A good mathematical notation is in my opinion a necessity when solving mathematical problems. It allows to capture and communicate mathematical reasoning in a concise, efficient and unambiguous way. Anybody doubting this should just try to read old mathematical texts from before the time when modern mathematical notation was common. Mathematical historians usually quote the old masters using up a whole book page worth of text and then translate that for us in modern mathematical notation using an equation in one line or so.
Choosing the right notation for the job is also very important. Expressions in a notation should be easy to manipulate, expressions should stay small but the notation should not hide properties of the mathematical objects involved. My dad always gives this example of a bad notation choice: In some schools EE students learn to use + for "or" and "*" for "and" in boolean expressions. This is bad because in boolean algebra "or" distributes over "and" and "and" distributes over "or". But we the chosen notation one would write A (B + C) = A B + A C and A + (B C) = (A + B) (A + C). The second one looks unnatural because it doesn't work in integer algebra so students are reluctant to use it for boolean expressions even though it's perfectly fine there.
I tried to make all of this lead to this paper by E.W. Dijkstra where he explains the notational conventions that he used and why. He is one of the first to take notation very seriously and he pushed for corrections to the modern mathematical notation (For example he criticized the familiar use of the classical summation symbol as sloppy and misleading, he repeatedly argued for numbering starting from zero, ...).
