To answer this question, we first have to examine: what is the goal and intention that A-level mathematics has to fill.
Right now: the goal is to provide the essential mathematical concepts and ideas necessary for all industries of math, including:
- Chemistry, Biology, Physics
- Computer Science, Engineering, Finance, Accounting, Economics.
There are a lot of industries that A-level math has to accommodate. That, is one of the fundamental issues with A-level Math and Further Math. It is too general to accommodate so many different areas, that it cannot focus on topics pertinent to University Mathematics, because it has to provide so much context to other subjects, like Chemistry and Physics, that's it's very difficult to compromise on changing A-level math, without some subject being negatively affected by the topics being removed/added.
Therefore, to change A-level mathematics, I propose that the idea is to change the other A-levels first.
Firstly, ask yourself the question: does it make sense for many degrees to depend on A-level Mathematics more than the A-level subject its self? If I want to study Chemistry, A-level mathematics is more important than Chemistry its self for the degree. If I want to get into a Physics degree, Mathematics is more important than the Physics qualification its self.
That needs to change.
I think that all qualifications that depend on A-level mathematics should have Core and Additional for each subject.
Core focusing on the mathematics for that subject.
Additional focusing on the subject using the mathematics from core.
Then, you can change A-level mathematics to be focused more on University level Mathematics; otherwise, in the current system, it's difficult to change the qualification unless some other subject/field suffers as a consequence.
So, as a concrete example:
A degree in Chemistry requires A-level mathematics. Instead, Chemistry Core A-level should have all the essential math from A-level math/Further Math included in the subject, while Chemistry Additional is all of the content from the current Chemistry A-level.
Then, you can change A-level mathematics, so we no longer need to include some topics of math that are applications of chemistry, or physics, engineering or computing, etc, and we can focus A-level math more on Pure Math, proofs, etc, as a result. That way, A-level math and Further Math, is focused more towards a university level qualification, rather than providing lots of applications of math, which shouldn't be the role of A-level math.
A subject that depends on A-level mathematics as part of its syllabus, should be divided into two A-level subjects:
- Core Subject A-level. Focuses on the math of the subject
- Additional Subject A-level. Uses the math in a theoretical/practical way.
The Core would take all the mathematics from the A-level Mathematics and Further Mathematics qualification, and teach it there. Additional would teach the current A-level for that subject.
- Core Physics A-level, teaches all the mechanics and mathematics pertinent to Physics from A-level Math and Further Math.
- Additional Physics A-level teaches all the current physics content, using the math from Core.
- Remove the Physics math in A-level Math/Further Math, and have A-level math focus more on Pure Math or university level preparation.
Physics A-level shouldn't depend on A-level mathematics to provide all the math. All the math from A-level mathematics, should be included in a Core Physics A-level, that teaches all the mechanical content of A-level Mathematics, while Additional Physics focuses on the current A-level content.
Personally, it doesn't make sense how a subject completely depends on another, to be considered for a degree. Unless other subjects are changed first, we cannot change A-level mathematics, as changing that subject could potentially remove essential math that is important for other fields, like Physics (Mechanics), computing (discrete math), Economics, Finance (Statistics), etc.
Hope that makes sense