Mathematics is taught from primary level up to A-level and sometimes beyond. Students have been as young as six years old but sometimes middle aged.

At the primary level, it might be necessary to teach to count and to recognise numbers but more usually multiplication tables, which are not always taught well at school. Most of our primary school students are 11+ candidates however. The mathematics needed for the 11+ examination does seem to go well beyond what most primary schools provide. Fractions, decimals, percentages, means, modes, medians and range… Many of our new primary students are unfamiliar with one or more of these and many other topics that may be needed. In preparing for the 11+ it is best to start at least a year before the exam. We do not like students to cram for the exam, as this can be very distressing for children of that age and does not give them a proper in-depth knowledge in any case. We advise students to start with us at least a year before the examination. The next level up is the Common Entrance examination.

We see fewer students who require this because generally they are moving from a good public school or private primary school upwards and have been well prepared. However, sometimes a student may have moved from abroad, for example and will not have this preparation. The level of maths needed for Common Entrance is not far from that for Foundation Level GCSE. For GCSE Mathematics there are two options: The Foundation Level and the Higher Level.

Although the Foundation Level questions are easier, there is a limit to how high a grade a candidate can be awarded. If an A or A* is needed, they would have to take the Higher Level and some of the questions are quite difficult, for example, those on vectors, exponential functions and harder trigonometry. Histograms seem to give a great deal of trouble too.Next up is A-level. We find the biggest problem is that students do not do enough work in the first term and the mock examinations are upon them before they know it. We would rather they came to see us in the first term even if they continue on their own after that. It's a big jump from GCSE Maths to A-level Maths and it's one that students often underestimate.

It appears it should be easier than GCSE because instead of doing nine or ten GCSEs they are now only doing four A-levels but the A-levels are each at least four times as difficult as were the GCSEs. A common mistake that students make with Mathematics is not to do enough questions to sharpen their skills and gain in depth knowledge of the various techniques that are needed for problem solving. Mathematics cannot be learned by just reading the instructions and looking at the examples.

It’s absolutely necessary to do as many questions as possible. Apart from sharpening a student’s skills, this also makes it less likely that a student will forget a topic. Beyond A level Maths is Further Pure. We recommend that a student should not take Further Pure unless they have a real talent for Mathematics. An interesting difference between Further Pure and A-level is that up to A-level the work is very ‘fiddly’, so to speak. It’s very easy to make little slips and cause a whole question to go wrong. For Further Pure, it’s more about trying to understand more difficult concepts. The questions are not so detailed or fiddly but are more difficult to understand. It's worth mentioning that in the cases of Mechanics and Statistics, the necessity of doing as many questions as possible is particularly important.

Knowing how to apply the formalism to a situation, usually expressed in ordinary language, needs some intuition and this takes much time and effort to acquire. A student should try to do as many questions of a varied nature as possible. Occasionally, we are contacted by students at university or college for help with the Mathematics. We should make it clear that we will not assist students with writing up course work, dissertations or theses.

PERCENTAGES QUESTIONS

- Change to fractions: a) 70% b) 37½%
- Change to decimals: a) 60% b) 62½%
- Change to %. a) 3/5 b) 7/8
- Change to %. a) 0.6 b) 0.375
- Find 65% of £270.
- Find 62½% of £816.
- Find 12% of 25%.
- 32% of a workforce of 500 are absent due to a dispute. How many is this?
- 33 1/3% duty is payable on an item listed at £66. How much is this?
- An item is listed at £6.80 but has 15% VAT added. What is the selling price?
- An article costing £84 is reduced by 20%. What is the selling price?
- On one day, 22% of the workforce of a factory are absent. 45% of those are women. What percentage of the workforce are women who are absent on that day?

**Answers**

- a) 7/10 1b) 3/8
- a) 0.6 2b) 0.625
- a) 60% 3b) 87½%
- a) 60% 4b) 37.5%
- £175.50
- £510
- 3%
- 160
- £22
- £7.82
- £67.20
- 9.9%