I don't have any research references handy, though I'm sure many exist for this question (particularly for the difference between a function and an expression in writing things like $x^3+1$). However, one I can vouch very clearly causes trouble in my own experience is the use of parentheses in American mathematics education.

• Parentheses mean grouping, which may or may not be associative: $2^{\left(2^3\right)}\neq \left(2^2\right)^3$
• Parentheses mean functions: $f(x)=x^2+1$
• Parentheses mean multiplication: $2(3)=6$

Unfortunately, especially the latter two are often confused - often enough that the "If I had a dollar for each time ..." cliche might actually hold true. Why? Some examples of miscomprehension I come across every year:

• If $f(x)=3x$ then $f=3$
• If $g(5)=30$ then $g(10)=60$, even if $g(x)=x^2+5$
• If $\ln(x)$ is the notation then $\ln(x+y)=\ln(x)+\ln(y)$ must be true.
• A related issue is pronunciation of "f of x" as "f x" or even "f times x".

I'm not saying these problems couldn't show up in other ways, but the implicit linearity one gets from the similarity of these notations is pretty aggravating, as an educator, and surely frustrating to students who are tripped up it. The function concept is hard enough (note that it took Euler to first use it sort of properly) without having its notation shared by something else important.

Side note: My understanding is that in a lot of the Commonwealth/UK the last meaning of parentheses (brackets in the UK?) is not anywhere near as common, writing instead $2.3$ or something. I'd love to see precise evidence of when the parenthesis convention became popular in the United States (and perhaps elsewhere).

I don't have any research references handy, though I'm sure many exist for this question (particularly for the difference between a function and an expression in writing things like $x^3+1$). However, one I can vouch very clearly causes trouble in my own experience is the use of parentheses in American mathematics education.

• Parentheses mean grouping, which may or may not be associative: $2^{\left(2^3\right)}\neq \left(2^2\right)^3$
• Parentheses mean functions: $f(x)=x^2+1$
• Parentheses mean multiplication: $2(3)=6$

Unfortunately, especially the latter two are often confused - often enough that the "If I had a dollar for each time ..." cliche might actually hold true. Why? Some examples of miscomprehension I come across every year:

• If $f(x)=3x$ then $f=3$
• If $g(5)=30$ then $g(10)=60$, even if $g(x)=x^2+5$
• If $\ln(x)$ is the notation then $\ln(x+y)=\ln(x)+\ln(y)$ must be true.
• A related issue is pronunciation of "f of x" as "f x" or even "f times x".

I'm not saying these problems couldn't show up in other ways, but the implicit linearity one gets from the similarity of these notations is pretty aggravating, as an educator, and surely frustrating to students who are tripped up by it. The function concept is hard enough (note that it took Euler to first use it sort of properly) without having its notation shared by something else important.

Side note: My understanding is that in a lot of the Commonwealth/UK the last meaning of parentheses (brackets in the UK?) is not anywhere near as common, writing instead $2.3$ or something. I'd love to see precise evidence of when the parenthesis convention became popular in the United States (and perhaps elsewhere).