### Fractions and Student Incredulity

Quiz #2 had an exercise: “Simplify ${2\over3}x + {6\over7} -{1\over4}x -{5\over8}''$. Now, everybody knows (or anyhow, knew on that day) that the appropriate move in getting started is to “collect like terms”. And in a few cases, that’s as far as it went: $({2\over3}-{1\over4})x +({6\over7}-{5\over8})$. But almost everyone also had the wit to realize that the two rational-number subtraction problems should also be carried out. That this procedure actually presents considerable difficulty to sizable portions of roomsful of (community) college students might strike readers as scandalous (and indeed, I’m inclined to agree). But the actual nature of their difficulties appears not to be at all well-understood … and since I’ve somehow found myself presenting suchlike problems to suchlike students for over a decade, I’m sort of curious about it.

So. Several students (from either class, in different parts of the room … not a case of an error propagated by students using “the eyeball your neighbor” method) had $5x+13$ in place of (the correct) ${5\over{12}}x+{{13}\over{56}}$. Let’s call this the “Ignore Denominators Fallacy”. The reason the IDF occurs at all probably isn’t that hard to find: in solving an equation (as opposed to simplifying an expression), one “clears fractions” by multiplying both sides (of the equation) by an appropiate constant. For example, to solve ${2\over3}x - {3\over4} = 0$, a reasonable first step would be to replace the given equation with $8x - 9 =0$; one has of course multiplied each side by 12 here; nobody’s trying to pretend ${2\over3}x - {3\over4} = 8x -9$.

Now, all of these students have heard phrases like “multiply both sides of the equation by [so-and-so]” manymany times before ever having met me; they’ve heard ’em a few times from me by now, too (along with “multiply in the numerator and denominator by [such-and-such]”). But they’ve decided that these are just (deliberately obfuscatory) math-head code for the actually correct “multiply everything in sight by [whatever]” (which, admittedly, will sometimes escape my mouth when I’m deliberately being informal).

Evidently at least a handful of students in any given 102 class find it easier to believe ${5\over{12}}=5''$than it is to believe that whatever technical language their teachers repeatedly and emphatically insist on using could actually ever mean anything. There’s just got to be something interesting going on here.

Actual deadlines loom; shutting up. When I started this “blog” business, I thought be doing a whole lot more rantage of this kind (not just the Jorn-like “all links, all the time” stuff I’d settled into before I quit even that). Watch this space.

1. Just found this post. I have a theory about why students would do stuff like this. It’s that they don’t understand the technical meaning of the word ‘simplify’. When they’re solving equations, each row of their procedure might have a left- and right-hand side that are different from the one above it; you really are changing the numbers. “Simplifying”, however, requires that each line is identically equal to the line above it. To many math students, this difference in goal has not been clearly articulated. They just think they have to “do stuff that I think is legal to come out with an answer at the end”. To us, it looks like they didn’t bother to check whether 5/12 really equals 5. To them, they didn’t even know those were SUPPOSED to be equal, so why would they even check it?

My solution? It doesn’t work all the time, of course, but I make them learn ‘solve’ and ‘simplify’ as actual vocabulary words, and make them learn exactly what it is that can’t change in a problem: in a ‘simplify’ problem, the value of the expression can’t change, and in a ‘solve’ problem, the truth of the equality (or inequality) can’t change.

2. yes. good move. i’ll add only that,
as indeed you may have hinted here,
the *resistance* to learning “vocabulary”
in a “math” class can be pretty fierce …

3. Its like you read my mind! You seem to know so

I think that you can do with some pics to drive the message home a bit, but instead of that,
this is magnificent blog. A great read. I
will certainly be back.

1. 1 The One-Year Anniversary Carnival of Mathematics « 360

[…] same time wonders, about errors that students commit (such as IDF – Ignore Denominators Fallacy) in Fractions and Student Incredulity and its follow-up Oh, P.S., over at Vlorbik on Math […]

• ## (Partial) Contents Page

Vlorbik On Math Ed ('07—'09)
(a good place to start!)