1) are notoriously difficult to grasp, both intuitively, conceptually and computationally; 2) are central and critical for furthering one's mathematical understanding; 3) eventually become quite easy ("How is it I could not understand it before?"), because we employ them so much that any mathematician will internalize them after a while.
Setting bar for these three criteria high, four threshold concepts come to my mind:
- basic algebra (it is well-known that many children struggle a lot with middle school maths when transitioning for arithmetics to algebra); - differentiation and integration (AFAIK, differentiation seems more difficult of these two for most students, because it makes them think about graphs in a novel way); - delta-epsilon arguments in real analysis (as 99% undergraduate students in maths can confirm :)); - forcing in advanced set theory (I know nothing about it myself, but I have read several places that it can be a backbreaker; but I am not sure whether it satisfies (3)).
Do you know other examples of such threshold concepts in mathematics?
Someone•8mo ago
It surfaces in such things as
- why do we consider 0.9̅ and 1 to represent the same number?
- why are there ‘as many’ even integers as there are integers, and ‘as many’ integers as there are natural numbers?
- why are not all infinites equal?
If you don’t know/accept these, you’ll keep using intuition that served you well for many, many problems on finite sets on problems involving infinite sets, with disastrous results.