Also if you, dear reader, have suggested #thresholdconcepts for #mathmatics, please reply with your suggestion. (I do already have a bunch of research on this, so things like "negative" numbers and sets are strong candidates for threshold concepts.)
Link about threshold concepts (in case this term is unfamiliar): https://www.ee.ucl.ac.uk/~mflanaga/thresholds.html
The idea of proof, the idea of proof by induction and the idea of proof by contradiction are #thresholdconcepts
The idea that mathematics is the study of pattern
The idea that numbers are just equivalence classes
Something about axioms too, as possible whimsical and arbitrary and not necessarily modelling a reality - that mathematics is a game, sometimes with applicability.
@EdS I agree! Proof hadn't made it on my list yet. Thanks! I am planning on using this research to write up some tutorials, one of the unifying themes is mathematics is about patterns and the other is mathematics is about problem solving through abstractions, i.e. reality to mathematical model to greater understanding of reality or application (inclusive or – almost an and, but hedging).
Here is my list of candidates (their level of troublesomeness varies with human development) so far:
number, numeral, digit
place value system
unknowns (@bstacey 's algebraic variables. Are these ideas different? I suppose it depends on what we mean by algebra)
continuity (orig. limit and derivative, but this seems the more foundational concept)
proof, esp. inductive (h/t @EdS )
Fermi estimation may be a #thresholdconcept - it has something to do with number sense, which is not unrelated to mathematics.
Calculating with symbols, certainly - a hurdle for some students when the numbers fall away from the maths lesson.
@thelibrarian applied math/mathematical modelling breakpoints in some order
spatial reasoning as an allegory (slide rules and number lines are actually useful teaching tools)
variables v. parameters
derivatives vs. functions
sensitivity analysis ("you can change parameters?!?")