- Algebra and Curves :
developing a deep understanding of polynomial factorisation methods and curve sketching is essential for STEP. It is often the A2 rather than further maths topics that are a problem on STEP. You need to know quadratic theory like the back of your hand, and cubics similarly, even it seems like you are having to go back to learn the basics in more depth.
- Trigonometry and Functions :
Trigonometry is the number one topic that students struggle with on STEP and school maths doesn’t generally currently provide a good basis for it, (largely down to the approach rather than the actual topics being learnt). One thing that makes Trig a harder topic on STEP is that it involves a lot of learning and procedures to follow, as well as difficult linking and problem solving, so if you are stuck, it can be hard to tell why. Trigonometry is also a good way to start understanding Function theory, which provides the underlying logical structure to much that is being tested on STEP.
[sometimes it’s just a case of ‘knowing’ what you don’t know’ (not always obvious), rather than thinking you can’t do a question].
- Calculus :
Learning similar methods together and practicing mixed up integral questions is very useful A thorough knowledge of all standard integration techniques is necessary (although not sufficient) for integration on STEP.
- Number theory :
Euclid’s simple proofs, and related methods are useful to learn for Number theory. Learn, understand and extend Euclid’s proofs that ‘sqrt(2) is irrational’ and there are an ‘Infinite number of Primes’, which is one of many ‘proofs’ that infinity does indeed exist, throwing up contradictions in logic, and, hence, driving many mathematicians mad (cf. Charles Seife’s amazing book – ‘Chapter Zero’). ‘Proof by contradiction’ is used a lot in Number theory, as well as ‘modulo’ and set-theoretic ideas.
It’s all about ‘Primes’
Unique Prime Factorisation in the Natural Numbers, as shown by Euclid, is a good place to start understanding the Natural numbers, (which is not the only whole number system by the way).
- Logic and Function theory :
Function theory is often an ‘unseen’ idea behind a question, or the ‘theme’ of a question.
Inequalities and application of non one-to-one functions, such as trigonometric functions, can test the logic of function theory. Students frequently think they have successfully answered a STEP question, whilst completely missing the underlying idea being tested here. It’s harder to know if you’re unaware you don’t know….
- Sequences and Series :
Learning general sequence relationships, rather than just specifics is useful for STEP.
Getting familiar with the ‘Fibonacci‘ Sequence is useful here.
This topic links to the Maclaurian and Taylor series, as well as the ideas of limit and convergence.
- Probability and Counting :
It is important to have an understanding of ‘Combinatorics’ and factorial algebra for pure maths as well as probability.
Probability formulae are generally not helpful without an understanding of them (which often negates the need for them in the first place!). The ideas of counting are needed for probability, especially when events are not independent, making it generally impossible to apply probability formulae.
The interesting thing about probability is that it can require you to change your perception of something entirely, and, of course, if you’re not 100% sure of something, how do you know you don’t have an entirely incorrect perception of it and you may think you are 99% sure of it when, in fact, from a different perspective, you are 0% sure of it ?
STEP Vector questions are the only questions I have so far found that actually ensure you have a good understanding of vectors, as opposed to just learning methods, (which are likely to get forgotten anyway), whilst this is one of the most important pure maths topics to understand for a mathematics degree.
Vectors are also important to understand for STEP III topics such as complex numbers.
One issue in STEP Vectors questions is not understanding the scalar product, nor knowing the rules of it, whilst others arise from lacking a proper grounding in coordinate geometry.
- Maclaurin and Taylor Series :
It’s useful to learn this topic as early as possible for STEP, as it links many important ideas.
Common differentiable and continuous functions that students need to sketch at this level are themselves power series, and differentiation of simple continuous functions is actually based on differentiation of these powers of x which form their Maclaurian series,
( eg. ‘Sin x ‘, ‘Cos x’, ‘Exp x’).