Mathematics is seen as a key skill at Ashbourne College and the faculty is fully resourced with an incredible teaching team. Specialist staff teach each strand of the course: physicists teach our students the mechanics modules; statisticians and ex-city workers teach Statistics; computer scientists teach the Decision Maths and pure mathematicians teach the Core Maths modules. This has led to consistently outstanding results with 80% of our students achieving A grades and 60% going on to achieve A grades at Further Maths.

## Why study Mathematics?

Mathematics is the bedrock of nearly all science and economic-based disciplines, from physicists and geographers to bankers and engineers. A student should always consider taking Mathematics to at least AS level to prove to future employers that they are comfortable dealing with many different types of figures and calculations.

## Which syllabus do we follow?

Ashbourne follows the Edexcel specification for Mathematics AS level and A level.

## What is covered in the course?

The syllabus comprises three important elements of mathematics: pure mathematics, applied mathematics (mechanics) and statistics. Very briefly pure mathematics at A level introduces and explores calculus, arguably the most powerful tool of modern mathematics. Mechanics develops an understanding of Newton’s laws (how many?) and, more interestingly, of mathematical models used in analysing the forces of nature. Statistics develops a deeper understanding of probability.

### A level (first year) and AS level

^{12}. Leaving the most feared to last, you will extend your understanding of sine, cosine and tangent so far that you will not recognise them and, as a bonus, get to know their graphs as well. You will learn about a new type of equation: a trigonometric equation of course (sometimes associated with madness). To be fair, trigonometry forms the basis for much of our understanding of the world, including MRI scanners at hospitals (x-rays too), sound boards in recording studios (equalisers) and starlight. Spinach is good for you and so is trigonometry.

Logarithms, natural logarithms and the mysterious number ‘e = 2.718626…’ are next along with exponential functions. They vie with trigonometry as the most hated topic in maths but, as you might guess, they are extremely important. If you are a clever scientist you will see that the work on exponential functions is dominated by its application to rates of change. You may thus relate it to processes such as radioactive decay in physics, rates of reaction in chemistry and the growth of populations in biology.

We finally arrive at calculus and learn about differentiation and integration which are related by the fundamental theorem of calculus. Despite its grand introduction earlier on, this is perhaps the easiest topic for this part of the course!

We complete this section with a review and extension of students’ understanding of vectors. This is fun and gives you an imaginative new way of looking at geometry.

We examine the analysis of data and will help you answer the following questions: 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 stock markets. In fact they will even sell you volatility if you have the appetite! This section also deals with the very important area of correlation and regression. For example, between the ages of zero to puberty, height is positively correlated to age; the older you are the taller you are. Does this mean that age causes height? If you have a good teacher (all at Ashbourne) they will help you understand 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?

We review and advance your understanding of probability and in particular extend your facility with tree diagrams by teaching you binomial probability; for example how to calculate the probability of 17 heads in 20 tosses of a coin. This may sound boring but at least you will get to apply your knowledge of (1 + 2x)^{12} from the pure maths part of this course. But more importantly you will have a basis for deciding whether the coin is fair or not (probably not). This is called hypothesis testing and forms the basis of the scientific method for medicine, psychology and economics.

Principally mechanics 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 Aristotle’s idea.

### A level (second year)

^{-1/2}. Of course there is more trigonometry, equations and identities.

Work on series continues with the study of both arithmetic and geometric series. This is very interesting and will introduce two crucial elements of pure mathematics: limits and infinity. With your understanding of an infinite geometric series you have pretty well all you need to understand interest rates, the value of stocks and bonds and how to make a fortune in the City.

You will learn a different approach to functions and how to transform the graphs of related functions, each to the other. You will learn many more interesting and challenging methods of differentiation and integration and so be able to solve many types of differential equations, the foundation of the physical sciences. You will carry forward your study of vectors and learn a few really clever things you can do with them.

This is a question in conditional 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 which will help you make loads of money in the City should you choose. It is also an example of discrete probability and you will use your understanding of calculus to understand continuous probability. A good example of continuous probability is the normal or bell curve which, for example, tells you about how ‘intelligence’ is distributed among any population as well as the distribution of marks of students in the UK on any maths exam.

The most important part of this module for A level Mathematics is the study of hypothesis testing and confidence intervals which are fundamental for psychologists, economists and medicine. Used properly these tests are extremely powerful and have with near perfection predicted the results of all recent elections in the United States.

## Who teaches this course?

### 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.

### Chella Nathan

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

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

### Sean Pillai

MA Civil Engineering with Business Management (Warwick University)

Sean began his career as an investment banker but quickly realised that teaching was his preferred profession. Sean is a keen sportsman who plays lacrosse, football and tennis in his spare time.

### 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 more 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.

### Seema Ali

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

Seema has been teaching Maths for more than 25 years in the UK, Netherlands, Canada and Mexico and has also been an A level examiner for more than 15 years. She joined Ashbourne in September 2016.

### Madeeha Saad

BSc Mathematical Sciences (Open Univerisity), MSc Medical Statistics (University of London)

Madeeha has been teaching Maths for more than 17 years in London and has been an assistant examiner for Edexcel.

### Imran Shah

BSc Physics (Imperial College, London)

Imran is an experienced Maths and Physics tutor who has been teaching at independent colleges in London since 1998.

## Beyond A level Mathematics

After studying Maths and Further Maths many of our students have chosen to read degrees in Engineering, Physics, Economics, Accountancy and Pure Mathematics. Our students have gone on to study both in London at the prestigious University College London and Imperial College London and further afield with many students being placed at Warwick, Nottingham and Cambridge.

## Any other information

Students intending to study Mathematics at AS and A2 level should ideally possess A*, A or B at GCSE. However, it is possible for a student with a grade C at intermediate level to be successful in their study of A level Mathematics.

Mathematics at AS and A2 level is both challenging and demanding. It is fundamentally different from the study of Mathematics at GCSE requiring strong skills in algebraic manipulation and a disciplined approach to study. Success is achieved by students who adopt a consistent work pattern. Students who have a tendency to ‘cram’ at the last minute may find this strategy unsuccessful in this subject.

## Suggested reading and resources

*Number, the Language of Science*by Tobias DanzigThis is easy-to-read classic was recommended by Albert Einstein.

*Against the Gods: The Remarkable story of Risk* by Peter Bernstein

This great book deals with the application of mathematics, in particular probability, to economics, risk and investment.

*A Brief History of Infinity* by Brian Clegg

For students beginning their study of mathematics this is a wonderful introduction to one of the most important and controversial topics.

*The Man Who Knew Infinity: The Life and Genius of Ramanujan* by Robert Knigel

This is a fascinating story of a self-taught Indian mathematician whose genius was discovered and developed by GH Hardy, one of the best mathematicians of the last century.

*A Mathematician’s Apology* by GH Hardy

This is a very brief autobiography of one of the best mathematicians of the last century.

*Spacetime Physics: An Introduction to Special Relativity* by John Archibald Wheeler

Mr Wheeler was one of the most brilliant physicists of the last century. This book does not require a background of complex mathematics but

does require a determined imagination.

Any books by **John Barrow, eg The Book of Nothing**. This is a great book about the relationship between zero and infinity.

## Textbooks

Authors – Greg Attwood, Jack Barraclough

*et al.*

Title – Edexcel AS and A level Mathematics Statistics and Mechanics 1

Authors – Greg Attwood, Ian Bettison *et al.*

Title – Edexcel A level Mathematics Pure Mathematics 2

Authors – Greg Attwood, Jack Barraclough *et al.*

Title – Edexcel A level Mathematics Statistics and Mechanics 2

Author – Harry Smith