I’d like to look at the stuff that is excluded in the Jun Kao but included in British teaching. I have some difficulties getting straight answers from Chinese students—a chronic problem every Chinese student should be both aware of and fighting against.
Highest on the list is everything to do with statistics. This is hardly taught at all in China. You know what a mean is (add all the numbers together and divide by how many numbers there were) and you might know a median (the middle value when they have been put into order) and a mode (the most common one). British Y11 students will have looked extensively at charts showing distributions of data and can talk about skew; they can find data from the distribution and can create a cumulative distribution from a frequency chart, including answering questions based on either chart. In particular they can identify the middle half of a distribution and the range that covers that half (the difference between the values of the 75% and the 25% data marks).
The advanced students (“Higher” level) can calculate and use the standard deviation and begin to make sense of conversations using these measures. Many of the students will have practical experience of generating statistics (collecting data themselves, generating some statistics and presenting this both in public and in written form). Quite a large number will take this experience of GCSE Higher statistics and turn it with only a little extra work into a module of their AS Maths course (Statistics 1, or S1).
Be clear about this: every Briton under 35 now has a fair grasp of basic statistics. This is true of many Europeans, too. That places Chinese at a disadvantage not only right now but in later life, as it is recognised that we are more likely to use statistics to make sense of numbers than we are to be using any arithmetical skills (that’s what you have a calculator for, isn’t it?). The most significant use of statistics in later life is to use it to ask sensible questions of any (every, perhaps) statistics presented to you.
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All British students have a grasp of symmetry¹ (linear & axial) and of transformations; the four basic ones of translation, enlargement, rotation and reflection and (at Higher Level), stretch and shear. Their Chinese counterparts have usually not met the concepts.
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Chinese students are generally good at arithmetic by hand; the British are relatively poor at this but an equivalent student is likely to use more functions of their calculator on a daily basis. [opinion: citation wanted, please]. Examinations have changed to challenge this attitude by the addition of “non-calculator” papers and the basic arithmetic skills are rising again [citation, please]. What is studied thoroughly in Britain is estimation—producing a quick idea of the size of an answer—and precision² – giving an answer (accurately) to an appropriate number of figures. That includes making sensible decisions for appropriateness of answers. [E.g. a cup of tea might be 250ml but to say it is 247.834ml is unlikely to be correct to the sixth significant figure³, especially after a calculation, unless all the figures used are given to a similar level of precision.] It also includes recognising the possible error bounds (some would prefer to describe this as the set of answers that fit the description). So the 250ml cup of tea, if correct to two sig.fig., has a value between 245 and 255 (and we will reject, conventionally, anything ‘exactly’ 255, because we would round that up to 260ml). At higher and intermediate levels it would be expected that a student can estimate the result of a product divided by a product, where the figures given may be up to 5 sig,fig (but probably three) and the answer can be given to one (or two) sig.fig. by approximating each number given to one (or two) sig.fig. See Precision.
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At GaoKao, the typical Chinese student has considered a range of Pure Maths topics only found in Britain at Further Maths. On the other hand, the typical A-level student (which will be less than half of all of the students of that age) has covered two or three applied topics typically amounting to half of the subject matter examined, although the Pure elements will be standard whatever the course taken. Thus all UK maths students will be consistently proficient at the Higher GCSE skills and have (for example) an appreciation of calculus, and of straight line algebraic geometry [list to add]. But on top of this they have a similarly provable knowledge of statistics beyond what they have already studied, with some mechanics similar to but treated differently form the same mechanics studied in Physics, or some Decision Maths (business mathematics) or something similar in applicability. [statistics on uptake of modules to insert as footnote] This raises the numeracy levels of the nation in a way that should, it is hoped, mean that the country’s people will use their appreciation of number and its place in their world by personal application. I have often said to A-level students that the least of the skills acquired tells them when they need to go employ a (better) mathematician.
DJS 20110103
1 Symmetry: Axial, or line symmetry, recognises that there is a line for which every point on one side of the line has a corresponding conjugate so that a line joining any conjugate pair has the line of symmetry as its perpendicular bisector. Thus minimum line symmetry is zero, no axes of symmetry.
Rotational, or circular symmetry, says that any point has a related set of points at the same radius from the centre of rotation and at equal angles of rotation around that centre. The order of symmetry is the number in each set per rotation. Thus the minimum rotational symmetry is unity (one).
2 Precision is what you have applied when you use three figures because four is inappropriate or misleading or not feasible. Accuracy is what you applied in deciding that three figures were appropriate; your answer was correct (accurate) to three figures, but you don’t think it will be accurate with more figures supplied.
A room in a house can possibly be measured to the nearest centimetre but not to the nearest millimetre because the walls are not flat enough or perpendicular enough. This distinction about appropriateness is what gives the sensible level of accuracy – the typical room cannot be given accurately to millimetric precision, which is the point.
3 Significant figures: Sig.fig., S.F., s.f.: The declared, assumed or implied level of precision. Start with the leftmost non-zero digit and count to the right until the correct number of digits is reached. Round this last digit off. See Lower School: Numeracy, and Lower School: Precision.