At the algebra level you are at, you must toss out many of your old ideas of what the symbols mean and use what they are currently defined as. Actually, I've seen many introductory algebra texts that start out with the operator $\star$ to rule out this old intuition.
It is common at a more mathematically mature level to abstract the idea of some concept with a new notation, symbol, or even a name. This allows for the manipulation of concepts at a level where prior knowledge might get in the way or to have students practice in thinking more abstractly. Just one brief example might be defining $\frac{a}{b} = (a,b)$ so that $(a,b) + (c,d) = (ad+bc,bd)$, highlighting a slightly different area of the concept (and allowing it to be on a ring product too e.g. $\mathbb{Z}\times\mathbb{Z}$). I have also seen professor's use uncommon notation or terminology when they want to discourage students from looking up the solutions to problems online.
That said, I don't think it would look at any lower maturity levels. Many students would grow flustered when you convert back to the more accustomed notations, thinking of it as one more thing to keep track of. They usually don't have the practice in abstraction to see that they are the same object. You can see this frustration when presenting a problem to beginning calculus students with just relabeling indices on a summation. They are the same object, but the look is just enough off than many might not realize it at first.
