I can think of a couple. limits at infinity are thoroughly hand wavy, at least at the level they're introduced in high school or lower division college calculus. And I would say this even projects backwards into geometry, and even the Blurred overloaded operations of rational numbers and simple fractions. the notion of being able to add up an infinity of nothing and get a quantifiable something is just sort of taken for granted without elaboration.
The equality operator ( = ) is also presented as an implicit expression of truth when introduced, when it clearly admits to shades of true at best. 1*1 = 1 = 1^1 = 1^2 is one kind of true and a bunch of kinds of false. and although at a young age kids may not be sensitive to this, letting it go without explanation or curious challenge can be the source of inarticulate senses of doubt about what's really going on when they get introduced to higher levels of math.
I can think of a couple. limits at infinity are thoroughly hand wavy, at least at the level they're introduced in high school or lower division college calculus. And I would say this even projects backwards into geometry, and even the Blurred overloaded operations of rational numbers and simple fractions. the notion of being able to add up an infinity of nothing and get a quantifiable something is just sort of taken for granted without elaboration.
The equality operator ( = ) is also presented as an implicit expression of truth when introduced, when it clearly admits to shades of true at best. 1*1 = 1 = 1^1 = 1^2 is one kind of true and a bunch of kinds of false. and although at a young age kids may not be sensitive to this, letting it go without explanation or curious challenge can be the source of inarticulate senses of doubt about what's really going on when they get introduced to higher levels of math.