A Level Further Mathematics

Further Mathematics is not that difficult but poses a problem even for those who enjoy maths: you have to study. With AS Maths, once you have confidently mastered a technique, say max/min problems, it is not really necessary to ‘revise’. For A level Maths you must spend a little time memorising trigonometric identities (at least you should) or refreshing your understanding of parametric integration. But in general, provided you practise a lot you will be fine come the exam. For Further Mathematics this process of refreshing is essential.

You will learn more about vectors, transformations and, if you choose to, hyperbolic functions which together provide a foothold on the mountain leading to Einstein’s general theory of relativity. You will also delve more deeply into the theory of complex numbers, culminating with an elegant introduction to complex transformations. Interestingly FP1 and FP2 or FP3 are the only compulsory modules; you have a wealth of choice and can avoid mechanics or statistics (but not both) should one or the other not butter your bread.

Unsurprisingly top universities such as Imperial College demand A* for undergraduate courses in Mathematics or Engineering; surprisingly the London School of Economics (LSE) now asks for it for Economics.

Generally studying Further Maths is a good idea if you like maths but a nightmare if you don’t.

Why study Further Mathematics?

If you are applying to mathematically related courses at one of the top universities (engineering for example) you are certain to require Further Maths. Such advice is, perhaps, a bit superfluous because if you are considering such courses you MUST enjoy Mathematics. If you do enjoy Maths why not study it in more depth? At the very least it will give you an insight into the difficulties, challenges and beauty of advanced Mathematics.

Which syllabus do we follow?

We follow Edexcel specification for Further Mathematics.

How many units are there?

The units in Mathematics are called modules. Students take six modules, two of which are compulsory. The compulsory modules are Further Pure Mathematics: FP1 and either FP2 or FP3. The remaining four optional modules and be chosen from: Mechanics – M1, M2, M3, M4; Statistics – S1, S2, S3; and Decision Mathematics – D1, D2. Students must choose optional modules that they have not previously studied.

What is each unit about?

Compulsory modules: Further Pure Mathematics

FP1FP2FP3
This module is most interesting because it introduces students to two completely new concepts: complex numbers and matrices. It is impossible to overestimate the importance of these to modern Mathematics and Physics: if you like maths you will be thrilled! You will also be introduced to the concept of proof. For example, try proving that there are an infinite number of prime numbers before trawling the Internet. Elegant and profound.
Arguably the most interesting aspects of this module are the investigation of transformations of complex numbers – (if you need to see a need, these may be used to design airplanes) – and the expression of functions as infinite series (for example ex = 1 + 2/2! + 3/3! + 4/4! + …. This is the way your calculator does it (almost)). Quite interesting as well are inequalities and advanced differential equations.
To be frank this module is for purists (get it) only; almost one half of our students avoid it. Nevertheless there are some juicy bits such as the vector cross product (a new operation to complement the vector dot product) and the diagonalisation of matrices. These topics are essential for modern Physics (Maxwell’s equations, eigenvectors and Quantum Mechanics) and will stretch and challenge even the most able.

Optional modules

Mechanics

M1M2
Once you ‘get it’ this module, M1, is for the most part quite easy. However, it is fair to say that students find M1 very difficult; and the reason they find it difficult is that their understanding of what forces act on everyday objects (say a book on a table or, more challenging, yourself, sitting on your sofa watching the X factor) is insecure – perhaps terrifying. A bit like Ghostbusters our job is to restore this confidence and make M1 a breeze (we can do it!).

Principally, M1 is about elementary applications of Newton’s laws (how many?) but, more interestingly, it is about the mathematical model used in analysing the forces of nature. Understanding the model is challenging but, if taken a little further than A level, will give anyone an appreciation of why Aristotle was wrong in thinking that all bodies require a force to keep moving in a straight line (the law of inertia). Equally you should wonder at the genius of Galileo, then Newton, then Einstein in overturning the Greek’s idea. The struggle to understand the model in M1 schools students in the patience necessary to understand Einstein’s Special Theory of Relativity.

Once you have mastered the M1 syllabus you can strut your stuff in M2 which by and large poses questions of technique. In other words how to solve more complex problems involving forces, turning forces (moments), momentum and motion under constant acceleration (projectiles). The questions are challenging but illustrate two ideas important for exam success:

1. Practice makes perfect.

2. Proficiency in exams depends on speed; the faster you are the more likely will be the reward of your efforts an A*. Speed depends on pattern recognition; you should be able to identify each and every problem you encounter as a ‘type’ which you have met before. You should rarely have to think in an exam because you have done your thinking beforehand.

M2 also introduces two new topics: centres of mass and conservation of energy. Of course all of these are related to physics. Sadly, much as we would otherwise prefer, most human beings seem genetically wired to silo mentalities: energy in mathematics is somehow different from energy in physics (where there is any energy at all). Still hope springs eternal.

Statistics

S1S2
Many people think the study of A level statistics is a sign of a lousy social life. They are wrong! Sixth form statistics is an essential tool for subjects as diverse as biology, psychology, economics and finance.

For example, suppose one person in 1000 actually has a particular disease. A pharmaceutical company develops a test which is 100% effective in identifying a disease; ie if you’ve got it, the technique will never fail to indicate that you’ve got it. Unfortunately, if you don’t have the disease, the test is wrong 1% of the time; ie even if you don’t have the disease, the test will say you’ve got it anyway one time in 100. You visit to your doctor’s surgery and test positive: what is the chance that you actually have the disease?

This example, of course, is a question in probability. There is an easy way to solve it and a hard way; and of course we teach you the hard way. The study of probability supplies you with an important way to understand risk. With little more than A level understanding of probability, it will certainly help you understand why Lehman Brothers’ bank collapsed as well as appreciating how careful you must be interpreting the results of trials for a new drug.

S1 also deals with the very important area of correlation and regression. If you have a good teacher (all at Ashbourne!) they will help you understand the true meaning of ‘regression to the mean’ and why correlation is not causation. An example of the latter is (ironically?): the more hours of private tuition students receives the worse their grades (negative correlation). Does this mean that teaching makes students stupid?

Finally S1 examines the analysis of data and will help you answer the following questions: 1. If Bill Gates walks into a pub, what happens to the mean (average) income of the patrons there? What happens to the modal income? Importantly it also deals with the concept of variance or standard deviation, both of which are used by financial boffins to measure risk and volatility in the markets. In fact they will even sell you volatility if you have the appetite!

P.S. You also get to learn quite a bit about bell (normal) distributions which apply to many things like IQs, exam results etc, but regrettably for bankers and the rest of us, not others like the price of shares and derivatives.

The most important part of the S2 module for A level Mathematics is the CLT (central limit theorem). This says that as the size of a sample from a population increases the mean (average) value of the sample follows the same distribution as for example, IQs (the bell or normal distribution). This means that if you could choose about 1000 adults at random (this random bit is not easy) whether or how they voted in the prelude to an election the result will with extremely high probability give you an excellent idea of the behaviour of all adults. This is obviously a powerful tool and in the right hands has with near perfection predicted the results of all recent elections in the United States.

This idea leads naturally to the topics of confidence intervals and hypothesis testing which are fundamental for psychologists, economists and doctors.

In S2 Mathematics you will also meet two very important distributions, namely the Poisson and Binomial. These are both extremely useful. For example, the Poisson arose from observing the number of officers in the Prussian cavalry in the nineteenth century who died each year because they were kicked by their horse. If someday you are bored waiting for your bus you might amuse yourself by counting the number of buses (not yours) that arrive at the bus stop each minute. More or less this illustrates a Poisson distribution.

Finally you will apply your knowledge of calculus to continuous distributions.

This module is not dense but is conceptually quite challenging. I suppose it indicates a general principle: in applied maths once you establish a rock solid understanding of the fundamental first principles you should have little difficulty afterwards.

Decision Mathematics

D1D2
Algorithms, algorithms on graphs, the route inspections problems; critical path analysis; linear programming; matchings.
Transportation problems; allocation (assignment) problems; the travelling salesman, game theory, further linear programming, dynamic programming; flows in networks.

Who teaches this course?

 

Chella Nathan

BEng Hons, (North London); BSc Hons (Sri Lanka) Mathematics and Physics

Chella has been teaching Mathematics and Physics for many years. His research interests are number theory and the development of renewable energy sources.

Dr Barry Rhule

PhD (London); BSc Hons (CNAA), Mathematics, Physics

Barry is a member of the London Mathematical Society and has been teaching Maths and Physics at Ashbourne for many years.

Dr Abdul Sami

BSc Hons, Pure Maths; MSc, Pure Maths (Queen Mary University of London); PhD, Algebra (Imperial College London)

Abdul taught Pure Maths at Michigan State University for three years before returning to the UK. Then he transcribed parts of Newton’s Principia at Sussex University before joining Ashbourne College. Some of his research is published in leading journals and he has given seminars and conference speeches in the UK, USA and Germany.

Pete Franklin

BSc, Actuarial Science (London School of Economics); PGCE, Secondary Mathematics (University of London).

Before retraining as a teacher in 2002, Pete worked at Channel 4 for 11 years, first as Marketing Manager and later writing and directing TV trailers. He joined Ashbourne in 2010.

Seema Ali

PGCE (University of Durham), BSc Maths (University of Durham)