What are the differences between Maths in NCEA and CIE?
Maths in NCEA and CIE are quite different.
Firstly in NCEA Statistics and Calculus are considered two completely separate subjects, whereas in CIE Maths you have to do a paper in both Statistics and Pure Mathematics.
Secondly, NCEA Maths has the potential to be broader. There are a lot of internals in a range of different topics, but most schools or classes won’t go over all of these internals. These include critical path analysis, linear programming, conic sections, systems of equations and trigonometry. In addition to this are the externals that most schools go over, differentiation, integration and complex numbers. CIE on the other hand, doesn’t touch these more obscure topics like critical path analysis, linear programming or conic sections (although note that linear programming is touched upon in IGCSE). Instead, CIE focuses on the more traditional Mathematics topics, such as calculus, complex numbers and vectors, as well as looking at some other topics like numerical methods of solving equations and binomial expansions. It also goes deeper into topics like Calculus, teaching more techniques of integration and things like implicit differentiation. Hence CIE can be considered a bit more in-depth.
Lastly, I should mention how Statistics differs between the two. Statistics at NCEA focuses on probability, probability distributions and evaluating statistical reports, as well as some internals on time series, formal inferences, bivariate data and experimental design. CIE statistics is much more different. There is still work with probability distributions, however it is more theoretical as well, with topics such as hypothesis testing, errors in hypothesis tests, sampling and estimation, combinations of linear variables, continuous random variables and the Central Limit Theorem. It is less about long written answers (like NCEA is) and more about calculations (although this is certainly a bit of a generalisation).