some of my students refer to there being an invisible $-1$ in front of the expression $-(x + 4)$ or in the exponent of $x$. While it is not phrased mathematically, I am ok with them saying this because it reminds them to distribute fully before simplifying etc. It got me thinking though is there a reason to not teach students to always write in $1$ wherever there is a single variable/unknown/etc such as $1(x^1+40)-1(3x^1 - 2)$? I know that it is not done in general because it is incredibly repetitive and annoying, but is there any reason why not to teach students to do this? Eventually they will be comfortable enough that they can imply the $1$ but I feel like a lot of my students would benefit from this especially when simplifying exponential expressions and distributing negative signs. Are there any downsides to this or is it just de facto mathematics education?
Actually, I don't think that it is quite right that there really "is an invisible $-1$" in front of $-x$. I would say that one defines the symbol $-x$ to be the one that fulfills $x + (-x) = 0$ and then one shows that $(-1) x =-x$.
Ok, you don't teach that at high-school level, but the message that $-x$ is a symbol on its own is important. If there were no such symbol $-x$ for any $x$ it would be rather strange to have it for the special case of $x=1$.
The reason for teaching this way is that mathematics is precisely about the accurate and concise description of facts. Leaving out unnecessary parts it a crucial point on the way to mathematical thinking.
As an example:
If n is an integer greater than 1, then either n is prime or n is a finite product of primes.
This formulation is not good in a mathematical sense, since it contains many unnecessary things. Much crisper and, I guess, even clearer, is:
A positive integer is a finite product of primes.
Writing concise, omitting everything unnecessary is a part of mathematical education.
I see this as two separate questions:
1) Are there any downsides?
2) Should I condone it for my students?
1) As mentioned in other answers, it can make the transition into deeper areas more difficult than need be. (Understanding for groups)
2) I think the answer to the questions can only be answered by the students you teach.
Some students have so little confidence in themselves that building this up should be a priority for any teacher. If leaving a $-1x$ is right for my students at that time then I will do it.
It depends what you mean by "teach them to do it". If you mean "insist that they do it, correct them every time they don't and take off marks on a test", then no. If you mean "let them know about this trick as a way to remember how to distribute correctly, but don't insist they do it if they can remember without it", then sure.
You wouldn't take off marks if someone didn't write "Soh Cah Toa" next to all their trig problems, and you wouldn't insist someone use training wheels if they're perfectly capable of riding a bike without them. But if you have students distributing $5-5(2+x)$ as $5-10+5x$ or something rather than $5-10-5x$, then I can't really think of a better way of explaining to them why this is wrong than the "invisible $-1$" notion.
The invisible $1$ makes sense when students are just getting started in using certain formulas.
In the equation $y = x + 4$, if students are new to the idea of $y = mx + b$, then it can help to ask the class, "if you had to put some number to the left of the $x$, what would it be?" Someone will probably say "$1$", and then you can sneak in the "$1$" to the left of the $x$ in the equation that you have written on the board: $y = 1x + 4$. Then you can ask what the slope of the line is, etc.
The same idea applies for the quadratic formula. Beginners can be encouraged to "sneak in the $1$" to the left of $x^2$ in $x^2 - 7x + 10 = 0$ or to the left of $x$ in $2x^2 - x - 15 = 0$
I do make clear that the invisible $1$ should be removed from any final answer. I teach that there is value in formatting an answer in a standard way so that it is likely to match precisely the answer given by a fellow student or by an answer key. In many cases, that means losing the $1$.
Like another poster, I also see two separate questions here, but different ones.
1) Should we explicitly write identity value? 1 is identity for multiplication and division, while 0 is identity for addition and subtraction. Adding 0 does not change the result, likewise dividing by 1 does not change the result. Which is why they are omitted for brevity, but if one wants to use it, it will not change the result, although when I was in school I would get half a grade off for extra constant or parentheses.
2) Should one discern subtraction of a positive value from addition of a opposite negative one? There was a custom to use inline + and - for addition and subtraction, and to specify the sign of a term above the term. But I guess because mathematicians are a lazy bunch, they use the property of "adding a number is the same as subtracting an opposite number," so two operations - binary subtraction and unary sign - have been condensed into one. This is, in fact, quite a hard idea for many students. I would not mind if some wrote -(5+x) as (-1) * (5+x) or something like this, if it helps.