The Librarian
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Also if you, dear reader, have suggested for , 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): ee.ucl.ac.uk/~mflanaga/thresho

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.

@thelibrarian

@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).

@thelibrarian Algebraic variables. What is this "x" that can have arithmetic done upon it?

Here is my list of candidates (their level of troublesomeness varies with human development) so far:
metaphor
symbol
collection
pattern
number, numeral, digit
addition
additive inverse
place value system
big-little principle
scaling/ratio
unknowns (@bstacey 's algebraic variables. Are these ideas different? I suppose it depends on what we mean by algebra)
sets
functions
slope
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.

brilliant.org/wiki/fermi-estim

@thelibrarian @bstacey

I can't believe how long it took me to realize I misspelled in the original post. 😖

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