From our earliest days in school, mathematics notation presents a problem. Do we really need so many ways to express multiplication?
2x3 is the first notation children see. Later versions include 2•3, 2∗3, 2(3), and (2)(3). As kids move into algebra, so many of them struggle, so maybe the notation is part of the problem. Bob Davis, one of the early pioneers in mathematics education, preferred the notation with missing numbers represented by empty boxes and triangles. It makes more sense to put 7 into an empty box, rather than into an x in 
.
An internet meme with a huge number of views is the arithmetic question 8÷2(2+2). As mathematics teachers, we would apply the order of operations to get a result of 16, however, many people believe the answer is 1. Are these people misinformed or might there be a bigger issue? I reached out to a friend who works in computing, and he said that multiplication written without a symbol is performed before other operations. Mathematicians recognize 8÷2(2+2) as 8x(1/2)x(2+2) whereas certain programming languages see it as 8/(2x(2+2)).
This debate raises many questions about other notation used in mathematics courses. Furthermore, if our community agrees that a certain notation is problematic, why does it never change?
As students move into math intensive courses, more challenges with notation arise. The most troubling may be sin2(x). This notation represents (sin x)2. How much time is saved by using the troubling notation? A related concern is sin-1x. Can we blame a student who thinks this notation represents 1/sin x = csc x?
By the way, PhotoMath is also confused by this. Similarly, isn’t sin x a problem? Recall that functions are typically written with the argument in parentheses. We write f(x) not fx, so shouldn't sin x be written as sin(x)? However, the function notation causes confusion too. Using the limit definition of the derivative, students must evaluate f(x+h).Each semester there are students who mistakenly evaluate this as fx + fh.
Mathematics is a language, so learning context is important to know what the notation means, but there must be a more efficient way to communicate these ideas.
There are many other issues with notation. For example, should the empty set be written as ∅ when writing { } is much clearer? How should composition of functions be written? f(g(x)) seems clearest, yet other notations can lead to confusion. When we say limx→2f(x) = ∞ we also say that limx→2f(x) does not exist. This must confuse students.
Has any notation changed? I had hoped that the proliferation of technology might lead to change, but instead, emerging technologies are “smart” enough to know that sin2x = (sin x)2. In my almost 40 years of teaching mathematics, the only change that I can think of is that I rarely see the quantifiers ∀ and ∃. I assume these were originally developed because a writer could put a paper upside down on a typewriter to get the symbol. Looking back, wouldn’t it have been easier to write, “for all” or “there exists.?”
Mathematics is challenging. Let’s be sure that the notation we use does not make it more so.