Mathematics as done by mathematicians at University is quite different in a lot of ways to mathematics done at School. There are two main differences, relating to rigour and abstraction.
The main objective of this course is to help you get a good idea of what it is that goes on in university mathematics, and in particular what rigour and abstraction are all about and why they are Good Things. This will be done via some directed reading though Gowers' book "Mathematics: a very short introduction". You will be expected to get a copy of this short book (143 pages, including index) and read it. You will be required to write two essays on what you learn, and there will be plenty of chance for feedback on your essay drafts before submission. There is no exam.
I expect that anyone with basic interest and ability in mathematics will succeed on this module and get a lot from it. An A-level (or equivalent) may help but will not be essential. There will not be enough time in the module for a large number of mathematical calculations or in-depth discussion of certain mathematical tecniques, nor is this necessary. Ultimately how far you take this will be up to you.
The web notes that go with the course are intended to be a commentary on Gowers' book. Sometimes the notes provide clarification or further explanation, sometimes I provide some alternatives to the conventions on mathematics that Gowers chooses. (As is perfectly reasonable for such a text, Gowers sometimes simplifies or takes a simplifying position when mainstream mathematics might take what he says in a slightly different way. I try to point this out where it happens.) Mostly, however, the notes provide extra information or additional examples for people interested in them.
The emphasis in mathematics at university is much more on rigour and proof. It is no longer enough to be able to guess an answer and wave your hands to say roughly why you think it is right. You must be able to prove rigorously that the answer is what you said it is.
Also, the objects of mathematics are regarded in a broader and different way. Even the idea of "what a number actually is" turns out to be surprisingly difficult. This leads to looking at different families of numbers and other mathematical objects with properties related to that of number.
These two shifts in direction arise from one another and support one another. For example, the concept of "real number" turns out to be difficult enough that it is easy to make mistakes, and proofs are required to ensure mistakes are not made. More tricky concepts follow, such as "set" and "function" and these cause other traps for the non-rigorous worker. Conversely, studying the properties of numbers (and other mathematical objects) results in a method of abstraction that makes the matter of writing proofs easier and provides a clearer view of what mathematics is all about.
This course will look at all of these things in generality. In particular we will look at the concept of number and what we want numbers to do, and will see problematic cases where more rigour and a broader view is needed, and we will see the beginnings of the very beautiful and useful mathematics that arises from this.