与生活里的量
入门级一的数字范围停在 20：能读写、排序并比较 20 以内的数，能数出 20 件以内的物品（含零），能做和在 20 以内的加法与相应的减法。符号只要求认出并解释加号、减号与等号。度量不做精确换算，而是用语言比较物品的大小、长度、宽度、高度、重量与容量。钱币按 20 以内的数额认出来，并用正确的符号写出。时间只到整点，能读 12 小时的指针钟与数字钟，并知道一周有几天、一年有哪些月份与季节。形状只认圆、正方体、长方形（含正方形）与三角形，位置与方向用左、右、前、后、下、上这类日常词描述。
At Entry Level 1 the number work stops at 20, where learners read, write, order and compare numbers, count up to 20 items including zero, and add or subtract within that limit, recognising the plus, minus and equals signs. Measures are compared in words rather than converted, covering size, length, width, height, weight and capacity, while coins and notes are recognised and written with the correct symbols for small amounts.
入门级二 200 以内的数 乘法表与度量单位
入门级二把范围推到 200，并要求可靠地数到 100、认出并排序 100 以内的奇偶数。四则运算的符号全部登场，两位数加减、两位数除以一位数并写出余数都在清单里，乘法表的要求是从 0 乘 0 到 12 乘 12。估算在这里第一次出现，做法是四舍五入到最近的十，再用结果检查答案。分数只到二分之一、四分之一与十分之一，小数只到一位，钱币处理到一英镑以内的便士与整英镑。度量单位开始成套出现，长度用毫米到千米，重量用克与千克，容量用毫升与升，温度只比较正值。形状清单加进五边形、六边形、圆柱、长方体、棱锥与球，位置词换成里、外、中间、下方、上面、向前与向后。信息处理在这一级出现，要求从清单、表格、图与条形图里取数，比较条形图高低，并按两个标准给物品分类。
Entry Level 2 stretches the number range to 200, with reliable counting to 100, odd and even numbers up to 100 and all four operation symbols in play; learners add and subtract two-digit numbers, divide two-digit numbers by single digits with remainders, and are expected to know times tables from 0 x 0 to 12 x 12. Rounding to the nearest 10 appears here as a checking habit, fractions stop at halves, quarters and tenths and decimals stop at one place, while measures now come in families from millimetres to kilometres, grams to kilograms and millilitres to litres. Data work begins with lists, tables, diagrams and bar charts, including sorting by two criteria and switching information from one format to another.
入门级三 三位数 分数与频数表
入门级三把数位推到三位数，要读写、排序并比较 1000 以内的数。三位数加减、三位数除以一位或两位数并处理余数、两位数乘一位或两位数都在要求之内。估算升格为四舍五入到最近的十或百，再用它检查计算结果。数列从整数延伸到含小数的数列，小数读到两位。分数出现三分之一、四分之一、五分之一与十分之一，并要求认识等值形式。钱用小数记账，金额可四舍五入到最近的 1 英镑或 10 便士；时间用上午与下午表述，指针钟与 24 小时数字钟都要读到分钟。度量开始做公制与英制的比较，形状按对称、长度、直角与角度分类，方向词加进八个罗盘方位与整圈、半圈、四分之一圈。数据部分要求把清单、表格、图与图中的信息整理成频数表，再用表格、图、简单折线图或条形图表达出来。
By Entry Level 3 the range reaches 1,000, with three-digit addition and subtraction, division of three-digit numbers by single and double digits, and multiplication of two-digit numbers by both; rounding to the nearest 10 or 100 becomes a way of checking results rather than a topic in itself. Fractions widen to thirds, quarters, fifths and tenths with equivalent forms, decimals run to two places, and money is written in decimal notation and rounded to the nearest pound or 10 pence. Direction work adds the eight compass points and whole, half and quarter turns, while data tasks ask learners to pull numbers out of lists, tables, diagrams and charts and rebuild them as frequency tables or simple line and bar charts.
一级 一百万 百分比与简单利息
一级的数域拉到一百万，正负数进入清单，乘除 10、100、1000 要求一眼看出结果。公式以文字形式出现，处理一到两步运算；一位数与两位数的平方、运算顺序也在要求内。分数与带分数要会读写、排序与比较，并会求一个数量的几分之几；小数读到三位、计算到两位，四舍五入到整数或一两位小数。百分比只要求整数百分比，并要会算由 5% 及其倍数构成的增减；等值换算、简单比与正比例开始出现。钱的部分要求会算由 5 的倍数构成的简单利息与折扣。数据部分要求把离散数据放进表格、图、饼图、条形图与折线图，会分组并用图表示，会算平均数与极差，并把概率放在 0 到 1 的刻度上用分数表达。
Level 1 opens the number range to one million and brings in positive and negative numbers, with multiplication and division by 10, 100 and 1,000 treated as routine. Formulae arrive in words for one or two-step operations, and fractions, mixed numbers, decimals to three places and percentages in whole numbers all have to be ordered, compared and used, with simple interest and discounts in multiples of five per cent sitting alongside ratio and direct proportion. Data work grows to include pie charts, grouped discrete data, the mean and range, and probability expressed as a fraction on a scale from 0 to 1.
二级 比例 复合度量与复利
二级要求处理任意大小的数，包括任意大小的正负整数。百万以内的计算要会做，并会用估算与近似检查答案。公式用文字与符号给出，要求会代入求值，运算顺序包含指数。分数、小数、百分数与比之间要能互相转换，百分比要会算一个量占另一个量的比例、任意幅度的增减，以及变动前的原值。比、正比例与反比例都在要求内。钱的部分出现复利、折扣、税与简单预算。度量要求用换算系数与换算图在公制与英制之间转换，并会算速度、密度与工资率这类复合度量；三角形与圆的周长面积公式要自己记住，其他二维与三维图形的公式会给出。图纸按比例尺还原实际尺寸、二维坐标、三维图形的平面表示、三视图与展开图、用角度与坐标解二维三维问题都在清单里。数据部分要算中位数与众数、由离散数据的分组频数表估计平均数、用平均数中位数众数与极差比较两组数据、算复合事件的概率并会用双向表、画并解释散点图并辨认正负相关。
Level 2 works with numbers of any size, including positive and negative integers, and expects calculations up to a million with estimation and approximation used as checks. Expressions and formulae arrive in words and symbols and have to be evaluated on substitution, with indices included in the order of operations, while fractions, decimals, percentages and ratios are interconverted and percentage change in either direction is calculated, including the original value. Compound interest, tax and budgeting sit alongside compound measures such as speed, density and rates of pay, and metric and imperial units are converted using both a factor and a conversion graph; data work reaches the mean of a grouped frequency distribution, two-way tables and scatter diagrams showing positive or negative correlation.
数据处理从条形图到散点图
数据处理在五个等级之间是一条连续的线，不是五张互相无关的清单。入门级二是从清单、表格、图与条形图里取数，并按两个标准分类。入门级三要求把信息整理成频数表，再用表格、图、简单折线图与条形图表达。一级加进饼图、离散数据的分组与图形表示，并引入平均数与极差。二级要求中位数与众数、从离散数据的分组频数表估计平均数，并用平均数、中位数、众数与极差比较两组数据。概率同样逐级加深，一级只在 0 到 1 的刻度上比较可能性、用等可能结果把概率写成分数。二级要算复合事件的概率，可以用图、表或双向表，并把概率写成分数、小数或百分数。散点图与正负相关是二级独有的要求。常见的错误是把统计当成背公式，考卷里的数据题先考的是读题与选工具。
Data handling reads as one continuous line rather than five separate lists: bar charts and sorting by two criteria start at Entry Level 2, frequency tables and simple line graphs appear at Entry Level 3, and Level 1 adds pie charts, grouped discrete data, the mean and range. Level 2 moves to the median and mode, an estimated mean from a grouped frequency table and comparisons of two data sets, while probability grows from a scale between 0 and 1 into combined events, two-way tables and answers written as fractions, decimals or percentages. Scatter diagrams and correlation belong to Level 2 alone, and the common mistake is to treat statistics as a set of formulae to memorise when the task first tests reading and choosing a tool.
考试结构 25% 无计算器与 75% 允许计算器
Ofqual 的数学条件文件规定，功能性技能数学每个资格只有一个 component，考试必须把 25% 的分值放在不许使用计算器的题目上，其余 75% 允许使用计算器。同一份文件还规定分值的 25% 用于考基础技能、75% 用于考问题解决，允许上下浮动两个百分点。基础技能的定义是不在问题情境里做数学的能力，官方要求它在有计算器和没有计算器的部分都能被单独考出来。一级与二级的总考试时长被限定在 1 小时 45 分钟到 2 小时 30 分钟之间，具体卷面总分与两个部分的分钟数由各颁证机构在这个区间里自定。同是二级，不同机构的卷子时长与题量并不相同。很多学生以为无计算器部分可以放弃，实际上它占满四分之一的分值。常见的错误是把复习时间全花在计算器题上，导致第一部分来不及做完。
Under Ofqual's conditions each functional skills maths qualification is a single component, and 25 per cent of the marks must sit in questions completed without a calculator while the remaining 75 per cent allow one. The same document fixes 25 per cent of marks for underpinning skills, defined as the ability to do maths when it is not part of a problem, and 75 per cent for problem solving, with a tolerance of two percentage points either way. Total assessment time at Levels 1 and 2 must fall between one hour 45 minutes and two hours 30 minutes, with each awarding organisation choosing its own marks and section timings inside that window, so two Level 2 papers can differ in length and question count.
2019 年改革改掉的与新增的
现行这套内容不是一开始就长这样。政府在 2015 年决定改革功能性技能，改革后的英语与数学从 2019 年 9 月 1 日起首次教学，此前注册旧版的学生最晚要在 2020 年 8 月 31 日之前拿到证书。改革没有提高等级本身的难度要求，改动落在内容上。共同内容变得更具体，基础技能的比重大幅提高，数学卷第一次出现无计算器部分，而在改革前所有功能性技能数学卷都允许使用计算器。Ofqual 后来专门做过一次改革后数学资格难度的评估，结论是整体难度没有偏离原有水平。数字化方向另有一条线，数字功能性技能在 2023 年 8 月开始首次教学，接替原来的信息与通信技术版本。写复习计划时要分清手上材料是改革后的还是旧版，2019 年之前的复习书里没有无计算器训练这类题目。
The current content comes from a reform decided by government in 2015, with the new English and maths qualifications first taught from 1 September 2019 and legacy registrations closed off by 31 August 2020. The reform was not intended to raise the level of demand, so the changes landed in the content, where common content became much more specific, underpinning skills gained weight and a non-calculator element appeared for the first time, since every paper before the reform allowed a calculator. Digital functional skills followed a separate timetable, first taught from August 2023 in place of the ICT version.
16 至 19 岁资助条件里的功能性技能二级
功能性技能数学在 16 岁以后的路线里有一个明确的位置。按 gov.uk 公布的 2026 至 2027 学年数学与英语资助条件，16 至 18 岁、持 EHC 计划的可到 25 岁的学生，如果在 11 年级结束时没有拿到 GCSE 数学 4 分及以上，并且课程在 150 小时以上，就必须继续学这门课。被认可的资格包括 GCSE、功能性技能二级，以及教育部批准的入门级与一级垫脚石资格。全日制学生如果 GCSE 数学只拿到 3 分，官方要求继续冲 GCSE；数学在 2 分或以下的学生才能把功能性技能二级当目标。二级通过之后这项资助条件就算满足，不需要再学数学。持有 EHC 计划并经学校评估确认无法学习 GCSE 或垫脚石资格的学生可获豁免。International GCSE 不满足这项资助条件。数学与英语的额外资助按 band 分档，band 4 及以上每人每科 418 英镑，band 3 与 band 2 为 255 英镑，宽容比例已从 5% 降到 2.5%。
Functional Skills maths has a defined place in post-16 routes; under the maths and English condition of funding published on gov.uk for 2026 to 2027, students aged 16 to 18, or up to 25 with an EHC plan, who leave Year 11 without a GCSE grade 4 in maths and follow a programme of 150 hours or more must carry on studying the subject. Approved qualifications are GCSEs, Functional Skills Level 2 and entry level or Level 1 stepping stones approved by the Department for Education, and a full-time student holding a grade 3 is directed towards GCSE while a grade 2 or below may aim for a Functional Skills Level 2 pass, which satisfies the condition with no further study. International GCSEs do not meet the condition, and extra funding for maths and English is paid at 418 pounds per subject for band 4 and above and 255 pounds for bands 3 and 2, with the tolerance cut from 5 to 2.5 per cent.
2025 年课程与评估审查的保留意见
2025 年 11 月 5 日公布的课程与评估审查最终报告对功能性技能给出了保留意见。审查承认，对一部分学生来说功能性技能资格确实帮他们重新投入到数学与英语的学习里。但多数受访机构反映，功能性技能并不适合作为通向二级的路径，理由是工作场景内容对考生陌生、通过或不及格的二元评分，以及家长、雇主与大学对它的认可度不足。报告没有建议取消 16 岁后继续学数学与英语的要求，反而认为这件事对人生机会的分量太重，应继续保留。它建议新增一年制的一级阶梯资格，面向 GCSE 数学或英语只拿到 1 分或 2 分的学生。这类资格采用模块化考试，允许把通过的模块累积起来，成绩最高相当于扎实的 GCSE 3 分。
The final report of the Curriculum and Assessment Review, published on 5 November 2025, kept its distance from Functional Skills while accepting that the qualifications re-engage some learners with maths and English. The panel heard from providers that the route does not work for most students, because work-related contexts feel unfamiliar, grading is a blunt pass or fail, and parents, employers and universities do not recognise it strongly. Rather than dropping the requirement to keep studying maths and English after 16, the review recommended new one-year level 1 stepped qualifications for those with a grade 1 or 2, assessed in modules that can be banked and graded up to a strong grade 3.
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