I was deliberately avoiding putting forward an answer to what the true nature of mathematics is. I think that it’s highly debatable. I tended to see it as axiomatic reasoning, but I know of professors of mathematics who describe maths as “about patterns” and plenty of professional (applied) mathematicians who would say that it’s about problem-solving. At some levels it’s almost a metaphysical question what mathematics is.

When we consider those who teach maths, who are overwhelmingly unlikely to have a maths degree at secondary, and are often without even a maths A-level at primary, then there is even less of a consensus. I commented mainly because I think that phrases like “appreciate the true nature of mathematics” are not terribly helpful when discussing maths teaching. Ultimately, maths is a body of knowledge to learnt and/or applied, and it is far from clear at what point anybody truly appreciates the nature of that body of knowledge as a whole. I’d just be happier if at every stage students were fluent enough, and unencumbered by misconceptions, to move smoothly onto the next level of mathematics rather than people try to pass on something more profound, that might actually not hold up at the next level.