I'd like to raise the issue of constant misuse of parentheses in the U.S., and I'm wondering if anybody else shares the same feelings, has had the same issues, knows any history behind it, and can offer some thoughts on the subject. Let me explain what I'm talking about.
In elementary algebraic notation, i.e. when writing algebraic expressions, parentheses mean two things: order of operations and function notation; http://mathworld.wolfram.com/Parenthesis.html seems to agree with me. (There are all the other things, such as interval notation, matrices, etc, but I'm not discussing those here.) But here in the U.S. it is used all the time to enclose each factor in multiplication, so $5\cdot2$ would be written as $(5)(2)$. And while it's technically speaking not wrong, as it still means the same thing, it's redundant and pedagogically terribly harmful, imho. This way of writing multiplication is so persistent from early on and then everywhere, that students grow up believing that that's what parentheses mean: that their purpose is to enclose each individual quantity in multiplication. Not for grouping, not for function notation, but mostly just for that. Almost any of my students here (and I've tried asking a whole class, so it's kinda experimentally confirmed) will write the product of $2$ and $x+1$ as "$(2)x+1$".
In my teaching I always try to correct my students' bad writing habits. Here's a recent example. Solving an example on the board during my "College Trigonometry" class, I actually came to the need of setting up a product of something like (don't remember exactly) $2$ and $\sin x+1$, which I intentionally wrote as "$2\cdot\sin x+1$". I turned to my class and said: "By the way, this is wrong. Can you correct what I wrote?" One of the best students in the class immediately said: "It misses parentheses."
"Halleluja! My efforts haven't been futile," I thought to myself... until he continued: "$2$ must be in parentheses because it's multiplied."
It would have been just a little notational nuisance, had it not been a cause of so many real mistakes. The most obvious one is when people don't distribute multiplication, since there's nothing indicating the need to distribute. I mean, as a quick example, when instead of $2(x+1)$ someone wrote $(2)x+1$, and then they need to plug e.g. $x=3$, they'll get $7$ instead of $8$. Another one, that I believe stems from the same misuse of parentheses, is misreading function notation. Again, I've done this literally with thousands of my students, and most read "$f(x)$" as "$f$ times $x$" or "$\sin(x)$" as "sine times $x$".
I only have the experience of working in two countries. In Russia, my homecountry, nobody ever writes $(2)(5)$. (And mind you, its educational system has a truckload of problems of its own, but at least this is not one of them.) In the U.S., it's not just about the students, but it's all over the books, textbooks, and in almost everybody's math writing, including teachers and professors — which is probably why all students pick up this habit. I honestly don't know about the rest of the world.
So, back to the questions that I asked in the beginning. Can anybody explain where this writing tradition comes from? Does anyone agree that it causes some real problems in students' learning of math?
Update: Let me clarify my concerns a little bit. As others pointed out below, $(2)(5)$ is weird but not wrong. And I agree: it's not wrong mathematically, but it is wrong pedagogically. And although it still irritates me (to put it mildly), that's not the problem in itself. But $(2)x+1$ is actually wrong when it stands instead of $2(x+1)$. I apologize for not being clear enough, but here's what I meant: I believe that seeing $(2)(5)$ results in students' writing $(2)x+1$, and that's wrong, and that's the problem.