Firstly, TSR is a good forum to check whether you have ability to explain clearly and whether your explanation is understood by students. I accidentally ended up in TSR when I was looking 9-1 maths stuff for helping some local students, free of charge or at negligible cost
. This is to make them realise, there is a value in what they are getting.
I selected challenging questions (well mentioned by TSR members) and answered them to see whether students can understand my explantion. The first hardest question I answered, which I did more than 25 years ago and I forgot it. However, after looking the details for about 5 minitues, I got it back. This was a GCSE stats question.
I have been explaining to both GCSE and A levels, Furher maths etc when I get time.
Though I am qualified up to MSc in engineering and computer science, I still learn higher mathematics using youtube, MIT courseware etc for keeping in the trend. For example, tensor calculas in Youtube's MathTheBeautiful channel. Maths can be taught and developed to some extent and I believe it is a natural talent as well as a skill, and this is not boasting.
Maths is some thing I enjoy, learning, teaching and researching both for fun and profit, although my profession is completely different.
Some of the methods, like mathematical deduction and observing mathematical patterns (espeicially when solving elipse with parameterised coordinates for further maths), I think may well have been lost in the teaching side. May be I am wrong because I have not been teaching.
I am still finding to correct point for teaching or helping students because I feel I must not teach them over the limit of GCSE or A levels, or for Furhter Maths I must not under teach them. However, one of the fundamental in Pure Mathamatics is understanding and application of first principles, esepcailly in calculas, algebra, trig etc.
I also agree with nontek about knowing examinations. However, as I previously said in one of the other post (see below in quotes copy of the text of my post) and this is purely my opinion after seeing 9-1 maths and Syllabus explanation by OCR in youtube for 9-1 maths.
"As I said, all examination boards are making maths learning and examinations to be more analytical in approach and original thinking and planning by students in both learning and examinations.
This is now confirmed by this year (9-1) maths GCSE, where students who found it relatively easier now find it examinations challenging, hard, tough, to various degree.
However, I think this is partly to do with previous style of teaching and approach in schools. That is more on mathematical fact & guidance based, and the students were guided through questions. In addition, questions progress with increasing difficulty. This has been generally true for all boards for previous maths syllabus.
Now the change is that students are expected to figure out approach & guidance, and that migh/will involve using more than two topics (like say, algebra, geometry, coordinate geometry, number analysis etc). You are now expected to remeber more formulae, which I think is difficult without analytical approach (i.e. how the formulae were derived in the first place).
I think this is major step for GCSE (and in some ways it is moving to old style examinations, although not in every ways)."
I do welcome any correction of my observation about 9-1 maths from those who are alreadt in-and-out of syllabus, teaching etc.
Hope this helps.