I recently gave an exam problem that roughly 2/3 of the class made much more difficult by using the identity $1/x+1/y=(x+y)/xy$. Their work would have been much simpler if they hadn't done this. It seems like a symptom of the type of intellectual laziness that causes people to do things by pattern recognition rather than because they have a logical reason for doing them.
When students are taught this identity, are they told a reason why they might want to use it? What would the reason be?
I will ask them why they did it when I return the exam, but I suspect I'll get answers to the effect of "that's how you do that kind of problem in algebra." Maybe this is similar to stuff like multiplying out an expression such as $(x+y+z)(a+b+c+d)$, which I think they do because they think of the notation as a set of orders that they need to carry out.
Is this reflex similar to the one that compels them to rationalize the denominator in an expression such as $1/\sqrt{2}$? That one at least I think I understand the original reason for. If you're doing decimal arithmetic and performing division using long division (rather than log tables, a slide rule, or an electronic calculator), then rationalizing the denominator is a big win, so that would have sort of made sense before slide rules became common. Would the justification for habitually doing $1/x+1/y=(x+y)/xy$ be that division is a more expensive operation than multiplication if you don't have a calculator, slide rule, or log tables?
Or maybe math teachers want answers like $1/\sqrt2$ in a canonical form so that it's easier to check students' work? (But this seems like an anachronism now that we have WeBWorK et al.) But I don't see how $(x+y)/xy$ is more canonical than $1/x+1/y$, and in general it's preferable not to have the same variable appearing more than once, e.g., if we want to reason about the dependence on that variable.
Questions:
(1) Why is this identity taught?
(2) Why do students use it in inappropriate situations, and what can we do to cure this problem?