- Algebra and Curves :
developing a deep understanding of polynomial factorisation methods and curve sketching is essential for STEP.
- Trigonometry and Functions :
Trigonometry is the number one topic that students struggle with.
School maths doesn’t generally currently provide a good basis for trigonometry on STEP.
One thing that makes Trig a harder topic for STEP is that it involves
a lot of learning and procedures to follow, as well as difficult linking and problem solving.
Also, Trigonometry is 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 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 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’ Series is useful here.
This topic links to the Maclaurian and Taylor series, and to 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 means you don’t need them anyway). The ideas of counting are needed for probability.
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 degree.
Vectors are also imporotant 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, and 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,
( eg. ‘Sin x ‘, ‘Cos x’, ‘Exp x’).