TITLE: Why do new rules cause students to forget and misapply older rules?
QUESTION [30 upvotes]: I teach at a community college. I have taught everything from arithmetic to linear algebra. I have also taught at 4-year schools, but at present, I'm devoting my energies to the problem of helping remedial algebra students to succeed. 
I have noticed a pattern in my remedial classes. It's disturbing. You see, my students first learn the concept of "combining alike terms" example:
$3x^2y - 5x^2y = -2x^2y$ 
I present this topic in a number of ways including manipulatives and concrete examples.  We then apply this concept in a variety of contexts including systems of linear equations, and in word problems. Basically, it appears that they are getting quite good at working with variables.
But, later when we study exponent rules such as:
$x^ax^b=x^{a+b}$
$(x^a)^b=x^{ab}$
$x^{-a}=\frac{1}{x^a}$, for $x \neq 0$
etc.
these new rules seem to displace and muddy the older rules in the minds of the students. A week ago they would have considered:
$2x + 5x = 7x$
to be "easy" and every single student (even the weakest) had mastered this type of problem (signed numbers and fractions could be another matter...but still) Yet, after teaching the exponent rules I notice students doing things like this:
$2x + 5x = 7x^2$
to me this indicates a fundamental disconnect in terms of how mathematics works, I know they are aware of the older "rules" but it is as if they expect each problem to have different set of rules. This has happened all three times that I have taught this course, despite my effort to teach it in a different way each time. 
I find that this type of error is much more common in remedial classes. Why is that? I have also taught elementary school algebra and I simply never saw mistakes like this.  Not, at least, with the frequency I'm finding them now, even among responsible students who are clearly intelligent people as evidenced by their work in other subjects and pursuits, students who are putting in large amounts of time studying, who take notes etc. And these erors are hard to fix, explanations don't seem help much. 
What is going on here? Is there a name for this? 
I honestly wonder what it is I've taught them in the past two months if each new concept displaces and corrupts the old concepts.
Eventually the students will master the new rules but I get the feeling many of them are working much harder than they should be to do so. It's like they are doing something that's more like memorizing a complex gymnastics routine than mathematics. Others become very frustrated, to them it must seeming like I'm just making up random stuff as I go along to vex them. 
But I know mathematics makes sense. That's why I love it.
How can I help them to see this?

REPLY [3 votes]: This is mostly speculative.
Consider doing exercises on when not to use a 'rule'.
I like to give an example from machine learning: if you design an AI to answer a particular question, and then give it a lot of training data for which the answer is "yes", then what the AI will learn is that every question should be given a "yes" answer. You have to give it enough examples where the answer is "no" in order to learn properly.
(aside: for the AI enthusiasts out there, I'm sure there are designs that the above doesn't apply to. I'm just trying to make a good example)
Of course, a human will realize that they're supposed to be learning some sort of pattern, and so they'll try and guess the pattern on their own. They'll get some feedback that their guess is wrong in the form "you did this problem wrong", but it will come slowly.
For students who are having trouble working out the pattern, seeing examples that don't fit the pattern may be far more informative than more examples that do. A variety of examples will surely help as well.
Consider explaining how to recognize when to use the rule
Rather than trying to learn the pattern through osmosis, it may help to explicitly describe what features they should be looking for in an expression that allow a rule to be used.
Explain how to break down expressions into terms and factors and such, and to see that one rule might be applied to combine terms of a sum (or to split one term into several), or another rule might be applied to combine the factors of a product (or to split one factor into several).
A full blown lesson on mathematical grammar might even be useful. I've always been curious if learning how to make parse trees out of an expression could help students.

On a related note, I think a lot of people (both on the learning and teaching sides) have trouble even recognizing the existence of this skill and how important it is -- students who focus on learning how to use tools without ever realizing they're supposed to be learning how to select tools (or maybe even how to pack a toolbox!), and teachers who think the latter skills will come automatically to anyone who learns the former.
I don't know how to teach it though, especially to a student to doesn't acknowledge it, or worse, rejects it because they think they need to focus on how to use tools. :(