Math notation in school

While tutoring pre-algebra today it occurred to me that many school kids may be confused by the notation used in mathematics more than by the underlying mathematical concepts. Correspondingly, I'm wondering whether undue emphasis is being placed on the operational aspects of mathematics (how to manipulate mathematical expressions) and too little on the semantics (what the expressions mean).

As an example, today in math class we were practicing how to manipulate exponents, e.g.:

The rule says, "if you multiply two numbers with the same base, add the exponents."

On the other hand:

because of the rule, "if you raise a number raised to a power to another power, you multiply the two powers."

From my interactions with my students, I had the impression that the students were executing the rules blindly, with no understanding of why they were correct. Didactically I found it helpful (and hopefully they did too) to expand the powers of x:

so that the "addition" rule of exponents emerges naturally.

Room for Improvement

I'm imagining a different way (from what I'm used to) of teaching what exponents are: First off, the reason why you would ever multiply a number by itself should be exemplified clearly (I'll sweep this issue under the rug for now). Second, start multiplying several variables by themselves many times, e.g.,

Then say, "instead of having to write out all these x's, let's create a 'shortcut': when I write x^4, this means xxxx." (The carat ^ symbol was used in place of superposition because the Blogger doesn't allow it; sorry.)

The idea here is to make the students feel the pain of not having the notation - it takes too long to write. Then, show them the shortcut notation and how it saves them time and effort. I think this might help make the computation that is based on the notation (e.g., add the exponents) seem less arbitrary when it's taught.