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?
Best Answer
One approach for dealing with this problem: make a three-part worksheet. Part one is "stuff using the formulas we just learned today." Part two is "review of old stuff." Part three is "mixed problems."
In essence, when you teach the first concept, the student isn't learning it as an if-then statement: "when you see this, then do this." The student is just learning "do this," because that's the only sort of problem the student sees. So when you introduce the second concept, really the student has twice as many new things to learn - how to solve the second type of problem, and how to distinguish the first type of problem from the second type of problem. Of course there's no getting around this. You have to introduce something first. But that's my theory for why the second thing you introduce makes things slower.