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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?

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    Instead of rules one might make them notice that they are *facts*; the rule of exponents is not something you told them, it is a fact of nature.2012-11-17
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    I don't know if that distinction would make sense to them. I have tried replacing x with numbers and showing them that $2x+5x=7x^2$ is simply not true for most values of x. But this seemed to be even more confusing. That fact that it *is* true for the solutions to that quadratic is just too complicated to even mention. The kind of "proof" that I find very helpful and convincing seems to sound like gibberish to them. I try to avoid talking too quickly or putting too many symbols on the board at once...2012-11-17
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    What I'm really looking for hear is a name for this kind of destabilization. Something I can look-up in math education research journals. Maybe there is a way to avoid it.2012-11-17
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    but you can turn their confusion about *some* numbers satisfying that equation into theproblem of finding which do, and thereby getting some motivation for the problem of solving the quadratic. A motivation they'll feel close to them.2012-11-17
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    I know a significant number of people who would rayher suggest you avoid the math education journas themselves :-)2012-11-17
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    Why are these taught as rules? From the first example, the rest can probably be derived. The tree-structure is the most illuminating part, I would imagine. Instead, they are shortcuts to the same result. I would crack under the pressure of remembering a long list of rules for an exam, but principals stick.2012-11-17
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    This is probably related to how kids have trouble learning conjugations of verbs. They start out learning irregular exceptions (eat -> ate), then they learn regular ones (turn -> turned), then they misapply it (eat -> eated or ated). It's just a matter of how we learn to begin with.2012-11-17
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    @Josh: The problem is that with students like these, principles **don’t** stick, because they aren’t understood as principles in the first place. To these students the subject **is** just a maze of complicated rules with no underlying system. Some of them even want it to be that way: they want rules to follow and actively resist the idea that understanding some underlying principles will make life easier. Rules are seen as easy, principles as hard.2012-11-17
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    @BrianM.Scott that should be the answer2012-11-18
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    @RoundTower: Thanks, but it doesn’t actually address the OP’s question of how to deal with the problem.2012-11-18
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    @BrianM.Scott I guess you're right and it wouldn't be good enough as an answer. But it's something a lot of people fail to understand, I believe.2012-11-18

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