“Clearing Fractions” Clarified

February 7, 2008 in Rants

Jerome Dancis remarks in “Toward Understanding and Remembering—How to do Hand Calculations with Fractions” that “It is not uncommon for students in college engineering calculus classes to still [be] doing calculations with fractions incorrectly. This reduces their chances for success.”, and offers several suggestions for dealing with this situation.

Most interestingly for my purposes right now, Dancis introduces what he calls “Epstein’s Rule”:

Epstein’s Rule. It is not all right to (or to teach students to) move numbers around in an equation.

“Subtracting 5 from both sides of an equation” uses the basic Rule: Equals minus equals are equals; instead of “moving the 5 to the other side” where it magically gets transformed to -5.

Violating Epstein’s Rule is an invitation and a common reason for creative mistakes.

“Cross-multiplying” is cited as an example: students “learn” to solve without understanding by “cross-multiplication” (“… which has the numbers 4 and 2 climb up from the cellar and walk across the equal sign, as if it were a bridge“); they then (and this is how we know about the “without understanding” part) invent bogus generalizations (in the case of , “… the most common creative error made was to incorrectly “cross-multiply” the two terms next to the equal sign while leaving the “1” alone, thereby obtaining the incorrect equation: 4x = 6x – 1.”).

This kind of mistake—or, more precisely, the persistence of such mistakes despite one’s efforts at clarification was the subject of two recent posts by me.

Which brings me to my purpose here today: to discuss a certain “with-understanding” method. One day it just sort of came to me.

They keep insisting on making the same mistakes over and over (quite often claiming “…but this is the way I was taught to do it”, as if a. at least one of their ex-teachers was a liar and b. that makes it alright to believe lies). I’m more-or-less duty bound to try to show them how to do it right. Nothing I’ve done so far ever seems to sink in. What haven’t I been doing? Well, the trouble seems mostly to arise in this “Clear Fractions” step … can I slow that down somehow and pin down more precisely where things are going wrong? Hmm … what if I use a Common Denominator and explicitly rewrite, e.g., as before “Multiplying on Both Sides” (by 4, to obtain 2x = 3x -4 … which all my current students can then solve easily)?

In effect, the idea here is to combine two of Dancis’s “understanding” methods (multiplying on both sides of the equation and using common denominators). The amazing thing is, I’ve actually had a certain amount of success with breaking things down this way: I’m reasonably sure that, for a example, a small handful of my present students have made some progress in this area because I showed them this “extra step”. So I recommend it to whatever teachers may be struggling with this issue. Or would if I thought any were reading this.