Module choices for AS and A Level Mathematics (code 9709)

In the AS Level Cambridge curriculum, Mathematics is unusual in the degree of choice offered to the student. This allows students to select modules according to their preferences, and to begin aligning their curriculum choices with their planned future careers. The good news is that there really is no bad choice for AS Level Maths, but it is nevertheless worthwhile thinking carefully about the modules that you will choose.

Cambridge requirements for AS Level Maths

For AS Maths, the formal requirements of the Cambridge exam board are as follows:

• Pure maths 1 is compulsory

The student must take a second module selected from:

• Pure maths 2
• Mechanics
• Statistics 1

The content of the AS Level Maths modules

Pure maths 1

Pure maths 1 provides a good introduction to some important topics in mathematics, including topics like quadratic equations, functions and graphs, trigonometry, sequences and series, and calculus (both differentiation and integration). These topics are essential for all future mathematical work and provide students whose careers will require maths with a good foundation. In addition to the specific topics covered, the Pure maths course 1 helps the student to develop a good degree of maturity in mathematical thinking  and problem solving. Exam questions require students to go beyond a mechanical application of the skills and techniques and to combine the skills in creative ways in order to solve interesting mathematical problems.

Unfortunately, there is one significant omission in the Pure maths 1 course, and that is the absence of any work on logarithms. Logarithms are an essential part of one's mathematical toolbox and it is a pity that a student can complete AS Level maths without ever encountering them.

Pure maths 2

With the above limitation in mind, Pure maths 2 has real value. It does cover logarithms and also develops the other topics from Pure maths 1 in a very useful way. Both trigonometry and calculus are developed in such a way as to provide the student with a real understanding of their essential nature, as well as an ability to apply them quite broadly to mathematical problems. For students who are not going to go on to A Level Maths,  Pure maths 2 rounds out their understanding of the essential topics in pure mathematics and provides them with a good foundation for all future mathematical work.

There are, nevertheless, two potential drawbacks to choosing Pure maths 2 for AS Level Maths. The first is that Pure Maths 1 must be completed before tackling Pure Maths 2. This means that it is more difficult (though not impossible) to complete the AS Level maths course in a short space of time. If one is aiming to complete AS Level Maths in one year, there are real advantages — from a scheduling point of view — to choose either Mechanics or Probability and Statistics 1, since they can be taken in parallel with Pure maths 1.

The second potential drawback to choosing Pure maths 2 is that Pure maths 3, which includes Pure maths 2, is compulsory for A Level. This means that, if a student has taken Pure maths 2 for AS Level and then decides to do the full A Level, he/she will lose the credit for Pure maths 2 (see below) and will have to take three modules instead of two to complete A Level Maths.

Mechanics

The mathematical discipline of mechanics can can be classified as a branch of applied maths. It deals very much with the principles of Newtonian physics, but the emphasis is on the mathematical aspect of their application to physical problems. The fundamental concepts are the same as what one would encounter in AS and A Level Physics, and those concepts need to be well understood, but the emphasis is on the maths.

The Mechanics module of AS Maths covers the key areas of Newtonian mechanics, including kinematics (the study of motion), dynamics (the study of forces), momentum, and work/energy.

Despite the mathematical emphasis in mechanics as a mathematical discipline, studying mechanics also sharpens one's understanding of the physical concepts beyond what is possible with a less mathematical approach. Choosing the Mechanics module for AS Maths alongside AS Physics is therefore an excellent option. What you learn in the Mechanics module of Maths will really enhance your understanding of Physics and will put you in a much better position to do better in Physics than you would have done without taking Mechanics. It will also give you a good foundation for university-level physics and aspects of applied maths.

Probability and statistics 1

While Mechanics is sharply focussed on the use of maths in a branch of physics, statistics is widely applicable in many fields, including social sciences, medicine (specifically epidemiology) and business — as well as in the hard sciences. As a mathematical discipline, statistics is really quite fascinating in the way it allows one to make sense of huge volumes of data. Given that computers have enabled us to collect data on an unprecedented scale, the ability to process it meaningfully is more important than ever.

The AS Probability and statistics 1 — Statistics 1 for short — module covers the basics of probability and statistics, including the processing and representation of data (mean, standard deviation, etc.), counting methods (permutations and combinations), and a few common probability distributions (including the geometric and normal distributions). Although the practical use of statistics generally requires knowledge beyond what is covered in the AS module, the module nevertheless provides an interesting and useful foundation on which one can build in the future.

A Level

The requirements for the A Level Maths course are as follows:

• Pure maths 1 (compulsory)
• Pure maths 3 (compulsory)
• Probability and statistics 1
• Mechanics or Probability and statistics 2

The most common way of taking an A Level subject is to use a staged approach, which means that the student writes the AS portion of the subject in one exam sitting and then the remaining papers within 13 months. (AS stands for "A subsidiary" and the second portion of A Level is known as A2. ) In the case of A Level Mathematics there are two papers for the AS portion and two papers for the A2 portion. One can also write all the A Level papers in one sitting, but this is less common.

If A Level Maths is taken using a staged approach, the options are as follows:

• Option 1
  • AS Level: Pure maths 1 and Mechanics
  • A2: Pure Maths 3 and Statistics 1
• Option 2 
  • AS Level: Pure maths 1 and Statistics 1
  • A2: Pure maths 3 and Mechanics
• Option 3
  • AS Level: Pure maths 1 and Statistics 1
  • A2: Pure maths 3 and Statistics 2

Content of the A Level modules

Pure maths 3

Pure maths 3 is a very interesting and useful course.  It goes well beyond Pure maths 1 and Pure maths 2 in rounding out one's mathematical foundations. It includes all the key topics from Pure maths 2, but then goes on to deal with more advanced techniques of differentiation and especially of integration. New topics, which are essential to advanced maths and its application to physics and engineering, include vectors, differential equations and complex numbers.

Needless to say, Pure maths 3 will be invaluable for anyone planning on a career in physical science or engineering, but its usefulness is not limited to these fields. Anyone who enjoys maths and desires a good grounding in it will benefit from Pure maths 3.

Probability and statistics 2

As Pure maths 2 and Pure maths 3 round out Pure maths 1, so Probability and statistics 2 (or just Statistics 2 for short) rounds out Statistics 1. It provides a more rigorous perspective on random variables and probability distributions and specifically includes the Poisson distribution. The course also deals with topics that are essential to the practical application of statistics to real-life problems, notably sampling and estimation, and hypothesis testing.

The combination of Statistics 1 and Statistics 2 will be invaluable for a wide variety of fields, including sociology,  psychology and epidemiology — not to mention the "hard" sciences.

Recommendations

 As noted at the beginning of this article, there are really no bad choices for AS and A Level Maths, provided you don't take Pure maths 2 when you want to keep open the possibility of upgrading your AS Level to A Level. No matter which modules you choose, if you move into a career that requires maths there will be a lot more to learn at a later stage, so leaving out one module or another in your AS/A Level studies is not critical. What your AS/A Level studies will accomplish, irrespective of the modules that you choose, is that they will give you a good grounding in most of the foundational topics of maths; they will also help you to develop a degree of maturity in your ability to understand maths and use it in solving various problems, mathematical and otherwise.

 That said, it is possible, nevertheless, to provide some guidance to students who are about to embark on their AS/A Level Maths studies. Here are my recommendations:

1. If you know that you are not going to do A Level Maths, Pure maths 2 is an excellent choice for AS Level, no matter which field you plan to go into after school.
2. Whether or not you are planning on doing A Level Maths, but know that you are heading for a career in physical science or engineering, then Mechanics is an excellent choice. It will sharpen up your understanding of Physics in a way that you wouldn't otherwise achieve very easily. I might add that Mechanics will also be helpful in biokinetics and even dancing!
3. Whether or not you are planning on doing A Level Maths, but know that you are aiming for a career in anything but physical science or engineering (e.g. biology, psychology, health sciences, business), then Statistics 1 is possibly your best option.
4. If you are going to do A Level, your only choice is between Mechanics and Statistics 2. Both options are good, but if you plan on a career in physical science or engineering it would help you a lot to take Mechanics. Statistics 2 will be helpful for students aiming for careers outside physical science and engineering, but the balance between Mechanics and Statistics 1 won't harm anyone.

Two final comments deal with practical matters.

1. The first concerns the scheduling. As noted above, taking Statistics 1 or Mechanics alongside Pure maths 1 for AS Level has the advantage of allowing you to take your two modules in parallel. This does relieve pressure and allows you the full period of your AS studies to grow in your understanding of both modules.
2. The second concerns the availability of tutors and/or classes for Statistics 2. It is relatively easy for a tutoring centre or school to offer Mechanics and Statistics 1, since both of these modules are popular with AS Maths students. Furthermore, it is possible to cover the full A Level syllabus with just these two modules in addition to Pure maths 1 and Pure maths 3. Statistics 2 is less sought-after, which means that it isn't always cost-effective for a tutoring centre or school  to offer it as an option. Nevertheless, if you are keen on statistics and are able either to study on your own or to find a tutor for Statistics 2, by all means, go ahead and choose it.

 At Imago Education we offer all the AS and A Level Maths modules except Statistics 2. See here for further details.

Enjoy your maths!