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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. –  Mariano Suárez-Alvarez Nov 17 '12 at 18:00
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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... –  a little don Nov 17 '12 at 18:09
    
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. –  a little don Nov 17 '12 at 18:13
    
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. –  Mariano Suárez-Alvarez Nov 17 '12 at 19:19
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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. –  Brian M. Scott Nov 17 '12 at 23:40

6 Answers 6

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.

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One article that immediately came to mind, and which I think you'd find very helpful for your purposes, is Memory and Mathematical Understanding (link to article abstract).

Also perhaps relevant is Working Memory and Mathematics. It is relatively recent (published in 2010) and has a long list of references, to explore further.

You might want to check out the Mathematical Association of America "special interest group"'s SIGMAA RUME website and this auxiliary site. (RUME = Research in Undergraduate Mathematics Education.) You will find additional resources and references on undergraduate math education, and perhaps connect with others sincerely interested in post-secondary teaching.

That said, I really think much of what you are observing is a "cognitive" phenomenon: not necessarily a phenomenon of "what is being learned" (i.e., mathematics), but more appropriately addressed in the domains of cognitive & educational psychology. Certainly, math cognition is, in many respects, qualitatively different than (or a more specific domain of) cognition in general. So I do not mean to imply that this question doesn't belong here. I suspect that there are a lot of people here with lots of experience teaching, and many of whom are also interested in improving students' comprehension of mathematics.

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Thanks, the link to 'Memory and mathematical understanding'. seems to be pointing to the wrong place? –  a little don Nov 17 '12 at 18:17
    
Updated with correct link. It's in pdf, which is always handy! (I think I lost the link when I added the first link). –  amWhy Nov 17 '12 at 18:19
    
Thanks. I'm going to think about this angle, though I'm not certain it's a memory problem. These students seem to have normal memories, some of them might have better memories than I do. When I ask they are aware that last week 2x+3x was equal to 5x... it's more like they no longer think this is valid because of the new rules... –  a little don Nov 17 '12 at 18:35
    
The article discusses "working memory" among other things. There may be some "cognitive competition" going on between working memory (that which is most recently learned, or that which with one is currently trying to grapple) and longer-term memory (being that which was previously learned). And it could be that "circuits" are crossing/clashing. –  amWhy Nov 17 '12 at 20:03
    
The links were corrected. In my original post, "Memory and Mathematical Understanding" was actually linked to "Working Memory and Mathematics," the former article being , I suspect, more relevant to the question at hand. –  amWhy Nov 18 '12 at 14:28

Here is a guess:

It is possible, that as remedial students, they have an expectation that the problems will be difficult. I.e. they are expecting to put a lot of mental strain and effort into the solution.

This instinct may be appropriate in some situations, e.g. when they are learning the problem. However, in other situations it may make them discard the simpler (correct) solution and employ a more complex (incorrect) solution, simply because it seems to be more complex, and therefore meets the level of mental effort that they are expecting to experience.

If this hypothesis is correct, then perhaps additional practice on problems that they are already comfortable with will help recalibrate their "maths-is-hard" mental parameter?

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If the needed rule is not the rule being directly studied, the student must first select the rule from their entire toolbox of rules that could be used. This is much harder than trying to apply the one rule that will probably be needed to solve all problems in the current problem set.

To put it another way, when students are studying a specific topic and they try to solve exercises in that topic's section, they are effectively given a really big hint that the problem probably has something to do with that topic. If any other topics are also needed, they must refer to all previous topics to consider the set of possible applicable rules.

You can see this phenomenon at work by simply suggesting the rule, like combining alike terms, when the student gets stuck. Often they can then immediately see that this rule can be applied. Explanation: They got their hint back.

Once you move out of the laboratory and into the real world, what seemed easy may now be much harder!

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I have similar experience in teaching high school algebra and trigonometry class. We stream students into three levels of math: honor, intermediate and elementary. The elementary level is where I observe similar problem.

After many years of teaching I am starting to suspect that some topics are not teachable (if that is a word) for some students. Is this a polite thing to say these days?

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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. :(

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